Detectability of Supermassive Dark Stars with the Roman Space Telescope

The dark ages of the universe end with the formation of the first stars. This event—the cosmic dawn—happens roughly 200–400 Myr after the Big Bang, when pristine, zero metallicity, molecular hydrogen clouds at the center of 10⁶–10⁸ M_⊙ minihaloes start to undergo gravitational collapse, a necessary first step toward the formation of stars. In the standard picture of the formation of Population III stars (Barkana & Loeb 2001; Abel et al. 2002; Bromm & Larson 2004; Yoshida et al. 2006, 2008; O'Shea & Norman 2007; Bromm et al. 2009), the collapsing hydrogen clouds eventually become protostars ∼10⁻³ M_⊙, which grow by accretion as long as radiative feedback does not halt accretion. They are thought to grow to 100–1000 M_⊙. ⁶ On the other hand, the large dark matter (DM) abundance at the centers of the minihalos may alter these conclusions. DM, whether it be weakly interacting massive particles (WIMPs), self-interacting dark matter (SIDM), or other candidates, may provide a heat source that halts the collapse of the hydrogen clouds (Spolyar et al. 2008) and leads to a dark star (DS), a star made almost entirely of hydrogen and helium but powered by DM heating (for a review, see Freese et al. 2016). These dark stars may grow to becomes very massive and bright supermassive dark stars (SMDSs; Freese et al. 2010), up to 10⁶ M_⊙ and 10¹⁰ L_⊙, and be visible in current and upcoming telescopes, including the James Webb Space Telescope (JWST; Ilie et al. 2012) and, as is the subject of the current paper, the upcoming Roman Space Telescope (RST). The goal of the present paper is to make predictions for the observability of DSs in RST and suggest how RST and JWST can be used in tandem to discover SMDS candidates and confirm them spectroscopically.

Before the era ushered in by the JWST, our understanding of the formation and properties of first luminous objects in the universe came only from a combination of theoretical and numerical models. Already, only a few months after becoming operational, photometric data from JWST are finding the most distant galaxy candidates to date (e.g., Finkelstein et al. 2022; Naidu et al. 2022a, 2022b; Curtis-Lake et al. 2023; Donnan et al. 2023; Labbé et al. 2023; Robertson et al. 2023) and stand to discover SMDSs. Those findings significantly strengthen the tension that begun to emerge during the Hubble Space Telescope (HST) era between the picture one gets via numerical simulations of the formation of the first stars and the assembly of the first galaxies in the universe (e.g., Gnedin 2016; Dayal & Ferrara 2018; Yung et al. 2019; Behroozi et al. 2020) and what nature offers us (e.g., Stefanon et al. 2019; Bowler et al. 2020; Bagley et al. 2022; Harikane et al. 2022; Labbé et al. 2023). Simply put, we are observing too many extremely bright galaxy candidates too early in the universe. If some of those objects were supermassive DSs (M ∼ 10⁶ M_⊙) instead of galaxies, this tension would be alleviated. Important theoretical uncertainties, such as how massive the first generation of stars can be, or if they can be powered by DM annihilations (dark stars) or only by nuclear fusion (Population III stars), still remain open questions, which are now more relevant than ever.

In a recent paper (Ilie et al. 2023), two of us showed that JWST may have already discovered dark stars, although differentiation from early galaxies is not yet possible, given the low signal-to-noise ratio (S/N) of the available spectral data for the SMDS candidates we identified based on photometric data, as well as the lack of resolution to tell if the detected objects are point or extended in nature. As yet, only ∼10 high redshift objects found by JWST have spectra, which is required to prove that the objects are indeed at z > 10 based on clean observation of the Lyman break. We examined the four high redshift objects in the JWST Advanced Extragalactic Survey (JADES) data (Curtis-Lake et al. 2023; Robertson et al. 2023), and showed that three of them (JADES-GS-z13-0, JADES-GS-z12-0, JADES-GS-z11-0) are consistent with being DSs. In addition to JWST, the upcoming RST will also observe some of the first galaxies, and potentially even the first stars. In this paper, we address the following question: how can one disambiguate between supermassive dark stars and the first galaxies made entirely of standard Population III stars with RST.

The standard picture of the formation of the first stars may be drastically changed due to the role of the large DM abundance at the center of the host minihalo, leading to a heat source for the collapsing molecular cloud. Spolyar et al. (2008) first considered the possibility that, if DM consists of WIMPs, their annihilation products would be trapped inside the collapsing molecular cloud, thermalize with the cloud, and heat it up. This DM heating can overcome the dominant hydrogen cooling mechanisms and thereby halt the collapse of the protostellar gas cloud. As such, DM heating could lead to the formation of a new phase in the stellar evolution, a dark star. These are actual stars, made almost entirely of hydrogen and helium, with DM providing only 0.1% of the mass of the star. The stars are in hydrostatic and thermal equilibrium, obeying all the equations of stellar structure. Initially ∼1 M_⊙, dark stars grow by accretion, and some can become very massive (>10⁶ M_⊙) and very bright (>10⁹ L_⊙). They are puffy diffuse objects, ∼10 au in size, with DM annihilation power spread uniformly throughout the star. They have low surface temperatures (T_eff ∼ 10⁴ K), leading to very little ionizing radiation and thus very little feedback preventing their further accretion. Thus, some of the dark stars can grow to be supermassive SMDSs.

In a recent paper (Wu et al. 2022), one of us considered the possibility that DM, rather than being comprised entirely of WIMPs, consists instead of a different type of particle, namely SIDM. In the case of SIDM, it was found that the deepening gravitational potential can speed up gravothermal evolution of the SIDM halo, yielding sufficiently high DM densities for dark stars to form. The SIDM-powered DSs can have similar properties, such as their luminosity and size, as dark stars predicted in WIMP DM models. Regardless of the nature of the DM particle powering an SMDS (WIMPs versus SIMPs versus any other potential candidates that could form a dark star), the final fate of SMDSs is the same: once the DM fuel runs out, the object eventually collapses to a supermassive black hole (SMBH), leading to an interesting possible explanation of the many puzzling SMBHs in the Universe found even at high redshifts.

In this work, we consider the possible detection of supermassive dark stars with the upcoming RST, which is set to launch in the mid-2020s. Some, but not all, supermassive dark stars versus early galaxies comprised only of standard Population III stars have different photometric signatures in JWST (see Zackrisson 2011; Ilie et al. 2012). In order to familiarize the reader with the subject of SMDS detection and their observable signatures, in the remainder of this paragraph, we summarize the results obtained by two of us (Ilie et al. 2012) regarding the predictions of SMDSs for JWST; many of the remarks here will apply also to RST. One of the main results of Ilie et al. (2012) was that at redshifts z_emi ∼ 10 SMDSs of mass 10⁶ M_⊙ or higher are bright enough to be detectable in all the bands of the NIRCam instrument on JWST, even without gravitational lensing. With photometry, distant objects (z ≳ 6) are detectable as dropouts, i.e., the presence of an image in a band and its absence in the immediately adjacent band probing shorter wavelengths. This is due to the efficient attenuation of Lyα photons by the neutral H present in abundance at redshifts higher than z ∼ 6. For instance, with JWST, SMDS will show up as either J-band (z_SMDS ∼ 10), H-band (z_SMDS ∼ 12), or K-band dropouts (z_SMDS ∼ 14). Moreover, based on null detection results with HST (therefore assuming SMDS survive until z ∼ 10 where HST bounds apply) in Ilie et al. (2012), we estimated that a multiyear deep parallel survey with JWST covering an area of 150 arcmin² can find anywhere between one and thirty unlensed 10⁶ M_⊙ SMDSs as either K-band or H-band dropouts. Of course, this number can be even higher ⁷ if SMDSs do not survive until z ∼ 10, and therefore, our bounds based on HST null detection do not apply. Compared to JWST, the wider effective field of view (FOV) of the RST will increase the probability of detection so that the predicted number of objects will be larger.

In principle, there are three techniques to differentiate between SMDSs and early galaxies dominated by Population III stars: (a) SMDSs are essentially point sources versus early galaxies are extended objects, (b) different locations in color–color plots, and (c) spectra including specific spectral lines. In somewhat more detail, first, SMDSs are point objects, whereas Population III galaxies are not. This can show as a point-spread function (PSF) difference, if the telescope has high enough resolution, and/or if the objects are gravitationally lensed. For RST, we apply this technique in Sections 6 and 7. Second, the photometric signatures in color–color plots can be quite different for SMDS for which nebular emission is negligible versus Population III galaxies. This was shown in Zackrisson (2011), Ilie et al. (2012) for the case of JWST, and for RST, we do this analysis in Sections 6 (see Figure 17) and 7 (see Figures 23 and 24). The differences are due to different spectral features, which, in turn, are converted into different colors in various bands. Third, the most precise way to disambiguate is to have a full spectral analysis. We use the TLUSTY code (Hubeny 1988) to obtain spectra for the SMDSs, as presented in Section 3 (see also Ilie et al. 2012 for previous work). Since SMDSs are stars, their spectra can roughly be modeled as blackbodies. The more detailed spectra obtained using TLUSTY also include the effects of the stellar atmospheres, while ignoring any possible effects of nebular emission. For comparison with spectra of Population III galaxies, we use simulations from the Yggdrasil model grids of early galaxies (Zackrisson et al. 2011). ⁸ SMDS are usually too cool to produce any nebular emission lines, whereas the spectrum of a Population III galaxy is usually dominated by nebular emission. ⁹ In particular, relevant for the JWST are the He-ii emission line at λ ∼ 1.6 μm (also called the He ii λ1640 line) and the Hα emission. Those would be telltale signatures of a Population III galaxy. By contrast, for SMDSs, at the same wavelength, we would have an absorption feature. Other prominent SMDS spectral features (that get redshifted into the region RST is sensitive to) are He-i absorption and the Lyα break for the case of the coolest SMDS (T_eff ≲ 2 × 10⁴ K), whereas for the hottest SMDS (T_eff ∼ 5 × 10⁴ K) the main spectral features will be He ii absorption lines. A detailed discussion of the SMDS spectra can be found in Section 3.

Regarding the spectra of dark stars. We have in this paper not included the effects of nebular emission, which could make differentiation from early galaxies more difficult. We expect negligible nebular emission for the cooler dark stars, those in which the DM is obtained only gravitationally via adiabatic contraction (AC). For the hotter DS formed via capture, we make the assumption of no nebular emission in the studies in this paper but recognize the need to do better in future work. In more detail, SMDS with T_eff ≲ 2 × 10⁴ K, such as all SMDSs formed via and powered by WIMPs more massive than 100 GeV (see Figure 1), are too cool to produce any nebular emission. Moreover, the neutral H present in abundance in their atmospheres absorbs very efficiently the vast majority of their photoionizing flux, as one can see in Figure 2. For the hotter ones (such as those formed via DM capture, or via AC and powered by WIMPs lighter than 100 GeV), there is a possibility that they might create an ionization bounded nebula, and therefore exhibit nebular emission (Zackrisson 2011). The main uncertainty in this scenario is how much gas is available around a supermassive dark star with T_eff ≳ 2 × 10⁴ K. In this paper, we consider only SMDSs of purely stellar spectra, i.e., the photometric differences and spectral differences we work out in this paper are only valid under the assumption of SMDSs without nebular emission. We emphasize the caveat that the neglect of nebular emission may alter the results for the hotter SMDS formed via capture considerably, as must be studied in future work. On the other hand, the spectrum of a Population III/II galaxy is usually dominated by nebular emission.

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The literature on the possibility that DM annihilation might have effects on stars dates back to the '80s and early '90s, with the initial work studying the effects on current day stars (e.g., Krauss et al. 1985; Press & Spergel 1985; Spergel & Press 1985; Gould 1988, 1992; Salati & Silk 1989; Gould & Raffelt 1990, to name a few). Regarding the DM heating effects on the first stars, we mention Spolyar et al. (2008, 2009), Freese et al. (2008a, 2008b, 2010), Taoso et al. (2008), Yoon et al. (2008), Iocco et al. (2008), Casanellas & Lopes (2009), Ripamonti et al. (2009, 2010), Gondolo et al. (2010), Hirano et al. (2011), Sivertsson & Gondolo (2011), Ilie et al. (2012), and Gondolo et al. (2022). For reviews, see Ch. 29 ("Dark Matter and Stars") of Bertone (2010), Tinyakov et al. (2021), and Freese et al. (2016).

We organize this paper as follows. We start, in Section 2 with a brief review of dark stars and their properties. A discussion of the simulated spectra of suppermassive dark stars is presented in Section 3. Then, in Section 4, we study the detectability of SMDSs with the RST and make predictions on observational results. In Section 6, we consider different possibilities for the observed high redshift (z ≳ 10) objects, such as Population III galaxies, and compare their observable properties to those of SMDSs. The effect of gravitational lensing on our results is discussed in Section 7. In Section 8, we present conclusions and summarize our study. We end three appendices: Appendix A where we present key parameters used for our simulated images, Appendix B where we estimate the maximum mass for an SMDS formed via DM capture that would have sufficient gas surrounding it to form an ionization bounded nebula, and Appendix C where we discuss the comparison between SMDSs and other types of primordial supermassive stars (SMSs) commonly considered in the literature. Throughout this paper, we will assume ΛCDM cosmology and use the following cosmological parameters: Ω_Λ = 0.73, Ω_M = 0.27, and H₀ = 72 km s⁻¹ Mpc⁻¹.

The first stars form as clouds of molecular hydrogen starting to collapse inside the DM-rich centers of 10⁶ M_⊙ minihalos at z ∼ 10–20. As first showed by Spolyar et al. (2008), the collapsing baryons gravitationally pull DM in with them, increasing even further the DM abundance. If the DM is made of WIMPs, they annihilate among themselves. The WIMP annihilation rate is ${n}_{\chi }^{2}\langle \sigma v\rangle$ where n_χ is the WIMP number density, the standard annihilation cross section is

$$\begin{eqnarray}&&\langle \sigma v\rangle \simeq 3\times {10}^{-26}\,{\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 1 }$$

and WIMP masses are in the range 1 GeV–10 TeV. WIMP annihilation produces energy at a rate per unit volume

$$\begin{eqnarray}&&{\hat{Q}}_{\mathrm{DM}}={n}_{\chi }^{2}\langle \sigma v\rangle {m}_{\chi }{c}^{2}=\langle \sigma v\rangle {\rho }_{\chi }^{2}{c}^{2}/({m}_{\chi }),\end{eqnarray} \tag{ 2 }$$

m_χ is the WIMP mass, ¹⁰ and ρ_χ is the WIMP energy density. The cross section in Equation (1) is the canonical value used for WIMP annihilation by the DM community: WIMPS annihilating with this (weakly interacting) value in the early Universe automatically lead to the correct relic DM abundance today to explain the DM. This value does have model-dependence, with detailed models predicting slightly different values of the annihilation cross section. Hence, it is important to emphasize that dark stars are produced for a wide variety of WIMP masses and annihilation cross sections. Indeed, the cross section may be several orders of magnitude larger or smaller than the canonical value in Equation (1), and yet, (roughly) the same DSs result. Further, we note that the canonical cross section used in the paper is in agreement with all experimental bounds including those from indirect detection. In previous work, we studied WIMP masses m_χ ∼ 1 GeV–10 TeV (as we will show below, see Figure 1, the lighter WIMPs provide more heating and therefore slightly different DSs since Q ∝ 1/m_χ ). Since the heating rate scales as $Q\propto {\rho }_{\chi }^{2}/{m}_{\chi }$, considering a variety of WIMP masses is equivalent to studying a variety of annihilation cross sections.

The annihilation products typically are electrons, photons, and neutrinos. The neutrinos escape the star, while the other annihilation products are trapped in the dark star, thermalize with the star, and heat it up. The luminosity from the DM heating is

$$\begin{eqnarray}&&{L}_{\mathrm{DM}}\sim {f}_{Q}\int {\hat{Q}}_{\mathrm{DM}}{dV}\end{eqnarray} \tag{ 3 }$$

where f_Q ∼ 1 is the fraction of the annihilation energy deposited in the star, and dV is the volume element.

Dark stars are born with masses ∼1 M_⊙. They are giant puffy objects (∼10 au), cool (surface temperatures <10,000 K), yet bright objects (Freese et al. 2008a). They reside in a large reservoir of baryons, i.e., ∼15% of the total minihalo mass 10⁶–10⁸ M_⊙. These baryons can start to accrete onto the dark stars. Dark stars can continue to grow in mass as long as there is a supply of DM fuel. Two mechanisms—discussed below—can provide enough DM fuel to potentially allow the DS to become supermassive (M_SMDS > 10⁵ M_⊙) and very luminous (L > 10⁹ L_⊙).

