Effect of the Modified Gravity on the Large-scale Structure Formation

Today we live in the phase of accelerated expansion (Perlmutter et al. 1997; Riess et al. 1998) of the universe, which has the dynamical effect similar to inflation in the very beginning. We require the matter that has sufficient negative pressure to explain accelerated expansion of the universe but the known matter does not have such negative pressure. We can handle this problem by adding exotic matter in the gravitational Lagrangian density. These models are known as dark energy (DE) models (Sahni & Starobinsky 2000; Padmanabhan 2003; Copeland et al. 2006; Sahni 2008; Verma 2010). We can also consider the modification in the Lagrangian density of the gravitational part of Einstein's theory of general relativity to explain the accelerated expansion of the universe, broadly called the modified theory of gravity (MG).

In the present work, we consider the f(R) theory of gravity as an effective theory and study the large-scale structure formation. Of course, in the f(R) theory of gravity, we replace Ricci scalar R with a general function of R in the Einstein–Hilbert action (Capozziello 2002) and f(R) reduced to the general relativity (GR) in the $R\gg {R}_{o}={H}_{o}^{2}$ regime.

The astrophysical structures existing today in the universe had started evolving before the inflation in the very early universe. The motherland of initial seeds of the large-scale structures is inflation and was created due to quantum fluctuation in the inflation field in this period. These seeds started growing after decoupling of the matter from the radiation but in different ways in the radiation-dominated (RD), matter-dominated (MD) and the late-time cosmic acceleration phases. If we use DE models and f(R) models to explain accelerated expansion, they provide the same results after tuning the free parameters but the growth of matter perturbations is different in both approaches (Starobinsky 1998). Matter density perturbations are used as a tool to distinguish the DE models in GR background Λ cold dark matter (ΛCDM) from f(R) gravity models (Starobinsky 1998; Gannouji et al. 2009; Narikawa & Yamamoto 2010). The matter density perturbations depend on the models of f(R) gravity; therefore, their behavior varies with the parameters of the models.

One of the most popular models in f(R) gravity (Amendola et al. 2007; Appleby & Battye 2007; Hu & Sawicki 2007; Percival et al. 2007; Starobinsky 2007; Tsujikawa 2008) is the Dvali–Gabadadze–Porrati (DGP) brane-world model (Dvali et al. 2000), beyond the ΛCDM model. In this paper we have used the power-law models (Capozziello et al. 2003; Carloni et al. 2005; Jaime 2013; Motohashi 2015; Sharma & Verma 2022a, 2022b; Sharma et al. 2020, 2021; Yadav & Verma 2019), the Starobinsky model (Starobinsky 1998), and the ΛCDM model (Sahni & Starobinsky 2000; Padmanabhan 2003) to study the matter density perturbations and compare their results with observations (Cole et al. 2005) in the linear regime.

This paper is organized in five sections. In Section 2 we give the basic introduction of the modified theory of gravity and its dynamical equations. We discuss the cosmological perturbations in the linear regime and matter density perturbations in Section 3. In Section 4, we investigate the behavior of the growth index and modified gravitational constant. We also discuss the nature of matter density perturbations in the asymptotic future. At the end, in Section 5 we give a summary of the results of our work.

(Throughout the paper, we have used c = 1 and κ² = 8π G).

A simple way to modify the gravity is to replace the Ricci curvature R with a function of R in the Einstein–Hilbert action (Amendola & Tsujikawa 2010; De Felice & Tsujikawa 2010; Sotiriou & Faraoni 2010; Tsujikawa 2018). Thus, the action in f(R) gravity is given as

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal A } & = & \displaystyle \frac{1}{2{\kappa }^{2}}\int {d}^{4}x\sqrt{-g}f(R)\\ & & +\int {d}^{4}x\sqrt{-g}{{ \mathcal L }}_{m}({g}_{\mu \nu },{{\rm{\Psi }}}_{m}),\end{array}\end{eqnarray} \tag{ 1 }$$

where ${{ \mathcal L }}_{m}$ is the matter field Lagrangian density minimally coupled with curvature. Taking the variation of Equation (1) with respect to metric tensor g_{μ ν} , we obtain the gravitational field equations as

$$\begin{eqnarray}&&\begin{array}{l}F(R){R}_{\mu \nu }-\displaystyle \frac{1}{2}f(R){g}_{\mu \nu }-{{\rm{\nabla }}}_{\mu }{{\rm{\nabla }}}_{\nu }F(R)\\ +\ {g}_{\mu \nu }\square F(R)={\kappa }^{2}{T}_{\mu \nu },\end{array}\end{eqnarray} \tag{ 2 }$$

where □ ≡ g^{μ ν} ∇_μ ∇_ν is the covariant d'Alembertian operator, F(R) = ∂f/∂R, ∇_μ is the covariant derivative, and T_{μ ν} is the energy-momentum tensor of matter filed and it should satisfy the energy-momentum conservation equation ${{\rm{\nabla }}}^{\mu }{T}_{\mu \nu }^{(m)}=0$ of the matter field. The trace of Equation (2) is

$$\begin{eqnarray}&&\begin{array}{l}3\square F(R)+F(R)R-2f(R)={\kappa }^{2}T,\end{array}\end{eqnarray} \tag{ 3 }$$

