Improved Calculation of the Primordial Gravitational Wave Spectrum in the Standard Model

TL;DR

The paper addresses how the primordial gravitational wave spectrum deviates from a simple scale-invariant form due to the evolution of the effective relativistic degrees of freedom $g_*(T)$ and neutrino free-streaming. It develops a transfer-function formalism for tensor modes, presents both heuristic and rigorous treatments of how $g_*$ changes affect the expansion rate and horizon re-entry, and computes the spectrum by numerically solving the GW equation with Standard Model data and neutrino damping. The main findings reveal distinct spectral features at frequencies corresponding to the QGP transition and electron–positron annihilation, as well as neutrino-induced damping and possible oscillatory imprints from rapid $g_*$ variations; these features could serve as probes of beyond-Standard-Model physics and the reheating epoch. Overall, the work provides a robust, SM-consistent prediction for the primordial GW spectrum and outlines how future detectors could leverage these features to constrain high-energy physics and the early-universe thermal history.

Abstract

We show that the energy density spectrum of the primordial gravitational waves has characteristic features due to the successive changes in the relativistic degrees of freedom during the radiation era. These changes make the evolution of radiation energy density deviate from the conventional adiabatic evolution, ρ_r~ a^{-4}, and thus cause the expansion rate of the universe to change suddenly at each transition which, in turn, modifies the spectrum of primordial gravitational waves. We take into account all the particles in the Standard Model of elementary particles. In addition, free-streaming of neutrinos damps the amplitude of gravitational waves, leaving characteristic features in the energy density spectrum. Our calculations are solely based on the standard model of cosmology and particle physics, and therefore these features must exist. Our calculations significantly improve the previous ones which ignored these effects and predicted a smooth, featureless spectrum.

Figures (11)

• Figure 1: The primordial gravitational wave spectrum at present, $\tau=\tau_0$, as a function of the comoving wavenumber, $k$ (or $kc$ in units of Hertz). The frequency of gravitational waves observed today is related to $k$ by $f_0=kc/2\pi$. The spectrum at large wavenumber is exactly scale-invariant as we have assumed de-Sitter inflation. In this figure we have not taken into account the effects of the change in effective relativistic degrees of freedom or neutrino free-streaming.
• Figure 2: Evolution of the effective number of relativistic degrees of freedom contributing to energy density, $g_*$, as a function of temperature. The solid and dashed lines represent $g_*$ in the Standard Model and in the minimal extension of Standard Model, respectively. At the energy scales above $\sim 1$ TeV, $g_*^{\rm{SM}}=106.75$ and $g_*^{\rm{MSSM}}\simeq 220$. At the energy scales below $\sim 0.1$ MeV, $g_*=3.3626$ and $g_{*s}=3.9091$; $g_*=g_{*s}$ otherwise.
• Figure 3: Evolution of $a'$ as a function of the conformal time. If $g_*$ and $g_{*s}$ were constant, $\rho\propto a^{-4}$ and $a'$ would also be constant.
• Figure 4: The primordial gravitational wave spectrum at present, $\Omega_h(\tau_0,k)/10^{-10}$, as a function of the comoving wavenumber, $k$ (or $kc$ in units of Hertz). The frequency of gravitational waves observed today is related to $k$ by $f_0=kc/2\pi$. We have assumed a scale-invariant primordial spectrum and $\Omega_m=1-\Omega_r$, $\Omega_r=4.15\times 10^{-5}h^{-2}$, $h=0.7$, and $E_{\rm{inf}}=10^{16}$ GeV. We have included the effects of the effective number of relativistic degrees of freedom and neutrino free-streaming. The dashed line shows the envelope of the previous calculations which ignored the change in the number of relativistic degrees of freedom and neutrino free-streaming (Fig. \ref{['fig1']}).
• Figure 5: A blow-up of Fig. \ref{['fig4']}. Note that density of vertical lines shows density of sampling points at which we evaluate $\Omega_h(\tau_0,k)$. The dashed line shows the envelope of the spectrum in the Standard Model of elementary particles.