Probing non-Gaussianity during reheating with SIGW in the LISA band

Abstract: We analyse the effects of a non-standard evolution of the Universe during the reheating epoch on the spectrum of scalar-induced gravitational waves (SIGWs) accounting for the presence of primordial non-Gaussianity. We show that given values of $w$ and $c_s<sup>2$ leave characteristic features in the spectrum which can be detectable by third generation interferometers like LISA. In addition, we argue that the specific reheating dynamics can suppress or even enhance the spectrum, with crucial consequences for its detectability. We perform a Fisher forecast for different values of $w$ and different scans to assess the detectability of the signal when different values of the amplitude and central frequency are considered.

• The paper presents a SIGW formalism that incorporates arbitrary reheating equations of state and primordial non-Gaussianity up to the trispectrum level.
• The analysis demonstrates that variations in the sound speed and EoS critically affect the IR slope and resonance peak of the SIGW spectrum, with Fisher forecasts predicting precise constraints.
• The study highlights that enhanced GW amplitudes for w > 1/3 enable robust separation of non-Gaussian and reheating effects, opening avenues for early-universe exploration.

Probing Primordial Non-Gaussianity during Reheating with Scalar-Induced Gravitational Waves in the LISA Band

Introduction

This work ("Probing non-Gaussianity during reheating with SIGW in the LISA band" (2604.08493)) presents a comprehensive analysis of how non-Gaussianity (NG) and the post-inflationary expansion history — specifically, the equation of state (EoS) and sound speed during reheating — imprint on the scalar-induced gravitational wave (SIGW) spectrum in the millihertz regime targeted by LISA and similar third-generation interferometers. The investigation goes significantly beyond the standard scenario (radiation domination, $w=1/3$), generalizing the SIGW calculation to arbitrary $w \in [0,1]$ and $c_s^2$, including primordial local-type NG. The detectability prospects of these signals, and the associated capability to probe microphysics in the early universe via LISA, are quantified with Fisher forecasts over a broad parameter space.

Scalar-Induced GW Formalism with General Expansion History and Non-Gaussianity

The paper develops the formalism for the calculation of the SIGW spectrum induced at second order by enhanced primordial scalar perturbations, accounting for general $w$ and $c_s^2$ during a reheating stage of arbitrary duration. The cosmic background is parametrized by $w$ and $c_s^2$, which can range from near matter-dominated, through standard radiation, up to stiff or kinetic-dominated phases. Several well-motivated microphysical models are mapped to benchmark $w$ values: e.g. cannibal dark matter ($w\approx0.1$), post-chaotic inflation preheating ($w\sim0.2-0.3$), power-law inflaton oscillations ($w \in [0,1]$0), and quintessential or stiff phases ($w \in [0,1]$1).

Crucially, the evolution of the scalar sources and the corresponding second-order tensor modes, and the matching conditions at the radiation-reheating transition, are treated analytically with constant $w \in [0,1]$2 and $w \in [0,1]$3. The impact of primordial NG, modeled via local $w \in [0,1]$4, is incorporated up to the trispectrum level, and the full expression for the SIGW energy density, including all Gaussian and non-Gaussian connected/disconnected contributions, is provided.

Spectral Signatures of Reheating Parameters and Non-Gaussianity

Numerical evaluation of the SIGW spectrum reveals several key dependencies:

• The infrared (IR) slope is highly sensitive to $w \in [0,1]$5, controlled by an analytic exponent $w \in [0,1]$6 or $w \in [0,1]$7 with $w \in [0,1]$8, and is independent of $w \in [0,1]$9. The IR scaling, if reconstructed, constitutes a robust probe of the EoS during reheating.
• The location and prominence of the resonance peak reflect $c_s^2$0: larger sound speeds shift the peak to higher frequencies and decrease its sharpness, with the resonance disappearing for $c_s^2$1.
• As $c_s^2$2 increases above $c_s^2$3, the overall amplitude of the SIGW spectrum is enhanced due to less dilution of GW energy, while for $c_s^2$4 the spectrum is suppressed.

Figure 1: SIGW spectral shapes for different values of $c_s^2$5 and $c_s^2$6. The amplitude and structure of Gaussian and non-Gaussian contributions are displayed, with the IR analytic behavior (dashed black) indicated for reference.