where $T={g}_{\mu \nu }{T}_{(m)}^{\mu \nu }=-{\rho }_{m}+3{p}_{m}$ and the suffix m denotes pressureless matter (including dark matter). The Ricci scalar in the flat FLRW background metric is $R=6(\dot{H}+2{H}^{2})$. The first derivative of the f(R) gives a new scalar degree of freedom that denotes the deviation from GR. If we consider the de Sitter universe, then the Hubble expansion rate is constant $(\dot{H}=0)$. Therefore, the Ricci scalar becomes constant and □F = 0 at the de Sitter point. We can rewrite Equation (3) for the de Sitter point as

$$\begin{eqnarray}&&\begin{array}{l}F(R)R-2f(R)={\kappa }^{2}T,\end{array}\end{eqnarray} \tag{ 4 }$$

and the solution of the above Equation (4) becomes

$$\begin{eqnarray}&&\begin{array}{l}{R}_{\min }=\displaystyle \frac{2f({R}_{\min })+{\kappa }^{2}T}{F({R}_{\min })},\end{array}\end{eqnarray} \tag{ 5 }$$

where R_min is the value of the Ricci scalar at de Sitter point.

Equation (2) can also be written as

$$\begin{eqnarray}&&\begin{array}{l}{G}_{\mu \nu }\equiv {R}_{\mu \nu }-\displaystyle \frac{1}{2}{{Rg}}_{\mu \nu }\\ =\,8\pi G\left[{T}_{\mu \nu }+{g}_{\mu \nu }\left(\displaystyle \frac{f(R)-{RF}}{2}\right)\right.\\ \,\left.+\,({{\rm{\nabla }}}_{\mu }{{\rm{\nabla }}}_{\nu }F(R)-{g}_{\mu \nu }\square F(R))\right].\end{array}\end{eqnarray} \tag{ 6 }$$

We can, in a straightforward way, compare the above Equation (6) with Einstein's equation R_{μ ν} − (1/2)Rg_{μ ν} = κ² T_{μ ν} in GR. We have modified the energy-momentum tensor as ${T}_{(m)}^{\mu \nu }+{T}_{f(R)}^{\mu \nu }$.

We obtain Friedmann equations in the f(R) gravity as (Huang 2014)

$$\begin{eqnarray}&&\begin{array}{l}3{H}^{2}=8\pi G{\rho }_{m}\\ \,+\,\left[\displaystyle \frac{{RF}-f(R}{2}-3H\dot{F}-3{H}^{2}(F-1)\right],\end{array}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\begin{array}{l}-2\dot{H}=8\pi G{\rho }_{m}+\left[\ddot{F}-H\dot{F}+2\dot{H}(F-1)\right],\end{array}\end{eqnarray} \tag{ 8 }$$

where $\ddot{a}/a=\dot{H}+{H}^{2}$ and $\dot{a}/a\equiv H$. Some extra terms appear in the Friedmann Equations (7) and (8) due to f(R) gravity, which act as pressure and energy density due to f(R) gravity. We have an effective equation of state in the f(R) gravity given as

$$\begin{eqnarray}&&\begin{array}{l}{w}_{\mathrm{eff}}=-1-\displaystyle \frac{2\dot{H}}{3{H}^{2}}.\end{array}\end{eqnarray} \tag{ 9 }$$

We obtain the effect of acceleration in the f(R) gravity without introducing an exotic form of matter as a DE component. The equation of state to explain the effect of the late-time acceleration is w = −1.03±0.03 in Aghanim et al. (2020). The expression of w_eff (Sharma & Verma 2022a) for model $f(R)=R+{R}^{1+\delta }/{R}_{c}^{\delta }$ is

$$\begin{eqnarray}&&\begin{array}{l}{w}_{\mathrm{eff}}=-1+\displaystyle \frac{2}{3}{\varepsilon }_{1}=-1+\displaystyle \frac{2(1-\delta )}{3\delta (1+2\delta )},\end{array}\end{eqnarray} \tag{ 10 }$$

where ε₁ ≃ (1 − δ)/[3δ(1 + 2δ)] and δ is a model parameter.

The value of w_eff ≃ −0.998 at δ = 0.98, which is consistent with Planck 2018 data (Aghanim et al. 2020). Figure 1 clearly shows that the value of w_eff never goes below −1.02 at δ = 2.2247. This value is inside the present observational limits. We can recover the ΛCDM model w_eff = −1 at δ = 1. The quantitative behavior of the model parameter δ for acceleration and inflation has been discussed in detail in our previous work (Sharma & Verma 2022a). We obtained the range of δ, which is 0.366 < δ < 1 for acceleration and for inflation (i.e., for a quasi–de Sitter universe) δ ≈ 0.98. Therefore, we restrict to this limit and use δ = 0.98 for late-time cosmic acceleration as well.

