Gravitational Wave Spectrum Induced by Primordial Scalar Perturbations

TL;DR

The paper demonstrates that the observed nearly scale-invariant primordial scalar perturbations generate a calculable second-order gravitational-wave spectrum through general relativity, independent of the mechanism that produced the scalars. It derives the evolution equations, constructs the source term from first-order scalars, and provides analytic estimates for horizon crossing and a nontrivial transfer function, all of which are confirmed by exact numerical integration. The resulting spectrum is nearly scale-invariant on small scales with a pronounced feature near the matter-radiation equality horizon, and it can dominate over first-order tensor modes in certain cosmologies (e.g., ekpyrotic/cyclic). This scalar-induced background constitutes a model-independent lower bound on early-Universe GWs and has clear implications for interpreting future GW observations and constraining cosmological history.

Abstract

We derive the complete spectrum of gravitational waves induced by primordial scalar perturbations ranging over all observable wavelengths. This scalar-induced contribution can be computed directly from the observed scalar perturbations and general relativity and is, in this sense, independent of the cosmological model for generating the perturbations. The spectrum is scale-invariant on small scales, but has an interesting scale-dependence on large and intermediate scales, where scalar-induced gravitational waves do not redshift and are hence enhanced relative to the background density of the Universe. This contribution to the tensor spectrum is significantly different in form from the direct model-dependent primordial tensor spectrum and, although small in magnitude, it dominates the primordial signal for some cosmological models. We confirm our analytical results by direct numerical integration of the equations of motion.

Figures (5)

• Figure 1: Spectra of first- and second-order gravitational waves: This schematic illustrates the conjectured form of $\Omega_{\rm GW}(k)$, the fraction of the critical density in gravitational waves per log-interval of wavenumber $k$, as derived in section \ref{['sec:analytic']}. The topmost curve represents the typical first-order inflationary tensor spectrum. With fine-tuning, it can be suppressed below the level of the second-order, scalar-induced tensor perturbations (bottom curves). The bottom curves represent a sequence of times: matter-radiation equality ($a_{\rm eq}$), redshift $z=100$, and today ($a_0$). The scalar-induced tensor spectra shown here are for a perfectly scale-invariant scalar input spectrum ($n_s = 1$). If the scalar spectrum is blue ($n_s > 1$) the induced tensor spectrum is enhanced on small scales (large $k$), while a red spectrum ($n_s < 1$) suppresses tensor fluctuations on small scales (see section \ref{['sec:discussion']} for cautionary remarks about extrapolating spectra to very small scales using the large-scale power law form of the scalar spectrum). $\Omega_{\rm GW}$ is of course ill-defined on superhorizon scales. On superhorizon scales (dashed lines) we therefore formally define the rescaled tensor power spectrum, $k^2 P_h(k)$, but do not interpret it as an energy density of gravitational waves (see section \ref{['sec:analytic']}).
• Figure 2: Evolution of scalar source and induced gravitational waves. Second-order tensors, $h$, are generated when the mode $k$ enters the horizon at $a_k$. If horizon entry occurs during the radiation dominated era, then the scalar source decays as $a^{-\gamma}$ until matter-radiation equality, $a_{\rm eq}$. During matter domination the scalar source terms remains at a constant value, ${\cal S}^{\rm (f)}$. Gravitational waves redshift like $a^{-1}$ as long as $h > {\cal S}^{\rm (f)}/k^2$, but remain at a constant amplitude maintained by the constant source term after that, $a > a^*_k$.
• Figure 3: Numerical spectra of scalar-induced gravitational waves (lower curves) and the scale-invariant primordial tensor spectrum for an inflationary model with tensor-to-scalar ratio $r=0.1$ (upper curve). The scalar-induced spectra are shown at three different epochs, $z+1= 3400, 100,$ and $1$. Each curve has been extended, for pedagogical reasons, to modes with small wavenumbers $k$ that lie outside the horizon at the given epoch (dotted range of the three lower curves). Note that current ($z+1=1$) scalar-induced contributions cross the primordial inflationary contribution at intermediate wavelengths, as suggested by the schematic in Figure \ref{['fig:1']}. The simulation assumes a flat $\Lambda$CDM cosmology with the following model parameters: $\Delta_{\cal R}^2(k_0=0.002\, {\rm Mpc}^{-1}) = 2.4 \times 10^{-9}$, $n_s = 1$, $n_t=0$, $r=0.1$, $\Omega_b h^2 = 0.022$, $\Omega_m h^2 =0.11$, $h=0.7$.
• Figure 4: Observational prospects. Shown are the theoretical predictions for the present spectrum of primordial inflationary gravitational waves and scalar-induced gravitational waves, as well as current (solid bars) and future (dashed bars) experimental bounds (figure modified from Latham). The range of amplitudes for the primordial tensors corresponds to minimally tuned models of inflation as described in Latham. With more fine-tuning this signal can be suppressed. In contrast, the amplitude of the scalar-induced tensors is fixed by the observed amplitude of scalar fluctuations and therefore provides an absolute lower limit on the stochastic gravitational wave background. The CMB constraints depend on assumptions about the transfer function for gravitational waves to extrapolate constraints obtained at decoupling to the current spectrum. Since second-order gravitational waves do not redshift on CMB scales, the CMB observations imply separate constraints on the current first- and second-order spectra. These constraints are labeled CMB(1) and CMB(2), respectively. The dashed section of the scalar-induced tensor spectrum illustrates extrapolation from CMB to direct-detection scales using a scale-invariant scalar spectrum ($n_s=1$). Important uncertainties in the extrapolation between CMB and BBO scales are discussed in the main text.
• Figure 5: Transfer functions for $\Phi$ and $\Psi$. Neutrino anisotropic stress leads to ${\cal O}(10)$% difference between $\Phi$ and $\Psi$Hu:1995fq.