# Inflation, (p)reheating and neutrino anomalies: production of sterile neutrinos with secret interactions

## Abstract

A number of experimental anomalies involving neutrinos hint towards the existence of at least an extra (a very light) sterile neutrino. However, such a species, appreciably mixing with the active neutrinos, is disfavored by different cosmological observations like Big Bang Nucleosynthesis (BBN), Cosmic Microwave Background (CMB) and Large Scale Structure (LSS). Recently, it was shown that the presence of additional interactions in the sterile neutrino sector via light bosonic mediators can make the scenario cosmologically viable by suppressing the production of the sterile neutrinos from active neutrinos via matter-like effect caused by the mediator. This mechanism works assuming the initial population of this sterile sector to be negligible with respect to that of the Standard Model (SM) particles, before the production from active neutrinos. However, there is fair chance that such bosonic mediators may couple to the inflaton and can be copiously produced during (p)reheating epoch. Consequently, they may ruin this assumption of initial small density of the sterile sector. In this article we, starting from inflation, investigate the production of such a sterile sector during (p)reheating in a large field inflationary scenario and identify the parameter region that allows for a viable early Universe cosmology.

## 1 Introduction

Several anomalies from different experiments measuring neutrino oscillations have hinted towards the existence of an additional sterile neutrino species. While LSND [1, 2] and MiniBooNE [3] reported an excess in \({\bar{\nu }}_{\mu }\rightarrow {\bar{\nu }}_e\) and the latter have also indicated an excess of \(\nu _e\) in the \(\nu _{\mu }\) beam. Within a 3+1 framework, MiniBooNE result hints towards the existence of a sterile neutrino with eV mass at 4.8\(\sigma \) significance, which raises to 6.1\(\sigma \) when combined with the LSND data. Further, Daya Bay [4], NEOS [5], DANSS [6] and other reactor experiments [7, 8, 9] probed the \(\nu _e\) disappearance in the \({\bar{\nu }}_{e}\rightarrow {\bar{\nu }}_e\) channel, whereas Gallium experiments [10, 11, 12] like GALLEX [13], SAGE [14] have performed similar measurements in the \(\nu _e\rightarrow \nu _e\) channel. The \({\bar{\nu }}_e\) disappearance data also hints in favour of sterile neutrinos at \(3\sigma \) level.

However, there are significant tension among different neutrino experiments. In particular, observed excess in the experiments measuring \(\nu _{\mu } ({\bar{\nu }}_{\mu }) \rightarrow \nu _{e} ({\bar{\nu }}_e)\) appearance (i.e. LSND and MiniBooNE) are in tension with strong constraints on \(\nu _{\mu }\) disappearance, mostly from MINOS [15] and IceCUBE [16], while attempting to fit together using a 3+1 framework [17]. Thus, the existence of a light (\({\mathcal {O}}(1)\) eV) sterile neutrino within a simple 3+1 framework, as a possible resolution to the \(\nu _e\) appearance anomalies, remains debatable.^{1} However, such a light additional sterile neutrino, with mixing \(\sin \theta \lesssim {\mathcal {O}}(0.1)\) with the active neutrino species, can be consistent with constraints from various terrestrial neutrino experiments.

In the early Universe, production of a light sterile neutrino, if exists, can be significant, thanks to its sizable mixing with the active neutrinos [19, 20, 21, 22, 23, 24]. Further, inflaton decays during re-heating, or any other heavy scalar particle can possibly decay into sterile neutrinos [25, 26, 27]. Due to its sizable mixing with the active neutrinos, thermalization with the SM particles are also ensured. However, several cosmological constraints disfavor the viability of such a scenario. In particular, constraints from Big Bang Nucleosynthesis (BBN) [28, 29, 30, 31] restricts the effective number of relativistic degrees of freedom. This is because it would enhance the expansion rate at the onset of BBN (\(\simeq 1\) MeV). Depending on its mass, such a species can contribute as either matter or radiation during matter-radiation equality epoch (for more explanation see [32]). Additional non-relativistic neutrinos can also affect late-time expansion rate. Thus, it can affect the position of the acoustic peaks. Further, such light species can lead to slow down of Dark Matter clustering and thanks to their large free-streaming length [33], can wash-out small-scale structure. Thus, Cosmic Microwave Background (CMB), together with Baryon Acoustic Oscillation (BAO) [34, 35, 36] and Ly-\(\alpha \) measurements [37, 38] put forward significant constraint on the total neutrino mass \(\varSigma m_{\nu }\) as well as on the number of relativistic degrees of freedom \(N_{\mathrm{eff}}\) [39]. Both of these constraints impact the viability of an additional light sterile neutrino species.

Planck [41], assuming three neutrinos with degenerate mass, Fermi-Dirac distribution and zero chemical potential, constrains the properties of neutrinos. If they become non-relativistic after recombination, they mainly affect CMB through the change of angular diameter distance, which is degenerate with \(H_0\). So, the cleanest signal is through lensing power spectra that in turn affect the CMB power spectra. Since neutrino mass suppresses lensing whereas CMB prefers higher lensing, neutrino mass is strongly constrained by CMB lensing data. Neutrinos with large mass, that become non-relativistic around recombination, can produce distinctive features in CMB (such as reducing the first peak height) and are thus ruled out. Planck constrains \(\varSigma m_{\nu }<0.12\) eV and \(N_{\mathrm{eff}}=2.99_{-0.33}^{+0.34}\) for the 2018 dataset [40, 41] Planck TT + TE + EE + lowE + Lensing + BAO at \(95\%\) confidence level. It also constrains the effective mass of an extra sterile neutrino \(m_{\nu , {\mathrm{sterile}}}^{\mathrm{eff}}<0.65\) eV with \(N_{\mathrm{eff}}<3.29\) for the same dataset and same confidence level (though this value depends on chosen prior).^{2}

