Post-Reheating Inflaton Production as a Probe of Reheating Dynamics

TL;DR

This paper explores how inflaton particles can be produced after reheating through a new mechanism, offering insights into reheating dynamics and potential explanations for dark matter abundance.

Contribution

It introduces a novel post-reheating inflaton production channel that can probe reheating processes and relate to dark matter.

Findings

Inflaton quanta can be regenerated after decay from the thermal bath.

The mechanism provides a new way to investigate reheating dynamics.

Potential explanation for dark matter abundance from inflaton production.

Abstract

Cosmological reheating bridges the inflationary epoch and the hot big bang phase, yet its underlying dynamics remain poorly understood. In this work, we investigate a minimal scenario in which the inflaton evolves under a simple power-law potential during reheating and interacts with other particles via renormalizable couplings. We show that inflaton quanta can be regenerated from the thermal bath even after the decay of the coherent inflaton field, unveiling a previously overlooked channel for inflaton particle production, which offers a novel window into probing reheating. Remarkably, this mechanism may also account for the observed dark matter abundance, providing a natural link between early Universe dynamics and present-day cosmological observations.

Equations

$$\varphi (x)$$

$${\mathcal{L}}_{\inf }$$

$$V(\overline{\varphi })$$

$$\lambda$$

$${\rho }_{\overline{\varphi }}(a)$$

$$m_{\varphi }(a)$$

$$⟨{\sigma }_{\chi \chi ,\delta \varphi \delta \varphi }{v}_{\mathrm{rel}}⟩$$

$$\frac{dY}{dz}$$

$${\Omega }_{\delta \varphi }{h}^2$$

$$-{\mathcal{L}}_{\mathrm{int}}$$

$$⟨\varphi ⟩$$

$$m_{\varphi }^2$$

$$V_{\mathrm{CW}}$$

$$m_{\varphi }^2$$

$$m_{\varphi }$$

$${\mathcal{L}}_{\mathrm{mix}}$$

$$\theta \simeq \frac{2\mu v}{m_h^2},$$

$$\mu \lesssim 10^{-3}\frac{m_h^2}{2v}\simeq 0.032\mathrm{GeV}.$$

Full text

Introduction.— The hot big bang Universe has been at a central place in modern cosmology since the success of the Big Bang Nucleosynthesis [1, 2, 3] and subsequent observations of the cosmic microwave background (CMB) radiation [4]. The cosmic inflation has been proposed to keep the success of and complete the hot big bang scenario by solving the horizon and flatness problems [5, 6, 7, 8], while seeding the primordial density perturbations that lead to the large-scale structure of the Universe [9, 10, 11, 12, 13]. Once inflationary expansion ends, the Universe is left cold and empty, and thus a heating mechanism, called reheating, is necessary to bring the Universe into a hot state.

The reheating era bridges inflation and the radiation-dominated Universe, yet its microphysics remains elusive. In conventional scenarios, the inflaton condensate decays completely [14, 15, 16], producing the Standard Model particles and reheating the Universe to a temperature $T_{\rm reh}$. However, this picture may be incomplete. There may be preheating regime prior to reheating, where the inflaton field oscillates about its minimum and decays into other particles through non-perturbative processes [17, 18, 19, 20]. Such a non-perturbative process may lead to parametrically enhanced superradiance of inflaton as well [21]. There may also be a possibility that the inflaton condensate fragments into inflaton particles before reheating completes [22, 23, 24, 25]. All these processes are yet to be fully understood and tested experimentally. Given the observational challenges associated with probing the reheating epoch, any potential signature from this phase is of significant interest.

In this letter, we propose that the inflaton quanta can be regenerated from thermal bath in the post-reheating era via the same coupling responsible for rehearing, provided the effective mass of the inflaton is smaller than the bath temperature. This condition is commonly realized when the inflaton potential takes a monomial form higher than a quadratic term: $V(\phi)\propto\phi^{k}$ with $k=4,6,\cdots$ near the bottom of the potential. The abundance of produced inflaton quanta can be non-negligible, even leading to overclose the Universe, depending on the size of the coupling for a relevant interaction, from which we may place constrains on the interactions responsible for reheating. This opens up a new avenue for probing reheating processes and offers a valuable guidance for model-building of reheating scenarios motivated by particle physics.

