Reconstructing $f(R)$ gravity from the spectral index

Recent cosmological observations are in good agreement with the scalar spectral index $n_s$ with $n_s-1\simeq -2/N$, where $N$ is the number of e-foldings. In the previous work, the reconstruction of the inflaton potential for a given $n_s$ was studied, and it was found that for $n_s-1=-2/N$, the potential takes the form of either $\alpha$-attractor model or chaotic inflation model with $\phi^2$ to the leading order in the slow-roll approximation. Here we consider the reconstruction of $f(R)$ gravity model for a given $n_s$ both in the Einstein frame and in the Jordan frame. We find that for $n_s-1=-2/N$ (or more general $n_s-1=-p/N$), $f(R)$ is given parametrically and is found to asymptote to $R^2$ for large $R$. This behavior is generic as long as the scalar potential is of slow-roll type.


Introduction
The latest Planck data [1] are in good agreement with the scalar spectral index n s with n s − 1 ≃ −2/N , where N is the number of e-foldings. Quadratic chaotic inflation model [2], Starobinsky model [3] and Higgs inflation with a nonminimal coupling [4,5] or α-attractor model connecting them with one parameter "α" [6][7][8] are typical examples which predict such a relation. What else are there any inflation models predicting such a relation? With this motivation, in [9], we studied such an inverse problem : reconstruct V (φ) from a given n s (N ) and found that for n s − 1 = −2/N , V (φ) is either tanh 2 (γφ/2) ("T-model") [6][7][8] or φ 2 (chaotic inflation) to the leading order in the slow-roll approximation.
This paper is a continuation of this project: reconstruct f (R) from a given n s (N ). Since f (R) gravity in vacuum is equivalent to a scalar field coupled to the Einstein gravity via the conformal transformation [10][11][12][13][14] and the spectral index is invariant under the conformal transformation [15,16] the problem is very simple: convert the reconstructed V (φ) into f (R). We provide such a procedure in Sec.2. We find that as long as the scalar potential is of slow-roll type f (R) is approximated by R 2 . In fact, for n s − 1 = −2/N (or more general n s − 1 = −p/N ), f (R) only asymptotes to R 2 for large R irrespective of V (φ). In Sec.3, we also provide the procedure to reconstruct f (R) without relying on the Einstein frame. Sec.4 is devoted to summary.
We study the action (Jordan frame action) which is given by where f (R) is a function of the Ricci scalar R and κ 2 = 8πG.
In order to determine f (R), we utilize the equivalence of f (R) gravity with the Einsteinscalar system [10][11][12][13][14] for which we already have reconstructed possible shapes of the potential for n s − 1 = −p/N [9]. To show the equivalence, first we note that Eq. (1) is equivalent to the action where f ψ = df /dψ. The variation with respect to the auxiliary field ψ gives ψ = R if f ψψ = 0 and the action Eq. (2) reduces to f (R) action Eq. (1) on-shell. Then, by the conformal transformation g µν = g E µν /f ψ and by introducing κφ = 3/2 ln f ψ , the action Eq. (2) can be rewritten as (so-called Einstein frame action) where the quantity with the subscript E denotes the one defined by the metric g E µν and φ and V (φ) are given using ψ = R by Therefore, once f (R) is given, V (φ) is determined by the above relation. The relation can be converted. Namely, once V (φ) is given, f (R) is determined by [17] where V ,φ = dV /dφ. In the following, we set κ = 1. We note one important consequence of the relations (6) and (7). Namely, if V ,φ is negligible and hence by solving this differential equation it follows that f is approximately proportional to R 2 . Therefore, R 2 gravity is quite common in slow-roll inflation.
2.1. n s − 1 = −2/N In our previous paper, we have found that n s can be written in terms of the derivative with respect to the e-fold number N as n s − 1 = (ln(V ,N /V 2 )) ,N and dφ/dN = V ,N /V , where V ,N = dV /dN . Therefore, for a given n s (N ), we can reconstruct V (φ).
In particular, for n s − 1 = −2/N , the potential is found to be either V (φ) = 1 2 m 2 φ 2 (chaotic inflation) or V (φ) = V 0 tanh 2 (γφ/2) (α-attractor model (T-model)) [9]. So, using the above relations (6) and (7), we immediately obtain the corresponding f (R). The power index of f (R) as a function of φ. Solid curves are for α-attractor model with γ = 5, 2/3, 1/5 from top to down. A dotted curve is for a quadratic potential. (6) and Eq. (7), we obtain f (R) parametrically in terms of φ: the Einstein gravity is recovered. In fact, the power index of the functional form of f (R) is calculated by Hence, since φ ∼ m −1 ∼ 10 6 at the beginning of the chaotic inflation, the power index is very close to 2 and f ∝ R 2 , and for small φ the index approaches unity. Since φ ≃ 2 √ N for large N , the index at the observationally relevant scale deviates from 2 (see Fig. 1).

