# Signatures of primordial gravitational waves in matter power spectrum

## Abstract

We simulate the evolution of a dust universe from \(z=1089\) to \(z=0\) by numerically integrating the Einstein’s equation for a spatially flat Friedmann–Lemaire–Robertson–Walker (FLRW) background spacetime with scalar perturbations which are derived from the matter power spectrum produced with the Code for Anisotropies in the Microwave Background (CAMB). To investigate the effects of primordial gravitational waves (GWs) on the inhomogeneity of the universe, we add an additional decaying, divergenceless and traceless primordial tensor perturbation with its initial amplitude being \(3\times 10^{-4}\) to the above metric. We find that this primordial tensor perturbation suppresses the matter power spectrum by about \(0.01\%\) at \(z=0\) for modes with wave number similar to its. This suppression may be a possible probe of a GWs background in the future.

## 1 Introduction

One of the most important predictions by inflation [1, 2] is that there is a stochastic gravitational waves (GWs) background. So far, people have made every endeavor to detect such a GWs background and test inflation scenario experimentally: the most promising one is the B-mode polarization of the cosmic microwave background (CMB) [3, 4, 5]; the complementary and even more sensitive one is the 21 cm HI emission from the dark ages [6, 7]; some not very competitive ones including weak lensing shear [8, 9] and other large-scale structure observables [10, 11]. The goal of this paper is to investigate the signatures of primordial GWs in matter power spectrum with numerical relativity, thereby proposing a possible probe of a GWs background.

The cosmological parameters predicted by Planck 2018 TT, TE, EE + lowE [36]

\(\Omega _b h^2\) | \(\Omega _c h^2\) | \(\Omega _m\) | \(H_0 \ [\text {km}~\text {s}^{-1}\,\text {Mpc}^{-1}]\) | \(n_s\) | \(10^9A_s\) | \(z_*\) | \(z_{\text {re}}\) |
---|---|---|---|---|---|---|---|

0.02236 | 0.1202 | 0.3166 | 67.27 | 0.9649 | 2.101 | 1089 | 7.68 |

So far, the power spectrum data from the Clustering of the Sloan Digital Sky Survey DR7 Luminous Red Galaxies ranges from \(k=0.02 \, h \text {Mpc}^{-1}\) to \(k=0.2 \, h \text {Mpc}^{-1}\) [19] and the power spectrum data from the WiggleZ Dark Energy Survey ranges from \(k=0.01\, h \text {Mpc}^{-1}\) to \(k=0.5 \, h \text {Mpc}^{-1}\) [20]. Due to their small *k* span, these data are not suitable to constrain the large-scale primordial fluctuations and inflation. However, the future high precision lensing and galaxy redshift surveys, such as the Large Synoptic Survey Telescope (LSST) [21, 22], will has a large enough *k* span to confirm the turnover in the power spectrum and constrain the large-scale primordial fluctuations. So, in this paper, we will consider primordial tensor perturbations with comparable wave number to the scale of turnover.

Three conversions between scale factor and redshift *z*. \(a^P\) follows the usual convention in cosmology. \(a^S\) is used during our simulations. \(a^F\) is a fiducial one which relates the former two

\(z+1=1090\) | \(z+1=10\) | \(z+1=1\) | |
---|---|---|---|

\(a^P\) | 0.00092 | 0.1 | 1 |

\(a^F\) | 0.0092 | 1 | 10 |

\(a^S\) | 1 | 109 | 1090 |

The comoving matter density \({\bar{\rho }}_m^I*={\bar{\rho }}_m^I(a^I)^3\) for three different universes, where \(I=P, F, S\) for our universe, fiducial universe and simulations respectively

\(a^P=1\) | \(a^F=1\) | \(a^S=1\) | |
---|---|---|---|

\({\bar{\rho }}_m^I*\) | \(6.0{\times }10^{-9}{\times }0.3166\) | \(6.0{\times }10^{-6}{\times }0.3166\) | \(6.0{\times }10^{-6}{\times }0.3166\) |

This paper is organized as follows. In Sect. 2, we give the initial conditions for background by rescaling the scale factor and perturbations by analyzing the matter power spectrum. In Sect. 3, we show the results of simulations. At last, a brief summary and discussion are included in Sect. 4. In this paper, we adopt the following conventions: Greek indices run in {0, 1, 2, 3}, Latin indices run in {1, 2, 3} and repeated indices implies summation.

## 2 Initial conditions

### 2.1 Initial conditions for background

Since we will perform large-scale cosmological simulations instead of the simulations of black-hole-binary-like astrophysical system, we modify the file EOS_Omni_Module. F90 in Einstein Toolkit to replace the default unit system: \(\text {M}_{\odot }=\text {G}=\text {c}=1\) with the new one: \(1\,\text {Mpc}=\text {G}=\text {c}=1\) [35]. Under this new unit system and with the cosmological parameters consistent with Planck 2018 results [36] as shown in Table 1, the matter density of our universe is \({\bar{\rho }}_m^P=6.0\times 10^{-9}\times 0.3166\) at \(z=0\), hence \({\bar{\rho }}_m^P=6.0\times 10^{-6}\times 0.3166\) at \(z=9\). Considering a fiducial universe whose matter density \({\bar{\rho }}_m^F(z)\) is equal to \({\bar{\rho }}_m^P(z)\), the scale factor of this fiducial one \(a^F=10a^P\) as shown in Table 2 means that the comoving matter density of it is \({\bar{\rho }}_m^F*=6.0\times 10^{-6}\times 0.3166\) as shown in Table 3. Here we will turn to a blown-up fiducial universe by 109 times to mimic our universe in simulations: setting the scale factor used during simulations as \(a^S=109a^F\) as shown in Table 2 and keeping the comoving matter density being \({\bar{\rho }}_m^S*=6.0\times 10^{-6}\times 0.3166\) as shown in Table 3. That is to say, simulations with \({\bar{\rho }}_m^S*=6.0\times 10^{-6}\times 0.3166\) from \(a^S=1\) to \(a^S=1090\) can give the evolution of our universe with \({\bar{\rho }}_m^P*=6.0\times 10^{-9}\times 0.3166\) from \(a^P=0.00092\) to \(a^P=1\) when we analyze the results from simulations taking this blowing-up by 109 times into consideration and regardless of the existence of dark energy and radiation.

