Probing Reheating Phase via Non-Helical Magnetogenesis and Secondary Gravitational Waves

TL;DR

The paper examines inflationary magnetogenesis with a non-helical $f^2( au)F_{\mu u}F^{\mu u}$ coupling and analyzes how the reheating phase, parameterized by $w_{ m re}$ and $T_{ m re}$, shapes both primary and secondary gravitational waves and the present-day magnetic field. By quantizing the gauge field and evolving the EM power spectra through inflation and reheating, the authors derive backreaction and strong-coupling bounds that constrain the coupling index $n$ to roughly $-2.2 leq n < 0$, while noting Faraday induction can boost magnetic fields in low-conductivity reheating. They compute SGWs sourced by EM fields across inflation, reheating, and radiation eras, revealing that SGW spectra exhibit distinct breaks and tilts sensitive to $w_{ m re}$ and $n_{ m E}$, with potential detectability by LISA, DECIGO, BBO, SKA, and PTA experiments. Through $r$- and $ riangle N_{ m eff}$-based constraints and PTA-informed MCMC analyses, the work shows that non-instantaneous reheating with negligible conductivity can realize observable magnetogenesis without violating current bounds, linking early-universe dynamics to forthcoming gravitational-wave observations and offering a pathway to probe reheating physics. The results underscore the synergy between inflationary magnetogenesis, reheating history, and gravitational-wave phenomenology as a testbed for high-energy cosmology.

Abstract

In the past two decades, significant advancements have been made in observational techniques to enhance our understanding of the universe and its evolutionary processes. However, our knowledge of the post-inflation reheating phase remains limited due to its small-scale dynamics. Traditional observations, such as those of the Cosmic Microwave Background (CMB), primarily provide insights into large-scale dynamics, making it challenging to glean information about the reheating era. In this paper, our primary aim is to explore how the generation of Gravitational Waves (GWs) spectra, resulting from electromagnetic fields in the early universe, can offer valuable insights into the Reheating dynamics. We investigate how the spectral shape of GWs varies across different frequency ranges, depending on the initial magnetic profile and reheating dynamics. For this, we consider a well-known non-helical magnetogenesis model, where the usual electromagnetic kinetic term is coupled with a background scalar. Notably, for such a scenario, we observe distinct spectral shapes with sufficiently high amplitudes for different reheating histories with the equation of state parametrized by ($w_{\rm re}$). We identify spectral breaks in the GW spectra for both $w_{\rm re}<1/3$ and $w_{\rm re}>1/3$ scenarios. We find that future GW experiments such as BBO, LISA, SKA, and DECIGO are well within the reach of observing those distinct spectral shapes and can potentially shed light on the underlying mechanism of the reheating phase.

Figures (5)

• Figure 1: In this figure, we present the lowest allowed reheating temperature (in GeV) as a function of the reheating equation of state parameter $w_{re}$, over a wide range. The three different colors correspond to three distinct values of the electric spectral index: $n_{\mathrm{E}} = 0.01$ (blue), $n_{\mathrm{E}} = 0.1$ (red), and $n_{\mathrm{E}} = 0.2$ (magenta). Solid lines represent the scenario with zero electrical conductivity (EC), while dashed lines correspond to the high EC limit. Throughout the analysis, we fix the inflationary energy scale at $H_{\rm inf} \simeq 10^{-5} M_{\rm Pl}$.
• Figure 2: In this figure, we show the present-day magnetic field strength evaluated at the scale $1\,\mathrm{Mpc}^{-1}$ as a function of the reheating equation of state parameter $w_{re}$. The three different colors represent three distinct values of the reheating temperature. Solid lines correspond to $n_{\mathrm{E}} = 0.01$, while dashed lines indicate $n_{\mathrm{E}} = 0.20$. The left panel illustrates the case of zero electrical conductivity during reheating, whereas the right panel corresponds to the infinite conductivity limit. In both cases, we impose the condition that the total electromagnetic (EM) energy density remains subdominant to the background energy density throughout reheating, thereby ensuring that the universe does not enter a magnetic field-dominated phase before the onset of Big Bang Nucleosynthesis (BBN).
• Figure 3: In this figure, we plot $\Omega_{\mathrm{gw}}h^2$ versus $f$ (in Hz) for $w_{re} = 0$ for two different scenarios. In the left panel, we show how the present-day GWs strength depends on the magnetogenesis parameter $n_{\mathrm{E}}(=4-2|n|)$ for a fixed reheating temperature $T_{\rm re}=10^3\,\hbox{GeV}$. Here, four different colors indicate four different values os the electric spectral index. Whereas in the right panel, we have shown the dependency of the reheating temperature $T_{\rm re}$, indicated through four different colors with a fixed electric spectral index $n_{\mathrm{E}}=0.6$.
• Figure 4: In the figures above, we have plotted $\Omega_{\mathrm{gw}}h^2$ as a function of frequency $f$(in Hz) for the coupling $n < -1/2$, assuming zero electrical conductivity during the reheating era for $w_{re}=3/5$ (stiff like fluid during reheating). In the left plot, we have shown the dependency of the GW spectrum as a function of the electrical spectral index $n_{\mathrm{E}}$ defined via four different colors for a fixed reheating temperature $T_{\rm re}=10^6\,\ GeV$. In the right panel, we have shown how the present-day GW strength and spectral shape are dependent on the reheating temperature, denoted by four different colors for a fixed electric al spectral index $n_{\mathrm{E}}=0.2$.
• Figure 5: In the figures presented above, we display the posterior distributions of the parameters governing the Stochastic Gravitational Waves (SGWs) induced by non-helical magnetic fields, generated using the PTARcade code 2023arXiv230616377M. The diagonal elements of the corner plot show the 1D marginalized distributions for the parameters $n_{\mathrm{E}}$, $w_{re}$, and $T_{\rm re}$. In the left figure, we consider a prior for the electric spectral index $n_{\mathrm{E}} \in (0,1.0)$, while in the right figure, we consider an extended prior range for $n_{\mathrm{E}} \in (0,1.5)$.