Cosmic growth in f(T) teleparallel gravity

Abstract

Physical evolution of cosmological models can be tested by using expansion data, while growth history of these models is capable of testing dynamics of the inhomogeneous parts of energy density. The growth factor, as well as its growth index, gives a clear indication of the performance of cosmological models in the regime of structure formation of early Universe. In this work, we explore the growth index in several leading f(T) cosmological models, based on a specific class of teleparallel gravity theories. These have become prominent in the literature and lead to other formulations of teleparallel gravity. Here we adopt a generalized approach by obtaining the Mészáros equation without immediately imposing the subhorizon limit, because this assumption could lead to over-simplification. This approach gives avenue to study at which k modes the subhorizon limit starts to apply. We obtain numerical results for the growth factor and growth index for a variety of data set combinations for each f(T) model.

Appendix A: Field equations functions

The system of first order perturbation field Eqs. (33–37) can be simplified using the continuity equation  (43), velocity equation (44), and gauge invariant comoving fractional matter perturbation (41) such that each component can be expressed as a function of variables in their Fourier transform form and their higher order time derivatives:

$$\begin{aligned} {W_{A}}^{\mu }{_{\text {components}}} = \mathcal {W}_{1}(\varphi , \dot{\varphi },\psi ,\dot{\psi },\ddot{\psi }, \omega , \dot{\omega }, \delta _{m}, \dot{\delta }_{m})\,. \end{aligned}$$

(A1)

While the off-diagonal spatial-spatial component, given by Eq. (36), provides an expression of $\omega$ in terms of the other variables, the rest of the components are reduced to the set of Eqs. (47–50) for which functions $\mathcal {F}_{i}$ ($i \in [1,4]$) are given by

$$\begin{aligned} \mathcal {F}_{1}&= \frac{2 \frac{k^{4}}{a^{4}} H^{2} (1+f_{T} -36 \frac{a^{2}}{k^{2}} H^{2} \dot{H}f_{TT}) }{\dot{H} \left( \frac{k^{2}}{a^{2}} - 3\dot{H}\right) } \varphi \nonumber \\&\quad + \frac{2 \frac{k^{2}}{a^{2}} (1+f_{T}) (- H^{2} + \dot{H}) }{\dot{H}} \psi + \kappa ^{2} \rho \delta _{m} \nonumber \\&\quad + \frac{6 H \left( 1+f_{T} - 12 H^{2} f_{TT}\right) }{\frac{k^{2}}{a^{2}} - 3 \dot{H}} \left( \tfrac{k^{2}}{a^{2}} \dot{\psi } - \dot{H} \dot{\delta }_{m}\right) \,, \end{aligned}$$

(A2)

$$\begin{aligned} \mathcal {F}_{2}&= \frac{6 \frac{k^{2}}{a^{2}} (1+f_{T} - 36 \frac{a^{2}}{k^{2}} H^{2} \dot{H} f_{TT})}{\frac{k^{2}}{a^{2}} - 3 \dot{H}} ( H \varphi + \dot{\psi }) \nonumber \\&\quad -72 H \dot{H} f_{TT} \psi -\frac{6 \dot{H} (1+f_{T} - 12 H^{2} f_{TT})}{\frac{k^2}{a^2} - 3 \dot{H}} \dot{\delta }_{m}\,, \end{aligned}$$

(A3)

$$\begin{aligned} \mathcal {F}_{3}&= -\frac{2 \frac{k^2}{a^2} H (1+f_{T})}{ \dot{H}} \psi \nonumber \\&\quad +\frac{2 \frac{k^{4}}{a^4} H (1 + f_{T} - 36 \frac{a^2}{k^2} H^2 f_{TT})}{\dot{H} (\frac{k^2}{a^2} - 3 \dot{H})} \varphi \nonumber \\&\quad + \frac{6 (1 + f_{T} - 12 H^2 f_{TT} ) }{ \frac{k^{2}}{a^2} - 3 \dot{H} } \left( \frac{k^2}{a^2}\dot{\psi } - \dot{H} \dot{\delta }_{m} \right) \,, \end{aligned}$$