(1) Extended adiabatic contraction (AC). The infall of baryons into the center of the minihalo provides a deeper potential well that increases the DM density. A simple approach toward this gravitational effect is the Blumenthal method for AC. We (Freese et al. 2009) and others (Iocco et al. 2008; Natarajan et al. 2009) have previously confirmed that this simple method provides DM densities accurate to within a factor of 2, sufficient for these studies. There remains the question of how long this process can continue. In the central cusps of triaxial DM halos, DM particles follow a variety of centrophilic orbits (box orbits and chaotic orbits; Valluri et al. 2010) whose population is continuously replenished, allowing DM annihilation to continue for a very long time (hundreds of millions of years or more). Freese et al. (2010) then followed the growth to SMDSs of mass M_DS > 10⁵ M_⊙.

(2) Capture. At a later stage, there is an additional mechanism that provides DM fuel.

Once the initial DM reservoir is exhausted (or the DS is kicked away from the DM-rich region), the star shrinks, its density increases, and subsequently, further DM is captured from the surroundings (Freese et al. 2008b; Iocco 2008; Sivertsson & Gondolo 2011) as it scatters elastically off of nuclei in the star. In this case, the additional particle physics ingredient of WIMP scattering is required. This elastic scattering is the same process that the direct detection experiments (e.g., Large Underground Xenon experiment (LUX/XENON), Super Cryogenic Dark Matter Search (SuperCDMS), DAMA, CRESST) rely upon in WIMP direct detection searches. The DM capture rate, and, in turn, the properties of the ensuing dark star, will depend on two key parameters: the ambient DM density (${\bar{\rho }}_{\chi }$) and the elastic scattering cross section σ. In what follows, we assume $\sigma {\bar{\rho }}_{\chi }={10}^{-40}\,{\mathrm{cm}}^{2}\times {10}^{14}$ GeV cm⁻³. The DM density assumed is consistent with estimates based on the AC prescription (Spolyar et al. 2008; Freese et al. 2009) and consistent with numerical simulations (Abel et al. 2002), while the scattering cross section is within the allowed region of parameter space for spin dependent interactions (Amole et al. 2019; Aprile et al. 2019).

2.1. Dark Star Properties

The properties of dark stars have been studied in a series of papers, first using polytropic models (Freese et al. 2008a; Spolyar et al. 2009), ¹¹ and then using the MESA stellar evolution code. For the case of polytropic models for supermassive dark stars, see Freese et al. (2010). Dark stars start as ∼1 M_⊙ convective stars in the thermal and hydrostatic equilibrium, powered exclusively by adiabatically contracted DM annihilations. As such, they can be well modeled by n = 3/2 polytropes. Accretion leads to their growth, and by the time they reach roughly 100–500M_⊙, they become radiation pressure dominated and can be well modeled by n = 3 polytropes. As any radiation pressure dominated star, dark stars more massive than ∼500M_⊙ will have a luminosity approximated well by the Eddington limit:

$$\begin{eqnarray}&&{L}_{\mathrm{Edd}}=\displaystyle \frac{4\pi {{cGM}}_{\star }}{{\kappa }_{\star }},\end{eqnarray} \tag{ 4 }$$

where G is the universal gravitational constant, c is the speed of light, M_⋆ is the mass of the star in question, and κ_⋆ is the stellar atmospheric opacity. To get an order of magnitude estimate, we can assume that the dominant opacity source in metal-free atmospheres of supermassive dark stars is due to Thompson electron scattering. This is a function of the hydrogen fraction (X) of the star: κ_⋆ ≃ κ_es = 0.2(1 + X) cm² g⁻¹. Further, assuming a Big Bang nucleosynthesis (BBN) composition of the stellar atmospheres allows one to recast the Eddington limit as follows:

$$\begin{eqnarray}&&{L}_{\mathrm{Edd}}\simeq 3.7\times {10}^{4}({M}_{\star }/{M}_{\odot }){L}_{\odot }.\end{eqnarray} \tag{ 5 }$$

For the case of adiabatically contracted DM powered dark stars, the polytropic approximation has been tested and confirmed by Rindler-Daller et al. (2015) using numerical solutions of the full stellar structure equations using the Modules for Experiments in Stellar Astrophysics (MESA) 1D stellar evolution code. By the time their mass becomes M_⋆ ∼ 10⁵–10⁷ M_⊙, for SMDS formed via either the extended AC (without capture) mechanism or DM capture, their luminosity is as high as L_DS ∼ 10⁹–10¹⁰ L_⊙. In Figure 1 (reproduced from Freese et al. 2010), we plot the evolutionary tracks in an H-R diagram for DSs that grow via accretion and reach a supermassive phase (SMDS) via either DM capture (labeled "with capture") or extended AC (labeled "without capture"). Those evolutionary tracks are obtained by solving numerically the equations of stellar structure under the simplifying assumption of a polytrope of variable index. From Equation (5), we can see that, for even more massive dark stars, the luminosity scales linearly with the stellar mass. Note that lighter DM particles are more efficient in heating up a dark star, in both the extended AC and the "with capture" cases, as L_DM ∼ 1/m_χ (this fact is easy to see for the extended AC case by combining Equations (2) and (3)). This effect is manifest in Figure 1 in two ways. First, at the same effective temperature (T_eff), an SMDS powered by a lighter WIMP will be brighter and puffier for both formation mechanisms considered. For a given luminosity, an SMDS powered by a lighter DM particle (in either scenario) will be cooler and puffier than those powered by heavier DM particles. We emphasize that the evolutionary tracks of Figure 1 are largely insensitive to the assumed accretion rate, which only dictates how fast a DS can grow from the moment it was formed (z_form) until the moment it emits light to be observed by a telescope (z_emi). As such, the observable properties of SMDSs are largely independent of the assumed formation redshift or accretion rate, and are instead controlled by the stellar mass, stellar composition, surface temperature, and radius. In turn, the latter two are fixed by DM parameters such as its mass (m_χ ), annihilation cross section (〈σ v〉), and, for the case "with capture," ambient DM density (ρ_χ ). For future reference, we list in Table 1 relevant parameters for SMDSs obtained via the polytropic approximation for the case of 100 GeV WIMPs and SMDS formed in 10⁸ M_⊙ DM halos at redshift z_form = 15.

Table 1. Parameters of SMDSs Used in This Paper

Formation Mechanism	M*	L*	R*	Teff	g
	$\left({M}_{\odot }\right)$	$\left({10}^{6}{L}_{\odot }\right)$	(au)	$\left({10}^{3}\,{\rm{K}}\right)$	(m s−2)
Extended AC	2.04 × 104	407	31	10	0.126
Extended AC	105	2.42 × 103	39	14	0.390
Extended AC	106	2.01 × 104	61	19	1.59
Capture	4.1 × 104	774	1.8	49	75.1
Capture	105	1.75 × 103	2.7	51	81.4
Capture	106	2.03 × 104	8.5	51	82.1

Note. The values listed here are adopted from Freese et al. (2010; see their Tables 3 and 4). We assume both type SMDSs are powered by annihilations of 100 GeV WIMPs and formed in 10⁸ M_⊙ DM halos at redshift z_form = 15 and grow via accretion, at a rate of $\dot{M}={10}^{-1}{M}_{\odot }{\mathrm{yr}}^{-1}$. For the case of an SMDS formed via DM capture, we further assume that the product between the ambient DM density and the DM–proton scattering cross section is: ρ_χ σ = 10¹⁴ GeV cm⁻³ × 10⁻⁴⁰ cm².

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2.2. Death of Dark Stars and the Formation of Supermassive Black Holes

Since the DS spectra control most observable properties (such as color in various bands, S/N, etc.), we discuss here the most prominent properties of the spectra, and the various differences in the two formation scenarios considered: with capture and extended AC. The spectra of SMDSs were first presented and discussed by two of us in Freese et al. (2010), Ilie et al. (2012).

To the zeroth order, the spectrum of a DS (like any other star) is a blackbody. The role of the SMDSs stellar atmospheres in reprocessing of photons and thus reshaping the blackbody spectrum approximation can be investigated by using the TLUSTY code (Hubeny 1988). To simplify our discussion here, we only consider SMDSs of three masses: ∼10⁴, 10⁵, and 10⁶ M_⊙. Lower mass SMDS have very little chance of detection, unless they are significantly gravitationally lensed, as we will show in Figure 22.

In Figure 2, we present the rest-frame spectra of the three different mass SMDSs considered for both formation scenarios. The left panel illustrates the spectra for the case of "With Capture," i.e., when WIMPs captured due to elastic scattering are included. The right panel illustrates the DS spectra for SMDSs in the case Extended AC, where only the WIMPs gravitationally pulled into the center of the minihalo provide the heat source. In this figure, we have taken the WIMP mass to be 100 GeV. In the remainder of this section, we discuss the wavelength, flux density, and most prominent spectral features (due to absorption by neutral H and He in the stellar atmospheres), since these will be important features in detecting SMDSs with RST as well as differentiating them from high redshift galaxies containing more standard Population III stars.

Peak wavelength. As discussed above, the DS formed "with capture" are hotter than those formed via extended AC (as a reminder, DSs "with capture" form at a later stage of stellar evolution, when the original gravitational DM has run out, and the star has collapsed via Kelvin–Helmholtz contraction to a denser object capable of capturing further WIMPs). As a consequence, the DSs "with capture" have spectra peaking at shorter wavelengths in the UV range at roughly 0.1 μm. The flux does not decrease substantially (with decreasing wavelength) until after the He ii (singly ionized He) absorption break (due to absorption by the stellar atmosphere, see discussion below) at wavelengths ranging between [0.023–0.03 μm] (see top left panel of Figure 2). In contrast, for SMDSs formed via the extended AC mechanism, the absorption by neutral H or He cuts off most of the UV flux at wavelengths shorter than the Lyα line (see top right panel of Figure 2). Yet SMDSs "with capture" have a steeper UV continuum slope β (f_λ ∝ λ^β ), which is one of the factors leading to those objects having a larger magnitude difference between neighboring IR bands relevant for both JWST and RST (and used to search for dropouts as described below).

Rest-frame flux density. First, we note that the flux density for SMDS formed via capture is roughly insensitive to the stellar mass since they have nearly equal surface temperatures. Second, by comparing the top two panels of Figure 2, one can see that the flux density around the peak for SMDS of the same mass formed via DM capture is about 100× larger than that formed via extended AC. This is due to the higher effective temperature (T_eff) in the case "with capture." However, as they reach supermassive status, both type SMDS will be Eddington limited, and therefore, their brightness is determined largely by the stellar mass, and to a lesser degree by their composition via the opacity of the stellar atmosphere (κ_⋆), as per Equation (5). The fact that different flux densities (extended AC versus capture) lead to the same total luminosity (Eddington limit) for both cases can be traced to their different radii: extended AC stars are more puffy since they never undergo a contraction phase.

We also note that the question of mass loss for dark stars, in view of possible super-Eddington winds or pulsations, was studied by Rindler-Daller et al. (2021). That work found that dark stars are not significantly affected by those effects. Hence, wind mass loss does not affect the spectral features of dark stars.

Spectral features. The most prominent spectral features of SMDSs, whenever nebular emission is negligible, are due to absorption by neutral H and He in the SMDSs stellar atmospheres (see Figure 2). This is due to their relatively cool surface temperatures (T_eff ∼ 10⁴ K). In contrast, the fluxes from Population III galaxies are, in many scenarios, dominated by nebular emission (Zackrisson et al. 2011). Going back to the features in the SMDSs spectra, below, we contrast them for the two formation scenarios considered. The relatively low surface temperature (T_eff ≲ 2 × 10⁴ K) for the SMDS formed via the extended AC scenario (right panels of Figure 2) leads to a larger fraction of neutral H and He in their stellar atmospheres, when compared to an SMDS of the same mass but formed via the DM capture mechanism. This explains the strong absorption lines at wavelengths corresponding to the Lyman series (0.1216–0.0912 μm), caused by neutral H, and, at shorter wavelengths (∼0.05–0.06 μm), the He i break. Additionally, the presence of neutral H in the cooler SMDSs formed via AC leads to a prominent Balmer break at λ ≃ 0.36 μm, followed by a sequence of strong absorption lines in the Balmer series, as can be seen in the lower right panel of Figure 2. Conversely, the higher surface temperature (T_eff ∼ 5 × 10⁴ K) of SMDS formed via DM capture implies a large ionization fraction for H; hence, the Lyman or Balmer absorption lines are weaker. The most prominent spectral energy distribution (SED) feature of SMDS formed via capture is the He ii (singly ionized He) break at wavelengths ranging between 0.023 and 0.030 μm. Common between the two cases are He i lines at wavelengths ∼[0.3–0.45 μm], the isolated He ii λ1640 absorption line at 0.1640 μm, a sequence of He ii lines at wavelengths ∼0.46 μm, and more He i lines at ∼[0.47–1.0 μm]. Note that JWST and RST will be sensitive to a relatively narrow bin of rest-frame wavelengths, as shown in the lower two panels of Figure 2. Thus, many of the spectral features discussed above will not be observable with either instrument. Since we expect SMDSs to be at z_emi ≳ 10, this implies that all the features to the left of the rest-frame Lyα line (i.e., 0.1216 μm) are going to be erased by the Gunn–Peterson trough. At the other end, the highest value of the rest-frame wavelength probed is just ${\lambda }_{\max }/(1+{z}_{\mathrm{emi}})$, with ${\lambda }_{\max }$ being the maximum wavelength to which each instrument is sensitive (i.e., 1.93 μm for the Grism spectrometer of RST and 5 μm for NIRSpec on board JWST). In turn, this implies that for objects at z_emi ≳ 10 (such as SMDSs) RST will only probe the SEDs spectroscopically up to λ_rest ≲ 0.1754 μm, whereas JWST will be sensitive to features in the SEDs up to λ_rest ≲ 0.44 μm. The single most intriguing spectral feature that can be potentially detected in the region of overlap of both instruments is the He ii λ1640 absorption line at the rest-frame wavelength 0.1640 μm. The observation of this feature would be a smoking gun for an SMDS, since at the same wavelength galaxies would typically exhibit a nebular emission line instead.

SMDSs versus compact clusters of Population III stars. Population III stars, or clusters thereof, are unlikely to have similar spectra with SMDSs, due to the differences in combinations of their surface gravity (g) and T_eff. Using the Population III parameters from Ilie et al. (2021), we compare g and T_eff to the values listed in Table 1 (for SMDSs). Population III stars more massive than 10M_⊙ will be hotter than the hottest SMDSs considered in our work. In fact, by the time they reach 100 M_⊙, Population III stars are as hot as 10⁵ K. As such, Population III stars occupy different locations in an H-R diagram, and have significantly different continuum of their SEDs. Moreover, the surface gravity g of SMDSs are less than 100 m s⁻² since they are puffy; whereas, for all Population III stars, g ≫ 100 m s⁻². The differences in T_eff and g would lead to very different spectral features.

Here, we examine the detectability of SMDSs with the RST. In this section, we assume that the objects are unlensed. ¹⁴ In Section 4.1, we estimate in which RST wavelength bands the SMDSs of various masses will be observable with exposure times ranging from 10⁴ to 10⁶ s. Then, in Section 4.2, we examine the use of photometric dropout criteria for SMDSs in RST. We will show that SMDSs of mass ∼10⁶ M_⊙ can be detected by RST with exposure times of 10⁶ s as J-, H-, and H/K-band dropouts, corresponding to emission redshifts of z ∼ 11, z ∼ 13, and z ∼ 14, respectively.

4.1. Detector Capability and SMDS Redshifted Spectra

Roman's wavelength coverage of visible and infrared light will span 0.5–2.3 μm. In Figure 3, we plot the projected sensitivity limits of the RST in its Wide Field Instrument (WFI) filters ¹⁵ (left panel) and contrast them to those of the NIRCam filters covering the same wavelength range (0.5–2.3 μm) on board the James Webb Space Telescope ¹⁶ (right panel). For both detectors, we show the flux necessary to achieve an S/N ≃ 5 for two exposure times considered: 10⁴ and 10⁶ s. The two telescopes have comparable sensitivity below ∼1.8 μm whereas at higher wavelengths RST loses sensitivity, as seen by the gradual increase in the necessary flux to achieve a detection (S/N = 5) in the F184 and F213 WFI bands of RST. This is in contrast with the NIRCam instrument, for which the sensitivity actually mildly improves for the filters probing those wavelengths. For instance, the projected detection limits of the F213 WFI RST filter are about an order of magnitude weaker than the corresponding ones for the F200W NIRCam filter. This, in turn, corresponds to JWST being able to detect, in the F200W filter, objects that are 10 times dimmer (i.e., 2.5 larger magnitude) than RST with the F213 WFI filter, assuming the same exposure time. For most other overlapping bands, the difference in their detector capability is minimal.

