Cosmological Perturbations via Quantum Corrections in M-Theory

Abstract

In the early universe, it is important to take into account the quantum effect of gravity to explain the feature of inflation. In this paper, we consider the M-theory effective action which consists of 11-dimensional supergravity and (Weyl)${}^4$ terms. The equations of motion are solved perturbatively, and the solution describes the inflation-like expansion in 4-dimensional spacetime. Equations of motion for tensor perturbations around this background are derived perturbatively. We also check that the equations of motion are obtained from the effective action up to the second order of the perturbations. Finally, we solve the equations of motion for the tensor perturbations perturbatively and obtain analytic expressions for them.

1. Introduction

Inflationary expansion of the early universe resolves problems of the hot Big Bang scenario, such as the horizon problem or flatness problem [1,2,3,4,5]. It also generates perturbations to the homogeneous background, which become initial conditions for the structure formation of the present universe. Recent observations of cosmological parameters restrict the quantities of these perturbations, and hence models of the inflation [6,7,8]. In order to realize the inflation via the 4-dimensional effective theory of gravity, it is usual to introduce an inflation field or some higher curvature terms1. The inflaton field causes the inflationary expansion while it slowly rolls down the potential [5,17,18,19,20]. The inflation via higher curvature term was first studied by Starobinsky [1], which is surprisingly consistent with the current observations.

Since the energy scale of the inflationary era will be near that of the grand unified theory, it is important to analyze this scenario by using ultraviolet completion of quantum gravity. Actually, there have been many attempts to explain the inflation from supergravity theory, which is a low-energy effective action of superstring theory. It is difficult, however, to obtain de Sitter (dS) vacua in the effective theory obtained by compactification of the supergravity theory, which is stated as a no-go theorem [21,22,23] or dS swampland conjecture [24,25,26]. Especially, the dS swampland conjecture says that low-energy effective theories which realize the inflation by the inflaton field cannot be consistent with quantum theory of gravity at Planck scale.

To evade the no-go theorem, we need to take into account corrections to the supergravity theory. One way is to introduce sources of branes in the superstring theory, and it was proposed in [27] that the de Sitter vacua can be constructed in the superstring theory with orientifold planes and flux compactification. Some recent consistency checks with the swampland conjecture are reported in [28]. Another way is to introduce higher curvature corrections in the superstring theory, and some earlier works in this direction can be found in [29,30,31,32,33].

In this paper, we pursue the possibility of the inflationary scenario in M-theory. The M-theory is defined as a strong coupling limit, or uplift to 11 dimensions, of type IIA superstring theory, and the low-energy limit is approximated by 11-dimensional supergravity. Since the type IIA superstring theory contains one-loop corrections, the uplift of these terms also gives corrections to the 11-dimensional supergravity. As for the metric, it gives products of four Weyl tensors, abbreviated as $W^4$, so it is important to understand the effect of these terms in relation to the inflationary scenario in the M-theory. Actually, it is shown in [34,35] that if the spacetime is divided into four dimensions and seven internal spatial directions, we obtain a perturbative solution where the three spatial directions are expanding and the internal ones are shrinking. Scalar perturbations around the inflationary background are discussed in [36]. The purpose of this paper is to analyze tensor perturbations around the inflation-like background. Equations of motion for tensor perturbations around this background are derived perturbatively. In addition, we also check that the equations of motion are obtained from the effective action up to the second order of the perturbations. Finally, we solve the equations of motion for the tensor perturbations perturbatively and obtain analytic expressions for them.

The organization of this paper is as follows. In Section 2, we briefly review the background geometry of [34]. In Section 3, we consider the tensor perturbations around the background and obtain equations of motion perturbatively. The effective action for the tensor perturbations are also derived. In Section 4, we solve the equations of motion for the tensor perturbations perturbatively and obtain analytic solutions. Conclusions and discussion are given in Section 5. The scalar perturbations and their solutions are summarized in Appendix A. Some discussion on the power spectrum of the cosmological perturbations are given in Appendix B and Appendix C.

2. Effective Action and Inflationary Solution in M-Theory

Low-energy effective action of the superstring theory is obtained by analyzing scattering amplitudes [37,38] or conformal invariance of loop corrections on the string world-sheet [39,40]. The low-energy effective action of the M-theory is described by 11-dimensional supergravity, and its leading correction is obtained by uplifting the results of the type IIA superstring theory. The bosonic part of the effective action of the M-theory is given by [41,42]

$$\begin{array}{cc}\hfill S_{11}& =\frac{1}{2{\kappa }_{11}^2}\int d^{11}xe\left(R+\mathrm{\Gamma }Z\right),\hfill \\ \hfill Z& \equiv 24(W_{abcd}{W}^{abcd}{W}_{efgh}{W}^{efgh}-64W_{abcd}{W}^{aefg}{W}^{bcdh}{W}_{efgh}\hfill \\ & +2W_{abcd}{W}^{abef}{W}^{cdgh}{W}_{efgh}+16W_{acbd}{W}^{aebf}{W}^{cgdh}{W}_{egfh}\hfill \\ & -16W_{abcd}{W}^{aefg}{W}^b{}_{ef}{}^h{W}^{cd}{}_{gh}-16W_{abcd}{W}^{aefg}{W}^b{}_{fe}{}^h{W}^{cd}{}_{gh}),\hfill \end{array}$$

(1)

where $a,b,c,\cdots =0,1,2,\cdots ,10$ are local Lorentz indices and $W_{abcd}$ is a Weyl tensor2. The 11-dimensional gravitational constant $2{\kappa }_{11}^2$ and a coefficient $\mathrm{\Gamma }$ are dimensionful parameters and expressed in terms of the 11-dimensional Planck length ${\ell }_p$ as

$$\begin{array}{c}\hfill 2{\kappa }_{11}^2={\left(2\pi \right)}^8{\ell }_p^9,\mathrm{\Gamma }=\frac{{\pi }^2{\ell }_p^6}{2^{11}{3}^2}.\end{array}$$

(2)

Note that the above action can be derived directly in 11 dimensions by imposing local supersymmetry [43,44,45,46,47,48,49]. The Z terms in the action (1) are related to a topological term via local supersymmetry and the numerical values in this action are protected against higher perturbations. Of course, there exist higher derivative terms of $\mathcal{O}\left({\mathrm{\Gamma }}^n\right)$ for $n\ge 2$, but those contributions are unknown so far. Some thoughts on the possible terms of $n=2$ can be found in [36].

In this paper, we solve the equations of motion perturbatively up to the linear order of $\mathrm{\Gamma }$. By varying the effective action (1), we obtain following equations of motion [50]3.

$$\begin{array}{cc}\hfill E_{ab}& \equiv R_{ab}-\frac{1}{2}{\eta }_{ab}R+\mathrm{\Gamma }\left\{-\frac{1}{2}{\eta }_{ab}Z+R_{cdea}{Y}^{cde}{}_b-2D_{(c}{D}_{d)}{Y}^c{}_{ab}{}^d\right\}=0.\hfill \end{array}$$

(4)

Here, $D_a$ is a covariant derivative for local Lorentz index, and tensors $X_{abcd}$ and $Y_{abcd}$ are defined as

$$\begin{array}{cc}\hfill X_{abcd}& =96(W_{abcd}{W}_{efgh}{W}^{efgh}-4W_{abce}{W}_{dfgh}{W}^{efgh}+4W_{abde}{W}_{cfgh}{W}^{efgh}\hfill \\ & -4W_{cdae}{W}_{bfgh}{W}^{efgh}+4W_{cdbe}{W}_{afgh}{W}^{efgh}+2W_{abef}{W}_{cdgh}{W}^{efgh}\hfill \\ & +4W_{ab}{}^{ef}{W}_{ce}{}^{gh}{W}_{dfgh}+4W_{cd}{}^{ef}{W}_{ae}{}^{gh}{W}_{bfgh}+8W_{aecg}{W}_{bfdh}{W}^{efgh}\hfill \\ & -8W_{becg}{W}_{afdh}{W}^{efgh}-8W_{abeg}{W}_{cf}{}^e{}_h{W}_d{}^{fgh}-8W_{cdeg}{W}_{af}{}^e{}_h{W}_b{}^{fgh}\hfill \\ & -4W^{ef}{}_{ag}{W}_{efch}{W}^g{}_b{}^h{}_d+4W^{ef}{}_{ag}{W}_{efdh}{W}^g{}_b{}^h{}_c+4W^{ef}{}_{bg}{W}_{efch}{W}^g{}_a{}^h{}_d\hfill \\ & -4W^{ef}{}_{bg}{W}_{efdh}{W}^g{}_a{}^h{}_c),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill Y_{abcd}& =X_{abcd}-\frac{1}{9}({\eta }_{ac}{X}_{bd}-{\eta }_{bc}{X}_{ad}-{\eta }_{ad}{X}_{bc}+{\eta }_{bd}{X}_{ac})+\frac{1}{90}({\eta }_{ac}{\eta }_{bd}-{\eta }_{ad}{\eta }_{bc})X,\hfill \end{array}$$

(6)

where $Y^c{}_{acb}=0$, $X_{ab}=X^c{}_{acb}$ and $X=X^a{}_a$.

