Scalar perturbations in $f(T)$ gravity using the $1+3$ covariant approach

We investigate the cosmological scalar perturbations of standard matter in the context of extended teleparallel $f(T)$ gravity theories using the 1 + 3 covariant formalism. We review the gravitational field equations of $f(T)$ gravity to introduce therein a gauge-invariant spatial gradient of the torsion fluid and obtain the linear perturbation equations. After performing the usual scalar and harmonic decompositions, we analyze the matter perturbations in the quasi-static approximation for two non-interacting fluids scenarios, namely torsion-dust and mixtures. We consider the $f(T)$ power-law paradigmatic classes of model $f(T) = \alpha(-T/T_0)^n$, for both torsion-dust and torsion-radiation scenarios. Under this scope, exact solutions of the matter perturbations are obtained. We examine the growth of the matter density contrast for these mixtures. In a similar manner, we also consider the long- and short-wavelength modes in the torsion-radiation case. We consider different values of n to explore the growth of matter density contrast with red-shift. For the case of n closer to one, our paradigmatic $f(T)$ gravity model is favored with the usual results of general relativity (GR) and general relativity with a cosmological constant ({\Lambda}CDM). However, for the case of n\gg1 the amplitude of the matter density fluctuation extremely high and unrealistic to compare with GR and {\Lambda}CDM results for these non-interacting fluids. Notably, in the torsion-radiation system, the behavior of the growth of matter density contrasts is abandoned and ruled-out for n \ll 1 in both wave modes. While our results show a richer set of possibilities that can help to constrain the model parameters using future observational data, they also accommodate currently known features of power spectrum in the large-scale structure in the general relativistic limit.


Introduction
The recent discovery of the accelerating expansion of the Universe [1,2] has put a big question mark on the General Relativity (GR)-based cosmology. Moreover, there are many outstanding open problems and active theoretical and observational research areas in physical cosmology such as the origin, evolution and formation of the primordial Universe [3,4,5,6], the anisotropy of the Cosmic Microwave Background radiation (CMB) [7,8,9], the accelerating expansion of the entire Universe, how cosmological perturbations [10,11,12,13,14] and the primordial fluctuations of the early Universe formed the large-scale structure [7,8,11,15], and the formation and evolution of the early type astrophysical objects like galaxies [16,17,18], to mention but a few. The current Concordance (ΛCDM) cosmological model takes into account the assumption that the Universe passed through different epochs such as inflation, radiation-dominated, matter-dominated and dark energy-dominated epochs. The growth of energy density fluctuations is the cause behind the inhomogeneity and formation of the large-scale structures in the Universe [19]. Of course, different aspects of cosmology can be explored in different extended theories of gravity including f (T ) gravity theory [20,21,22,23,24,25,26,27,28,29]. The study of linear cosmological perturbations in f (T ) gravity theory using the 1 + 3 covariant formalism is the main issue in this manuscript. Basically, there are two mainstream formalisms to study cosmological perturbations, namely, the metric formalism [12,30,31] and the 1 + 3 covariant gauge-invariant formalism [10,32,33,34,35], for GR and extended gravity approaches. In the 1 + 3 covariant formalism, the perturbations defined describe true physical degrees of freedom and no physical gauge modes exist. In recent years, there has been active research on cosmological perturbations theory for both GR [36,37,38,39] and different extended gravity theories [32,33,40] using the 1 + 3 covariant formalism where the recent observations constrained considered gravity models.
Within the f (T ) gravity framework, we will resort to the energy-momentum tensor (EMT) of the torsion [fluid] in addition to the EMT of the physical standard matter fluids [41,42] to derive the perturbation equations. After deriving the perturbation evolution equations for genetic f (T ) theories, we will consider the power-law [43,44,45] f (T ) = T 0 (−T /T 0 ) n paradigmatic model admitting an ansatz scale factor of the field equation for matter dominated Universe [46] as a = a 0 (t/t 0 ) m , for further analysis using the well-known approximation technique dubbed quasi-static approximation [21,33,47,48,49]. For instance, in [21] the validation of such an approximation technique was considered to explore the so-called effective field theory approach to torsional modified gravities by considering the k 2 /a 2 H 2 1 mode.
In this paper, we shall apply this approximation method and assume very slow temporal fluctuations in the perturbations of both the torsion energy density and momentum compared with the fluctuations of matter energy density. As such, the time derivative terms of the fluctuations of torsion density and momentum are neglected. Finally, for comparison, we depict that the growth of the energy density fluctuations with red-shift for both GR and f (T ) gravity approaches. The energy density contrasts have done for GR approach in dust-dominated and radiation-dominated Universe, and for f (T ) gravity approach in torsion-dust and torsion-radiation systems for both modes, namely short-and long-wavelength modes.
The road-map of this paper is as follows: in the following section, we review the 1 + 3 covariant gauge-invariant cosmological perturbations formalism within the f (T ) gravity framework. The kinematic feature of the Universe and the general fluid description are studied in the presence of an effective torsion fluid in Secs. 3 and 4 respectively. In Section 5, we derive the linear evolution equations for matter and torsion perturbations, the scalar decomposition of which will be carried out in Section 6. In Section 7, we discuss the harmonic decomposition of the scalar perturbations and figure out how to analyze the growth of the matter energy density perturbations. We explore the growth of energy density fluctuations in Sec. 8 for dust and radiation fluids in the GR context and in Sec. 9, for the torsion-dust and torsion-radiation systems for f (T ) gravity approach. The growth of the energy density contrast f (T ) and GR is covered in this section. Finally, we wrap up with the main results of the manuscript in Section 10.

