Hiding an elephant: heavy sterile neutrino with large mixing angle does not contradict cosmology

TL;DR

This paper proposes a model where keV-scale sterile neutrinos with large mixing angles can exist without conflicting with cosmological constraints, especially when coupled to a hidden sector, enabling new experimental detection possibilities.

Contribution

It introduces a hidden sector framework that evades standard cosmological bounds on sterile neutrinos, allowing for laboratory detection of heavy sterile neutrinos with large mixing angles.

Findings

Hidden sector coupling suppresses sterile neutrino production in the early Universe.

Sterile neutrinos can be massless or superheavy in the early Universe within this model.

Oscillation-based generation of sterile neutrinos is significantly suppressed.

Abstract

We study a model of a keV-scale sterile neutrino with a relatively large mixing with the Standard Model sector. Usual considerations predict active generation of such particles in the early Universe, which leads to constraints from the total Dark Matter density and absence of X-ray signal from sterile neutrino decay. These bounds together may deem any attempt of creation of the keV scale sterile neutrino in the laboratory unfeasible. We argue that for models with a hidden sector coupled to the sterile neutrino these bounds can be evaded, opening new perspectives for the direct studies at neutrino experiments such as Troitsk $\nu$-mass and KATRIN. We estimate the generation of sterile neutrinos in scenarios with the hidden sector dynamics keeping the sterile neutrinos either massless or superheavy in the early Universe. In both cases the generation by oscillations from active neutrinos in plasma is suppressed.

Figures

Equations

$$T_{\max }\sim 133\text{MeV}{\left(\frac{M}{1\text{keV}}\right)}^{1/3}.$$

$$\mathcal{L}=i\bar{N}\hat{\partial }N+\frac{M}{2}\overline{N^c}N+y_{\nu }H{\bar{\nu }}_{a}N+\text{h.c.},$$

$$HT{\left(\frac{\partial f_N}{\partial T}\right)}_{E/T}=\left[\frac{1}{2}{\sin }^2(2{\theta }_M)\right]\frac{{\Gamma }_A}{2}{f}_A,$$

$${\sin }^2(2{\theta }_M)=\frac{m_D^2}{m_D^2+[c{\Gamma }_{A}E/M+M/2{]}^2},{\Gamma }_A\approx 1.27\times G_F^2{T}^{4}E,$$

$$\frac{f_N}{f_A}=\frac{2.9}{g_{\ast }^{1/2}}\left(\frac{{\theta }^2}{10^{-6}}\right)\left(\frac{M}{\text{keV}}\right){\int }_x^{x_c}\frac{yd{x}^{\prime }}{(1+y^2{x}^{\prime 2}{)}^2},$$

$$\frac{f_N(y)}{f_A(y)}\simeq 0.13\times {\theta }^2{\left(\frac{10.75}{g_{\ast }}\right)}^{1/2}{\left(\frac{T_c}{\text{MeV}}\right)}^3\cdot y.$$

$$h^2{\Omega }_N\simeq 4.3\times {\theta }^2{\left(\frac{10.75}{g_{\ast }}\right)}^{1/2}{\left(\frac{T_c}{\text{MeV}}\right)}^3\left(\frac{M}{\text{keV}}\right)$$

$$\frac{{\theta }^2}{10^{-4}}=\frac{1\text{keV}}{M}\frac{m}{0.1\text{eV}}.$$

$$\mathcal{L}=\frac{f}{2}\varphi \overline{N^c}N+\text{h.c.},$$

$$⟨⟨\varphi ⟩⟩{|}_{T_h>\xi T_c}=0,$$

$$⟨⟨\varphi ⟩⟩{|}_{T_h<\xi T_c}=v_{\varphi },M=f{v}_{\varphi }.$$

$$\frac{f_{S,\text{in}}}{f_A}\simeq \frac{m_D^2}{4y^2{T}_c^2}\simeq \frac{0.25\times 10^{-6}{\theta }^2}{y^2}{\left(\frac{M}{\text{keV}}\right)}^2{\left(\frac{\text{MeV}}{T_c}\right)}^2,$$

$$h^2{\Omega }_{N,\text{in}}=\frac{M{T}_{\nu ,0}^3}{({\rho }_c/h^2)}\frac{2}{2{\pi }^2}{\int }_0^{\infty }dy{y}^2{f}_{S,\text{in}}(y)\approx 10^{-6}{\theta }^2{\left(\frac{M}{\text{keV}}\right)}^3{\left(\frac{\text{MeV}}{T_c}\right)}^2.$$

