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Higher Order Corrections to the Primordial Gravitational Wave Spectrum and its Impact on Parameter Estimates for Inflation

We study the impact of the use of the power series expression for the primordial tensor spectrum on parameter estimation from future direct detection gravitational wave experiments. The spectrum approximated by the power series expansion may give large deviation from the true (fiducial) value when it is normalized at CMB scale because of the large separation between CMB and direct detection scales. We derive the coefficients of the higher order terms of the expansion up to the sixth order within the framework of the slow-roll approximation and investigate how well the inclusion of higher order terms improves the analytic prediction of the spectrum amplitude by comparing with numerical results. Using the power series expression, we consider future constraints on inflationary parameters expected from direct detection experiments of the inflationary gravitational wave background and show that the truncation of the higher order terms can lead to incorrect evaluation of the parameters. We present two example models; a quadratic chaotic inflation model and mixed inflaton and curvaton model with a quartic inflaton potential.


I. INTRODUCTION
Inflation [1] is a successful paradigm not only for solving the horizon and flatness problems, but also for explaining the origin of density perturbations in the Universe. Inflation predicts adiabatic and almost scale-invariant primordial fluctuations, which are in excellent agreement with current observations such as cosmic microwave background (CMB) and so on. However, no direct evidence of inflation has yet been found. During inflation, the gravitational waves could also be produced [2], whose detection can give a direct evidence of inflation and would be a key test of inflation.
Early detection of the inflationary gravitational wave background may be achieved through its unique signature in the polarization of the CMB [3,4]. The ongoing satellite mission, Planck [5], can detect such indirect signal of gravitational waves if the tensor-toscalar ratio is r 0.05. The next-generation experiment, such as CMBpol [6] and Cosmic Origins Explorer (COrE) [7], are designed to reach r ∼ 10 −3 . Moreover, the direct detection may be possible with space-based laser interferometers such as the DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) [8,9] and Big-Bang Observer (BBO) [10], which would provide independent information about inflation.
While CMB polarization experiments observe large-scale gravitational waves (k ∼ O(0.001) Mpc −1 ), space-based laser interferometers measure gravitational waves at millihertz frequencies (k ∼ O(10 13 ) Mpc −1 ). This millihertz frequency band is the most prospective region for direct detection of the inflationary gravitational wave background. The detection becomes easier at lower frequencies, since interferometer with longer arms can obtain larger displacement signals by gravitational waves. On the other hand, frequencies below a millihertz would be contaminated by the gravitational wave background generated from white dwarf binaries [11].
We should note that there are also many other mechanisms which may generate a gravitational wave background around the millihertz frequency, such as preheating [12][13][14][15], bubble collisions during a first-order phase transition [16], self-ordering scalar fields following a global phase transition [20,21], second order effects from enhanced scalar perturbations [22][23][24][25], topological defects [26][27][28][29][30][31][32], supernova explosions of population III first stars [33][34][35], gamma-ray bursts [36], and so on. However, their amplitude and frequency strongly depend on their unknown physics, so the millihertz band is still a window to search for the inflationary gravitational wave background. In this paper, we focus on the gravitational wave background from inflation and do not consider other sources which may contaminate the millihertz band.
The large difference between CMB and direct detection scales means that these two types of observations enable us to look at different periods of inflation, which would greatly help to investigate the inflaton potential [37][38][39][40][41][42][43]. However, we should carefully choose the method to connect the two separate scales. A common method is to use a power-law extrapolation from CMB scales to direct detection scales for describing the primordial tensor spectrum.
Yet recent works [44][45][46] have pointed out that the Taylor expansion around the CMB scale is no longer valid at the direct detection frequency and it causes an incorrect estimation of the amplitude of the inflationary gravitational wave background.
One way to avoid the wrong estimation of the spectrum is to resort to a full numerical calculation to obtain the gravitational wave spectrum. However, the power-law extrapolation is much simpler and easier than the numerical method and, in principle, its precision can be improved by including higher order terms in the Taylor expansion as much as possible.