Evolution of inflaton condensate.— We introduce the inflaton condensate as a zero-mode of $\phi(x)$:

[TABLE]

where $\overline{\phi}$ and $\delta\phi$ denote the zero mode and the non-zero momentum modes (or inflaton quanta), respectively. Note that since $\delta\phi$ does not have the zero-momentum mode, $\overline{\phi}$ and $\delta\phi$ are orthogonal to each other.

The Lagrangian of $\overline{\phi}$ relevant for inflation is given by

[TABLE]

where, during reheating ($\overline{\phi}\ll M_{P}$), we consider a simple power-law potential

[TABLE]

with $k$ an even integer and $M_{P}$ the reduced Planck mass, $M_{P}\simeq 2.4\times 10^{18}$ GeV. The inflaton potential coupling $\lambda$ is determined by CMB observation when the above term is somehow relevant during inflation. For definiteness, we suppose that the inflaton potential during inflation is given by an extended version of the T-model [26],

[TABLE]

In this case, $\lambda$ is approximately estimated by

[TABLE]

where the amplitude of scalar perturbation is given by $A_{S}\simeq 2.1\times 10^{-9}$ at the Planck pivot scale $k_{\rm pivot}=0.05~{\rm Mpc}^{-1}$ [27]. The number of $e$-folds depends on models of inflation, and for definiteness we take $N=50$ in our analysis, by which we obtain $\lambda=4.1\times 10^{-12},6.9\times 10^{-13},1.2\times 10^{-13},1.9\times 10^{-14}$ for $k=4,6,8,10$, respectively.

After inflation ends at the scale factor $a=a_{e}$, the energy density of $\overline{\phi}$ evolves as

[TABLE]

where $\rho_{e}\equiv\rho_{\overline{\phi}}(a_{e})$. As studied in detail in Refs. [28, 29, 30], while its evolution resembles dust-like energy density for $k=2$, it redshifts away faster for $k\geq 4$. Remarkably, whereas the inflaton mass is constant for $k=2$, it decreases with time when $k\geq 4$ and becomes negligible in later times. As we will argue, because the negligible inflaton mass is crucial in regenerating the inflaton quanta, we focus on $k\geq 4$ in the rest of our analysis.

Inflaton interactions.— Inflaton must have some coupling(s) through which radiation-dominated Universe is eventually realized such that the Universe can be described by the standard big bang cosmology. Although the process of reheating is generally model-dependent, if we limit ourselves to the renormalizable interactions, we can consider only the following type: $y\phi\overline{f}f$, $\mu\phi\chi\chi$, $\sigma\phi^{2}\chi^{2}$, where $\chi$ and $f$ are bosonic and fermionic field constituting the thermal plasma, respectively and $y,\mu$ and $\sigma$ are the coupling parameters. In the following, we assume that the reheating is somehow completed via one of the interactions above at $a=a_{\rm reh}$.

As mentioned above, we consider the potential (3) with $k\geq 4$,which leads to a time-dependent inflaton mass that decreases as the Universe expands:

[TABLE]

Note that $a_{e}$ is defined by the end of the slow-roll regime, specifically when the slow-roll parameter $\epsilon(\phi)$ satisfies $\epsilon(\phi)=2M_{P}^{2}(\partial_{\phi}H)^{2}/H^{2}=1$ with $H$ being the Hubble parameter. Although $a_{e}$ serves as a reference point, physical quantities such as the inflaton mass at $a_{e}$ are determined by the field value $\overline{\phi}_{e}\equiv\overline{\phi}(a_{e})$. For instance, the inflaton mass at $a=a_{e}$ is given by $m_{\phi}(a_{e})=\sqrt{\lambda k(k-1)(\overline{\phi}_{e}/M_{P})^{k-2}}M_{P}=5.1\times 10^{13}~{\rm GeV},1.7\times 10^{14}~{\rm GeV},6.0\times 10^{14}~{\rm GeV},2.2\times 10^{15}~{\rm GeV}$ for $k=4,6,8,10$, respectively.

