Second-order cosmological perturbations produced by scalar–scalar coupling during inflation stage

Abstract

We study the perturbations up to the 2nd-order for a power-law inflation driven by a scalar field in synchronous coordinates. We present the 1st-order solutions, and analytically solve the 2nd-order perturbed Einstein equation and scalar field equation, give the 2nd-order solutions for all the scalar, vector, and tensor metric perturbations, as well as the perturbed scalar field. During inflation, the 1st-order tensor perturbation is a wave and is decoupled from other perturbations, the scalar metric perturbation and the perturbed scalar field are coupled waves, propagating at the speed of light, differing from those in the dust and relativistic fluid models. The 1st-order vector perturbation is not wave and just decreases during inflation. The 2nd-order perturbed Einstein equation is similar in structure to the 1st-order one, but various products of the 1st-order perturbations occur as the effective source, among which the scalar–scalar coupling is considered in this paper. The solutions of all the 2nd-order perturbations consist of a homogeneous part similar to the 1st-order solutions, and an inhomogeneous part in a form of integrations of the effective source. The 2nd-order vector perturbation is also a wave since the effective source is composed of the 1st-order waves. We perform the residual gauge transformations between synchronous coordinates up to the 2nd-order, and identify the 1st-order and 2nd-order gauge modes.

Graphic abstract

Appendices

Perturbed stress tensor

The nonvanishing Christoffel symbols, the perturbed Ricci tensor, the perturbed Einstein tensors up to 2nd order, are listed in Ref. [107] for a general flat RW spacetime in synchronous coordinate. In the following we list the stress tensor and the conservation equation of the inflaton scalar field with a general potential up to the 2nd order.

The stress tensor of the scalar field is

$$\begin{aligned} T_{\mu \nu } = \varphi _{, \mu } \varphi _{, \nu } -g_{\mu \nu }\left[ \frac{1}{2} g^{\alpha \beta } \varphi _{, \alpha } \varphi _{, \beta } + V(\varphi )\right] , \end{aligned}$$

(A.1)

the energy density $\rho = - T^{0}\, _{0}$ and the pressure $p=\frac{1}{3} T^{i}\, _{i}$. The 0th-order is

$$\begin{aligned} T^{(0)}_{00} =&\frac{1}{2} \varphi ^{(0)'}\varphi ^{(0)'} +a^2 V(\varphi ^{(0)}) , \end{aligned}$$

(A.2)

$$\begin{aligned} T^{(0)}_{0i} =&0, \end{aligned}$$

(A.3)

$$\begin{aligned} T^{(0)}_{ij} =&\big [ \frac{1}{2} \varphi ^{(0)'}\varphi ^{(0)'} - a^2 V(\varphi ^{(0)}) \big ]\delta _{ij}, \end{aligned}$$

(A.4)

the 0th-order energy density $\rho = - T^{(0)0}\,_{0} = \frac{1}{2} a^{-2} \varphi ^{(0)'}\varphi ^{(0)'} + V(\varphi ^{(0)})$, and the 0th-order pressure $p=\frac{1}{3} T^{(0)i}\, _{i} = \frac{1}{2} a^{-2} \varphi ^{(0)'}\varphi ^{(0)'} - V(\varphi ^{(0)})$. The 1st-order perturbed stress tensor is

$$\begin{aligned} T^{(1)}_{00} =&\varphi ^{(0)'}\varphi ^{(1)'} +a^2\varphi ^{(1)} V(\varphi ^{(0)})_{,\varphi } , \end{aligned}$$

(A.5)

$$\begin{aligned} T^{(1)}_{0i} =&\varphi ^{(0)'} \varphi ^{(1)}_{, i} , \end{aligned}$$

(A.6)

$$\begin{aligned} T^{(1)}_{ij} =&\left[ \varphi ^{(0)'}\varphi ^{(1)'} -a^2\varphi ^{(1)} V(\varphi ^{(0)})_{,\varphi } \right] \delta _{ij} \nonumber \\&+ \left[ \frac{1}{2} \left( \varphi ^{(0)'}\right) ^{2} -a^2 V(\varphi ^{(0)})\right] (-2 \phi ^{(1)}) \delta _{ij} \nonumber \\&+\left[ \frac{1}{2}\left( \varphi ^{(0)'}\right) ^{2} -a^2 V(\varphi ^{(0)})\right] D_{ij}\chi ^{||(1)} \nonumber \\&+\left[ \frac{1}{2}\left( \varphi ^{(0)'}\right) ^{2} -a^2 V(\varphi ^{(0)})\right] \chi ^{\perp (1)}_{ij} \nonumber \\&+\left[ \frac{1}{2}\left( \varphi ^{(0)'}\right) ^{2} -a^2 V(\varphi ^{(0)})\right] \chi ^{\top (1)}_{ij} . \end{aligned}$$

(A.7)

Raising up an index of (A.7) gives

$$\begin{aligned} T^{(1)i}\, _j&\simeq a^{-2}(\delta ^{i\mu } -\gamma ^{(1)i\mu }) T_{\mu j} \nonumber \\&= \left[ a^{-2} \varphi ^{(0)'}\varphi ^{(1)'} - \varphi ^{(1)} V(\varphi ^{(0)})_{,\varphi } \right] \delta ^i_j \, , \end{aligned}$$

(A.8)

agreeing with the expression (25) of Ref. [7]. The 1st-order perturbed energy density and pressure are

$$\begin{aligned} \rho ^{(1)} = \frac{1}{2} a^{-2} \varphi ^{(0)'}\varphi ^{(1)'} +\varphi ^{(1)} V(\varphi ^{(0)})_{\, ,\varphi } \, , \end{aligned}$$

(A.9)

$$\begin{aligned} p^{(1)} = \frac{1}{2} a^{-2} \varphi ^{(0)'}\varphi ^{(1)'} - \varphi ^{(1)} V(\varphi ^{(0)})_{\, ,\varphi } \, . \end{aligned}$$

(A.10)

Notice that $\rho ^{(1)}=0=p^{(1)}$ for the exact de Sitter inflation, since $\varphi ^{(0)'}=0= V(\varphi ^{(0)})_{,\varphi }$.

The 2nd-order perturbed stress tensor is

$$\begin{aligned} T^{(2)}_{00} =&\frac{1}{2}\varphi ^{(0)'} \varphi ^{(2)'} +\frac{1}{2}a^{2} V(\varphi ^{(0)})_{,\varphi } \varphi ^{(2)} \nonumber \\&+\frac{1}{2}\varphi ^{(1)'}\varphi ^{(1)'} +\frac{1}{2}\varphi ^{(1),l}\varphi ^{(1)}_{,l} +\frac{1}{2}a^{2}\varphi ^{(1)}\varphi ^{(1)} V(\varphi ^{(0)})_{,\varphi \varphi } , \end{aligned}$$

(A.11)

$$\begin{aligned} T^{(2)}_{0i} =&\frac{1}{2}M_\textrm{Pl}\frac{\sqrt{2(\beta +1) (\beta +2)}}{\tau }\varphi ^{(2)}_{, i} +\varphi ^{(1)}_{, i}\varphi ^{(1)'}, \end{aligned}$$

(A.12)

$$\begin{aligned} T^{(2)}_{ij} =&\Big [ -\frac{1}{2} ( \varphi ^{(0)'})^2 +a^2 V(\varphi ^{(0)}) \Big ]\phi ^{(2)}\delta _{ij} +\Big [ \frac{1}{2}\varphi ^{(0)'}\varphi ^{(2)'} -\frac{1}{2}a^2V(\varphi ^{(0)})_{,\, \varphi }\varphi ^{(2)} \Big ]\delta _{ij} \nonumber \\&+ \Big [ \frac{1}{4} ( \varphi ^{(0)'})^2 -\frac{1}{2}a^2 V(\varphi ^{(0)}) \Big ]\chi _{ij}^{(2)} \nonumber \\&+\Big [ \frac{1}{2}\varphi ^{(1)'}\varphi ^{(1)'} - \frac{1}{2} \varphi ^{(1),k}\varphi ^{(1)}_{, k} -\frac{1}{2}a^2 V(\varphi ^{(0)})_{,\, \varphi \varphi } \varphi ^{(1)}\varphi ^{(1)} \nonumber \\&-2 \varphi ^{(0)'}\varphi ^{(1)'}\phi ^{(1)} +2a^2 V(\varphi ^{(0)})_{,\, \varphi }\varphi ^{(1)}\phi ^{(1)} \Big ]\delta _{ij} \nonumber \\&+\varphi ^{(1)}_{, i}\varphi ^{(1)}_{, j} -a^2 V(\varphi ^{(0)})_{,\, \varphi } \varphi ^{(1)}\chi _{ij}^{(1)} +\varphi ^{(0)'}\varphi ^{(1)'}\chi _{ij}^{(1)} . \end{aligned}$$

(A.13)

The 2nd-order perturbed stress tensor are generally nonzero even for the de Sitter inflation. These stress tensors generally will change under residual gauge transformations, and we shall not discuss in details. See Ref. [127] for relevant discussions for a general metric.