The 1 + 3 covariant formalism in f (T ) gravity
In the 1 + 3 covariant decomposition formalism, it is assumed that a fundamental observer slices space-time into temporal and spacial hyper-surfaces [50]. Given the fact that the matter components in the Universe would define a physically motivated preferred motion, it is usual to choose the CMB frame, where the radiation dipole vanishes, as the natural reference frame in Cosmology [51,52]. For the unperturbed (background) Universe, we define the tangent space-time by the tetrad field e a 0 = u a , where u a is the four-velocity vector of the observer. The preferred world-line is given in terms of local coordinates x a in the general coordinates x a = x a (τ ) and we define the four-vector velocity u a as where τ is measured along the fundamental world-line. According to the reason above, the component of any vector X a parallel to the 4-velocity vector u a becomes where U a b is the projection tensor into the one-dimensional tangent line and satisfies the following relations: Moreover, we define h ab as another projection tensor into the three-dimensional, orthogonal to u µ and it satisfies the following properties: The action of modified teleparallel gravity (TG) one is given by [44] S where e is the determinant of the tetrad field e a µ i.e., e = det|e a µ | = √ −g and the coupling constant κ 2 = 8πG/c 4 1 . Note that, TG and GR could be recover for the limiting case of f (T ) = T , whereas, we restore GR with cosmological constant so-called ΛCDM for the case of of f (T ) = T + 2Λ [29]. The field equations for f (T ) gravity [53] yield where g ab is the metric, denotes the usual EMT of standard matter (m) fields, and S d ab denotes a super-potential term [22,23,24] S d ab = where K d ab is the contortion tensor and T c cb is the torsion tensor. It is straightforward to see that the above field equations can be written in the more compact form as where we have defined the EMT of the torsion (T ) fluid as [20] Θ (T ) All thermodynamic quantities, such as the total energy density ρ, isotropic pressure p, heat flux q a and anisotropic stress tensor π ab for matter (m) and torsion (T ) fluids are extracted from the total EMT Θ ab as follows: whereas the respective quantities for both matter and torsion components can similarly be extracted from their corresponding EMTs, such that From Eq. (6), the Friedmann equations of the effective fluid are presented in [20,53] as follows: 2Ḣ where H(t) ≡ȧ(t)/a(t) is the Hubble parameter defined in terms of the scale factor a(t) and its time derivative. One can directly obtain the corresponding thermodynamic quantities such as the effective energy density of the fluid and the effective pressure of the fluid respectively. It is easy to show that the Friedmann equations (14)- (15) can be re-expressed as where we have introduced the following new variables: It is worth noting here that if Ω d and Ω r are the fractional energy densities of the dust (d) and radiation (r) fluids respectively, then Ω m = Ω d + Ω r . From the above extracted basic quantities, namely the energy density ρ and pressure p of the fluid, in the following we shall investigate the growth of energy density linear perturbations in 1 + 3 gauge-invariant approach in both the matter and radiation-dominated cosmological epochs. For perfect fluids, the energy flux (12) and the anisotropic stress (13) are identically zero.
In this paper, we consider the non-interacting perfect fluids and the energy flux and anisotropic stress zero in our case. Obviously, in the case of a gravitational Lagrangian f (T ) ≡ T [41,54], the physical quantities in Eqs. (9), (16) and (17) reduce to the usual GR limit. In such a limit, the linear cosmological perturbations have been widely studied in [55,56].