The coherent oscillation of $\overline{\phi}(t)$ ceases when reheating is completed, and thus the inflaton can be effectively massless, unless $\chi$ develops a vacuum expectation value (VEV) to generate the mass for $\delta\phi$ if $\phi^{2}\chi^{2}$ coupling is present111Note that we do not assume a bare mass term for inflaton. Its effect on reheating has been studied in Ref. [31]. . This indicates that the same interaction relevant for the reheating regenerates inflaton quanta during post-reheating era, and its abundance can become significant to affect the evolution of the Universe, from which we can constrain the coupling responsible for reheating, and probe the reheating process.

Here we consider the interaction $-{\cal L}_{\rm int}=(1/2)\sigma\phi^{2}\chi^{2}$ as an explicit example to demonstrate how post-reheating regeneration of the inflaton quanta allows us to study the reheating process. Since this interaction has $Z_{2}$ symmetry222In general, this symmetry may not be respected in the inflaton potential., making it stable (see e.g., [32] for some particular mechanism), $\delta\phi$ may become dark matter or dark radiation, depending on the VEV of $\chi$. The abundance can be calculated by the standard procedure using the Boltzmann equation. The details of the analysis will be given elsewhere [33]. The thermally-averaged production rate is given by

[TABLE]

where $n_{\chi}^{\rm eq}$ is the equilibrium number density of $\chi$, and the final state $\delta\phi$ is approximated as massless, appearing in the Boltzmann equation:

[TABLE]

with $z\equiv m_{\chi}/T$, $Y=n_{\delta\phi}/s$ using the entropy density $s$, and $Y^{\rm eq}$ being its equilibrium value. By solving Eq. (9) under the assumption that $\chi$ develops a VEV $v$ at some epoch, thereby generating the mass for the inflaton as $m_{\phi}=\sqrt{\sigma}v$, we evaluate the final abundance of $\delta\phi$ given by

[TABLE]

where $\rho_{c0}$ is the today’s critical density, and we define $Y_{0}=Y(T_{0})$ with $T_{0}$ the temperature at the present Universe.

Figure 1 shows the $\delta\phi$ abundance for $v=10~{\rm GeV}$, $246~{\rm GeV}$ (the case $\chi$ is the Standard Model Higgs), and $10~{\rm TeV}$. The gray region, where $\Omega_{\delta\phi}h^{2}>0.12$ [34], is excluded, yielding an upper bound on $\sigma$. We assume a vanishing initial density $n_{\delta\phi}$, though this may be modified by preheating and inflaton fragmentation. In the fragmentation regime $\rho_{\overline{\phi}}\simeq\rho_{\delta\phi}$[24, 35, 25], while the evolution of $Y$ during reheating depends on entropy production during preheating and after fragmentation. As a result, $Y_{0}$ may become excessive even for small $\sigma$, depending on the value of $Y$ at the end of reheating, and thus, our bound should be viewed as conservative.

Since we assume the potential given in Eq. (3) with $k\geq 4$, i.e., having the inflaton self-interaction, the homogeneous inflaton field $\overline{\phi}$ can eventually fragment, depending on the coupling $\sigma$. If reheating has not yet completed before fragmentation occurs, the resulting inflaton quanta $\delta\phi$ dominate the total energy density of the Universe [25]. To ensure that reheating completes before fragmentation, the coupling $\sigma$ needs to be greater than certain values, as discussed in Ref. [25]. Notice that the inflaton dissipation rate during reheating is characterized by $\sigma_{\rm eff}$, which accounts for the time-averaged inflaton oscillations and is related to the bare coupling $\sigma$ [29, 30]. In Ref. [25], bounds on the effective coupling $\sigma_{\rm eff}$ is provided, which can be translated to bounds on the bare coupling as $\sigma_{\rm eff}/\sigma=3.6,7.0,11,15$ for $k=4,6,8,10$, respectively. These bounds are illustrated in Fig. 1 for the cases with $k=4$ and $6$, which correspond to $\sigma<2.0\times 10^{-6}$ and $\sigma<1.5\times 10^{-10}$, respectively. From the figure, we also observe that the regions with $\sigma<3.4\times 10^{-3},3.0\times 10^{-2}$ and $6.1\times 10^{-1}$ for $v=10~{\rm GeV},246~{\rm GeV}$ and $10~{\rm TeV}$, respectively, are disfavored due to either the fragmentation or the overclosure of the Universe. Importantly, just above these values, $\delta\phi$ can constitute dark matter in the Universe, and therefore the reheating process can also be relevant to the later evolution of the Universe through the regeneration of inflaton quanta.

