Primordial Gravitational Wave- and Curvature Perturbation-Induced Energy Density Perturbations

Abstract

We study the second-order scalar and density perturbations generated by Gaussian curvature perturbations and primordial gravitational waves in the radiation-dominated era. After presenting all the possible second-order source terms, we obtain the explicit expressions of the kernel functions and the power spectra of the second-order scalar perturbations. We show that the primordial gravitational waves might affect second-order energy density perturbation ${\delta }^{\left(2\right)}=\delta {\rho }^{\left(2\right)}/{\rho }^{\left(0\right)}$ significantly. The effects of primordial gravitational waves are studied in terms of different kinds of primordial power spectra.

1. Introduction

The cosmological perturbations which are originated from the quantum fluctuations during inflation will inevitably induce higher-order perturbations. These induced higher-order perturbations can also affect the evolution of the universe [1].

Cosmological perturbations can be decomposed as scalar, vector, and tensor perturbations on account of the $SO\left(3\right)$ symmetry of the Friedmann–Robertson–Walker (FRW) spacetime [2,3,4]. Recently, the higher-order perturbations induced by the primordial perturbations have been attracting a lot of interest because of their rich phenomenology [5]. Among tensor perturbations, higher-order induced tensor perturbations are known as induced gravitational waves (GWs) [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Regarding higher-order induced scalar perturbations [32,33,34], a higher-order energy density perturbation ${\delta }^{\left(n\right)}={\rho }^{\left(n\right)}/{\rho }^{\left(0\right)}$ can be calculated in terms of these scalar perturbations. And ${\delta }^{\left(n\right)}$ can affect primordial black hole (PBH) formation [35,36] and the large-scale structure (LSS) of the universe [37,38]. Higher-order induced vector perturbations can also affect many cosmological observations [5,39,40,41,42,43,44,45,46,47,48], such as redshift-space distortions [42] and weak lensing [45,46,47].

The source terms of high-order induced perturbations originate from the primordial perturbations generated during inflation. Since vector perturbations decay as $1/a^2$ [39], we typically neglect primordial vector perturbations. At large scales (≥1 Mpc), the amplitude of a primordial scalar perturbation $A_{\zeta }$ is constrained by observations of the cosmic microwave background (CMB) and large-scale structures to be about $A_{\zeta }\simeq 2^{-9}$. For primordial tensor perturbations at large scales (≥1 Mpc), the tensor-to-scalar ratio ($r=A_h/A_{\zeta }$) is constrained to be less than 0.06 [49], where $A_h$ is the amplitude of the primordial tensor perturbation. Therefore, when studying higher-order induced perturbations at large scales (≥1 Mpc), primordial tensor perturbations can be neglected compared with primordial scalar perturbations.

At small scales (≤1 Mpc), the constraints of primordial scalar and tensor perturbations are significantly weaker than those at large scales [50]. Over the past few years, primordial scalar perturbations with large amplitude at small scales have been attracting a lot of interest. They are closely related to primordial black holes and scalar induced GWs [51,52,53,54,55,56]. In primordial tensor perturbations at small scales, their amplitude could also be much larger than it is at large scales. Large primordial tensor perturbations at small scales can be realized using many models of early universe, such as $G^2$-inflation [57], spectator field [58], and so on [35]. Recently, the power spectra of second-order tensor perturbations induced by primordial scalar and tensor perturbations with large amplitudes were studied in Refs. [17,59,60,61]. They considered the log-normal primordial scalar and tensor power spectra at small scales and found that primordial tensor perturbations have a very important effect on second-order induced tensor perturbations.

Second-order induced scalar and energy density perturbations have been studied for many years [32,33,34,35,36,37,38,62]. However, a complete study of second-order induced scalar perturbations has not been presented. The importance of scalar–tensor coupling source term ${\mathcal{S}}_2\sim \varphi h$ has been neglected in previous studies. In this paper, we consider the second-order energy density perturbation induced by primordial curvature and tensor perturbations during the RD era systematically. Second-order scalar perturbations can be generated by scalar–scalar, scalar–tensor, and tensor–tensor coupling source terms ${\mathcal{S}}_1\sim \varphi \varphi$, ${\mathcal{S}}_2\sim \varphi h$, and ${\mathcal{S}}_3\sim hh$. A second-order energy density perturbation ${\mathcal{P}}_{\delta }^{\left(2\right)}$ can be calculated in terms of these second-order induced scalar perturbations. The explicit expressions of second-order scalar and energy density perturbations are presented in this work.

This paper is organized as follows: In Section 2, second-order scalar perturbations are studied. The explicit expressions of the second-order power spectra are presented. In Section 3, we investigate the second-order power spectrum in terms of monochromatic primordial power spectra. ${\mathcal{P}}_{\delta }^{\left(2\right)}$ induced by log-normal primordial power spectra are studied in Section 4. The conclusions and discussions are summarized in Section 5.

2. Second-Order Scalar Perturbations

In this section, we study the equations of motion and the kernel functions of the second-order scalar perturbations induced by primordial curvature and tensor perturbations in the RD era.