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JWST is able to probe the universe deeper than RST, i.e., to higher redshifts, due to it being sensitive to larger wavelengths (the Mid-Infrared Instrument, hereafter MIRI, is sensitive up to 28 μm). However, RST excels in its FOV capability. For instance, each of the RST deep field images will cover an area of the sky roughly equivalent to the apparent size of a full moon. Conversely, Webb's First Deep Field image covers an area of the sky smaller than the apparent size of a grain of sand as viewed at arm's length. As such, the probability of detecting SMDSs at redshifts as high as z_emi ∼ 14 using wide field surveys is much larger for RST ¹⁷ than JWST, simply because each survey will cover a much larger area.

Given the sensitivity limits of RST and JWST, we can then overlay the relevant DS spectra to determine their detectability of an individual DS in these instruments. We take the spectra in the rest frame of the star obtained in Section 3 and redshift them appropriately:

$$\begin{eqnarray}\begin{array}{rcl}{F}_{v}({\lambda }_{\mathrm{obs}};{z}_{\mathrm{emi}}) & = & \displaystyle \frac{(1+{z}_{\mathrm{emi}})4\pi {R}_{* }^{2}{F}_{v}\left({\lambda }_{\mathrm{emi}}\right)}{4\pi {D}_{{\rm{L}}}^{2}({z}_{\mathrm{emi}})},\\ {\lambda }_{\mathrm{obs}} & = & (1+{z}_{\mathrm{emi}}){\lambda }_{\mathrm{emi}},\end{array}\end{eqnarray} \tag{ 6 }$$

where λ_obs and λ_emi represent the observed and emitted wavelength, R_* is the radius of SMDSs, D_L is the luminosity distance, and ${F}_{v}\left({\lambda }_{\mathrm{emi}}\right)$ and F_v (λ_obs; z_emi) represent the original and shifted spectra. To be concrete, we assume a 10⁶ M_⊙ SMDS formed via either mechanism survives to various redshifts, as shown in Figure 4. Moreover, we add the sensitivity limits discussed above, in order to assess the potential observability of SMDSs with RST. Note that the SMDSs formed via AC (right panel) will appear brighter in all bands covered by RST, when compared to SMDSs of the same mass formed via DM capture (left panel). This is a combination of two factors. First, the SMDSs formed via DM capture are typically more compact, having undergone a Kelvin–Helmholtz contraction phase. Therefore, the ${R}_{\star }^{2}$ enhancement factor (see Equation (6)) is milder. Moreover, the slope of the UV continuum for the rest-frame fluxes of SMDSs formed via DM capture is steeper, as noted in our discussion of Figure 2. This, in turn, will lead to a faster decrease with the wavelength of their redshifted fluxes, as one can see from comparing the left and right panels of Figure 4. Unlensed 10⁶ M_⊙ SMDSs formed via extended AC could be observed if emitting at z_emi ∼ 10 (green line in right panel of Figure 4) even with 10⁴ s exposure time with RST in the following bands: F129, F146, F184, F158, F184. For the same objects, a greater exposure time is needed for observation in the F213 band, as the detector loses sensitivity in that higher wavelength filter. On the other hand, SMDSs of the same mass formed via DM capture (left panel of Figure 4) require an exposure time longer than 10⁴ s (varying in the range 10⁴–10⁶ depending on the band) in order to be detected at the level of S/N = 5.

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One common feature between all spectra presented in Figure 4 is the sharp cutoff of the flux at wavelengths shortward of the redshifted Lyα line, which, in turn, will allow for the use of the photometric dropout detection technique for SMDSs or high redshift galaxies. Photons emitted from distant sources can be efficiently scattered by the neutral H in the intergalactic medium (IGM). For sources at redshifts z ≲ 6, this effect is minimal, since it is roughly around that redshift that the universe has become fully reionized. In contrast, for more distant objects (z ≳ 6), the redshifted photons that reach a cold gas cloud with neutral H can get resonant-scattered out of the line of sight very efficiently if their redshifted wavelengths as they reach the neutral H have a value of 1216 Å or, equivalently, 0.1216 μm (the Lyα line). As a result, for any object emitting from z_emi ≳ 6, the observed fluxes shortward of the redshifted Lyα line, i.e., for λ_obs ≲ (1 + z_emi)1216 Å, are highly suppressed. ¹⁸ Depending on the model assumed for reionization, the effect of the attenuation of Lyα photons by neutral H in the IGM will be slightly different. However, for quasars emitting at z_emi ≳ 6, observations show that most of the flux shortward of the redshifted Lyα line is completely attenuated. In turn, this implies that the redshifted SEDs of SMDSs are cut off to the left of the horizontal lines plotted in Figure 4. In summary, the SMDSs formed via AC will appear brighter in all bands covered by RST, when compared to SMDSs of the same mass formed via DM capture. Unlensed SMDSs at z ∼ 10, of 10⁶ M_⊙, formed via AC can be detected at the level of S/N = 5 with 10⁴ s of exposure time in RST.

4.2. Photometric Dropout Criteria for Roman Space Telescope

In this subsection, we turn the information regarding the redshifted fluxes of SMDSs into dropout criteria of their potential observation with RST. For luminous objects at z ≳ 6, photometry alone, i.e., color magnitudes in various bands, is sufficient to give a rough estimate of the emission redshift via the so-called "dropout technique" pioneered by Steidel et al. (1996). This photometric redshift determination method requires a 5σ detection of an object in one band but a nondetection in an adjacent band of lower wavelength. The absence of emission in the latter bands is assumed to occur due to Lyα attenuation by hydrogen clouds in between the source and us, allowing for an approximate estimate of the redshift of the object. More specifically, we take as our dropout criterion

$$\begin{eqnarray}&&{\rm{\Delta }}{m}_{\mathrm{AB}}\geqslant 1.2\end{eqnarray} \tag{ 7 }$$

where Δm_AB is the difference in apparent magnitude between adjacent bands of observation.

For instance, for the 10⁶ M_⊙ SMDS at z ∼ 10 and formed via the extended AC mechanism considered in the right panel (green line) of Figure 4, there will be a sharp increase in magnitudes (decrease in flux) as observed in any filter with a central wavelength shorter than ∼1.34 μm. This, in turn, will lead to the SMDSs at z ∼ 10 appearing as J-band dropouts.

The dropout technique has been applied extensively to J- and H-band observations of the Hubble Ultra Deep Field (HUDF). For example, Bouwens et al. (2011) used it to detect the first galaxy at z ∼ 10 as a "J-band dropout." This object was observed in the 1.60 μm (H band) but was not seen in the 1.15 μm (Y band) or 1.25 μm (J band). Since then, this technique has been used successfully to identify many high redshift luminous objects. For instance, the pre-JWST record holder as the most distant galaxy candidate was HD1, at a whopping z ∼ 13, i.e., only 300 Myr after the Big Bang. This object (HD1) has been recently identified as a H-band dropout by Harikane et al. (2022). JWST data already broke this record multiple times (e.g., Castellano et al. 2022; Naidu et al. 2022a, 2022b; Donnan et al. 2023; Robertson et al. 2023), and all those new candidates for the title of "most distant galaxy ever observed" were first detected as photometric dropouts. It is worth emphasizing that spectroscopy is the only available tool to confirm such high z candidates. ¹⁹ If confirmed as galaxies, those objects could pose challenges to our current standard models of the formation of the first stars and galaxies, as they imply a much faster star formation rate (SFR) and growth of galaxies in the cosmic dawn era than any of the available numerical simulations predict.

In Figures 5 and 6, we plot the predicted AB magnitudes in different Roman WFI filters as a function of the emission redshift for SMDSs of three various masses: ∼10⁴ M_⊙, 10⁵ M_⊙, and 10⁶ M_⊙. Those stars are powered by annihilations of 100 GeV WIMPs and are assumed to have formed via either the extended AC (Figure 6) or the DM capture mechanism (Figure 5). The apparent AB magnitude (m_AB) in any band (filter) is calculated by the following prescription (Oke & Gunn 1983):

$$\begin{eqnarray}&&\begin{array}{l}{m}_{\mathrm{AB}}\\ =\,-2.5\mathrm{log}\left[\displaystyle \frac{{\int }_{{\lambda }_{\min }}^{{\lambda }_{\max }}T({\lambda }_{\mathrm{obs}}){F}_{\nu }({\lambda }_{\mathrm{obs}};{z}_{\mathrm{emi}})d{\lambda }_{\mathrm{obs}}/{\lambda }_{\mathrm{obs}}}{{\int }_{{\lambda }_{\min }}^{{\lambda }_{\max }}T({\lambda }_{\mathrm{obs}})d{\lambda }_{\mathrm{obs}}/{\lambda }_{\mathrm{obs}}}\right]+31.4,\end{array}\end{eqnarray} \tag{ 8 }$$

where T(λ_obs) is the throughput curve of the filter considered, ²⁰ and F_ν (λ_obs; z_emi) is the redshifted flux density, obtained via Equation (6). By comparing Figures 5 and 6, we find, just as we did in Figure 4, that at the same mass an SMDS formed via the extended AC mechanism has a larger observed flux, and therefore a smaller AB magnitude, rendering it easier to detect. This is mainly due to their larger radii, and therefore larger observed fluxes (see Equation (6)). However, at around z_emi ∼ 12, for SMDSs of 10⁵ M_⊙ and below, their magnitudes fall below the S/N = 5 detection limit even with 10⁶ s of exposure. Therefore, for the case of unlensed SMDSs, in the subsequent sections, we will limit our discussion to the case of a hypothetical 10⁶ M_⊙ SMDS.

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A trend clearly seen in Figures 5 and 6 is the increase of AB magnitudes with the redshift of emission. There are two reasons for this behavior. First, and foremost, the same object viewed from farther away will be dimmer. This very intuitive fact can be easily explained by an increase of the luminosity distance d_L (z) with redshift, and therefore a larger suppression of the observed flux as per Equation (6). Further, the sharp increase in the AB magnitudes at redshifts z ≳ 6 corresponds to the Gunn–Peterson trough affecting different filters at different wavelengths. For instance, the WFI F106 filter, with a center wavelength of 1.06 μm and covering the [0.927–1.192 μm] band will start to be affected by the Gunn–Peterson trough at z = 0.927/0.1216 − 1 ∼ 6.6, and by z = 1.192/0.1216 − 1 ∼ 8.8, the entire flux in the F106 band is completely suppressed by attenuation due to neutral H along the line of sight. Those estimated values of the redshift where the F106 band would be affected by the Gunn–Peterson trough can be confirmed by looking at the third panel on the top row of either Figure 5 or Figure 6.

In the next three subsections, we apply the dropout criterion in Equation (7) to show that SMDSs at z ∼ 11 could be observed with RST as J₁₂₉ dropouts, whereas those at z ∼ 13 would appear as H₁₅₈ dropouts, and those at z ∼ 14 would appear as F₁₈₄ (H/K) dropouts. ²¹

4.2.1. Detection of SMDSs with RST at z ∼ [10, 12] as J₁₂₉-band Dropouts

Here, we show that M ∼ 10⁶ M_⊙ SMDSs at z ∼ 11 can be found as J₁₂₉-band dropouts in RST data. Figure 7 illustrates this result, in the left panel for SDMSs formed via AC and in the right panel for those formed with capture. For the two filters F158 and F129, the figure shows the predicted SMDS magnitudes versus redshift as solid lines, and the S/N = 5 detection limits of RST assuming 10⁴ and 10⁶ s exposures. For a 10⁶ M_⊙ supermassive dark star, the difference in the magnitudes in the two bands will satisfy the dropout criterion at z ≳ 10, independent of the formation mechanism, as marked by the vertical green dashed line that corresponds to the redshift where the magnitudes differ by 1.2. The Lyα attenuation will completely cut off the flux in the RST J₁₂₉ band at z ≳ 11, which is slightly larger than the corresponding value (z ≳ 9.5) for JWST at J₁₁₅ band. This difference can be accounted for given the fact that the Roman J-band filter (J₁₂₉) covers wavelengths up to 1.454 μm whereas the corresponding JWST J-band filter (J₁₁₅) only covers wavelengths up to 1.282 μm. In turn, this implies that J-band dropouts with RST have a slightly larger estimated photometric redshift than those detected with JWST. Of course, the actual redshift of any object can only be accurately determined via spectroscopy. We point out that, for a ∼10⁶ M_⊙ SMDS formed via capture, an exposure time longer than ∼10⁴ s is necessary in order for them to show as J₁₂₉-band dropouts in RST. For the same mass SMDSs formed via extended AC, even slightly lesser exposure times would suffice, as seen from contrasting the left and right panels of Figure 7.

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4.2.2. Detection of SMDSs with RST at z ∼ [12, 14] as H₁₅₈-band Dropouts

Objects of redshift higher than z ∼ 11 would show as photometric dropouts in filters of increasingly higher central wavelength. For instance, an H-band dropout usually implies a photometric redshift of z ∼ 13, a K-band dropout implies a photometric redshift of z ∼ 15, and so on. In this subsection, we demonstrate that SMDSs of mass M ∼ 10⁶ M_⊙ at z ∼ 13 are bright enough to be detected as H₁₅₈-band dropout with the RST by comparing its AB magnitudes as a function of redshift in the F158 (H band) and the F184 (H/K-band) filters (see Figure 8).

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In both panels of Figure 8, the Lyα attenuation by the IGM starts to take effect at z ≳ 14 for the F₁₈₄ (H/K) band and at z ≳ 12 for the H₁₅₈. The dropout criterion is satisfied at z ∼ 12.5, as shown by the green dashed vertical lines, and the Gunn–Peterson trough will completely suppress the flux in the H₁₅₈ band at z ≳ 14. In both cases, one will need more than 10^4s of total exposure time to capture this effect. In the following sections, we will consider 10⁶ M_⊙ SMDSs emitting at z_emi ∼ 12 as our canonical case to contrast against other most likely possible luminous objects observable with RST at the same redshift (i.e., Population III/II galaxies). Based on Figures 7 and 8, we would expect complete cutoff of the SMDSs fluxes in F129 filter, and the image in F184 filter to be brighter than that in F158 band of the RST.

4.2.3. Detection of SMDSs with RST at z ∼ [14, 15] as F₁₈₄- (H/K-) band Dropouts

In this subsection, we consider the possibility to detect SMDSs as photometric dropouts at z ∼ 14. Specifically, we will demonstrate that SMDSs of mass M ∼ 10⁶ M_⊙ are bright enough to be detected as F184- (H/K-) band dropout (z_emi ∼ 14) with the RST. In Figure 9, we are comparing the AB magnitudes of 10⁶ M_⊙ SMDSs formed via either mechanism (DM capture or extended AC) as a function of redshift in the F184 (H/K band) and the F213 (K-band) filters. For SMDSs formed via extended AC (right panel of Figure 9), the dropout criterion is satisfied at z ∼ 14.2 (green dashed line) for 10⁶ s exposure times. For the case of the dimmer SMDSs formed via DM capture (left panel), longer exposure times would be needed in order to detect them at z ∼ 14 as F₁₈₄ dropouts. We point out that z ∼ 14 is the highest redshift where SMDSs can be detected (even with infinite exposure time) as photometric dropouts (F184 H/K dropouts) with RST. This is because RST lacks an L band, and, as such, no K-band dropout is possible, i.e., no dropout detection at z_emi ≳ 15.

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Having shown that SMDSs are bright enough to be observable as photometric dropouts with RST, at redshifts as high as z ∼ 14, we move on next to a discussion of the objects that may look very similar to SMDS in RST and JWST data: early galaxies formed of regular, nuclear burning stars (i.e., Population III and Population II galaxies).

Early galaxies containing many Population III or Population II stars are the competitors to SMDS in observations of high redshift data taken with RST or JWST. Indeed, according to simulations, large numbers of early galaxies are expected to be found at z ≳ 10 and may be hard to differentiate from SMDS. Before discussing the comparison between SMDS and galaxies in data, in this section, we describe the relevant parameters and observable properties of Population III and Population II galaxies.