Let us consider the solution of effective action (1) in the early universe. We assume that the 10-dimensional space directions are divided into a 3-dimensional flat homogeneous space and a 7-dimensional internal space compactified on the 7-dimensional flat torus. Then, the ansatz of the metric is expressed as

$$\begin{array}{cc}\hfill d{s}^2& =-d{t}^2+a{\left(t\right)}^{2}d{x}_i^2+b{\left(t\right)}^{2}d{y}_m^2,\hfill \end{array}$$

(7)

where $i=1,2,3$ and $m=4,\cdots ,10$. $a\left(t\right)$ and $b\left(t\right)$ are scale factors for the 3-dimensional space and the 7-dimensional internal one, respectively. By inserting the above ansatz into Equation (4), we obtain differential equations for $H\left(t\right)=\frac{\dot{a}\left(t\right)}{a\left(t\right)}$ and $G\left(t\right)=\frac{\dot{b}\left(t\right)}{b\left(t\right)}$. Here, the dot represents the time derivative. The solution up to linear order of $\mathrm{\Gamma }$ is given by [34]

$$\begin{array}{cc}& H\left(\tau \right)=\frac{H_{\mathrm{I}}}{\tau }+\frac{c_h\mathrm{\Gamma }{H}_{\mathrm{I}}^7}{{\tau }^7}+\mathcal{O}\left(\frac{{\mathrm{\Gamma }}^2{H}_{\mathrm{I}}^{13}}{{\tau }^{13}}\right),\hfill \\ & G\left(\tau \right)=\frac{-7+\sqrt{21}}{14}\frac{H_{\mathrm{I}}}{\tau }+\frac{c_g\mathrm{\Gamma }{H}_{\mathrm{I}}^7}{{\tau }^7}+\mathcal{O}\left(\frac{{\mathrm{\Gamma }}^2{H}_{\mathrm{I}}^{13}}{{\tau }^{13}}\right),\hfill \end{array}$$

(8)

with

$$\begin{array}{cc}& c_h=\frac{13824(477087-97732\sqrt{21})}{8575}\sim 47111,\hfill \\ & c_g=-\frac{41472(532196-110451\sqrt{21})}{60025}\sim -17996.\hfill \end{array}$$

(9)

Here, $H_{\mathrm{I}}$ is an integral constant and $\tau$ is defined by

$$\begin{array}{cc}\hfill \tau & =\frac{(-1+\sqrt{21})H_{\mathrm{I}}t+2}{2}.\hfill \end{array}$$

(10)

Notice that $\tau$ is the dimensionless parameter and takes the range of $1\le \tau$. By integrating Equation (8), the scale factors $a\left(\tau \right)$ and $b\left(\tau \right)$ are written as

$$\begin{array}{cc}\hfill log\left(\frac{a}{a_{\mathrm{E}}}\right)& =\frac{1+\sqrt{21}}{10}log\tau -\frac{1+\sqrt{21}}{60}{c}_h\mathrm{\Gamma }{H}_{\mathrm{I}}^6\frac{1}{{\tau }^6}+\mathcal{O}\left(\frac{{\mathrm{\Gamma }}^2{H}_{\mathrm{I}}^{12}}{{\tau }^{12}}\right),\hfill \\ \hfill log\left(\frac{b}{b_{\mathrm{E}}}\right)& =-\frac{3\sqrt{21}-7}{70}log\tau -\frac{1+\sqrt{21}}{60}{c}_g\mathrm{\Gamma }{H}_{\mathrm{I}}^6\frac{1}{{\tau }^6}+\mathcal{O}\left(\frac{{\mathrm{\Gamma }}^2{H}_{\mathrm{I}}^{12}}{{\tau }^{12}}\right),\hfill \end{array}$$

(11)

respectively. Here, $a_{\mathrm{E}}$ and $b_{\mathrm{E}}$ are integral constants and correspond to scale factors around the end of the inflation or the deflation, respectively. Since the string theory suppresses the divergence in the UV region, we expect that the scale factor $a\left(\tau \right)$ and $b\left(\tau \right)$ become convergent functions in the region $1<\tau$. This means that coefficients of the higher-order terms will become the same order. In Figure 1, we show plots of $\frac{1+\sqrt{21}}{10}log\tau$, $\frac{1+\sqrt{21}}{60}{c}_h\mathrm{\Gamma }{H}_{\mathrm{I}}^6\frac{1}{{\tau }^6}$ and $\frac{1+\sqrt{21}}{60}{c}_h^{\prime }{\left(\mathrm{\Gamma }{H}_{\mathrm{I}}^6\right)}^2\frac{1}{{\tau }^{12}}$ for $\mathrm{\Gamma }{H}_{\mathrm{I}}^6=0.014$ and $c_h^{\prime }=\frac{c_h}{\mathrm{\Gamma }{H}_{\mathrm{I}}^6}$, respectively. From this, we see that higher-order corrections $\frac{1}{{\tau }^{6n}}(2\le n)$ are suppressed in the region $1.5<\tau$, and the inflationary expansion or deflation is realized by each $1/{\tau }^6$ term in Equation (11).

Figure 1. Plots of $\frac{1+\sqrt{21}}{10}log\tau$(blue), $\frac{1+\sqrt{21}}{60}{c}_h\mathrm{\Gamma }{H}_{\mathrm{I}}^6\frac{1}{{\tau }^6}$(orange) and $\frac{1+\sqrt{21}}{60}{c}_h^{\prime }{\left(\mathrm{\Gamma }{H}_{\mathrm{I}}^6\right)}^2\frac{1}{{\tau }^{12}}$(green). Parameters are chosen as $\mathrm{\Gamma }{H}_{\mathrm{I}}^6=0.014$ and $c_h^{\prime }=\frac{c_h}{\mathrm{\Gamma }{H}_{\mathrm{I}}^6}$.

The motivation for the inflation is to resolve the horizon problem. This requires that the particle horizon $\int \frac{dt}{a\left(t\right)}$ during the inflationary era is almost equal to that after the radiation-dominated era. The particle horizon during the inflationary era is given by

$$\begin{array}{cc}\hfill \frac{\sqrt{21}+1}{10H_{\mathrm{I}}}{\int }_1^2\frac{d\tau }{a\left(\tau \right)}& =\frac{\sqrt{21}+1}{10a_{\mathrm{E}}{H}_{\mathrm{I}}}{\int }_1^{2}d\tau {\tau }^{-\frac{1+\sqrt{21}}{10}}{e}^{\frac{1+\sqrt{21}}{60}{c}_h\mathrm{\Gamma }{H}_{\mathrm{I}}^6\frac{1}{{\tau }^6}}.\hfill \end{array}$$

(12)

On the other hand, if we approximate the scale factor as $a=a_{\mathrm{E}}{\tau }^{\frac{1+\sqrt{21}}{10}}$ after $\tau =2$, the particle horizon during this era is evaluated as

$$\begin{array}{cc}\hfill \frac{\sqrt{21}+1}{10H_{\mathrm{I}}}{\int }_2^{{\tau }_0}\frac{d\tau }{a\left(\tau \right)}& =\frac{\sqrt{21}+1}{10a_{\mathrm{E}}{H}_{\mathrm{I}}}{\int }_2^{{\tau }_0}d\tau {\tau }^{-\frac{1+\sqrt{21}}{10}}\sim \frac{\sqrt{21}+1}{10a_{\mathrm{E}}{H}_{\mathrm{I}}}\frac{9+\sqrt{21}}{6}{\tau }_0^{\frac{9-\sqrt{21}}{10}},\hfill \end{array}$$

(13)

where ${\tau }_0$ is the value at current time $t_0$. We also obtain $a\left(\tau \right)H\left(\tau \right)=a_{\mathrm{E}}{H}_{\mathrm{I}}{\tau }^{\frac{-9+\sqrt{21}}{10}}$.

Now we define the e-folding number as $N_{\mathrm{e}}=log\frac{a\left({\tau }_0\right)}{a\left(2\right)}$. This means that ${\tau }_0=2e^{\frac{\sqrt{21}-1}{2}{N}_{\mathrm{e}}}$. By equating Equation (12) with Equation (13), we obtain

$$\begin{array}{cc}\hfill {\int }_1^{2}d\tau {\tau }^{-\frac{1+\sqrt{21}}{10}}{e}^{\frac{1+\sqrt{21}}{60}{c}_h\mathrm{\Gamma }{H}_{\mathrm{I}}^6\frac{1}{{\tau }^6}}& \sim \frac{9+\sqrt{21}}{6}{2}^{\frac{9-\sqrt{21}}{10}}{e}^{\frac{\sqrt{21}-3}{2}{N}_{\mathrm{e}}}.\hfill \end{array}$$

(14)

This gives a relation between $\mathrm{\Gamma }{H}_{\mathrm{I}}^6$ and $N_{\mathrm{e}}$, and we obtain $\mathrm{\Gamma }{H}_{\mathrm{I}}^6\sim 0.014$ for $N_{\mathrm{e}}=69$, for example.

We also obtain $a\left({\tau }_0\right)H\left({\tau }_0\right)=a_{\mathrm{E}}{H}_{\mathrm{I}}{\tau }_0^{\frac{-9+\sqrt{21}}{10}}\sim e^{-55}{a}_{\mathrm{E}}{H}_{\mathrm{I}}$ for $N_{\mathrm{e}}=69$.

3. Tensor Perturbations in M-Theory

3.1. Equations of Motion for the Tensor Perturbations

In this section, we investigate the tensor perturbations around the background geometry (7) up to the linear order of $\mathrm{\Gamma }$. Since the action (1) contains complicated $W^4$ terms, we employ a Mathematica code to obtain the results here, even though these expressions are analytic. We deal with equations of motion for perturbations order by order, in the spirit of [51]. We also consult the calculations of cosmological perturbations in the modified gravity [52,53].

