Constraints on the Primordial Curvature Power Spectrum and Reheating Temperature from the NANOGrav 15-Year Dataset

Abstract

The stochastic signal observed by collaborations such as NANOGrav, PPTA, EPTA +InPTA, and CPTA may originate from gravitational waves induced by primordial curvature perturbations during inflation. This study investigates small-scale properties of inflation and reheating, assuming a log-normal form for the power spectrum of the primordial curvature and a reheating phase equation of state $w=1/9$. Inflation and reheating scenarios are thoroughly examined using Bayesian methods applied to the NANOGrav 15-year dataset. The analysis establishes constraints on the reheating temperature, suggesting $T_{\mathrm{rh}}\gtrsim 0.1\mathrm{Gev}$, consistent with Big Bang nucleosynthesis constraints. Additionally, the NANOGrav 15-year dataset requires the amplitude (A∼0.1) and width ($\mathsf{\Delta }\lesssim 0.001$) of the primordial curvature power spectrum to be within specific ranges. A notable turning point in the energy density of scalar-induced gravitational waves occurs due to a change in the equation of state w. This turning point signifies a transition from the reheating epoch to radiation domination. Further observations of scalar-induced gravitational waves could provide insights into the precise timing of this transition, enhancing our understanding of early Universe dynamics.

1. Introduction

The detection of gravitational waves (GWs) resulting from compact binary mergers by the LIGO–Virgo–KAGRA collaboration [1,2,3] represents a groundbreaking achievement in astrophysics and cosmology, significantly advancing our understanding of the diverse population of GW sources [4,5,6,7,8,9,10,11,12,13,14,15]. This milestone has opened avenues for exploring stochastic GW backgrounds, which promise profound insights into various astrophysical and cosmological phenomena [16,17], such as dark matter [18,19,20] and modified gravity [21,22,23,24,25,26].

Recent observations from multiple pulsar timing array (PTA) collaborations—including at the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) [27,28], Parkes Pulsar Timing Array (PPTA) [29,30], European Pulsar Timing Array (EPTA) in combination with Indian Pulsar Timing Array (InPTA) [31,32], and Chinese Pulsar Timing Array (CPTA) [33]—have revealed a common spectrum signal characterized by the Hellings–Downs angular correlation feature inherent in GWs. This distinctive signal, resembling a fiducial characteristic strain spectrum of $f^{-2/3}$, is regarded as an ensemble of inspiraling supermassive black hole binaries by default [27,30,32,33]. However, alternative interpretations are also possible, and uncovering the precise origin of this interesting signal remains an active area of investigation [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69]. For more about the physical processes generating GWs with the PTA band, see [70,71,72,73,74,75,76,77,78,79,80].

Scalar-induced gravitational waves (SIGWs), linked to the formation of primordial black holes (PBHs), originate from secondary orders of linear scalar perturbations during the inflationary epoch, and these perturbations are seeded by primordial curvature perturbations, as extensively discussed in the literature [5,7,9,11,12,13,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124]. Generating significant SIGWs requires amplifying the amplitude of the primordial curvature power spectrum. This amplification is often realized by inflationary models that incorporate transitional ultra-slow-roll phases [91,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166]. The frequencies of the peak of SIGWs are intricately linked to the scales of the peak in the power spectrum of primordial curvature perturbations. Specifically, the frequencies of SIGWs in nanohertz (nHz) correspond to the scales characterized by wave numbers on the order of $k_p$∼10⁷ Mpc⁻¹ in the power spectrum of primordial curvature perturbations. This relationship between frequency and scale is critical for understanding the detectability and observational implications of SIGWs in the universe’s early history.

This study is centered around analyzing the PTA signal associated with SIGWs, specifically during the reheating epoch, with the equation of state being $w=1/9$ before the radiation-dominated era. This particular equation of state emerges within the framework of a scalar field and offers a solution to the problem of excessive PBH production observed when fitting PTA data with SIGWs. Recent investigations have underscored challenges linked to this issue, highlighting the significance of non-Gaussianity in primordial curvature perturbations [38,39,167]. Our study aims to rigorously constrain inflationary and reheating scenarios by employing Bayesian methods applied to the NANOGrav 15-year dataset within the context of the equation of state $w=1/9$.