Whereas in the case of an supermassive dark star the flux is largely given by the SED of a single object, the supermassive dark star in question, the SEDs of galaxies are due to an interplay between the flux coming from all the stars inside the observed galaxy and nebular emissions, which are especially important for young galaxies that actively form stars. For the first type of galaxies, those effects have been simulated by Zackrisson et al. (2011) using the Yggdrasil code. In this paper, we use their model grids for the integrated spectra of first galaxies, available at https://www.astro.uu.se/~ez/yggdrasil/yggdrasil.html. The relevant input parameters and the possible choices are enumerated below:

• 1.  
  Initial mass function (IMF). The initial distribution of stellar mass population of Population III stars for the single stellar populations (SSP) is in Yggdrasil. There are three types of initial mass functions (IMFs) we consider, following Zackrisson et al. (2011), Ilie et al. (2012), corresponding to three types of Population III galaxies: Population III.1, Population III.2, and Population III Kroupa IMF. Population III.1 galaxies have a top-heavy IMF and an SSP from Schaerer (2002), and we impose a cutoff on the lower mass end of 50M_⊙. Note that "top heavy" refers to a stellar IMF that, at the high mass end, has a less steep slope than the local Salpeter IMF (Salpeter 1955); this implies a larger fraction of heavy stars than the "local" Salpeter distribution. ²² In our case, we model one of the extreme scenarios where all stars are formed within the mass range 50–500M_⊙; henceforth, we call this scenario extreme top heavy where the word extreme refers to the lower mass cutoff rather than to the slope. Specifically, for Population III.1 galaxies, the simulation adopts a power-law IMF with a Salpeter slope (${dN}/{dM}\propto {M}^{-2.35}$) through the 50–500M_⊙ stellar mass range. Population III.2 galaxies are characterized by a moderately top-heavy IMF with an SSP from Tumlinson (2006), Raiter et al. (2010). Specifically, we model Population III.2 galaxies as having a log-normal IMF extending from 1 to 500M_⊙ with a characteristic mass M_c = 10 M_⊙, and distribution width σ = 1. In view of recent simulations, the mass of Population III stars might be lower. Therefore, we also include the case of the Kroupa (2001) IMF, usually describing Population II/I galaxies. The stellar masses range in the 0.1–100M_⊙ and the SSP is a rescaled version of the one used in Schaerer (2002).
• 2.  
  Metallicity (Z). This index is used to describe the relative abundance of all elements heavier than helium. For reference, the metallicity of the Sun is Z_⊙ ∼ 0.02. The characteristic value of 0.02 represents typical Population I galaxies while 0.0004 is average for Population II galaxies. If one were to include intermediate cases, the following values are available in Yggdrasil model grids for the metallicity of galaxies: Z = 0 (zero-metallicity), 0.0004, 0.004, 0.008, 0.02. Since our lowest redshift of interest lies around z ∼ 10, previous works (e.g., Jaacks et al. 2018, 2019; Liu & Bromm 2020) find via simulations that at such high redshift the metal enrichment process would not be sufficient to make the transition from Population III to Population II galaxies. The resulting estimation gives an upper-bound of the mean metallicity: Z < 10^−2.5 Z_⊙ ≃ 0.00006. This value is about an order of magnitude smaller than 0.02 Z_⊙, which corresponds to the transition to Population II galaxies. Therefore, in order to be conservative, we will also include Population II galaxies (Z = 0.0004) in addition to Population III galaxies (Z = 0) in our comparisons.
• 3.  
  Gas covering factor (f_cov). This parameter determines the relative contribution of the nebular emissions to the integrated stellar SED. Depending on how compact the ionized hydrogen (H ii) region is, the escape fraction (f_esc) for ionizing radiation (Lyman continuum) from the galaxy into the IGM can vary anywhere from 0 to 1. Moreover, the escape fraction f_esc is related to the gas covering factor via the following: f_esc = 1 − f_cov. Hence, we consider two extreme cases for the gas covering factor: f_cov = 1 (Type A galaxies; maximal nebular contribution and no escape of Lyman continuum photons), and f_cov = 0 (Type C galaxies; no nebular emission).
• 4.  
  Star formation history. Yggdrasil model grids include instantaneous burst and constant SFR lasting for t = 10, 30, or 100 Myr. Following Ilie et al. (2012), Rydberg et al. (2015), we restrict our comparison to the instantaneous burst case, which produces a single age stellar population. We assume that all the stars are formed at t = 1 Myr, as measured from the formation of the galaxy (defined to be at t = 0), the same as argued in Ilie et al. (2012), Rydberg et al. (2015).

For future reference, we summarize in Table 2 the relevant parameters and the choices made that differentiate between the various Population III or Population II galaxies we considered in our comparison with supermassive dark stars.

5.1. Appropriate Choice of Which Early Galaxies to Compare with SMDS

In this section, we will imagine a z ∼ 12 object has been detected with RST as a photometric H₁₅₈ dropout, and we will investigate our ability to differentiate whether this object is an SMDS or an early galaxy, using photometry alone.

In previous work on differentiating galaxies versus SMDSs in JWST, we (two of us) fixed the stellar masses (M_⋆) of all Population II/III galaxies to be equal to the mass of the particular SMDSs considered (Ilie et al. 2012). However, for objects detected with RST as photometric H₁₅₈ dropouts, if a spectral analysis is not available, the mass and nature of the object in question would still be uncertain. The only available observables, in this scenario, would be the AB magnitudes for this object in the various bands it is detected, and, if resolved, an estimation of its effective radius.

Hence, we address the following question: Would it be possible, using only photometry, S/Ns, and image morphology, to differentiate between a supermassive dark star and a Population III/II galaxy as potential candidates for this object? Our approach is to compare SDMSs and Population III/II galaxies with the same absolute magnitude in the F₁₈₄ band as the SMDSs we compare them against. The reason we chose this band is that it is bridging the gap between the H₁₅₈ and the K₂₁₃ band, which involve the only three RST filters in which the flux is not completely attenuated by neutral hydrogen in the IGM, for objects emitting at z ≳ 12. Moreover, for objects at z ∼ 12, which are our primary targets in this paper, the F₁₈₄ band is not affected by any emission or absorption features in the spectra of galaxies or SMDSs, respectively (see Figure 11). Lastly, for a H₁₅₈ photometric detection, the object would typically be brightest (have lowest AB magnitude) in the F₁₈₄ filter. In Figure 10, we plot the corresponding stellar mass we found for each of the Population III/II galaxies considered in such a way that their AB magnitudes in the F₁₈₄ band match those of a 10⁶ M_⊙ SMDSs formed via AC (right panel) or via DM capture (left panel), assuming z_emi ∼ 12 for all objects. In order to find the stellar mass (M_⋆) for each galaxy, we assumed, following Zackrisson et al. (2011), that the total brightness of the galaxy is proportional to the stellar mass L ∝ M. ²³

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For galaxies older than ∼3 Myr, their stellar mass required in order to match the observed flux of an SMDS monotonically increases with time, as can be seen in Figure 10. This is due to the fact that by ∼3 Myr after the initial instantaneous burst stars begin to die as they exhaust their nuclear fuel, thus rendering a galaxy dimmer. Since we are requiring a constant flux (AB magnitude) in the F₁₈₄ band, as per our comparison criterion with SMDSs, this dimming of a galaxy of constant mass will be compensated by an increase in the stellar mass of the galaxy. ²⁴

An important takeaway of Figure 10 is that, at the same mass, an SMDS is typically much brighter than a galaxy. For this reason, SMDSs are a natural solution to the problem posed for the standard cosmological model of ΛCDM by the JWST data, in terms of too much stellar light coming from high redshift sources, if those are interpreted as giant early galaxies.

To reiterate, we will compare SMDSs to sets of early galaxies, defined as young (∼1 Myr) and older (3.6 Myr for Population III.1 galaxies and ∼100 Myr for Population III.2 or Population III.Kroupa galaxies). We compare SMDS and galaxies that have the same observable magnitude (in the F184 band) and the same redshift.

In the remainder of this paper, we present a detailed comparison of the properties of supermassive dark stars versus those of Population III/II galaxies, assuming both types of objects could be observed with the upcoming RST as photometric dropouts at z ≳ 10. We have shown this to be the case for SMDSs in Sections 4.2.1–4.2.3. For Population III galaxies, see, for instance, Vikaeus et al. (2022), who demonstrate that Population III galaxies can be photometrically detected even in blind surveys with RST, JWST, ²⁵ and Euclid. Here, we perform an analysis regarding the possibility to disambiguate between those two classes of objects, if observed with the RST. In order to do so, we use a spectral analysis (spectroscopy), color magnitudes analysis (photometry), and image morphology (point source/extended objects). In Section 6.1, we discuss the potential to use spectroscopy to tell apart SDMSs from Population III or Population II galaxies. In Section 6.3, we present simulated images of SMDSs versus Population III galaxies in various RST filters. The potential to disambiguate between Population III galaxies and SMDSs via photometry alone is discussed in Sections 6.2 (AB magnitudes in various RST bands) and 6.4 (color–color plots technique). The effects of gravitational lensing are discussed in Section 7.

6.1. Spectroscopy to Differentiate SMDS versus Early Galaxies

In this subsection, we investigate the possibility to differentiate SMDS versus early galaxies based on their spectra. We will first investigate the difference between SMDS and early galaxy spectra for young (1 Myr old) Population III/II galaxy candidates (see Figure 11) and then turn to the same study for the oldest possible galaxies (see Figure 12); in both cases, the galaxies are assumed to have the same AB magnitude in the F₁₈₄ band as SDMSs counterparts. For older galaxies, the spectra is computed at an age of ∼100 Myr (measured since the formation of the galaxy), with the exception of the extreme top-heavy Population III.1 galaxies, which burn through their stars in ∼3.6 Myr, leading to an abrupt end of their evolutionary stage (see Figure 10). All cases have been chosen such that they could be discovered as H₁₅₈-band dropouts.

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We begin with a study of comparison between young galaxies versus SMDSs, assuming negligible nebular emission for the latter. In Figure 11, we compare the redshifted spectra of 10⁶ M_⊙ SMDSs formed via the DM capture mechanism (top two panels) or those formed via the extended AC mechanism (lower two panels) to the SEDs of all the young (1 Myr old) Population III/II starburst galaxy candidates considered. An emission redshift of z_emi ≃ 12 is assumed; however, the results and conclusions from this plot will not change significantly even if we took any other emission redshift in our interval of interest: z_emi ∈ [10 − 14]. The most striking finding is that the spectra of young Type C galaxies (no nebular emission) is essentially identical to that of SMDSs formed via the DM capture mechanism, as one can see from the top right panel of Figure 11. The only difference is the absorption feature at ∼2.1 μm, present only for SMDSs. This feature corresponds to a He ii absorption line, at a rest-frame wavelength of ∼1.6 μm, i.e., the He ii λ1640 line. However, as shown in Vikaeus et al. (2022), the He ii emission/absorption lines are unlikely to be detected in blind surveys, and dedicated spectroscopic follow-ups are necessary. Therefore, in photometric surveys, SMDSs formed via DM capture can masquerade for young Population III/II galaxies with little to no nebular emission. In the lower right panel of Figure 11, we compare redshifted SEDs for Type C Population III/II galaxies to those of SMDSs formed via extended AC. In this case, SEDs can be differentiated, in principle, based on their distinct slopes of the continuum. The SEDs of young Type A galaxies (maximum nebular emission) are different from those of SMDSs, formed via either formation mechanism, as can be seen from the left two panels of Figure 11. The absence of any emission lines for the SEDs of SMDSs is in stark contrast to the two prominent emission features of the Type A galaxies at ∼1.6 and ∼2.1 μm. Those two correspond to rest-frame wavelengths of ∼0.12 μm (the Lyα line) and ∼0.16 μm (the He ii λ1640 line). The strength of the Lyα line leads to a higher integrated flux in the F158 band for the galaxies, when contrasted to the SMDSs. ²⁶ We have also considered the possibility that SMDSs and young Population III/II galaxies have the same AB magnitude in the F158 band, instead of our choice F184, which was explained in our discussion of Figure 10. In this case, the galaxies will look much dimmer than SMDSs in all the other bands, since their flux in the F158 band is primarily driven by the prominent Lyα line, which is absent from purely stellar SMDSs spectra. For this reason, in what follows, our criterion when selecting galaxies to contrast against SMDSs is to require the same integrated flux in the F184 band. In conclusion, young Type C Population II/III galaxies without nebular emission (upper right panel) cannot be differentiated from SMDSs on the basis of their spectra; in all other cases, the young galaxies can in fact be differentiated from SMDSs.

In Figure 12, we contrast the SEDs of SMDSs to those of older Population III/II galaxies. For all, except the Population III.1 galaxies, we chose their age to be 100 Myr. Population III.1 galaxies have a top-heavy IMF, and, as we mentioned in our discussion of Figure 10, all their stars go dim when the galaxy is roughly 3.5 Myr old, which is the age we assumed for those types of galaxies (Population III.1) in Figure 12. Note how the spectra of all 100 Myr old galaxies considered contain no distinctive features at the wavelengths of interest. On the other hand, the Population III.1A galaxies (age 3.5 Myr) are distinctive due to the strong Lyα nebular emission line, which should be easily detectable with spectroscopy and also drives up the integrated flux in the F158 to become higher than the corresponding one for the SMDSs. The main takeaway from Figure 12 is that old (100 Myr) Population III/II galaxies are great SMDSs chameleons. The SEDs of SMDSs formed via DM capture (top row of Figure 12) are nearly identical, up to the He ii λ1640 absorption line, to the SEDs of all 100 Myr old galaxies considered (i.e., Population III.2 A/C, Population III.Kroupa A/C, Population II.Kroupa A/C). When contrasting SEDs of SMDSs formed via AC to those of old (100 Myr) Population III/II galaxies (lower row of Figure 12), we find that the spectra of Population II.Kroupa galaxies (dotted green lines) are very similar to those of SMDSs formed via AC. However, the two cases differ in that only the SMDS exhibit a He ii λ1640 absorption line, for masses M ≳ 10⁵ M_⊙ (see lower right panel of Figure 2). In principle, one can detect this line with dedicated spectroscopic studies. However, for objects at z > 10.7, the line is outside of the wavelength range probed by RST and would require follow-up studies (e.g., with JWST).

We end the discussion of Figures 11 and 12 by pointing out that the hottest SMDSs (T_eff ≳ 5 × 10⁴ K, typically reached by SDMSs formed via DM capture) could actually have significant nebular emission, as shown by Zackrisson (2011). This would further complicate the prospects of disambiguation between SMDSs and the first galaxies based on their SEDs. Namely, for each SMDS for which nebular emission is significant, one could identify, in principle, a galactic counterpart with a similar photometry of spectral signatures. Moreover, such objects (SMDSs surrounded by an ionization bounded nebula) could, in principle, be resolved as compact extended objects. We leave the detailed investigation of the possible role of nebular emission from the hottest dark stars on their detectable signatures for a future study. Our expectations are that, whenever SMDSs form an ionization bounded nebula, the disambiguation between such an object as compact (i.e., unresolved or poorly resolved) as an early galaxy would be nearly impossible with RST. For a more in depth discussion, see Appendix B.

In this subsection, we have investigated the ability to distinguish SMDSs and early galaxies on the basis of their spectra. There is one clear signature of an SMDS in the spectra: if a He ii λ1640 (i.e., rest-frame wavelength 0.1640 μm) absorption line is seen, then the object is a DS and not a galaxy. However, the He ii emission/absorption lines are unlikely to be detected in blind surveys, and dedicated spectroscopic follow-ups will be necessary. The Grism spectrometer that will be part of the WFI on board RST is only sensitive to wavelengths as high as 1.93 μm. Note that at z_emi ∼ 12 the only significant spectral feature within the range of the RST Grism is the possible Lyα nebular emission of galaxies (see Figures 11 and 12). Thus, until follow-up spectroscopic data are available (possibly from JWST, which is sensitive to much higher wavelengths), it will be impossible to differentiate z ∼ 12 SMDS versus old Population III/II galaxies (or those with little nebular emission) on the basis of their spectra with RST. In the remainder of the paper, we investigate how one could differentiate between SMDSs and Population III/II galaxy candidates in the absence of spectroscopic data.