While within the paradigm of standard cosmology all these constraints impact on the viability of a light sterile neutrino with sizable active-sterile mixing, it has been shown that the CMB constraints can be partly relaxed going beyond the standard cosmology modifying the primordial power spectrum at small scales [42], within the paradigm of modified gravity [43], within Beyond Standard Model (BSM) physics with time-varying Dark Energy component [44], by light dark matter [45], from large lepton asymmetry [46]. It has been also pointed out that additional interactions in the sterile neutrino sector can also render such a scenario viable [47, 48, 49, 50, 51, 52]. The presence of such interaction leads to the suppression of the active-sterile mixing angle and thus delay the production. This significantly reduce the sterile neutrino abundance in the early Universe. In addition, it provides a mechanism to cut-off the free-streaming length of the sterile neutrinos at late time, and opens up an annihilation channel for the same. However, suppression of this production alone does not suffice to constrain the energy density in the sterile neutrino sector. In addition, one also needs to assume that post-inflationary production of sterile neutrino and the light mediator, at least from the inflaton decay, remains small. During the re-heating epoch, considering perturbative decay of the inflaton, this can be ensured by simply assuming that the branching ratio of the inflaton into the sterile sector particles remain insignificant compared to the Standard Model (SM) particles. However, even this additional consideration does not serve the purpose when a bosonic mediator is invoked. The reason is that post-inflationary particle production can be significant during preheating [53, 54]. While light fermions, which couples to the inflaton, are not produced in abundance during this epoch, the same does not hold for bosons. A (light) boson, which couples to the inflaton (via a quartic and/or tri-linear coupling, say) can be copiously produced during this non-perturbative process, thanks to the large Bose enhancement. Thus, while attempting to suppress the production of the sterile neutrino with secret interactions, the possibility of producing the bosonic mediator during (p)re-heating, and therefore, that of the sterile neutrinos can not be ignored.

In this article, we have considered a minimal renormalizable framework, consisting of an inflaton, the Higgs boson, and the light mediator (interacting with the sterile neutrinos) as only scalar particles to explore issues which are essential to make such a sterile neutrino sector cosmologically viable, starting from inflation. Within this framework all renormalizable terms are sketched out and their roles have been explored. Generally, the inflaton couples to the light mediator which can give rise to large effective mass during the inflationary epoch. Consequently, this prevents the light field to execute jumps of order \(\dfrac{H}{2 \pi }\), *H* being the Hubble parameter during inflation [55]. Thus, it ensures any additional contribution to the energy density from the light scalar remains negligible, which in turn, evades stringent constraint from non-observation of iso-curvature perturbation by CMB missions [56]. However, the same term can lead to the possibility of production during the preheating epoch. While the presence of a light scalar field with negligible coupling to the inflaton have been considered in literature [55], and stringent constraint on the quartic coupling of the light field have been put [57], aspects of the non-perturbative production, especially with a small quartic self-coupling, during the preheating epoch have not been considered in details. This may lead to serious issues which may destroy inflationary cosmology altogether. In this article, we have explored the production of the scalar mediator during (p)reheating and subsequent production of \(\nu _s\), explicitly stressing on regions of the parameter space where the production of the light mediator, and consequently, the light sterile neutrino can be significant at the on-set of BBN, and also the regions where such a sterile sector may be viable. We then discuss about some benchmark parameter values elaborating the same.

The paper is organized in the following order. In Sect. 2 the model has been discussed. In the following Sect. 3 constraints on the relevant model parameters from inflation, stability of the potential has been described. Further, the constraints already present in the literature on such secret interactions has been sketched. Subsequently, in Sect. 4 we discuss the production of the light pseudoscalar during preheating and estimate the abundance of the sterile neutrinos in details. Finally, in Sect. 5 we summarize our findings.

## 2 Construction of the minimal framework

*H*is given by,

*H*or \(\chi \) to drive inflation will be discussed in Sect. 3.1.1. The final energy fraction transferred to \(\chi \) or

*H*through \(\lambda _{\phi \chi }\phi ^2\chi ^2\) or \(\lambda _{\phi H}\phi ^2 H^\dagger H\) interaction is weakly dependent on the \(\lambda _{\phi \chi }\) or \(\lambda _{\phi H}\) couplings if the couplings are greater than a certain threshold value and energy is evenly distributed in \(\phi \), \(\chi \) and

*H*fields after preheating. Since back scattering \(\phi \) particles to \(\chi \) or

*H*is not much effective for energy transfer from the energy density left in inflaton to other sectors [58], we consider trilinear term(s) for energy transfer typical of reheating. Inflaton-sterile neutrino coupling (of the form \(\phi \nu _s\bar{\nu _s}\)) does not help in the case we are concerned about, because it results in the total energy of the inflaton flowing into the sterile neutrino sector making the energy density in sterile sector and SM sector comparable, which ruins the \(N_{\mathrm{eff}}\) bound at BBN. So we consider the trilinear terms \(\frac{\sigma _{\phi \chi }}{2} \phi \chi ^2 \) and \( \frac{\sigma _{\phi H}}{2} \phi H^{\dagger }H\). We have neglected \(\frac{\sigma _{\chi H}}{2} \chi H^{\dagger }H\) term at tree level, since it may give rise to mixing between

*H*and \(\chi \) once Higgs get vev after EWPT. The inflaton decay rate arising due to a trilinear term \(\frac{\sigma _{\phi \chi }}{2} \phi \chi ^2\) is given by,

*i*by decay of the inflaton depends on the branching ratio defined by, \({\mathcal {B}}_i=\frac{\varGamma _{i}}{\varSigma \varGamma _i}\).

## 3 Cosmology and light sterile neutrinos: initial constraints

### 3.1 Parameters of the scalar potential

As observed in Sect. 2, there are independent parameters in the minimal potential required for a viable cosmological scenario but in no way they can be of arbitrary values. Rather, we expect them to be tightly constrained due to impositions by a successful inflationary paradigm as well as by phenomenological requirements of the current framework.