The equations of covariant conservation are given by (2.12). By calculation, we get the 1st-order energy conservation $[T^{ 0\nu }\,_{; \, \nu }]^{(1)}=0$ as the following

$$\begin{aligned} \varphi ^{(0)'}\varphi ^{(1)''} + 2\frac{a'}{a} \varphi ^{(0)'}\varphi ^{(1)'} -\varphi ^{(0)'}\nabla ^2\varphi ^{(1)} +a^{2}V(\varphi ^{(0)})_{,\, \varphi \varphi } \varphi ^{(0)'} \varphi ^{(1)}\nonumber \\ -3 (\varphi ^{(0)'} )^2\phi ^{(1)'} =0 , \end{aligned}$$

(A.14)

which is equivalent to the equation of the 1st-order perturbed scalar field (C.27). The 1st-order momentum conservation $[T^{\, i\nu }\,_{; \, \nu }]^{(1)}=0$ leads to an identity $0=0$ because the scalar field $\varphi$ contains no curl part. This property of the stress tensor of the scalar field $\varphi$ is similar to that of a dust [105, 106], but in contrast to the relativistic fluid for which the momentum conservation is not an trivial identity [107, 108].

The 2nd-order energy conservation $[T^{ 0\nu }\,_{; \, \nu }]^{(2)}=0$ gives

$$\begin{aligned} 0=&\frac{1}{2}a^{-2} \varphi ^{(0)'} \varphi ^{(2)''} +a'a^{-3}\varphi ^{(0)'}\varphi ^{(2)'} -\frac{1}{2}a^{-2}\varphi ^{(0)'}\nabla ^2\varphi ^{(2)} +\frac{1}{2}V_{,\, \varphi \varphi } \varphi ^{(0)'}\varphi ^{(2)} \nonumber \\&+\frac{1}{4}a^{-2}(\varphi ^{(0)'})^{2}\gamma ^{(2)'k}_k +\frac{1}{2}V_{,\varphi \varphi \varphi } \varphi ^{(0)'} \varphi ^{(1)}\varphi ^{(1)} +\frac{1}{2}a^{-2}\varphi ^{(0)'}\varphi ^{(1)'}\gamma ^{(1)'k}_{k} \nonumber \\&+a^{-2}\varphi ^{(0)'}\gamma ^{(1)kl}_{,l}\varphi _{,k }^{(1)} +a^{-2}\varphi ^{(0)'}\gamma ^{(1)kl}\varphi _{,kl }^{(1)} -\frac{1}{2}a^{-2}\varphi ^{(0)'}\gamma ^{(1)k}_{k,l}\varphi ^{(1),\,l} \nonumber \\&-\frac{1}{2}a^{-2}(\varphi ^{(0)'})^{2}\gamma ^{(1)kl}\gamma ^{(1)'}_{kl} , \end{aligned}$$

(A.15)

which can be written as

$$\begin{aligned}&\varphi ^{(2)''}_S +2 \frac{a'}{a} \varphi ^{(2)'}_S -\nabla ^2\varphi ^{(2)}_S +a^2 V(\varphi ^{(0)})_{,\, \varphi \varphi }\varphi ^{(2)}_S = - A_S + 3\varphi ^{(0)'}\phi ^{(2)'}_S , \end{aligned}$$

(A.16)

where the subindex “S" indicates the scalar–scalar coupling, and

$$\begin{aligned} A_S \equiv&\, a^2V_{,\varphi ^{(0)}\varphi ^{(0)}\varphi ^{(0)}}\varphi ^{(1)}\varphi ^{(1)} -6\varphi ^{(1)'}\phi ^{(1)'} +2\varphi _{,k}^{(1)}\phi ^{(1),k} -4\phi ^{(1)}\nabla ^2\varphi ^{(1)} \nonumber \\&-12\varphi ^{(0)'}\phi ^{(1)} \phi ^{(1)'} +\frac{4}{3}\varphi _{,k }^{(1)}\nabla ^2\chi ^{||(1),k} +2\varphi _{,kl}^{(1)}\chi ^{||(1),kl} \nonumber \\&-\frac{2}{3}\nabla ^2\varphi ^{(1)}\nabla ^2\chi ^{||(1)} -\varphi ^{(0)'}\chi ^{||(1),kl}\chi ^{||(1)'}_{,kl} +\frac{1}{3}\varphi ^{(0)'}\nabla ^2\chi ^{||(1)}\nabla ^2\chi ^{||(1)'}. \end{aligned}$$

(A.17)

We have also checked that the 2nd-order momentum conservation $[T^{\, i\nu }\,_{; \, \nu }]^{(2)}=0$ is also a trivial identity, $0=0$, as the 1st-order one.

Gauge transformations from Synchronous to Synchronous

The formulae for the transformations between two general coordinates for a flat RW spacetime, as well as between two synchronous coordinates, have been given in Appendix C in Ref.  [105]. Here we add the transformation of the scalar field $\varphi$.

First consider the 1st-order transformation. In the synchronous coordinate, for the power-law inflation, the transformation vector is the following

$$\begin{aligned} \xi ^{(1)0}(\tau , \textbf{x})= & {} \frac{A^{(1)}(\textbf{x})}{a(\tau )} = \frac{A^{(1)}(\textbf{x})}{(-\tau )^{\beta +1}}, \end{aligned}$$

(B.1)

$$\begin{aligned} \xi ^{(1)i} (\tau , \textbf{x})= & {} A^{(1)}(\textbf{x})^{,i} \int ^\tau \frac{d\tau ' }{a(\tau ')} + C^{(1)i}(\textbf{x}) \nonumber \\= & {} A^{(1)}(\textbf{x})^{,i} \int ^\tau \frac{d\tau ' }{(-\tau ')^{\beta +1}} + C^{(1)i}(\textbf{x}), \end{aligned}$$

(B.2)

where $A^{(1)}$ and $C^{(1)i}$ are small, arbitrary functions depending on $\textbf{x}$ only, and the constant $l_0^{-1}$ from $a(\tau )$ has been absorbed into $A^{(1)}$ for notational simplicity. $C^{(1)i}$ can be decomposed into

$$\begin{aligned} C ^{(1)i} (\textbf{x}) = C\, ^{ ||(1),\,i}(\textbf{x}) + C\, ^{ \bot (1) i}(\textbf{x}), \end{aligned}$$

(B.3)

where the transverse part satisfies $\partial _i C^{ \bot (1)\, i}=0$. Corresponding to (B.1) (B.2), the residual gauge transform of the metric perturbations between the synchronous coordinates is

$$\begin{aligned} {\bar{\phi }}^{(1)}&= \phi ^{(1)} + \frac{1}{3}\nabla ^2 A^{(1)} \int ^\tau \frac{d\tau }{(-\tau )^{\beta +1}} + \frac{1}{3}\nabla ^2C^{||(1)} + \frac{a'}{a^2} A^{(1)} , \end{aligned}$$

(B.4)

$$\begin{aligned} {\bar{\chi }}^{||(1)}&= \chi ^{||(1)} - 2 A^{(1)} \int ^\tau \frac{d\tau }{(-\tau )^{\beta +1}} -2C^{||(1)}, \end{aligned}$$

(B.5)

$$\begin{aligned} {\bar{\chi }}^{\perp (1)}_{ij}&= \chi ^{\perp (1)}_{ij} -C^{\perp (1)}_{i,j} -C^{\perp (1)}_{j,i}, \end{aligned}$$

(B.6)

$$\begin{aligned} {\bar{\chi }}^{\top (1)}_{ij}&= \chi ^{\top (1)}_{ij} , \end{aligned}$$

(B.7)

and the combined scalar $\zeta$ of (C.36) transforms as

$$\begin{aligned} {\bar{\zeta }} =&\zeta -2 \frac{a'}{a^2} A^{(1)} . \end{aligned}$$

(B.8)

We see that the 1st-order tensor modes remains invariant under the residual transformation. The 1st-order perturbed scalar field transforms between the synchronous coordinates as

$$\begin{aligned} {\bar{\varphi }} ^{(1)}&= \varphi ^{(1)} - \varphi ^{(0)'} \frac{A^{(1)}}{a} \end{aligned}$$

(B.9)

$$\begin{aligned}&= \varphi ^{(1)} +M_\textrm{Pl}\sqrt{2(\beta +1) (\beta +2)} A^{(1)} (-\tau )^{-\beta -2} , \end{aligned}$$

(B.10)

It is seen that $\varphi ^{(1)}_{gi}$ of (D.3) is invariant under the transformation (B.5) (B.9), $\Psi$ of (D.1) is invariant under the transformation (B.4) (B.5), and $Q_\varphi$ of (2.28) and ${{\mathcal {R}}}$ of (D.4) are also invariant as combinations of $\varphi ^{(1)}_{gi}$ and $\Psi$.