### 2.2 Initial conditions for perturbations

*t*for FLRW background spacetime and they will become space-dependent in an inhomogeneous spacetime. For the latter case, we still take them as background quantities by taking the average of them across the simulation box. Also, we can give the evolutions of perturbations according to the solutions to first-order Einstein equations for (1)

The last plot in Fig. 1 shows the distribution of spatial curvature perturbations \(\Phi (x^i)\) (or \(f(x^i)\)) at \(a^S=1\). In fact, we use the function make_gaussian_random_field in c2raytools [37] to generate the density perturbations \(\delta (x^i)\) (the second plot in Fig. 1) from the matter power spectrum at \(z+1=1090\) (the first plot in Fig. 1) produced by the Code for Anisotropies in the Microwave Background (CAMB) [38] with parameters listed in Table 1 first. And then we derive \(f(x^i)\) from \(\delta (x^i)\) according to the Fourier version of (4), hence \(\Phi (x^i)\), \(\Psi (x^i)\) and \(v^i(x^i)\) (the third plot in Fig. 1). As for tensor perturbations, we here only consider one single mode with \(k=\frac{2\pi }{L}\) and the space distribution as \(\cos (\frac{2\pi }{L} z)\), where \(L=1000\) is the length of one side of our simulation box with \(x^i\) in \([-500, 500]\). And we set its initial amplitude \(h_{\frac{2\pi }{L}}^s(0)=10^{-3}\), but it has crossed inside the horizon and decayed by \(70\%\) when \(\eta _0\simeq 2\sqrt{\frac{3}{8\pi {\bar{\rho }}^S_{m,\text {init}}}}\).

## 3 Results

*E*and \(p_i\), we can specify the form of the Hamiltonian constraint violation and the momentum constraint violation as

## 4 Summary and discussion

We simulate a dust universe from \(a^S=1\) (or \(z=1089\)) to \(a^S=1090\) (or \(z=0\)) by numerically integrating the Einstein’s equation whose solution at \(a^S=1\) is a spatially flat FLRW metric with scalar perturbations which are derived from the matter power spectrum produced with CAMB. Then we add an additional decaying, divergenceless and traceless primordial tensor perturbation with its initial amplitude being \(3\times 10^{-4}\) to the metric as shown in Fig. 2. Simulations at \(160^3\), \(80^3\) and \(40^3\) resolution converge and show that this primordial tensor perturbation suppresses the matter power spectrum by about \(0.01\%\) at \(z=0\) for modes with wave number \(k\sim 0.05\) as shown in Fig. 4.

In the linear perturbation theory, scalar and tensor perturbations are supposed to be totally decoupled. However, there are some non-linear coupling terms between scalar and tensor perturbations for the full Einstein equations which are used in our simulations. Even though we turn to the first-order perturbed Einstein equations for the initial data, they satisfy the full Einstein constraints of early universe just with tiny deviations. That is to say, Einstein Toolkit takes the all possible terms of the full Einstein equations into our consideration. Therefore, this suppression results from the fully relativistic treatment for Einstein equations. Although there are nonlinear structures formed at the end of simulations (\(a^S=1090\)) as shown in Fig. 3, their scales are smaller than tensor perturbations’. So this suppression sown before the tensor perturbations died out and amplified with time is still in linear regime.

There are two caveats. First the production of monochromatic single mode gravitational wave seems unrealistic in cosmology and most inflation models predict a scale-invariant spectrum of gravitational waves. Here we only consider a monochromatic gravitational wave because primordial gravitational waves enter the horizon one by one. Given the comoving length of one side of our simulation box \(L=1000\) and the initial matter density, the modes with wave length \(>1000\) are initially outside the simulation box and will never enter it during simulations. As for modes with wave length \(<1000\), they entered the horizon earlier and almost died out. Therefore, if we want to study scale dependence of the results, we must perform simulations under other larger *L*, which results in high computational cost. Here we just make our results as a first step to more comprehensive studies. Also it’s necessary to include the dark energy if one want to compare the results with observations. Because dark energy is supposed to affect the very late-time growth factor by about \(10\%\). So far, however, people can’t simulate dark energy in Einstein Toolkit. Here we just keep it in mind.

This suppression may be a possible probe of a GWs background in the future only if the matter power spectrum is measured in high enough precision. Undoubtedly, by the time LSST is in full operation, the required precision for detection of such suppression is still far beyond reach. However, this suppression is an unique signature put by primordial GWs.

## Notes

### Acknowledgements

We would like to thank You-Jun Lu for his helpful discussions and advices on this paper. This work is partly supported by the National Natural Science Foundation of China under Grant no. 11690024, the Strategic Priority Program of the Chinese Academy of Sciences (Grant no. XDB 23040100).

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