#### 3.1.1 Inflation, quantum corrections and threat to flatness of potential

We begin with exploring the requirements for a successful inflationary paradigm. In this setup, primarily we have two scalar fields, namely, the SM Higgs and \(\chi \). So, at first we argue why we are using a separate inflaton field rather than using the scalar fields we already have, namely Higgs and \(\chi \).

Inflation with the Higgs field is a well studied subject [59]. It has been shown that Higgs as inflaton requires a large non-minimal coupling of order \(\xi \sim 50{,}000\), which can result in so called unitarity violation [60]. Moreover, if \(\lambda _{ H}\) becomes negative at high field values (which is certainly the case during inflationary energy scales), the non-minimal coupling is of no help to drive inflation. However, some non-standard cases has been explored recently that enables the Higgs to be the inflaton. It has been found out that successful Higgs inflation can take place even if the SM vacuum is not absolutely stable [61]. It is also to be noted that if the action may be extended by an \(R^2\) term on top of the Higgs non-minimal coupling to *R*; the Higgs field may drive inflation [62]. However, in this work we do not get into these scenarios and hence do not consider the Higgs to be the inflaton candidate.

Using \(\chi \) as inflaton is also problematic because sizable \(\chi -H\) coupling is needed in order to have enough energy flow to the Higgs sector (and thus to the SM sector) but at the same time, this will induce a mass term to the \(\chi \) field and consequently \(\chi \) cannot be light enough as required to evade \(\varSigma m_\nu \) bound [51]. Therefore we consider a separate inflaton field \(\phi \).

At high energy scales (\(\sim 10^{-5}\) \(\hbox {M}_{\mathrm{Pl}} \)), affected mostly by quantum corrections from top quark Yukawa coupling, the Higgs quartic coupling \(\lambda _H\) becomes negative [67]. However, a positive value of \(\lambda _H\) is required for a successful preheating phase, and will be shown in Sect. 4 that the energy flow to the Higgs sector during preheating explicitly depends on its value during preheating. Higgs stability is a well studied subject [67, 68, 69, 70] and ways to resolve the consequences during inflation are also well known [71, 72, 73, 74, 75, 76, 77]. As in this work we are only interested in a successful preheating dynamics, we shall assume that \(\lambda _H\) stays positive during this era. This can be easily achieved with help of another scalar coupled to Higgs, and as a result of this new coupling the new scalar thermalises with SM sector. But we do not explicitly introduce the scalar in the discussion and simply assume \(\lambda _H\) to be positive during preheating.

#### 3.1.2 Requirement of small mass for \(\chi \) and \(\nu _s\)

If inflation is driven by quartic potential along with a trilinear term involving inflaton and another field (e.g. \(\chi \)) it gets vev at the end of the inflation. This is problematic from model-building perspective as the vev of inflaton and \(\chi \) result in mass terms for \(\chi \) and \(\nu _s\) respectively but we want the particles \(\chi \) and \(\nu _s\) to be of small masses \({\mathcal {O}}(eV)\). So, we would need extreme fine tuning in this case. On the other hand if the inflaton has a quadratic term then it is possible to have the minima of the potential at 0 field values. This is why we mainly consider quadratic inflation for our case. Nevertheless, even if we choose to ignore the quartic term at some energy scale (i.e. set it to 0), it will become nonzero at other energy scales due to RGE running.

#### 3.1.3 Iso-curvature perturbations and stability of light fields during inflation

Presence of light scalar fields during inflation can lead to iso-curvature perturbation [55]. The relaxation time scale for a quantum fluctuation to roll back down to its minima is \(m^{-1}\) (Note that during inflation, a field coupled to inflaton has mass depending on the expectation value of the inflaton \(\phi _0\), if its bare mass is negligible), whereas the time scale for the evolution of the universe is given by \(H^{-1}\). So, if \(m>H\), then the field is stable and the curvature perturbations due to that field may be neglected to be in agreement with constraint from non-observation of iso-curvature perturbation by CMB missions [56]. This condition translates to \(\sigma _{\phi \chi }\langle \phi _0\rangle +\lambda _{\phi \chi }\langle \phi _0^2\rangle >m_{ \phi }^2\phi _0^2/M_{Pl}^2\) for quadratic inflation. Keeping \(\sigma _{\phi \chi }/M_{Pl}\) or \(\lambda _{\phi \chi }\) substantially larger than \(m_{\phi }^2/M_{Pl}^2\) can stabilize those fields during inflation. This is a condition we ensure while choosing these parameter values.

### 3.2 Interaction parameters and bounds on \(m_{\chi }-g_s\) plane

*C*is the collision operator. We are interested in finding the spectrum of the sterile neutrinos through collision operators corresponding to \(\chi \chi \longrightarrow \nu _s \nu _s\) and oscillation from active neutrinos. The Boltzman equation with the entire spectrum \(f_i\) is difficult to solve even numerically, so it is assumed that the \(\chi \) particles should follow a thermal distribution, i.e. \(\chi \) particles are thermalized among themselves. For this assumption to be true just we need a parameter region of \(\lambda _{\chi }\) estimated by the relation:

*H*is the Hubble parameter, \(\sigma \) is the interaction cross-section, \(v_{\mathrm{mol}}\) is the Moller velocity of \(\chi \) and n is the total number density. For our model, during radiation dominated epoch, for \(\chi ~\chi \rightarrow \chi ~\chi \) scattering, \(n_{\chi }=\frac{3}{4}\frac{\zeta (3)}{\pi ^2}T_{\chi }^3 \) and \(\sigma =\frac{4\pi }{64\pi ^2 s}36 \lambda _{ \chi }^2\sim \frac{\lambda _{ \chi }^2}{T_{\chi }^2}\), gives \(\varGamma \sim T_{\chi }\lambda _{ \chi }^2\); whereas the Hubble parameter is given by \(H=\sqrt{\frac{1}{3 M_{Pl}^2}\frac{\pi ^2}{30}g_\star } T_{SM}^2\). As temperature of any relativistic species goes down at same rate 1 /

*a*, even if the temperature of \(\chi \) and SM are different, \(\varGamma \) eventually becomes lower than H and gets thermal distribution. An estimate of the thermalisation temperature shows \(\lambda _{ \chi }\sim 10^{-8}\) gives \(T_{\mathrm{Ther}}\sim 10^{-16}M_{Pl}\) (assuming \(T_{SM}\sim T_{\chi }\), as even much lower energy density means temperature difference of order (density)\(^{1/4}\)), which is far before BBN.