Now we list the 2nd-order synchronous-to-synchronous gauge transformations. (See Appendix C in Ref. [107] for a general RW spacetime.) For the power-law inflation and the scalar–scalar coupling, we give the 2nd-order transformation vector

$$\begin{aligned} \xi ^{(2)0}&=\frac{ A^{(2)}({\textbf{x}})}{(-\tau )^{1+\beta }} \, , \end{aligned}$$

(B.11)

$$\begin{aligned} \xi ^{(2)i }&= \partial ^i \beta ^{(2)} + d^{(2)i}, \end{aligned}$$

(B.12)

with

$$\begin{aligned} \beta ^{(2)}=&\nabla ^{-2}\Big [ \nabla ^2 A^{(1)} \int ^\tau \frac{4\phi ^{(1)}(\tau ',\textbf{x})}{(-\tau ')^{1+\beta }}d\tau ' +A^{(1)}_{,i}\int ^\tau \frac{4\phi ^{(1)}(\tau ',\textbf{x})^{,i}}{(-\tau ')^{1+\beta }}d\tau ' \nonumber \\&-A^{(1) ,\, ij}\int ^\tau \frac{2\chi ^{(1)}_{ij}(\tau ',\textbf{x})}{(-\tau ')^{1+\beta }}d\tau ' -A^{(1) ,\, j}\int ^\tau \frac{2\chi ^{(1)}_{ji}(\tau ',\textbf{x})^{,\,i}}{(-\tau ')^{1+\beta }}d\tau ' \nonumber \\&+ 2 A^{(1),\, ij}C^{||(1)}_{,\, ij}\int ^\tau \frac{d\tau '}{(-\tau ')^{1+\beta }} + 2 A^{(1) ,\, i}\nabla ^2C^{||(1)}(\textbf{x})_{,i} \int ^\tau \frac{d\tau '}{(-\tau ')^{1+\beta }}\Big ] \nonumber \\&-\frac{1}{2(-\tau )^{2+2\beta }}A^{(1)} A^{(1)} + A^{(1),\, i} A^{(1)}_{,\, i} \int ^\tau \frac{d\tau '}{(-\tau ')^{1+\beta }} \int ^{\tau '}\frac{d\tau '' }{(-\tau '')^{1+\beta }} \nonumber \\&+A^{(2)} \int ^\tau \frac{d\tau '}{(-\tau ')^{1+\beta }} +C^{||(2)} \,, \end{aligned}$$

(B.13)

$$\begin{aligned} d^{(2)}_i =&\partial _i\nabla ^{-2}\Big [ -\nabla ^2 A^{(1)} \, \int ^\tau \frac{4\phi ^{(1)}(\tau ',\textbf{x})}{(-\tau ')^{1+\beta }}d\tau ' -A^{(1)}_{, j}\int ^\tau \frac{4\phi ^{(1)}(\tau ',\textbf{x})^{,j}}{(-\tau ')^{1+\beta }}d\tau ' \nonumber \\&+2A^{(1) ,\, lj}\int ^\tau \frac{ \chi ^{(1)}_{lj}(\tau ',\textbf{x})}{(-\tau ')^{1+\beta }}d\tau ' +2A^{(1),\, l}\int ^\tau \frac{\chi ^{(1)}_{ lj }(\tau ',\textbf{x})^{,\, j}}{(-\tau ')^{1+\beta }}d\tau ' \nonumber \\&- 2 A^{(1),\, lj }C^{||(1)}(\textbf{x})_{,\, lj }\int ^\tau \frac{d\tau '}{(-\tau ')^{1+\beta }} - 2 A^{(1),\, j} \nabla ^2C^{||(1)}_{,j } \int ^\tau \frac{d\tau '}{(-\tau ')^{1+\beta }}\Big ] \nonumber \\&+ 4A^{(1)}_{,\, i}\int ^\tau \frac{\phi ^{(1)}(\tau ',\textbf{x})}{(-\tau ')^{1+\beta }}d\tau ' -2A^{(1)\, ,\, l}\int ^\tau \frac{\chi ^{(1)}_{li}(\tau ',\textbf{x})}{(-\tau ')^{1+\beta }}d\tau ' \nonumber \\&+ 2 A^{(1)\, ,\, l}C^{||(1)}_{,\, li}\int ^\tau \frac{d\tau '}{(-\tau ')^{1+\beta }} + C^{\perp (2)}_i . \end{aligned}$$

(B.14)

where $A^{(2)}$ is an arbitrary function of 2nd order, $C^{(2)}_i$ is an arbitrary 3-vector of 2nd order and can be decomposed into $C^{(2)}_i = C^{||(2)}_{,\,i} +C^{\perp (2)}_{\,i}$. (See also Ref.  [105].) From these we obtain the residual transformations of the 2nd-order metric perturbations (5.1)–(5.4), and of the 2nd-order perturbed scalar field (5.5) in the context, where we have omitted the 1st-order curl vector.

1st-order perturbations

Here we show detailed calculations for the 1st-order perturbations.

The (00) component of 1st-order perturbed Einstein equation gives the 1st-order energy constraint,

$$\begin{aligned} -6\frac{a'}{a}\phi ^{(1)'} +2\nabla ^2\phi ^{(1) } +\frac{1}{3}\nabla ^2\nabla ^2\chi ^{||(1)} =8\pi G \Big [ \varphi ^{(0)'}\varphi ^{(1)'} +a^2 V(\varphi ^{(0)})_{,\varphi } \varphi ^{(1)} \Big ] ,\nonumber \\ \end{aligned}$$

(C.1)

and for the power-law inflation,

$$\begin{aligned}&-6\frac{\beta +1}{\tau }\phi ^{(1)'} +2\nabla ^2\phi ^{(1) } +\frac{1}{3}\nabla ^2\nabla ^2\chi ^{||(1)} \nonumber \\&= \frac{1}{M_\textrm{Pl}}\frac{\sqrt{2(\beta +1) (\beta +2)}}{\tau }\varphi ^{(1)'} -\frac{1}{M_\textrm{Pl}}\frac{ \sqrt{2(\beta +1) (\beta +2)} (2 \beta +1)}{\tau ^{2}} \varphi ^{(1)} , \end{aligned}$$

(C.2)

which contains no second order time derivative. The (0i) component of 1st-order perturbed Einstein gives the 1st-order momentum constraint

$$\begin{aligned} 2\phi ^{(1)' }_{,i} + \frac{1}{2} D_{ij}\chi ^{||(1)',j } + \frac{1}{2} \chi ^{\perp (1)',j }_{ij} = 8\pi G \varphi ^{(0)'} \varphi ^{(1)}_{, i}, \end{aligned}$$

(C.3)

which, for the power-law inflation, is

$$\begin{aligned} 2\phi ^{(1)' }_{,i} + \frac{1}{2} D_{ij}\chi ^{||(1)',j } + \frac{1}{2} \chi ^{\perp (1)',j }_{ij} = \frac{1}{M_\textrm{Pl}}\frac{\sqrt{2(\beta +1) (\beta +2)}}{\tau }\varphi ^{(1)}_{, i}. \end{aligned}$$

(C.4)

The vector mode is formed from a curl vector as (2.7) and can be separated out from (C.4)

$$\begin{aligned} \chi ^{\perp (1)', \, j }_{ij}=0, \end{aligned}$$

(C.5)

and the scalar part in (C.4) can be rewritten as (by dropping the spatial differentiation)

$$\begin{aligned} 2\phi ^{(1)' } + \frac{1}{3} \nabla ^2\chi ^{||(1)'} = 8\pi G \varphi ^{(0)'} \varphi ^{(1)}, \end{aligned}$$

(C.6)

(which can be written as $\varphi ^{(1)} =\frac{1}{8\pi G \varphi ^{(0)'} } ( 2\phi ^{(1)' } + \frac{1}{3} \nabla ^2\chi ^{||(1)'} ) = - \frac{1}{8\pi G \varphi ^{(0)'} } \zeta '$.) i.e.,

$$\begin{aligned} 2\phi ^{(1)' } + \frac{1}{3} \nabla ^2\chi ^{||(1)'} = \frac{1}{M_\textrm{Pl}}\frac{ \sqrt{2(\beta +1) (\beta +2)} }{\tau } \varphi ^{(1)}. \end{aligned}$$

(C.7)