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Now, we study matter density perturbations in the linear regime, which enables us to distinguish DE models in f(R) gravity from the other theories of DE in GR. We can also distinguish the ΛCDM model from f(R) theories. We take a metric perturbation about the flat FLRW background (De Felice & Tsujikawa 2010):

$$\begin{eqnarray}&&\begin{array}{l}{{ds}}^{2}=-(1+2\alpha ){{dt}}^{2}-2a(t)[{\partial }_{i}\beta -{S}_{i}]{{dtdx}}^{i}\\ \,+\,{a}^{2}(t)[{\delta }_{{ij}}+2\psi {\delta }_{{ij}}+2{\partial }_{i}{\partial }_{j}\gamma \\ \,+\,2{\partial }_{j}{F}_{i}+{h}_{{ij}}]{{dx}}^{i}{{dx}}^{j},\end{array}\end{eqnarray} \tag{ 11 }$$

where α, β, γ, and ψ are scalar perturbations, S_i and F_i are vector perturbations, and h_ij is the tensor perturbation. In this paper we focus on the scalar perturbations only. After using the longitudinal gauge, the line element without vector and tensor perturbations is given (De Felice & Tsujikawa 2010) as

$$\begin{eqnarray}&&\begin{array}{l}{{ds}}^{2}=-(1+2{\rm{\Phi }}){{dt}}^{2}+{a}^{2}(t)\\ \,\times \,(1-2{\rm{\Psi }}){\delta }_{{ij}}{{dx}}^{i}{{dx}}^{j}.\end{array}\end{eqnarray} \tag{ 12 }$$

We include perturbations in the nonrelativistic matter with negligible pressure, p_m = 0, in the f(R) background (De Felice & Tsujikawa 2010):

$$\begin{eqnarray}&&\begin{array}{l}\delta {\dot{\rho }}_{m}+3H\delta {\rho }_{m}={\rho }_{m}\left(A-3H\alpha -\displaystyle \frac{{k}^{2}}{{a}^{2}}v\right),\end{array}\end{eqnarray} \tag{ 13 }$$

where $A\equiv 3(H\alpha -\dot{\psi })+\tfrac{{k}^{2}}{{a}^{2}}\chi$, $\chi \equiv a(\beta +a\dot{\gamma })$. After taking the longitudinal gauge α = Φ, ψ = Ψ and considering quasi-static approximation in the f(R) background (Boisseau et al. 2000; Copeland et al. 2006; Tsujikawa 2007; Tsujikawa et al. 2008), when the wavenumber k keeps deep inside the Hubble radius (k ≫ aH):

$$\begin{eqnarray}&&\begin{array}{l}\left[\displaystyle \frac{{k}^{2}}{{a}^{2}}| {\rm{\Phi }}| ,\displaystyle \frac{{k}^{2}}{{a}^{2}}| {\rm{\Psi }}| ,\displaystyle \frac{{k}^{2}}{{a}^{2}}| \delta F| ,{M}^{2}| \delta F| \right]\\ \gg \,\,({H}^{2}| {\rm{\Phi }}| ,{H}^{2}| {\rm{\Psi }}| ,{H}^{2}| B| ,{H}^{2}| \delta F| ),\end{array}\end{eqnarray} \tag{ 14 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}| \dot{X}| \lesssim | {HX}| ,\end{array}\end{eqnarray} \tag{ 15 }$$

where $\dot{X}={\rm{\Phi }},{\rm{\Psi }},F,\dot{F},\delta F,\dot{\delta F}$. Thus the equation for matter density perturbations becomes (De Felice & Tsujikawa 2010)

$$\begin{eqnarray}&&\begin{array}{l}{\ddot{\delta }}_{m}+2H{\dot{\delta }}_{m}-4\pi {G}_{\mathrm{eff}}{\rho }_{m}{\delta }_{m}\simeq 0,\end{array}\end{eqnarray} \tag{ 16 }$$

where δ_m = δ ρ_m /ρ_m and G_eff is the effective gravitational coupling given in Tsujikawa (2007) and Tsujikawa et al. (2008) as

$$\begin{eqnarray}&&\begin{array}{l}{G}_{\mathrm{eff}}=\displaystyle \frac{G}{F}\displaystyle \frac{4{k}^{2}/{a}^{2}+3{M}^{2}}{3({k}^{2}/{a}^{2}+{M}^{2})},\end{array}\end{eqnarray} \tag{ 17 }$$

where M² is the mass of the new degree of freedom given as

$$\begin{eqnarray}&&\begin{array}{l}{M}^{2}=\displaystyle \frac{1}{3}\left[\displaystyle \frac{F({R}_{\min })}{{R}_{\min }{F}_{R}({R}_{\min })}-{R}_{\min }\right],\end{array}\end{eqnarray} \tag{ 18 }$$

with F_R ≡ ∂² f(R)/∂R². In Equation (16), the second term has the Hubble parameter. It carries the information of expansion of the universe and G_eff contains the information about the modification of the gravity through f(R). In the high-density region (${R}_{\min }\gg \,R$) Equation (18) becomes M² ≃ 1/3F_R (Tsujikawa et al. 2008). For viable f(R) gravity models, we have assumed ∣F(R)∣ ≪ 1 and ∣F_R ∣ ≪ 1. Then G_eff/G can be further written in the (Gannouji et al. 2009; Narikawa & Yamamoto 2010) simplified way as

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{G}_{\mathrm{eff}}}{G}\simeq 1+\displaystyle \frac{1}{3}\displaystyle \frac{{k}^{2}/{a}^{2}}{\left({a}^{2}/{k}^{2}+1/3{f}_{{RR}}\right)},\end{array}\end{eqnarray} \tag{ 19 }$$

where f_RR ≡ ∂² f(R)/∂R². Above relation shows that the G_eff depends on the wavenumber k and on the f(R) model parameter. In the regime where k/a ≪ M the effective gravitational constant becomes G_eff ≃ G, and mimics GR. In the scalar-tensor regime k/a ≫ M, G_eff becomes ≃ 4G/3, from Equation (19). The transition from GR to scalar-tensor regime occurs during the MD phase at k/a = M.