The tensor perturbations around the background metric are chosen as follows:

$$\begin{array}{c}\hfill d{s}^2=-d{t}^2+a^2({\delta }_{ij}+h_{ij})d{x}^{i}d{x}^j+b^{2}d{y}_m^2.\end{array}$$

(15)

In addition, $h_{ij}$ can be divided into two polarization modes, $h_{+}$ and $h_{\times }$. In the vielbein formalism, it is expressed as

$$\begin{array}{cc}\hfill e^a{}_{\mu }& =\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& \cdots & 0\\ 0& a(1+\frac{1}{2}{h}_{+})& 0& 0& 0& \cdots & 0\\ 0& a{h}_{\times }& a(1-\frac{1}{2}{h}_{+})& 0& 0& \cdots & 0\\ 0& 0& 0& a& 0& \cdots & 0\\ 0& 0& 0& 0& b& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & 0\\ 0& 0& 0& 0& 0& \cdots & b\end{array}\right),\hfill \end{array}$$

(16)

up to the linear order of the perturbation. Linearized equations of motion for the metric perturbations are obtained by varying Equation (4). The result is given as follows:

$$\begin{array}{cc}\hfill \delta E_{ab}& =\delta R_{ab}-\frac{1}{2}{\eta }_{ab}\delta R+\mathrm{\Gamma }\{-\frac{1}{2}{\eta }_{ab}\delta R_{cdef}{Y}^{cdef}+\delta R_{cdea}{Y}^{cde}{}_b+R^{cde}{}_a\delta Y_{cdeb}\hfill \\ & -2\delta e^{\mu }{}_c{D}_{\mu }{D}_d{Y}^c{}_{\left(ab\right)}{}^d-2\delta {\omega }_c{}^c{}_e{D}_d{Y}^e{}_{\left(ab\right)}{}^d+2\delta {\omega }_{ce(a}{D}^d{Y}^{ce}{}_{b)d}\hfill \\ & +2\delta {\omega }_{ce(a}{D}^d{Y}^c{}_{b)}{}^e{}_d-2D_c{D}_d\delta Y^c{}_{\left(ab\right)}{}^d-2D_c(\delta e^{\nu }{}_d{D}_{\nu }{Y}^c{}_{\left(ab\right)}{}^d\hfill \\ & +\delta {\omega }_d{}^c{}_e{Y}^e{}_{\left(ab\right)}{}^d-\delta {\omega }_{de(a}{Y}^{ce}{}_{b)}{}^d-\delta {\omega }_{de(a}{Y}^c{}_{b)}{}^{ed}+\delta {\omega }_d{}^d{}_e{Y}^c{}_{\left(ab\right)}{}^e\left)\right\}=0.\hfill \end{array}$$

(17)

In this paper, we will solve the above equations of motion in the momentum space for the tensor perturbations. $h_{\alpha }(\alpha =+,\times )$ are expanded as

$$\begin{array}{cc}& h_{\alpha }(t,x,y)=\int d^{3}k\sum _l\left\{h_{\alpha }(t,k,l)e^{i{k}_i{x}^i+i{l}_m{y}^m}+h_{\alpha }{(t,k,l)}^{*}{e}^{-i{k}_i{x}^i-i{l}_m{y}^m}\right\},\hfill \end{array}$$

(18)

where $k_i$ is a momentum in the 3 spatial directions, and $l_m$ is a discretized momentum on the internal 7-dimensional torus. The internal momentum appears in the equations of motion with a factor of $\frac{l^2}{b^2}$, so these terms decouple from the equations of motion as b becomes quite small during the inflation [36]. Hence, we neglect the contributions of internal momentum in this paper.

By inserting the expression (16) into the equations of motion (17), we obtain following differential equation for $h_{\alpha }$:

$$\begin{array}{c}\hfill 0={\ddot{h}}_{\alpha }+(3H+7G){\dot{h}}_{\alpha }+\frac{k^2}{a^2}{h}_{\alpha }+\mathrm{\Gamma }\left(A_0{h}_{\alpha }+A_1{\dot{h}}_{\alpha }+A_2{\ddot{h}}_{\alpha }+A_3{\stackrel{⃛}{h}}_{\alpha }+A_4{\stackrel{⃜}{h}}_{\alpha }\right),\end{array}$$

(19)

where $A_i(i=0,1,2,3,4)$ are functions of H and G. The explicit forms of $A_i$ are derived by employing Mathematica codes, which are reported in [54], and the results are written as