The (ij) component of 1st-order perturbed Einstein equation gives the 1st-order evolution equation

$$\begin{aligned}&2\phi ^{(1)''} \delta _{ij} +4\frac{a'}{a}\phi ^{(1)'}\delta _{ij} +\phi ^{(1) }_{,ij} -\nabla ^2\phi ^{(1) }\delta _{ij} +\left[ 4\frac{a''}{a}-2(\frac{a'}{a})^2\right] \phi ^{(1)}\delta _{ij} \nonumber \\&+\frac{1}{2} D_{ij}\chi ^{||(1)''} +\frac{a'}{a}D_{ij}\chi ^{||(1)'} +\left[ (\frac{a'}{a})^2-2\frac{a''}{a}\right] D_{ij}\chi ^{||(1)} \nonumber \\&+\frac{1}{6}\nabla ^2D_{ij}\chi ^{||(1)} -\frac{1}{9}\delta _{ij}\nabla ^2\nabla ^2\chi ^{||(1) } \nonumber \\&+\frac{1}{2} \chi ^{\perp (1)''}_{ij} +\frac{a'}{a}\chi ^{\perp (1)'}_{ij} +\left[ (\frac{a'}{a})^2-2\frac{a''}{a}\right] \chi ^{\perp (1)}_{ij} \nonumber \\&+\frac{1}{2} \chi ^{\top (1)''}_{ij} +\frac{a'}{a} \chi ^{\top (1)'}_{ij} +\left[ (\frac{a'}{a})^2 -2\frac{a''}{a}\right] \chi ^{\top (1)}_{ij} - \frac{1}{2}\nabla ^2\chi ^{\top (1) }_{ij} \nonumber \\ =&8\pi G \Big \{ \left[ -\left( \varphi ^{(0)'}\right) ^{2} +2a^2 V(\varphi ^{(0)})\right] \phi ^{(1)}\delta _{ij} +\left[ \varphi ^{(0)'}\varphi ^{(1)'} -a^2\varphi ^{(1)} V(\varphi ^{(0)})_{,\varphi ^{(0)}} \right] \delta _{ij} \nonumber \\&+\left[ \frac{1}{2}\left( \varphi ^{(0)'}\right) ^{2} -a^2 V(\varphi ^{(0)})\right] D_{ij}\chi ^{||(1)} \nonumber \\&+\left[ \frac{1}{2}\left( \varphi ^{(0)'}\right) ^{2} -a^2 V(\varphi ^{(0)})\right] \chi ^{\perp (1)}_{ij} \nonumber \\&+\left[ \frac{1}{2}\left( \varphi ^{(0)'}\right) ^{2} -a^2 V(\varphi ^{(0)})\right] \chi ^{\top (1)}_{ij} \Big \} \end{aligned}$$

(C.8)

which is valid for a general RW spacetime with a general $a(\tau )$. Using (2.14) to cancel some terms on both sides, the vector and tensor parts of (C.8) are the following

$$\begin{aligned} \chi ^{\perp (1)''}_{ij} + 2 \frac{a'}{a}\chi ^{\perp (1)'}_{ij} =0 , \end{aligned}$$

(C.9)

$$\begin{aligned} \chi ^{\top (1)''}_{ij} +2 \frac{a'}{a} \chi ^{\top (1)'}_{ij} - \nabla ^2\chi ^{\top (1) }_{ij} =0 . \end{aligned}$$

(C.10)

The scalar part of (C.8) is the following

$$\begin{aligned}&2\phi ^{(1)''} \delta _{ij} +4\frac{a'}{a}\phi ^{(1)'}\delta _{ij} +D_{ij}\phi ^{(1)} +\frac{1}{3} \delta _{ij} \nabla ^2 \phi ^{(1)} -\nabla ^2\phi ^{(1) }\delta _{ij} -\frac{1}{9}\delta _{ij}\nabla ^2\nabla ^2\chi ^{||(1) } \nonumber \\&\qquad +\frac{1}{2} D_{ij}\chi ^{||(1)''} +\frac{a'}{a}D_{ij}\chi ^{||(1)'} +\frac{1}{6}\nabla ^2D_{ij}\chi ^{||(1)} \nonumber \\&\quad = 8\pi G \left[ \varphi ^{(0)'}\varphi ^{(1)'} -a^2\varphi ^{(1)} V(\varphi ^{(0)})_{,\varphi ^{(0)}} \right] \delta _{ij} , \end{aligned}$$

(C.11)

which is also split into the trace part

$$\begin{aligned}{} & {} 2\phi ^{(1)''} +4\frac{a'}{a}\phi ^{(1)'} - \frac{2}{3} \nabla ^2 \phi ^{(1)} -\frac{1}{9} \nabla ^2\nabla ^2\chi ^{||(1) } \nonumber \\{} & {} \quad = 8\pi G \left[ \varphi ^{(0)'}\varphi ^{(1)'} -a^2 V(\varphi ^{(0)})_{,\varphi ^{(0)}} \varphi ^{(1)} \right] , \end{aligned}$$

(C.12)

and the traceless part

$$\begin{aligned}&D_{ij}\phi ^{(1)} +\frac{1}{6}\nabla ^2D_{ij}\chi ^{||(1)} +\frac{1}{2} D_{ij}\chi ^{||(1)''} +\frac{a'}{a}D_{ij}\chi ^{||(1)'} = 0 . \end{aligned}$$

(C.13)

Dropping off $D_{ij}$ in the traceless further gives

$$\begin{aligned} \phi ^{(1)} +\frac{1}{6}\nabla ^2\chi ^{||(1) } +\frac{1}{2} \chi ^{||(1)''} +\frac{a'}{a}\chi ^{||(1)'} =0. \end{aligned}$$

(C.14)

For the power-law inflation the above equations are

$$\begin{aligned}&2\phi ^{(1)''} \delta _{ij} +4\frac{\beta +1}{\tau }\phi ^{(1)'}\delta _{ij} +\phi ^{(1) }_{,ij} -\nabla ^2\phi ^{(1) }\delta _{ij} \nonumber \\&+\frac{1}{2} D_{ij}\chi ^{||(1)''} +\frac{\beta +1}{\tau }D_{ij}\chi ^{||(1)'} +\frac{1}{6}\nabla ^2D_{ij}\chi ^{||(1)} -\frac{1}{9}\delta _{ij}\nabla ^2\nabla ^2\chi ^{||(1) } \nonumber \\&+\frac{1}{2} \chi ^{\perp (1)''}_{ij} +\frac{\beta +1}{\tau } \chi ^{\perp (1)'}_{ij} \nonumber \\&+\frac{1}{2} \chi ^{\top (1)''}_{ij} +\frac{\beta +1}{\tau } \chi ^{\top (1)'}_{ij} - \frac{1}{2}\nabla ^2\chi ^{\top (1) }_{ij} \nonumber \\ =&\frac{1}{M_\textrm{Pl}}\frac{\sqrt{2(\beta +1) (\beta +2)}}{\tau }\varphi ^{(1)'}\delta _{ij} +\frac{1}{M_\textrm{Pl}} \frac{\sqrt{2(\beta +1) (\beta +2)} (2 \beta +1)}{\tau ^2}\varphi ^{(1)}\delta _{ij}. \end{aligned}$$

(C.15)

which involves all the scalar, vector and tensor modes. The trace part of (C.15) is

$$\begin{aligned}&\phi ^{(1)''} +2\frac{\beta +1}{\tau }\phi ^{(1)'} -\frac{1}{3}\nabla ^2\phi ^{(1) } -\frac{1}{18}\nabla ^2\nabla ^2\chi ^{||(1) } \nonumber \\ =&\frac{1}{M_\textrm{Pl}}\frac{\sqrt{2(\beta +1) (\beta +2)}}{2\tau }\varphi ^{(1)'} +\frac{1}{M_\textrm{Pl}} \frac{\sqrt{2(\beta +1) (\beta +2)} (2 \beta +1)}{2\tau ^2}\varphi ^{(1)}, \end{aligned}$$

(C.16)

which is an inhomogeneous equation with $\varphi ^{(1)}$ as a source. The traceless part of (C.15) is

$$\begin{aligned}&D_{ij}\phi ^{(1) } +\frac{1}{2} D_{ij}\chi ^{||(1)''} +\frac{\beta +1}{\tau }D_{ij}\chi ^{||(1)'} +\frac{1}{6}\nabla ^2D_{ij}\chi ^{||(1) } \nonumber \\&+\frac{1}{2} \chi ^{\perp (1)''}_{ij} +\frac{\beta +1}{\tau } \chi ^{\perp (1)'}_{ij} \nonumber \\&+\frac{1}{2} \chi ^{\top (1)''}_{ij} +\frac{\beta +1}{\tau } \chi ^{\top (1)'}_{ij} - \frac{1}{2}\nabla ^2\chi ^{\top (1) }_{ij} =0, \end{aligned}$$

(C.17)

which is a homogeneous equation, and can be further decomposed into scalar, vector, tensor parts as

$$\begin{aligned}{} & {} D_{ij}\phi ^{(1)} +\frac{1}{6}\nabla ^2D_{ij}\chi ^{||(1)} +\frac{1}{2} D_{ij}\chi ^{||(1)''} +\frac{\beta +1}{\tau }D_{ij}\chi ^{||(1)'} =0, \end{aligned}$$