Now, Equation (16), after changing the $d/{dt}\to d/{dN}$ and a growth parameter $f\equiv d\mathrm{ln}{\delta }_{m}/{dN}={{\rm{\Omega }}}_{m}^{\gamma }$ (where γ is the growth index) becomes

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{df}}{d\mathrm{ln}a}+{f}^{2}+\left(2+\displaystyle \frac{\dot{H}}{{H}^{2}}\right)f=\displaystyle \frac{3}{2}{{\rm{\Omega }}}_{m}\displaystyle \frac{{G}_{\mathrm{eff}}}{G},\end{array}\end{eqnarray} \tag{ 20 }$$

where $N\equiv \mathrm{ln}a$ and Ω_m = 8π G ρ_m /(3H²). We have found the constant growing mode of the matter density perturbation deep inside the MD universe from Equation (20) if f = p (Gannouji & Polarski 2008; Gannouji et al. 2009):

$$\begin{eqnarray}&&\begin{array}{l}{\delta }_{m}={{Aa}}^{{P}_{1}}+{{Ba}}^{{P}_{2}},\end{array}\end{eqnarray} \tag{ 21 }$$

where ${P}_{1}=(1/4)(-1+\sqrt{1+24C})$, ${P}_{2}=(1/4)(-1-\sqrt{1+24C})$, and C ≡ (G_eff/G)Ω_m = constant. This solution is also valid for z < 1. In the regime where k/a ≪ M we have C ≈ 1 and C ≈ (4/3) for k/a ≫ M. We get δ_m in Starobinsky (2007) as

$$\begin{eqnarray}&&\begin{array}{l}{\delta }_{m}\propto a\propto {t}^{2/3}\,\mathrm{if}\,k/a\ll \,M,\\ {\delta }_{m}\propto {a}^{\tfrac{\sqrt{33}-1}{4}}\propto {t}^{\tfrac{\sqrt{33}-1}{6}}\,\mathrm{if}\,k/a\gg \,M.\end{array}\end{eqnarray} \tag{ 22 }$$

If we consider the growth index γ, as a function of redshift z, Equation (20) can be expressed in terms of Ω_m , w_DE, γ, and $d\gamma /{dz}=\gamma ^{\prime}$, using $f={{\rm{\Omega }}}_{m}^{\gamma (z)}$ as

$$\begin{eqnarray}&&\begin{array}{l}-(1+z)(\mathrm{ln}{{\rm{\Omega }}}_{m}){\gamma }^{{\prime} }+{{\rm{\Omega }}}_{m}^{\gamma }+\displaystyle \frac{1}{2}\left(1+3(2\gamma -1){w}_{\mathrm{eff}}\right)\\ =\,\displaystyle \frac{3}{2}\displaystyle \frac{{G}_{\mathrm{eff}}}{G}{{\rm{\Omega }}}_{m}^{1-\gamma },\end{array}\end{eqnarray} \tag{ 23 }$$

and ${\gamma }_{o}^{{\prime} }$ at redshift z = 0 is

$$\begin{eqnarray}&&\begin{array}{l}{\gamma }_{o}^{{\prime} }=\displaystyle \frac{{{\rm{\Omega }}}_{{mo}}^{{\gamma }_{o}}+3({\gamma }_{o}-0.5){w}_{\mathrm{eff}}-\tfrac{3}{2}\tfrac{{G}_{\mathrm{eff}}}{{G}_{N}}{{\rm{\Omega }}}_{{mo}}^{1-{\gamma }_{o}}-\tfrac{1}{2}}{\mathrm{ln}{{\rm{\Omega }}}_{{mo}}},\end{array}\end{eqnarray} \tag{ 24 }$$

where Ω_m is the matter density parameter and o indicates the present values of the parameters. G_eff/G = 1 in the ΛCDM model in GR background and the above expression becomes

$$\begin{eqnarray}&&\begin{array}{l}{\gamma }_{o}^{{\prime} }=\displaystyle \frac{{{\rm{\Omega }}}_{{mo}}^{{\gamma }_{o}}+3({\gamma }_{o}-0.5){w}_{\mathrm{eff}}-\tfrac{3}{2}{{\rm{\Omega }}}_{{mo}}^{1-{\gamma }_{o}}-\tfrac{1}{2}}{\mathrm{ln}{{\rm{\Omega }}}_{{mo}}}.\end{array}\end{eqnarray} \tag{ 25 }$$

We have found the constraint on the growth index as $f({\gamma }_{o},{\gamma }_{o}^{{\prime} },{{\rm{\Omega }}}_{{mo}},{w}_{\mathrm{eff}})=0$ and the growth index in the DE models in GR given in Wang & Steinhardt (1998) and Nesseris & Perivolaropoulos (2008) as

$$\begin{eqnarray}&&\begin{array}{l}\gamma =\displaystyle \frac{3\left({w}_{\mathrm{eff}}-1\right)}{\left(6{w}_{\mathrm{eff}}-5\right)}\\ \,+\,\displaystyle \frac{3}{125}\displaystyle \frac{\left(1-{w}_{\mathrm{eff}}\right)\left(1-3{w}_{\mathrm{eff}}/2\right)}{{\left(1-6/5{w}_{\mathrm{eff}}\right)}^{3}}\left(1-{{\rm{\Omega }}}_{m}\right),\end{array}\end{eqnarray} \tag{ 26 }$$

where γ takes constant value ∼0.56. However, we considered the dependence of γ on the redshift z beyond the GR. Therefore we can expand it around z as $\gamma (z)\simeq {\gamma }_{o}+{\gamma }_{o}^{{\prime} }z$ for 0 ≤ z ≤ 0.5 in Polarski & Gannouji (2008).