$$T_{c,\text{min}}\simeq 0.05\text{MeV}{\left(\frac{M}{\text{keV}}\right)}^{2/5}.$$

$$h^2{\Omega }_{N,\text{min}}\simeq h^2{\Omega }_{N,T<T_C}+h^2{\Omega }_{N,\text{in}}\simeq 0.9\times 10^{-3}{\theta }^2{\left(\frac{M}{\text{keV}}\right)}^{11/5}.$$

$${\mathcal{L}}_{DS}=\frac{1}{2}(\partial \varphi {)}^2-\frac{1}{2}{m}_{\varphi }^2{\varphi }^2.$$

$$M_{N,i}=f{\varphi }_i\gg T.$$

$$H_{\text{osc}}\simeq m_{\varphi },$$

$$\varphi \propto a^{-3/2},$$

$$\left(\frac{T_{\mathrm{osc}}}{100\text{eV}}\right)\left(\frac{2.73\text{K}}{T_0}\right)\gtrsim {\left(\frac{M_{N,i}}{1\text{TeV}}\right)}^{2/3}{\left(\frac{1\text{keV}}{M}\right)}^{2/3}.$$

$${\rho }_{\varphi ,0}=\frac{1}{2}{m}_{\varphi }^2{\varphi }_i^2(1+z_{\text{osc}}{)}^{-3},$$

$${\rho }_{\varphi ,0}={ϵ}_{\varphi }{\Omega }_{\text{DM}}{\rho }_c,$$

$$m_{\varphi }^2=H_{\text{osc}}^2=H_0^2{\Omega }_{\text{rad}}{\left(1+z_{\text{osc}}\right)}^4=H_0^2{\Omega }_{\text{rad}}\frac{T_{\text{osc}}^4}{T_0^4},$$

$${\varphi }_i^2=\frac{6M_{\text{Pl}}^2}{8\pi }\frac{{\Omega }_{\mathrm{DM}}{ϵ}_{\varphi }}{{\Omega }_{\mathrm{rad}}}\frac{1}{1+z_{\text{osc}}},$$

$$1+z_{\text{osc}}=\frac{1}{{\Omega }_{\mathrm{rad}}^{1/4}}{\left(\frac{m_{\varphi }}{H_0}\right)}^{1/2}.$$

$$m_{\varphi }\sim H_{\text{osc}}=\frac{T_{\mathrm{osc}}^2}{M_{Pl}^{\ast }}\ll \frac{T_{\max }^2}{M_{Pl}^{\ast }}\lesssim 10^{-11}\text{eV}.$$

$${\sin }^{2}2\theta (M)<\frac{{\Omega }_{\mathrm{DM}}}{{\Omega }_N}{\sin }^{2}2{\theta }_{\text{X-ray}}(M).$$

$${\theta }_{\text{maxX-ray}}^2(M)=\sqrt{{\theta }_{\text{X-ray}}^2(M){\theta }_{\text{NRP}}^2(M)}.$$

$$m_{\text{NRP}}=\frac{⟨⟨p⟩⟩{|}_{f_A}}{⟨⟨p⟩⟩{|}_{f_N}}M=\frac{3.1}{4.1}M,$$

Full text

1 Introduction

Sterile neutrinos appear in the vast majority of extensions of the Standard Model (SM), see [1] for a review. In this article we concentrate on intermediate, 1–100 keV, mass range of the sterile neutrinos [2]. It is usually assumed that such neutrinos have an extremely small mixing angle with the active neutrinos, $\sin^{2}(2\theta)\ll 10^{-7}$. If this is not the case, multiple astrophysical considerations contradict the existence of such particles. The main problem is the production in the early Universe through the mixing with active neutrinos [3, 4], leading to over-closure of the Universe. At the same time, direct laboratory bounds [5, 6, 7] on the active-sterile mixing in this mass range is reaching at most the mixing angles $\sin^{2}(2\theta)\gtrsim 10^{-3}$.