Kinematic quantities in the presence of torsion
As stated previously, the kinematics of the four-velocity vector u a determines the geometry of the fluid flow. Any tensor V ab can be expressed as a sum of its symmetric V (ab) and anti-symmetric V [ab] parts as In this formalism, the covariant derivative of u b is split into the kinematic quantities [57] as whereθ is the fluid expansion,σ ab is the shear tensor,u a is the four-acceleration of the fluid andω ab is the vorticity tensor in the presence of torsion. Notice that a tilde represents torsion-dependent physical parameters and a non-tilde represents Levi-Civita connection-dependent parameters. The detailed expressions of torsion dependent kinematic quantities such as expansion of the fluid, shear tensor, the vorticity tensor and the relativistic acceleration vector are presented in Refs. [25,26,27]. The expansion of the fluid flow in the presence of torsion is given byθ where the vector torsion T b can be either space-like, time-like or light-like and this three different types of vector torsion is discussed in [25]. Here we have defined the Hubble expansion parameter 3H ≡ θ and θ = u b ; b is the volume-expansion. The shear tensor denotes the change of distortion of the matter flow with time and it is given asσ and the vorticity tensor denotes the rotation of the matter relative to the non-rotating (Fermi-propagated) frame and it is given asω Also, the relativistic acceleration vector describes the degree of matter to move under forces other than gravity plus inertia, namelyu which vanishes for free-falling matter. The general expression for the torsion-based Raychaudhuri equation is given by [25,26,27] In this paper, we assume that the world-line is tangent to u c but parallel tou c , i.e., u cu c = 0. Moreover, ω cb = 0 =σ cb in the case of non-rotational and shear-free fluids and from the covariant approach of the field equation, R cb u c u b = 1/2 (ρ + 3p) for relativistic fluid [58,59]. Then, Eq. (27) becomeṡ For a space-like torsion vector the inner product of the torsion and four-velocity vectors of the fluid u b T b is vanished identically [27].

General fluid description
Here, we assume the non-interacting matter fluid (ρ m ≡ ρ r + ρ d ) with torsion fluid in the entire Universe and the growth of the matter energy density fluctuations has a significant role for formation of large-scale structures.

Matter fluids
Let us consider a homogeneous and isotropic expanding (FLRW) cosmological background and define spatial gradients of gauge-invariant variables such as those of the energy density X a , pressure Y a and volume expansion of the fluid Z a as follows [33,38,61,62]: Those two gradient variables are a key points to examine evolution equation for matter density fluctuations.

Torsion fluids
Analogously to the 1 + 3 cosmological perturbations treatment for f (R) gravity theory [33], let us define extra key variables resulting from spatial gradients of gauge-invariant quantities which are connected with the torsion fluid for f (T ) gravity. Accordingly, we define the quantities F a and B a as to characterize the fluctuations in the torsion density and momentum respectively.
All the quantities listed in Eqs. (30) -(33) will be considered to develop the system of cosmological perturbation equations for f (T ) gravities in the 1 + 3 covariant formalism. Moreover, for each non-interacting fluid, the following conservation equations, considered in [37,38] The speed of sound c 2 s = δp δρ would play an important role since it allows us to relate the perturbed pressure with the energy density of the fluid. Also, the time derivative of equation of state parameterẇ =ṗ m /ρ m can be expressed with speed of sound [33], and the non-constant equation of state parameter for all fluids aṡ and this equation of state parameter is the generalized one for all matter fluids. In fact, for non-interacting fluids, in the following we shall consider the equation of state parameter to be independent of time, thusẇ = 0. In this approach, the speed of sound becomes equivalent to the equation of state parameter w = c 2 s [63]. For perfect fluid the energy flux and anisotropic-stress are zero (Ψ a = Π a = 0).

Linear evolution equations
Here we derive the first-order evolution equations for the above-defined gauge-invariant gradient variables. In the energy frame of the matter fluid, these evolution equations for the perturbations are given as: In the following section, we will see how to decompose the evolution of the above vector gradient variables (38) -(41) into those of scalar variables by applying the scalar decomposition method outlined.

Scalar decomposition
It is generally understood that the large-scale structure formation follows a spherical clustering mechanism, and that only the scalar (non-solenoidal) parts of the above gradient vectors (38) -(41) assist in the clustering. As a result, we extract the scalar part of a vector I a by taking its divergence as [33] where I =∇ a I a , and Σ I ab = I (ab) − The last two terms of Eq. (42) describe shear and vorticity effects, respectively. To extract the (scalar) density contrast, the vorticity vanishes and only the shear part is considered. From vector quantities, one can further extract the scalar gradient quantities of our cosmological perturbations, believed to be responsible for the spherical clustering of large-scale structure [33,64]. Let us now define our scalar gradient variables as follows: It can be shown that these quantities evolve as: Finally, the second-order scalar evolution equations can be derived by differentiating the above first-order evolution equations with respect to time. For instance, from Eqs. (48) and (49) we obtain whereas from Eqs. (50) and (51) we geẗ The scalar gradient variables (38)) -(53) we take as an input to study the energy density fluctuations in different cosmological era by applying the harmonic decomposition of these variables in the next section.