Although the above general consideration has already provided substantial insight into the reheating process, we can obtain more constraints on the reheating coupling by specifying the model further. Given that the Standard Model particles should be eventually produced, one of the simplest scenarios would be to identify $\chi$ as the Standard Model Higgs boson, which we are going to investigate in the following.

Higgs portal interactions.— The idea of Higgs portal interactions to particles beyond the Standard Model has been widely studied in the context of dark matter [36, 37, 38, 39], and in the context of inflation, the effect of the Higgs portal couplings to inflationary models and subsequent reheating has been studied in Refs. [40, 41, 42, 43, 44, 45]. Here we identify $\chi$ as the Higgs boson, being responsible for reheating. At the renormalizable level, $\phi$ cannot have a coupling to the Standard Model fermions due to gauge symmetry, and thus we only consider the bosonic coupling below, which we label by b-I and b-II:

[TABLE]

where $H$ is the Higgs doublet. Now we discuss what information can be obtained for each case in the following.

Case b-I.— This case is essentially the same as the one discussed above for the interaction $-{\cal L}_{\rm int}=(1/2)\sigma\phi^{2}\chi^{2}$, but with the VEV $v=246~{\rm GeV}$. Thus the abundance of $\delta\phi$ produced from this interaction is given by red line in Fig. 1. Since we identify $\chi$ as the Higgs boson, collider search can also bring us more insight into the coupling. Indeed the Higgs invisible decay measurement excludes the parameter $\sigma>2.5\times 10^{-3}$ by requiring the branching ratio to be less than 20 % [46], which is also shown in Fig. 1. It should be noted that, in the region where $m_{\phi}\simeq m_{h}/2$, the Higgs resonance may enhance the reaction rate and suppress the abundance considerably, which happens when $\sigma\simeq 0.065$. However, this region has already been excluded by the Higgs invisible decay measurement mentioned above, and thus it is irrelevant to our discussion. We also note that the other $s$-channel processes of pair-annihilating $\delta\phi$ into a pair of the Standard Model particles can also be included, however, its rate is suppressed by $m_{\phi}^{2}/m_{h}^{2}$ (in non-relativistic regime) for $m_{\phi}\ll m_{h}$ [38] and hence it is neglected in our analysis.

Case b-II.— A tadpole term develops after the Higgs field acquires a non-zero VEV, yielding a non-zero VEV of $\phi$,

[TABLE]

which induces a mass of $\delta\phi$ given by

[TABLE]

However, such a contribution is negligibly small in the parameter space of our interest, compared to the radiatively induced mass as discussed in the following. Therefore, we drop the tadpole contribution to $\langle\phi\rangle$ in the following analysis.

The Coleman-Weinberg potential for a general $\chi$ field arising from its one-loop contributions is given by

[TABLE]

where $\Lambda$ is the renormalization scale and $c_{\chi}$ is the number of degrees of freedom. For the Higgs boson case, we take $c_{\chi}=4$. By minimizing $V+V_{\rm CW}$, we find the effective mass of inflaton as

[TABLE]

where the dots include $\Lambda$-dependent terms in the form of Lambert $W$ function. Given that $\Lambda$ may take at an arbitrary scale, we may set such terms smaller than unity, so that the effective mass is simply given by

[TABLE]

The physical Higgs field $h$ mixes with the inflaton particle $\delta\phi$ through

[TABLE]

which induces a mixing angle

[TABLE]

under the assumption $m_{\phi}\sim\mu\ll m_{h}$, with the Higgs mass $m_{h}=125$ GeV. The inflaton mass is set by $\mu$ in the present case, and hence we obtain $\theta\simeq m_{\phi}/10~{\rm GeV}$. Restricting to $m_{\phi}>50$ MeV then gives $\theta\gtrsim 5\times 10^{-3}$, which is excluded by meson-decay measurements [47]. For smaller masses and mixing, stringent astrophysical and cosmological bounds apply [48] (see also [49] for further details).