The thermalization process within some sector starts much before the interaction rate (calculated from scattering of the particles) becomes comparable to the Hubble rate. It is well known that at the start of the preheating epoch, modes with only some specific wave numbers gets excited exponentially governed by Mathieu equation. But, it has been observed from the LATTICEEASY simulation that, even if at the start of preheating stage, only some specific range of infrared momentum modes gets excited, as time progresses, the energy gets distributed to higher momentum modes. This observation can be interpreted as start of thermalisation process at the end of preheating [78].

The thermalisation of \(\chi \) and \(\nu _s\) is governed by the interaction \(\chi ~\chi \rightarrow \nu _s~\nu _s\), having \(\varGamma =\frac{3}{4}\frac{\zeta (3)}{\pi ^2}T_{\chi }^3 \frac{g_s^4}{8\pi T_{\chi }^2}\). This means for \(g_s\sim 10^{-4}\) the thermalisation happens at 1 GeV [50].

*Bounds on* \(m_{\chi }-g_s\) *plane*

**(i) From BBN**The standard way to parameterize the radiation energy density (\(\rho _R\)) is like [79],

^{3}

**(ii) From CMB and LSS** The physical condition for the observed power spectrum in CMB and LSS is not only to have the active neutrinos free-streaming, but also another sterile neutrino species (with \({\mathcal {O}}(eV)\) mass) not to be free-streaming. If this new species is of with similar number density as that of the active neutrinos, then there is much suppression in the power spectrum than that observed in CMB or LSS. So, to satisfy the CMB and LSS constraints of \(\varSigma m_{\nu }\) (which basically quantifies the total mass of free-streaming species, assuming the same number density as that of active neutrinos), any massive species must annihilate or decay into lower mass particles to evade the mass constraints all together or they must interact among themselves or with some other species in order to cut off the free-streaming length scale. Reference [51] used the first recipe^{4} to evade the mass bounds coming from CMB and LSS observations. They chose the mediator mass to be \({\mathcal {O}}(<0.1~eV)\) which meant that the interaction \(\nu _s~\nu _s \rightarrow \chi ~\chi \) goes only in the forward direction once the temperature of the Universe goes below \({\mathcal {O}}(1~eV)\), i.e. the backward interaction becomes kinematically inaccessible due to Hubble expansion dominating over this process. Thus, for this suitable choice, the free-streaming length scale (as per CMB and LSS) need not be cut off since the sterile neutrinos annihilates into \(\chi \) particles only, thereby not hurting the mass bound. The gray region to the right of \(m_{\chi }=0.1\) eV in Fig. 1 shows the excluded region by such arguments [51].

**(iii) From Supernova** Constraints from SNe observations [85, 86] also does not allow the couplings to be \(g_s>10^{-4}\) [52]. So, this leaves us with a patch of parameter space at the lower left corner of the BBN allowed region as depicted in Fig. 1.

## 4 Production of sterile sector from (p)reheating

Initial stage that can be treated mostly analytically.

Back-reaction dominated stage for which one needs a detailed numerical treatment.

### 4.1 Initial stage of (p)reheating

*z*.

Mathieu equation has well known unstable exponential solutions for instability regions of \(A-q\) parameter space and hence for specific *k* values. Modes corresponding to these *k* values grow exponentially, which is interpreted as exponential particle production in those modes.

The statements up to now is only true for the initial stages of preheating. The growth of the fluctuations give rise to a mass term \(\lambda _{\phi \chi }\langle \chi ^2\rangle \) to the *e.o.m* of the zero mode of the inflaton and as a result affects other modes through Eq. 12. This phenomenon is known as the back-reaction effect [54]; after the back-reaction effect starts, Eq. 12 does not hold true. The back-reaction effect is usually estimated by the Hartree approximation, but still it does not take care of effects like re-scattering. So the only way to fully solve the* e.o.m* of the fluctuations throughout the preheating period is by lattice simulation. These simulations solve the classical field equations in lattice points numerically and give far accurate results than approximate analytic solutions.

### 4.2 Numerical evolution for back-reaction dominated stage

To simulate preheating evolution numerically we use the publicly available code LATTICEEASY [87]. In this section we start from the simplest potential and discuss the results, motivating towards the final model. In each case we discuss the pros and cons of the potential in hand. As we proceed we add new terms to the potential and clarify the implicit assumptions (as mentioned earlier) needed to reconcile the model with cosmological observations. It is to be noted that, we neglect the effect of non-minimal coupling to gravity on the potential during the (p)reheating era, as a small coupling suffices to bring the inflationary scenario to the sweet spot of \(n_s-r\) plane for a quadratic potential of inflaton. This is a logical assumption, as the potential remains unchanged near the minima of the inflaton for small non-minimal coupling, and the preheating is efficient when the inflaton oscillates about its minima.

^{5}but also on the self-quartic coupling of the field. A self-quartic coupling blocks the energy flow to that sector, as evident from the plots (Fig. 2). This salient feature is also in agreement with the studies in Refs. [88, 89]. The reason for this phenomenon is the extra energy cost due to the potential term, which blocks the modes to grow. It can also be interpreted from the following perspective: as the Fourier modes of \(\chi \) grow, the quartic behaves like an effective mass term \(\frac{1}{2}\lambda _{\phi \chi }\langle \chi ^2\rangle \chi ^2\), making it difficult for the particle to be produced. This piece of information is very vital for our work as we control the flow to \(\chi \) sector by this self quartic term.