In this paper, we derive the slow-roll expression for the primordial tensor power spectrum up to the sixth order in the Taylor expansion and examine how much the inclusion of the higher order terms improves the estimation of the amplitude at direct detection scales by comparing with the full numerical computation [46]. Furthermore, we discuss the impact of the truncation of the higher order terms in the power series expansion of the tensor spectrum by presenting constraints on inflationary parameters expected from future direct detection experiments, which is an example that such a poor estimation of the spectrum amplitude causes a problem. This paper is organized as follows: In Sec. II, we give a formula of the power series expression for the primordial tensor spectrum including up to the sixth order in the Taylor expansion. Next, in Sec. III, we discuss whether the expression given in Sec. II well describes the tensor power spectrum by comparing those with numerically obtained spectra. We consider two example models for the comparison, the chaotic inflation model with quadratic and quartic potentials. Although the quartic chaotic inflation is already excluded by observations such as CMB, by adding the contribution from another source of fluctuations such as the curvaton, the quartic inflation model can be allowed due to the existence of the curvaton fluctuations, which is sometimes called mixed inflaton and curvaton model [47][48][49][50]. Note that this kind of mixed scenario can give sizable tensor-to-scalar ratio as well as large non-Gaussianity, which might be interesting from the viewpoint of near future observations. In Sec. IV, we give expected constraints on the inflationary parameters for the above mentioned two models. In passing, we discuss to what extent the truncation of the tensor spectrum expression at some (lower) order leads to incorrect evaluation of the inflationary parameters. Finally, we conclude in Sec. V.

II. SLOW-ROLL FORMALISM AND POWER SERIES EXPANSION
In the standard picture of the early universe, a scalar field φ, the inflaton, drives superluminal cosmic expansion, the inflation. The equation of motion for φ is given bÿ where the dot and prime denote the derivative with respect to t and φ, respectively. The dynamics of inflation is often characterized by the slow-roll parameters. In this paper, we work with the slow-roll parameters which are defined in terms of the inflaton potential V and its derivatives as [51] where the subscript (n) denotes the n-th derivative with respect to φ. Inflation lasts as long as ǫ V , |η V | ≪ 1, which are called the slow-roll conditions, and it ends when this condition is violated, max{ǫ V (φ end ), η V (φ end )} = 1. In the slow-roll limit, the evolution of the Hubble parameter H(t) is given by H 2 ≃ V /(3M 2 Pl ), where M Pl = 1/ √ 8πG is the reduced Planck mass. The duration of inflation is characterized by the e-folding number, N ≡ ln(a end /a), which can be rewritten in terms of the potential, Within the slow-roll approximation, the primordial power spectra of scalar and tensor perturbations are given by [52,53] where C = −2 + ln 2 + γ ≃ −0.73 with γ being the Euler constant and ǫ H and η H are the Hubble slow-roll parameters, ǫ H ≡ 2M 2 Pl (H ′ /H) 2 and η H ≡ 2M 2 Pl H ′′ /H. Hereafter, we only consider the leading order for the slow-roll parameters. Then the power spectra are given in terms of the inflaton potential as They are evaluated at the moment when each Fourier mode k crosses the Hubble horizon, as indicated by the subscript "k = aH." It is often assumed that the values of V and its derivatives evolve so slowly during inflation that the spectra can be parametrized by using the Taylor expansion in terms of the logarithm of the wave number, where the coefficients are the parameters characterizing a deviation from the scale-invariant spectrum, The expression for the scalar power spectrum is the same except that the coefficient of the first term is (n S⋆ − 1). The subscript ⋆ denotes quantities evaluated at the pivot scale, which is commonly taken to be the scale of the CMB, k ⋆ = 0.002Mpc −1 . The coefficients can be given in terms of the slow-roll parameters as The amplitude of the tensor perturbation at the CMB scale is often parametrized by the tensor-to-scalar ratio: As we will show in the next section, the inclusion of the higher order terms in the Taylor expansion up to the 6th order seems to be in very good agreement with a full numerical calculation, which indicates that the above expressions would be precise enough to give correct tensor power spectra for many inflation models and can be used for parameter estimation from observational data.

III. OVERESTIMATION OF THE TENSOR POWER SPECTRUM
In most works, it is common to simply adopt the power-law extrapolation from CMB scales to direct detection scales for describing the gravitational wave background spectrum.