6.2. Photometry Alone Cannot Differentiate Unlensed SMDSs versus Early Galaxies in RST

Early data are likely to be photometric in nature, with detailed spectra only obtained later. Hence, in this section, we examine the capability to differentiate SMDS versus early galaxies in photometric data. We convert the information included in the direct spectroscopic comparison of the SEDs of SMDSs and Population III/II galaxies (discussed in Section 6.1) in a comparison in terms of photometry with RST. We start with Figure 13, where we plot the calculated AB magnitudes (left vertical axes) and average fluxes (right vertical axes) of SMDSs and young Population III/II galaxies. This figure is the photometric representation of Figure 11, meaning all the parameters are the same as in Figure 11. Specifically, in order to get m_AB, we convolved, according to Equation (8), the redshifted SEDs with the RST throughput curves for each of the following bands: J129, H158, F184, and K213. In order to estimate the uncertainties in m_AB, we used Pandeia to obtain the S/N for each object, assuming 10⁶ s exposures. The same conclusion we drew from Figure 11, in terms of spectra, can be easily justified in terms of photometry from Figure 13. Specifically, SMDSs formed via DM capture will have almost identical photometric signatures when compared with Population III/II galaxies without nebular emission (top right panel of Figure 13). Moreover, now, we find that in terms of photometry even the SMDSs formed via AC are very similar compared to Population III/II galaxies without nebular emission (lower right panel of Figure 13).

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In Figure 14, we repeat the same analysis done in Figure 13, considering now old galaxies instead. From the left two panels, we can conclude that SMDSs formed via either formation mechanism will have colors (AB magnitudes) that are indiscernible from old Population III/II galaxies without nebular emission. And, in fact, most old galaxies should be expected to have little to no nebular emission. The only exception to this are old Population III 1 galaxies, which, just before the end of their evolutionary tracks at ∼3.5 Myr, can still exhibit quite a lot of nebular emission. This can be explicitly seen in the comparison of left to right panels, where the only galaxies that have different colors in the left (Type A, i.e., maximal nebular emission) to the right (Type C, i.e., no nebular emission) are Population III 1 galaxies, i.e., those with the most extreme top-heavy IMF. From them, the strong Lyα nebular emission drives the averages fluxes up in the H158 band (assuming, as we did here, z_emi = 12).

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To sum up, we find there is a clear difference between SMDSs formed via either formation mechanism versus Type A young galaxies based on their vastly different colors in the H158 band (see left panels Figure 13), but not versus Type C young galaxies. Comparing SMDSs with old (100 Myr old) galaxies, we find that there is no way to tell them apart from Population III/II galaxies, using RST photometry (Figure 14). Since we do not know ab initio what type of galaxy is actually in the sky, we have to conclude that SMDSs cannot be differentiated from all possible galaxies via photometry alone. Thus, RST photometric data will not be sufficient to uniquely determine that a ∼10⁶ M_⊙ SMDS has been discovered. ²⁷

6.3. RST Image Simulations Using Pandeia for Unlensed SMDS versus Early Galaxies: Morphology Alone Cannot Distinguish in RST

In this section, we address the following question: is it possible, based on image morphology—i.e., distinguishing between point and extended objects—to tell apart unlensed 10⁶ M_⊙ SMDSs from their counterpart Population III/II galaxies (those that have the same observed flux in the F184 RST band)? We will assume objects are observed with RST as dropouts with a photometric redshift z ≳ 10. We start this analysis by estimating the effective angular size of high redshift galaxies as a function of stellar mass (see Equation (10)). By comparing the effective size of galaxies to the size of an RST pixel we will soon show that unlensed galaxies will barely cover a few pixels. Thus, just as SMDSs, unlensed galaxies (counterparts to SMDSs) will appear unresolved, and as such, are indistinguishable based on image morphology alone, when observed with RST.

To reinforce this point, we will also present RST image simulations of unlensed SMDSs and Population III/II galaxies obtained using the Pandeia exposure time calculator developed at STScI (Pontoppidan et al. 2016). Among other things, Pandeia allows the user to simulate the effects of the PSF of different instruments on board JWST or RST. As such, one can perform high-fidelity modeling of the image(s), as seen with JWST or RST, of any hypothetical source(s) for which the rest-frame SEDs are known. For our comparison of simulated images for unlensed SDMSs versus galaxies, as viewed in various RST bands, see Figures 15 and 16.

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As stated above, we begin by estimating the effective angular size of galaxies and contrast it against the size of an RST pixel. The main ingredient in this calculation is the effective radii (r_eff) of those objects. The size of Population III/II galaxies are largely uncertain (due to the lack of observational data) and somewhat model dependent. However, JWST already offers a glimpse into the size of very luminous galaxies at high redshift. For instance, one of most distant objects found with JWST, Maisie's galaxy, at z ∼ 14, has an estimated half-light radius of r_eff = 330 ± 30 pc (Finkelstein et al. 2022). This is in excellent agreement with what one expects from extrapolating results from lower redshift observed galaxies. For instance, Figure 15 of Kawamata et al. (2018) presents the redshift evolution of the average size of bright galaxies observed up to 2018, in the redshift range 2–12. Using their best-fit function and extrapolating to z = 14, we find, remarkably, a value of r_eff = 330 pc, a perfect match to Maisie's galaxy size.

In what follows, we will proceed to estimate the galactic stellar mass dependence of the angular size of high redshift galaxies. Recall that, for galaxies with a stellar mass not much greater than 10⁹ M_⊙, the luminosity is directly proportional to the stellar mass. Therefore, by using the size–luminosity relationship from observed data, we can also make an estimate of how the effective size scales with the galactic stellar mass: ${r}_{\mathrm{eff}}\sim {L}^{\beta }\sim {M}_{\star }^{\beta }$, where β is the scaling coefficient. The mass–size relationship for lensed galaxies found with HST at high redshift (z ∼ 6–9) can be found in Figures 2 and 3 of Bouwens et al. (2021). In what follows, we take the value of r_eff ∼ 20 pc as the typical size of galaxies of 10⁶ M_⊙ stellar mass (inferred from their figures), and a value of β = 0.5, as found by Bouwens et al. (2021). Thus,

$$\begin{eqnarray}&&{r}_{\mathrm{eff}}\approx 20\,\mathrm{pc}{\left(\displaystyle \frac{{M}_{\star }}{{10}^{6}{M}_{\odot }}\right)}^{0.5}.\end{eqnarray} \tag{ 9 }$$

Since the redshift range considered by Bouwens et al. (2021; i.e., z ∈ [6 − 8]) does not cover our range of interest (z ≳ 10), we expect the linear fit given by Bouwens et al. (2021) to be mildly overpredicting the size of galaxies at higher redshifts, since, at the same stellar mass, the higher redshift galaxies are typically more compact, according to simulations and recent JWST observations. For Maisie's galaxy, M_⋆ ≈ 10^8.5 M_⊙ (Finkelstein et al. 2022), leading to a predicted r_eff ≈ 355 pc, which is within the observed r_eff = 330 ± 30 pc, albeit mildly overpredicted, as discussed. For the most massive galaxies we consider (i.e., M_⋆ ∼ 10⁹ M_⊙), we get, using Equation (9), an estimated size of r_eff ∼ 0.6 kpc, which agrees very well with the recent results from JWST observations of z ≳ 10 galaxies (e.g., Finkelstein et al. 2022; Naidu et al. 2022b). This further validates our method of estimating r_eff for a given mass galaxy (Equation (9)). Knowing the effective radius of a galaxy allows us to compute its angular size via the following prescription:

$$\begin{eqnarray}&&{\theta }_{\mathrm{eff}}\simeq 2\times 20\,\mathrm{pc}\times \displaystyle \frac{{(1+{z}_{\mathrm{emi}})}^{2}}{{D}_{{\rm{L}}}\left({z}_{\mathrm{emi}}\right)}{\left(\displaystyle \frac{{M}_{\star }}{{10}^{6}{M}_{\odot }}\right)}^{0.5}.\end{eqnarray} \tag{ 10 }$$

For z ∈ [12 − 14], an r_eff of 0.6 kpc corresponds to θ_eff ∈ [016–018]. The size of the pixel for RST is ∼011. In view of this, even the most massive galaxies we consider (M_⋆ ∼ 10⁹ M_⊙) have an unlensed size that is below two pixels. However, inherent diffraction always spreads out light over neighboring pixels, so even a point object (such as an SMDS) will end up covering an area of the detector that is a few pixels across (see top panels of Figures 15 and 16). Therefore, it will be difficult to distinguish SMDSs from Population III galaxies based on morphology alone, unless the objects of interest are gravitationally lensed, which is a possibility we consider in Section 7.

Next, we investigate, using the Pandeia engine (Pontoppidan et al. 2016), possible differences between SMDSs and Population III/II galaxies based on image morphology and/or S/N values. Of course, based on the discussion above, we already know that, for unlensed objects, the disambiguation based on image morphology between SDMSs and Population III/II galaxies will be challenging. The full power of the Pandeia engine will become transparent when we study lensed objects (see Section 7); however, we use it here as well, as a warm-up exercise, and to confirm our expectations obtained in the paragraphs above. The relevant parameters used in our image simulations can be found in Appendix A. For simplicity, we only assume one object in each scene, although in a real scenario one might expect other luminous sources in the adjacent field. Following the previous sections, we still use the 10⁶ M_⊙ SMDSs and take 10⁶ s of total exposure time, as fiducial values.

Simulated images of SMDS versus young (1 Myr) galaxies for the case of no lensing. In Figure 15, we present the simulated RST images of a 10⁶ M_⊙ SMDSs formed via capture (top row) versus those of select young (1 Myr) galaxies for the case of no lensing. ²⁸ A similar plot for SMDSs formed via extended AC (top panels) can be found in Figure 16. The angular diameter of the galaxies is given by Equation (10), with the mass selected in such a way that the AB magnitude in the F184 band is the same for all objects considered. In order to be conservative, in our simulation, the galaxies are assigned a spherical shape, such that it would be harder to tell the difference between the extended objects and stellar object based on the image morphology alone. However, RST will be largely insensitive to the degree of symmetry of the object, at the redshifts of interest for us. At z ∼ 12, even the 10⁹ M_⊙ galaxies have an effective size that barely covers a few pixels, which renders shape extraction impossible. From the simulation, we see that most of the flux received by the detector is concentrated in the central pixel of our target object. As predicted in the previous sections, we find that all of flux in the J₁₂₉ has been attenuated by the neutral H present in the IGM, leading to detection as photometric dropouts. In the F213 band, due mostly to the decrease in the capability of the detector, all objects appear extremely dim. The strong Lyα emission of young Type A galaxies boosts their fluxes in the F158 band (as can be seen from Figure 11), which leads to an enhanced S/N in the same band (see second and fourth rows of Figure 15). Note the almost perfect match between the simulated images for Population III.1C galaxies and SMDSs formed via DM capture (see top and third from the top rows of Figure 15), thus further demonstrating that SMDSs can pose as young Population III Type C galaxies (i.e., with little to no nebular emission). This is due primarily to their almost identical spectra (see top right panel of Figure 11) and the fact that at those high redshifts, without gravitational lensing, even the most massive galaxies are not resolved by RST (see Figure 19 for the effect of gravitational lensing).

In Figure 16, we compare the Pandeia simulated images of SMDSs formed via AC to those of young (1 Myr) Population III/II galaxies, selected following the same prescription followed for Figure 15, again for the case of no lensing. All objects are assumed to emit at z_emi = 12. All Type A young galaxies will have a much higher S/N in the F158 band, when compared to SMDSs with purely stellar spectra. This fact is due, again, to the strong Lyα nebular emission line present in the F158 filter at that redshift. Note that even if the slope of the UV continuum in the SEDs of SMDSs formed via AC and those of Type C galaxies is different (lower right panel of Figure 11) RST will not be sensitive enough to detect this using photometry alone, as can be seen from comparing the top and third from the top rows of Figure 16, and as we have explicitly seen in the lower right panel of Figure 13. Therefore, young Population III galaxies without significant nebular emission can also pose as SMDSs formed via the AC mechanism.

Image morphology of SMDS versus older (3–100 Myr) galaxies for the case of no lensing. Here, we discuss the image morphology of the older Population III/II galaxies when contrasted to that of SMDSs, as they would be seen by RST. As explained in our discussion of Figure 12, old (100 Myr) galaxies no longer produce nebular emission, rendering the distinction between Type C and Type A irrelevant. Old Population III/II galaxies have SEDs that are very similar to those of purely stellar SMDSs. However, the older a galaxy is, the more massive it has to be in order for it to match the brightness of an SMDS of a fixed mass (see Figure 10). Even for the most massive galaxies considered (M_⋆ ∼ 10⁹ M_⊙), the effective area in RST is below the size of 2 × 2 pixels, which renders them either impossible to resolve or with S/Ns less than 5 in each pixel, if unlensed, as can be seen in Figure 20.

In summary, we find that unlensed SMDSs can be easily mistaken for unlensed Population III.1C galaxies throughout their evolution. For the case of Population III.1A (maximum nebular emission) galaxies, the strong Lyα line in the F158 band will be a telltale signature, even in photometry, that can set them apart from SMDSs with purely stellar spectra.

We end this section with the same cautionary note mentioned at the end of Section 5 regarding the disambiguating between SDMSs and the first galaxies. As pointed out by Zackrisson (2011), the hottest SMDSs (those formed via DM capture) can have a detectable nebular emission contribution to their SEDs. This, in turn, would lead them to appear as compact, but extended objects, with SEDs that are similar to those of Type A galaxies. In a future publication, we plan to analyze this possibility in detail. In this paper, we restrict our attention to SMDSs with purely stellar spectra, which is the most likely scenario, as discussed in Appendix B.

In conclusion, for unlensed objects, morphology alone will not be enough to clearly identify an observed object as an SMDS versus early galaxies in RST data. In other words, one cannot differentiate cleanly between an SMDS as a point object versus galaxies as extended objects. Some types of early galaxies will look different from SMDSs. However, for any image in RST data that matches predictions for an SMDS, there is always some galaxy type that could produce a virtually indistinguishable image. We will address the effects of gravitational lensing on this question in Section 7.1.

6.4. Color–Color Index Comparison for the Case of No Lensing

In this section, we investigate the ability to differentiate unlensed SMDSs from Population III/II galaxies using the color–color method, following Zackrisson et al. (2010), Ilie et al. (2012). In principle, different objects could occupy different locations in a color–color plot, thereby allowing a way to identify the nature of the object in the data. As before, we focus on z_emi ∼ 12 objects found as J₁₂₉ dropouts. Specifically, we will plot the color–color diagram in F₁₈₄ − K₂₁₃ versus H₁₅₈ − F₁₈₄ for different objects, as those are the only three RST bands available for objects at z ∈ [12 − 13.5]. By z ≃ 13.5, the Gunn–Peterson trough will cover the entire F158 band, and as such, for z_emi ≳ 13.5, one can no longer use the color index method, as it requires detection in at least three photometric bands.

In Figure 17, we present the color–color diagram for unlensed 10⁶ M_⊙ SMDSs ²⁹ formed via DM capture (red circles) or AC (red stars). On the left/right panels, we contrast the colors of SMDSs to those of unlensed Population III/II galaxies with maximal/no nebular emission (Type A/C), as they evolve from young (1 Myr; lowest points along the evolutionary tracks) to old (100 Myr; highest points along the tracks) galaxies. The error bars of the magnitude difference ${\sigma }_{{m}_{1}-{m}_{2}}$ are calculated by the following formula:

$$\begin{eqnarray}&&{\sigma }_{{m}_{1}-{m}_{2}}=\sqrt{{\sigma }_{{m}_{1}}^{2}+{\sigma }_{{m}_{2}}^{2}}=\sqrt{\displaystyle \frac{1}{{\rm{S}}/{{\rm{N}}}_{1}^{2}}+\displaystyle \frac{1}{{\rm{S}}/{{\rm{N}}}_{2}^{2}}},\end{eqnarray} \tag{ 11 }$$

where ${\sigma }_{{m}_{1}}$ and ${\sigma }_{{m}_{2}}$ represent the standard deviation of the magnitude in the neighboring filters, which is given by the inverse of the S/N value that corresponds to the image simulation results calculated by Pandeia. Thus, the color–magnitude for SMDS formed via capture is m₁₅₈ − m₁₈₄ = 0.61 ± 0.17, and m₁₈₄ − m₂₁₃ = − 0.15 ± 0.43, and those for SMDS formed via AC are as follows: m₁₅₈ − m₁₈₄ = 0.76 ± 0.07, and m₁₈₄ − m₂₁₃ = 0.07 ± 0.13. Note that the colors of unlensed SMDSs are well within one standard deviation of the evolutionary tracks, for Population III/II galaxies, with or without nebular emission.