The growth of the matter perturbation changes after entering into the scalar-tensor regime until the time t_acc, which is the epoch at which universe enters the late-time accelerated phase.

We can distinguish the DE models in GR background and beyond the GR on the basis of the power spectrum of the growth of matter perturbations Δn_s , growth index γ, growth factor f, and G_eff.

4.1. ΛCDM Model

In the ΛCDM model, the Λ term is used to explain the late-time cosmic acceleration with cold dark matter. In this model, the equation of state is constant w_Λ = −1 and G_eff = G_N because df/dR = 1. We have Δn_s = 0 here, and the nonzero Δn_s indicates a theory beyond the GR.

In previous works (Peebles 1984; Lahav et al. 1991), several authors obtained approximately constant growth index γ with negligible dependence on the Ω_m . Some papers (Wang & Steinhardt 1998; Gannouji & Polarski 2008; Gannouji et al. 2009) show γ_o = 6/11 ≈ 0.5454.

However, some authors realized that the growth index might be a function of redshift and therefore, the deviation from constant γ provides a new constraint on the theories beyond GR. This information is stored in the ${\gamma }^{{\prime} }$. For ΛCDM, (Gannouji & Polarski 2008) the growth index and its derivative are 0.554 ≤ γ_o ≤ 0.558 and $-0.0195\leqslant {\gamma }_{o}^{{\prime} }\leqslant -0.0157$ for 0.2 ≤ Ω_mo ≤ 0.35. Many other models in the GR background (Gannouji & Polarski 2008) have $| d\gamma /{dz}\equiv {\gamma }_{o}^{{\prime} }(z=0)| \lesssim 0.02$ with slight dependence on Ω_mo . But for slowly varying w_DE, and Ω_m = 0.3 the growth index is slightly higher γ_o = 0.555 in Gannouji et al. (2009). A higher and positive value of the ${\gamma }_{o}^{{\prime} }$ is a clear signal of the departure from the ΛCDM model. Authors have found the bound on the Δn_s conservatively in Gannouji et al. (2009) as

$$\begin{eqnarray}&&\begin{array}{l}{n}_{s}^{\mathrm{gal}}-{n}_{s}^{\mathrm{CMB}}={\rm{\Delta }}{n}_{s}\lt 0.05.\end{array}\end{eqnarray} \tag{ 27 }$$

However, the observations (Tegmark et al. 2006) do not allow for any significant difference in between the slopes of the two power spectra.

4.2. Starobinsky Model

In this part of the paper, we have used the Starobinsky model in the f(R) gravity beyond the GR background, gives as

$$\begin{eqnarray}&&\begin{array}{l}f(R)=R-\lambda {R}_{c}\left[1-{\left(1+\displaystyle \frac{{R}^{2}}{{R}_{c}^{2}}\right)}^{-n}\right]+\alpha {R}^{2},\end{array}\end{eqnarray} \tag{ 28 }$$

where λ, n > 0 and β R_c ≃ 2Λ(∞) (Gannouji et al. 2009). In the above model, the second term is used to explain the late-time cosmic acceleration and the third term for inflation (very early acceleration).

In the Starobinsky model, one finds (Gannouji et al. 2009) a distinction from the ΛCDM model on basis of the evolution of matter perturbations. There appears to be a difference in the spectral index n_s ^CMB of the primordial power spectrum $P(k)\propto {k}^{{n}_{s}^{\mathrm{CMB}}}$ and spectral index n_s ^gal, calculated from the galaxy power spectrum $P(k)\propto {k}^{{n}_{s}^{\mathrm{gal}}}$. We obtained Δn_s in Starobinsky (2007) after dropping the α R² from the model given by Equation (28). In the MD era at k/a ≪ M

$$\begin{eqnarray}&&\begin{array}{l}a\propto {t}^{2/3}\,{R}_{o}=\displaystyle \frac{4}{3t};\\ {f}_{,{RR}}\propto {t}^{4n+4};\,M(R)\propto {t}^{-2n-2},\end{array}\end{eqnarray} \tag{ 29 }$$

and δ_m grows as t^2/3 and it changes after k = aM(R) as ${t}^{(\sqrt{33}-1)/6}$ and to t_acc, the end of the MD epoch (Starobinsky 2007; De Felice & Tsujikawa 2010). Thus,

$$\begin{eqnarray}&&\displaystyle \frac{{P}_{{\delta }_{m}}({t}_{\mathrm{acc}})}{{P}_{{\delta }_{m}}^{{\rm{\Lambda }}\mathrm{CDM}}({t}_{\mathrm{acc}})}={\left(\displaystyle \frac{{t}_{\mathrm{acc}}}{{t}_{k}}\right)}^{2\left(\tfrac{\sqrt{33}-1}{6}-\displaystyle \frac{2}{3}\right)}\propto \,{k}^{\tfrac{\sqrt{33}-5}{6n+4}},\end{eqnarray} \tag{ 30 }$$

where time t = t_acc corresponds to the end of the MD phase and at that epoch the universe enters into the later-time acceleration phase. Now, we have Δn_s (Starobinsky 2007; De Felice & Tsujikawa 2010):

$$\begin{eqnarray}&&\begin{array}{l}{n}_{s}^{\mathrm{gal}}-{n}_{s}^{\mathrm{CMB}}=\displaystyle \frac{\sqrt{33}-5}{2(3n+2)}.\end{array}\end{eqnarray} \tag{ 31 }$$

