Non-thermal Higgs Spectrum in Reheating Epoch:
Primordial Condensate vs. Stochastic Fluctuation

Despite the discovery of the Higgs boson at the Large Hadron Collider at CERN [1, 2], the least understood in the Standard Model of particle physics is the Higgs sector. This is simply because, in the absence of sufficient experimental data, many possibilities can fit into the theory of electroweak symmetry breaking, which necessarily include cosmological implications; see e.g. Ref. [3] for a recent review. For instance, electroweak phase transition in the early universe has been extensively studied and many connections to other subjects including baryogenesis [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and gravitational waves [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] have been found. The Higgs sector may also be responsible for producing dark matter, known as the Higgs portal scenario [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]; see also Ref. [49] for a pedagogical introduction on particle dark matter.

The recent development of observation technologies allows us to address the cosmological history at much earlier times than when the electroweak phase transition takes place. The inflationary scenario is one of the main targets of the recent cosmological/astrophysical observations; see e.g. Refs. [50, 51, 52, 53, 54, 55, 56] for reviews. Indeed, cosmic microwave background (CMB) observation has set constraints on the scalar and tensor power spectra, by which many inflationary models have already been ruled out [57]. In such early times during the inflation epoch, the Higgs field may play an important role. One such example is the Higgs inflation where the Standard Model Higgs is identified as the inflaton [58, 59]; see also Refs. [60, 61, 62, 63, 64, 65, 66, 67, 68, 69] for related topics and a review.

In the case that the Higgs itself is not the inflaton and the inflation is driven by another field, the Higgs is regarded as a light spectator field during inflation/reheating. In this case, the Higgs vacuum stability during inflation and reheating is an important subject. The mass measurement of the Higgs and top quark has brought up an indication that the vacuum becomes unstable at high energies, whose cosmological implications have also been extensively studied [70, 71, 72, 73, 74, 75, 76, 77]. The Higgs field may also affect in many ways the temperature evolution after the end of inflation. For instance, the highest temperature of the universe is affected by the gravitational production of the Higgs during reheating [78, 79] or preheating through thermalization of weak gauge bosons produced by the oscillation of primordial Higgs condensate [80, 81, 82, 83].

In discussing the cosmological role of the Higgs, especially during and right after inflation, there are two distinctive contributions: the primordial condensate and the stochastic fluctuations. Sometimes their difference is not well recognized or even mixed up. This is partly because when one does not need to care about the momentum distribution of the primordial Higgs field, such refined separation is not necessary. However, when one needs to care, e.g. in considering the decay of the Higgs, then a closer look at the difference is needed.

In the present work, we focus on the non-thermal phase space distribution arising from two different origins, namely, the primordial Higgs condensate and the stochastic fluctuation. These are both seen as a displacement of the Higgs field value during inflation, hence are hardly distinguishable from each other in the field space. On the other hand, they have different phase space distributions after the end of inflation. We will explicitly show such a non-thermal spectrum of the Standard Model Higgs and, as a possible implication, will discuss their subsequent decay.

The paper is organized as follows. In Sec. 2, we first present an explicit model of the Higgs and inflaton sectors to set up the stage for our discussion. We explain the dynamics of the Higgs during inflation in Sec. 3, where we first clarify what the primordial condensate means and then briefly review the stochastic fluctuations. The non-thermal Higgs spectrum after the end of inflation is discussed in Sec. 4, which is then applied to the non-thermal Higgs decay in Sec. 5. The summary of our findings is given in Sec. 6.

Before discussing the detail of the Higgs dynamics during inflation, we make our framework explicit in this section, including the Higgs and inflaton sectors. While we will consider the Higgs sector in its conventional form (the quadratic and quartic terms in the potential), the choice of the inflationary sector is arbitrary to some extent. Since what we will argue in the subsequent sections does not quite depend on the inflationary models, our choice here should be considered as an example for the sake of concreteness.

Our focus is on the evolution of the Standard Model Higgs field during inflation, where the Higgs is considered as a spectator field whose energy density is assumed to be subdominant compared with that of the inflaton.

