The cosmological gravitational wave background from primordial density perturbations

TL;DR

The paper investigates the stochastic gravitational wave background produced at second order by primordial density perturbations during the radiation era. It derives the second-order GW evolution equation with a source term quadratic in first-order scalar perturbations and solves it via Green's functions to compute the GW power spectrum for both delta-function and power-law primordial spectra. It identifies resonant features for narrow spectral inputs and shows that, although small, the induced background could constrain the primordial power spectrum on scales inaccessible to electromagnetic observations with future detectors like LISA, DECIGO, and BBO. It also explores how the scalar tilt affects the GW amplitude and highlights the complementary potential of GW measurements to probe early-universe physics beyond CMB and large-scale structure.

Abstract

We discuss the gravitational wave background generated by primordial density perturbations evolving during the radiation era. At second-order in a perturbative expansion, density fluctuations produce gravitational waves. We calculate the power spectra of gravitational waves from this mechanism, and show that, in principle, future gravitational wave detectors could be used to constrain the primordial power spectrum on scales vastly different from those currently being probed by large-scale structure. As examples we compute the gravitational wave background generated by both a power-law spectrum on all scales, and a delta-function power spectrum on a single scale.

Figures (6)

• Figure 1: The function $\mathcal{F}_{\delta}(x)$ for different values of $k_{in}/k=u=v$, with $x_0=0$.
• Figure 2: The function $\mathcal{F}_{\delta}(x)$ from early times (super-Hubble scales) to late times (small scales) for $k=k_{in}$ (i.e. $u=v=1$), $x_0=0$. For large $x$ the function oscillates with a constant amplitude.
• Figure 3: The function $\mathcal{F}_{\delta}(x)$ from early times (super-Hubble scales) to late times (small scales), for the resonant case of $k_{in}/k=u=v=\sqrt{3}/2$. For large $x$ the power in the generated GW continues to grow logarithmically.
• Figure 4: The power at late times, $x=10^{20}$, for all scales, showing a resonance at $u=\sqrt{3}/2$ and a power-law tail (due to modes on larger scales being uncorrelated). Detail of the resonance is shown in the inset revealing a wobbly structure.
• Figure 5: $\mathcal{F}_{n_{s}}(x)$ for a scale invariant input power spectrum. For this case the power spectrum of gravitational waves is also scale invariant owing to the scale invariant conversion factor between $\mathcal{F}_{n_{s}}$ and $\mathcal{P}_{h}$ given by Eq. (\ref{['ph-fn']}).