2.1. Equation of Motion

The perturbed metric in the flat FRW spacetime with Newtonian gauge is given by

$$\begin{array}{cc}\hfill \mathrm{d}{s}^2& =a^2\left[-\left(1+2{\varphi }^{\left(1\right)}+{\varphi }^{\left(2\right)}\right)\mathrm{d}{\eta }^2+\left(\left(1-2{\psi }^{\left(1\right)}-{\psi }^{\left(2\right)}\right){\delta }_{ij}+h_{ij}^{\left(1\right)}\right)\mathrm{d}{x}^i\mathrm{d}{x}^j\right],\hfill \end{array}$$

(1)

where ${\varphi }^{\left(n\right)}$ and ${\psi }^{\left(n\right)}$$\left(n=1,2\right)$ are first-order and second-order scalar perturbations, $h_{ij}^{\left(1\right)}$ is the first-order tensor perturbation. Here, we neglect the first-order vector perturbation, since the vector modes decay as $1/a^2$ after they leave the Hubble horizon during inflation. We use the xPand package to study the perturbations of the Einstein equation, as xPand package can help us to obtain and simplify the equations of motion of cosmological perturbations [63]. The equations of motion of second-order scalar perturbations are given by

$$\begin{array}{c}\hfill {\mathsf{\Psi }}^{\left(2\right)}-{\mathsf{\Phi }}^{\left(2\right)}=-2{\Delta }^{-1}\left({\partial }^i{\Delta }^{-1}{\partial }^j-\frac{1}{2}{\mathcal{T}}^{ij}\right){\mathcal{S}}_{ij}\left(\mathbf{x},\eta \right),\end{array}$$

(2)

$$\begin{array}{c}\hfill {\partial }_{\eta }^2{\mathsf{\Psi }}^{\left(2\right)}+3\mathcal{H}{\partial }_{\eta }{\mathsf{\Psi }}^{\left(2\right)}-\frac{5}{6}\Delta {\mathsf{\Psi }}^{\left(2\right)}+\mathcal{H}{\partial }_{\eta }{\mathsf{\Phi }}^{\left(2\right)}+\frac{1}{2}\Delta {\mathsf{\Phi }}^{\left(2\right)}=-\frac{1}{2}{\mathcal{T}}^{ij}{\mathcal{S}}_{ij}\left(\mathbf{x},\eta \right),\end{array}$$

(3)

where ${\mathcal{T}}^{ij}={\delta }^{ij}-{\partial }^i{\Delta }^{-1}{\partial }^j$ is the transverse operator. During the RD era, the conformal Hubble parameter can be expressed as $\mathcal{H}=1/\eta$. For convenience, we use the symbols ${\varphi }^{\left(1\right)}\equiv \varphi$ and $h_{ij}^{\left(1\right)}\equiv h_{ij}$. As shown in Figure 1, source term ${\mathcal{S}}_{ij}\left(\mathbf{x},\eta \right)$ is composed of three parts: ${\mathcal{S}}_{ij}\left(\mathbf{x},\eta \right)={\mathcal{S}}_{ij,1}+{\mathcal{S}}_{ij,2}+{\mathcal{S}}_{ij,3}$.

The explicit expressions of the source terms are given in Appendix A. By substituting Equation (2) into Equation (3), we obtain the equation of motion of second-order scalar perturbation ${\mathsf{\Psi }}^{\left(2\right)}$ in the RD era:

$$\begin{array}{cc}\hfill & {\partial }_{\eta }^2{\mathsf{\Psi }}^{\left(2\right)}+\frac{4}{\eta }{\partial }_{\eta }{\mathsf{\Psi }}^{\left(2\right)}-\frac{1}{3}\Delta {\mathsf{\Psi }}^{\left(2\right)}\hfill \\ \hfill & =-\frac{1}{2}{\mathcal{T}}^{ij}{S}_{ij}-2{\Delta }^{-1}\left({\partial }^j{\Delta }^{-1}{\partial }^i-\frac{1}{2}{\mathcal{T}}^{ij}\right)\left(\frac{1}{2}\Delta +\frac{1}{\eta }{\partial }_{\eta }\right){\mathcal{S}}_{ij}\left(\mathbf{x},\eta \right)\hfill \\ \hfill & =\left(-\frac{1}{2}{\mathcal{T}}^{ij}-\left({\partial }^j{\Delta }^{-1}{\partial }^i-\frac{1}{2}{\mathcal{T}}^{ij}\right)-2{\Delta }^{-1}\left({\partial }^j{\Delta }^{-1}{\partial }^i-\frac{1}{2}{\mathcal{T}}^{ij}\right)\frac{1}{\eta }{\partial }_{\eta }\right)S_{ij}\left(\mathbf{x},\eta \right)\hfill \\ \hfill & =-\left({\partial }^j{\Delta }^{-1}{\partial }^i+2{\Delta }^{-1}\left({\partial }^j{\Delta }^{-1}{\partial }^i-\frac{1}{2}{\mathcal{T}}^{ij}\right)\frac{1}{\eta }{\partial }_{\eta }\right)\sum _{a=1}^3{\mathcal{S}}_{ij,a}\left(\mathbf{x},\eta \right).\hfill \end{array}$$

(4)

2.2. Kernel Functions

In order to solve the equation of motion of second-order scalar perturbations, we rewrite Equation (4) in momentum space as

$$\begin{array}{c}\hfill {\mathsf{\Psi }}^{{\left(2\right)}^{″}}\left(\mathbf{k},\eta \right)+\frac{4}{\eta }{\mathsf{\Psi }}^{{\left(2\right)}^{\prime }}\left(\mathbf{k},\eta \right)+\frac{k^2}{3}{\mathsf{\Psi }}^{\left(2\right)}\left(\mathbf{k},\eta \right)=\sum _{a=1}^3{\mathcal{S}}_a\left(\mathbf{k},\eta \right),\end{array}$$

(5)

where

$$\begin{array}{c}\hfill {\mathcal{S}}_a\left(\mathbf{k},\eta \right)={\mathcal{D}}_{\psi }^{ij}{\mathcal{S}}_{ij,a}\left(\mathbf{k},\eta \right),{\mathcal{D}}_{\psi }^{ij}=-\left(\frac{k^i{k}^j}{k^2}-\left(\frac{3k^i{k}^j}{k^4}-\frac{{\delta }^{ij}}{k^2}\right)\frac{k^2}{x}{\partial }_x\right).\end{array}$$

(6)