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Therefore, it would be impossible to differentiate between unlensed SMDSs and most Population III/II galaxies using the color–color plot alone; the inclusion of the error bars for the evolutionary lines of galaxies, which we omitted here for simplicity, would only strengthen this conclusion. The only exception is Population III.1A (i.e., top-heavy IMF with maximal nebular emission) galaxies (blue solid lines in the left panel of Figure 17). Due to them becoming dim at an age of ∼3.5 Myr, their evolutionary tracks never reach the loci occupied in the color–color plot by SMDSs. In the next section, we will consider the effects of gravitational lensing on both image morphology and color index for the prospects of telling apart SMDSs from the first galaxies.

6.5. Conclusions for Unlensed SMDSs in RST

Our discussion in Section 5 above only pertained to unlensed objects, both SMDS and early galaxies. Now, we turn to the case where these objects are gravitationally lensed by foreground material. The method of strong lensing as a tool to find high redshift objects has been successfully applied and refined in recent years. The detectability of dark stars via lensing was first explored by Zackrisson et al. (2010), who found that a magnification factor μ ≳ 160 is needed in order to observe dark stars (at z_emi ≃ 10 ) with low masses ∼100M_⊙ with JWST. For the RST, we find (see Figures 5 and 6) that unlensed SMDSs of mass smaller than ∼10⁶ M_⊙ are too faint to be detected even with exposure times of ∼10⁶ s. Therefore, SMDSs of M ≲ 10⁶ M_⊙ can only be detected if they are gravitationally lensed, as we will explicitly show in this section.

Lensing by massive foreground galactic clusters at z ∼ 0.5 (as described by Zitrin et al. 2009 for example) would lead to large magnification factors (μ > 10). For example, Coe et al. (2013), Lam et al. (2019) describe the discovery of high redshift candidate galaxies (z ∼ 11) through lensing. Perhaps, one of the most striking examples of strong gravitational lensing is the recent discovery of the most distant star (z ∼ 6.2) ever observed: Earendel (Welch et al. 2022a, 2022b). The light from the 50–500M_⊙ stellar object has been emitted when the universe was merely 900 million years old. ³⁰ The lensing effect from the foreground cluster boosts the stellar flux by 1000–40,000 times, providing a great example on how strong gravitational lensing allows the observation of very distant individual stars. Additionally, soon after JWST released its first images, the list of high redshift galaxy candidates has been growing almost on a weekly basis, with the record for the "most distant candidate galaxy" being broken multiple times (e.g., Castellano et al. 2022; Finkelstein et al. 2022; Naidu et al. 2022a, 2022b; Donnan et al. 2023). If those objects are indeed at the redshifts indicated by photometry, then at least some of them must have been gravitationally lensed in order to have been observed. In a separate publication, we plan to analyze if the available data from any of those candidates can be well-fit by SMDSs SEDs. Recently, we have shown that, out of the four z ≳ 10 spectroscopically confirmed "galaxies" found by Robertson et al. (2023), Curtis-Lake et al. (2023), the three most distant ones (specifically JADES-GS-z13-0, JADES-GS-z12-0, and JADES-GS-z11-0) are each consistent with a supermassive dark star interpretation (Ilie et al. 2023).

In this section, we discuss the effects of gravitational lensing on the prospects of telling apart SMDSs from the first galaxies using RST by using image morphology (Section 7.1) and color–color indices (Section 7.2). Below, we remind the reader what are some reasonable amounts of lensing we expect based on a nonexhaustive list of previous observations. Besides the extreme case of Earandel, discussed above, we mention other examples of greatly lensed stars: Kaurov et al. (2019) finds a star with μ ∼ 200 at low redshift z ∼ 0.9; Meena et al. (2023), one star with μ ∼ 200 and one star with μ ∼ 50 at high redshift z ∼ 4.8. For galaxies, Hsiao et al. (2022) finds a triply lensed z ∼ 11 candidate, with combined μ ∼ 15; Bradley et al. (2022) find z ∼ 9–13 galaxies lensed by μ ≲ 10; Adams et al. (2023), one lensed by μ ∼ 6 with z ∼ 9.

7.1. Image Morphology

Gravitational lensing magnification would increase the apparent size of an object, but for nearly point sources (stars), this translates to the magnification in the flux density by a factor μ, called the magnification factor (for details, see Schneider et al. 1992, for example). This factor is given by the following (e.g., Kilbinger 2015, for a review):

$$\begin{eqnarray}&&\mu =\displaystyle \frac{1}{\left[{(1-\kappa )}^{2}-{\gamma }^{2}\right]}\end{eqnarray} \tag{ 12 }$$

where κ is determined by the mass field of the lens as a function of the position vector, κ ∝ D_L D_LS/D_S; where D_L, D_S, and D_LS refer to the angular distance to the lens, the source, and their distance in between, respectively. γ is the shear factor that determines the distortion of the source object. In a strong-lensing survey out of the catalog of 12 galaxy clusters from the MAssive Cluster Survey (MACS), 9 of them have a large area ($\gt 2.5\,{\square }^{{\prime} }$) ³¹ of high lensing magnification (μ > 10) for source objects at z ∼ 8 (Zitrin et al. 2011). Since our targeted redshift is z ∼ 12, we find a simultaneous increase in D_LS and D_S, thus an increase in the value of κ. Therefore, for the same relative position in the sky, a higher redshift corresponds to a higher μ, which means we have an even larger area for high lensing magnification. And since this area is much larger than the size our target sources (order of ∼□^''), ³² it is possible for multiple Population III galaxies and/or SMDSs to fit in this magnification area.

It is likely to observe multiple images or giant arcs from strong lensing (e.g., Welch et al. 2023), but given the relatively small size of galaxies at very high redshift, they might still appear as spatially unresolved (e.g., Coe et al. 2013; Lam et al. 2019). For simplification, and in order to be the most conservative, we assume a spherical shaped image and that γ = 0 throughout which means the magnification would not distort the shape of the object for each individual image (although this simplification is unlikely in practice as it requires homogeneous distribution of mass). The reason for this choice is that a distorted image would be a telltale sign of an extended object (galaxy), rather than a point object (SMDS with no nebular emission). As a result of lensing, the surface area is increased by a factor known as the convergence factor. Assuming a uniform convergence, as justified above, this factor simply becomes 1/(1 − κ)² = μ. Furthermore, we consider our sources to be close to the critical curve to a degree such that a high magnification factor (μ > 10) could be achieved without generating a spatially resolved arc.

The main goal of this subsection is to investigate how gravitational lensing could be used to tell apart first galaxies from supermassive dark stars, as observed with RST. The former are extended objects, with radii ranging from a few tens of parsecs to hundreds of parsecs while the latter are puffy stars, with radii of the order of ∼10 au. As we have seen in Section 6.3, RST's angular resolution is not sufficient to tell those two kinds apart conclusively, without gravitational lensing. For each galaxy considered, we determined their magnified angular size by using the Pandeia simulator. Our procedure is exemplified in Figure 18, where we extract the effective lensed sizes for two objects of interest magnified by μ = 100: a 10⁶ M_⊙ SMDSs formed via AC (left panel) and a young (1 Myr) Population III.1C galaxy (right panel). Note how now the light from the galaxy covers a significantly larger area of the detector, indicating an extended object. In order to estimate the effective size of each object, we use the standard full width at half-maximum (FWHM) histogram technique. Namely, we slice the images along a direction. In the figure, this is represented by the red horizontal band that goes to the left and the right central pixel of each simulated image. The unit flux values in pixels along this band are then plotted as a histogram (blue lines in the bottom panels of Figure 18). For SMDSs, as they are spherically symmetric stellar objects, we apply the conventional Gaussian fit (e.g., Mighell 2005). Conversely, for Population III galaxies, we simply interpolate the flux values at each pixel. The corresponding FWHM values are extracted and plotted as black lines in each of the bottom panels of Figure 18. For instance, for the SMDSs, we find that its effective size will be roughly 017, and for the Population III galaxy chosen here, the estimated effective size is ∼041. For the band considered (F184), the angular resolution, according to the Rayleigh criterion (1.22λ/D), where D is the diameter of the telescope, leads to a minimum angular size of ∼019 for an object to be resolved. Thus, the SMDSs, even if lensed, are still unresolved, whereas the Population III.1C galaxy will show as an extended object. It is important to note here that the estimated effective size for all objects that remain unresolved is merely an upper-bound of the actual size of the object.

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Using the procedure described above, in Figure 19, we estimate the apparent sizes as a function of lensing magnification factor (μ) for young (1 Myr) Population III/II galaxies (upward trending lines, as labeled) that match the integrated F184 flux of supermassive dark stars (top/bottom rows for 10⁶/10⁵ M_⊙ SMDSs) formed via either formation mechanism (left/right columns correspond to DM capture/AC). In the lower panels, the evolutionary lines break sharply downward, signifying that, below the magnification where this break happens, the S/N in each pixel of the F184 band becomes lower than the minimum S/N = 5 value required for detection. In each of the four panels, the black dashed horizontal line (labeled Rayleigh criterion) is the angular resolution of the RST in the band considered here (F184), estimated to be ∼019. Note that SMDSs at z_emi ∼ 12 are unresolved even at μ ≫ 100. Thus, for all intents and purposes, SMDSs without nebular emission will remain point objects, even with extreme magnifications, and even with telescopes with much better angular resolution than RST. The galaxies considered will become resolved once μ becomes larger than a critical value that can be read off the plot at the intersection of the horizontal dotted line (labeled as Rayleigh criterion) and the upward trending curve corresponding to the galaxy in question. Young Population III.Kroupa Type C galaxies will be the easiest to resolve, in view of their large mass needed in order to match the flux of the SMDSs in the F184 band (see Figure 10). In contrast, Population III.1 Type A galaxies need the smallest stellar mass to match the flux of the SMDSs, and are more difficult to resolve. When contrasted against a 10⁶ M_⊙ SMDS formed via DM capture, a Population III.1A galaxy will need to be magnified by at least μ ∼ 125 in order to have an angular size barely above the resolution limit and, as such, somewhat distinguishable, based on image morphology from the SMDS. For the case of SMDSs formed via AC, the corresponding Population III/II galaxies that match their F184 flux are easier to resolve, as it can be seen from contrasting the left and right panels of Figure 19. In the lower two panels of Figure 19, we consider 10⁵ M_⊙ SMDSs formed via either mechanism. In this case, even at μ ∼ 100, there are still many Population III/II galactic counterparts that will not be resolved. Thus, for SMDSs with M ≲ 10⁵ M_⊙, the differentiation from young (∼1 Myr old) Population III/II galactic sources based on image morphology is only possible at μ ≫ 100. However, the color–color technique can be very useful in telling apart a M ≳ 10⁵ M_⊙, formed via AC, from Population III/II galaxies (see Figures 23 and 24).

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The main takeaway of Figure 19 is that SMDSs will appear as point objects, independent of the magnification factor. In contrast, all young Population III/II galactic counterparts of a 10⁶ M_⊙ SMDS formed via AC would be resolved by RST with μ ≳ 40 thus making disambiguation possible based on image morphology whenever μ ≳ 40. For 10⁶ M_⊙ SMDSs formed via capture, the same would be possible with a higher magnification factor of μ ≳ 125, but at μ ∼ 100, it will be hard to disambiguate them from young Population III.1A galaxies. For lower mass SMDSs (M ≲ 10⁵ M_⊙), independent of the formation mechanism, one needs μ ≫ 100 to be able to use image morphology as an RST differentiating tool from young Population III/II galaxies.

In Figure 20, we repeat the same analysis done in Figure 19, this time for older galaxies, with age of 100 Myr. ³³ The top two panels of Figure 20 demonstrate that most old galaxies that match the flux of a 10⁶ M_⊙ SMDSs in the F184 band will be above the resolution limit as soon as they are detected at an S/N = 5 level. The only exception is the Population III.1 galaxies (blue lines), which, when contrasted against a 10⁶ M_⊙ SMDSs formed via DM capture (top left panel), can be detected as unresolved, as long as it is magnified by μ ≲ 5. With this in mind, we conclude that old galaxies would be, in principle, differentiable from 10⁶ M_⊙ SMDSs based on image morphology. Regarding the 10⁵ M_⊙ SMDSs, all of their old Population III/II galactic counterparts are large enough to be resolved at μ ranging from a few to a few tens. In particular, for the case of the 10⁵ M_⊙ SMDS formed via DM capture (lower left panel of Figure 20) by μ ∼ 50, even the more compact Population III.1A (blue solid line) galaxies are resolved. When contrasting 10⁵ M_⊙ SMDSs formed via extended AC (lower right panel), we note that at μ ≳ 20 all Population III/II galaxies will show as extended objects, in contrast to the SMDS. Thus, differentiation based on image morphology of a ∼10⁵ M_⊙ SMDS, formed via either mechanism, from Population III/II old galaxies is possible with magnifications μ ≳ 50.

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Since the age of an object detected with photometry cannot be estimated exactly, one should view our two extreme cases considered in Figures 19 and 20 as the two limits in between which any observation would lie. Moreover, we point out that the size of the galaxies or other lensed objects will depend on the assumed lensing model and is usually associated with a large error bar, as shown for example in Figure 8 of Lam et al. (2019). Thus, one should be careful in using the Rayleigh criterion as a disambiguation between galaxies (extended objects) and SMDSs with purely stellar spectra (point objects), especially whenever the objects are barely resolved.

In summary, in this section, we have demonstrated that, with sufficient magnification (μ ≳ 40), a 10⁶ M_⊙ SMDS formed via AC can be differentiated via image morphology from all Population III/II galaxies considered. The galaxies would all appear as extended objects whereas the SMDSs are still unresolved, even for μ ≫ 100 (see top right panels of Figures 19 and 20). In contrast, for all μ ≲ 100, the image morphology cannot be used to conclusively differentiate between galaxies and M ≲ 10⁵ M_⊙ SMDS (lower panel of Figure 19) or between galaxies and a 10⁶ M_⊙ SMDS formed via DM capture (top left panel of Figure 19). In the next section, we will explore if lensed SMDSs can be differentiated from Population III/II galaxies based on their locations in color–color diagrams.

7.2. S/N and Color–Color Diagram

In this subsection, we discuss the effects of gravitational lensing on the prospect of detecting smaller mass SMDSs and the possibility to use color–color diagrams to differentiate those SMDSs from Population III/II galaxies. We begin our analysis by asking the following question: what is the required magnification factor needed for an SMDS of a given mass in order to be detectable with RST at the level of S/N = 5? In order to answer this question, we plot in Figure 21 the values of the S/N in the F184 band as a function of magnification (μ) for SMDS of various masses (as labeled) formed either via DM capture (left panel) or via AC (right panel). The S/N values are estimated using Pandia and are selected from the pixel where most of the photons from the SMDS will generate a signal, in a hypothetical 10⁶ s exposure. We noted before that at the same mass SMDSs formed via AC look brighter than their counterparts formed via DM capture, which can also be seen in the slightly larger S/N values on the curves in the right panel (SMDS AC) of Figure 21. The main takeaway of Figure 21 is that the magnification factors of μ ≳ 10 are sufficient to lead to SMDSs of M ∼ 10⁴ M_⊙ (formed via either mechanism) to be detectable with 10⁶ s exposures by RST.

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In Figure 22, we repeat the same exercise done in Figure 21, but this time, we consider dark stars of lower masses, i.e., with M ≲ 10³ M_⊙. We assume z_emi ≃ 12, and a heat source from 100 GeV WIMPs powering the DSs. The properties of DS models used here (such as T_eff and R) are taken from Table 2 of Spolyar et al. (2009). In the left panel of Figure 22, we show the S/N values in the F184 RST band, whereas the right panel depicts the S/N values in the F213 RST band. In both cases, an exposure time of 10⁶ s is assumed. The weakened sensitivity in the F213 filter leads to higher μ values required for a detection at the same S/N level. Further, we can comment on the difference between the cases of SMDS formed by AC (the 479 M_⊙ and 716 M_⊙ SMDSs shown in the figure) versus the DS formed via capture (the 756M_⊙ and 787M_⊙ DSs shown in the figure). SMDSs formed via capture are hotter, leading to a peak in the SED at shorter wavelengths. Thus, in the two filters considered here, the 479M_⊙ and 756M_⊙ DSs (formed via capture) are somewhat dimmer and require more magnification (a larger μ) than the 716M_⊙ and 787M_⊙ DSs formed via AC.