We argue that in the models with additional hidden sector coupled to sterile neutrinos, the their generation in the early Universe can be arbitrarily suppressed. The key ingredient is a special dynamics of the hidden sector which changes sterile neutrino parameters responsible for their production. In the early Universe the most active production of sterile neutrinos of mass $M$ by oscillations of SM neutrinos happens at plasma temperatures [3, 8]

[TABLE]

Thus, if at this temperature the oscillations do not happen,111Possibly the most radical but rather constrained solution is to start the Hot Big Bang stage of the early Universe evolution with temperatures (much) below (1.1), see e.g. ref. [9]. the final abundance of sterile neutrino will be suppressed. In this letter we suggest two mechanisms to suppress oscillations at temperatures above some critical temperature $T_{c}\ll T_{\mathrm{max}}$.

In section 2 we outline the calculation of the sterile neutrino abundance and velocity distribution when oscillations are active only for $T\lesssim T_{c}$. Then in section 3 we consider how to suppress the sterile neutrino production. We concentrate on the natural idea that the suppression of oscillations can be achieved by making the mass of the sterile neutrino different in the early Universe (cf. variable neutrino mass models [10, 11]). Section 3.1 describes the hidden sector which undergoes phase transition, and keeps the sterile neutrino massless at high temperatures. On the contrary, in section 3.2 we make the neutrino very heavy at early times due to the coupling with a very light and extremely feebly interacting scalar, which was frozen since inflation, and started oscillations only at $T\sim T_{c}$.

Finally, in section 4 we outline the overall compatibility of the suggested framework with astrophysical observations. Section 5 contains the summary.

2 Generation of a variable mass sterile neutrino

The calculation of generation of sterile neutrino in the model of the form

[TABLE]

can be made following refs. [3, 4]. The Yukawa term in (2.1) gives the Dirac mass ${m_{D}=y_{\nu}\langle H\rangle}$, leading to the active neutrino mass $m=\theta^{2}M$ and the active-sterile mixing angle $\theta\simeq m_{D}/M$. In the primordial plasma this mixing provides oscillations of the active neutrinos into sterile ones. The analytic approximation of [3] breaks if number of d.o.f. $g_{*}$ changes during production, but for the most interesting region of parameters in the present paper we examine production only at $T\ll 100\,\text{MeV}$, so this approximation is completely justified.222Strictly speaking, everywhere in the present paper except subsection 3.2 by the temperature of the active sector we understand the temperature of the active neutrinos $T_{\nu}$. With this convention we do not have to worry about changes of the number of d.o.f. after neutrino freeze-out. Moreover, we assume that the number of active neutrinos is not depleted significantly by conversion to the sterile neutrinos. For small mixing angles $\theta$ this is true both above and below neutrino freeze-out temperature $T_{\nu,f}\sim 2\,\text{MeV}$. With this assumptions the distribution function $f_{N}(T,E/T)$ of the sterile neutrino evolves according to the equation

[TABLE]

where the effective angle and rate of active neutrino scattering in matter, in assumption of mixing only between the sterile and electron neutrino333The overall approach described here can be applied to the case of mixing with muon and tau neutrino as well, with the appropriate change of the numerical factors in $\Gamma_{A}$ and $c$. For more detailed information see e.g. [12]. [13], reads

[TABLE]

with $c\approx 63$ [12]. The solution to equation (2.2) in the expanding Universe with constant $g_{*}$ is

[TABLE]

where $y\equiv E/T$, $x\equiv 148(T/\,\text{GeV})^{3}(\,\text{keV}/M)$ with $T$ corresponding to the observation moment and $T_{c}$ to the critical temperature. Here $f_{A}(y)=1/(\mathrm{e}^{y}+1)$ is the distribution function of the active neutrinos, and $g_{*}=10.75$ is the number of d.o.f. just before the active neutrino freeze-out. At late times we can put the lower integration limit to $x=0$, and for $T_{c}\ll 100\,\text{MeV}$ we can expand the expression for small $x_{c}$, arriving at sterile neutrino distribution

[TABLE]

This is a warm distribution with average momentum $\langle\!\langle p\rangle\!\rangle=4.1T$, as compared to the usual thermal average $\langle\!\langle p\rangle\!\rangle=3.1T$. The number density for the distribution (2.5) gives the present sterile neutrino contribution to the matter density

[TABLE]

which is strongly suppressed for small $T_{c}$. In figure 1 we show the results of direct integration of (2.2) with temperature dependent $g_{*}$ and $\Gamma_{A}$ following refs. [8, 14] (for improved calculations at $T\sim T_{\mathrm{max}}$ see [14]). This demonstrates the applicability of (2.6) for the interesting range of $T_{c}$.