$$\begin{array}{cc}\hfill A_0& =\frac{3584k^2}{10125a^2}(164G^6-156051G^{5}H+552087G^4{H}^2+126177G^4\dot{H}-169557\dot{G}{G}^4\hfill \\ & -634247G^3{H}^3+42420G^3\ddot{H}+383772\dot{G}{G}^{3}H-227652G^{3}H\dot{H}-45450G^3\ddot{G}\hfill \\ & +237309G^2{H}^4-3501\dot{G}{G}^2{H}^2-145749G^2{H}^2\dot{H}+131070G^{2}H\ddot{G}-118800G^{2}H\ddot{H}\hfill \\ & +3030G^2\stackrel{⃛}{H}-21708G^2{\dot{H}}^2+39906\dot{G}{G}^2\dot{H}-3030G^2\stackrel{⃛}{G}-15018{\dot{G}}^2{G}^2-954G{H}^5\hfill \\ & +29592\dot{G}{H}^4-10710H^3\ddot{G}-240306\dot{G}G{H}^3+243966G{H}^3\dot{H}-74910G{H}^2\ddot{G}\hfill \\ & +59460G{H}^2\ddot{H}-6210H^2\stackrel{⃛}{G}-30111{\dot{G}}^2{H}^2-12588\dot{G}{H}^2\dot{H}-24720G\dot{H}\ddot{G}\hfill \\ & -27900\dot{G}G\ddot{H}+21540G\dot{H}\ddot{H}+26160\dot{G}H\ddot{G}-32520H\dot{H}\ddot{G}-29340\dot{G}H\ddot{H}\hfill \\ & -6360\ddot{G}\ddot{H}+9240GH\stackrel{⃛}{G}-9240GH\stackrel{⃛}{H}-3180\dot{H}\stackrel{⃛}{G}-3180\dot{G}\stackrel{⃛}{H}+185049{\dot{G}}^{2}GH\hfill \\ & +115749GH{\dot{H}}^2-307158\dot{G}GH\dot{H}-10737\dot{G}{\dot{H}}^2-6753{\dot{G}}^2\dot{H}+31080\dot{G}G\ddot{G}\hfill \\ & +3180{\ddot{G}}^2+3180\dot{G}\stackrel{⃛}{G}+8081{\dot{G}}^3+1692H^6+3258H^4\dot{H}+16920H^3\ddot{H}+6210H^2\stackrel{⃛}{H}\hfill \\ & +45879H^2{\dot{H}}^2+3180{\ddot{H}}^2+35700H\dot{H}\ddot{H}+3180\dot{H}\stackrel{⃛}{H}+9409{\dot{H}}^3)\hfill \\ & -\frac{14336k^4}{675a^4}(92G^4-322G^{3}H+401G^2{H}^2-46G^2\dot{H}+46\dot{G}{G}^2-204G{H}^3+72\dot{G}{H}^2\hfill \\ & -118\dot{G}GH+118GH\dot{H}+26\dot{G}\dot{H}-13{\dot{G}}^2+33H^4-72H^2\dot{H}-13{\dot{H}}^2),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill A_1& =\frac{3584}{3375}(56G^7+82155H{G}^6-286878H^2{G}^5+88465\dot{G}{G}^5-63454\dot{H}{G}^5+243124H^3{G}^4\hfill \\ & -83314H\dot{G}{G}^4+26141\ddot{G}{G}^4+3967H\dot{H}{G}^4-24251\ddot{H}{G}^4+33174H^4{G}^3+119992{\dot{G}}^2{G}^3\hfill \\ & +80084{\dot{H}}^2{G}^3-369471H^2\dot{G}{G}^3-52736H\ddot{G}{G}^3+369900H^2\dot{H}{G}^3-199896\dot{G}\dot{H}{G}^3\hfill \\ & +45356H\ddot{H}{G}^3+1930\stackrel{⃛}{G}{G}^3-1930\stackrel{⃛}{H}{G}^3-61587H^5{G}^2-213335H{\dot{G}}^2{G}^2\hfill \\ & -37713H{\dot{H}}^2{G}^2+303948H^3\dot{G}{G}^2-25977H^2\ddot{G}{G}^2+39778\dot{G}\ddot{G}{G}^2-180903H^3\dot{H}{G}^2\hfill \\ & +241268H\dot{G}\dot{H}{G}^2-37578\ddot{G}\dot{H}{G}^2+32157H^2\ddot{H}{G}^2-38678\dot{G}\ddot{H}{G}^2+36478\dot{H}\ddot{H}{G}^2\hfill \\ & -4690H\stackrel{⃛}{G}{G}^2+4690H\stackrel{⃛}{H}{G}^2-7776H^{6}G+19057{\dot{G}}^{3}G-25346{\dot{H}}^{3}G-124134H^2{\dot{G}}^{2}G\hfill \\ & +2020{\ddot{G}}^{2}G-192414H^2{\dot{H}}^{2}G+75909\dot{G}{\dot{H}}^{2}G+2020{\ddot{H}}^{2}G+55728H^4\dot{G}G\hfill \\ & +43998H^3\ddot{G}G-86454H\dot{G}\ddot{G}G-110772H^4\dot{H}G-69620{\dot{G}}^2\dot{H}G+335568H^2\dot{G}\dot{H}G\hfill \\ & +69734H\ddot{G}\dot{H}G-41778H^3\ddot{H}G+78094H\dot{G}\ddot{H}G-4040\ddot{G}\ddot{H}G-61374H\dot{H}\ddot{H}G\hfill \\ & +510H^2\stackrel{⃛}{G}G+2020\dot{G}\stackrel{⃛}{G}G-2020\dot{H}\stackrel{⃛}{G}G-510H^2\stackrel{⃛}{H}G-2020\dot{G}\stackrel{⃛}{H}G+2020\dot{H}\stackrel{⃛}{H}G\hfill \\ & -2268H^7-24056H{\dot{G}}^3-25095H{\dot{H}}^3+17277H^3{\dot{G}}^2-5100H{\ddot{G}}^2-50157H^3{\dot{H}}^2\hfill \\ & +19974H\dot{G}{\dot{H}}^2+7841\ddot{G}{\dot{H}}^2-5100H{\ddot{H}}^2+4644H^5\dot{G}+8574H^4\ddot{G}+1681{\dot{G}}^2\ddot{G}\hfill \\ & -21084H^2\dot{G}\ddot{G}-18738H^5\dot{H}+29177H{\dot{G}}^2\dot{H}+23460H^3\dot{G}\dot{H}+35604H^2\ddot{G}\dot{H}\hfill \\ & -9522\dot{G}\ddot{G}\dot{H}-11484H^4\ddot{H}-3221{\dot{G}}^2\ddot{H}-9381{\dot{H}}^2\ddot{H}+28344H^2\dot{G}\ddot{H}+10200H\ddot{G}\ddot{H}\hfill \\ & -42864H^2\dot{H}\ddot{H}+12602\dot{G}\dot{H}\ddot{H}+2250H^3\stackrel{⃛}{G}-5100H\dot{G}\stackrel{⃛}{G}+5100H\dot{H}\stackrel{⃛}{G}-2250H^3\stackrel{⃛}{H}\hfill \\ & +5100H\dot{G}\stackrel{⃛}{H}-5100H\dot{H}\stackrel{⃛}{H})+\frac{7168k^2}{675a^2}(259G^5+863G^{4}H-3235G^3{H}^2-1744G^3\dot{H}\hfill \\ & +2010\dot{G}{G}^3+2713G^2{H}^3-266G^2\ddot{H}-1746\dot{G}{G}^{2}H+1142G^{2}H\dot{H}+266G^2\ddot{G}-468G{H}^4\hfill \\ & +528\dot{G}{H}^3+72H^2\ddot{G}-792\dot{G}G{H}^2+1202G{H}^2\dot{H}-338GH\ddot{G}+338GH\ddot{H}-194\dot{H}\ddot{G}\hfill \\ & -194\dot{G}\ddot{H}+1017G{\dot{H}}^2-2228\dot{G}G\dot{H}-241{\dot{G}}^{2}H+288\dot{G}H\dot{H}+194\dot{G}\ddot{G}+1211{\dot{G}}^{2}G\hfill \\ & -132H^5-600H^3\dot{H}-72H^2\ddot{H}+194\dot{H}\ddot{H}-47H{\dot{H}}^2),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill A_2& =-\frac{3584}{3375}(1952G^6+27187G^{5}H+30271G^4{H}^2-23579G^4\dot{H}+34969\dot{G}{G}^4-112291G^3{H}^3\hfill \\ & -10190G^3\ddot{H}+163296\dot{G}{G}^{3}H-138316G^{3}H\dot{H}+11110G^3\ddot{G}+21957G^2{H}^4\hfill \\ & -50623\dot{G}{G}^2{H}^2-11037G^2{H}^2\dot{H}+29850G^{2}H\ddot{G}-29530G^{2}H\ddot{H}-920G^2\stackrel{⃛}{H}\hfill \\ & +44556G^2{\dot{H}}^2-139692\dot{G}{G}^2\dot{H}+920G^2\stackrel{⃛}{G}+98216{\dot{G}}^2{G}^2+24228G{H}^5-19524\dot{G}{H}^4\hfill \\ & -15690H^3\ddot{G}-128118\dot{G}G{H}^3+130938G{H}^3\dot{H}-25270G{H}^2\ddot{G}+21870G{H}^2\ddot{H}\hfill \\ & -2160H^2\stackrel{⃛}{G}+7367{\dot{G}}^2{H}^2-68874\dot{G}{H}^2\dot{H}-40460G\dot{H}\ddot{G}-43540\dot{G}G\ddot{H}+37380G\dot{H}\ddot{H}\hfill \\ & +27300\dot{G}H\ddot{G}-33460H\dot{H}\ddot{G}-30380\dot{G}H\ddot{H}-6160\ddot{G}\ddot{H}+1240GH\stackrel{⃛}{G}-1240GH\stackrel{⃛}{H}\hfill \\ & -3080\dot{H}\stackrel{⃛}{G}-3080\dot{G}\stackrel{⃛}{H}+130037{\dot{G}}^{2}GH+126477GH{\dot{H}}^2-262674\dot{G}GH\dot{H}-8001\dot{G}{\dot{H}}^2\hfill \\ & -15099{\dot{G}}^2\dot{H}+46620\dot{G}G\ddot{G}+3080{\ddot{G}}^2+3080\dot{G}\stackrel{⃛}{G}+12733{\dot{G}}^3+6696H^6+41994H^4\dot{H}\hfill \\ & +17850H^3\ddot{H}+2160H^2\stackrel{⃛}{H}+64587H^2{\dot{H}}^2+3080{\ddot{H}}^2+36540H\dot{H}\ddot{H}+3080\dot{H}\stackrel{⃛}{H}\hfill \\ & +10367{\dot{H}}^3)+\frac{7168k^2}{675a^2}(37G^4+118G^{3}H-479G^2{H}^2-266G^2\dot{H}+266\dot{G}{G}^2+456G{H}^3\hfill \\ & +72\dot{G}{H}^2-338\dot{G}GH+338GH\dot{H}-194\dot{G}\dot{H}+97{\dot{G}}^2-132H^4-72H^2\dot{H}+97{\dot{H}}^2),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill A_3& =-\frac{28672}{675}(14G^5+272G^{4}H-229G^3{H}^2-284G^3\dot{H}+330\dot{G}{G}^3-315G^2{H}^3\hfill \\ & -46G^2\ddot{H}+686\dot{G}{G}^{2}H-670G^{2}H\dot{H}+46G^2\ddot{G}+159G{H}^4-348\dot{G}{H}^3\hfill \\ & -108H^2\ddot{G}-668\dot{G}G{H}^2+498G{H}^2\dot{H}+62GH\ddot{G}-62GH\ddot{H}-154\dot{H}\ddot{G}\hfill \\ & -154\dot{G}\ddot{H}+477G{\dot{H}}^2-1108\dot{G}G\dot{H}+293{\dot{G}}^{2}H-740\dot{G}H\dot{H}+154\dot{G}\ddot{G}\hfill \\ & +631{\dot{G}}^{2}G+99H^5+456H^3\dot{H}+108H^2\ddot{H}+154\dot{H}\ddot{H}+447H{\dot{H}}^2),\hfill \end{array}$$

(23)

and

$$\begin{array}{cc}\hfill A_4& =-\frac{14336}{675}(2G^4+38G^{3}H-49G^2{H}^2-46G^2\dot{H}+46\dot{G}{G}^2-24G{H}^3-108\dot{G}{H}^2\hfill \\ & +62\dot{G}GH-62GH\dot{H}-154\dot{G}\dot{H}+77{\dot{G}}^2+33H^4+108H^2\dot{H}+77{\dot{H}}^2).\hfill \end{array}$$

(24)

Since we solve the equations of motion up to the linear order of $\mathrm{\Gamma }$, we expand H, G and $h_{\alpha }$ as

$$\begin{array}{c}\hfill H=H_0+\mathrm{\Gamma }{H}_1,G=G_0+\mathrm{\Gamma }{G}_1,h_{\alpha }=h_0+\mathrm{\Gamma }{h}_1.\end{array}$$

(25)

Notice that $H_0=\frac{H_{\mathrm{I}}}{\tau }$ and $G_0=\frac{-7+\sqrt{21}}{14}\frac{H_{\mathrm{I}}}{\tau }$ are the leading parts of the background (8), and satisfy $G_0=\frac{-7+\sqrt{21}}{14}{H}_0$ and ${\dot{H}}_0=\frac{1-\sqrt{21}}{2}{H}_0^2$. We substitute these into Equation (19) and expand it up to the linear order of $\mathrm{\Gamma }$. Finally, with the aid of Mathematica code [54], we obtain the explicit forms of the equations of motion. The equation of motion for $h_0$ is written as

$$\begin{array}{cc}\hfill 0& ={\ddot{h}}_0+(7G_0+3H_0){\dot{h}}_0+\frac{k^2}{a^2}{h}_0={\ddot{h}}_0+\frac{1}{2}\left(\sqrt{21}-1\right)H_0{\dot{h}}_0+\frac{k^2}{a^2}{h}_0.\hfill \end{array}$$