Here, we have defined $x\equiv k\eta$. The explicit expressions of ${\mathcal{S}}_{ij,a}\left(\mathbf{k},\eta \right)$$(a=1,2,3)$ are given in Appendix A. By substituting Equations (A4)–(A6) into Equation (6), we obtain the expressions of ${\mathcal{S}}_a\left(\mathbf{k},\eta \right)$:

$$\begin{array}{ccc}\hfill {\mathcal{S}}_1& =& {\mathcal{D}}_{\psi }^{ij}{\mathcal{S}}_{ij,1}=\int \frac{d^{3}p}{{\left(2\pi \right)}^{3/2}}{k}^2{f}_1\left(u,v,x\right)\frac{4}{9}{\zeta }_{\mathbf{k}-\mathbf{p}}{\zeta }_{\mathbf{p}},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\mathcal{S}}_2& =& {\mathcal{D}}_{\psi }^{ij}{\mathcal{S}}_{ij,2}=\int \frac{d^{3}p}{{\left(2\pi \right)}^{3/2}}{k}^2{f}_2^{{\lambda }_1}\left(u,v,x\right)\frac{2}{3}{\zeta }_{\mathbf{k}-\mathbf{p}}{\mathbf{h}}_{\mathbf{p}}^{{\lambda }_1},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\mathcal{S}}_3& =& {\mathcal{D}}_{\psi }^{ij}{\mathcal{S}}_{ij,3}=\int \frac{d^{3}p}{{\left(2\pi \right)}^{3/2}}{k}^2{f}_3^{{\lambda }_1{\lambda }_2}\left(u,v,x\right){\mathbf{h}}_{\mathbf{k}-\mathbf{p}}^{{\lambda }_1}{\mathbf{h}}_{\mathbf{p}}^{{\lambda }_2},\hfill \end{array}$$

(9)

where ${\lambda }_1$ and ${\lambda }_2$ are the polarization indices and the spatial indices of ${\mathcal{S}}_{ij,a}$$(a=1,2,3)$ are contracted. By substituting Equations (7)–(9) into Equation (5), we solve Equation (5) using the Green’s function method:

$${\mathsf{\Psi }}^{\left(2\right)}=\sum _{a=1}^3{\mathsf{\Psi }}_a^{\left(2\right)},(a=1,2,3),$$

(10)

where

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_1^{\left(2\right)}& =& \int \frac{d^{3}p}{{\left(2\pi \right)}^{3/2}}{I}_1\left(u,v,x\right)\frac{4}{9}{\zeta }_{\mathbf{k}-\mathbf{p}}{\zeta }_{\mathbf{p}},\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_2^{\left(2\right)}& =& \int \frac{d^{3}p}{{\left(2\pi \right)}^{3/2}}{I}_2^{{\lambda }_1}\left(u,v,x\right)\frac{2}{3}{\zeta }_{\mathbf{k}-\mathbf{p}}{\mathbf{h}}_{\mathbf{p}}^{{\lambda }_1},\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_3^{\left(2\right)}& =& \int \frac{d^{3}p}{{\left(2\pi \right)}^{3/2}}{I}_3^{{\lambda }_1{\lambda }_2}\left(u,v,x\right){\mathbf{h}}_{\mathbf{k}-\mathbf{p}}^{{\lambda }_1}{\mathbf{h}}_{\mathbf{p}}^{{\lambda }_2}.\hfill \end{array}$$

(13)

In Equations (11)–(13), the kernel functions $I_a(u,v,x)$$(a=1,2,3)$ are defined as

$$\begin{array}{cc}\hfill I_1& ={\int }_0^x\mathrm{d}\overline{x}{\left(\frac{\overline{x}}{x}\right)}^2\left\{-\frac{x\overline{x}}{\sqrt{3}}\left[j_1(x/\sqrt{3})y_1(\overline{x}/\sqrt{3})-j_1(\overline{x}/\sqrt{3})y_1(x/\sqrt{3})\right]\right\}{f}_1(u,v,\overline{x}),\hfill \\ \hfill I_2^{{\lambda }_1}& ={\int }_0^x\mathrm{d}\overline{x}{\left(\frac{\overline{x}}{x}\right)}^2\left\{-\frac{x\overline{x}}{\sqrt{3}}\left[j_1(x/\sqrt{3})y_1(\overline{x}/\sqrt{3})-j_1(\overline{x}/\sqrt{3})y_1(x/\sqrt{3})\right]\right\}{f}_2^{{\lambda }_1}(u,v,\overline{x}),\hfill \\ \hfill I_3^{{\lambda }_1,{\lambda }_2}& ={\int }_0^x\mathrm{d}\overline{x}{\left(\frac{\overline{x}}{x}\right)}^2\left\{-\frac{x\overline{x}}{\sqrt{3}}\left[j_1(x/\sqrt{3})y_1(\overline{x}/\sqrt{3})-j_1(\overline{x}/\sqrt{3})y_1(x/\sqrt{3})\right]\right\}{f}_3^{{\lambda }_1,{\lambda }_2}(u,v,\overline{x}).\hfill \end{array}$$

(14)

At the end of this section, we calculate the kernel functions in Equation (14). We present these three kinds of kernel functions ${\left(I_a(u=1,v=1,x)\right)}^2$$(a=1,2,3)$ as functions of $x=k\eta$ in Figure 2.

As shown in Figure 2, kernel function $I_1$ is much larger than other kernel functions.