A remarkable thing is that with oscillations suppressed at high temperatures the sterile neutrino can be involved into the seesaw type I mechanism [15, 16, 17, 18, 19] of giving mass to the active neutrino. This is contrary to the cases with usual non-resonant or resonant dark matter sterile neutrino production (see e.g. ref. [2] for details). Indeed, typical values of the mixing angle following from the seesaw expression of active neutrino mass $m$ are

[TABLE]

For active neutrino masses of order of solar $\sqrt{\delta m_{\mathrm{sol}}^{2}}\simeq 0.9\times 10^{-2}\,\text{eV}$ or atmospheric $\sqrt{\delta m_{\mathrm{atm}}^{2}}\simeq 0.05\,\text{eV}$ neutrino masses and $T_{c}$ in MeV range, the sterile neutrino contribution to Dark Matter (2.6) can be small enough, $\Omega_{N}\ll 1$, to make astrophysical constraints irrelevant (see section 4 for details). Note, that with multiple sterile neutrinos in the model mixing angle $\theta$ can be larger than (2.7) due to cancellations in the mass matrix.

3 Possibilities with a hidden sector coupled to sterile neutrinos

To suppress the oscillations (2.2) in the early Universe plasma one must reduce the effective mixing angle in matter $\theta_{M}$ (2.3). This can be achieved by introducing a hidden sector coupled to the sterile neutrinos. Then two options are straightforward: arrange the dynamics of the hidden sector and its coupling to neutrinos in such a way that the sterile neutrinos in the early Universe become either massless or super-heavy. The first option may be realized with late-time phase transition in the hidden sector, which makes the sterile neutrino massive. This option is considered in section 3.1. The second option may be realized by either sufficiently strong interaction of sterile neutrino with hidden sector plasma (which cools down while the Universe expands) or by neutrino coupling to a classical field frozen in the early Universe and forming Bose–Einstein condensate at present. The later realization is studied in section 3.2, while investigation of the former is deferred to the future.

3.1 Phase transition in the hidden sector

This scenario allows to change the Majorana mass of sterile neutrino $M$ from zero in the early Universe (so that the sterile and active neutrino pair forms a pure Dirac fermion) to the present day value at keV scale. In this scenario we assume existence of an extended hidden sector which is connected to the active sector (SM) only via sterile neutrinos (neutrino portal). We also assume that the hidden sector has its own temperature below $T_{h}=\xi T$, with $\xi\ll 1$ (a setup similar to the one adopted in Mirror World models, for review see e.g. [20]). With the hidden sector temperature being below that of the active sector we are not concerned with the hidden sector contribution to the energy density of the Universe starting from the generation of sterile neutrinos and until today. We also assume that one of the hidden scalars444If $\phi$ is a complex scalar rather it carries the lepton charge and becomes a Majoron, leading to more complicated phenomenology, see e.g. ref. [21]. For the main purpose of our paper this is not important, and we proceed with the real scalar. $\phi$ interacts with the sterile neutrino

[TABLE]

so that the Majorana mass of the sterile neutrino entering Lagrangian (2.1) is generated by its expectation value, $M=f\langle\!\langle\phi\rangle\!\rangle$.

The required behaviour of the sterile neutrino mass is achieved in the situation with phase transition happening in the hidden sector at temperature $T_{h,c}=\xi T_{c}$, so that $\phi$ is in the symmetric state at high temperatures,

[TABLE]

and the symmetry is broken at low temperatures,

[TABLE]

With an instant phase transition in the hidden sector, the spectrum of sterile neutrinos produced at later stages of evolution is given by eq. (2.4).

We find in (2.6) that abundance of sterile neutrino can be strongly suppressed by decreasing the critical temperature $T_{c}$. Let us find the maximal suppression we can expect. In eq. (2.5) we assumed that the abundance of sterile neutrino is exactly zero at $T_{c}$. Note, that in this scenario $M$ is zero at higher temperatures, so we can approximate neutrinos as Dirac particles of mass $m_{D}$, and expect the admixture of right handed components in the primordial plasma at the level of $m_{D}^{2}/4p^{2}=m_{D}^{2}/4y^{2}T^{2}$. Therefore, in addition to (2.5), there is also an initial contribution to the sterile neutrino spectrum

[TABLE]

where we expressed $m_{D}^{2}=\theta^{2}M^{2}$ using the present day values of the mixing angle and Majorana mass. This effect makes the contribution to the sterile neutrino fraction in the energy density of the present Universe

[TABLE]

Equating the estimates for the initial (3.3) and late (2.6) contributions we get that they become equal at critical temperature555In formula (2.6) we used here $g_{*}\approx 13$ valid for the critical temperature of the active neutrino sector below $1\,\text{MeV}$.

