Reconstructing the inflaton potential from the spectral index

Recent cosmological observations are in good agreement with the scalar spectral index $n_s$ with $n_s-1\sim -2/N$, where $N$ is the number of e-foldings. Quadratic chaotic model, Starobinsky model and Higgs inflation or $\alpha$-attractors connecting them are typical examples predicting such a relation. We consider the problem in the opposite: given $n_s$ as a function of $N$, what is the inflaton potential $V(\phi)$. We find that for $n_s-1=-2/N$, $V(\phi)$ is either $\tanh^2(\gamma\phi/2)$ ("T-model") or $\phi^2$ (chaotic inflation) to the leading order in the slow-roll approximation. $\gamma$ is the ratio of $1/V$ at $N\rightarrow \infty$ to the slope of $1/V$ at a finite $N$ and is related to"$\alpha$"in the $\alpha$-attractors by $\gamma^2=2/3\alpha$. The tensor-to-scalar ratio $r$ is $r=8/N(\gamma^2 N +1) $. The implications for the reheating temperature are also discussed. We also derive formulas for $n_s-1=-p/N$. We find that if the potential is bounded from above, only $p>1$ is allowed. Although $r$ depends on a parameter, the running of the spectral index is independent of it, which can be used as a consistency check of the assumed relation of $n_s(N)$.


I. 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] or α-attractor connecting them with one parameter "α" [5][6][7] are typical examples which predict such a relation. What else are there any inflation models predicting such a relation? In this note, we consider such an inverse problem : reconstruct V (φ) from a given n s (N ). In Sec.II, we describe the procedure for reconstructing V (φ) from n s (N ). In Sec. III, the case of n s − 1 = −2/N is studied. We shall find that known examples exhaust the possibility: for n s − 1 = −2/N , V (φ) is either tanh 2 (γφ/2) ("T-model") [5][6][7] or φ 2 (chaotic inflation) to the leading order in the slow-roll approximation. We also examine the case of n s − 1 = −p/N in Sec.IV. We discuss the implications for the reheating temperature for the case of n s − 1 = −2/N in Sec.V. Sec.VI is devoted to summary.
Related studies are given in preceding works [8,9]. In [8] the slow-roll parameter is given as a function of N to construct V (φ). In [9] the slow-roll parameters and η are given as function of N to construct V (N ) and compute r. Related results are found in [10,11]. In particular, n s − 1 = −p/N case is studied in [11] by solving for the slow-roll parameter . We study the same problem by solving for the potential directly. Our approach is similar in spirit to [8]; the only difference is that our starting point is n s rather than the slow-roll parameter . Moreover, we clarify the meaning of the integration constants and find a relation between n s (N ) and V (φ).
We use the units of 8πG = 1.
We explain the method to reconstruct V (φ) for a given n s (N ). We study in the framework of a single scalar field with the canonical kinetic term coupled to the Einstein gravity. To do so, we first introduce the e-folding number N and then the scalar spectral index n s . The e-folding number N measures the amount of inflationary expansion from a particular time t until the end of inflation t end N = ln(a(t end )/a(t)) = where φ end = φ(t end ) and the slow-roll equation of motion, 3Hφ = −V , is used in the fourth equality. We assume N ∼ O(10) ∼ O(10 2 ) under the slow-roll approximation. For the standard reheating process, N 50 ∼ 60 corresponds to the comoving scale k probed by CMB experiments first crossed the Hubble radius during inflation (k = aH). In terms of the slow-roll parameters n s is written as (to the first order in the slow-roll approximation), The program to reconstruct V (φ) from n s (N ) is to (i) first construct V (N ) from Eq. (3) and then to (ii) rewrite N as a function of φ from Eq. (1). So, we first need to rewrite the slow-roll parameters as a function of N . From Eq. (1), dN = (V /V )dφ. Hence, we have Therefore, we obtain from which V ,N > 0 is required: V is larger in the past in the slow-roll approximation. We note that this inequality also follows fromḢ < 0 which holds as long as the weak energy condition is satisfied: where the slow-roll equation of motion is used in the second equality.
Assuming V > 0, from Eq. (4), we have Similarly, we obtain Thus, Eq. (3) becomes and dN = (V /V )dφ becomes Eq. (9) and Eq. (10) are the basic equations for reconstructing V (φ) from n s (N ). We also give the formulae for r and the running of the spectral index: where we have used d ln k = d ln aH = −dN under the slow-roll approximation.
III. n s − 1 = −2/N As a warm up, we consider the famous relation which is integrated to give where α is the integration constant and should be positive from V ,N > 0. This equation is where β( = 0) is the second integration constant. The case of β = 0 is to be considered separately. By taking the inverse of V , the meaning of the two integration constants is clear: β is the value of 1/V at N → ∞ and α is related to the value of (1/V ) ,N = −α/N 2 .
Given V (N ), we now proceed to the second step: rewrite N as a function of φ. From Eq.
(10), we have which can be integrated depending on the sign of β. We first consider the case of β > 0.
For β > 0, where C is the integration constant corresponding to the shift of φ. Putting this into Eq.
(16), we finally obtain where we have introduced γ = |β|/α which is the ratio of The potential is in fact the same as that of "T-model" [5], as one might have expected, although V is only accurate for large γφ since we have used the slow-roll approximation and hence N is large. So, V is approximated as The parameter "α" in the α-attractor model [5] corresponds to 2/(3γ 2 ). The Starobinsky model corresponds to γ = 2/3.
On the other hand, for β < 0, V has a pole at −α/β = γ −2 and N is restricted in the range N < γ −2 . Large N is possible for small γ. Putting this into Eq. (16), we obtain However, from the slow-roll condition, N γ −2 is required. Therefore, the potential reduces to which is nothing but the quadratic potential.
We also give the predictions for the tensor-to-scalar ratio r and the running of the spectral index from Eq. (11) and Eq. (12): IV. n s − 1 = −p/N Next, we consider a more general relation where p > 0 and p = 2 are assumed. First, from Eq. (9), V (N ) is written as where α(> 0) and β are the integration constants and we assume p = 1. We will consider the case of p = 1 separately later. From Eq. (10), we have and the integration can be performed using the Gauss hypergeometric function, but the result is not illuminating. However, without using the hypergeometric function, we can see the asymptotic form of V (φ) for all cases of β for large N .
A. β > 0 First, we consider several cases of p for β > 0. For p > 1, N p dominates over N in Eq.
Although the functional form of V (φ) is the same, the behavior of φ for large N is different depending on whether 1 < p < 2 or p > 2 : For 1 < p < 2, from Eq. (31), we find that φ increases as N increases without bound, and V (φ) is of "Starobinsky" type (in the sense that the potential asymptotes to a constant from below for large φ). On the other hand, for p > 2, φ asymptotes to C as N → ∞, and V (φ) is of symmetry-breaking/hilltop type.
1 For p = 2, the integral gives ln N as given in the previous section.
For the case of β < 0 and p = 1, a positive V is possible only for p > 1. In this case, from Eq. (29), V (N ) has a pole at N * = 1/(γ 2 (p − 1)) 1/(p−1) with γ = |β|/α and N is restricted in the range N N * . Large N is possible for small γ. Then, V ,N /V can be approximated as assuming N N * . Hence, Eq. (10) is integrated to give and V (φ) can be written as which is again the power-law potential.
Finally, we consider the case of p = 1. In this case, from Eq. (9) V (N ) is written as where α and β are the integration constants and are both positive. 2 So, V (N ) has a pole atln N = β/α ≡ γ 2 and we can only consider the range ln N γ 2 . Large N is possible for large γ. Then, V ,N /V can be approximated as assuming γ is large. Hence, Eq. (10) is integrated to give and V (φ) can be written as and V (φ) is of logarithmic type.
The schematic shape of the potential for each case of p is shown in Fig. 1. We find that if V (φ) is bounded from above, only p > 1 is allowed.
We also give the predictions for the tensor-to-scalar ratio r and the running of the spectral index from Eq. (11) and Eq. (12): r varies from 8(p − 1)/N (chaotic inflation model) to 8/γ 2 N p (modified T-model).