However, as we will show below, such a power-law extrapolation may not be valid and lead to an incorrect estimation of the spectrum amplitude at direct detection scales. In Figure 1, we show the gravitational wave spectra calculated using the Taylor expansion truncating at some order and the one obtained from full numerical computations. Here we consider quadratic (φ 2 ) and quartic (φ 4 ) chaotic inflation models. For the φ 4 model, we consider a mixed inflaton and curvaton model where fluctuations from the curvaton [54][55][56] also contribute to cosmic density perturbations. This is because the quartic chaotic inflation model predicts too large tensor-to-scalar ratio which is already excluded by current observational data. In addition, the curvaton model can generate large non-Gaussianity, thus such a mixed model would be interesting to investigate since it can produce both sizable gravitational wave amplitude and large non-Gaussianity detectable in the near future observations 1 .
As seen from the figure, the power series expression overestimates the amplitude of the spectrum because of the large separation between the two scales. The spectra are plotted using Eq. (8) by truncating the Taylor expansion at each order, respectively. The exact spectrum, which is obtained from a numerical calculation [46], is also plotted for comparison.
The truncation of the higher order terms in Eq. (8) is the cause of the overestimation because the contribution of the higher order terms is non-negligible as they are boosted by the n-th power of ln(k 0.2Hz /k ⋆ ) ≃ 38.7, even though the coefficients of the n-th terms are suppressed as ǫ n . The overestimation of the spectrum amplitude can be avoided if the slow-roll parameters are much smaller than [ln(k 0.2Hz /k ⋆ )] −1 ≃ 2.58 × 10 −2 , but this is not the case, in particular, for chaotic inflation models.
where ρ c,0 ≡ 3M 2 Pl H 2 0 is the critical density of the Universe today and ρ GW is the energy density of the gravitational waves. The primordial tensor spectrum P T (k) can be converted to the present-day density parameter by using the transfer function as The transfer function is given by (1 + 1.57x eq + 3.42x 2 eq ), (13) where [41,58]. The effective number of degrees of freedom is given as g * (T hc ) = g * s (T hc ) = 106.75, when the contribution from the relativistic standard model particles are taken into account. The values at present are g * 0 = 3.36 and g * s0 = 3.90. If we assume the cosmological parameters to be Ω Λ = 0.734, Ω m h 2 = 0.1334, h = 0.710 (taken from the WMAP 7-year mean values [59]), the amplitude of the primordial spectrum at the direct detection frequency f = 0.2 Hz is given by with the assumption that P T is determined at both the CMB k ⋆ and direct detection scales Below we give some detailed discussion for models considered here: the quadratic chaotic inflation and a mixed inflaton and curvaton model with a quartic inflaton potential.
A. φ 2 model In the case of the chaotic inflation with a quadratic potential, the slow-roll parameters are given as and Eq. (3) gives From Eq. (6) we obtain and from Eq. (7) The spectrum is calculated assuming that the e-folding number corresponding to the CMB scale is N ⋆ = ln(a end /a ⋆ ) = 60, which gives ǫ V ⋆ = 8. If we consider the chaotic inflation model with a quartic potential, the slow-roll parameters are given as and Eq. (3) gives We again take the value of the e-folding number as N ⋆ = 60, which gives ǫ V ⋆ = 1.63 × 10 −2 and r = 0.26. This large tensor-to-scalar ratio is already excluded by current observational constraints, but it can be avoided by introducing the curvaton fluctuations.
In the curvaton scenario, the fluctuations in the curvaton field σ produce the scalar perturbations, which results in a different expression for the tensor-to-scalar ratio [47][48][49][50].
Since our interest is in the case where the detection of the gravitational wave background is possible, we assume here that the tensor-to-scalar ratio is r = 0.1. In this case, fluctuations both from the inflaton and the curvaton contribute to the primordial curvature perturbation.