To organize our analysis effectively, we structure our discussion as follows: Section 2 offers a succinct review of the energy density of SIGWs. Section 3 presents our findings and interpretations derived from the analysis. Finally, Section 4 concludes our study, summarizing key insights and implications. Throughout this work, we adopt natural units, where the speed of light $c=1$.

2. Scalar-Induced Gravitational Waves

In this section, we delve into a cosmological scenario featuring a reheating epoch with the equation of state parameter satisfying $w=1/9$, which precedes the onset of Big Bang nucleosynthesis (BBN). Specifically, we focus on a scenario where the transition from the reheating era with $w=1/9$ to the standard radiation-dominated (RD) era is instantaneous, with the temperature of the Universe at this transition denoted by $T_{\mathrm{rh}}$. The w = 1/9 reheating scenario was chosen for this study because it represents a non-standard equation of state that can arise in certain inflationary models with non-canonical kinetic terms or non-trivial interactions [168,169]. Investigating this scenario allows us to explore the potential observable consequences of alternative reheating dynamics and broaden our understanding of the range of possible post-inflationary behaviors. Previous investigations of SIGWs in similar scenarios have been conducted [170], where the authors fixed the reheating temperature $T_{\mathrm{rh}}$ and the width of the primordial curvature power spectrum. In contrast, our study loosens the constraints on the reheating temperature $T_{\mathrm{rh}}$ and the width of the peak in the power spectrum of primordial curvature perturbations. Utilizing Bayesian methods and data from the NANOGrav 15-year dataset, we rigorously constrain both the primordial curvature power spectrum and $T_{\mathrm{rh}}$. This method enables us to examine a wider range of parameter space, providing insights into the implications for cosmological models that incorporate SIGWs within these scenarios.

For a general constant equation of state w and constant sound speed $c_s$, the energy density in a SIGW can be expressed as [86]

$${\mathsf{\Omega }}_{\mathrm{GW},\mathrm{rh}}={\left(\frac{k}{k_{\mathrm{rh}}}\right)}^{-2b}{\int }_0^{\infty }dvs.{\int }_{|1-v|}^{1+v}du\mathcal{T}(u,v,b,c_s){\mathcal{P}}_{\mathcal{R}}\left(ku\right){\mathcal{P}}_{\mathcal{R}}\left(kv\right),$$

(1)

where the subscript “$\mathrm{rh}$” denotes the “reheating”, $b\equiv (1-3w)/(1+3w)$, and ${\mathcal{P}}_{\mathcal{R}}$ is the power spectrum of the primordial curvature perturbations. Here, $u=|\mathbf{k}-\tilde{\mathbf{k}}|/k$ and $v=\tilde{k}/k$ are dimensionless variables, where the auxiliary wave number $\tilde{\mathbf{k}}$ comes from the Fourier transform of the second-order source, $k=\left|\mathbf{k}\right|$ and $\tilde{k}=\left|\tilde{\mathbf{k}}\right|$ [84]. The transfer function $\mathcal{T}(u,v,b,c_s)$ is   

$$\begin{array}{cc}\mathcal{T}(u,v,b,c_s)=\hfill & \mathcal{N}(b,c_s){\left(\frac{4v^2-{(1-u^2+v^2)}^2}{4u^2{v}^2}\right)}^2|1-y^2{|}^b\hfill \\ & \times \{{\left({\mathsf{P}}_b^{-b}\left(y\right)+\frac{b+2}{b+1}{\mathsf{P}}_{b+2}^{-b}\left(y\right)\right)}^2\mathsf{\Theta }(c_s(u+v)-1)\hfill \\ & +\frac{4}{{\pi }^2}{\left({\mathsf{Q}}_b^{-b}\left(y\right)+\frac{b+2}{b+1}{\mathsf{Q}}_{b+2}^{-b}\left(y\right)\right)}^2\mathsf{\Theta }(c_s(u+v)-1)\hfill \\ & +\frac{4}{{\pi }^2}{\left({\mathcal{Q}}_b^{-b}(-y)+2\frac{b+2}{b+1}{\mathcal{Q}}_{b+2}^{-b}(-y)\right)}^2\mathsf{\Theta }(1-c_s(u+v))\}.\hfill \end{array}$$