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The main takeaway of Figure 22 is that dark stars as distant as z_emi ≃ 12, with masses as low as M ≃ 800M_⊙ can be observed with RST, if magnified by μ of a few hundreds or more. This is consistent with the findings of Zackrisson et al. (2010), who shows that for μ ≃ 160 DS at z_emi ≃ 12, with masses as low as 700M_⊙ can be detected at the S/N = 5 level in JWST, assuming exposure times of 100 hr.

In the remainder of this subsection, we focus on the role of gravitational lensing on differentiating SMDSs from Population III/II galaxies using the color–color technique discussed in Section 6.4. We start with Figure 23, where the color index analysis in the m₁₈₄ − m₂₁₃ versus m₁₅₈ − m₁₈₄ plane presented in Figure 17 is redone, assuming now that all objects are magnified with a lensing factor μ = 100. In the left/right panel, we compare the color indices of SMDSs formed via DM capture (red circles) and AC (red star symbols) to the evolutionary tracks of Population III/II Type A/C. For each SMDS, we calculate the size of the error bar according to Equation (11). The boosted S/Ns lead to a significant reduction of the uncertainty in the color indices of SMDSs, as can be seen by comparing Figures 17 and 23. Moreover, since SMDSs formed via DM capture have roughly equal temperature, independent of stellar mass, their SEDs (and therefore colors) do not change significantly as the SMDS grows. For this reason, all the color indices of the SMDSs formed via DM capture are stacked upon each other at a point that is very close to the evolutionary tracks of both types of galaxies. This reinforces once more that SMDS formed via DM capture can be easily mistaken for early galaxies if neither of those objects is resolved. We do not add here error bars for the points along the evolutionary tracks of galaxies, as those uncertainties will be dependent on the S/N (via Equation (11)), and, as such, on the mass of the SMDSs counterpart. However, we revisit this in Figure 24 where we plot separately each SMDS formed via AC, which allows for the inclusion of uncertainties in color indices of both galaxies and SMDSs.

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Figure 23 reinforces the point that SMDSs formed via capture are great Population III/II galactic chameleons. Independent of their mass, they will occupy the same spot in the m₁₅₈ − m₁₈₄ versus m₁₈₄ − m₂₁₃ color–color plot, a spot that is along the evolutionary lines of both Type A (left panel) and Type C (right panel) galaxies. A natural question arises: can one, in absence of spectral data, tell apart an SMDS formed via DM capture from the Population III/II galaxies' counterparts that occupy roughly the same location in the color–color plot of Figure 23? First, notice that the color–color technique is very valuable, as it clearly disambiguates between both types of SMDSs and Population III.1A galaxies. Therefore, considering the case of a 10⁶ M_⊙ SMDS formed via DM capture, for μ ≳ 40, the image morphology can break the degeneracy presented in Figure 23, as at those magnifications the galactic counterparts (that also occupy the same location in the color–color plot) will be all resolved, whereas the SMDSs will still be well under the resolution limit (see Figure 19). Naturally, for lower mass SMDSs, a larger magnification is required to achieve the same effect.

When contrasting SDMSs formed via extended AC (red star symbols in Figure 23) to Population III/II galaxies, the situation becomes more promising, as SMDS of smaller and smaller mass deviate more and more from the evolutionary tracks of both Types A and C of galaxies. All SMDSs formed via this mechanism, boosted by μ ∼ 100, will be more than 2σ away from the evolutionary tracks of galaxies. The one caveat here is that Figure 23 does not include the uncertainty in the color indices of galaxies, as explained above. We address this point in Figure 24, where we present the color index comparison between SMDSs formed via AC of 10⁵ and 10⁶ M_⊙ to the evolutionary tracks of Type A/C Population III/II galaxies (top/bottom panels) that are selected in such a way that they match the F184 flux of the SMDS, as per our comparison criterion. For all objects, a gravitational lensing magnification factor of μ = 100 is assumed. In order to be conservative, and avoid unnecessarily cluttering the plots, we only keep the Kroupa IMF type galaxies in this figure, since their evolutionary tracks lie closest to the SMDSs. However, our conclusions are unaffected by this esthetic simplification.

For the magnification μ = 100 assumed in Figure 24, we find that 10⁵ and 10⁶ M_⊙ SMDS formed via extended AC are separated at the 2σ level (in each direction) from the evolutionary tracks of the Population III/II galaxies' counterparts that have the same F184 flux. For smaller SMDS, the distinction between SMDS and galaxies is even smaller. Hence, a larger magnification would be required to cleanly differentiate the type of object in a color–color plot. If objects with high enough magnification are found to unambiguously identify an SMDS, such a discovery could indirectly confirm the nature of the DM.

DSs can form at redshift z ∼ 10–20. Made almost entirely of hydrogen and helium, they are fueled by DM heating. As they are relatively cool (T_eff ≲ 50,000 K), dark stars can grow, via accretion, to become supermassive (SMDS) and extremely bright. In this paper, we examined the capability of the upcoming RST to detect SMDSs. First, we showed that the sensitivity of RST does indeed allow for detection of SMDS, and then, we turned to the question of whether or not they can be differentiated in the data from high redshift galaxies.

RST will be an excellent observatory in the search for dark stars. Due to its large FOV, it has a great potential for the discovery of new objects at redshifts z < 16. In this paper, we considered supermassive dark stars in the mass range 10⁴–10⁶ M_⊙ formed via two possible mechanisms: extended AC and capture. Here, extended AC refers to the case in which the DM is accreted gravitationally only; the c apture mechanism relies upon an additional ingredient, capture of further DM via the scattering of nuclei in the star. As our fiducial values, we have studied SMDS emitting at z_emi = 12, and DM mass m_χ = 100 GeV.

SMDS detectability in RST. First, we compared predictions of SMDS spectra (simulated with TLUSTY as shown in Figure 2) against the sensitivity of RST filters to determine which SMDSs at which redshifts would be detectable by RST, using the standard dropout techniques. We found that SMDS candidates formed via either the capture or extended AC formation mechanism could indeed be detected by RST using photometry alone.

Figures 5 and 6 show the comparison of AB magnitudes of unlensed SMDSs at different redshifts, in contrast to the detection limit (S/N = 5) for each RST filter assuming exposure times of 10⁴ and 10⁶ s. For SMDSs with masses ∼10⁶ M_⊙, formed via AC or capture, we find that at emission redshift z_emi ∼ 10–15 they would be observable by RST with ≲10⁶ s of exposure time, even if unlensed. With lensing factors of μ ≳ 10, RST can easily observe SMDSs with masses as low as M ≲ 10⁴ M_⊙ (see Figure 21). SMDSs will appear as J-band dropouts at z_emi ∼ 10 and H-band dropouts at z_emi ∼ 13, and as F₁₈₄ (H/K) dropouts at z ∼ 14. In order to be able to also include the color index technique, which requires photometric data in at least three bands, we chose z_emi ∼ 12, as our fiducial value throughout this paper.

Differentiating SMDS from high redshift galaxies. A key question is whether or not SMDS can be differentiated in the data from high redshift (zero (Population III) or very low metallicity (Population II)) galaxies. For these galaxies, the spectral evolution with respect to age, nebular emission, IMF, and metallicity were obtained using the YggDrasil (Zackrisson et al. 2011) simulated model grids, and the relevant choice of parameters are given in Table 5. For the purposes of comparison to SDMS, we chose Population III/II galaxies that have the same integrated observed F184 flux as the DSs in question. We used several approaches to see if we can differentiate SDMSs from early galaxies: SEDs in photometric data, location in color–color plots, and image morphology (point versus extended objects).

Given the resolution of RST, SMDS will be point objects whereas galaxies may be resolved. Without lensing, many galaxies will remain unresolved so that one cannot tell the difference from SMSs. The higher the magnification, the more likely that galaxies are resolved whereas SMDS remain point sources. We estimate that the magnification μ ∼ 100 is sufficient to resolve the all galaxies we have investigated, but upcoming observations will determine this number more accurately. We note that airy patterns, being specific only for point objects, would definitively identify them as such. However, very long exposure times ${ \mathcal O }$(year) would be needed for their detection, since the first ring is only 1.75% as bright as the central spot. Therefore, the discovery of such a pattern in RST is unlikely for dark stars, yet it is a signature of SMDS that future telescopes would be able to see.

For objects only seen with RST photometry (rather than spectroscopy), SEDs of both SMDS and Population III/II galaxies can equally well match observations. One approach to differentiate SMDS versus galaxies would be their location in color–color plots. We used the Pandeia engine to simulate RST observations of different candidate objects (e.g., Figures 15 and 16). ³⁴ Specifically., we use the F − K versus H − F color indices for all of the objects of interest (see Figures 17, 23, and 24).

In Table 3, we summarize the potential to detect SMDSs and then to disambiguate them via image morphology and/or color index techniques from their respective Population III/II galactic counterparts. The third and fourth columns take binary (yes/no) values and address the question of detection at S/N = 5 level, without lensing (column three) or with μ = 100 lensing (column 4). Similarly, in the fourth and fifth columns, we summarize the answers to the following question: can an SMDS be differentiated from all possible galactic counterparts based on image morphology, i.e., point source (SMDSs) versus potentially resolved extended object (galaxy)? The last two columns summarize the potential to use the m₁₅₈ − m₁₈₄ versus m₁₈₄ − m₂₁₃ color diagram to separate SMDSs of a given mass versus galaxies.

Table 3. For the SMDSs of Stellar Mass and Formation Mechanism as Indicated, the Table Summarizes the Detectability Results Based on Drop-out, Color–Color, Morphology Methods, and Presence of the He ii λ1640 Absorption Line for SMDSs

Formation Mechanism	${M}_{* }\left({M}_{\odot }\right)$	Drop-out	(S/N > 5)	Morphology	Morphology	Color–color	Color–color	He ii 1640 Absorp. Line
		No Lensing	μ ∼ 100	No Lensing	μ ∼ 100	No Lensing	μ ∼ 100
Extended AC	2.04 × 104	No	Yes	No	No	No	No	No
Extended AC	105	No	Yes	No	No	No	At 2σ	No
Extended AC	106	Yes	Yes	No	Yes	No	At 2σ	Yes
Capture	4.1 × 104	No	Yes	No	No	No	No	Yes
Capture	105	No	Yes	No	No	No	No	Yes
Capture	106	Yes	Yes	No	Yes	No	No	Yes

Note. We show results for the case of no lensing as well as with lensing (magnification factors μ listed in table header). In all cases, an exposure time of 10⁶ s is assumed.

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Differentiating SMDS with masses up to 10⁶ M_⊙ against early galaxies without spectroscopy proves to be difficult. As mentioned above, both types of SMDS as well as early galaxies have SEDs that match RST data ∼equally well. For SMDS formed via extended AC, for SMDS masses with M ∼ 10⁵, 10⁶ M_⊙, the loci occupied by the SMDSs in the color–color plots deviate from the evolutionary lines of Population III/II galaxies at the 2σ level for the case of μ ≳ 100 (see Figure 24). For smaller SMDS or for higher statistical significance of the difference between the objects, even larger magnifications would be required. For SMDS formed via capture, the locations in color–color plots cannot be differentiated from early galaxies; the only differentiating characteristic would then be that these are point objects whereas galaxies can be resolved at sufficient magnifications. For instance, for the 10⁶ M_⊙ SMDS via DM capture, by μ ≳ 100, all the galactic counterparts are resolved (top left panels of Figures 19 and 20).

There is a smoking gun for detection of SMDS once the spectroscopy is available: a telltale spectroscopic signature, the He ii λ1640 absorption line. While RST does not cover the wavelength band required to find this line (for z_emi ≳ 10), JWST does. Hence, the two detectors can be used together in identifying SMDS. SMDS formed via capture would produce this line; for the cooler SMDS formed via extended AC, this feature is present only for extremely massive (M ≳ 10⁶ M_⊙) SMDSs (see Figure 2). However, there is a caveat: as yet, we have not modeled the nebula surrounding the star, which in the case of the hotter SMDS formed via capture could wash out the absorption line and/or produce emission lines. The cooler SMDSs formed via extended AC are not likely to be affected. In follow-up work, we plan to use the CLOUDY code to add the effects of the nebula to the stellar spectra produced by TLUSTY. In any case, should a He ii λ1640 absorption line be found, it would be a strong indicator that an object found in RST is an SMS rather than an early galaxy.

While the He ii λ1640 line is a very useful spectroscopic tool, since it is isolated from other neighboring lines (see Figure 2), we note here that SMDSs differ from Population III/II galaxies with respect to their spectral signatures in other important ways. First of all, the Balmer break, i.e., the sudden jump in the flux at around 0.35 μm (rest frame), can be seen in the lower two panels of Figure 2 as a hump-like feature. This feature will be more pronounced for Population III/II galaxies with nebular emission, but most importantly, the sequence of lines that follows the Balmer break, i.e., the Balmer series, are emission lines for galaxies with nebular emission versus absorption lines for SMDS with purely stellar spectra. Thus, in addition to the He ii λ1640 line, as a spectroscopic disambiguation tool, one can also use lines in the Balmer series as well.

On heavier 10⁷ M_⊙ SMDS. In this paper, we have focused on SMDS with masses up to 10⁶ M_⊙. However, in principle, SMDS could grow to become even heavier, e.g., if they formed in 10⁸ M_⊙ minihalos or if smaller minihaloes merged together and the DS remains in a DM-rich region. We therefore consider below the detectability of 10⁷ M_⊙ SMDS. Since total luminosity scales linearly with mass, these objects would be 10 times brighter than 10⁶ M_⊙ SMDS and thus easier to detect. We note, however, that the differentiation in color–color plots against early galaxies would actually be more difficult (see Figure 21). On the plus side, galaxies this bright should be easier to resolve. Thus, the differentiation of these very heavy SMDS versus early galaxies should be possible. Further, the He ii λ1640 line would be more pronounced (deeper) for these hotter objects; such a line would be detectable in RST only for z_emi < 10 while JWST could find it even for higher redshift objects. In summary, 10⁷ M_⊙ SMDS are much brighter, therefore easier to detect as well as differentiate from early galaxies.

Using RST and JWST together. The primary advantage of RST is that it will have a much larger FOV than that of JWST. Thus, at the redshifts RST is able to probe, it will have a larger probability of successfully finding SMDSs. Two of us (Ilie et al. 2012) have previously studied SMDS in JWST data (including estimating the numbers of SMDS one could find with JWST). As shown in Figure 3, RST and JWST have comparable detector sensitivity for wavelengths up to ∼1.8 μm, including the J and H bands relevant for identifying objects as J- and H-band dropouts, corresponding to z ∼ 11 and z ∼ 13 respectively. JWST is far more sensitive at higher wavelengths and thus can find higher redshift objects than RST can, e.g., K-band dropouts at z ≳ 15. JWST can probe, via photometry wavelengths up to 28 μm (NIRCam up to 5 μm and MIRI from 5 to 28 μm), whereas the WFI on board RST will be sensitive only up to 2.3 μm. In terms of spectroscopy, the Grism on RST will be probing up to 1.93 μm, whereas NIRSpec on board JWST is sensitive up to 5 μm. As such, RST will be essentially blind with respect to any objects at z_emi ≳ 18. At those high redshifts, the Gunn–Peterson trough will cut off the entirety of the redshifted flux in the region that RST is potentially sensitive to. Moreover, even at the redshift of interest to us in this paper (z_emi ∼ 12), there is quite important information one can gain from possible redshifted spectral features that fall outside of the sensitivity range of RST, such as, for example, the Balmer Break, or other H or He absorption lines present in the SMDS SEDs. Moreover, with a much larger aperture, JWST has a better angular resolution. Therefore, once a candidate DS object is identified by RST, follow-up observations with JWST could be used to further verify and validate the dark star hypothesis. As mentioned above, a smoking gun for SMDS is the existence of a He ii λ1640 line, which would not be expected for the case of early galaxies. While RST wavelengths are insensitive to this line, spectroscopy with JWST may be able to detect it.

We note that the ability of NIRSpec on JWST to obtain clean spectra is remarkable, see, e.g., GNz11 in Bunker et al. (2023). This z = 10.6 galaxy has a spectrum seen to m_AB ∼ 26 in the H band, which would be comparable to the brightness of a M_⋆ ≃ 4 × 10⁶ M_⊙ SMDS formed via AC; see Figure 5 in Ilie et al. (2012). While GNz11 is clearly not a dark star, as it is resolved, and moreover shows numerous metal emission lines (see Figure 1 of Bunker et al. 2023), based on the relatively short total exposure time (∼17 hr) needed to generate its exquisite spectrum with NIRSpec, we would expect spectroscopy with JWST is a viable tool for an SMDS. In summary, RST will be an excellent instrument for finding SMDS candidates, with follow-up observations from JWST potentially providing a definitive determination of dark star discovery.

