Gravitational waves from an early matter era

TL;DR

The paper shows that an early matter-dominated epoch can enhance a second-order gravitational-wave background generated by linear density perturbations, because the tensor source remains constant while density perturbations grow on sub-Hubble scales. The resulting spectrum can be strongly boosted by the duration of the matter era, via an enhancement factor $F^2=(k_{\rm dom}/k_{\rm dec})^2$, but nonlinear evolution imposes a cut-off at $k_{\rm NL}$ that limits the amplitude and shifts the peak to lower wavenumbers. Predictions for present-day energy density $\Omega_{\mathrm{GW},0}(k)$ depend on the small-scale amplitude of primordial perturbations $\triangle_{\mathcal{R}}^2$ and the reheating temperature $T_{\rm dec}$, with potential detectability by LIGO/LISA requiring unusually large small-scale power, or by future detectors such as BBO under favorable conditions. The authors emphasize that nonlinear (numerical) calculations are necessary to reliably assess the GW background when the MD era lasts long enough to drive density perturbations into the nonlinear regime. Overall, the work suggests that gravitational-wave measurements could constrain the pre-BBN thermal history and the small-scale primordial power spectrum, provided accurate nonlinear modeling is employed.

Abstract

We investigate the generation of gravitational waves due to the gravitational instability of primordial density perturbations in an early matter-dominated era which could be detectable by experiments such as LIGO and LISA. We use relativistic perturbation theory to give analytic estimates of the tensor perturbations generated at second order by linear density perturbations. We find that large enhancement factors with respect to the naive second-order estimate are possible due to the growth of density perturbations on sub-Hubble scales. However very large enhancement factors coincide with a breakdown of linear theory for density perturbations on small scales. To produce a primordial gravitational wave background that would be detectable with LIGO or LISA from density perturbations in the linear regime requires primordial comoving curvature perturbations on small scales of order 0.02 for Advanced LIGO or 0.005 for LISA, otherwise numerical calculations of the non-linear evolution on sub-Hubble scales are required.

Figures (2)

• Figure 1: The power spectrum of gravitational waves, shown as a function of wavenumber $k$, generated from scalar perturbations during a matter dominated era, ${\cal P}_h(k,\eta_{\rm dec})$. The solid line shows the prediction using the linear matter power spectrum down to $k_{\rm dom}=10^3k_{\rm dec}$, the comoving Hubble scale at the start of matter domination. The dotted line shows the prediction when the matter power spectrum is truncated at $k_{\rm cut}=200k_{\rm dec}$. $k_{\rm dec}$ denotes the Hubble scale at the end of the matter era.
• Figure 2: The present energy density of gravitational waves, $\Omega_{GW,0}$, generated during a matter dominated era shown as a function of wavenumber $k$. In this example $F=(k_{\rm dom}/k_{\rm dec})^2=10^6$. The solid line shows the result predicted using the linear matter power perturbation for $k<k_{\rm dom}$, while the dotted line shows the result using the matter power spectrum truncated at $k>k_{\rm cut}=200k_{\rm dec}$.