(2)

Here, the variable $y\equiv 1-[1-c_s^2{(u-v)}^2]/\left(2c_s^{2}uv\right)$, the function $\mathsf{\Theta }\left(x\right)$ is the Heaviside theta function, and the function $\mathcal{N}(b,c_s)$ is

$$\begin{array}{c}\hfill \mathcal{N}(b,c_s)\equiv \frac{4^{2b}}{3c_s^4}\mathsf{\Gamma }{(b+\frac{3}{2})}^4{\left(\frac{b+2}{2b+3}\right)}^2{\left(1+b\right)}^{-2(1+b)},\end{array}$$

(3)

where the function $\mathsf{\Gamma }\left(x\right)$ is the Gamma function.

The Ferrers functions ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$ and ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$, along with Olver’s function ${\mathcal{Q}}_{\nu }^{\mu }\left(x\right)$ in the transfer function (2), can be expressed using hypergeometric functions. Specifically, they are defined as follows:

$$\begin{array}{c}\hfill {\mathsf{P}}_{\nu }^{\mu }\left(x\right)={\left(\frac{1+x}{1-x}\right)}^{\mu /2}\mathbf{F}(\nu +1,-\nu ;1-\mu ;\frac{1}{2}-\frac{1}{2}x),\end{array}$$

(4)

$$\begin{array}{cc}{\mathsf{Q}}_{\nu }^{\mu }\left(x\right)=\hfill & \frac{\pi }{2sin\left(\mu \pi \right)}(cos\left(\mu \pi \right){\left(\frac{1+x}{1-x}\right)}^{\mu /2}\mathbf{F}(\nu +1,-\nu ;1-\mu ;\frac{1}{2}-\frac{1}{2}x)\hfill \\ \hfill & -\frac{\mathsf{\Gamma }(\nu +\mu +1)}{\mathsf{\Gamma }(\nu -\mu +1)}{\left(\frac{1-x}{1+x}\right)}^{\mu /2}\mathbf{F}(\nu +1,-\nu ;1+\mu ;\frac{1}{2}-\frac{1}{2}x)),\hfill \end{array}$$

(5)

$$\begin{array}{cc}{\mathcal{Q}}_{\nu }^{\mu }\left(x\right)=\hfill & \frac{\pi }{2sin\left(\mu \pi \right)\mathsf{\Gamma }(\nu +\mu +1)}({\left(\frac{x+1}{x-1}\right)}^{\mu /2}\mathbf{F}(\nu +1,-\nu ;1-\mu ;\frac{1}{2}-\frac{1}{2}x)\hfill \\ & -\frac{\mathsf{\Gamma }(\nu +\mu +1)}{\mathsf{\Gamma }(\nu -\mu +1)}{\left(\frac{x-1}{x+1}\right)}^{\mu /2}\mathbf{F}(\nu +1,-\nu ;1+\mu ;\frac{1}{2}-\frac{1}{2}x)).\hfill \end{array}$$

(6)

and

$$\begin{array}{c}\hfill \mathbf{F}(a,b;c;x)=\frac{1}{\mathsf{\Gamma }\left(c\right)}F(a,b;c;x),\end{array}$$

(7)

where $F(a,b;c;x)$ is Gauss’s hypergeometric function. By utilizing the correspondence between the evolution of the GW’s energy density and that of radiation, we can calculate the energy density in the present-day GW. This calculation is

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_{\mathrm{GW},0}{h}^2\approx & 1.62\times {10}^{-5}\left(\frac{{\mathsf{\Omega }}_{r,0}{h}^2}{4.18\times {10}^{-5}}\right)\left(\frac{g_{\ast r}\left(T_{\mathrm{rh}}\right)}{106.75}\right){\left(\frac{g_{\ast s}\left(T_{\mathrm{rh}}\right)}{106.75}\right)}^{-\frac{4}{3}}{\mathsf{\Omega }}_{\mathrm{GW},\mathrm{rh}},\hfill \end{array}$$

(8)

where $g_{\ast r}$ and $g_{\ast s}$ are the effective energy and entropy degrees of freedom at the generation of the SIGWs and ${\mathsf{\Omega }}_{r,0}{h}^2$ is the energy density parameter of the present-day radiation, respectively.