Harmonic decomposition of variables
From the results of the previous section, we clearly see that the linear cosmological evolution equations of the scalar variables are second-order differential equations, complicated to solve. Thus, in order to obtain the eigenfunctions and the corresponding wave-numbers from those second-order differential equations, we shall apply the separation-of-variables technique. Then we shall use the standard harmonic decomposition of the evolution equations for cosmological perturbations [33,61,65] for further details on this technique). All the above linear evolution equations (48) -(53) have a similar structure as the harmonic oscillator equation and the second-order differential evolution equations for any functions X and Y can be represented schematically as [33] where the terms A, B, and C represent the damping oscillator or frictional force, restoring force and source force respectively. Then by applying the separation-of-variables technique, we express where k is the wave-number and Q k (x) is the eigenfunctions of the covariant derivative. The wave-number k represent the order of the harmonic oscillator and relate with the scale factor as k = 2πa λ , where λ is the wavelength of the perturbations. Here, we define eigenfunctions of the covariant derivative with the Laplace-Beltrami operator for FLRW space-time as Armed with all this machinery, the first and second-order evolution equations (48) -(53) are expressed as: In the following, we shall apply the aforementioned quasi-static approximation in which time fluctuations in the perturbations of the torsion energy density F k and momentum B k are assumed to be constant with time, i.e., one is allowed to takeḞ k =F k =Ḃ k ≈ 0. Under this approximation, the first-order linear evolution equations (57)-(58) reduce to: Also, from Eqs. (59) and (62) results the relation By using the latter Eq. (65) together with the quasi-static approximation itself, Eq. (61) for matter energy density perturbations yields For the case of f (T ) = T + 2Λ , Eq. (66) is reduced to the well-known evolution equation of ΛCDM : where the energy density of cosmological constant fluid Λ = 3H 2 0 Ω Λ . In this context, the matter and cosmological constant fluids have involved in the growth of the energy density fluctuations and formation of large-scale structures. Also, for the paradigmatic case of f (T ) = T [41] GR is exactly recovered and the evolution equation (66) coincides with GR as [32] As we shall see in the following sections, Eq. (66) remains a key equation for analyzing the growth of energy density fluctuations capable of explaining the formation of large-scale structures. For the sake of simplicity and with the aim at illustrating the versatility of our analysis, we shall consider paradigmatic power-law f (T ) gravity models where α and n are dimensionless constants, and T 0 = −6H 2 0 is the present-day value of the torsion scalar. We assume the ansatz for the scale factor solution [46,66,67] where m is a positive constant, and as usual the scale factor is related to the cosmological red-shift as a = a 0 /(1 + z) 2 . For convenience, we also transform any time derivative functions f and H into a red-shift derivative as follows:ḟ .
Before solving the linear evolution equations (63) -(66), let us point out that for f (T ) gravity model (69) and scale factor (70), the background quantities X and Y as defined in Eq. (20) become From Eq. (18), we redefine the normalized energy density parameter for non-interacting torsion-matter fluids as 1 = Ω d + Ω T , i.e. Ω T ≡ X is the normalized energy density parameter of torsion fluid. Consequently, the normalized energy density parameter for matter yields For the case of n < 0.5, the normalized matter energy density parameter has a negative sign which shows an unphysical mode. Based on relation (78), With this definition, it is possible to know the amount matter fluid in the non-interacting system and analyze the growth of matter fluctuations with red-shift (we will see in detail for torsion-dust and torsion-radiation cases in Sec. 9). As an example: for n = 1, the matter fluid is large enough in the system and the torsion fluid becomes negligible. In this case, we obtain the matter dominated Universe and our generalized evolution Eq. (66) reduces to Eq. (68). For n ≥ 1, Ω m ≥ 1 and Ω T ≤ 0, in this situation the matter fluid is a major component of the Universe and the contributions of torsion like fluid with a negative energy density are the same as of a cosmological constant. However, the effective energy density of the fluid becomes ρ = ρ m + ρ T ≥ 0 [20] as presented in Eq. (16).
Here, we define the normalized energy density contrast for matter fluid as where the subscript in refers the initial value of ∆ m (z) at initial red-shift z in 3 .
Indeed, the variation of CMB temperature detected observationally is of the order of 10 −5 [68] and this variation strongly supports the gravitational perturbations initially through their red-shifting effect on the CMB [69,70]. Also, we shall assume the following initial conditions as ∆ in ≡ ∆ k (z in = 1100) = 10 −5 anḋ ∆ in ≡∆ k (z in = 1100) = 0, for every mode k to deal with the growth of matter fluctuations (similar analysis is done in [71]). The energy density fluctuations ∆ m (z in ) = ∆ in = 10 −5 for all n at the initial red-shift z in = 1100. At this red-shift the value of the normalized energy density perturbations of the matter fluid presented in Eq.
(79) becomes one (δ(z in ) = 1). Moreover, we define the relative difference of matter energy density fluctuations between results for f (T ) gravity and GR as where analogously to (79) For the case of f (T ) ≡ T , f (T ) gravity coincides with TEGR. In this limit, δ k (z) = δ k GR and δ(z) can be reduced to the GR limit and (z) = 0.