In this model, the scalar power spectrum is given by where P S is given by The scalar spectral index is also modified as Although the expressions for the scalar perturbation quantities are modified in this kind of mixed models, the formulae for the tensor perturbation spectrum P T and the parameters for its scale dependence, n T , α T , β T , γ T , δ T , θ T , are not modified from the usual inflationary predictions without the curvaton. The tensor spectrum in the φ 4 chaotic inflation model is obtained from Eq. (7) as Since the scalar power spectrum is modified as in Eq. (23), the tensor-to-scalar ratio is given by Assuming N ⋆ = 60, we obtain n T ⋆ ≃ −3.25 × 10 −2 , α T ⋆ ≃ −5.29 × 10 −4 , β T ⋆ ≃ −1.72 × Even if fluctuations from the inflaton also contribute to the density fluctuations in the Universe, as far as the curvaton also generates some fraction of the fluctuations, f local NL can be large. Furthermore, large tensor-to-scalar ratio is also possible in this model, which can be detectable in the near future. Note that, when the curvaton is the only source of density fluctuations, which is usually assumed in many works, the tensor-to-scalar ratio becomes very small. However, this kind of mixed model can give sizable f NL and r.
In this mixed scenario where local-type non-Gaussianity is generated, f NL is given by [50,60] Here f can be very large. Thus, by tuning these parameters, the case of α ⋆ = 1.6 (and r = 0.1), which is assumed in this section, can also give large f NL . Hence, once the gravitational waves and (large) non-Gaussianity are detected, this kind of scenario would be worth investigating carefully [61].

IV. IMPACT ON PARAMETER ESTIMATION
Now, in this section, we study the influence of the poor estimation of the gravitational spectrum amplitude when one adopts the Taylor approximation truncated at some order. If direct detection determines the amplitude of the inflationary gravitational wave background, one may try to extract information on the inflaton potential and the e-folding number [41] by combining observations of CMB [62][63][64] and other complementary experiments [65,66].
However, as shown in the previous section, when the power series expression of the spectrum 2 Current constraints on local-equilateral-and orthogonal-types of f NL are (95 % C.L.) [59]: −10 < f local NL < 74, −214 < f equil NL < 266 and −410 < f equil NL < 6, respectively. is adopted, one would overestimate the amplitude of the gravitational wave spectrum at the direct detection scale if one truncates the expression at some lower order. Here we present how such overestimation of the amplitude affects the determination of the inflationary parameters by investigating future constraints. In this section, we again consider the models discussed in the previous section. Note that, in this paper, we do not consider effect of reheating which may change the shape of the inflationary gravitational wave background around the direct detection frequency [67][68][69].
A. φ 2 model If the quadratic chaotic inflation is the model realized in the nature, the inflationary gravitational wave background could be directly detected with Ω GW,0.2Hz = 1.54 × 10 −16 (taken from the numerical result, given in Table I), which is obtained assuming N ⋆ = 60 and the scalar perturbation being normalized as P S⋆ = 2.43 × 10 −9 . With the power-law approximation, one can describe the amplitude of the gravitational wave background at direct detection scale as where we have used Eqs. (8), (10), (14), (16) and ln(k 0.2Hz /k ⋆ ) = 38.7. Notice that, from the above expression, the relation between P T ⋆ and ǫ V ⋆ can be provided once the value of Ω GW,0.2Hz is determined. The values of P T ⋆ and ǫ V ⋆ directly give information on the e-folding number N ⋆ and the mass of the inflaton m via the following relations, which follow from Eqs. (17) and (19).
In Figure 4,  In particular, for the case of truncation at first or second order, the deviation is not negligible even if the error in measuring Ω GW is taken into account. To present this clearly, we also plot the expected error in future direct detection experiments in Fig. 5. We assume that future experiments determine the value of Ω GW with an accuracy of [70][71][72] σ Ω GW = 8.0 × 10 −18 10 −16 which is derived from the sensitivity of the BBO experiment (Detailed values for computing the noise spectrum are given in Ref. [69]). The Fabry-Perot type DECIGO has a similar sensitivity. Thus, the region within the error band would be similar to parameter space allowed by constraints from direct detection by DECIGO or BBO. Therefore, the use of power series expression of the spectrum may lead to incorrect parameter constraints from direct detection experiments, when one truncates it at lower order. However, if we includes up to the sixth order term, the estimate almost coincides with the true value. tions. In the same way as in Sec. III B, we assume the tensor-to-scalar ratio to be r = 0.1.