In this study, we adopt the log-normal form for the power spectrum of primordial curvature perturbations, represented as:

$${\mathcal{P}}_{\mathcal{R}}\left(k\right)=\frac{A}{\sqrt{2\pi }\mathsf{\Delta }}exp\left(-\frac{{ln}^2(k/k_{\ast })}{2{\mathsf{\Delta }}^2}\right),$$

(9)

where $\mathsf{\Delta }$ controls the width of the spectrum, $k_{\ast }$ determines the peak position in the spectrum, and A denotes the amplitude of the spectrum. This paper focuses on narrow peak spectra with $\mathsf{\Delta }\le 0.1$, a condition supported by our results presented below. For an extremely narrow peak spectrum with $\mathsf{\Delta }\to 0$, the log-normal form power spectrum (9) reduces to a $\delta$-function form, ${\mathcal{P}}_{\mathcal{R}}=A{k}_{\ast }\delta (k-k_{\ast })$. For a log-normal spectrum with a finite width, the corresponding energy density parameter ${\mathsf{\Omega }}_{\mathrm{GW},0}^{\left(\mathsf{\Delta }\right)}{h}^2$ for SIGWs is given by [100]

$${\mathsf{\Omega }}_{\mathrm{GW},0}^{\left(\mathsf{\Delta }\right)}{h}^2=\mathrm{Erf}\left[\frac{1}{\mathsf{\Delta }}{sinh}^{-1}\frac{k}{2k_{\ast }}\right]{\mathsf{\Omega }}_{\mathrm{GW},0}^{\left(\delta \right)}{h}^2.$$

(10)

Here, ${\mathsf{\Omega }}_{\mathrm{GW},0}^{\left(\mathsf{\Delta }\right)}{h}^2$ denotes the present energy density in SIGWs from the primordial curvature power spectrum with a finite-width log-normal form, ${\mathsf{\Omega }}_{\mathrm{GW},0}^{\left(\delta \right)}{h}^2$ is the energy density induced by the $\delta$-function form, and the function $\mathrm{Erf}\left(x\right)$ denotes the error function.

At the horizon crossing for a wavenumber k, the temperature T is approximately related by:

$$k\simeq \frac{1.5\times {10}^7}{\mathrm{Mpc}}{\left(\frac{g_{\ast r}\left(T\right)}{106.75}\right)}^{\frac{1}{2}}{\left(\frac{g_{\ast s}\left(T\right)}{106.75}\right)}^{-\frac{1}{3}}\left(\frac{T}{\mathrm{GeV}}\right).$$

(11)

The corresponding frequency f related to the scale k is:

$$f=\frac{k}{2\pi }\simeq 1.6\mathrm{nHz}\left(\frac{k}{{10}^6{\mathrm{Mpc}}^{-1}}\right).$$

(12)

Combining Equations (11) and (12), we establish the relationship between the frequency f and the temperature T as:

$$f\simeq 24\mathrm{nHz}{\left(\frac{g_{\ast r}\left(T\right)}{106.75}\right)}^{\frac{1}{2}}{\left(\frac{g_{\ast s}\left(T\right)}{106.75}\right)}^{-\frac{1}{3}}\left(\frac{T}{\mathrm{GeV}}\right).$$

(13)

It is important to note that there is a minimum requirement for the reheating temperature during BBN, which is $T_{\mathrm{rh}}\ge 4\mathrm{MeV}$, as indicated by various studies [171,172,173,174]. Consequently, this imposes an upper limit on the reheating frequency $f_{\mathrm{rh}}$, ensuring that $f_{\mathrm{rh}}\le 0.1\mathrm{nHz}$, which falls below the sensitivity range of pulsar timing arrays.