Matter density fluctuations in ΛCDM and GR Limits
In this section, we analyze the growth of the energy density fluctuations for dust and radiation fluids in ΛCDM and GR limits from Eqs. (67) and (68) respectively.

Dust-dominated Universe
If, we assume that the Universe is dominated by dust fluid only, then the equation of state parameter is w d ≈ 0. Consequently, Eq. (67) and Eq. (68) read∆ By applying Eq. (71) this equation becomes and admits the solution where C 1 and C 2 are integration constants and we determine these constants by imposing the above initial conditions.
In the dust-dominated Universe, the input parameter Ω d is a key point to determine the magnitude of matter fluctuations with red-shift. For instance, the numerical result of Eq. (84) is presented in Fig. 7 (see Appendix 11.1 for Ω d = 1 and it shows the growth of energy density fluctuations of dust fluid in the dust-dominated Universe 4 . On the other hand, we fill in for the observed value of Ω d = 0.32 [72] in Eq. (84) to see the effect on the growth of matter density contrast and present the numerical plot in Fig. 8 (see Appendix 11.1. With this plot, the matter density contrasts is also growing up with red-shift.

For ΛCDM Limit
Here, we study the growth of energy density perturbations of the radiation fluid, by assuming the Universe has two non-interacting cosmic fluid components, namely radiation and the cosmological constant. In this assumption, the equation of state parameter w r ≈ 1/3 and the normalized energy density parameter is Ω Λ = 1 − Ω r . Then, Eq. (67) in red-shift space becomes The exact solution for the short-wavelength mode, k 2 /a 2 H 2 1, becomes whereas in the long-wavelength mode, k 2 /a 2 H 2 1, the exact solution reads 5 ∆(z) = log(1 + z) C 1 sinh 2(Ω r + Ω Λ ) + C 2 cosh 2(Ω r + Ω Λ ) . (87)

For GR Limit
For the radiation-dominated Universe, the red-shift dependent of evolution Eq. (68) becomes The exact solution of Eq. (89) for short-wavelength mode is 4 For all figures, we use the black dotted line to shows the energy density fluctuations growth (at z 0 = 0) and λ = 0.05Mpc for plotting in this manuscript. 5 Note that we have used the relation k 2 a 2 H 2 = 16π 2 λ 2 (1 + z) 4 . and for long-wavelength mode we have Notice that, the difference between ΛCDM and GR limits is the parameter Ω Λ which is not appeared in GR or Ω Λ = 0 in GR solutions 6 . The numerical results of the growth of matter density contrast for Eqs. (86) and (89) and also Eqs. (87) and (90) are depicted in Figs. 10 and 9 respectively (see Appendix 11.1).
The current observational data indicates that the growth of energy density fluctuations of radiation fluid is almost negligible at large scale structure and the normalized energy density of the radiation Ω r ≈ 4.48 × 10 −5 [73]. In this situation, the numerical results of Eq. (89) are presented in Figs. 11 and 12 (see Appendix 11.1) for short-and long-wavelength modes respectively.
From all above plots, we see that the energy density fluctuations of a radiation fluid are growing-up in a very small scales with red-shift.

Matter density fluctuations in f (T ) gravity approach
Here, we consider the cosmic medium as a mixture of two non-interacting fluids as a torsion-dust and torsionradiation mixture.