(C.18)

$$\begin{aligned}{} & {} \quad \chi ^{\perp (1)''}_{ij} +\frac{2\beta +2}{\tau } \chi ^{\perp (1)'}_{ij} =0, \end{aligned}$$

(C.19)

$$\begin{aligned}{} & {} \quad \chi ^{\top (1)''}_{ij} +\frac{2\beta +2}{\tau } \chi ^{\top (1)'}_{ij} -\nabla ^2 \chi ^{\top (1) }_{ij} =0. \end{aligned}$$

(C.20)

Taking off $D_{ij}$ from (C.18), we have

$$\begin{aligned} \phi ^{(1)} +\frac{1}{6}\nabla ^2\chi ^{||(1) } +\frac{1}{2} \chi ^{||(1)''} +\frac{\beta +1}{\tau }\chi ^{||(1)'} =0. \end{aligned}$$

(C.21)

Thus, the 1st-order vector modes evolve independently, so do the 1st-order tensor modes, and their solutions are simple to give. The evolution equation (C.19) of the vector mode is not a hyperbolic equation, so the vector mode is not a wave, and does not propagate. The vector equation (C.19) is not hyperbolic and its solution is of the form

$$\begin{aligned} \chi ^{\perp (1)}_{ij} \propto \tau ^{1-2(\beta +1)}, ~ \tau ^0 \end{aligned}$$

(C.22)

which is not an oscillator. So the vector perturbation is not a wave. Moreover, the constant mode $\propto \tau ^0$ is a gauge mode and can be removed by the gauge transformation (B.6), and the mode $\tau ^{1-2(\beta +1)} \sim a^{-3}$ for $\beta \sim -2$ is decaying away during inflation. Thus, we can set

$$\begin{aligned} \chi ^{\perp (1)}_{ij}=0, \end{aligned}$$

(C.23)

like in the dust and fluid models [105,106,107,108].

The equation (C.20) of the tensor perturbation is a hyperbolic 2nd-order partial differential equation which describes the 1st-order gravitational wave propagating at the speed of light. The solution of (C.20) is the RGW and has been studied in Refs. [25,26,27, 29, 30, 39,40,41,42,43,44,45,46]. It is written as

$$\begin{aligned} \chi ^{\top (1)}_{ij} ( \textbf{x},\tau )= \frac{1}{(2\pi )^{3/2}} \int d^3k e^{i \,\textbf{k}\cdot \textbf{x}} \sum _{s={+,\times }} {\mathop \epsilon \limits ^s}_{ij}(k) ~ {\mathop h\limits ^s}_k(\tau ), \end{aligned}$$

(C.24)

where ${\mathop \epsilon \limits ^s}_{ij}(k)$ are the polarization tensors, ${\mathop h\limits ^s}_k(\tau )$ with $s= {+,\times }$ are two modes of RGW and can be assumed to be statistically equivalent, so that the superscript s can be dropped,

$$\begin{aligned} h_k(\tau ) =&\frac{1}{a(\tau )}\sqrt{\frac{\pi }{2}} \sqrt{\frac{-\tau }{2}} \left[ b_1 H_{\beta +\frac{1}{2}}^{(1)}(-k \tau ) +b_2 H_{\beta +\frac{1}{2}}^{(2)}(-k \tau )\right] , \end{aligned}$$

(C.25)

where $H^{(1)}_{\beta +\frac{1}{2}}$, $H^{(2)}_{\beta +\frac{1}{2}}$ are the Hankel functions, and the constant coefficients $b_1$ and $b_2$ can be fixed by the initial condition. At high k, $H_{\beta +\frac{1}{2}}^{(1)}(-k\tau ) \simeq \sqrt{\frac{2}{\pi }} \frac{1}{\sqrt{- k\tau }} e^{ -i k\tau -\frac{i \pi (\beta +1) }{2}}$ and

$$\begin{aligned} h_k(\tau ) \sim \frac{1}{a(\tau )} \big [ b_1 \frac{1}{ \sqrt{2k} } e^{ -i k\tau -\frac{i \pi (\beta +1) }{2}} + b_2 \frac{1}{ \sqrt{2k} } e^{ i k\tau +\frac{i \pi (\beta +1) }{2}} \big ]. \end{aligned}$$

(C.26)

For the Bunch-Davies vacuum state during inflation and assuming the quantum normalization condition that for each k mode and each polarization of tensor, there is a zero point energy $\frac{1}{2} \hbar k$ in high frequency limit, we obtain $b_1= \frac{2}{M_\textrm{Pl}} e^{ \frac{i \pi (\beta +1) }{2}}$, $b_2=0$. The primordial spectrum of tensor perturbation is $\Delta ^2_t \simeq 0.5 \frac{H^2}{M_{Pl}} k^{2\beta +4}$ at low $k|\tau |\ll 1$ [39,40,41,42,43,44,45,46].

We also need the 1st-order perturbed scalar field. By expanding (2.8), we get the 1st-order perturbed field equation

$$\begin{aligned} \varphi ^{(1)'' } + 2 \frac{a'}{a} \varphi ^{(1)'} - \nabla ^2 \varphi ^{(1)} + a^2 V(\varphi ^{(0)})_{,\varphi \varphi } \, \varphi ^{(1)} = 3 \varphi ^{(0)'} \phi ^{(1)'} , \end{aligned}$$

(C.27)

which is not closed with the metric perturbation $\phi ^{(1)'}$ on the rhs. (The equation (C.27) is equivalent to the 1st-order energy conservation (A.14) in Appendix A.) For the power-law inflation, (C.27) becomes the following

$$\begin{aligned} \varphi ^{(1)''} +\frac{2(\beta +1)}{\tau }\varphi ^{(1)'} +\frac{ 2 (2 \beta +1)(\beta +2)}{\tau ^{2}}\varphi ^{(1)} -\nabla ^2\varphi ^{(1)} \nonumber \\= 3M_\textrm{Pl}\frac{\sqrt{2(\beta +1)(\beta +2)}}{\tau } \phi ^{(1)'}. \end{aligned}$$

(C.28)

Combining (C.28) and the constraints (C.2) (C.7), we get a third-order, closed equation

$$\begin{aligned} \varphi ^{(1)'''}&+\frac{3 \beta +4 }{\tau }\varphi ^{(1)''} +\frac{2 \beta ^2+2\beta -2 }{\tau ^2}\varphi ^{(1) '} \nonumber \\&-\frac{4 \beta ^2+10 \beta +4}{\tau ^3} \varphi ^{(1)} -\nabla ^2 \varphi ^{(1) '} - \frac{ \beta +2}{\tau } \nabla ^2 \varphi ^{(1)} =0 . \end{aligned}$$

(C.29)

It has a solution

$$\begin{aligned} \varphi ^{(1)}(\tau ,{\textbf{x}}) \propto \tau ^{-\beta -2}, \end{aligned}$$

(C.30)

which is a gauge mode and can be dropped by the gauge transform (B.9). We look for other two solutions of (C.29). Introduce

$$\begin{aligned} {\hat{\varphi }}^{(1)} = (-\tau ) ^{\beta +1}( \varphi ^{(1) '} +\frac{ \beta +2}{\tau }\varphi ^{(1)} ). \end{aligned}$$

(C.31)

Then (C.29) becomes the following second-order wave equation

$$\begin{aligned} {\hat{\varphi }}^{(1)}\, ^{''} -\frac{(\beta +2)(\beta +1)}{\tau ^2} {\hat{\varphi }}^{(1)} -\nabla ^2 {\hat{\varphi }}^{(1)} =0, \end{aligned}$$

(C.32)

which has the solutions

$$\begin{aligned} {\hat{\varphi }}^{(1)} (\textbf{x},\tau ) \equiv \frac{1}{(2\pi )^{3/2}} \int d^3 k e^{-i \,\textbf{k}\cdot \textbf{x}} {\hat{\varphi }}^{(1)} _k (\tau ), \end{aligned}$$

(C.33)

with the Fourier modes

$$\begin{aligned} {\hat{\varphi }}^{(1)} _k (\tau ) = \sqrt{-\tau }\Big [ d_1 H^{(1)}_{\beta +\frac{3}{2}}(-k \tau ) +d_2 H^{(2)}_{\beta +\frac{3}{2}}(-k \tau ) \Big ], \end{aligned}$$

(C.34)

where $d_1$, $d_2$ are the constant coefficients, and will be determined by the initial conditions later. From (C.31) (C.33), we obtain the solution of 1st-order perturbed scalar field as the following

$$\begin{aligned} \varphi ^{(1) }_k (\tau ) =&k^{-1}(-\tau ) ^{-\beta -\frac{1}{2}} \Big [ d_1 H_{\beta +\frac{1}{2}}^{(1)}(-k\tau ) +d_2 H_{\beta +\frac{1}{2}}^{(2)}(-k\tau ) \Big ] \nonumber \\&+(\beta +2)k^{-1} (-\tau ) ^{-\beta -2}\nonumber \\ {}&\int ^\tau d\tau _1 \sqrt{-\tau _1 } \Big [d_1H_{\beta +\frac{1}{2}}^{(1)}(-k\tau _1 ) +d_2 H_{\beta +\frac{1}{2}}^{(2)}(-k\tau _1 ) \Big ], \end{aligned}$$

(C.35)

which contains no explicit residual gauge mode. The perturbed scalar field $\varphi ^{(1)}$ is also waves propagating with the speed of light, like the tensor modes.

