Cosmological Backgrounds of Gravitational Waves

TL;DR

This review surveys the rich landscape of cosmological gravitational-wave backgrounds, detailing their theoretical origins—from quantum vacuum fluctuations during inflation to complex post-inflationary dynamics like preheating, first-order phase transitions, and cosmic defects—and how they imprint stochastic signals across detector bands. It presents a unified framework for GW generation, propagation, and statistical characterization in expanding spacetimes, and maps current observational bounds from BBN, CMB, PTAs, and ground- and space-based interferometers. The authors catalog a broad array of mechanisms that can yield testable signals in the near term (e.g., LISA, PTA networks) or in future facilities (e.g., BBO/DECIGO, ET), highlighting distinctive features such as chiral or non-Gaussian backgrounds and blue-tilted spectra. The work underscores the potential for GW observations to probe high-energy physics inaccessible to colliders, offering a complementary window into the early Universe and fundamental interactions.

Abstract

Gravitational waves (GWs) have a great potential to probe cosmology. We review early universe sources that can lead to cosmological backgrounds of GWs. We begin by presenting definitions of GWs in flat space-time and in a cosmological setting, and discussing the reasons why GW backgrounds from the early universe are of a stochastic nature. We recap current observational constraints on stochastic backgrounds, and discuss some of the characteristics of present and future GW detectors including advanced LIGO, advanced Virgo, the Einstein Telescope, KAGRA, LISA. We then review in detail early universe GW generation mechanisms proposed in the literature, as well as the properties of the GW backgrounds they give rise to. We classify the backgrounds in five categories: GWs from quantum vacuum fluctuations during standard slow-roll inflation, GWs from processes that operate within extensions of the standard inflationary paradigm, GWs from post-inflationary preheating and related non-perturbative phenomena, GWs from first order phase transitions (related or not to the electroweak symmetry breaking), and GWs from topological defects, in particular from cosmic strings. The phenomenology of early universe processes that can generate a stochastic background of GWs is extremely rich, and some backgrounds are within the reach of near-future GW detectors. A future detection of any of these backgrounds will provide crucial information on the underlying high energy theory describing the early universe, probing energy scales well beyond the reach of particle accelerators.

Figures (9)

• Figure 1: Black line: the characteristic GW frequency of Eq. (\ref{['f0']}) as a function of temperature (the corresponding redshift is shown above). Shaded regions: the frequency ranges detectable by several GW experiments, from right to left respectively $1\,{\rm Hz}\lesssim f\lesssim 10^3\,{\rm Hz}$ for ground-based interferometers, $10^{-5}\,{\rm Hz}\lesssim f\lesssim 0.1\,{\rm Hz}$ for LISA, $3\times 10^{-9}\,{\rm Hz}\lesssim f\lesssim 10^{-6}\,{\rm Hz}$ for Pulsar Timing Arrays, and $3.4\times 10^{-19}\,{\rm Hz}\lesssim f\lesssim 7\times 10^{-18}\,{\rm Hz}$ for the CMB.
• Figure 2: The power spectrum of Eq. (\ref{['eq:omegaKnt']}) for $r=0.07$ and $n_T=-r/8$ (black dashed line), and $n_T=0.2$ (black solid line), together with the constraint given by Planck+BICEP2+Keck Array data given in Eq. (\ref{['eq:rbound']}), i.e. setting $r=0.07$ and $n_T=0$ (green solid line), and the reach of current and future GW detectors: PTA (magenta, solid), advanced LIGO at the first run and at design sensitivity (blue, solid) and LISA (orange, solid).
• Figure 3: The Power Law-Integrated Curves of current and future GW detectors. From left to right: for PTA (blue) we show the predictions for the International Pulsar Timing array and for a network monitored by the SKA, taken from Moore:2014lga; for LISA (red), we show the expected power law-integrated curve adapted from Ref. Audley:2017drz; for Advanced LIGO/Virgo (green) we show the sensitivities given in TheLIGOScientific:2016wyq for the first (O1) and second (O2) runs, and at design sensitivity (O5).
• Figure 4: Left panel: the scale factor for a radiation-dominated universe (blue curve), a matter-dominated one (red curve), and a universe performing the transition from radiation to matter domination (black curve). The green, dashed line represents the scale factor at equality $a_{\rm eq}= a_0 (\Omega_{\rm rad}/\Omega_{\rm mat})$. Right panel: the GW energy density power spectrum, normalised to the primordial inflationary spectrum $\mathcal{P}_h(k)$, as a function of normalised wave-number $k/k_{\rm eq}$. The aim is to show the effect of the transfer function when modes enter the horizon (hence the normalisation). The orange curve represents Eq. (\ref{['eq:rhogwT']}), where we have inserted the correct transfer function, found by numerically integrating the GW evolution equation through the radiation-matter transition. The blue curve is the analytical approximation, i.e. Eq. (\ref{['eq:rhogwT']}) inserting Eq. (\ref{['eq:Tfunct']}). The black curve is approximation (\ref{['eq:omegaK']}), where we have accounted for oscillations by inserting a factor $1/2$. The black, dashed curve shows the result of Ref. Turner:1993vb, i.e. using Eq. (\ref{['eq:TWL']}) for the transfer function.
• Figure 5: Scenario of sustained particle production: numerical spectrum of GWs today $h^2\Omega_{\rm GW}$ for a model of quadratic inflaton potential $V(\phi) = {1\over2}m^2\phi^2$, with inflaton - gauge field coupling $\Lambda = M_{Pl} / 35$ (continuous line) versus local parametrization $h^2\Omega_{\rm GW} \propto (f/f_*)^{n_T}$, evaluated at various pivot frequencies $f_*$ and with spectral tilt obtained from successive approximations to $n_T$. This figure is taken from Ref. Bartolo:2016ami, and also shows the Power Law-Integrated Curves of six LISA configurations that were still being considered at that time, c.f. discussion in Bartolo:2016ami.