Gravitational waves from two scalar fields unifying the dark sector with inflation

TL;DR

The paper examines a two-scalar-field cosmological model that unifies inflation with the dark sector (dark matter and dark energy) and computes the full gravitational-wave background generated throughout cosmic history using the continuous Bogoliubov-coefficient formalism. It shows that inflation dominates the GW production, with secondary contributions arising during the transitions between inflation, radiation, matter, and dark-energy domination. The resulting spectrum, expressed as $\Omega_{GW}(f)$, is compared to projected sensitivities of next-generation detectors (e.g., LISA, BBO, CE, ET, DECIGO), highlighting frequency windows (notably around $0.01$–$1$ Hz) where the signal could be detectable. The study demonstrates how multi-field and modified-gravity cosmologies imprint distinctive GW spectra, offering a potential observational probe to constrain early-Universe dynamics and dark-sector physics. Future work could extend perturbation analysis, reheating details, and non-Gaussian signatures to further sharpen detector prospects.

Abstract

We investigate the gravitational-wave background predicted by a two-scalar-field cosmological model that aims to unify primordial inflation with the dark sector, namely late-time dark energy and dark matter, in a single and self-consistent theoretical framework. The model is constructed from an action inspired by several extensions of general relativity and string-inspired scenarios and features a non-minimal interaction between the two scalar fields, while both remain minimally coupled to gravity. In this context, we derive the gravitational-wave energy spectrum over wavelengths ranging from today's Hubble horizon to those at the end of inflation. We employ the continuous Bogoliubov coefficient formalism, originally introduced to describe particle creation in an expanding Universe, in analogy to the well-established mechanism of gravitational particle production and, in particular, generalized to gravitons. Using this method, which enables an accurate description of graviton creation across all cosmological epochs, we find that inflation provides the dominant gravitational-wave contribution, while subdominant features arise at the inflation-radiation, radiation-matter, and matter-dark energy transitions, i.e., epochs naturally encoded inside our scalar field picture. The resulting energy density spectrum is thus compared with the sensitivity curves of the planned next-generation ground- and space-based gravitational-wave observatories. The comparison identifies frequency bands where the predicted signal could be probed, providing those windows associated with potentially detectable signals, bounded by our analyses. Consequences of our recipe are thus compared with numerical outcomes and the corresponding physical properties discussed in detail.

Figures (6)

• Figure 1: Evolution of the dissipation ratio $Q$ during the inflationary period, which extends from $u\approx-135.3$ to $u\approx-65.3$ ($70$ e-folds of expansion), and the transition from inflation to the radiation-dominated era, which extends from $u\approx-65.3$ to $u\approx-64.3$. Throughout the entire inflationary period, a strong dissipative regime is maintained. At the end of this period, the dissipation coefficients $\Gamma_\xi=\Gamma_\phi$ are exponentially suppressed, resulting in a sharp decrease of $Q$.
• Figure 2: Evolution of the ratios $T/H$, $\rho_\texttt{R}/\rho_\xi$, and $\rho_\texttt{R}/\rho_\phi$ during the inflationary period. The end of inflation occurs when $\rho_\texttt{R}/\rho_\xi=1$. From then on, radiation dominates the Universe's evolution.
• Figure 3: The minimum angular frequency of a gravitational wave corresponds to a wavelength equal, today, to the Hubble distance, $\omega_\texttt{min} = 2\pi c/d_\texttt{Hub}(u_0)e^{u_0} \approx 2\pi H_0=10^{-16.86}\, \hbox{rad/s}$, where the present value of the Hubble parameter $H_0\equiv H(u_0=0)$ is taken to be $2.17\times10^{-18}\, \hbox{s}^{-1}$ in our base scenario. The maximum angular frequency corresponds to a wavelength equal to the Hubble distance at the end of the inflationary period $u_\texttt{inf}$, afterwards redshifted by the expansion of the Universe, $\omega_\texttt{max} \approx 2\pi H(u_\texttt{inf}) e^{u_\texttt{inf}} = 10^{9.20} \, \hbox{rad/s}$ for the base scenario.
• Figure 4: Evolution of $|\beta_k|^2$ as a function of $u$ for the base scenario and $\omega_0=10^{-16}$ rad/s. During inflation, which lasts from $u=-135.3$ to $u=-65.3$, gravitational waves are generated copiously. During the transition from inflation to the radiation-dominated era (lower inset) and, as well as, during the transition to a matter-dominated and then dark energy-dominated eras (upper inset), more gravitational waves are produced, but incomparably fewer than during the inflationary period. During the radiation-dominated era, no gravitational waves are produced.
• Figure 5: Full gravitational-wave energy spectrum for the base scenario of the two-scalar-field cosmological model, superimposed on the sensitivity curves of planned next-generation ground- and space-based gravitational-wave detectors, LISA, BBO, CE, ET, SKA, IPTA, and DECIGO Schmitz:2020syl (for the repository with the sensitivity curves see Ref. repository). While the main plot displays the energy spectrum obtained for the values of $\alpha$ and $\beta$ adopted in the base scenario, the inset shows the envelope containing the spectra generated when the error bars for these parameters are taken into account (see Eqs. \ref{['v-alpha']} and \ref{['v-beta']}).