Now we derive the scalar metric perturbations $\phi ^{(1)}$ and $\chi ^{||(1)}$. Denote

$$\begin{aligned} \zeta \equiv -2\phi ^{(1)} -\frac{1}{3}\nabla ^2 \chi ^{||(1)}. \end{aligned}$$

(C.36)

From the constraints (C.1) (C.6) and the equations (2.13) (2.14) (2.16), we obtain the third-order, closed equation

$$\begin{aligned} \zeta '''&+ \big (a H - \frac{H'}{H} -\frac{H''}{H'} \big )\zeta '' + \big (\frac{H''}{H}-\frac{a H H''}{H'} +\frac{\left( H''\right) ^2}{\left( H'\right) ^2} -\frac{H ''' }{H'} \big ) \zeta ' \nonumber \\&- \nabla ^2 \zeta ' + \frac{a^2 H' }{a'} \nabla ^2 \zeta =0 \, , \end{aligned}$$

(C.37)

which is valid for a general $a(\tau )$. For the power-law inflation, (C.37) reduces to

$$\begin{aligned} \zeta ''' + \frac{3 (\beta +2)}{\tau } \zeta '' + \frac{ 2 (\beta +1) (\beta +3)}{\tau ^2} \zeta ' - \nabla ^2 \zeta ' - \frac{(\beta +2)}{ \tau }\nabla ^2 \zeta =0. \end{aligned}$$

(C.38)

It has a solution

$$\begin{aligned} \zeta (\tau ,{\textbf{x}}) \propto \tau ^{-\beta -2} \propto \frac{ a'}{a^2}, \end{aligned}$$

which is a gauge mode associated with the gauge mode (C.30) and can be dropped. Other two solutions can be derived in the same procedure as for (C.29). Actually the moment constraint (C.7) gives a simple relation

$$\begin{aligned} \zeta ' = - \frac{1}{M_\textrm{Pl}}\frac{\sqrt{2(\beta +1) (\beta +2)}}{\tau } \varphi ^{(1)}, \end{aligned}$$

(C.39)

Since $\varphi ^{(1)}$ is known by (C.35), integrating (C.39) gives

$$\begin{aligned} \zeta _{ k} = \frac{\sqrt{2(\beta +1)(\beta +2)}}{M_\textrm{Pl}\,k}(-\tau ) ^{-\beta -2} \nonumber \\\int ^{\tau } \sqrt{-\tau _1 }\Big [d_1 H^{(1)}_{\beta +\frac{1}{2}}(-k \tau _1 ) +d_2 H^{(2)}_{\beta +\frac{1}{2}}(-k \tau _1 ) \Big ] \, d\tau _1. \end{aligned}$$

(C.40)

where $d_1$ and $d_2$ are the same constants in (C.35), and represent two independent solutions.

The separate solutions of $\phi ^{(1)}$ and $\chi ^{||(1)}$ can be given as follows. By use of the relation (C.39) we write (C.2) in the k-space as

$$\begin{aligned} \phi ^{(1)'}_{ k} = \frac{k^2 }{6(\beta +1)} \tau \zeta _{ k} -\frac{1}{6 M_\textrm{Pl}}\frac{\sqrt{2(\beta +1) (\beta +2)}}{(\beta +1)}\varphi ^{(1)'}_{ k} -\frac{2 \beta +1}{6(\beta +1)}\zeta _{ k}^{\,'}. \end{aligned}$$

(C.41)

Since $\zeta _{k}$ and $\varphi ^{(1)}_{k}$ are known, integrating (C.41) gives

$$\begin{aligned} \phi ^{(1)}_{ k} =&-\frac{\sqrt{2(\beta +1) (\beta +2)}}{(\beta +1)} \frac{(-\tau )^{-\beta -\frac{1}{2}}}{6 M_\textrm{Pl}k} \Big [ d_1 H_{\beta +\frac{1}{2}}^{(1)}(-k\tau ) +d_2 H_{\beta +\frac{1}{2}}^{(2)}(-k\tau ) \Big ] \nonumber \\&-\frac{1}{M_\textrm{Pl}}\frac{\sqrt{2(\beta +1) (\beta +2)}}{(\beta +1)} \Big ( \frac{\beta +1}{2 \,k} (-\tau ) ^{-\beta -2} + \frac{k}{6 \beta } (- \tau )^{-\beta } \Big ) \nonumber \\&\times \int ^{\tau } d\tau _1 \sqrt{-\tau _1 }\Big [ d_1 H^{(1)}_{\beta +\frac{1}{2}}(-k \tau _1 ) +d_2 H^{(2)}_{\beta +\frac{1}{2}}(-k \tau _1 ) \Big ] \nonumber \\&+\frac{\sqrt{2(\beta +1) (\beta +2)}}{(\beta +1)\beta } \frac{k}{6 M_\textrm{Pl}} \int ^\tau d\tau _1 (-\tau _1)^{-\beta + \frac{1}{2} } \nonumber \\&\times \Big [ d_1 H^{(1)}_{\beta +\frac{1}{2}}(-k \tau _1 ) +d_2 H^{(2)}_{\beta +\frac{1}{2}}(-k \tau _1 ) \Big ] . \end{aligned}$$

(C.42)

Then by the relation (C.36) we get

$$\begin{aligned} \chi ^{||(1)}_{ k} =&-\frac{\sqrt{2(\beta +1)(\beta +2)}}{ M_\textrm{Pl}(\beta +1) k^3(-\tau ) ^{\beta +\frac{1}{2}}} \Big [ d_1 H_{\beta +\frac{1}{2}}^{(1)}(-k\tau ) +d_2 H_{\beta +\frac{1}{2}}^{(2)}(-k\tau ) \Big ] \nonumber \\&-\frac{\sqrt{2(\beta +1) (\beta +2)}}{ M_\textrm{Pl}(\beta \!+\!1)k} \frac{1}{\beta }(-\tau )^{-\beta } \!\int ^\tau d\tau _1 (-\tau _1)^{\frac{1}{2}} \Big [d_1H_{\beta +\frac{1}{2}}^{(1)}(-k\tau _1 ) \!+\!d_2 H_{\beta +\frac{1}{2}}^{(2)}(-k\tau _1 ) \Big ] \nonumber \\&+\frac{\sqrt{2(\beta +1) (\beta +2)}}{ M_\textrm{Pl}(\beta +1)k} \frac{1}{\beta } \int ^{\tau }d\tau _1 (-\tau _1)^{-\beta + \frac{1}{2} } \Big [ d_1 H_{\beta +\frac{1}{2}}^{(1)}(-k\tau _1 ) +d_2 H_{\beta +\frac{1}{2}}^{(2)}(-k\tau _1 ) \Big ] . \end{aligned}$$

(C.43)

A constant gauge mode has been dropped from (C.42) (C.43). In absence of an anisotropic stress tensor, the two scalars $\phi ^{(1)}$ and $\chi ^{||(1)}$ are not independent. Moreover, $\phi ^{(1)'}$, $\chi ^{||(1)}$ and $\varphi ^{(1)}$ are all related, and one set of coefficients $(d_1, d_2)$ fixes all the scalar perturbations.