If there is no trilinear coupling of the fields to inflaton, the inflaton cannot fully decay to other fields, thus contributing a significant amount to the total energy density of the Universe [58], as mentioned in Sect. 2. Therefore, in order to direct the flow of the energy density stored in inflaton to other species, we introduce trilinear couplings of inflaton to other scalars, as evident from the choice of potential in Eq. 2. In the subsections below we will discuss the plausible scenarios case by case.

#### 4.2.1 Trilinear interactions of inflaton with Higgs only

^{6}; also \(g_{sterile}^{final}=(\frac{7}{8}\times 2+1)=2.75\) and \(g_{SM}^{final}=3.36\). Further, we have not taken into account the details of the thermalization post-preheating (it has been shown in [78] that the thermalization process starts at the end of preheating), and assumed comoving entropy conservation. In Figs. 4 and 5 (left panel) we plot \(\triangle N_{\mathrm{eff}}\) for some benchmark values of the parameters (here \(\triangle N_{\mathrm{eff}}\) corresponds to the whole sterile sector, i.e. pseudoscalar and sterile neutrino, where the sterile neutrino and pseudoscalar are thermalised, which is indeed the case for \(g_s \sim 10^{-4}\). Note that in Fig. 1, we considered only contributions from sterile neutrinos in \(\triangle N_{\mathrm{eff}}\)) and try to demonstrate the parameter values which satisfy the \(N_{\mathrm{eff}}\) constraints at BBN. Note that, in this case we have trilinear coupling of inflaton only to the Higgs, making sure that decay of inflaton does not populate \(\chi \) sector. It is observed that even if a set of parameter values does not satisfy the \(N_{\mathrm{eff}}\) constraints (Fig. 5 (left panel), top plot for small \(\lambda _{ \chi }\)), increasing the self-quartic coupling \(\lambda _{ \chi }\) only can give a viable cosmological scenario again.

#### 4.2.2 Trilinear interactions of inflaton with \(\chi \) only

Inflaton with trilinear interactions with only \(\chi \) is trivially not satisfied by the \(N_{\mathrm{eff}}\) constraints as in this case the total inflaton energy after preheating flows into the \(\chi \) sector.

#### 4.2.3 Trilinear interactions with both Higgs and \(\chi \)

*H*and \(\chi \) sectors will thermalise with each other through a \(HH\rightarrow \chi \chi \) scattering mediated through inflaton, resulting in a fully thermalised \(\chi \) species with the SM, thereby trivially not respecting the \(N_{\mathrm{eff}}\) bounds of BBN.

### 4.3 Allowed benchmark parameter values and additional constraint on \(m_{\chi }-g_s\) plane

## 5 Conclusions and outlook

\(\lambda _{\chi }\) needs to be kept large to suppress the preheating production of \(\chi \).

\(\sigma _{\phi H}\) needs to be small in order to have a non-relativistic phase of inflaton before it decays and dilute the relic of \(\chi \) from preheating. \(\sigma _{\phi \chi }\) needs to be much smaller to prevent \(\chi \) population during inflaton perturbative decay.

\(\lambda _{\chi H}\) needs to be negligible to prevent SM from thermalizing with the BSM sector.

\(\lambda _{\phi H}\) and \(\lambda _{\phi \chi }\) cannot be large, or else the flatness of inflaton potential will be ruined due to RG running.

Future CMB missions like LiteBIRD [91], COrE [92], PIXIE [93], CMB S4 [94], CMB Bharat [95] aim at constraining the inflationary observables and the other cosmological parameters further and hence will be an important probe for this model. Results from the neutrino experiments MicroBooNE [96] may decidedly prove the existence of the sterile neutrino. Sterile neutrinos with secret interactions have been proposed to be looked for in IceCube experiment [97]. In CMB polarization observations of BICEP the sterile neutrinos may also be a relevant signature to look for as per Ref. [98].

## Footnotes

- 1.
Recently a new idea involving sterile neutrinos altered dispersion relations was shown to satisfy all the existing anomalies [18].

- 2.
In Planck 2015 results higher neutrino mass was allowed, though not favored. The effect of higher neutrino mass was nullified by larger primordial power spectrum amplitude \(A_s\). This was allowed in Planck 2015 because of the degeneracy between optical depth \(\tau \) and \(A_s\), as a result of missing data points in low-multipole EE spectra, which could break the degeneracy. Planck 2018 results constrain the \(A_s\) to lower values, disfavoring higher \(\varSigma m_{\nu }\).

- 3.
The decay of \(\chi \) of mass \(\sim 0.1\) eV occurs after the decay time scale \(\gamma \varGamma ^{-1}\) (where \(\varGamma \) is the decay width of \(\chi \) in its rest frame and \(\gamma \) factor comes due to time dilation for a relativistic particle) and the age of the universe at some epoch is \(\sim H^{-1}\). Therefore, for \(\chi \) to be still present during BBN, we require \(\gamma \varGamma ^{-1}>H_{BBN}^{-1}\), i.e. \(\gamma >9.26\times 10^{13} \frac{m_{\chi }}{eV} g_s^2\). Picking a conservative \(g_s\sim 10^{-4}\) gives \(\gamma >10^5\) which condition is easily satisfied for a 0.1 eV particle during BBN (\(\sim \) MeV). It is also to be noted that in the calculation we have ignored the mixing angle term, which will suppress the decay width further.