Torsion-dust system
In this fluid mixture, we assume that the Universe hosts two dominant cosmological fluids, namely a torsion-like fluid and the usual dust (w d = 0) matter. In this case, evolution equation (66) reduces tö We note that for the case of n = 1, the parameter Y = 0 and Eq. (92) reduces to the well-known evolution equation of energy density of dust fluid in the GR limit namelÿ In red-shift space, it can be shown that Eq. (92) yields with the exact solution given as Our free parameters Ω d , m and n have a significant role to present the numerical solution of Eq. 95, and explore the growth of energy density fluctuations with red-shift.
Due to cosmic expansion, the background energy density of the dust fluid decreases with the scale factor of Universe, ρ = ρ 0 a −3 and it is proportional to the red-shift z. Then, the scale factor becomes a(t) = a 0 (t/t 0 ) 2/3(1+w d ) .
Here, has two complex and one real possible roots such as r 1,2 ≈ 0.54522 ± i0.57031 and n ≈ 2.4096. However, for illustrative purposes, we use different values of n for all numerical analysis and show that the f (T ) gravity model under consideration is an alternative approach to study the growth of the matter fluctuations in dust-dominated Universe and make a comparison with the well-known theory of gravity GR limit as well 7 .
In order to fix the parameter Ω d , we use either the definition of Eq.(78) for n ≥ 0.5 or Ω d = 0.32 for any values of n. In the following, we apply these two options and explore the energy density contrast in f (T ) gravity approach systematically. For any case at n = 1, the numerical solution of δ(z) coincides with GR limit.
Case I: Ω d = 2n−1 n , n ≥ 0.5 To provide the parameter Ω d , we use the definition from Eq.(78). From this definition, it is possible to determine the fractional amount of the normalized energy density parameters of the torsion (Ω T ) and dust (Ω d ) in the system. For instance, at n = 1, Ω d = 1 and Ω T reads zero. In this case, the numerical solution is reduced to GR limit Eq. (83). For n ≥ 1, Ω d ≥ 1 and Ω T ≤ 0, then, the dust fluid is the major component of the Universe and we note the contributions of torsion like fluid with negative energy density but ρ = ρ d + ρ T ≥ 0. For n = 0.9, Ω d ≈ 0.88 and Ω T ≈ 0.12, meaning that the Universe has relatively more dust fluid than torsion fluid and at a particular n ≈ 0.5953, the value of the normalized energy density parameter of dust fluid is favored with the observed value Ω d ≈ 0.32 in SNIa data [72], consequently, Ω T = 0.6800001, and close to the observed value of Ω Λ = 0.68 [72]. With this values, we suggest that at n ≈ 0.595, the growth of energy density fluctuations is occurred in the present torsion-dust era. For the case of n = 0.5, Ω d reads zero and Ω T = 1. In this condition, the torsion fluid is large enough and the dust fluid is negligible in the torsion-dust system.
From the below Fig. 1 and Appendix 11.2 Figs. 13 and 14, we observe that the growth of the energy density fluctuations is nearly homogeneous initially, and growing-up with red-shift till near-future epoch for different values of n ≥ 0.5.
For the case of n = 1 (red solid line) in Fig. 1, the growth of density fluctuations is the same as GR which is presented in Fig. 7. This once more indicates that our paradigmatic model is reduced to GR. Even if, at n ≈ 0.595 (black solid line), the fluctuation of the energy density is growing-up with cosmic-time and similarly with the observed value of Ω d presented in Fig. 8. In this situation, we present the numerical result of Eq. (97) in Fig. 2 for different values of n. For n = 1 (red solid line) in Fig. 2 the density contrast has the same result as GR which is presented in Fig. 8. However, the growth the fluctuations is proportional to n values.  We also study the matter density contrasts between f (T ) and GR approaches by using the definition in Eq.
(80) and present the numerical results in Figs. 15 -17 for both cases accordingly in the Appendix 11.2. In those figures, the comparison of dust perturbations between f (T ) gravity and GR is done by considering all above different values of n.
In general, we depict that our paradigmatic f (T ) gravity model is an alternative approach to study matter density contrast and the formation of large-scale. It explains the fluctuations of energy density in both cases and the fluctuations are growing-up for different intervals of n in the torsion-dust Universe. From all plots, we observe that our f (T ) gravity model has much gained an attention to explain the matter density contrast with range of n ≤ 1 and favored with the usual results of GR. For instance, at n = 1, the result is exactly similar behavior with one of GR (dust-dominated universe), and for n ≈ 0.595, it is favored with the observed value of Ω d ≈ 0.32. However, the growth of the fluctuations is nearly constant with red-shift at n = 0.5 for the Case I alone. Whereas, at n 1, the amplitude of the matter density contrast is very high and unrealistic to compare with the GR limit (see Fig . 13).

Torsion-radiation system
Here, we assume that the Universe is dominated by a torsion fluid and radiation (w r = 1/3) mixture as a background, consequently the energy density of dust matter contribution is negligible. In such a system, perturbations would evolve according to the following equation (66) i.e., ∆ m ≈ ∆ r and ρ m = ρ r . By applying our paradigmatic f (T ) gravity model and the power scale factor associated with Eqs. (76) and (77), it can be shown that the second-order evolution equation (98) of the energy density for torsion-radiation system can be re-written as For n = 1, this equation reduces to the well-known GR limit [32]: Using Eqs. (71) -(74), we can re-write Eq. (99) in red-shift space as In the following two sub-sections, we further analyze the growth of energy density fluctuations from the solution of Eq. (101) in short-and long-wavelength modes.