3. Methodology and Results

In this section, we proceed under the assumption that the signal in the NANOGrav 15-year dataset originates from gravitational waves induced by the finite-width log-normal form of the primordial curvature power spectrum given in Equation (9). We utilize Bayesian methods on the NANOGrav 15-year dataset to derive constraints on both the primordial curvature power spectrum and the reheating temperature $T_{\mathrm{rh}}$. The analysis involves leveraging the information from the 14 frequency bins in the NANOGrav 15-year dataset [27,34] to infer the posterior distribution of $T_{\mathrm{rh}}$ and the parameters within the primordial curvature power spectrum parameterized as Equation (9). We employ the Bilby code [175], which implements the dynesty algorithm for nested sampling [176], to compute the posterior distribution. To establish the log-likelihood function, we begin by calculating the energy density of scalar-induced gravitational waves at frequencies corresponding to the 14 bins in our analysis. Subsequently, we create the logarithm of the probability density functions using 14 independent kernel density estimates. Finally, we multiply the probabilities from these 14 bins to construct the comprehensive likelihood function [39,43,46,47,177,178]. This formulation is expressed as

$$\mathcal{L}\left(\mathsf{\Lambda }\right)=\prod _{i=1}^{14}{\mathcal{L}}_i\left({\mathsf{\Omega }}_{\mathrm{GW}}\left(f_i,\mathsf{\Lambda }\right)\right).$$

(14)

Here, $\mathsf{\Lambda }=\{A,\mathsf{\Delta },f_{\ast },T_{\mathrm{rh}}\}$ encompasses all the model parameters, covering both those associated with the primordial curvature power spectrum (9) and the reheating temperature $T_{\mathrm{rh}}$. The priors for these parameters are specified in

Table 1. Summary of the prior distributions, median posterior values, and $90\%$ credible interval bounds for the reheating temperature ($T_{\mathrm{rh}}$) and the parameters associated with the primordial curvature power spectrum, as described in Equation (9). The priors are represented by uniform distributions, denoted by $\mathcal{U}$. The median posterior values provide the most probable estimates for each parameter, while the credible interval bounds of $90\%$ indicate the range within which the true parameter values are likely to lie, based on the given data and the assumed model.

Parameter	${log}_{10}({\mathit{f}}_{\ast }/\mathbf{Hz})$	${log}_{10}\mathbf{\Delta }$	${log}_{10}\mathit{A}$	${log}_{10}({\mathit{T}}_{\mathbf{rh}}/\mathbf{GeV})$
prior	$\mathcal{U}(-10,-2)$	$\mathcal{U}(-6,-1)$	$\mathcal{U}(-5,1)$	$\mathcal{U}(-10,-7)$
posterior	$-6.{26}_{-1.16}^{+1.20}$	$-3.6_{-2.2}^{+2.3}$	$-0.{28}_{-1.20}^{+1.16}$	$-0.{12}_{-1.51}^{+0.64}$

, where $\mathcal{U}$ indicates a uniform distribution.

Figure 1 presents the posterior distributions for the parameters linked to the primordial curvature power spectrum defined in Equation (9), including the reheating temperature denoted as $T_{\mathrm{rh}}$. Table 1 compiles the median posterior values and the boundaries of the $90\%$ credible intervals for these parameters. Within the posterior distributions, the relationship between the scale $k_{\ast }$ and the corresponding frequency $f_{\ast }$ is governed by Equation (12). At a confidence level of $90\%$, the NANOGrav 15-year dataset requires the width of the power spectrum of primordial curvature to be $\mathsf{\Delta }\le 0.001$, which supports the narrow peak assumption made in Section 2. Moreover, the lower bound on the reheating temperature, $T_{\mathrm{rh}}\ge 0.1\mathrm{Gev}$, conforms to the constraints imposed by BBN. The constraints on the amplitude A and the peak scale frequencies $f_{\ast }$ of the primordial curvature power spectrum are consistent with those reported in a publication by the NANOGrav Collaboration investigating new physical phenomena [34].

Using the optimal parameter values obtained from Table 1, we computed the energy density in SIGWs. The resulting energy density is depicted by the blue line in Figure 2, and the blue region denotes the 1-$\sigma$ confidence intervals. In particular, around a frequency of approximately f∼10^−8.1 Hz, we observe a notable turning point in the energy density of SIGWs. This turning point corresponds to a change in the equation of state w, which affects the evolution of SIGWs and leads to a distinctive feature in the energy density spectrum.