The above expression depends on the model parameter n. If n increases then Δn_s shifts toward the (Tegmark et al. 2006) bound given by Equation (27). Therefore, the constraint on n is n ≥ 2. For n = 1, Δn_s = 0.074, and for n = 2, Δn_s = 0.047 at k = 3000a_o H_o , z_acc = 0.7 and z_k (t_k ) = 2.5.

In this model, the growth index γ_o (z = 0) ≃ 0.41 is smaller than the ΛCDM model in the GR background. We have also found out that γ_o ≃ 0.4, ${\gamma }_{o}^{{\prime} }\simeq -0.25$ for Ω_mo = 0.32 and γ_o ≃ 0.43, ${\gamma }_{o}^{{\prime} }\simeq -0.18$ for Ω_mo = 0.23 in Gannouji & Polarski (2008) and Gannouji et al. (2009). The matter density of the universe is lower when γ_o and the value of ${\gamma }_{o}^{{\prime} }$ move closer to the predicted value even though it still remains far from it.

We have found the significant effect of the f(R) model on the source term as

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{G}_{\mathrm{eff}}}{{G}_{N}}{{\rm{\Omega }}}_{m}\propto {a}^{3(2n+2)+3w-2},\end{array}\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{G}_{{\rm{e}}{\rm{f}}{\rm{f}}}}{{G}_{N}}{{\rm{\Omega }}}_{m}\propto a;\,if\,n=0,\\ \displaystyle \frac{{G}_{{\rm{e}}{\rm{f}}{\rm{f}}}}{{G}_{N}}{{\rm{\Omega }}}_{m}\propto {a}^{7};\,if\,n=1.\end{array}\end{eqnarray} \tag{ 33 }$$

We have used different values of n at which we consider the de Sitter type background expansion, i.e., w = −1 and $\dot{H}/{H}^{2}=0$.

We have found that the source terms always dominates over the expansion term. Thus, G_eff/G_N remains 4/3 in the asymptotic future and the growth of the matter perturbations never ends. We can also see it from Figure 2 where growth in matter density is large at the present epoch but far back in the MD phase, and the growth rate was smaller than the present rate.

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4.3.  $f(R)=R\,+\,{R}^{1+\delta }/{R}_{c}^{\delta }$ Model

We aim to study the evolution of the matter density perturbations in our model given by Equation (34) and compare it with the ΛCDM model in the GR background and the Starobinsky model in the f(R) background. Certainly, the evolution of the matter density perturbations δ_m or scalar perturbations begins at the preinflationary universe and further enters into the radiation and MD eras. Indeed, at the end, these perturbations enter into the late-time comic acceleration phase. These perturbations got frozen during inflation and started growing after reentering the Hubble horizon. Growth of the matter density perturbation far inside the MD era is δ_m ∝ t^2/3 and after the transition from GR to the scalar-tensor regime at k = aH = M(R)a, δ_m evolves differently, which depends on the model parameters. Our f(R) model is given as

$$\begin{eqnarray}&&\begin{array}{l}f(R)=R+\displaystyle \frac{{R}^{1+\delta }}{{R}_{c}^{\delta }}.\end{array}\end{eqnarray} \tag{ 34 }$$

Now we calculate scale factor a, f(R)_,RR , and M² from Equation (34) as

$$\begin{eqnarray}&&\begin{array}{l}a\propto {t}^{2/3};\,f{(R)}_{,{RR}}\,=\,\,\displaystyle \frac{\delta (1+\delta )}{{R}_{c}^{\delta }}{\left(\displaystyle \frac{4}{3}\right)}^{\delta -1}{t}^{-2\left(\delta -1\right)},\end{array}\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&\begin{array}{l}{M}^{2}(R)\simeq \displaystyle \frac{1}{3{f}^{{\prime\prime} }}\,=\,\displaystyle \frac{{R}_{c}^{\delta }}{3\delta (1+\delta )}{\left(\displaystyle \frac{3}{4}\right)}^{\delta -1}{t}^{2(\delta -1)}\propto {t}^{2(\delta -1)}.\end{array}\end{eqnarray} \tag{ 36 }$$

Further using Equation (36) to obtain the ratio ${P}_{{\delta }_{m}}({t}_{\mathrm{acc}})/{P}_{{\delta }_{m}}^{{\rm{\Lambda }}\mathrm{CDM}}({t}_{\mathrm{acc}})$ we have