V. REHEATING TEMPERATURE
Once V (N ) is given, it is possible to connect N and the reheating temperature T RH [12][13][14]. 3 For simplicity we consider the case of n s − 1 = −2/N with β ≥ 0.
For the mode with wavenumber k, the comoving Hubble scale when this mode exited the horizon (k = aH) is related to that of the present time, a 0 H 0 by where a end = a(t end ) and a RH is the scale factor at the end of reheating. Here, by definition, a/a end = e −N . Moreover, assuming that during the reheating phase the effective equation of state of the universe is matter-like due to coherent oscillation of the inflaton, the energy density at the end of inflation ρ end is related to the energy density at the end of reheating where g RH is the effective number of relativistic degrees of freedom at the end of reheating. Further, assuming the conservation of entropy, g S,RH a 3 RH T 3 RH = (43/11)a 3 0 T 3 0 , where g S,RH is the effective number of relativistic degrees of freedom for entropy at the end of reheating. Finally, the Hubble parameter during inflation is related to the scalar amplitude A s by H 2 = V /3 = (π 2 /2)rA s , where A s 2.14 × 10 −9 from Planck [1]. Plugging these relations into Eq. (44), we obtain where h denotes the dimensionless Hubble parameter and we have set g RH = g S,RH . From the requirement that the energy density at the end of reheating should be smaller than the energy density at the end of inflation, we have an upper bound on the reheating temperature For the potential Eq. (16), the ratio is estimated as where the factor 4/3 comes from the contribution of the inflaton kinetic energy to ρ end and we have defined the end of inflation by = V ,N /2V = 1. Therefore, T RH can be written as T RH 10 9 GeV = e 3(N −56.9) k a 0 H 0 where we have assumed g RH = 106.75.
In Fig. 2 [14]. The upper bounds are significantly relaxed for the pivot scale k = 0.002Mpc −1 because T RH depends on k 3 (see Fig. 3).

VI. SUMMARY
Motivated by the relation n s − 1 −2/N indicated by recent cosmological observations, we derive the formulae to derive the inflaton potential V (φ) from n s (N ). Applied to n s −1 = −2/N , to the first order in the slow-roll approximation, we find that the potential is classified into two categories depending on the value of 1/V (N → ∞): T-model type (tanh 2 (γφ/2)) for 1/V (N → ∞) > 0 or quadratic type (φ 2 ) for 1/V (N → ∞) ≤ 0. γ is the ratio of 1/V (N → ∞) to −N 2 (1/V ) ,N . We have calculated the reheating temperature versus the tensor-scalar ratio diagram. We have found that for fixed N the reheating temperature slightly increases as γ increases. For the pivot scale k = 0.05Mpc −1 , the upper bound on the reheating temperature puts the upper bound on the e-folding number, N < 60.
We find that although r depends on the ratio of the integration constant γ, the running of the spectral index does not (by construction). Therefore, the measurement of r can be used to discriminate the model and to narrow down the shape, while the measurement of the running is used as a consistency check of the assumed form of n s (N ).