The inclusion of local primordial NG modifies the spectral shape in the following ways:

• Non-Gaussian connected trispectrum components can introduce additional peaks (notably in the UV), and contribute negative-valued portions due to the sign structure of the generalized kernel, though the total GW energy density remains manifestly positive.
• For fixed $c_s^2$7 and $c_s^2$8, the interplay of NG amplifies distinctive features enabling separation of the background EoS and NG signatures, provided sufficient experimental precision.

Figure 2: Left: Gaussian SIGW spectra for varying $c_s^2$9, demonstrating systematic evolution of shape. Right: Variation with $w$0 at fixed $w$1; even in the absence of resonant features, $w$2 imprints observable deviations.

Detectability Forecasts and Parameter Constraints with LISA

A suite of Fisher matrix forecasts is performed, assuming a lognormal input primordial spectrum and four years of LISA integration time. Analyses span the $w$3 parameter space for three representative reheating scenarios ($w$4, $w$5, $w$6).

• For $w$7 ($w$8), the SIGW signal is suppressed compared to radiation domination, degrading the achievable constraints on all parameters unless the amplitude is very large. Parameter degeneracies remain prominent, particularly involving $w$9, $c_s^2$0, and $c_s^2$1, while the IR tail still offers leverage for $c_s^2$2.

Figure 3: Fisher forecast for $c_s^2$3, showing weak parameter constraints and strong degeneracies, due to suppressed GW production during this reheating scenario.

• For $c_s^2$4 ($c_s^2$5), the GW amplitude is enhanced, and all parameters, including $c_s^2$6 and $c_s^2$7, can in principle be determined to high fidelity (fractional errors below $c_s^2$8 or better), even for moderate primordial amplitudes. The boost grows with larger $c_s^2$9 and lower reheating transition frequency.

Figure 4: Fisher forecast for $w$0, showing greatly improved constraints as GWs are enhanced for such expansion histories.

Figure 5: Fisher forecast for $w$1, illustrating even more precise reconstruction enabled by the significant GW amplitude boost in this regime.

The Signal-to-Noise Ratio (SNR) is mapped over the full $w$2 plane for each $w$3. The detectability region, subject to Big Bang Nucleosynthesis constraints, is maximized for large $w$4 and optimal reheating frequencies.

Figure 6: SNR scan in the $w$5 parameter space for $w$6 with fixed $w$7. The GW signal detectability is delineated for multiple reheating scenarios, with BBN limits shaded.

Implications and Outlook

This study demonstrates that SIGW observations in the LISA band can constrain the EoS and NG of the primordial universe during reheating. The spectral features induced by $w$8 and $w$9 (specifically, the IR tail and resonance structure), and the modulation from $c_s^2$0, provide orthogonal handles for disentangling early-universe physics. Crucially, it is shown that parameter inference for NG is not strongly degenerate with that for $c_s^2$1; allowing $c_s^2$2 to float in the analysis does not erase sensitivity to NG, contrary to some suggestions in the literature.

From a practical perspective, in scenarios where $c_s^2$3, SIGWs can be detected with high SNR for much lower primordial amplitude compared to the standard scenario. This enables SIGW probes even in regimes where primordial black hole (PBH) overproduction is not expected, decoupling GW and PBH observational constraints for a broad class of models.

The theoretical ramifications are notable: the methodology outlined is extendable to arbitrary trispectrum shapes and backgrounds with nontrivial reheating histories, making it a framework for model-independent early-universe inference from GW experiments. Future studies incorporating model-specific NG, varying $c_s^2$4, and extended reheating profiles (rather than sharp transitions) are a natural extension.

Conclusion

The paper delivers a rigorous generalization of the SIGW phenomenology to arbitrary post-inflationary thermal histories and local-type primordial non-Gaussianity, demonstrating that third-generation GW experiments will have the capability to reconstruct key properties of the early universe — EoS, sound speed, and primordial NG — with high precision, contingent on the underlying scenario. The work sets a benchmark for the use of gravitational wave backgrounds in reconstructing microphysics immediately after inflation and provides the theoretical underpinning for SIGW-targeted analyses of future GW data.