We turn our discussion to the phase space distribution of the Higgs after the end of inflation. In the following argument, we suppose that the thermalization of the Higgs takes a very long time such that it does not complete and that the non-thermal spectrum persists. This setup is justified e.g. when the reheating temperature is sufficiently low compared with the effective Higgs mass, which can easily be realized unless $\lambda$ is extremely small.

The phase space distribution for the primordial condensate is particularly simple, which is by definition given by

where $p$ is the physical momentum, and the normalization $n_{\rm cond}$ is determined by

By denoting the initial squared amplitude of the Higgs by $|\Phi_{0}|^{2}$, the effective mass is given by $m_{h,{\rm eff}}^{2}=\lambda|\Phi_{0}|^{2}$, and the initial energy density $\rho_{h,0}$ is determined accordingly. Note that $\rho_{h,{\rm cond}}$ keeps to be $\rho_{h,{\rm cond}}\sim\rho_{h,0}$ until $a_{\rm osc}$ that satisfies $H(a=a_{\rm osc})\simeq m_{h,{\rm eff}}$. Afterward, $\rho_{h,{\rm cond}}\simeq\rho_{h,0}(a/a_{\rm osc})^{-4}$ assuming that the quartic term in the potential dominates during the Higgs oscillation.

It becomes more involved to take into account other Higgs spectra than the primordial condensate. As we will discuss in the following, there are three distinctive regimes in the phase space density for the stochastic fluctuation.

As an implication of the non-thermal Higgs spectrum, we consider the perturbative decay of the Higgs into the Standard Model particles. This can be possible whenever $m_{h,{\rm eff}}$ is larger than the mass of the decay products. In such kinematic reasoning, one should keep in mind that the decay products may also acquire their mass from the effective Higgs vacuum expectation value. Therefore, for instance, the (perturbative) decay into the weak gauge bosons does not occur when $\sqrt{\lambda}\lesssim g$, where $g$ is the SU(2) gauge coupling. This is normally the case if we extrapolate the renormalization group running of the couplings up to the inflationary scale within the Standard Model [80, 81].⁹⁹9 See also, e.g., Refs. [82, 83] for implications of non-perturbative Higgs decay. The Higgs energy density, however, cannot dissipate at all through the parametric resonance, since the instability terminates when the energy density of the produced particles reaches the Higgs energy density. Thus the complete dissipation of the energy density needs a perturbative decay [101]. However, such running is affected by, for instance, the error in the top quark mass measurements and/or the appearance of new physics at certain high scales [102]. Therefore, we take rather an agnostic approach to the choice of $\lambda$, regarding it as a free parameter at high scales.

In the same spirit, we take the Higgs decay width $\Gamma_{h}$ as a free parameter and will keep track of how the Higgs decays. To do this, instead of looking at the decay on a mode-by-mode basis, we consider the integrated Boltzmann equation given by

where the decay rate $R_{h\to AB}$ is given by

in which

with $f_{h}(p,t)=4f(p,t)$ by taking four degrees of freedom in $\Phi$ into account, or $f_{h}(p,t)=f_{\rm cond}(p,t)$ for the primordial condensate.¹⁰¹⁰10 All the four modes acquire the effective mass since it comes from the quartic coupling.

In the case of the primordial condensate (30), the Higgs decay is particularly easy. By using $\rho_{h}=m_{h,{\rm eff}}n_{h}$ from Eq. (31), we may rewrite the Boltzmann equation as

after the onset of the Higgs oscillation, approximating that the quartic coupling dominates in the potential during the oscillation [103]. The solution of Eq. (43) is given by

and thus the decay time can readily be estimated as¹¹¹¹11 Here, we have neglected the time dependence in $\Gamma_{h}$, though the effective Higgs mass may potentially be time-dependent during the Higgs oscillation epoch. To see this effect, a more careful study of every possible decay channel is needed, which is left for future work. For a related study, see Refs. [103, 104].

where we have used $t=2/3H\simeq\left(2/3H_{\text{e}}\right)\left(a/a_{\text{e}}\right)^{3/2}$.

