Probing the early universe with inflationary gravitational waves

TL;DR

The paper develops a comprehensive tensor transfer function $T_h(k,\tau)=C_1 C_2 C_3$ to translate the inflationary primordial tensor spectrum into the present-day GWB, explicitly incorporating time-varying dark energy w(z), tensor anisotropic stress from free-streaming relativistic particles, and nonstandard radiation-era physics. It decomposes the transfer function into redshift-suppression ($C_1$), horizon-crossing ($C_2$), and damping due to anisotropic stress ($C_3$), deriving analytic expressions and highlighting non-stationary phase coherence in the GWB. The work shows how post-inflationary effects can imprint on interferometer scales while leaving CMB scales relatively unaffected, enabling a probe of the primordial dark age between inflation and the electroweak transition. It also discusses implications for future missions like BBO/DECIGO, including how different observational outcomes could validate inflation, constrain post-inflationary physics, or challenge the inflationary consistency relations.

Abstract

Near comoving wavenumber k, the gravitational-wave background (GWB) from inflation carries information about the physical conditions near two moments in cosmic history: the moment when k ``left the horizon'' during inflation, and the moment when it ``re-entered the horizon'' after inflation. We investigate the extent to which this information can be extracted if the GWB is measured by a combination of cosmic-microwave-background (CMB) polarization experiments on large scales and space-based laser-interferometer experiments on small scales. To disentangle this information, we derive a new gravitational-wave transfer function that incorporates a number of physical effects that were treated less accurately, less generally, or were missing altogether in previous treatments. In particular, it incorporates: (i) dark energy with time-varying equation-of-state w(z); (ii) tensor anisotropic stress due to free-streaming relativistic particles in the early universe; and (iii) a variety of physical effects that cause deviations from the standard equation-of-state w=1/3 during the radiation era. Based on this transfer function, we consider the degree to which the GWB can be used to test inflation and to probe the ``primordial dark age'' between the end of inflation and the electroweak phase transition.

Figures (3)

• Figure 1: $C_{3}$ is the transfer function factor that accounts for the damping of the tensor power spectrum due to tensor anisotropic stress. The factor depends on the fraction $f$ of the background (critical) energy density contained in free-streaming relativistic particles. The figure plots this dependence for $0<f<1$.
• Figure 2: The black solid curve represents the present-day gravitational-wave energy spectrum, $\Omega_{gw}^{}(f,\tau_{0})$, for the inflationary model $V(\phi)=(1/2)m^{2}\phi^{2}$. The red dotted curve shows the damping effect due to (three ordinary massless species of) free-streaming neutrinos. The green dot-dashed curve shows the damping effect which arises if free-streaming particles make up fifty percent of the background energy density at the time $\tau_{BBO}^{}$ when the modes probed by BBO/DECIGO re-enter the horizon. As shown in the figure, the particles begin free-streaming sometime before $\tau_{BBO}^{}$, and decay sometime after $\tau_{BBO}^{}$, but prior to electroweak symmetry breaking. Finally, the blue dashed curve shows the effect of a conformal anomaly in the early universe that slightly reduces the equation of state from $w=0.33$ to $w=0.31$ above the electroweak phase transition. The spectrum will also be modified on comoving scales that re-enter the horizon during the reheating epoch after inflation; but the range of scales affected by reheating is unknown. Finally, note that the correlated BBO interferometer proposal claims a sensitivity that extends beyond the bottom of the figure (down to roughly $\Omega_{gw}^{}\sim10^{-17}$) in the frequency range from $10^{-1}$ Hz to $10^{0}$ Hz.
• Figure 3: The precise predictions of the primordial tensor power spectrum for two inflationary models whose parameters have been chosen so that they make identical predictions for the scalar fluctuation amplitude and spectral index on CMB scales. The two models are the quartic monomial potential $V(\phi)=\lambda\phi^{4}$ (top), and the axion (or "natural inflation") potential $V(\phi)=M_{I}^{4} [1+{\rm cos}(\phi/\mu)]$.