2.3. Initial Second-Order Perturbation

As we mentioned in Refs. [2,32], the contributions from the initial second-order perturbation also need to be considered. More precisely, second-order scalar perturbations are composed of two parts, the second-order scalar perturbations induced by the primordial perturbations and the initial second-order perturbation. The second-order curvature perturbation in the Newtonian gauge is given by [2]

$$\begin{array}{c}\hfill -{\zeta }^{\left(2\right)}={\mathsf{\Psi }}^{\left(2\right)}+\frac{\mathcal{H}}{{\rho }^{\left(0\right)\prime }}\left[\delta {\rho }^{\left(2\right)}-\frac{\delta {\rho }^{\left(1\right)\prime }}{{\rho }^{\left(0\right)\prime }}\delta {\rho }^{\left(1\right)}\right]-\frac{1}{4}{\chi }_{\delta \rho k}^k+\frac{1}{4}{\Delta }^{-2}{\partial }^i{\partial }^j{\chi }_{ij\delta \rho },\end{array}$$

(15)

where

$$\begin{array}{cc}\hfill {\chi }_{ij\delta \rho }& \equiv -2\frac{\mathcal{H}}{{\rho }^{\left(0\right)\prime }}\left[2\mathcal{H}\left(\frac{\delta {\rho }^{\left(1\right)2}}{{\rho }^{\left(0\right)\prime }}\right)-\frac{\delta {\rho }^{\left(1\right)\prime }}{{\rho }^{\left(0\right)\prime }}\delta {\rho }^{\left(1\right)}\right]{\delta }_{ij}-\frac{2}{{\rho }^{{\left(0\right)}^{\prime }2}}{\partial }_i\delta {\rho }^{\left(1\right)}{\partial }_j\delta {\rho }^{\left(1\right)}\hfill \\ \hfill & +4\left[-\frac{\delta {\rho }^{\left(1\right)}}{{\rho }^{\left(0\right)\prime }}\left[\left(-{\mathsf{\Psi }}^{\left(1\right)\prime }{\delta }_{ij}+\frac{1}{2}{h}_{ij}^{\left(1\right)\prime }\right)+2\mathcal{H}\left(-{\mathsf{\Psi }}^{\left(1\right)}{\delta }_{ij}+\frac{1}{2}{h}_{ij}^{\left(1\right)}\right)\right]\right],\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\chi }_{\delta \rho k}^k& \equiv -6\frac{\mathcal{H}}{{\rho }^{\left(0\right)\prime }}\left[2\mathcal{H}\left(\frac{\delta {\rho }^{\left(1\right)2}}{{\rho }^{\left(0\right)\prime }}\right)-\frac{\delta {\rho }^{\left(1\right)\prime }}{{\rho }^{\left(0\right)\prime }}\delta {\rho }^{\left(1\right)}\right]-\frac{2}{{\rho }^{{\left(0\right)}^{\prime }2}}{\partial }^k\delta {\rho }^{\left(1\right)}{\partial }_k\delta {\rho }^{\left(1\right)}\hfill \\ \hfill & +4\left[-\frac{\delta {\rho }^{\left(1\right)}}{{\rho }^{\left(0\right)\prime }}\left[\left(-3{\mathsf{\Psi }}^{\left(1\right)\prime }+\frac{1}{2}{h}_k^{\left(1\right)k\prime }\right)+2\mathcal{H}\left(-3{\mathsf{\Psi }}^{\left(1\right)}+\frac{1}{2}{h}_k^{\left(1\right)k}\right)\right]\right].\hfill \end{array}$$

(17)

In the superhorizon limit ($k\eta \ll 1$), the second-order curvature perturbation can be approximated as

$$\begin{array}{cc}\hfill {\zeta }^{\left(2\right)}(k,\eta )\simeq & \left\{-{\mathsf{\Psi }}^{\left(2\right)}(k,\eta )+\frac{1}{4}{\delta }^{\left(2\right)}-\frac{1}{8}{\delta }^{\left(1\right)2}-{\delta }^{\left(1\right)}{\mathsf{\Psi }}^{\left(1\right)}\right.\hfill \\ \hfill & \left.+\frac{{\delta }^{\left(1\right)}{h}_k^{\left(1\right)k}}{4}-\frac{{\delta }^{\left(1\right)}{\nabla }^{-2}{\partial }_i{\partial }_j{h}^{\left(1\right)ij}}{4}\right\}{T}_{\mathsf{\Psi }}\left(k\eta \right),\hfill \end{array}$$

(18)

where

$$\begin{array}{c}\hfill {\delta }^{\left(1\right)}=\frac{\delta {\rho }^{\left(1\right)}}{{\rho }^{\left(0\right)}}=\frac{-6\mathcal{H}\left(\mathcal{H}{\varphi }^{\left(1\right)}+{\psi }^{\left(1\right)\prime }\right)+2▵{\psi }^{\left(1\right)}}{3{\mathcal{H}}^2}\simeq -2{\varphi }^{\left(1\right)}=-\frac{4}{3}{\zeta }^{\left(1\right)}.\end{array}$$

(19)

We assume local-type non-Gaussianity here, which is parametrized as ${\zeta }^{\left(2\right)}=2a_{\mathrm{NL}}{\left({\zeta }^{\left(1\right)}\right)}^2$ in the superhorizon limit, and $a_{\mathrm{NL}}=1$ for Gaussian perturbations [64,65,66]. By substituting the condition of local-type non-Gaussianity into Equation (18), we obtain the contributions from the initial second-order perturbation:

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{\mathrm{in}}^{\left(2\right)}\left(\mathbf{k}\right)& =\frac{1}{3}\left(\frac{3k^i{k}^j}{k^4}-\frac{{\delta }^{ij}}{k^2}\right){\mathcal{S}}_{ij}+\int \frac{d^{3}p}{{\left(2\pi \right)}^{3/2}}\left(\left(-\frac{4}{3}{a}_{\mathrm{NL}}+\frac{28}{27}\right){\zeta }_{\mathbf{k}-\mathbf{p}}{\zeta }_{\mathbf{p}}\right.\hfill \\ \hfill & \left.-\frac{2}{9}{\epsilon }_i^{\lambda ,i}\left(\mathbf{p}\right){\zeta }_{\mathbf{k}-\mathbf{p}}{\mathbf{h}}_{\mathbf{p}}^{\lambda }+\frac{2}{9}\frac{p^i{p}^j}{p^2}{\epsilon }_{ij}^{\lambda }\left(\mathbf{p}\right){\zeta }_{\mathbf{k}-\mathbf{p}}{\mathbf{h}}_{\mathbf{p}}^{\lambda }\right).\hfill \end{array}$$