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{P}_{{\delta }_{m}}({t}_{\mathrm{acc}})}{{P}_{{\delta }_{m}}^{{\rm{\Lambda }}\mathrm{CDM}}({t}_{\mathrm{acc}})}\,=\,{\left(\displaystyle \frac{{t}_{\mathrm{acc}}}{{t}_{k}}\right)}^{2\left(\tfrac{\sqrt{33}-1}{6}-\displaystyle \frac{2}{3}\right)}\propto {k}^{\tfrac{\sqrt{33}-5}{1-3\delta }}.\end{array}\end{eqnarray} \tag{ 37 }$$

Thus, we can obtain the Δn_s from Equation (37) by using the definition ${dP}/d\mathrm{ln}k\equiv {n}_{s}$ (Amendola & Tsujikawa 2010) as

$$\begin{eqnarray}&&\begin{array}{l}{n}_{s}^{\mathrm{gal}}-{n}_{s}^{\mathrm{CBM}}=\displaystyle \frac{\sqrt{33}-5}{(1-3\delta )}.\end{array}\end{eqnarray} \tag{ 38 }$$

Equation (38) depends only on the model parameter δ. Figure 3 shows that when δ ≤ 1 then Δn_s < 0, meaning ${n}_{s}^{\mathrm{gal}}\lt {n}_{s}^{\mathrm{CBM}}$ and if 1 < 3δ, or 1/3 < δ then Δn_s > 0. The value of Δn_s ≃ −0.383795 at δ = 0.98, which is much larger than the lower bounds (Tegmark et al. 2006) given by Equation (27).

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We have found a discrepancy between the spectral index obtained from the galaxy survey n_s ^gal, and the primordial power spectrum from the Cosmic Microwave Background (CMB), n_s ^CMB , beyond the GR background.

We have used the growth parameter f, the growth index γ, and the variation of the γ with redshift (${\gamma }^{{\prime} }(z)$) to distinguish f(R) models from the ΛCDM model. Information about the modification in GR is stored in the G_eff and equation of state w_eff in Equations (18) and (9), respectively. At a large redshift G_eff → G/F ≃ 1 and at a low redshift G_eff → 4G/3F ≃ 1.33. G_eff in the model given by Equation (34) is

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{G}_{\mathrm{eff}}}{{G}_{N}}=1+\displaystyle \frac{1}{3}\\ \,\times \,\displaystyle \frac{{\left(1+z\right)}^{2}{\left(k/{a}_{o}{H}_{o}\right)}^{2}{H}_{o}^{2}}{{\left(1+z\right)}^{2}{\left(k/{a}_{o}{H}_{o}\right)}^{2}{H}_{o}^{2}+\tfrac{{R}_{c}^{\delta }}{3\delta \left(1+\delta \right){R}^{\delta -1}}}.\end{array}\end{eqnarray} \tag{ 39 }$$

Now, we use ${R}_{c}\simeq {\rm{\Lambda }}\simeq {H}_{o}^{2}$ and $R\simeq 3{H}_{o}^{2}[{{\rm{\Omega }}}_{{mo}}{a}^{-3}+(1-3{w}_{\mathrm{eff}}){{\rm{\Omega }}}_{{DEo}}{a}^{-3(1+{w}_{\mathrm{eff}})}]$ in the above equation to obtain

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{G}_{\mathrm{eff}}}{{G}_{N}}=1+\displaystyle \frac{1}{3}\\ \,\times \,\displaystyle \frac{{K}^{2}{a}^{-2}}{{K}^{2}{a}^{-2}+\tfrac{{[{{\rm{\Omega }}}_{{mo}}{a}^{-3}+(1-3{w}_{\mathrm{eff}}){{\rm{\Omega }}}_{{DEo}}{a}^{-3(1+{w}_{\mathrm{eff}})}]}^{1-\delta }}{{3}^{\delta }\delta (1+\delta )}},\end{array}\end{eqnarray} \tag{ 40 }$$

where K ≡ k/(a_o H_o ) and 1 + z = 1/a.

The value of G_eff does not seem to reach unity in the large redshift (during the MD era) regime. This means that the Newtonian gravitational constant is always modified in this model.

Further, we focus on the source term (last term) and expansion term (third term) or growth parameter at present and in the asymptotic future. We have obtained the modified gravitational constant from Equation (40) and the matter density parameter, G_eff/G_N ∝ a^{3(1−δ)−2} and ${{\rm{\Omega }}}_{m}\propto {a}^{3{w}_{\mathrm{eff}}}$ (Linder & Polarski 2019), respectively. Therefore, the combined expression becomes

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{G}_{\mathrm{eff}}}{{G}_{N}}{{\rm{\Omega }}}_{m}\propto {a}^{3(1-\delta )+3{w}_{\mathrm{eff}}-2}.\end{array}\end{eqnarray} \tag{ 41 }$$

The value w_eff in Equation (34) are ∼−0.988 and ∼−1.02 (Sharma & Verma 2022a). Thus, we consider the equation of state for background accelerated expansion to be w_eff = − 1 in Equation (41) leading to

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{G}_{{\rm{e}}{\rm{f}}{\rm{f}}}}{{G}_{N}}{{\rm{\Omega }}}_{m}\propto {a}^{-2};\,if\,{\delta }=0\\ \displaystyle \frac{{G}_{{\rm{e}}{\rm{f}}{\rm{f}}}}{{G}_{N}}{{\rm{\Omega }}}_{m}\propto {a}^{-5};\,if\,{\delta }=1.\end{array}\end{eqnarray} \tag{ 42 }$$

Here we notice that (G_eff/G_N )Ω_m decreases in the asymptotic future and the expansion term dominates over it and the growth rate of the matter perturbation ends after ∼5e folds. However, at the present accelerated epoch, the source term dominates over the expansion term. We can see it from Figure 6.