n s − 1 = −p/N
For n s − 1 = −p/N (p = 2 and p > 0 is assumed), the reconstructed potential is [9] V (φ) = where in the first case φ ≤ 0(≥ 0) for p > 2(< 2). f (R) is constructed using Eq. (6) and Eq. (7). In the left panel of Fig. 2, the power index of f (R) as a function of φ is shown for p = 1/2 (solid), p = 1 (logarithmic: dashed), p = 4 (chaotic: dotted). In the right panel, the index for p = 3 for the first case potential in Eq. (16) is shown. We assumed V 0 = V 1 . We find that in all cases f (R) approaches R 2 . It is interesting to note that although in terms of the scalar field potential we have a wide variety of the functional form of V (φ): power-law, exponential and logarithmic, in terms of f (R), for the same n s (N ), the functional form is very limited: only asymptotes to R 2 for large R.

Analysis in the Jordan frame
Finally, for completeness, we provide the analysis in the Jordan frame, in which the action is given by Eq. (1).
The equations of motion in a flat Friedmann universe in vacuum are given by where F = df /dR and the dot denotes the derivative with respect to the cosmic time t, and R = 6(2H 2 +Ḣ).

4/7
We introduce the following slow-roll parameters [18,19] We assume that these parameters are small in the slow-roll approximation. In terms of these parameters, the scalar spectral index n s is given by [18,19] Note that the expression in the first line is exact as long as the slow-roll parameters can be regarded as constants. In the second line the expression is expanded up to the second order in the slow-roll parameters. We also note that from Eq. (18) we have The e-folding number N , which measures the amount of inflationary expansion from a particular time t until the end of inflation t end , is defined by We assume that N is large (say N ∼ O(10) ∼ O(10 2 )) under the slow-roll approximation.
In terms of N , from dN = −Hdt, the slow-roll parameters are rewritten as where the subscript N denotes the derivative with respect to N . Now we introduce a bookkeeping rule to assign the order of smallness to the quantities in the slow-roll parameters according to the number of derivatives with respect to N : Then, up to O(2), the spectral index is given by 3.1. n s − 1 = −2/N As an example, let us consider the case of n s − 1 = −2/N . The analysis for n s − 1 = −p/N is similar.
Therefore, we need O(2) terms to calculate n s . Eq. (25) up to O(2) becomes Since O(2) term is proportional to N −2q , q = 1/2 is required. 3 Then, from Eq. (33), we find s = 2/3 and F is determined as where C is a constant. Eq. (17) becomes up to O(1) The solution is given by Up to O(1), the solution is and f (R) is given by These solutions Eq. (37) and Eq. (38) agree with Eq. (8) and Eq. (9) for large R.

Summary
In this paper, motivated by the relation n s − 1 ≃ −2/N indicated by recent cosmological observations, we derived f (R) (the Lagrangian density of f (R) gravity) from n s (N ) in the slow-roll approximation. We introduced two approaches to the problem. The first approach is to utilize the equivalence of f (R) gravity with the Einstein-scalar system and to determine f (R) from the scalar field potential V (φ) which is already known [9]. The second approach is to derive f (R) directly from n s (N ). In the first approach, we found that if V ,φ is negligible compared with V , then f (R) is approximated by R 2 . R 2 gravity is quite common in slow-roll inflation.
For n s − 1 = −2/N , we found that f (R) is determined parametrically in terms of either φ (Einstein frame case) or F = f R (Jordan frame case). The results of two approaches agree. The reconstructed f (R) has a common feature: f (R) ∝ R 2 for large R. The results for n s − 1 = −p/N are similar. In order to recover general relativity at the present time, f (R) is required to satisfy f (R) ≃ R at small R. Therefore, for the same n s , a rather restricted functional form of f (R) is allowed, although a wide variety of functional form of V (φ) is possible.