In this case, the amplitude of the gravitational wave background would be determined to be Ω GW,0.2Hz = 4.15 × 10 −17 by direct detection experiments. The determination of Ω GW,0.2Hz provides a relation between P T ⋆ and ǫ V ⋆ via where we have used Eqs. (8), (10), (14), (21) and ln(k 0.2Hz /k ⋆ ) = 38.7. It can be converted to the information on N ⋆ and λ by which follows Eqs. (22) and (26).
In Fig. 6, the values of λ and N ⋆ obtained from the determination of Ω GW,0.2Hz are shown for different order truncation of Eq. (33). The lines in the λ − N ⋆ plane are plotted by Eqs. (34) and (35) with parameters P T ⋆ and ǫ V ⋆ satisfying Eq. (33). For the same reason as described in the previous subsection, the truncation of the higher order terms gives an incorrect estimation of the values of λ and N ⋆ . The deviation from the true value is larger than the φ 2 case, because of the larger overestimation of the spectrum as presented in Sec.
III. In this case, the overestimation may come not only from the truncation of the higher order terms of the power series expansion in terms of ln k, but also those of the slow-roll approximation. In our numerical calculation, the slow-roll parameter is ǫ V ≃ 4.56 × 10 −2 when the mode corresponding to 0.2 Hz exits the horizon during inflation. This means the second order slow-roll correction in P T,0.2Hz (see Eq. (5)) can be a few percent around direct detection scales. Note that this cannot be improved even if we take into account the second order slow-roll correction as long as the spectrum is extrapolated from CMB scales, since the second order slow-roll correction is still small (ǫ V ⋆ ≃ 1.63 × 10 −2 ) when the modes corresponding to CMB scales exit the horizon.
In Fig. 7, the lines are shown with the experimental error of direct detection experiments, estimated by Eq. (32). The larger error than the φ 2 case is because of the smaller amplitude of the tensor spectrum due to the reduced normalization. Furthermore, since the CMB constraints are obtained marginalizing over not only the cosmological parameters but also α ⋆ characterizing the contribution of the curvaton fluctuations, the uncertainty becomes larger compared to the case for the φ 2 chaotic inflation model without the curvaton.

V. CONCLUSION
Inflation robustly predicts a stochastic gravitational wave background with a nearly scaleinvariant spectrum. The detection of such gravitational waves is one of the next challenges in observational cosmology. If both CMB polarization and direct detection experiments achieve the detection, the independent information from the two different scales would provide a breakthrough in understanding the underlying physics of inflation.
Since the two different experiments measure gravitational waves at wavelengths separated by 16 orders of magnitude, the deviation from the scale-invariant spectrum, which is traditionally expressed by the power series expansion of ln k, causes a large difference in amplitude of the primordial spectrum between two scales. The difference comes not only from the first order term of the power-law expansion, so-called the spectral index, but also from higher order terms, so-called runnings. We have presented that, in the case of chaotic inflation, the truncation of the running terms leads to the overestimation of the spectrum amplitude at the direct detection frequency. The overestimation is more prominent in the case where inflation predicts large slow-roll parameters. If we consider a single-field inflation model, large slow-roll parameters correspond to a large tensor-to-scalar ratio, in case of which we expect to detect the inflationary gravitational waves. Therefore, the overestimation of the tensor power spectrum should be carefully taken into consideration in case we achieve detection of the inflationary gravitational wave background.
Furthermore, we have investigated how the overestimation affects the determination of inflationary parameters. We have considered parameter constraints obtainable from future direct detection experiments, assuming a specific form of the inflation potential. We have presented two examples: a quadratic chaotic inflation model and mixed inflation and curvaton model with a quartic inflaton potential. In both cases, the use of truncated power-law spectrum causes an incorrect estimation of the inflationary parameters and it can be improved by adding higher order terms. For correct estimation of inflationary parameters, we need to take into account higher order terms, perform a numerical calculation, or develop a new parametrization of the spectrum to connect the two separate scales.