We have γ in the f(R) gravity (Narikawa & Yamamoto 2010) as

$$\begin{eqnarray}&&\begin{array}{l}\gamma (a)=\displaystyle \frac{-41+24\xi +\sqrt{1+24\xi }}{-70+48\xi }\\ \,+\,\displaystyle \frac{1}{8\left(-143+24\xi \right){\left(-35+24\xi \right)}^{2}}\\ \,\times \,[\left(-41+24\xi +\sqrt{1+24\xi }\right)(431+\sqrt{1+24\xi }\\ \,+\,24\xi (-13+\sqrt{1+24\xi })-36(2+\sqrt{1+24\xi })\\ \,+\,6\left(-2+3\sqrt{1+24\xi }\right))](1-{{\rm{\Omega }}}_{m})\\ \,+\,{ \mathcal O }\left({\left(1-{{\rm{\Omega }}}_{m}\right)}^{2}\right),\end{array}\end{eqnarray} \tag{ 43 }$$

where G_eff/G = ξ. Now we use Ω_mo ≃ 0.32 and w_eff from Equation (10) for calculating the γ_o at z = 0 from Equation (43). This is given as

$$\begin{eqnarray}&&\begin{array}{l}{\gamma }_{o}=0.547.\end{array}\end{eqnarray} \tag{ 44 }$$

Hence, we can say that our results are very close to the ΛCDM model. After putting the value of the G_eff/G_N from Equation (40) in Equation (24) to calculate the value of ${\gamma }_{o}^{{\prime} }$ for the model $f(R)=R\,+\,{R}^{1+\delta }/{R}_{c}^{\delta }$, we obtain

$$\begin{eqnarray}&&\begin{array}{l}{\gamma }_{o}^{{\prime} }=0.205595.\end{array}\end{eqnarray} \tag{ 45 }$$

The large value of the ${\gamma }_{o}^{{\prime} }$ shows clear distinction from the DE in the GR background and from the ΛCDM model. The parameter ${\gamma }^{{\prime} }$ increases the transparency in the different types of DE models. The variation of ${\gamma }_{o}^{{\prime} }$ at different values of matter density Ω_mo with δ in Figure 4 shows ${\gamma }_{o}^{{\prime} }=0$ if δ is around 0.3. This means that we find the signature of the f(R) in this model for δ > 0.3. This gives a large range of the model parameter δ in large-scale structure formation and Figure 5 also shows the large value of ${\gamma }_{o}^{{\prime} }$.

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In this study we conclude that the growth rate is influenced due to the modified gravity models and it also depends on the f(R) form in the models. We find the value of γ_o for the model given by Equation (34) with δ = 0.98 and Ω_mo = 0.32 to be 0.547. This value is in very close agreement with the ΛCDM model. But the value of $| \gamma {{\prime} }_{o}| =0.206$, which is larger than the ΛCDM value, where $\gamma {{\prime} }_{o}=-0.015$ in the GR background. This allows us to distinguish this model from the ΛCDM model. We notice a the small dependence of ${\gamma }_{o}^{{\prime} }$ on the value of Ω_mo and also on the model parameter δ from Figure 4. In Figure 5, we show that our model gives the value of γ_o within the range of observational values from the Dark Energy Survey/Planck/Joint Lightcurve Analysis/Baryon Acoustic Oscillations (Abbott et al. 2018) is γ_o = 0.640 ± 0.076 and from the Planck/TT+TE+EE+Low+lensing (Aghanim et al. 2020) is γ_o = 0.68 ± 0.089.

There is an another quantity, the Newtonian gravitational constant, which is potentially modified due to the f(R) model given by Equation (34). Its modification alters the behavior of the matter density perturbations.

We obtained the w ≃ −0.998 in our previous paper (Sharma & Verma 2022a). Thus, we assumed the de Sitter type background expansion and so the Hubble friction term of Equation (20) remains almost constant. Growth in the asymptotic future in our model, which is given by Equation (34), will end because G_eff/G_N Ω_m ∝ a⁻⁵ at δ = 1 decreases faster than the expansion term. On the other hand, the Starobinsky model shows the continuous growth in the matter density in the asymptotic future. At the present accelerated epoch, our model and the Starobinsky model both show an enhanced growth of the matter density at a low redshift i.e., z < 1 because G_eff/G_N Ω_m is dominating over the expansion term $(2+\dot{H}/{H}^{2})$. We can see that G_eff is higher than unity from Figures 2 and 6. Our model also supports the results (Linder & Polarski 2019) where the cosmic growth ends in the asymptotic future in the f(R) gravity and also in the ΛCDM model in GR. Therefore, our model and the ΛCDM model have the identical behavior of the growth of the matter density perturbations. Indeed, the real information about the model beyond GR is encoded in the form of G_eff, ${\gamma }^{{\prime} }$ and the evolution of the cosmic growth.

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The authors are thankful to IUCAA, Pune for support under the associateship program where most of the work was done. A.K.S. is also thankful to Vipin Sharma, Bal Krishna Yadav, and Vaishakh Prasad for the useful discussions on various aspects of matter density perturbations in the modified theory of gravity.