Gravitational waves from stochastic scalar fluctuations

We present a novel mechanism for gravitational wave generation in the early Universe. Light spectator scalar fields during inflation can acquire a blue-tilted power spectrum due to stochastic effects. We show that this effect can lead to large curvature perturbations at small scales (induced by the spectator field fluctuations) while maintaining the observed, slightly red-tilted curvature perturbations at large cosmological scales (induced by the inflaton fluctuations). Along with other observational signatures, such as enhanced dark matter substructure, large curvature perturbations can induce a stochastic gravitational wave background (SGWB). The predicted strength of SGWB in our scenario, ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2\simeq {10}^{-20}–{10}^{-15}$, can be observed with future detectors, operating between ${10}^{-5}\mathrm{Hz}$ and 10 Hz. We note that, in order to accommodate the newly reported NANOGrav observation, one could consider the same class of spectator models. At the same time, one would need to go beyond the simple benchmark considered here and consider a regime in which a misalignment contribution is also important.

Figure 1

Schematic of the mechanism. The comoving horizon $1/(aH)$ decreases during inflation and increases after that. Any $k$-mode carries a fluctuation of order $H/(2\pi )$ at the time of mode exit. However, modes with larger $k$ (red) exit the horizon later and encounters less dilution compared to modes with smaller $k$ (blue), since $t_{*}>{\tilde{t}}_{*}$. Consequently, modes with larger $k$ source stronger gravitational waves upon horizon reentry (shown via square box). We also depict the fact that $\sigma$ carries an energy density $\propto H^4$ during inflation, and dilutes as matter (for our benchmark choices) after inflation ends.

Figure 2

Time evolution of scalar field energy density ${\rho }_{\sigma }(t)$. In scenarios where the quartic term dominates the initial evolution (dashed red), the field dilutes as radiation (dot-dashed olive), ${\rho }_{\sigma }(t)\propto 1/a(t{)}^4$. Eventually, the mass term becomes important, and the behavior becomes ${\rho }_{\sigma }(t)\propto 1/a(t{)}^3$. The benchmark choices in this work will mimic the blue curve where the evolution of ${\rho }_{\sigma }(t)$ is always dominated by the mass term with a matter-like dilution. For both the blue and the red curves, $t=1$ corresponds to the moment when the Hubble scale is approximately equal to the effective mass and the field starts oscillating.

Figure 3

Power spectrum of curvature perturbations for the benchmarks discussed above. Stochastic effects lead to a blue-tilted spectrum of $\sigma$, with larger $m$ and $\lambda$ corresponding to larger tilts, leading to faster decay as $k$ gets smaller. The blue-tilt is eventually cut off at $k_d$, the $k$-mode that reenters the horizon at the time of $\sigma$ decay. For $k$ larger than $k_d$, the fractional energy density in $\sigma$ at the time of mode-reentry is smaller. Correspondingly, ${\mathrm{\Delta }}_{\zeta }^2$ gets suppressed. Eventually, for very large $k$, the effects of $\sigma$ become negligible, and ${\mathrm{\Delta }}_{\zeta }^2$ reverts back to its standard, slightly red-tilted behavior. A smaller value of $f_{\sigma }(k_d)$, the fractional energy density at the time $\sigma$ decay, suppresses the effect of $\sigma$ to ${\mathrm{\Delta }}_{\zeta }^2$, and hence leads to a suppressed peak. This mechanism predicts signatures in CMB spectral distortion measurements [48], especially in Super-PIXIE [49], along with Pulsar Timing Array (PTA) probes for enhanced DM substructure [50], and precision astrometry probes (AstroM) [51]. We also show constraints from FIRAS [52] and nonobservation of primordial black holes (PBH) [5].

Figure 4

(a) The kernel function from Eq. (76). We note a clear resonance contribution from $t\simeq 0.7$ corresponding to $u+v\simeq \sqrt{3}$. (b) The transfer function $T_{\mathrm{\Phi }}$. (c) Function $f(p,q,\tau )$ as in Eq. (71). We see that for the scalar modes that enter the horizon earlier, with $p,q>k$, this function is more suppressed as expected from the behavior of the transfer function.

Figure 5

Gravitational wave spectrum for the benchmarks discussed in Fig. 3. We notice that the number of $e$-folds after CMB-observable modes exited the horizon determines the peak frequency of the spectrum, and correspondingly, different detectors can be sensitive to the signal. Although a similarly peaked spectrum would appear in the context of cosmological phase transitions (PT), the low-frequency tail of this GW spectrum is different from the usual $f^3$ tail. While in the context of PT the $f^3$ scaling originates due to causality and superhorizon behavior of fluctuations, in our scenario, the $f$-scaling is determined by $\sigma$ mass. The differing frequency dependence can then be used to discriminate between the two classes of signals.