Short-wavelength mode
Here, we discuss the growth of fractional energy density fluctuations within the horizon, where k 2 /a 2 H 2 1. In this regime, the Jeans wavelength λ J is much larger than the wavelength of the mean free path of the photon λ p and the wavelength of the non-interacting fluid, i.e., λ << λ p << λ J (see similar analysis: [37] for GR and [33] for f (R) gravity theory approaches).
For further processing, we have to fix four basic parameters m Ω r , n and λ in Eq. (101) systematically. Here, we apply the same reason to fix the first parameter m as Sec. 9.1 for expanding Universe ρ = ρ 0 a −4 , and the scale factor becomes a(t) = a 0 (t/t 0 ) 2/3(1+wr) . Explicitly, we can choose our input parameter m = 2/3(1 + w) = 1/2 for the scale factor exponent in Eq. (70) assuming we are in a radiation-dominated epoch. In this context our leading Eq. (101) reads Let us define the following parameters as: then, our evolution equation Eq. (102) reads In the following, we consider three different cases to fix Ω r and λ, and to see the effect of these parameters on the growth of fluctuations in the torsion-radiation system: 1. Using the definition of Ω r presented in (78); 2. Using the observed value of Ω r for different n; and 3. Assuming γ 16π 2 3λ 2 (1+z) 4 for small λ values.
In the following we consider the above three cases and discuss the growth of fluctuations in radiation-torsion system for short-wavelength range.
Case I: Ω r = 2n−1 n , n ≥ 0.5 In this case, we apply the definition of Ω r presented in Eq. (78), and it is an essential point to know the amount of torsion and radiation fluids in the torsion-radiation system. In this situation, the above parameters β and γ read β ≡ 4(n − 1) and the solution of the second-order evolution equation (103) admits where For more clarity, the BesselJ and BesselY presented in Eq. (104) have increasing and decreasing behavior with red-shift respectively. For small values of n and λ, the second terms of the right hand-side Eq. (104) is decreasing with red-shift, in other words, increasing with cosmic-time and vis-versa for the first term of this equation. For n = 1, Eq. (103) is reduced to GR limit presented in Eq. (89).
Based on the definition of Ω r , we consider n ≥ 0.5 for numerical plotting and in these intervals of n, the value of ξ is always real. Apparently, at n = 1 the value of Ω r becomes unity, Ω T reads zero and r = 2 √ 2, consequently, Eq. (104) reduces to radiation dominated case in GR limit.
For the case of n ≈ 0.5000112, the value of Ω r ≈ 4.48 × 10 −5 and is closer to the observed value which presented in [73]. At n = 0.5, Ω r = 0 and Ω T = 1, the torsion fluid is the major component in torsion-radiation system.
For more clarity, we present the numerical results of the growth of energy density fluctuations for n 1, 0.5 ≤ n ≤ 1 and n is closer to one in the below Fig. 3 Then, ξ has two complex and three real roots such as r 1,2 ≈ 0.90521 ± 7.87304i, r 3 ≈ 0.123664 r 4 ≈ 2.56002 and r 5 ≈ 1.0059. In this situation, the values of ξ is changed by n dramatically. For example, from 0 < n ≤ 0.123, the value of ξ reads an imaginary value, and for n > 0.123, it reads real values, consequently, the numerical results of Eq. (104) is highly sensitive to the value of n. Explicitly, for n = 1, the numerical solution is exactly the same as the GR one presented in Fig. 11 and Fig. 4 (red solid line) for this case. For illustrative purposes, we shall assume n 1, n 1 and n is closer to GR in Fig. 4, and Fig. 21, 22 and 23 in the Appendix 11.3.1 accordingly.
Case III: For γ For more simplicity, we assume a small value of λ and γ 16π 2 3λ 2 (1+z) 4 , the leading Eq. (103) reads and the solution is In this limiting case, the parameter ξ becomes For the numerical results, we apply both cases I and II to fix Ω r , and present the numerical results of Eq. (108) in Figs. 24 and 25 (see Appendix 11.3.1) for different values of n respectively. From these figures, we observe that the fluctuations grow in the past and decay in the present to near-future epoch.
In this sub-section, the energy density contrasts between f (T ) gravity and GR is done based on the definition of Eq. (80) in torsion-radiation system with short-wavelength mode and the numerical results is presented in Appendix 11.3.1 for Eq. (104) and (108) in Figs. 28 -33 accordingly.
The detailed analysis of the growth of matter density fluctuations in torsion-radiation system for short-wavelength mode is made and it is convoluted for both cases and ranges of n, i.e., for n 1, n 1 and n closer to GR limits from Figs. 19 -25 (see Appendix 11.3.1).
For instance, the growth of matter density contrasts in below Fig. 3 Fig. 23 (see Appendix 11.3.1) is unity at initial redshift and then decaying with red-shift till today which is totally ruled-out and unrealistic.