(26)

In addition, the equation of motion for $h_1$ is given by

$$\begin{array}{cc}\hfill 0& ={\ddot{h}}_1+\frac{1}{2}\left(\sqrt{21}-1\right)H_0{\dot{h}}_1+\frac{k^2}{a_0^2}{h}_1\hfill \\ & +\frac{512(16940714\sqrt{21}-85692179)}{8575}{H}_0^7{\dot{h}}_0+\frac{256(613929\sqrt{21}-2861099)}{245}{H}_0^6{\ddot{h}}_0\hfill \\ & +\frac{6144(121\sqrt{21}-521)}{7}{H}_0^5{\stackrel{⃛}{h}}_0+\frac{2048(20\sqrt{21}-101)}{7}{H}_0^4{\stackrel{⃜}{h}}_0\hfill \\ & +\frac{k^2}{a_0^2}\{\left(-2{\overline{a}}_1-\frac{768(64897\sqrt{21}-270367)}{245}{H}_0^6\right)h_0+\frac{1024(26\sqrt{21}-383)}{7}{H}_0^5{\dot{h}}_0\hfill \\ & +\frac{512(25\sqrt{21}-43)}{7}{H}_0^4{\ddot{h}}_0\}-\frac{k^4}{a_0^4}\frac{2048(7\sqrt{21}-16)}{7}{H}_0^4{h}_0.\hfill \end{array}$$

(27)

Here, ${\overline{a}}_1=-\frac{1+\sqrt{21}}{60}{c}_h{H}_0^6$, which comes from $\frac{k^2}{a^2}{h}_0$. Note that $a_0=a_{\mathrm{E}}{\tau }^{\frac{1+\sqrt{21}}{10}}$ is the leading part of the scale factor a. We will solve Equation (26) first and then substitute the solution of $h_0$ into Equation (27). The solutions will be derived in Section 4.

3.2. Effective Action for the Tensor Perturbations

In this subsection, we show an effective action for the tensor perturbations up to the second order of $h_{\alpha }$. This is important to fix the normalization of the tensor perturbations as well as to perform a consistency check of the equations of motion.

The effective action for $h_{\times }$ is obtained by substituting the perturbation (16) into the action (1). The results are derived by using the Mathematica code [54], and the effective action for $h_{\times }$ is obtained as

$$\begin{array}{cc}\hfill S_{\mathrm{pt}}^{\left(2\right)}& =\frac{1}{2{\kappa }_{11}^2}\int d^{11}x{a}^3{b}^7\left[\frac{1}{2}{\dot{h}}_{\times }^2-\frac{k^2}{2a^2}{h}_{\times }^2+\mathrm{\Gamma }\left(B_0{h}_{\times }^2+B_1{\dot{h}}_{\times }^2+B_2{\ddot{h}}_{\times }^2\right)\right].\hfill \end{array}$$

(28)

In addition, explicit forms of $B_i(i=0,1,2)$ are given by

$$\begin{array}{cc}\hfill B_0& =-\frac{1792k^2}{10125a^2}(164G^6-156051G^{5}H+552087G^4{H}^2+126177G^4\dot{H}-169557\dot{G}{G}^4\hfill \\ & -634247G^3{H}^3+42420G^3\ddot{H}+383772\dot{G}{G}^{3}H-227652G^{3}H\dot{H}-45450G^3\ddot{G}\hfill \\ & +237309G^2{H}^4-3501\dot{G}{G}^2{H}^2-145749G^2{H}^2\dot{H}+131070G^{2}H\ddot{G}-118800G^{2}H\ddot{H}\hfill \\ & +3030G^2\stackrel{⃛}{H}-21708G^2{\dot{H}}^2+39906\dot{G}{G}^2\dot{H}-3030G^2\stackrel{⃛}{G}-15018{\dot{G}}^2{G}^2-954G{H}^5\hfill \\ & +29592\dot{G}{H}^4-10710H^3\ddot{G}-240306\dot{G}G{H}^3+243966G{H}^3\dot{H}-74910G{H}^2\ddot{G}\hfill \\ & +59460G{H}^2\ddot{H}-6210H^2\stackrel{⃛}{G}-30111{\dot{G}}^2{H}^2-12588\dot{G}{H}^2\dot{H}-24720G\dot{H}\ddot{G}\hfill \\ & -27900\dot{G}G\ddot{H}+21540G\dot{H}\ddot{H}+26160\dot{G}H\ddot{G}-32520H\dot{H}\ddot{G}-29340\dot{G}H\ddot{H}\hfill \\ & -6360\ddot{G}\ddot{H}+9240GH\stackrel{⃛}{G}-9240GH\stackrel{⃛}{H}-3180\dot{H}\stackrel{⃛}{G}-3180\dot{G}\stackrel{⃛}{H}+185049{\dot{G}}^{2}GH\hfill \\ & +115749GH{\dot{H}}^2-307158\dot{G}GH\dot{H}-10737\dot{G}{\dot{H}}^2-6753{\dot{G}}^2\dot{H}+31080\dot{G}G\ddot{G}\hfill \\ & +3180{\ddot{G}}^2+3180\dot{G}\stackrel{⃛}{G}+8081{\dot{G}}^3+1692H^6+3258H^4\dot{H}+16920H^3\ddot{H}\hfill \\ & +6210H^2\stackrel{⃛}{H}+45879H^2{\dot{H}}^2+3180{\ddot{H}}^2+35700H\dot{H}\ddot{H}+3180\dot{H}\stackrel{⃛}{H}+9409{\dot{H}}^3)\hfill \\ & +\frac{7168k^4}{675a^4}(92G^4-322G^{3}H+401G^2{H}^2-46G^2\dot{H}+46\dot{G}{G}^2-204G{H}^3\hfill \\ & +72\dot{G}{H}^2-118\dot{G}GH+118GH\dot{H}+26\dot{G}\dot{H}-13{\dot{G}}^2+33H^4-72H^2\dot{H}-13{\dot{H}}^2),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill B_1& =\frac{1792}{3375}(8G^6+11733G^{5}H-46011G^4{H}^2-10741G^4\dot{H}+12631\dot{G}{G}^4+54451G^3{H}^3\hfill \\ & -1930G^3\ddot{H}-25696\dot{G}{G}^{3}H+18316G^{3}H\dot{H}+1930G^3\ddot{G}-18597G^2{H}^4\hfill \\ & -15477\dot{G}{G}^2{H}^2+21657G^2{H}^2\dot{H}-4690G^{2}H\ddot{G}+4690G^{2}H\ddot{H}+8824G^2{\dot{H}}^2\hfill \\ & -18748\dot{G}{G}^2\dot{H}+9924{\dot{G}}^2{G}^2-828G{H}^5+1824\dot{G}{H}^4+2250H^3\ddot{G}+26718\dot{G}G{H}^3\hfill \\ & -24498G{H}^3\dot{H}+510G{H}^2\ddot{G}-510G{H}^2\ddot{H}-3147{\dot{G}}^2{H}^2+13554\dot{G}{H}^2\dot{H}\hfill \\ & -2020G\dot{H}\ddot{G}-2020\dot{G}G\ddot{H}+2020G\dot{H}\ddot{H}-5100\dot{G}H\ddot{G}+5100H\dot{H}\ddot{G}\hfill \\ & +5100\dot{G}H\ddot{H}-23717{\dot{G}}^{2}GH-15357GH{\dot{H}}^2+39074\dot{G}GH\dot{H}+2741\dot{G}{\dot{H}}^2\hfill \\ & -1201{\dot{G}}^2\dot{H}+2020\dot{G}G\ddot{G}-113{\dot{G}}^3-756H^6-4734H^4\dot{H}-2250H^3\ddot{H}\hfill \\ & -10407H^2{\dot{H}}^2-5100H\dot{H}\ddot{H}-1427{\dot{H}}^3)\hfill \\ & +\frac{3584k^2}{675a^2}(37G^4+118G^{3}H-479G^2{H}^2-266G^2\dot{H}+266\dot{G}{G}^2+456G{H}^3\hfill \\ & +72\dot{G}{H}^2-338\dot{G}GH+338GH\dot{H}-194\dot{G}\dot{H}+97{\dot{G}}^2-132H^4-72H^2\dot{H}+97{\dot{H}}^2),\hfill \end{array}$$

(30)

and

$$\begin{array}{cc}\hfill B_2& =\frac{7168}{675}(2G^4+38G^{3}H-49G^2{H}^2-46G^2\dot{H}+46\dot{G}{G}^2-24G{H}^3-108\dot{G}{H}^2\hfill \\ & +62\dot{G}GH-62GH\dot{H}-154\dot{G}\dot{H}+77{\dot{G}}^2+33H^4+108H^2\dot{H}+77{\dot{H}}^2).\hfill \end{array}$$

(31)

While not shown explicitly, it is also possible to write down the effective action for $h_{+}$, which is slightly different from the above expression. We remark, however, that the effective action for $h_{+}$ is the same as that of $h_{\times }$ after substituting the background metric.