The gauge invariant 1st-order scalar perturbations

The above 1st-order scalar metric perturbations and perturbed scalar field are generally subject to changes under the transformations between synchronous coordinates (see (B.4) $\sim$ (B.9) in Appendix B.) Therefore, the gauge-invariant 1st-order scalar perturbations are often used in cosmology, and mostly expressed in the Poisson coordinates with off-diagonal metric perturbations [4, 128, 131, 132, 137,138,139]. Here in the synchronous coordinates we give a complete treatment of four gauge-invariant scalars as the following

$$\begin{aligned}{} & {} \Psi = \phi ^{(1)} + \frac{1}{6} \nabla ^2 \chi ^{||(1)} + \frac{ a H}{2} \chi ^{||(1)'} , \end{aligned}$$

(D.1)

$$\begin{aligned}{} & {} \quad Q_\varphi = \varphi ^{(1)} + \frac{\varphi ^{(0)'} }{H a} \big (\phi ^{(1)} +\frac{1}{6}\nabla ^2 \chi ^{||(1)} \big ), \end{aligned}$$

(D.2)

$$\begin{aligned}{} & {} \quad \varphi ^{(1)}_{gi} = \varphi ^{(1)} - \frac{1}{2} \varphi ^{(0)'}\chi ^{||(1)'} = Q_\varphi - \frac{\varphi ^{(0)'}}{a H } \Psi , \end{aligned}$$

(D.3)

$$\begin{aligned}{} & {} \quad {{\mathcal {R}}} = \frac{a H }{\varphi ^{(0)'}} Q_\varphi . \end{aligned}$$

(D.4)

These scalars are invariant under the 1st-order gauge transformation (B.1) (B.2). $\Psi$ and $\varphi ^{(1)}_{gi}$ correspond to respectively (3.13) (6.8) in Ref. [128]. $Q_\varphi$ is often used in the presence of a scalar field [139, 142]. ${\mathcal {R}}$ amounts to a rescaling of $Q_\varphi$ and is called the comoving curvature perturbation [133, 134, 143,144,145]. Their equations for a general inflationary model in synchronous coordinates are given by

$$\begin{aligned}&\Psi '' + \big ( a H - \frac{H''}{H'} \big ) \Psi ' + \big ( a^2 H^2 + 2 a H' - \frac{a H H''}{H'} \big ) \Psi - \nabla ^2 \Psi =0 , \end{aligned}$$

(D.5)

$$\begin{aligned}&\quad Q_\varphi '' +2 a H Q_\varphi ' - \nabla ^2 Q_\varphi + \Big [ \frac{5 a^2 H^2}{4}+ \frac{3}{2} a H' -\frac{2 \left( H'\right) ^2}{H^2} - \frac{a H H''}{2 H'} \nonumber \\&\quad +\frac{2 H''}{H} +\frac{\left( H''\right) ^2}{4 \left( H'\right) ^2} - \frac{1}{2} \frac{H'''}{H'} \Big ] Q_\varphi = 0 \, , \end{aligned}$$

(D.6)

$$\begin{aligned}&\quad \varphi ^{(1)'' }_{gi} + 2 a H \varphi ^{(1)'}_{gi} - \nabla ^2 \varphi ^{(1)}_{gi} + \Big [ \frac{5 a^2 H^2}{4} -\frac{5 a H'}{2} -\frac{a H H''}{2 H'} \nonumber \\&\quad +\frac{\left( H''\right) ^2}{4 \left( H'\right) ^2} -\frac{H '''}{2 H'} \Big ] \varphi ^{(1)}_{gi} = \varphi ^{(0)'} \Big [ 4 \Psi ' + \big (5 a H + \frac{H''}{H'} \big ) \Psi \Big ] , \end{aligned}$$

(D.7)

$$\begin{aligned}&\quad {{\mathcal {R}}}'' + \big ( a H -2 \frac{H'}{H} + \frac{H''}{H'} \big ) {{\mathcal {R}}}' -\nabla ^2 {{\mathcal {R}}} =0 \, . \end{aligned}$$

(D.8)

For the power-law inflation, these equations reduce to

$$\begin{aligned}&\Psi _k \, '' + \frac{2\beta +4}{\tau } \Psi _k \, ' + k^2 \Psi _k =0 , \end{aligned}$$

(D.9)

$$\begin{aligned}&Q\, '' _{\varphi \, k} + \frac{2(\beta +1)}{\tau } Q\, ' _{\varphi \, k} + k^2 Q_{\varphi \, k} =0 \, , \end{aligned}$$

(D.10)

$$\begin{aligned}&\varphi ^{(1)'' }_{gi} + \frac{2(\beta +1)}{\tau } \varphi ^{(1)'}_{gi} +k^2 \varphi ^{(1)}_{gi} + \frac{2 (\beta +2) (2 \beta +1)}{\tau ^2} \varphi ^{(1)}_{gi} \nonumber \\&\quad = M_\textrm{Pl}\sqrt{2(\beta +1) (\beta +2)} \tau ^{-1} \big ( 4\Psi \, ' + \frac{4 \beta +2}{\tau } \Psi \big ) , \end{aligned}$$

(D.11)

$$\begin{aligned}&{{\mathcal {R}}}''_k +\frac{2 (\beta +1)}{\tau } {{\mathcal {R}}}'_k + k^2 {{\mathcal {R}}}_k=0 . \end{aligned}$$

(D.12)

The equation (D.11) of $\varphi ^{(1)}_{gi\, k}$ is not closed and contains inhomogeneous terms. Observe that Eq.(D.9) of $\Psi$ differs from Eq.(C.20) of the tensor $\chi ^{\top (1) }_{ij}$. The solutions are given by

$$\begin{aligned}&\Psi _{k} = \frac{\sqrt{2 (\beta +1) (\beta +2)}}{2 k^2 M_{\text {Pl}}} (-\tau )^{-\beta -\frac{3}{2}} \Big [ d_1 H^{(1)}_{\beta +\frac{3}{2}}(-k \tau ) +d_2 H^{(2)}_{\beta +\frac{3}{2}}(-k \tau ) \Big ] , \end{aligned}$$

(D.13)

$$\begin{aligned}&Q_{\varphi \, k} = k^{-1}(-\tau ) ^{-\beta -\frac{1}{2}} \Big [ d_1 H_{\beta +\frac{1}{2}}^{(1)}(-k\tau ) +d_2 H_{\beta +\frac{1}{2}}^{(2)}(-k\tau ) \Big ] \, , \end{aligned}$$

(D.14)

$$\begin{aligned}&\varphi ^{(1)}_{gi\,k} = k^{-1}(-\tau ) ^{-\beta -\frac{1}{2}} \Big [ d_1 H_{\beta +\frac{1}{2}}^{(1)}(-k\tau ) +d_2 H_{\beta +\frac{1}{2}}^{(2)}(-k\tau ) \Big ] \nonumber \\&\quad -k^{-2}(\beta +2) (-\tau )^{-\beta -\frac{3}{2}} \Big [ d_1 H^{(1)}_{\beta +\frac{3}{2}}(-k \tau ) +d_2 H^{(2)}_{\beta +\frac{3}{2}}(-k \tau ) \Big ], \end{aligned}$$

(D.15)

$$\begin{aligned}&{{\mathcal {R}}}_k = \sqrt{\frac{\beta +1}{2 (\beta +2)}} \frac{1 }{M_\textrm{Pl}} Q_{\varphi \, k} . \end{aligned}$$

(D.16)

Note that these solutions can be used to solve the 1st-order perturbations through the relation (D.1)–(D.4), and the results are the same as those given in Appendix C.

These gauge-invariant scalar solutions are simple, contain no integration terms, and can be also obtained from the scalar solutions (C.35) (C.42) (C.43). Among these, $\Psi$ and $Q_{\varphi }$ are sufficient to describe the 1st-order gauge invariant scalar perturbations.

The coefficients $d_1, d_2$ in (D.13) – (D.16) are the same as those in (C.35) (C.42) (C.43), and can be fixed as the following. We consider the scalar field $Q_\varphi$ as a quantum field

$$\begin{aligned} Q_{\varphi } (\textbf{x},\tau )&\equiv \int \frac{d^3 k}{(2\pi )^{3/2}} [ a_k Q_{\varphi \, k } (\tau ) e^{-i \,\textbf{k}\cdot \textbf{x}} + a^\dagger _k Q^* _{\varphi \, k } (\tau )e^{i \,\textbf{k}\cdot \textbf{x}}] . \end{aligned}$$

(D.17)

where the mode $Q_{\varphi \, k }$ is the positive frequency part of (D.14) and $d_2= 0$ is taken, and $a_\textbf{k}$ and $a_\textbf{k}^{\dag }$ are the annihilation and creation operators

$$\begin{aligned}{}[ a_\textbf{k},a_\mathbf{k'}^{\dag } ] = \delta ^3_{{\textbf{k}}, {\textbf{k}}'} . \end{aligned}$$

(D.18)

At high k, $Q_\varphi$ is approximately described as a harmonic oscillator with the energy density

$$\begin{aligned} \rho _Q = \frac{1}{2 a^2} \big [ (Q' _{\varphi })^2 + (\nabla Q_{\varphi })^2 \big ], \end{aligned}$$

(D.19)

(see the expression (A.1) with the potential V being neglected at high k.) Consider the Bunch-Davies vacuum state [136] during inflation

$$\begin{aligned} a_k|0\rangle =0. \end{aligned}$$

The energy density in the vacuum state is found to be

$$\begin{aligned} \langle 0| \rho _Q |0\rangle = \int \frac{d k }{k} \rho _{Q\, k} , \end{aligned}$$

(D.20)

where the spectral energy density is

$$\begin{aligned} \rho _{Q\, k}&= \frac{ k^3 }{4\pi ^2 a^2} \frac{1}{L^3} \Big ( | Q' _{\varphi \, k }|^2 + k^2 |Q_{\varphi \, k }|^2 \Big ) \nonumber \\&\simeq \frac{ k^3 }{4\pi ^2 a^2} \frac{1}{L^3} \Big |d_1 \frac{2 l_0}{k \sqrt{\pi } } e^{ -\frac{i \pi \beta }{2} -i\pi } \Big |^2 \frac{1}{a^2} k ~~~\hbox { at high}\ k . \end{aligned}$$