(20)

After considering the effects of the initial second-order perturbation, second-order scalar perturbation ${\mathsf{\Psi }}^{\left(2\right)}$ can be written as

$$\begin{array}{c}\hfill {\mathsf{\Psi }}^{\left(2\right)}\left(\mathbf{k}\right)={\mathsf{\Psi }}_{\mathrm{in}}^{\left(2\right)}\left(\mathbf{k}\right)+\sum _{a=1}^3{\mathsf{\Psi }}_a^{\left(2\right)},(a=1,2,3).\end{array}$$

(21)

We study the second-order scalar and density perturbations generated by the Gaussian curvature and tensor perturbations. Therefore, we set $a_{\mathrm{NL}}=1$ in this paper.

2.4. Power Spectra

In this section, the power spectra of second-order scalar and density perturbations are investigated. We assume that the two-point function $\langle {\zeta }_{{\mathbf{k}}_1}{\mathbf{h}}_{{\mathbf{k}}_2}^{\lambda }\rangle$ = 0 for arbitrary ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ [17]. Therefore, we only need to consider three kinds of four-point functions. The explicit expressions of these four-point functions are given in Appendix C. The power spectra of second-order scalar perturbation is defined as

$$\begin{array}{c}\hfill \langle {\mathsf{\Psi }}^{\left(2\right)}\left(\mathbf{k}\right){\mathsf{\Psi }}^{\left(2\right)}\left({\mathbf{k}}^{\prime }\right)\rangle =\delta \left(\mathbf{k}+{\mathbf{k}}^{\prime }\right)\frac{2{\pi }^2}{k^3}{\mathcal{P}}_{\psi }^{\left(2\right)}.\end{array}$$

(22)

By substituting Equations (11)–(13) into Equation (22), we obtain

$$\begin{array}{c}\hfill \langle {\mathsf{\Psi }}^{\left(2\right)}\left(\mathbf{k}\right){\mathsf{\Psi }}^{\left(2\right)}\left({\mathbf{k}}^{\prime }\right)\rangle =\sum _{i=1}^3\langle {\mathsf{\Psi }}_i^{\left(2\right)}\left(\mathbf{k}\right){\mathsf{\Psi }}_i^{\left(2\right)}\left({\mathbf{k}}^{\prime }\right)\rangle ,\end{array}$$

(23)

where

$$\begin{array}{ccc}\hfill \langle {\mathsf{\Psi }}_1^{\left(2\right)}\left(\mathbf{k}\right){\mathsf{\Psi }}_1^{\left(2\right)}\left({\mathbf{k}}^{\prime }\right)\rangle & =& \int \frac{d^{3}p}{{\left(2\pi \right)}^{3/2}}\frac{d^3{p}^{\prime }}{{\left(2\pi \right)}^{3/2}}{I}_1\left(|\mathbf{k}-\mathbf{p}|,p,\eta \right)I_1\left(|{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }|,p^{\prime },\eta \right)\hfill \\ & & \times \frac{16}{81}\langle {\zeta }_{\mathbf{k}-\mathbf{p}}{\zeta }_{\mathbf{p}}{\zeta }_{{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }}{\zeta }_{{\mathbf{p}}^{\prime }}\rangle ,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill \langle {\mathsf{\Psi }}_2^{\left(2\right)}\left(\mathbf{k}\right){\mathsf{\Psi }}_2^{\left(2\right)}\left({\mathbf{k}}^{\prime }\right)\rangle & =& \int \frac{d^{3}p}{{\left(2\pi \right)}^{3/2}}\frac{d^3{p}^{\prime }}{{\left(2\pi \right)}^{3/2}}{I}_2^{{\lambda }_1}\left(|\mathbf{k}-\mathbf{p}|,p,\eta \right)I_2^{{\lambda }_1^{\prime }}\left(|{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }|,p^{\prime },\eta \right)\hfill \\ & & \times \frac{4}{9}\langle {\zeta }_{\mathbf{k}-\mathbf{p}}{\mathbf{h}}_{\mathbf{p}}^{{\lambda }_1}{\zeta }_{{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }}{\mathbf{h}}_{{\mathbf{p}}^{\prime }}^{{\lambda }_1^{\prime }}\rangle ,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill \langle {\mathsf{\Psi }}_3^{\left(2\right)}\left(\mathbf{k}\right){\mathsf{\Psi }}_3^{\left(2\right)}\left({\mathbf{k}}^{\prime }\right)\rangle & =& \int \frac{d^{3}p}{{\left(2\pi \right)}^{3/2}}\frac{d^3{p}^{\prime }}{{\left(2\pi \right)}^{3/2}}{I}_3^{{\lambda }_1{\lambda }_2}\left(|\mathbf{k}-\mathbf{p}|,p,\eta \right)I_3^{{\lambda }_1^{\prime }{\lambda }_2^{\prime }}\left(|{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }|,p^{\prime },\eta \right)\hfill \\ & & \times \langle {\mathbf{h}}_{\mathbf{k}-\mathbf{p}}^{{\lambda }_1}{\mathbf{h}}_{\mathbf{p}}^{{\lambda }_2}{\mathbf{h}}_{{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }}^{{\lambda }_1^{\prime }}{\mathbf{h}}_{{\mathbf{p}}^{\prime }}^{{\lambda }_2^{\prime }}\rangle .\hfill \end{array}$$

(26)

The corresponding Feynman diagrams of these three kinds of two-point functions are given in Figure 3.