The decay of the stochastic fluctuation is more involved, compared with the primordial condensate. Figure 3 shows each contribution of the spectrum in the decay rate (normalized by $\Gamma_{h}H_{\text{e}}^{3}$) as a function of the scale factor $a$ with $\lambda=0.01$ and $H_{\text{e}}=m_{\phi}$. As seen in the figure, although $\gamma_{\rm soft,NR}$ is the greatest contribution at early times, the other two soon dominate and can be the main contribution in the Higgs decay. Depending on the value of $\lambda$, they can be comparable, and in the large-$a$ limit we find

where we have used Eq. (36) for $\gamma_{\rm soft,R}$. Note that the soft non-relativistic contribution takes a simple form

where $k_{\rm IR}$ is an IR cutoff which we take $k_{\rm IR}=0.05$ Mpc^-1 in the figure. A slight change in $k_{\rm IR}$, however, does not change the order of magnitude of the decay rate, since it affects the result only logarithmically.

Since $\gamma_{\rm hard}+\gamma_{\rm soft,R}\propto a^{-3}$, we write $\gamma\simeq\overline{\gamma}\left(a/a_{\text{e}}\right)^{-3}$. Then, Eq. (40) can be solved to find

where, from Eq. (29), we have

Supposing that $\gamma\sim\gamma_{\rm soft,R}\Rightarrow\overline{\gamma}\simeq\left(3/32\pi^{2}\right)\left(H_{\text{e}}^{4}/m_{h,{\rm eff}}\right)$ and requiring $n_{h}(a_{\rm dec})=0$, we find the decay time as

It is interesting that this is almost the same as the decay time of the primordial condensate given in Eq. (45).

The cosmological role of the Higgs, especially during and immediately after inflation, has two distinctive contributions, one being the primordial condensate and the other being the stochastic fluctuations. However, the difference between the two is not well recognized and sometimes confused. This is partly because such an elaborate separation is not necessary when one does not need to care about the momentum distribution of the primordial Higgs field. However, if, for example, the Higgs decay is taken into account, this difference needs to be looked at more closely.

In this paper, we focused on non-thermal phase space distribution arising from these two different sources: primordial Higgs condensate and stochastic fluctuation. Both of these components are considered as displacement of the Higgs field value during inflation and are therefore hardly distinguishable in the field space. On the other hand, once inflation is over, these components will have different phase space distributions.

The primordial condensate can be viewed as the initial field strength of the Higgs field. It persists during inflation and starts oscillating when the Hubble rate becomes equal to the condensate effective frequency (or mass) after the end of inflation. The stochastic fluctuation originates from the UV modes flowing into the IR regime. It is a secular effect if the Higgs was massless, namely, the variance of the field grows forever. However, the existence of the potential including the self-interaction works as a drift term that tries to restore the original position of the field, namely, the origin. The balance between these two effects leaves the finite variance for the Higgs field during inflation, by which the Higgs spectrum after inflation is affected.

The primordial Higgs condensate has the delta function-like shape of the spectrum localized at the zero momentum state. On the other hand, the stochastic fluctuation has a nontrivial phase space distribution which consists of three different regimes depending on the comoving momentum $k$, namely, $k>m_{\phi}$, $k_{*}<k<m_{\phi}$, and $k<k_{*}$. We derived an analytic estimate and showed that the estimate is in good agreement with our previous numerical result.

As an implication for the non-thermal Higgs spectrum, we discussed the Higgs decay after the inflation, where we assumed that the perturbative decay is always possible, while the Higgs remains non-thermal until it decays. We found that the decay times of the primordial condensate and the stochastic fluctuation almost coincide with each other, namely, both contributions decay away almost at the same time.

Finally, we emphasize that implication of the non-thermal Higgs spectrum is not limited to Higgs decays. Since the Higgs sector is linked to new physics in many ways, the non-thermal Higgs spectrum can have many opportunities to be relevant to Higgs cosmology.

The authors would like to thank Sung-Mook Lee and Kenji Nishiwaki for the helpful discussion at the early stages of the project. This work is in part supported by the JSPS KAKENHI Grant Nos. 19H01899 (KK and KO) and 21H01107 (KO). KK acknowledges support by Institut Pascal at Université Paris-Saclay during the Paris-Saclay Astroparticle Symposium 2021, with the support of the P2IO Laboratory of Excellence (program “Investissements d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038), the P2I axis of the Graduate School Physics of Université Paris-Saclay, as well as IJCLab, CEA, IPhT, APPEC, the IN2P3 master projet UCMN and EuCAPT.