The energy density spectrum of SIGWs, as shown in Figure 2, exhibits a sharp decrease in the ultraviolet region around the frequency of f∼10⁻⁶. This feature is a characteristic of SIGWs originating from a primordial curvature power spectrum with a narrow peak. In the case of a $\delta$-function form of the primordial curvature power spectrum, ${\mathcal{P}}_{\zeta }\left(k\right)=A_{\zeta }\delta [ln(k/k_{\ast })]$, which represents an extremely narrow peak, the energy density of SIGWs cuts off at scales of $k\ge 2k_{\ast }$ [90]. Our data analysis requires a very narrow peak in the primordial curvature power spectrum, with constraints indicating $\mathsf{\Delta }<0.001$. This narrow peak directly results in the observed sharp decrease in the energy density of SIGWs. Therefore, the feature in Figure 2 is a consequence of the primordial curvature power spectrum having a very narrow peak, as necessitated by the PTA data.

4. Conclusions and Discussion

The stochastic signal observed by collaborations such as NANOGrav, PPTA, and EPTA, along with InPTA and CPTA, likely originates from gravitational waves induced by primordial curvature perturbations generated during inflation. Following inflation, the Universe undergoes a reheating phase aimed at increasing its temperature. Information about the small-scale properties of inflation and reheating can be encoded in the energy density of SIGWs. This study explores the small-scale characteristics of inflation and reheating and assumes that the equation of state during the reheating phase is characterized by $w=1/9$. During the oscillation of the inflaton, $w=\frac{p-2}{p+2}$ for a power-law potential $V\left(\varphi \right)\propto {\varphi }^p$ [179,180]. The choice $w=1/9$ corresponds to a scalar field potential of the form $V\left(\varphi \right)\sim {\varphi }^{5/2}$ near its minimum, which exhibits a flatter minimum compared to the standard quadratic case, leading to distinct reheating dynamics. The ${\varphi }^{5/2}$ potential form can arise in inflationary models with non-canonical kinetic terms, such as k-inflation [181] or DBI inflation [182], or be motivated by considering higher-order corrections to the inflaton potential [183]. The total potential model under consideration includes terms that allow for a smooth transition from the inflationary phase to this specific reheating scenario [184].

To model the power spectrum of the primordial curvature perturbations, a log-normal form is adopted. By applying Bayesian methods and analyzing the NANOGrav 15-year dataset, we obtain posterior distributions for the key parameters. These include the characteristic frequency $f_{\ast }$, the width parameter $\mathsf{\Delta }$, the amplitude A of the primordial curvature power spectrum and the reheating temperature $T_{\mathrm{rh}}/\mathrm{GeV}$. The obtained posterior values are ${log}_{10}(f_{\ast }/\mathrm{Hz})=-6.{26}_{-1.16}^{+1.20}$, ${log}_{10}\mathsf{\Delta }=-3.6_{-2.2}^{+2.3}$, ${log}_{10}A=-0.{28}_{-1.20}^{+1.16}$ and ${log}_{10}(T_{\mathrm{rh}}/\mathrm{GeV})=-0.{12}_{-1.51}^{+0.64}$.

It is important to acknowledge that the $w=1/9$ reheating scenario is one among many possibilities for post-inflationary dynamics. The more commonly assumed scenario is $w=0$, which corresponds to a quadratic potential minimum [185,186]. However, exploring alternative scenarios, such as $w=1/9$, allows us to investigate the potential observational signatures of non-standard reheating dynamics and test the robustness of our conclusions. The choice of equation of state during reheating can have significant effects on observable signatures, such as the energy density of scalar-induced gravitational waves [86,101,187]. In the $w=1/9$ scenario, distinct reheating dynamics can lead to modifications in the spectral shape and amplitude of the gravitational wave background compared to the standard $w=0$ case. Exploring these alternative scenarios provides valuable insights into the sensitivity of observables to the details of the reheating phase and can help constrain inflationary models based on observational data [188].

Furthermore, the energy density of SIGWs displays a distinctive turning point, which corresponds to a change in the equation of state w. This turning point signifies the transition from the reheating epoch to the radiation-dominated era, highlighting a significant phase in the evolution of the early Universe. Future observational data on scalar-induced gravitational waves hold the potential to refine our understanding further, potentially pinpointing the precise time when the reheating phase transitioned into the radiation-dominated era. Such insights could shed light on fundamental aspects of cosmological evolution during the early Universe and aid in the refinement of theoretical models of inflation and reheating.