In conclusion. Dark stars are very bright sources. In fact, a million solar mass dark star can emit as much light as a billion solar mass galaxy so that the two look very similar in RST and JWST observations. Hence, the conundrum posed by the many observed, bright high redshift objects in JWST, in the context of the standard ΛCDM cosmological model, could be alleviated by the existence of dark stars containing much less baryonic mass than would be required of a comparably bright galaxy.

In this paper, we showed that SMDSs candidates formed via either the capture or extended AC formation mechanism and emitting at redshifts up to z_emi = 14 could be found by RST using photometry alone. We examined how those candidates could be differentiated from high redshift Population III or Population II galaxies, and summarized our results in this section. There are a number of potential signatures, including their location in color–color plots and image morphology. SDMSs with purely stellar spectra would appear as point objects, very bright (L ≳ 10⁹ L_⊙), yet relatively cool (T_eff ∼ 10⁴ K) sources. The detection of a He ii absorption line at 0.1640 μm rest-frame wavelength (the He ii λ1640 line) with follow-up spectroscopic observations would be a clear signature of supermassive dark stars, as this feature is absent in any galactic sources. The identification of supermassive dark stars would be truly remarkable, as it would imply the existence of a novel heat source powering stars, such as the DM annihilation we have studied. Further, such a discovery could open up the possibility of DM parameter estimations based on observed properties of dark stars. In contrast, whenever the nebular emission becomes significant for SMDSs, ³⁵ the differentiation between those objects and young compact galaxies dominated by zero metallicity stars would become nearly impossible. In turn, this could further help alleviate the too many too massive too soon mystery posed by the JWST data, as many of those galaxies could actually be SMDSs disguising under the veil of a nebula.

K.F. and S.Z. are grateful for support from the Jeff and Gail Kodosky Endowed Chair in Physics held by K.F. at the Univ. of Texas, Austin. K.F. and S.Z. acknowledge funding from the U.S. Department of Energy, Office of Science, Office of High Energy Physics program under award No. DE-SC0022021. K.F. acknowledges support by the Vetenskapsradet (Swedish Research Council) through contract No. 638-2013-8993 and the Oskar Klein Centre for Cosmoparticle Physics at Stockholm University. C.I. is grateful for the financial support, in the form of a publication expense grant, received from Colgate University's Research Council. We would acknowledge the useful discussions with James Rhoads, and help from Steve Finkelstein, Volker Bromm through personal communication. We are especially grateful for Jonathan Levine's help with the error ellipses and for him sharing with us the code he uses to generate such ellipses for his research. S.Z. would like to give special thanks to his roommate Changhan Ge who offered help with the Pandeia simulation installation.

In this appendix, we present the key parameters we have used for image simulation, as described in Tables 4–6. The complete input dictionary can be found on the Pandeia reference paper (Pontoppidan et al. 2016), as well as at the following link: https://outerspace.stsci.edu/display/PEN/Pandeia+Engine+Input+API. Below, we explain some of the parameters particularly relevant to our work and possible input choices for those.

• 1.  
  Sérsic index. The morphology of Population III galaxies remains largely unknown due to lack of observations. Since the masses of target galaxies are relatively small, we would expect them to have a shape similar to stellar clusters or irregular galaxies. The choice of this index n = 1 (exponential) reflects that the flux is not overly concentrated at the center. For those galaxies with a stellar mass on the order of M_⋆ ∼ 10⁹ M_⊙, we use the recent argument on the shape of the first galaxies from Park et al. (2022), where they have simulated that galaxies of this mass will have Sérsic index n ≲ 1.5. Also, in the recent results from JWST, Naidu et al. (2022b) fitted the Sérsic index of the observed two galaxies, and the results are both n ≲ 1, which is consistent with the simulation. Therefore, we still choose n = 1 throughout our paper when we scale up the mass. Though, for smaller galaxies, this might not be an accurate description, but it would not significantly change our conclusion since its angular size is relatively small to be resolved.
• 2.  
  Group/integration/exposure number. For simplicity, the readout pattern is chosen as the default for Roman (Deep2). To avoid saturation, the maximal allowed group number (N_group) for this readout pattern is 20, and the integration time (t_int) is a linear function of the group number. For that case of Deep2, it is given by t_int = t_frame × (20N_groups − 18), where t_frame = 10.73677 s as the sample exposure time. Since our desired exposure time would be ∼10⁶ s, we choose the maximum value for the group number as N_group = 20. ³⁶ We also have the integration number per exposure (N_int) and exposure number (${N}_{\exp }$), which multiply the integration time to give the total exposure time: ${t}_{\exp }={N}_{\exp }\times {N}_{\mathrm{int}}\times {t}_{\mathrm{int}}$. Since there is no special preference on the choice of these two numbers, ³⁷ we choose N_int = 29, and ${N}_{\exp }=30$ such that the total exposure time is closest to 10^6s.
• 3.  
  Background level. The relevant background properties and spectrum are described at https://jwst-docs.stsci.edu/jwst-general-support/jwst-background-model. For the range of wavelengths of interest (∼[1–2] μm), the background noise contribution from zodiacal lights and galactic dust dominates. To acquire optimal observation results and for simplicity, we choose the minimum zodiacal background as minzodi and set the background level to be its benchmark level.

In the three tables below, we list input parameters specific to SMDSs (Table 4), Population III/II galaxies (Table 5), and lastly common to both of those objects (Table 6).

Table 4. Input Dictionary Unique to the Point Objects—SMDSs

Aperture Size	Sky Annulus
02	[04, 06]

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Table 5. Input Dictionary Unique to Extended Objects—Population III/II Galaxies

Major	Minor	Geometry	Sérsic Index	Norm Method	Aperture Size	Sky Annulus
μ1/2 × θeff	μ1/2 × θeff	Sérsic	1	integ infinity	μ1/2 × 01	μ1/2 × [015, 02]

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For SMDSs formed via DM capture, the nebular emission could be significant, and thus potentially render their disambiguation from compact early galaxies nearly impossible. In this appendix, we explore the conditions under which such SMDSs will power an ionization bounded nebula. As we shall soon see, the mass of the nebula is inversely proportional to the number density of H atoms in its region. In turn, since the DM halo contains a finite baryonic mass, there is an upper-bound to the mass of the nebula, which translates into a lower-bound on the number density of H. When comparing this lower-bound to analytic or semianalytic models of H gas densities within primordial DM halos, we can find the maximum SMDS mass for which we expect the nebular emission to be significant.

Under the assumption of an equilibrium between ionization and recombination rates, one can estimate the radius of a nebula powered by a hot star as the so-called Strömgren radius:

$$\begin{eqnarray}&&{r}_{{\rm{S}}}={\left(\displaystyle \frac{3{Q}_{* }}{4\pi \alpha {n}_{{\rm{H}}}^{2}}\right)}^{1/3},\end{eqnarray} \tag{ B1 }$$

where Q_* represents the ionizing flux from the star, α is a temperature dependent recombination coefficient, and n_H represents the number density of protons (or electrons) inside the ionized H region. When considering the mass of the H ii (i.e., ionized) gas enclosed within the nebula, ${M}_{{\rm{H}}\,{\rm\small{II}}}=\tfrac{4\pi }{3}{r}_{{\rm{S}}}^{3}{n}_{{\rm{H}}}{m}_{{\rm{H}}}$, one can see that indeed M_{H II} ∼ 1/n_H, as alluded to before.

The upper-bound to M_{H II} is given by deducting the total mass of H in the star from that in the halo:

$$\begin{eqnarray}&&{M}_{{\rm{H}}\,{\rm\small{II}}}\leqslant {f}_{{\rm{H}}}\left({f}_{{\rm{B}}}{M}_{\mathrm{halo}}-{M}_{* }\right),\end{eqnarray} \tag{ B2 }$$

where f_H represents the H mass fraction, which for primordial DM halos can be assumed to be 75% according to BBN, and f_B represents the baryonic mass fraction within the DM halo, which we assumed to be ∼10%. The inequality saturates whenever one completely ignores the H i and H₂ regions around the star, and any other molecular H clouds that could be present within the DM halo. Therefore, the upper-bound in Equation (B2) should be viewed as strict. This, in turn, implies that all the results we below here are to be viewed as conservative.

Using the Strömgren radius from Equation (B1), we can convert the upper-bound from Equation (B2) into a lower-bound on n_H:

$$\begin{eqnarray}&&{n}_{{\rm{H}}}\geqslant \displaystyle \frac{{Q}_{* }({M}_{* }){m}_{{\rm{H}}}}{\alpha {f}_{{\rm{H}}}({f}_{B}{M}_{\mathrm{halo}}-{M}_{* })}.\end{eqnarray} \tag{ B3 }$$

Throughout, m_H represents the mass of a proton, and M_halo represents the mass of the DM halo within which the DS is embedded. Q_*(M_*) is the ionization photon flux from a star of mass M_*, which we calculate numerically for our SMDSs formed via DM capture models, from their fluxes (F_ν ) via the following:

$$\begin{eqnarray}&&{Q}_{* }={\int }_{h\nu =13.6\mathrm{eV}}^{\infty }4\pi {R}_{* }^{2}\displaystyle \frac{{F}_{\nu }(\nu )}{h\nu }d\nu .\end{eqnarray} \tag{ B4 }$$

Assuming, as typically done, that the equilibrium temperature for the gas within the H ii region is ∼10⁴ K, we can use the following value for the recombination coefficient: α = 2.6 × 10¹³ cm⁻³ s⁻¹. For SMDSs formed via DM capture considered in this paper, we tabulate in Table 7 the lower-bounds on n_H, under the assumption that M_halo = 10⁸ M_⊙. If instead we were to assume a 10⁷ M_⊙ DM halo, which could also in principle host an SMDS as massive as 10⁶ M_⊙, the lower-bound values n_H would increase by roughly 1 order of magnitude for the ∼10⁴ and 10⁵ M_⊙ SMDSs, whereas for the case of the 10⁶ M_⊙ SMDSs it would approach ∞. The physical significance ${n}_{{\rm{H}},\min }\to \infty$ in this context is simple: there is no gas left in the halo; therefore, no nebula can be powered.

Table 7. For SMDSs Formed via DM Capture

Formation Mechanism	M*	L*	R*	Teff	${n}_{{\rm{H}},\min }$
	$\left({M}_{\odot }\right)$	$\left({10}^{6}{L}_{\odot }\right)$	(au)	$\left({10}^{3}\,{\rm{K}}\right)$	(cm−3)
Capture	4.1 × 104	774	1.8	49	20
Capture	105	1.75 × 103	2.7	51	60
Capture	106	2.03 × 104	8.5	51	670

Note. Parameters are described in columns (2)–(5), and in column (6) we tabulate the lower-bound on n_H obtained according to Equation (B3). We assumed M_halo = 10⁸ M_⊙.

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There are several ways to estimate the hydrogen number density in a halo. If one assumes gas with uniform density in a virialized DM halo, then, as shown by Tegmark et al. (1997), the spherical collapse model leads to the following:

$$\begin{eqnarray}&&{n}_{{\rm{H}}}\simeq 1.01{\left(\displaystyle \frac{1+z}{31}\right)}^{3}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ B5 }$$

Note that at the redshifts of interest for us, i.e., z ∈ [10, 20] the estimated values of n_H obtained via Equation (B5) are all much smaller than the lower-bound for n_H tabulated in Table 7. This would indicate that SMDSs formed via DM capture do not have sufficient material surrounding them to power an ionization bounded nebula. If n_H is enhanced to ∼100 cm⁻³ (more than 2 orders of magnitude when compared to the value predicted by Equation (B5)), then one expects nebular emission from SMDSs with M ≲ 10⁵ M_⊙, as long as they form in a 10⁸ M_⊙ DM halo. For the same SMDSs formed inside halos with M_halo ≲ 10⁷ M_⊙, one would require an enhancement to n_H ≳ 1000 cm⁻³ in order to have significant nebular emission.

The main limitation of the analysis above is the assumption that the gas surrounding the star has uniform density, which is embedded even in our starting point, Equation (B1). For hydrogen density profiles that are cored, such as an isothermal sphere, or a truncated isothermal sphere, we expect that sufficient enhancements of n_H (with respect to its values predicted by Equation (B5)) are possible in the region where an equilibrium between recombination and ionization is attained. We leave the detailed exploration of nebular emission from DSs for a future dedicated study, which will also include the effects of realistic density profiles for n_H. However, the expectation is that, whenever nebular emission becomes significant, SMDSs will essentially look just as compact young galaxies, thus rendering any prospects for disambiguation nearly impossible.

We present below a brief discussion comparing the SEDs of SMDSs with those of another class of SMSs considered in the literature: Population III SMSs. Those are precursor to the so-called direct collapse black holes (DCBHs) and could form in atomic cooling DM halos that have been exposed to a sufficiently high level of Lyman–Werner (LW) radiation (Loeb & Rasio 1994; Begelman et al. 2006; Lodato & Natarajan 2006; Natarajan et al. 2017). This radiation can be generated abundantly by the first generation of stars (i.e., nuclear burning Population III stars or dark stars). As a consequence of the LW radiation, H₂ is dissociated, and the stellar formation is inhibited in nearby DM halos, until those reach M_halo ≳ 10⁸ M_⊙, when atomic H cooling becomes efficient. Simulations show that at this stage a catastrophic baryonic runaway collapse could build up a star at rates of up to ∼1M_⊙ yr⁻¹. By the time they reach ∼10⁵ M_⊙, those Population III SMSs quickly enter a gravitational instability regime, and collapse to what are typically called DCBHs. For a review of the DCBH scenario, the interested reader can consult Inayoshi et al. (2020).

We do note, however, that SMDSs can also form in the case of atomic hydrogen cooling in 10⁸ M_⊙ DM haloes. This type of SMDS is reviewed in Rindler-Daller et al. (2015) for the case called large minihaloes (see Equation (6) in that paper). Indeed, it is possible that the heat from DM annihilation will inhibit the SMS formation and instead lead to a supermassive dark star. In the remainder of the appendix, we compare the observable consequences of Population III SMS versus SMDS.

Population III SMSs follow two possible evolutionary tracks: cool (red) tracks at T_eff ≲ 10⁴ K (e.g., Surace et al. 2018; Vikaeus et al. 2022) or hot (blue) tracks at T_eff ≳ 10⁴ K (e.g., Surace et al. 2019). Here, we begin a simple comparison between Population III SMSs on each of those tracks with SMDSs in terms of observability. Initially, we take the case of no nebular emission for either type of object. With this assumption, it should be possible to distinguish SMSs on either track from SMDSs (of either formation via purely gravitational effects or via capture) based on their colors. At the same brightness, an SMS on the red supergiant track will be cooler (redder) than an SMDS. Conversely, at the same luminosity, an SMS on the blue supergiant track will be hotter (bluer) than an SMDS.

However, the situation changes when nebular emission is taken into account. For most Population III SMSs, processing of the stellar SEDs by the accretion disk surrounding them is expected, because of their formation from a disk undergoing runaway collapse. In other words, nebular emission is likely for Population III SMSs. They exhibit features such as reprocessing of photons and redistribution toward wavelengths longwards of the Lyα line, and the emergence of emission lines (see, for example, bottom panels of Figure 2 in Surace et al. 2019). As yet, we have not included the effects of nebular emissions on DSs and will study these effects in future work using CLOUDY. Hence, we reserve the detailed comparison of Population III SMSs SEDs processed by their gas envelope to that of SMDSs with nebular emission (see Appendix B) for a future study. We do expect nebular emission to be important for some but not all SMDS, and that the processing of the stellar SEDs by the gas envelope would lead to making a differentiation between Population III SMSs and SMDSs in those cases for which nebular emission is important to be quite difficult, if not impossible. As discussed in Appendix B, we expect the lower mass (M ≲ 10⁵ M_⊙) SMDS formed via capture to experience nebular emission. However, for the most massive ones, there is not sufficient H gas to form an ionization bound nebula. Further, we typically expect SMDS formed via AC to avoid nebular emission, due to the large amount of neutral hydrogen in the atmosphere that absorbs most of the ionizing flux, as can be seen in the upper right panel of Figure 2 (see the nosedive to the left of the Lyman edge in the figure). Thus, we expect SMDS formed via AC as well as the heavier (M > 10⁵ M_⊙) SMDS formed via capture to have unique signatures clearly differentiable from all other objects in JWST.