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

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.

• The paper presents a two-scalar-field model that unifies inflation, dark matter, and dark energy while computing its stochastic gravitational-wave spectrum.
• It employs the continuous Bogoliubov coefficient formalism to track graviton production across cosmic epochs, ensuring smooth transitions between inflationary and post-inflationary phases.
• The analysis highlights robust parameter dependencies, offering clear predictions within the sensitivity range of forthcoming gravitational-wave observatories.

Gravitational Waves from Two Scalar Fields Unifying the Dark Sector with Inflation

Introduction and Theoretical Framework

This paper presents a comprehensive analysis of the stochastic gravitational-wave background generated in a cosmological model featuring two scalar fields, designed to unify primordial inflation, dark matter, and dark energy within a single theoretical framework. The model is constructed from an action inspired by extensions of general relativity and string-theoretic scenarios, incorporating a non-minimal interaction between the scalar fields $\phi$ and $\xi$, while both remain minimally coupled to gravity. The potential $V(\xi)$ is chosen to be quadratic, $V(\xi) = V_a + \frac{1}{2} m^2 \xi^2$, facilitating a warm inflationary phase and subsequent transitions to radiation, matter, and dark energy domination.

The dynamical equations are derived for both the inflationary and post-inflationary epochs, with dissipation coefficients $\Gamma_\xi$ and $\Gamma_\phi$ parameterized by temperature-dependent power laws and exponential suppression, enabling a smooth transition from inflation to radiation domination. The model parameters $\alpha$ and $\beta$ are constrained via MCMC analysis against cosmological data, yielding $\alpha = 0.36^{+0.18}_{-0.26}$ and $\beta = 0.01^{+0.34}_{-0.24}$, with the base scenario adopting their mean values.

Continuous Bogoliubov Coefficient Formalism

The gravitational-wave spectrum is computed using the continuous Bogoliubov coefficient formalism, which tracks the evolution of graviton creation and annihilation operators in an expanding Universe. This approach avoids the need for sudden transitions between cosmological epochs and provides a unified framework for calculating the full spectrum of gravitational waves. The tensor perturbations to the FLRW metric are expanded in plane waves, and the mode functions $\chi$ satisfy a parametric oscillator equation. The Bogoliubov coefficients $\alpha_k$ and $\beta_k$ evolve according to a coupled system of ODEs, with initial conditions corresponding to the Bunch–Davies vacuum.

The number of gravitons produced is given by $|\beta_k|^2$, and the spectral energy density parameter is

$$\Omega_{GW}(\omega_0) = \frac{8\hbar G}{3\pi c^5 H_0^2} \omega_0^4 (|\beta_k|^2)_0$$

where $\omega_0$ is the present-day angular frequency. The system is reformulated in terms of the variable $u = -\ln(a_0/a)$ to match the cosmological evolution equations.

Gravitational-Wave Spectrum: Numerical Results

The gravitational-wave energy spectrum is obtained by numerically integrating the Bogoliubov equations across the full range of allowed frequencies, from $\omega_{\min} \sim 10^{-17}$ rad/s (corresponding to the present Hubble radius) to $\omega_{\max} \sim 10^9$ rad/s (set by the Hubble scale at the end of inflation).

Figure 1: The minimum angular frequency of a gravitational wave corresponds to a wavelength equal, today, to the Hubble distance, $$\omega_{\min} = 2\pi c/d_{\mathrm{Hub}}(u_0) e^{u_0}$$.

The numerical analysis reveals that the dominant production of gravitational waves occurs during the inflationary epoch, with subdominant features arising at the transitions between inflation and radiation, radiation and matter, and matter and dark energy. The evolution of $|\beta_k|^2$ as a function of $u$ demonstrates copious graviton production during inflation, with negligible generation during the radiation-dominated era.

Figure 2: Evolution of $|\beta_k|^2$ as a function of $u$ for the base scenario and $\omega_0 = 10^{-16}$ rad/s, highlighting graviton production during inflation and transitions.

The full gravitational-wave energy spectrum for the base scenario is superimposed on the sensitivity curves of planned next-generation detectors (LISA, BBO, CE, ET, SKA, IPTA, DECIGO). The spectrum exhibits a broad peak in the $10^{-2}$–$1$ Hz range, accessible to BBO and DECIGO, with marginal detectability at lower frequencies by SKA.

Figure 3: Full gravitational-wave energy spectrum for the base scenario, compared with sensitivity curves of future detectors.

Parameter Dependence and Model Robustness

The paper systematically explores the dependence of the gravitational-wave spectrum on the model parameters $\alpha$, $\beta$, $p$, and $q$. The spectra remain qualitatively similar across the parameter space, with quantitative differences in $\Omega_{GW}$ limited to less than an order of magnitude. The parameters $\beta$ and $p$ have the most significant impact, as they directly affect the inflationary dynamics and energy transfer rates, while $\alpha$ and $q$ play a secondary role.

Figure 4: Full gravitational-wave energy spectra for different parameter choices, illustrating robustness and sensitivity to $\alpha$, $\beta$, $p$, and $q$.

The envelope of all spectra corresponding to the considered parameter sets is shown, confirming the model's predictions are robust against reasonable variations in the underlying parameters.

Implications and Future Directions

The results demonstrate that multi-field cosmological models, such as the two-scalar-field scenario analyzed here, introduce additional dynamical degrees of freedom and new channels for gravitational-wave production, including parametric resonance, isocurvature-to-curvature conversion, and nonlinear emission from quasi-particles. The predicted gravitational-wave background provides a testable signature for future detectors, offering a means to discriminate between single-field and multi-field inflationary models and to constrain the physics of the dark sector.

The analysis is computationally intensive but tractable, and the predictions are stable under variations in cosmological parameters such as $H_0$. The approach can be extended to include more general potentials, couplings, and reheating dynamics, as well as higher-order quantum corrections and non-Gaussian features in the gravitational-wave spectrum.

Conclusion

This work provides a detailed calculation of the stochastic gravitational-wave background in a two-scalar-field cosmological model unifying inflation, dark matter, and dark energy. The continuous Bogoliubov coefficient formalism enables a precise determination of the gravitational-wave spectrum across all cosmological epochs. The dominant contribution arises from inflation, with subdominant features at cosmic transitions. The predicted spectrum is robust against parameter variations and falls within the sensitivity range of future gravitational-wave observatories, particularly BBO and DECIGO. These results underscore the potential of gravitational-wave astronomy to probe multi-field cosmological models and the physics of the early Universe, motivating further theoretical and observational investigations.