These constraints can be relaxed if the inflaton mass is not tied to $\mu$. A simple possibility is to introduce a $\sigma\phi^{2}|H|^{2}$ interaction in addition to the $\mu\phi|H|^{2}$ term, which generates a $\mu$-independent contribution to $m_{\phi}$. In such a setup, the $\sigma$ coupling can control inflaton production while the $\mu$ coupling governs inflaton decay, potentially leading to richer phenomenology. Alternatively, one may simply add a bare mass term, $m_{\phi}^{2}\phi^{2}/2$. In any case, treating $\theta$ and $m_{\phi}$ as independent parameters, we obtain a constraint on $\mu$ as

[TABLE]

from the dark Higgs search [47].

The limit from the inflaton fragmentation is also provided for the effective coupling $\mu_{\rm eff}$ which corresponds to the bare coupling $\mu$ as $\mu_{\rm eff}/\mu=1.4,1.7,1.9,2.1$ for $k=4,6,8,10$ [30, 25]. The lower bound on $\mu_{\rm eff}$ for $k=4,6$ is well above the EW scale [25], hence our analysis is not relevant for these limits. On the other hand, for $k=8,10$, the lower bound is given by $\mu=0.33~{\rm GeV},7.1\times 10^{-8}~{\rm GeV}$, respectively. Notice that the case with $k=8$ is already disfavored by the dark Higgs search.

If we neglect the fragmentation, we may analytically estimate the reheating temperature as a function of $\mu$, following the approach in Ref. [30]. This approximation is justified since fragmentation would have only a minor impact on reheating for this type of coupling, as discussed in Ref. [25]. Figure 2 shows the constraint from the dark Higgs search on the reheating temperature, which places an upper bound as $T_{\rm reh}<2.4\times 10^{5}~{\rm GeV},2.0\times 10^{7}~{\rm GeV},1.0\times 10^{8}~{\rm GeV},2.4\times 10^{8}~{\rm GeV}$ for $k=4,6,8,10$, respectively. Note again that our analysis may apply for only the case of $m_{\phi}\ll m_{h}$.

Conclusion.— We have shown that a significant regeneration of inflaton quanta during post-reheating era can serve as a powerful probe into the reheating process. Such a regeneration of inflaton quanta typically occurs when the (effective) mass of the inflaton is smaller than the temperature of the thermal bath, as is often the case in inflation models where a self-interaction term dominates near the potential minimum. By examining specific interaction examples, we demonstrated how such regeneration can be used to constrain couplings relevant to reheating.

Furthermore, we have investigated a scenario in which the Standard Model Higgs boson is coupled to the inflaton, enabling reheating via the Higgs portal. In this framework, collider constraints, such as the Higgs invisible decay and the dark Higgs search, can be combined with the consideration of the inflaton quanta regeneration to further refine our understanding of reheating dynamics.

A more comprehensive analysis of inflaton quanta regeneration is warranted in future work, as it holds promise for deepening our understanding of the reheating epoch [33].

Acknowledgments.— We would like to thank Yann Mambrini and Michael Schmidt for useful discussion. This work was made possible by Institut Pascal at Université Paris-Saclay with the support of the program “Investissements d’avenir” ANR-11-IDEX-0003-01, the P2I axis of the Graduate School of Physics of Université Paris-Saclay, as well as IJCLab, CEA, IAS, OSUPS, and APPEC. This work was also supported by JSPS KAKENHI Grant Numbers 25K01004 (TT) and MEXT KAKENHI 23H04515 (TT), 25H01543 (TT). KK would like to express special thanks to the Mainz Institute for Theoretical Physics (MITP) of the Cluster of Ecellence PRISMA+ (Project ID 390831469), for its hospitality and support.