By substituting Equation (25) into the action (28), we obtain the effective action for the tensor perturbation up to the second order of $h_0$ and $h_1$. The result is

$$\begin{array}{cc}\hfill S_{\mathrm{pt}}^{\left(2\right)}& =\frac{1}{2{\kappa }_{11}^2}\int d^{11}x{a}_0^3{b}_0^7[\frac{1}{2}{\dot{h}}_0^2-\frac{k^2}{2a_0^2}{h}_0^2+\mathrm{\Gamma }\{\frac{k^4}{a_0^4}\frac{1024(7\sqrt{21}-16)}{7}{H}_0^4{h}_0^2\hfill \\ & +\frac{k^2}{a_0^2}\left(\left(-\frac{1}{2}{\overline{a}}_1-\frac{7}{2}{\overline{b}}_1+\frac{384(64897\sqrt{21}-270367)}{245}{H}_0^6\right)h_0^2+\frac{256(25\sqrt{21}-43)}{7}{H}_0^4{\dot{h}}_0^2-h_0{h}_1\right)\hfill \\ & +\left(\frac{3}{2}{\overline{a}}_1+\frac{7}{2}{\overline{b}}_1+\frac{128(74649\sqrt{21}-289019)}{245}{H}_0^6\right){\dot{h}}_0^2-\frac{1024(20\sqrt{21}-101)}{7}{H}_0^4{\ddot{h}}_0^2+{\dot{h}}_0{\dot{h}}_1\left\}\right].\hfill \end{array}$$

(32)

Here, ${\overline{a}}_1=-\frac{1+\sqrt{21}}{60}{c}_h{H}_0^6$ and ${\overline{b}}_1=-\frac{1+\sqrt{21}}{60}{c}_g{H}_0^6$.

It is easy to see that the equation of motion (26) is obtained from the variation of $h_1$,

$$\begin{array}{cc}\hfill {\delta }_{h_1}{S}_{\mathrm{pt}}^{\left(2\right)}& =-\frac{1}{2{\kappa }_{11}^2}\int d^{11}x{a}_0^3{b}_0^7\mathrm{\Gamma }\left[\ddot{h_0}+\frac{\sqrt{21}-1}{2}{H}_0\dot{h_0}+\frac{k^2}{a_0^2}{h}_0\right]\delta h_1.\hfill \end{array}$$

(33)

Here, we used the relation of the background $G_0=\frac{-7+\sqrt{21}}{14}{H}_0$. Note also the relations of ${\dot{H}}_0=\frac{1-\sqrt{21}}{2}{H}_0^2$, ${\dot{\overline{a}}}_1=c_h{H}_0^7$ and ${\dot{\overline{b}}}_1=c_g{H}_0^7$. By using these relations, it is possible to evaluate the variation of the effective action with respect to $h_0$ as

$$\begin{array}{cc}& {\delta }_{h_0}{S}_{\mathrm{pt}}^{\left(2\right)}\hfill \\ & =-\frac{1}{2{\kappa }_{11}^2}\int d^{11}x{a}_0^3{b}_0^7[{\ddot{h}}_0+\frac{\sqrt{21}-1}{2}{H}_0{\dot{h}}_0+\frac{k^2}{a_0^2}{h}_0+\mathrm{\Gamma }\{-\frac{k^4}{a_0^4}\frac{2048(7\sqrt{21}-16)}{7}{H}_0^4{h}_0\hfill \\ & +\frac{k^2}{a_0^2}\left(\left({\overline{a}}_1+7{\overline{b}}_1-\frac{768(64897\sqrt{21}-270367)}{245}{H}_0^6\right)h_0+\frac{512(25\sqrt{21}-43)}{7}\left(H_0^4{\ddot{h}}_0-\frac{1+3\sqrt{21}}{2}{H}_0^5{\dot{h}}_0\right)\right)\hfill \\ & +(\frac{\sqrt{21}-1}{2}{H}_0\left(3{\overline{a}}_1+7{\overline{b}}_1+\frac{256(74649\sqrt{21}-289019)}{245}{H}_0^6\right)\hfill \\ & +\left(3c_h+7c_g+\frac{256(74649\sqrt{21}-289019)3(1-\sqrt{21})}{245}\right)H_0^7){\dot{h}}_0\hfill \\ & +\left(3{\overline{a}}_1+7{\overline{b}}_1+\frac{256(74649\sqrt{21}-289019)}{245}{H}_0^6+\frac{2048(20\sqrt{21}-101)6(11-\sqrt{21})}{7}{H}_0^6\right){\ddot{h}}_0\hfill \\ & +\frac{2048(20\sqrt{21}-101)}{7}\left(3(1-\sqrt{21})H_0^5{\stackrel{⃛}{h}}_0+H_0^4{\stackrel{⃜}{h}}_0\right)+{\ddot{h}}_1+\frac{\sqrt{21}-1}{2}{H}_0{\dot{h}}_1+\frac{k^2}{a_0^2}{h}_1\left\}\right]\delta h_0\hfill \\ & =-\frac{1}{2{\kappa }_{11}^2}\int d^{11}x{a}_0^3{b}_0^7[{\ddot{h}}_0+\frac{\sqrt{21}-1}{2}{H}_0{\dot{h}}_0+\frac{k^2}{a_0^2}{h}_0+\mathrm{\Gamma }\{-\frac{k^4}{a_0^4}\frac{2048(7\sqrt{21}-16)}{7}{H}_0^4{h}_0\hfill \\ & +\frac{k^2}{a_0^2}\left(\left(-2{\overline{a}}_1-\frac{768(64897\sqrt{21}-270367)}{245}{H}_0^6\right)h_0+\frac{1024(26\sqrt{21}-383)}{7}{H}_0^5{\dot{h}}_0+\frac{512(25\sqrt{21}-43)}{7}{H}_0^4{\ddot{h}}_0\right)\hfill \\ & +\frac{512(16940714\sqrt{21}-85692179)}{8575}{H}_0^7{\dot{h}}_0+\frac{256(613929\sqrt{21}-2861099)}{245}{H}_0^6{\ddot{h}}_0\hfill \\ & +\frac{6144(121\sqrt{21}-521)}{7}{H}_0^5{\stackrel{⃛}{h}}_0+\frac{2048(20\sqrt{21}-101)}{7}{H}_0^4{\stackrel{⃜}{h}}_0\hfill \\ & +{\ddot{h}}_1+\frac{\sqrt{21}-1}{2}{H}_0{\dot{h}}_1+\frac{k^2}{a_0^2}{h}_1+\left({\ddot{h}}_0+\frac{\sqrt{21}-1}{2}{H}_0{\dot{h}}_0+\frac{k^2}{a_0^2}{h}_0\right)(3{\overline{a}}_1+7{\overline{b}}_1)\left\}\right]\delta h_0.\hfill \end{array}$$

(34)

From this, we see that the equation of motion (27) is correctly derived if we use Equation (26). Although we rely on the Mathmatica codes for the calculations, this gives a non-trivial check between Equations (26) and (27) and the effective action (32).

4. Solution for the Tensor Perturbations

Let us analytically solve Equations (26) and (27) in this section. In order to solve these equations, we introduce a new time coordinate $\eta$ instead of t, which is defined by $dt=\frac{1+\sqrt{21}}{10H_{\mathrm{I}}}d\tau =a_{0}d\eta$. This means that $\eta$ behaves like a conformal time after the inflationary expansion. Now, $\eta$ is expressed in terms of $\tau$ as

$$\begin{array}{cc}\hfill \eta & =\frac{1+\sqrt{21}}{10H_{\mathrm{I}}}\int \frac{d\tau }{a_0}=\frac{3+\sqrt{21}}{6a_{\mathrm{E}}{H}_{\mathrm{I}}}{\tau }^{\frac{9-\sqrt{21}}{10}},\hfill \end{array}$$

(35)

and $a_0$ and a Hubble parameter ${\mathcal{H}}_0$ with respect to $\eta$ are given by

$$\begin{array}{c}\hfill a_0=a_{\mathrm{E}}{\left(\frac{\sqrt{21}-3}{2}{a}_{\mathrm{E}}{H}_{\mathrm{I}}\eta \right)}^{\frac{3+\sqrt{21}}{6}},{\mathcal{H}}_0=\frac{a_0^{\prime }}{a_0}=\frac{3+\sqrt{21}}{6}\frac{1}{\eta }.\end{array}$$

(36)

Here, the prime ${}^{\prime }$ represent $\frac{d}{d\eta }$. Then, by defining $h_0=a_0^{\frac{3-\sqrt{21}}{4}}{u}_0$ and multiplying $a_0^2$ to Equation (26), we obtain

$$\begin{array}{cc}\hfill 0& =h_0^{″}+\frac{\sqrt{21}-3}{2}{\mathcal{H}}_0{h}_0^{\prime }+k^2{h}_0=a_0^{\frac{3-\sqrt{21}}{4}}\left[u_0^{\prime \prime }+\left(k^2+\frac{1}{4{\eta }^2}\right)u_0\right].\hfill \end{array}$$

(37)

In addition, the above differential equation for $u_0$ can be solved as

$$\begin{array}{c}\hfill u_0=c_1\sqrt{k\eta }{J}_0\left(k\eta \right)+c_2\sqrt{k\eta }{Y}_0\left(k\eta \right).\end{array}$$

(38)

Here, $J_0$ and $Y_0$ are Bessel functions of the first and second kind, respectively. $c_1$ and $c_2$ are integral constants which have the dimension of mass and depend on k. In order to fix the ratio of $\frac{c_2}{c_1}$, we demand that $u_0$ behaves like $e^{-ik\eta }$ as $\eta$ goes to infinity. This means that the tensor perturbation is approximated by the free field as $\eta$ goes to infinity. Since $\sqrt{x}{J}_0\left(x\right)\sim \sqrt{\frac{2}{\pi }}cos(x-\frac{\pi }{4})$ and $\sqrt{x}{Y}_0\left(x\right)\sim \sqrt{\frac{2}{\pi }}sin(x-\frac{\pi }{4})$ as $x\to \infty$, we choose $\frac{c_2}{c_1}=-i$ and $u_0$ is given by

$$\begin{array}{c}\hfill u_0=c_1\sqrt{k\eta }{H}_0^{\left(2\right)}\left(k\eta \right).\end{array}$$