By substituting Equations (A15)–(A17) into Equations (24)–(26), we obtain the explicit expressions of the power spectra:

$$\begin{array}{c}\hfill {\mathcal{P}}_{\psi }^{\left(2\right)}={\mathcal{P}}_1^{\left(2\right)}+{\mathcal{P}}_2^{\left(2\right)}+{\mathcal{P}}_3^{\left(2\right)},\end{array}$$

(27)

where

$$\begin{array}{ccc}\hfill {\mathcal{P}}_1^{\left(2\right)}& =& \frac{1}{2}{\int }_0^{\infty }dv{\int }_{|1-v|}^{|1+v|}\frac{du}{v^2{u}^2}{I}_1\left(u,v,x\right){\mathcal{P}}_{\zeta }\left(uk\right){\mathcal{P}}_{\zeta }\left(vk\right)\hfill \\ & & \times \frac{16}{81}\left[I_1\left(|{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }|,p^{\prime },\eta \right){|}_{{\mathbf{p}}^{\prime }=-\mathbf{p}}+I_1\left(|{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }|,p^{\prime },\eta \right){|}_{{\mathbf{p}}^{\prime }=\mathbf{p}-\mathbf{k}}\right]{|}_{{\mathbf{k}}^{\prime }\to -\mathbf{k}},\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill {\mathcal{P}}_2^{\left(2\right)}& =& \frac{1}{2}{\int }_0^{\infty }dv{\int }_{|1-v|}^{|1+v|}\frac{du}{v^2{u}^2}{I}_2^{{\lambda }_1}\left(u,v,x\right){\delta }^{{\lambda }_1{\lambda }_1^{\prime }}{\mathcal{P}}_{\zeta }\left(uk\right){\mathcal{P}}_h\left(vk\right)\hfill \\ & & \times \frac{4}{9}{I}_2^{{\lambda }_1^{\prime }}\left(|{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }|,p^{\prime },\eta \right){|}_{{\mathbf{p}}^{\prime }=-\mathbf{p},{\mathbf{k}}^{\prime }=-{\mathbf{k}}^{\prime }}\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\mathcal{P}}_3^{\left(2\right)}& =& \frac{1}{2}{\int }_0^{\infty }dv{\int }_{|1-v|}^{|1+v|}\frac{du}{v^2{u}^2}{I}_3^{{\lambda }_1{\lambda }_2}\left(u,v,x\right){\mathcal{P}}_h\left(uk\right){\mathcal{P}}_h\left(vk\right)\hfill \\ & & \times \left[{\delta }^{{\lambda }_1{\lambda }_1^{\prime }}{\delta }^{{\lambda }_2{\lambda }_1^{\prime }}{I}_3^{{\lambda }_1^{\prime }{\lambda }_2^{\prime }}\left(|{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }|,p^{\prime },\eta \right){|}_{{\mathbf{p}}^{\prime }=-\mathbf{p}}\right.\hfill \\ & & \left.+{\delta }^{{\lambda }_1{\lambda }_2^{\prime }}{\delta }^{{\lambda }_2{\lambda }_1^{\prime }}{I}_3^{{\lambda }_1^{\prime }{\lambda }_2^{\prime }}\left(|{\mathbf{k}}^{\prime }-{\mathbf{p}}^{\prime }|,p^{\prime },\eta \right){|}_{{\mathbf{p}}^{\prime }=\mathbf{p}-\mathbf{k}}\right]{|}_{{\mathbf{k}}^{\prime }=-\mathbf{k}}.\hfill \end{array}$$

(30)

In Equations (28)–(30), the substitutions of ${\mathbf{k}}^{\prime }$ and ${\mathbf{p}}^{\prime }$ come from the three-dimensional delta functions in the Wick’s expansions of four-point functions. Since we have assumed that the two-point function $\langle {\zeta }_{{\mathbf{k}}_1}{\mathbf{h}}_{{\mathbf{k}}_2}^{\lambda }\rangle$ = 0, three kinds of source terms in Equations (A1)–(A3) are decoupled. More precisely, the two-point functions $\langle {\mathsf{\Psi }}_i^{\left(2\right)}\left(\mathbf{k}\right){\mathsf{\Psi }}_j^{\left(2\right)}\left({\mathbf{k}}^{\prime }\right)\rangle =0$ if $i\ne j$. As shown in Equations (28)–(30), the power spectra of second-order scalar perturbations are composed of three parts, which come from source terms ${\mathcal{S}}_1\sim \varphi \varphi$, ${\mathcal{S}}_2\sim \varphi h$, and ${\mathcal{S}}_3\sim hh$, respectively.

The energy density perturbation can be calculated as

$$\begin{array}{cc}\hfill {\delta }^{\left(2\right)}& =\frac{\delta {\rho }^{\left(2\right)}}{{\rho }^{\left(0\right)}}=-\frac{1}{3{\mathcal{H}}^4}(\frac{1}{4}{\mathcal{H}}^2{h}_{ij}^{\prime }\left(h^{{}^{\prime }ij}+8\mathcal{H}{h}^{ij}\right)+6{\mathcal{H}}^4\left(-4{\varphi }^2+{\mathsf{\Phi }}^{\left(2\right)}\right)+6{\mathcal{H}}^3{\mathsf{\Psi }}^{{\left(2\right)}^{\prime }}+4\mathcal{H}{\partial }_i{\varphi }^{\prime }{\partial }^i\varphi \hfill \\ \hfill & +2{\partial }_i{\varphi }^{\prime }{\partial }^i{\varphi }^{\prime }+{\mathcal{H}}^2\left(-2\left(3{\varphi }^{\prime 2}+8\varphi \Delta \varphi +\Delta {\mathsf{\Psi }}^{\left(2\right)}+2{\partial }_i\varphi {\partial }^i\varphi +\frac{1}{2}{h}^{ij}\left(-{\partial }_i{\partial }_j\varphi +\Delta h_{ij}\right)\right)\right.\hfill \\ \hfill & \left.+\frac{1}{4}\left(2{\partial }_l{h}_{ij}-3{\partial }_j{h}_{il}\right){\partial }^j{h}^{il}\right)).\hfill \end{array}$$