(A4)

$$\begin{aligned} \mathcal {F}_{4}&=0 \frac{k^2}{a^2} \left( (1 + f_{T})\left( 1 + \frac{H^2}{\dot{H}} - \frac{H \ddot{H}}{\dot{H}^{2}}\right) \right. \nonumber \\&\quad \left. - 12 H^{2} f_{TT} \right) (\psi - \varphi ) \nonumber \\&\quad + \left( \frac{k^2}{a^2} \frac{H}{\dot{H}} (1+f_{T}) + 3 H (1 + f_{T} - 12 H^2 f_{TT}) \right) \left( \dot{\psi } -\dot{\varphi } \right) \nonumber \\&\quad - \left( 6 \dot{H} (1+f_{T}) + 9 H^2 (1 + f_{T} - 20 \dot{H} f_{TT}) - 108 H^{4}(f_{TT} - 4 \dot{H} f_{TTT}) \right) \varphi \nonumber \\&\quad - 12 H \left( 1 + f_{T} - (12 H^{2} + 9 \dot{H}) f_{TT} + 36 H^2 \dot{H} f_{TTT}\right) \dot{\psi } - 3 (1 + f_{T}\nonumber \\&\quad - 12 H^2 f_{TT}) \ddot{\psi }\,. \end{aligned}$$

(A5)

Hence, equations for the zeroth order derivative variables can be obtained in terms of higher-order derivatives and $\delta _{m}$. One choice is that of deriving the expressions by simultaneously solving Eqs. (48) ($\mathcal {F}_{2}$) and (49) ($\mathcal {F}_{3}$) such that

$$\begin{aligned} \varphi :=\mathcal {G}_{1}&= \frac{ \dot{H} (1 + f_{T} - 36 \frac{a^2}{k^2} \dot{H}^{2} f_{TT}) (1 + f_{T} - 12 H^2 f_{TT})}{ \frac{k^2}{a^2} H (1 + f_{T} - 12 \dot{H} f_{TT}) [ 1+f_{T} - 36 \frac{a^2}{k^2} H^2 \dot{H} f_{TT} ]} \dot{\delta }_{m} \nonumber \\&\qquad - \left( \frac{ (1 + f_{T}) }{ H (1 + f_{T} - 12 \dot{H} f_{TT})} - \frac{ 36 \frac{a^2}{k^2} \dot{H}^2 f_{TT} (1 + f_{T} - 12 H^2 f_{TT}) }{ H (1 + f_{T} - 12 \dot{H} f_{TT})( 1 + f_{T} - 36 \frac{a^2}{k^2} H^2 \dot{H} f_{TT})} \right) \dot{\psi }\,, \end{aligned}$$

(A6)

$$\begin{aligned} \psi :=\mathcal {G}_{2}&= -\frac{1 + f_{T}}{H (1 + f_{T} - 12 \dot{H} f_{TT})} \dot{\psi } + \frac{\frac{a^2}{k^2} \dot{H} (1 + f_{T} - 12 H^2 f_{TT}) }{H (1 + f_{T} - 12 \dot{H} f_{TT})} \dot{\delta }_{m}\,. \end{aligned}$$

(A7)

The rest of the field equations, along with the continuity and velocity equations and with their time derivatives, are used to express the time derivatives of variables in terms of higher-order derivatives and $\delta _{m}$ to ultimately obtain an equation as a function of $\delta _{m}$ and its derivatives:

$$\begin{aligned} 0 = \mathcal {W}_{2}(\delta _{m}, \dot{\delta }_{m},\ddot{\delta }_{m})\,, \end{aligned}$$

(A8)

resulting in the Mészáros equation (59). It should be noted that these equations are derived without assuming the subhorizon limit $k \gg aH$, such that we have a more general Mészáros equation when considering the growth of CDM.