Figure 1

Spectra of first- and second-order gravitational waves: This schematic illustrates the conjectured form of ${\Omega }_{\mathrm{GW}}(k)$, the fraction of the critical density in gravitational waves per log-interval of wave number $k$, as derived in Sec. 3. 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 ($a_{\mathrm{eq}}$), 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 Sec. 5 for cautionary remarks about extrapolating spectra to very small scales using the large-scale power law form of the scalar spectrum). ${\Omega }_{\mathrm{GW}}$ 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 Sec. 3).Reuse & Permissions

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, $a_{\mathrm{eq}}$. During matter domination the scalar source terms remain at a constant value, ${\mathcal{S}}^{(\mathrm{f})}$. Gravitational waves redshift like $a^{-1}$ as long as $h>{\mathcal{S}}^{(\mathrm{f})}/k^2$, but remain at a constant amplitude maintained by the constant source term after that, $a>a_k^{*}$.Reuse & Permissions

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 wave numbers $k$ 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 Fig. 1. The simulation assumes a flat $\Lambda \mathrm{CDM}$ cosmology with the following model parameters: ${\Delta }_{\mathcal{R}}^2(k_0=0.002{\mathrm{Mpc}}^{-1})=2.4\times {10}^{-9}$, $n_s=1$, $n_t=0$, $r=0.1$, ${\Omega }_b{h}^2=0.022$, ${\Omega }_m{h}^2=0.11$, $h=0.7$.Reuse & Permissions

Figure 4

Observational prospects. Shown are the theoretical predictions for the spectrum of scalar-induced versus primordial inflationary gravitational waves as predicted for the present epoch, along with the current (solid bars) and future (dashed bars) experimental bounds (figure modified from 8). The band marked “primordial tensors” represents the inflationary predictions for minimally tuned models, as classified in 8; with further fine-tuning, this spectrum can be shifted downwards. 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 scalar-induced gravitational waves do not redshift on CMB scales, the CMB observations imply separate constraints on the current primordial and scalar-induced 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.Reuse & Permissions

Figure 5

Transfer functions for $\Phi$ and $\Psi$. Neutrino anisotropic stress leads to $\mathcal{O}(10)\%$ difference between $\Phi$ and $\Psi$ [18].Reuse & Permissions