(31)

The power spectrum of second-order energy density perturbation ${\mathcal{P}}_{\delta }^{\left(2\right)}$ is defined by

$$\begin{array}{c}\hfill \langle {\delta }^{\left(2\right)}\left(\mathbf{k}\right){\delta }^{\left(2\right)}\left({\mathbf{k}}^{\prime }\right)\rangle =\delta \left(\mathbf{k}+{\mathbf{k}}^{\prime }\right)\frac{2{\pi }^2}{k^3}{\mathcal{P}}_{\delta }^{\left(2\right)},\end{array}$$

(32)

where the energy density (${\delta }^{\left(2\right)}$) can be calculated in terms of Equation (21) and Equation (31).

3. Monochromatic Primordial Power Spectra

As mentioned, the constraints of primordial curvature and tensor perturbations at small scales are significantly weaker than those at large scales; the tensor-to-scalar ratio (r) can be very large at small scales. Therefore, we start with a toy model of a $\delta$-peak. In this case, the primordial scalar and tensor perturbations are very large at small scale. Since we consider the second-order scalar and density perturbations generated by the Gaussian curvature and tensor perturbations, we have set $a_{\mathrm{NL}}=1$ in Equation (20).

3.1. Monochromatic Primordial Power Spectra with the Same $k_{\ast }$

We consider monochromatic primordial power spectra, namely,

$$\begin{array}{c}\hfill {\mathcal{P}}_{\zeta }=A_{\zeta }{k}_{\zeta \ast }\delta \left(k-k_{\zeta \ast }\right),{\mathcal{P}}_h=A_h{k}_{h\ast }\delta \left(k-k_{h\ast }\right),k_{\zeta \ast }=k_{h\ast }=k_{\ast },\end{array}$$

(33)

where $k_{\ast }$ is the wavenumber at which the power spectrum has a $\delta$-function peak. In Figure 4, we plot the three kinds of power spectra for second-order perturbations ${\mathsf{\Phi }}^{\left(2\right)}$, ${\mathsf{\Psi }}^{\left(2\right)}$, and ${\delta }^{\left(2\right)}$. Here, we use the symbols ${\mathcal{P}}_1^{\left(2\right)}$, ${\mathcal{P}}_2^{\left(2\right)}$, and ${\mathcal{P}}_3^{\left(2\right)}$ to represent the contributions of source terms ${\mathcal{S}}_1\sim \varphi \varphi$, ${\mathcal{S}}_2\sim \varphi h$, and ${\mathcal{S}}_3\sim hh$, respectively. As shown in Figure 4, for tensor-to-scalar ratio $r=A_h/A_{\zeta }=1$, the second-order perturbations sourced by ${\mathcal{S}}_1\sim \varphi \varphi$ dominate the total power spectra of ${\mathsf{\Phi }}^{\left(2\right)}$, ${\mathsf{\Psi }}^{\left(2\right)}$, and ${\delta }^{\left(2\right)}$.

In order to study the effects of the large tensor-to-scalar ratio ($r=A_h/A_{\zeta }$) at small scales, we calculate the total power spectra of ${\mathsf{\Phi }}^{\left(2\right)}$, ${\mathsf{\Psi }}^{\left(2\right)}$, and ${\delta }^{\left(2\right)}$ for different $r=A_h/A_{\zeta }$. In Figure 5, the total power spectra of second-order scalar perturbations for different r are presented. For tensor-to-scalar ratio $r\ll 1$, the effects of primordial tensor perturbations become negligible, and the total power spectra ${\mathcal{P}}_{\varphi }^{\left(2\right)}$, ${\mathcal{P}}_{\psi }^{\left(2\right)}$, and ${\mathcal{P}}_{\delta }^{\left(2\right)}$ reduce to the results in Ref. [32]. For $r\gg 1$ at small scales, the effects of primordial tensor perturbations become obvious, and the total power spectra reduce to the results in Refs. [35,37,38].

3.2. Monochromatic Primordial Power Spectra with Different $k_{\ast }$

The monochromatic primordial power spectra with different $k_{\ast }$ can be written as

$$\begin{array}{c}\hfill {\mathcal{P}}_{\zeta }=A_{\zeta }{k}_{\zeta \ast }\delta \left(k-k_{\zeta \ast }\right),{\mathcal{P}}_h=A_h{k}_{h\ast }\delta \left(k-k_{h\ast }\right),k_{\zeta \ast }=k_{\ast }\ne k_{h\ast }.\end{array}$$

(34)

In Figure 6, we plot the three kinds of power spectra ${\mathcal{P}}_a^{\left(2\right)}$$(a=1,2,3)$ in Equations (28)–(30) for second-order energy density perturbation ${\delta }^{\left(2\right)}=\delta {\rho }^{\left(2\right)}/{\rho }^{\left(0\right)}$ with different $k_{h\ast }$.

As shown in Figure 6, for $k_{h\ast }\ne k_{\zeta \ast }$, the behaviors of power spectrum ${\mathcal{P}}_2^{\left(2\right)}$ sourced by ${\mathcal{S}}_2\sim \varphi h$ are different from the case of $k_{h\ast }/k_{\zeta \ast }=1$. More precisely, for $k_{h\ast }/k_{\zeta \ast }=n>1$, the domain of definition of ${\mathcal{P}}_2^{\left(2\right)}$ is $[(n-1)k/k_{\ast },(n+1)k/k_{\ast }]$. For $k_{h\ast }/k_{\zeta \ast }=n\gg 1$, power spectrum ${\mathcal{P}}_2^{\left(2\right)}$ becomes a small peak near $n\times k/k_{\ast }$. In this case, the contributions of power spectrum ${\mathcal{P}}_2^{\left(2\right)}$ can be neglected. And for $k_{h\ast }/k_{\zeta \ast }=n<1$, the domain of definition of ${\mathcal{P}}_2^{\left(2\right)}$ is $[(1-n)k/k_{\ast },(1+n)k/k_{\ast }]$. For $k_{h\ast }/k_{\zeta \ast }=n\ll 1$, power spectrum ${\mathcal{P}}_2^{\left(2\right)}$ sourced by ${\mathcal{S}}_2\sim \varphi h$ becomes a large peak near $k/k_{\ast }$.