(39)

$H_0^{\left(2\right)}\left(x\right)$ is the Hankel function of the second kind. $c_1$ is proportional to $\frac{1}{\sqrt{2k}}$ and the coefficient is determined by the normalization of the action (32). From this, $h_0$ is expressed as

$$\begin{array}{cc}\hfill h_0& =c_1{a}_0^{\frac{3-\sqrt{21}}{4}}\sqrt{k\eta }{H}_0^{\left(2\right)}\left(k\eta \right)={\overline{c}}_1{H}_0^{\left(2\right)}\left(k\eta \right),\hfill \\ \hfill {\overline{c}}_1& \equiv c_1{a}_{\mathrm{E}}^{\frac{3-\sqrt{21}}{4}}{\left(\frac{\sqrt{21}+3}{6}\frac{k}{a_{\mathrm{E}}{H}_{\mathrm{I}}}\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(40)

If we take $k\eta \to \infty$, $h_0$ approaches to ${\overline{c}}_1\sqrt{\frac{2}{\pi k\eta }}{e}^{i\frac{\pi }{4}}{e}^{-ik\eta }$.

Next, we investigate the equation of motion (27) for $h_1$. By multiplying $a_0^2$ to Equation (27) and using $\eta$, we obtain

$$\begin{array}{cc}\hfill 0& =h_1^{″}+\frac{1}{2}\left(\sqrt{21}-3\right){\mathcal{H}}_0{h}_1^{\prime }+k^2{h}_1\hfill \\ & +\frac{1}{a_0^6}[\frac{256(4950813\sqrt{21}-33216993)}{8575}{\mathcal{H}}_0^7{h}_0^{\prime }-\frac{768(484793-93483\sqrt{21})}{245}{\mathcal{H}}_0^6{h}_0^{″}\hfill \\ & +\frac{6144(81\sqrt{21}-319)}{7}{\mathcal{H}}_0^5{h}_0^{‴}-\frac{512(404-80\sqrt{21})}{7}{\mathcal{H}}_0^4{h}_0^{″″}\hfill \\ & +k^2\{\left(-\frac{768(64897\sqrt{21}-270367)}{245}{\mathcal{H}}_0^6-2a_0^6{\overline{a}}_1\right)h_0+\frac{1536(9\sqrt{21}-241)}{7}{\mathcal{H}}_0^5{h}_0^{\prime }\hfill \\ & -\frac{512(43-25\sqrt{21})}{7}{\mathcal{H}}_0^4{h}_0^{″}\}-k^4\frac{2048(7\sqrt{21}-16)}{7}{\mathcal{H}}_0^4{h}_0].\hfill \end{array}$$

(41)

In order to solve the above equation, we redefine $h_1$ as $h_1=a_0^{\frac{3-\sqrt{21}}{4}}{u}_1$. Then, a differential equation for $u_1$ is expressed as

$$\begin{array}{cc}\hfill 0& =u_1^{″}+\left(k^2-\frac{3(\sqrt{21}-5)}{8}{\mathcal{H}}_0^2\right)u_1\hfill \\ & +\frac{1}{a_0^6}[\frac{576(279929593\sqrt{21}-1282921273)}{8575}{\mathcal{H}}_0^8{u}_0+\frac{4608(11177551\sqrt{21}-51407611)}{8575}{\mathcal{H}}_0^7{u}_0^{\prime }\hfill \\ & -\frac{768(1625723-338133\sqrt{21})}{245}{\mathcal{H}}_0^6{u}_0^{″}+\frac{8192(101\sqrt{21}-420)}{7}{\mathcal{H}}_0^5{u}_0^{‴}-\frac{512(404-80\sqrt{21})}{7}{\mathcal{H}}_0^4{u}_0^{‴\prime }\hfill \\ & +k^2\{\left(-\frac{768(55797\sqrt{21}-234982)}{245}{\mathcal{H}}_0^6-2a_0^6{\overline{a}}_1\right)u_0+\frac{1024(43\sqrt{21}-525)}{7}{\mathcal{H}}_0^5{u}_0^{\prime }\hfill \\ & -\frac{512(43-25\sqrt{21})}{7}{\mathcal{H}}_0^4{u}_0^{″}\}-k^4\frac{2048(7\sqrt{21}-16)}{7}{\mathcal{H}}_0^4{u}_0]\hfill \\ & =u_1^{″}+\left(k^2+\frac{1}{4{\eta }^2}\right)u_1+\frac{2^8}{3·5^2{7}^3}{\left(\frac{\sqrt{21}+3}{6}\right)}^{3+\sqrt{21}}\frac{\sqrt{k\eta }}{a_{\mathrm{E}}^6{\eta }^8{\left(a_{\mathrm{E}}{H}_{\mathrm{I}}\eta \right)}^{3+\sqrt{21}}}\hfill \\ & [\left(73500(47\sqrt{21}+217)k^3{\eta }^3+6(1032913\sqrt{21}+4661457)k\eta \right)\left(c_1{J}_1\left(k\eta \right)+c_2{Y}_1\left(k\eta \right)\right)\hfill \\ & +\left(-44100(8\sqrt{21}+37)k^4{\eta }^4+(6318421\sqrt{21}+29265657)k^2{\eta }^2\right)\left(c_1{J}_0\left(k\eta \right)+c_2{Y}_0\left(k\eta \right)\right)].\hfill \end{array}$$

(42)

In the second line, we substituted Equations (36) and (38). A particular solution for the above equation is given by

$$\begin{array}{c}\hfill u_1=-\frac{576(999-218\sqrt{21})}{60025}\frac{H_{\mathrm{I}}^6\sqrt{k\eta }}{{\left(\frac{\sqrt{21}-3}{2}{a}_{\mathrm{E}}{H}_{\mathrm{I}}\eta \right)}^{9+\sqrt{21}}}\left(c_1{u}_{11}+\sqrt{\pi }{c}_2{u}_{12}\right).\end{array}$$

(43)

Here, the explicit forms of $u_{11}$ and $u_{12}$ are given by

$$\begin{array}{cc}\hfill u_{11}& =7\sqrt{\frac{\pi }{k\eta }}{J}_0\left(k\eta \right)\{-14700(146\sqrt{21}+669)k^3{\eta }^3{G}_{3,5}^{2,2}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{5}{4},\frac{2\sqrt{21}+17}{4},\frac{1}{4}\\ \frac{3}{4},\frac{3}{4},\frac{1}{4},\frac{3}{4},\frac{2\sqrt{21}+13}{4}\end{array}\right)\hfill \\ & +24500(857\sqrt{21}+3927)k^2{\eta }^2{G}_{3,5}^{2,2}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{5}{4},\frac{2\sqrt{21}+19}{4},\frac{3}{4}\\ \frac{5}{4},\frac{5}{4},\frac{1}{4},\frac{1}{4},\frac{2\sqrt{21}+15}{4}\end{array}\right)\hfill \\ & +(38465701\sqrt{21}+176254865)k\eta G_{3,5}^{2,2}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{5}{4},\frac{2\sqrt{21}+21}{4},\frac{1}{4}\\ \frac{3}{4},\frac{3}{4},\frac{1}{4},\frac{3}{4},\frac{2\sqrt{21}+17}{4}\end{array}\right)\hfill \\ & +6(6206377\sqrt{21}+28445153)G_{3,5}^{2,2}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{5}{4},\frac{2\sqrt{21}+23}{4},\frac{3}{4}\\ \frac{5}{4},\frac{5}{4},\frac{1}{4},\frac{1}{4},\frac{2\sqrt{21}+19}{4}\end{array}\right)\}\hfill \\ & +\pi {\left(k\eta \right)}^2{Y}_0\left(k\eta \right)\{-51450(61\sqrt{21}+279)k^2{\eta }^2{}_2{F}_3\left(\frac{1}{2},-\frac{\sqrt{21}}{2}-\frac{5}{2};1,1,-\frac{\sqrt{21}}{2}-\frac{3}{2};-k^2{\eta }^2\right)\hfill \\ & +85750\left(179\sqrt{21}+819\right)k^2{\eta }^2{}_2{F}_3\left(\frac{3}{2},-\frac{\sqrt{21}}{2}-\frac{5}{2};2,2,-\frac{\sqrt{21}}{2}-\frac{3}{2};-k^2{\eta }^2\right)\hfill \\ & +(46502521\sqrt{21}+213002167){}_2{F}_3\left(\frac{1}{2},-\frac{\sqrt{21}}{2}-\frac{7}{2};1,1,-\frac{\sqrt{21}}{2}-\frac{5}{2};-k^2{\eta }^2\right)\hfill \\ & +3(7499743\sqrt{21}+34391077){}_2{F}_3\left(\frac{3}{2},-\frac{\sqrt{21}}{2}-\frac{7}{2};2,2,-\frac{\sqrt{21}}{2}-\frac{5}{2};-k^2{\eta }^2\right)\},\hfill \end{array}$$