- 4.
Reference [83] followed the second route, i.e., they used strong interaction strength for the sterile neutrinos to self-interact via a secret mediator to cut off the free-streaming length. But, recent studies [84] have shown that this scenario generates interactions between the active neutrinos too, through flavor mixing (note that the suppression in oscillation is lifted off for most of the parameter space for eV scale and below), leading to a higher amplitude in power spectrum than the vanilla model itself. So, having a large interaction strength (with help of large coupling and/or small mediator mass) is perilous for the cosmological observables, if one relies only on cutting off the free-streaming length.

- 5.
If the coupling is lower than a certain threshold value, the preheating becomes inefficient (see Ref. [88] for details), whereas for values of that coupling over the threshold, the amount of energy flow is weakly dependent on the coupling.

- 6.
In this work, for computational simplicity, we have considered only one Higgs scalar degree of freedom (d.o.f), as is the case in the unitary gauge. However, we have checked for some points in parameter space, that considering four d.o.f of Higgs does not change our results significantly. This is because, in this case, the energy density in each d.o.f of the Higgs doublet after preheating is lower than the scenario where only one scalar Higgs d.o.f is considered. The suppression in energy density in each Higgs d.o.f is caused by additional blocking from the cross-terms in the latter case. It should also be noted that, the fraction of Inflaton decaying into Higgs in the four d.o.f case results in a Higgs sector with lower temperature. So, although there are four d.o.f, total energy density in the Higgs sector is not significantly different from the one d.o.f case, and hence our result does not change significantly (for example, for the parameter values \(m_{\phi }=10^{-6}~M_{Pl},~\lambda _{\phi }=10^{-14},~\lambda _{H}=10^{-4},~\sigma _{\phi H}=10^{-8}~M_{Pl},~\lambda _{\phi \chi }=\lambda _{\phi H}=10^{-6},~\lambda _{ \chi }=10^{-9}\), \(\triangle N_{\mathrm{eff}}\) changes from 4.8 to 3.5 for the four d.o.f case).

## Notes

### Acknowledgements

The authors gratefully acknowledge the use of the publicly available code, LATTICEEASY and also thank the computational facilities of Indian Statistical Institute, Kolkata. A.P. thanks CSIR, India for financial support through Senior Research Fellowship (File no. 09/093 (0169)/2015 EMR-I). A.C. acknowledges support from Department of Science and Technology, India, through INSPIRE faculty fellowship (Grant no: IFA 15 PH-130, DST/INSPIRE/04/2015/000110). A.G. is indebted to Davide Meloni for various support and encouragement and the hospitality of Indian Statistical Institute. Authors like to thank Gary Felder, Anupam Mazumdar, Alessandro Strumia and Subhendra Mohanty for discussions.