Long-wavelength mode
In the long-wavelength range where k 2 /a 2 H 2 1, all cosmological fluctuations begin and remain inside the Hubble horizon. With the k-dependences dropped, Eq. (103) admits an exact solution of the form For small values of n, the second term of the right hand-side of Eq. (109) is decaying with red-shift mean growing with cosmic-time and vis-versa for the first term of this equation. In the following, we also consider two cases for fixing the parameter Ω r in long-wavelength range.
Case I: Ω r = 2n−1 n , n ≥ 0.5 We apply the definition of Eq. (78) and present the numerical plots of Eq. (109) for n 1 and 0.5 ≤ n ≤ 1 in below Fig. 5 and Appendix 11.3.2 in Fig. 35 accordingly for long-wavelength mode. In the Figs. 35 -36 (see in the Appendix 11.3.2), the energy density fluctuations of radiation fluid are growing with red-shift in long-wavelength mode but the amplitude of δ(z) is very high and unrealistic for the case of n 1 to compare with either GR or ΛCDM results. Moreover, the result presented in Fig. 37 (see Appendix 11.3.2) is totally ruled-out and lost the reality of perturbations for the ranges of n 1, because it is decaying with redshift. Fortunately, for n closer to GR limit, the growth of the fluctuations has similar behavior as the GR, see Figs. 5 and 6 with Figs. 10 and 12 in the Appendix 11.3.2 for detail. We also present the matter density contrast between f (T ) and GR for long-wavelength mode in the Appendix 11.3.2 for all cases accordingly.
In general, f (T ) gravity theory has gained much attention for different cosmological implications and it is shown that the f (T ) gravity can be an alternative approach to study the growth of energy density fluctuations for torsion-dust and torsion-radiation systems with 1 + 3 covariant formalism by applying the paradigmatic f (T ) gravity model in Eq. (69) with power scale factor. We presented the numerical results of Eq. (66), for analyzing the growth of energy density fluctuations from past to near-future in both systems.
Unquestionably, our paradigmatic f (T ) gravity model is an alternative approach to explain cosmological perturbations and formation of large-scale structures for n closer to GR limit. However, it is not favored for n 1 (see Figs 23 and 37) (see Appendix 11.3.1 and 11.3.2 respectively), in these intervals of n, the value

Conclusions
This paper presented a detailed analysis of scalar cosmological perturbations in f (T ) gravity theory using the 1 + 3 covariant gauge-invariant formalism. We defined the gauge-invariant variables and derived the corresponding evolution equations. Then, the harmonic decomposition technique was applied to make the equations manageable for analysis. From that, we obtained exact solutions of the evolution equations for both torsionradiation and torsion-dust two-fluid systems after considering the quasi-static approximation, and computed the growth of the fractional energy density perturbations δ(z) and the deviation from GR values, (z), for the well-known power-law f (T ) gravity model and the power-law cosmological scale factor. For the torsion-dust system, we studied the behavior of dust perturbations and observed that δ(z) is growing with cosmic time. In the torsion-radiation system, we considered short-wavelength and long-wavelength modes. It is observed that the growth of matter density fluctuations for both modes and the matter density contrast change dramatically for different ranges of n considered, i.e., for n 1, n 1 and n closer to GR limits. The matter density contrasts in our paradigmatic f (T ) gravity model are consistent with GR predictions for n closer to one, these models are not favored in the range n 1 as the linearity of the perturbations is broken. It is evident from our preliminary results that our f (T ) model results contain a richer set of possibilities whose model parameters can be constrained using up-and-coming observational data and can accommodate currently known features of the large-scale structure power spectrum in the general relativistic and ΛCDM limits. We envisage to undertake this aspect of the task for more realistic f (T ) models in a multi-fluid cosmological fluid setting in a subsequent work.

Dust-torsion system
Here, we also present the energy density contrasts in f (T ) gravity approach. The energy density contrast

Short-wavelength
In the following, the matter density contrasts is presented in torsion-radiation system for short-wavelength range for different range of n and for different cases.

Long-wavelength
The matter density contrasts is presented in torsion-radiation system for long-wavelength for different cases.