As shown in Figure 7, for $k_{h\ast }/k_{\zeta \ast }=0.1$, power spectrum ${\mathcal{P}}_2^{\left(2\right)}$ can be larger than ${\mathcal{P}}_1^{\left(2\right)}$ sourced by ${\mathcal{S}}_1\sim \varphi \varphi$, even in the case of $r=0.1$. Note that the contributions of source term ${\mathcal{S}}_2\sim \varphi h$ in Equation (A2) are completely neglected in previous studies [32,35,37]. Here, we point out that the contributions of source term ${\mathcal{S}}_2\sim \varphi h$ are very important for the monochromatic primordial power spectra with $k_{h\ast }/k_{\zeta \ast }\ll 1$.

4. Log-Normal Primordial Power Spectra

Since monochromatic primordial power spectra have infinitesimal width, it is necessary to consider a more realistic model, such as log-normal primordial power spectra. We consider the log-normal power spectra for primordial scalar and tensor perturbations. Here, we concentrated on the effects of source term ${\mathcal{S}}_2\sim \varphi h$ and the corresponding second-order power spectra, ${\mathcal{P}}_2^{\left(2\right)}$. The log-normal primordial power spectra are given by

$$\begin{array}{c}\hfill {\mathcal{P}}_{\zeta }=\frac{A_{\zeta }}{\sqrt{2\pi {\sigma }_{\ast }^2}}exp\left(-\frac{{ln}^2\left(\frac{k}{k_{\zeta \ast }}\right)}{2{\sigma }_{\ast }^2}\right),{\mathcal{P}}_h=\frac{A_h}{\sqrt{2\pi {\sigma }_{\ast }^2}}exp\left(-\frac{{ln}^2\left(\frac{k}{k_{h\ast }}\right)}{2{\sigma }_{\ast }^2}\right).\end{array}$$

(35)

Here, we concentrate on the large peak of ${\mathcal{P}}_2^{\left(2\right)}$ with $k_{h\ast }/k_{\zeta \ast }\ll 1$. As mentioned in Section 3.2, for $k_{h\ast }/k_{\zeta \ast }=n\ll 1$, power spectrum ${\mathcal{P}}_2^{\left(2\right)}$ sourced by ${\mathcal{S}}_2\sim \varphi h$ has a large peak near $k/k_{\ast }$. For the log-normal primordial power spectra, we calculate the power spectra of second-order density perturbation ${\delta }^{\left(2\right)}$ with $k_{h\ast }/k_{\zeta \ast }=0.1$. As shown in Figure 8, for log-normal primordial power spectra, the peak of ${\mathcal{P}}_2^{\left(2\right)}$ becomes larger during the process of ${\sigma }_{\ast }\to 0$.

For comparison, we plot ${\mathcal{P}}_1^{\left(2\right)}$ and ${\mathcal{P}}_2^{\left(2\right)}$ for second-order density perturbation ${\delta }^{\left(2\right)}$ with tensor-to-scalar ratio $r=0.1$ in Figure 9. The figure shows that the contributions of ${\mathcal{P}}_2^{\left(2\right)}$ become smaller when ${\sigma }_{\ast }$ increases. Namely, the effects of source term ${\mathcal{S}}_2\sim \varphi h$ become more obvious when ${\sigma }_{\ast }$ and n become smaller.

5. Conclusions and Discussion

In this paper, we systematically studied second-order density perturbations induced by primordial gravitational waves and primordial scalar perturbations. Since the constraints of primordial curvature and tensor perturbations at small scales are significantly weaker than those at large scales, we considered large tensor-to-scalar ratios (r) at small scales. As shown in Figure 5, the effects of primordial tensor perturbations become obvious for $r\gg 1$. For tensor-to-scalar ratios $r\ll 1$, the effects of primordial tensor perturbations become negligible, and our results of power spectra ${\mathcal{P}}_{\varphi }^{\left(2\right)}$, ${\mathcal{P}}_{\psi }^{\left(2\right)}$, and ${\mathcal{P}}_{\delta }^{\left(2\right)}$ reduce to the previous results in Ref. [32].

We give the explicit expressions for the power spectra of primordial scalar and tensor induced scalar and density perturbations in Equations (28)–(32). Specifically, for a given primordial scalar and tensor power spectra, the power spectra of second-order induced scalar and density perturbations can be calculated using these equations. In this paper, we considered the primordial power spectra following delta and log-normal at small scales. It is essential to explore more general forms of primordial power spectra, such as log-normal primordial scalar power spectra and power-law primordial tensor power spectra [67].

Moreover, the second-order induced density perturbations offer a window to understand small-scale primordial gravitational waves and primordial curvature perturbations. More precisely, primordial scalar and tensor power spectra can be calculated in terms of a given inflation model. Using Equations (28)–(32), we can calculate the power spectra of second-order induced scalar and energy density perturbations. The induced density perturbations will affect many physical processes at small scales, such as the formation of PBH [68] and high-order GW background [59]. By observing PBHs or higher-order GWs, we can constrain the power spectra of second-order induced density perturbations, thereby constraining inflationary models and the physical properties of primordial scalar and gravitational waves at small scales. Related research might be given in future work.