(D.21)

where $L^3$ is a normalization volume. By the requirement of quantum normalization that each k mode in the vacuum contributes a zero point energy $\frac{k}{2 a(\tau )}$ (with the unit $\hbar =1$), the spectral energy density is given by $\rho _{Q\, k}= \frac{ k^4 }{4\pi ^2\,L^3 a^4}$, and the coefficient is determined by

$$\begin{aligned} d_1 = \frac{ \sqrt{\pi } k}{2 l_0} e^{\frac{i \pi \beta }{2}+i\pi }, \end{aligned}$$

(D.22)

(which also leads to $Q_{\varphi \, k} \simeq \frac{1}{a(\tau )} \frac{1}{\sqrt{2k} } e^{-i k\tau }$ at high k). Thus, all the scalar modes (D.13) – (D.16) are fixed, and we can give their power spectra. Consider the auto-correlation function of $Q_\varphi$ in the BD vacuum

$$\begin{aligned} \langle 0|Q_\varphi (\tau ,\textbf{x}) Q_\varphi (\tau ,\textbf{x}) |0\rangle = \int ^\infty _0 \Delta _Q^2 \frac{d k }{k}, \end{aligned}$$

(D.23)

where the vacuum power spectrum of $Q_\varphi$ is

$$\begin{aligned} \Delta _Q^2(k, \tau )&=\frac{k^3}{2 \pi ^2}\left| {Q}_{\varphi \, k}(\tau )\right| ^2 . \end{aligned}$$

(D.24)

We are more interested in the spectrum at the low k range

$$\begin{aligned} \Delta _Q^2(k, \tau ) = \Big | \frac{\pi +i \sin (\pi \beta ) \Gamma \left( -\beta -\frac{1}{2}\right) \Gamma \left( \beta +\frac{3}{2}\right) }{2^{\beta + \frac{3}{2}} \pi ^{\frac{1}{2}} \Gamma \left( \beta +\frac{3}{2}\right) }\Big |^2 l_0^{-2} k^{2\beta +4}, \end{aligned}$$

(D.25)

which is independent of time (“conserved" on large scales). Using $l_0^{-2}\simeq H^2 /k_*^{2\beta +4}$ where $k_*= a H$ evaluated at the horizon exit ($k_* |\tau |=1$), (D.25) is written as

$$\begin{aligned} \Delta _Q^2(k, \tau ) = A_Q (\frac{k}{k_*})^{2\beta +4} , \end{aligned}$$

(D.26)

with the amplitude

$$\begin{aligned} A_Q = \Big | \frac{\pi +i \sin (\pi \beta ) \Gamma \left( -\beta -\frac{1}{2}\right) \Gamma \left( \beta +\frac{3}{2}\right) }{2^{\beta + \frac{3}{2}} \pi ^{\frac{1}{2}} \Gamma \left( \beta +\frac{3}{2}\right) }\Big |^2 H^{2} \simeq \frac{1}{2} H^{2} ~~~\hbox { for}\ \beta \simeq -2. \end{aligned}$$

(D.27)

Analogously, the spectrum of $\Psi$ at low k is also independent of time,

$$\begin{aligned} \Delta _\Psi ^2(k, \tau ) = \frac{k^3}{2 \pi ^2} \left| \Psi _{k}(\tau )\right| ^2 = A_{\Psi } (\frac{k}{k_*})^{2\beta +4} \end{aligned}$$

(D.28)

Fig. 1

The four primordial spectra $\Delta _Q^2 /H^2$, $\Delta _{\varphi _{gi} }^2 /H^2$, $\Delta _R^2$, $\Delta _\Psi ^2$ are shown, where $\Delta _Q^2$ and $\Delta _{\varphi _{gi}}^2$ have been rescaled by $1/H^2$ for a clear illustration. $\Delta _R^2$ is about 2700 higher than $\Delta _\Psi ^2$. The model $\beta =-2.02$ and $H/M_{Pl}=2\times 10^{-3}$, and at a fixed time $|\tau |=1$ for illustration

with the dimensionless amplitude

$$\begin{aligned} A_{\Psi }&= \frac{|(\beta +1) (\beta +2)| }{ 2^{2\beta + 7} \pi } \Big | \frac{1}{\Gamma \left( \beta +\frac{5}{2}\right) } -\frac{i \sin (\pi \beta ) \Gamma \left( -\beta -\frac{3}{2}\right) }{\pi } \Big |^2 \big (\frac{H}{M_{\text {Pl}}}\big )^2 \nonumber \\&\simeq \frac{|\beta +2|}{8\pi ^2} \big (\frac{H}{M_{\text {Pl}}}\big )^2 ~~~~ \hbox { for}\ \beta \simeq -2 . \end{aligned}$$

(D.29)

The spectrum of $\varphi _{gi}$ is

$$\begin{aligned} \Delta _{\varphi _{gi} }^2(k, \tau ) = \frac{k^3}{2 \pi ^2} \left| \varphi _{gi\, k}(\tau )\right| ^2 \simeq \Delta _Q^2(k, \tau ) ~~~~ \hbox { for}\ \beta \simeq -2 , \end{aligned}$$

(D.30)

which is approximately equal to $\Delta _Q^2$, since $\varphi _{gi\, k}$ is dominated by the first part of (D.15) for $\beta \simeq -2$. And the spectrum of ${{\mathcal {R}}}$ at low k (by the relation (D.16)) is

$$\begin{aligned} \Delta _R^2(k, \tau )&\simeq A_R \big (\frac{k}{k_*}\big )^{2\beta +4} , \end{aligned}$$

(D.31)

with the dimensionless amplitude

$$\begin{aligned} A_R \simeq \frac{1}{4|\beta +2|} \big (\frac{H}{M_{\text {Pl}}}\big )^2 ~~~~ \hbox { for}\ \beta \simeq -2. \end{aligned}$$

(D.32)

These spectra at low k have a common power index $(2\beta +4)$, but with different amplitudes. $\Delta _\Psi ^2$ and $\Delta _R^2$ are dimensionless, whereas $\Delta _Q^2$ and $\Delta _{\varphi _{gi} }^2$ have the dimension of square mass. Figure 1 shows that $\Delta _Q^2$ and $\Delta _{\varphi _{gi}}^2$ are almost overlapping, $\Delta _R^2$ has a shape similar to $\Delta _Q^2$ and $\Delta _{\varphi _{gi}}^2$, and these three spectra are $\propto k^2$ at high k. But $\Delta _\Psi ^2$ is flat $\propto k^0$ at high k. For the inflation $\beta =-2.02$, the ratio $\Delta _R^2/\Delta _\Psi ^2 \sim 2.7 \times 10^3$. We enlarge $\Delta _\Psi ^2$ in Fig. 2 to show its slope $(2\beta +4)$ at low k.

Fig. 2

The spectrum $\Delta _\Psi ^2$ of Fig. 1 is enlarged to show the slope at low k

It should be mentioned that these spectra are actually UV divergent at high k. The UV divergences can be removed by appropriate regularization, and we shall not discuss this issue here. See Refs. [45, 46, 146,147,148,149,150]. In literature, the spectrum $\Delta _R^2$ is sometimes used to compare with the CMB observation [134]. However, ${\mathcal {R}}$ is made of the scalar field and the metric perturbation, and $\Psi$ is the scalar metric perturbations which directly generates CMB anisotropies, and is more appropriate to compare with the CMB observation [151]. This will allow for a higher H of inflation for a given set of observational date of CMB. For an accurate description, one should solve the Boltzmann equation [19,20,21,22] of CMB photons that contains the Sachs-Wolfe term $\gamma \, ' _{ij}e^i e^j$, where $\gamma _{ij}$ is the metric perturbations and $e^i$ is the direction vector of photon propagation.

The above results of 1st-order gauge invariant scalar perturbations are valid for a general power-law inflation model. In the special case $\beta =-2$ of exact de Sitter inflation, the scalar metric perturbations $\phi ^{(1)}$ and $\chi ^{||(1)}$ in (C.42) (C.43) and their gauge invariant combination $\Psi$ in (D.13) are zero, and the 1st-order perturbed energy density and pressure, (A.9) (A.10), are also zero consistently, as also mentioned in Ref. [7]. The perturbed scalar field $\varphi ^{(1)}$ of (C.35), $Q_{\varphi }$ of (D.14), and $\varphi ^{(1)}_{gi}$ of (D.15) are nonzero, whereas the comoving curvature perturbation ${\mathcal {R}}$ of (D.16) becomes singular and is improperly defined for the de Sitter inflation. Nevertheless, the tensor perturbation (C.25) is regular for the de Sitter inflation [39,40,41,42,43,44,45,46].