## References

- 1.C. Athanassopoulos et al. [LSND Collaboration], Phys. Rev. Lett.
**75**, 2650 (1995). arXiv:nucl-ex/9504002 - 2.A. Aguilar-Arevalo et al. [LSND Collaboration], Phys. Rev. D
**64**, 112007 (2001). arXiv:hep-ex/0104049 - 3.A.A. Aguilar-Arevalo et al. [MiniBooNE Collaboration], Phys. Rev. Lett.
**121**(22), 221801 (2018). arXiv:1805.12028 [hep-ex] - 4.F.P. An et al. [Daya Bay Collaboration], Phys. Rev. Lett.
**117**(15), 151802 (2016). arXiv:1607.01174 [hep-ex] - 5.Y.J. Ko et al. [NEOS Collaboration], Phys. Rev. Lett.
**118**(12), 121802 (2017). arXiv:1610.05134 [hep-ex] - 6.I. Alekseev et al., JINST
**11**(11), P11011 (2016). arXiv:1606.02896 [physics.ins-det] - 7.G. Mention, M. Fechner, T. Lasserre, T.A. Mueller, D. Lhuillier, M. Cribier, A. Letourneau, Phys. Rev. D
**83**, 073006 (2011). arXiv:1101.2755 [hep-ex]ADSGoogle Scholar - 8.
- 9.P. Huber, Phys. Rev. C
**84**, 024617 (2011) [Erratum: Phys. Rev. C**85**, 029901 (2012)]. arXiv:1106.0687 [hep-ph] - 10.M. Laveder, Nucl. Phys. Proc. Suppl.
**168**, 344 (2007)ADSGoogle Scholar - 11.
- 12.
- 13.F. Kaether, W. Hampel, G. Heusser, J. Kiko, T. Kirsten, Phys. Lett. B
**685**, 47 (2010). arXiv:1001.2731 [hep-ex]ADSGoogle Scholar - 14.J.N. Abdurashitov et al. [SAGE Collaboration], Phys. Rev. C
**80**, 015807 (2009). arXiv:0901.2200 [nucl-ex] - 15.P. Adamson et al. [MINOS Collaboration], arXiv:1710.06488 [hep-ex]
- 16.M.G. Aartsen et al. [IceCube Collaboration], Phys. Rev. Lett.
**117**(7), 071801 (2016). arXiv:1605.01990 [hep-ex] - 17.
- 18.D. Doring, H. Pas, P. Sicking, T.J. Weiler, arXiv:1808.07460 [hep-ph]
- 19.R. Barbieri, A. Dolgov, Nucl. Phys. B
**349**, 743 (1991)ADSGoogle Scholar - 20.K. Enqvist, K. Kainulainen, J. Maalampi, Nucl. Phys. B
**349**, 754 (1991)ADSGoogle Scholar - 21.R. Barbieri, A. Dolgov, Phys. Lett. B
**237**, 440 (1990)ADSGoogle Scholar - 22.K. Kainulainen, Phys. Lett. B
**244**, 191 (1990)ADSGoogle Scholar - 23.
- 24.A. Merle, A. Schneider, M. Totzauer, JCAP
**1604**(04), 003 (2016). arXiv:1512.05369 [hep-ph]ADSGoogle Scholar - 25.
- 26.
- 27.
- 28.R. Cooke, M. Pettini, R.A. Jorgenson, M.T. Murphy, C.C. Steidel, Astrophys. J.
**781**(1), 31 (2014). arXiv:1308.3240 [astro-ph.CO]ADSGoogle Scholar - 29.M. Tanabashi et al. [Particle Data Group], Phys. Rev. D
**98**, 030001 (2018)Google Scholar - 30.R.J. Cooke, M. Pettini, C.C. Steidel, Astrophys. J.
**855**(2), 102 (2018). arXiv:1710.11129 [astro-ph.CO]ADSGoogle Scholar - 31.G. Mangano, P.D. Serpico, Phys. Lett. B
**701**, 296–299 (2011). arXiv:1103.1261 [astro-ph.CO]ADSGoogle Scholar - 32.J. Lesgourgues, G. Mangano, G. Miele, S. Pastor,
*Neutrino Cosmology*(Cambridge University Press, Cambridge, 2013)zbMATHGoogle Scholar - 33.J.R. Bond et al., Phys. Rev. Lett.
**45**(1980)Google Scholar - 34.S. Alam et al. [BOSS Collaboration], Mon. Not. R. Astron. Soc.
**470**(3), 2617 (2017). arXiv:1607.03155 [astro-ph.CO] - 35.P. Zarrouk et al., Mon. Not. R. Astron. Soc.
**477**(2), 1639 (2018). arXiv:1801.03062 [astro-ph.CO]ADSMathSciNetGoogle Scholar - 36.
- 37.N. Palanque-Delabrouille et al., JCAP
**1511**(11), 011 (2015). arXiv:1506.05976 [astro-ph.CO]ADSGoogle Scholar - 38.C. Yeche, N. Palanque-Delabrouille, J. Baur, H. du Mas des Bourboux, JCAP
**1706**(06), 047 (2017). arXiv:1702.03314 [astro-ph.CO]ADSGoogle Scholar - 39.G. Mangano, G. Miele, S. Pastor, M. Peloso, Phys. Lett. B
**534**, 8 (2002). arXiv:astro-ph/0111408 ADSGoogle Scholar - 40.S. Vagnozzi, E. Giusarma, O. Mena, K. Freese, M. Gerbino, S. Ho, M. Lattanzi, Phys. Rev. D
**96**(12), 123503 (2017). arXiv:1701.08172 [astro-ph.CO]ADSGoogle Scholar - 41.N. Aghanim et al. [Planck Collaboration], arXiv:1807.06209 [astro-ph.CO]
- 42.N. Canac, G. Aslanyan, K.N. Abazajian, R. Easther, L.C. Price, JCAP
**1609**(09), 022 (2016). arXiv:1606.03057 [astro-ph.CO]ADSGoogle Scholar - 43.A.S. Chudaykin, D.S. Gorbunov, A.A. Starobinsky, R.A. Burenin, JCAP
**1505**(05), 004 (2015). arXiv:1412.5239 [astro-ph.CO]ADSGoogle Scholar - 44.E. Giusarma, M. Archidiacono, R. de Putter, A. Melchiorri, O. Mena, Phys. Rev. D
**85**, 083522 (2012). arXiv:1112.4661 [astro-ph.CO]ADSGoogle Scholar - 45.C.M. Ho, R.J. Scherrer, Phys. Rev. D
**87**(6), 065016 (2013). arXiv:1212.1689 [hep-ph]ADSGoogle Scholar - 46.
- 47.K.S. Babu, I.Z. Rothstein, Phys. Lett. B