(44)

and

$$\begin{array}{cc}\hfill u_{12}& =\sqrt{k\eta }{J}_0\left(k\eta \right)\{51450\left(61\sqrt{21}+279\right)\sqrt{\pi }{\left(k\eta \right)}^{\frac{7}{2}}{}_2{F}_3\left(\frac{1}{2},-\frac{\sqrt{21}}{2}-\frac{5}{2};1,1,-\frac{\sqrt{21}}{2}-\frac{3}{2};-k^2{\eta }^2\right)\hfill \\ & -85750\left(179\sqrt{21}+819\right)\sqrt{\pi }{\left(k\eta \right)}^{\frac{7}{2}}{}_2{F}_3\left(\frac{3}{2},-\frac{\sqrt{21}}{2}-\frac{5}{2};2,2,-\frac{\sqrt{21}}{2}-\frac{3}{2};-k^2{\eta }^2\right)\hfill \\ & -\left(46502521\sqrt{21}+213002167\right)\sqrt{\pi }{\left(k\eta \right)}^{\frac{3}{2}}{}_2{F}_3\left(\frac{1}{2},-\frac{\sqrt{21}}{2}-\frac{7}{2};1,1,-\frac{\sqrt{21}}{2}-\frac{5}{2};-k^2{\eta }^2\right)\hfill \\ & -3\left(7499743\sqrt{21}+34391077\right)\sqrt{\pi }{\left(k\eta \right)}^{\frac{3}{2}}{}_2{F}_3\left(\frac{3}{2},-\frac{\sqrt{21}}{2}-\frac{7}{2};2,2,-\frac{\sqrt{21}}{2}-\frac{5}{2};-k^2{\eta }^2\right)\hfill \\ & -205800(146\sqrt{21}+669)k^3{\eta }^3{G}_{3,5}^{3,1}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{2\sqrt{21}+15}{4},\frac{3}{4},\frac{3}{4}\\ \frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{3}{4},\frac{2\sqrt{21}+11}{4}\end{array}\right)\hfill \\ & +343000\left(857\sqrt{21}+3927\right)k^2{\eta }^2{G}_{3,5}^{3,1}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{2\sqrt{21}+17}{4},\frac{1}{4},\frac{1}{4}\\ -\frac{1}{4},-\frac{1}{4},\frac{3}{4},\frac{1}{4},\frac{2\sqrt{21}+13}{4}\end{array}\right)\hfill \\ & +14\left(38465701\sqrt{21}+176254865\right)k\eta G_{3,5}^{3,1}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{2\sqrt{21}+19}{4},\frac{3}{4},\frac{3}{4}\\ \frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{3}{4},\frac{2\sqrt{21}+15}{4}\end{array}\right)\hfill \\ & +84\left(6206377\sqrt{21}+28445153\right)G_{3,5}^{3,1}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{2\sqrt{21}+21}{4},\frac{1}{4},\frac{1}{4}\\ -\frac{1}{4},-\frac{1}{4},\frac{3}{4},\frac{1}{4},\frac{2\sqrt{21}+17}{4}\end{array}\right)\}\hfill \\ & +\frac{7}{\sqrt{k\eta }}{Y}_0\left(k\eta \right)\{14700(146\sqrt{21}+669)k^3{\eta }^3{G}_{3,5}^{2,2}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{5}{4},\frac{2\sqrt{21}+17}{4},\frac{1}{4}\\ \frac{3}{4},\frac{3}{4},\frac{1}{4},\frac{3}{4},\frac{2\sqrt{21}+13}{4}\end{array}\right)\hfill \\ & -24500\left(857\sqrt{21}+3927\right)k^2{\eta }^2{G}_{3,5}^{2,2}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{3}{4},\frac{2\sqrt{21}+19}{4},-\frac{1}{4}\\ \frac{1}{4},\frac{5}{4},-\frac{1}{4},\frac{1}{4},\frac{2\sqrt{21}+15}{4}\end{array}\right)\hfill \\ & -\left(38465701\sqrt{21}+176254865\right)k\eta G_{3,5}^{2,2}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{5}{4},\frac{2\sqrt{21}+21}{4},\frac{1}{4}\\ \frac{3}{4},\frac{3}{4},\frac{1}{4},\frac{3}{4},\frac{2\sqrt{21}+17}{4}\end{array}\right)\hfill \\ & -6\left(6206377\sqrt{21}+28445153\right)G_{3,5}^{2,2}\left(k\eta ,\frac{1}{2}|\begin{array}{c}\frac{3}{4},\frac{2\sqrt{21}+23}{4},-\frac{1}{4}\\ \frac{1}{4},\frac{5}{4},-\frac{1}{4},\frac{1}{4},\frac{2\sqrt{21}+19}{4}\end{array}\right)\}.\hfill \end{array}$$

(45)

Here, the function $G_{p,q}^{m,n}\left(z,r|\begin{array}{c}{a}_1,\cdots ,a_n,a_{n+1},\cdots ,a_p\\ b_1,\cdots ,b_m,b_{m+1},\cdots ,b_q\end{array}\right)$ is the generalized Meijer G-function, and the function ${}_p{F}_q\left(a_1,\cdots ,a_p;b_1,\cdots ,b_q;z\right)$ is the generalized hypergeometric function. The ratio of $\frac{c_2}{c_1}$ should be fixed by $\frac{c_2}{c_1}=-i$ as explained before. Thus, we have solved Equation (41) as

$$\begin{array}{cc}\hfill h_1& =-\frac{576(999-218\sqrt{21})}{60025}{c}_1{a}_0^{\frac{3-\sqrt{21}}{4}}\frac{H_{\mathrm{I}}^6\sqrt{k\eta }}{{\left(\frac{\sqrt{21}-3}{2}{a}_{\mathrm{E}}{H}_{\mathrm{I}}\eta \right)}^{9+\sqrt{21}}}\left(u_{11}-i\sqrt{\pi }{u}_{12}\right)\hfill \\ & =-\frac{576(999-218\sqrt{21})}{60025}{\overline{c}}_1\frac{H_{\mathrm{I}}^6}{{\tau }^6}\left(u_{11}-i\sqrt{\pi }{u}_{12}\right).\hfill \end{array}$$

(46)

Here, we used ${\tau }^6={\left(\frac{\sqrt{21}-3}{2}{a}_{\mathrm{E}}{H}_{\mathrm{I}}\eta \right)}^{9+\sqrt{21}}$.

In conclusion, we derived analytic form of the tensor perturbations which are given by Equations (40) and (46). If we know the effective action of the M-theory beyond the linear order of $\mathrm{\Gamma }$, we could derive the solution up to the same order by using the method developed in this paper. Some discussions on the power spectrum of the perturbations are given in Appendix B.

5. Conclusions and Discussion

In this paper, we examined the inflationary solution via higher derivative corrections in the M-theory and examined tensor perturbations around such a background. As a result, we have obtained the solutions of tensor perturbations analytically up to the linear order of $\mathrm{\Gamma }$. Although our calculations are limited up to the linear order of $\mathrm{\Gamma }$, our method developed in this paper is applied to higher-order effective action once its form is revealed.

The effective action of the M-theory contains (Weyl)${}^4$ terms. If we assume that the 10-dimensional space is divided into a 3-dimensional homogeneous space and a 7-dimensional internal one, it is possible to solve the equations of motion perturbatively with respect to $\mathrm{\Gamma }$ and obtain a inflationary solution due to the presence of (Weyl)${}^4$ terms. The tensor perturbations are analyzed around this inflationary background. These are expanded up to the linear order of $\mathrm{\Gamma }$ such as $h_{\alpha }=h_0+\mathrm{\Gamma }{h}_1$, and the equation of motion for $h_0$ is written by Equation (26) and that for $h_1$ is given by Equation (27). On the other hand, the effective action for the tensor perturbation for the cross mode is evaluated up to the linear order of $\mathrm{\Gamma }$. In addition, the effective action with respect to $h_0$ and $h_1$ is given by Equation (32). We could derive the same equations of motion from this action, and this gives a self-consistency check of our calculations, which are mainly done by using Mathematica codes. Although we omitted the calculations, the effective action for the plus mode is the same as that of the cross mode once we insert the background metric.

The equations of motion for $h_{\alpha }$ are solved perturbatively. The solution of $h_0$ is given by the linear combinations of Bessel functions. Then, we put the solution of $h_0$ into the equation of motion for $h_1$. The analytic form of $h_1$ is given by Equation (46). In this way, we developed a method to calculate tensor perturbations by using Mathematica codes. If we know the effective action of the M-theory beyond the linear order of $\mathrm{\Gamma }$, we could derive the solution up to the same order by using the method developed in this paper [55].

Since we know the analytic solution, at least perturbatively, it is straightforward to evaluate the power spectrum of the tensor perturbations. However, the vacuum of our inflationary solution is highly interacting, it is different from usual Bunch–Davies vacuum. So, in order to evaluate the power spectrum, we need to put an assumption on the amplitude of the power spectrum. Some discussions on the evaluation of the power spectrum are given in Appendix B.

As a future work, it is interesting to apply the method developed here to more a complicated internal geometry, such as the $G_2$ manifold [56]. It is also interesting to apply the analyses of this paper to the heterotic superstring theory with nontrivial internal space, which contains $R^2$ corrections [57], and reveal several problems in string cosmology [58]. Unification of the inflationary expansion and late time acceleration in modified gravity, such as $f\left(R\right)$ gravity or mimetic gravity, is an interesting direction to be explored [59,60,61,62].