**275**, 112 (1992)ADSGoogle Scholar - 48.K. Enqvist, K. Kainulainen, M.J. Thomson, Phys. Lett. B
**280**, 245 (1992)ADSGoogle Scholar - 49.S. Hannestad, R.S. Hansen, T. Tram, Phys. Rev. Lett.
**112**(3), 031802 (2014). arXiv:1310.5926 [astro-ph.CO]ADSGoogle Scholar - 50.M. Archidiacono, S. Hannestad, R.S. Hansen, T. Tram, Phys. Rev. D
**91**(6), 065021 (2015). arXiv:1404.5915 [astro-ph.CO]ADSGoogle Scholar - 51.M. Archidiacono, S. Hannestad, R.S. Hansen, T. Tram, Phys. Rev. D
**93**(4), 045004 (2016). arXiv:1508.02504 [astro-ph.CO]ADSGoogle Scholar - 52.M. Archidiacono, S. Gariazzo, C. Giunti, S. Hannestad, R. Hansen, M. Laveder, T. Tram, JCAP
**1608**(08), 067 (2016). arXiv:1606.07673 [astro-ph.CO]ADSGoogle Scholar - 53.L. Kofman, A.D. Linde, A.A. Starobinsky, Phys. Rev. Lett.
**73**, 3195 (1994). arXiv:hep-th/9405187 ADSGoogle Scholar - 54.L. Kofman, A.D. Linde, A.A. Starobinsky, Phys. Rev. D
**56**, 3258 (1997). arXiv:hep-ph/9704452 ADSGoogle Scholar - 55.A.A. Starobinsky, J. Yokoyama, Phys. Rev. D
**50**, 6357 (1994). arXiv:astro-ph/9407016 ADSGoogle Scholar - 56.Y. Akrami et al. [Planck Collaboration], arXiv:1807.06211 [astro-ph.CO]
- 57.K. Enqvist, S. Nurmi, T. Tenkanen, K. Tuominen, JCAP
**1408**, 035 (2014). arXiv:1407.0659 [astro-ph.CO]ADSGoogle Scholar - 58.J.F. Dufaux, G.N. Felder, L. Kofman, M. Peloso, D. Podolsky, JCAP
**0607**, 006 (2006). arXiv:hep-ph/0602144 ADSGoogle Scholar - 59.F.L. Bezrukov, M. Shaposhnikov, Phys. Lett. B
**659**, 703 (2008). arXiv:0710.3755 [hep-th]ADSGoogle Scholar - 60.
- 61.F. Bezrukov, J. Rubio, M. Shaposhnikov, Phys. Rev. D
**92**(8), 083512 (2015). arXiv:1412.3811 [hep-ph]ADSGoogle Scholar - 62.
- 63.R. Kallosh, A. Linde, D. Roest, Phys. Rev. Lett.
**112**(1), 011303 (2014). arXiv:1310.3950 [hep-th]ADSGoogle Scholar - 64.A. Linde, M. Noorbala, A. Westphal, JCAP
**1103**, 013 (2011). arXiv:1101.2652 [hep-th]ADSGoogle Scholar - 65.P. Ghosh, A.K. Saha, A. Sil, Phys. Rev. D
**97**(7), 075034 (2018). arXiv:1706.04931 [hep-ph]ADSGoogle Scholar - 66.Y. Ema, M. Karciauskas, O. Lebedev, S. Rusak, M. Zatta, arXiv:1711.10554 [hep-ph]
- 67.M. Sher, Phys. Rep.
**179**(5–6), 273–418 (1989)ADSGoogle Scholar - 68.J. Elias-Miro, J.R. Espinosa, G.F. Giudice, G. Isidori, A. Riotto, A. Strumia, Phys. Lett. B
**709**, 222 (2012). arXiv:1112.3022 [hep-ph]ADSGoogle Scholar - 69.G. Degrassi, S. Di Vita, J. Elias-Miro, J.R. Espinosa, G.F. Giudice, G. Isidori, A. Strumia, JHEP
**1208**, 098 (2012). arXiv:1205.6497 [hep-ph]ADSGoogle Scholar - 70.D. Buttazzo, G. Degrassi, P.P. Giardino, G.F. Giudice, F. Sala, A. Salvio, A. Strumia, JHEP
**1312**, 089 (2013). arXiv:1307.3536 [hep-ph]ADSGoogle Scholar - 71.J.R. Espinosa, G.F. Giudice, A. Riotto, JCAP
**0805**, 002 (2008). arXiv:0710.2484 [hep-ph]ADSGoogle Scholar - 72.K. Enqvist, T. Meriniemi, S. Nurmi, JCAP
**1407**, 025 (2014). arXiv:1404.3699 [hep-ph]ADSGoogle Scholar - 73.A. Shkerin, S. Sibiryakov, Phys. Lett. B
**746**, 257 (2015). arXiv:1503.02586 [hep-ph]ADSGoogle Scholar - 74.
- 75.J. Kearney, H. Yoo, K.M. Zurek, Phys. Rev. D
**91**(12), 123537 (2015). arXiv:1503.05193 [hep-th]ADSGoogle Scholar - 76.J.R. Espinosa, G.F. Giudice, E. Morgante, A. Riotto, L. Senatore, A. Strumia, N. Tetradis, JHEP
**1509**, 174 (2015). arXiv:1505.04825 [hep-ph]ADSGoogle Scholar - 77.J. Elias-Miro, J.R. Espinosa, G.F. Giudice, H.M. Lee, A. Strumia, JHEP
**1206**, 031 (2012). arXiv:1203.0237 [hep-ph]ADSGoogle Scholar - 78.
- 79.S. Dodelson,
*Modern Cosmology*(Academic Press, London, 2003)Google Scholar - 80.B.H.J. McKellar, M.J. Thomson, Phys. Rev. D
**49**, 2710 (1994)ADSGoogle Scholar - 81.G. Sigl, G. Raffelt, Nucl. Phys. B
**406**, 423 (1993)ADSGoogle Scholar - 82.T.D. Jacques, L.M. Krauss, C. Lunardini, Phys. Rev. D
**87**(8), 083515 (2013) [Erratum: Phys. Rev. D**88**(10), 109901 (2013)]. arXiv:1301.3119 [astro-ph.CO] - 83.
- 84.X. Chu, B. Dasgupta, M. Dentler, J. Kopp, N. Saviano, arXiv:1806.10629 [hep-ph]
- 85.
- 86.
- 87.G.N. Felder, I. Tkachev, Comput. Phys. Commun.
**178**, 929 (2008). arXiv:hep-ph/0011159 ADSGoogle Scholar - 88.
- 89.
- 90.A.D. Dolgov, S. Pastor, J.C. Romao, J.W.F. Valle, Nucl. Phys. B
**496**, 24 (1997). arXiv:hep-ph/9610507 ADSGoogle Scholar - 91.A. Suzuki et al. [LiteBIRD Collaboration], arXiv:1801.06987 [astro-ph.IM]
- 92.E. Di Valentino et al. [CORE Collaboration], JCAP
**1804**, 017 (2018). arXiv:1612.00021 [astro-ph.CO] - 93.A. Kogut et al. [PIXIE Collaboration], arXiv:1105.2044 [astro-ph.CO]
- 94.K.N. Abazajian et al. [CMB-S4 Collaboration], arXiv:1610.02743 [astro-ph.CO]
- 95.
- 96.R. Acciarri et al. [MicroBooNE Collaboration], JINST
**12**(02), P02017 (2017). arXiv:1612.05824 [physics.ins-det] - 97.B. Chauhan, S. Mohanty, arXiv:1808.04774 [hep-ph]
- 98.S. Roy Choudhury, S. Choubey, arXiv:1807.10294 [astro-ph.CO]
- 99.J.M. Cline, Phys. Rev. Lett.
**68**, 3137 (1992)ADSGoogle Scholar - 100.D. Notzold, G. Raffelt, Nucl. Phys. B
**307**, 924 (1988)ADSGoogle Scholar - 101.J.F. Nieves, S. Sahu, arXiv:1808.01629 [hep-ph]
- 102.C. Quimbay, S. Vargas-Castrillon, Nucl. Phys. B
**451**, 265 (1995). arXiv:hep-ph/9504410 ADSGoogle Scholar - 103.H.A. Weldon, Phys. Rev. D
**26**, 2789 (1982)ADSGoogle Scholar

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