[TABLE]

Thus, the minimal sterile neutrino abundance at given mass can be estimated as

[TABLE]

The final spectrum of sterile neutrinos in the model is given by the sum of generated (2.5) and initial (3.2) contributions, $f_{N}=f_{S,T<T_{C}}+f_{S,\text{in}}$. The former spectrum is warm, while the latter is cooled as compared to the thermal Fermi–Dirac spectrum. The presence of a cool component weakens the structure formation bounds on the DM. However, as we will see in section 4, this effect is insufficient to allow for generation of all of the DM by the described mechanism.

We do not provide a full and complete realization of the hidden sector in the present note, but rather list here several requirements that should be satisfied by the hidden sector model. The reader is invited to formulate a concrete realization. First, the phase transition should be fast enough. Second, it is preferable to have the interactions in the hidden sector in the weak coupling regime. This could put non-trivial constraints on the mass spectrum of the sector. Third, our analysis is applicable only if the interactions due to the Yukawa term in (3.1) do not significantly modify the oscillation picture, that is the rate of sterile neutrino scattering is be much smaller than $\Gamma_{A}$.

3.2 Feebly interacting scalar

Let us investigate the opposite dynamics in the scalar sector: instead of rapidly interacting sector, we use a free massive scalar which has only a feeble Yukawa coupling (3.1) to sterile neutrino, $f\ll 1$. Let the scalar be extremely light. The model Lagrangian for the dark sector contains, in addition to (3.1),

[TABLE]

The radiatively induced scalar self-coupling $\sim f^{4}/(4\pi)^{2}$ is negligibly small, and we consider again only the real field $\phi$. Due to the Hubble friction, the scalar field remains constant in the hot Universe, and its field value $\phi_{i}$ is determined by the pre-Big-Bang history: either inflation or preheating. In particular, as far as the scalar is effectively massless at inflation, its value can be more or less arbitrary at inflationary stage in the observed Hubble patch of the Universe.

If the initial field is large, the sterile neutrinos can be very heavy. Let us assume the Yukawa coupling $y_{\nu}$ in (2.1) is sufficiently small to prevent the direct sterile neutrino production in the hot plasma of the early Universe. Then after the EW transition the sterile neutrino production can be forbidden by kinematics, if sterile neutrinos are heavy enough,

[TABLE]

For example, sterile neutrinos of $M_{N,i}=200\,\text{GeV}$ will not be produced. Much later, when the Hubble rate drops below the scalar field mass,

[TABLE]

the scalar field starts to oscillate with decreasing amplitude,

[TABLE]

and hence the scalar contribution to neutrino mass drops as well. Thus, at late times the scalar contribution (3.1) to the sterile neutrino mass becomes negligible, and the present day mass is given by the bare mass term in (2.1).666Another option to have neutrino massive after the oscillation start is non-renormalizable interaction (instead of he Yukawa coupling) $f\frac{\phi^{2}}{\Lambda}\bar{N}^{c}N$ with an interesting option of gradually decreasing sterile neutrino mass including the present epoch; we do not study this option here, see however ref. [22, 23] for the case of similar mechanism for active neutrino.

With the temperature of the onset of scalar field oscillations $T_{\text{osc}}$ being sufficiently high, but lower than $T_{\mathrm{max}}$ (1.1), we can face the situation when the present day contribution of (3.1) to sterile neutrino mass is negligible, and sterile neutrino production via oscillations in the early Universe (2.2) is strongly suppressed as well.

Let us estimate the minimal value of critical temperature $T_{c}$ leading to the described scenario. For this, we assume that the interaction (3.1) does not contribute significantly to the present mass, $f\phi_{0}\ll M$. Together with the assumption of MD scaling of the field (3.8) during its oscillations this leads to the inequality777Hereafter in this subsection $T$ denotes the photon temperature.

[TABLE]

This means that the described picture breaks only for $T_{\text{osc}}$ below $100\,\text{eV}$. One concludes that the sterile neutrino mass can start from TeV scale and reach keV range by the present epoch. In this case sterile neutrinos always remain non-relativistic in plasma and hence completely avoid production in oscillations: no contribution to the relic abundance from eq. (2.6). Thus, the sterile neutrino production by oscillations, which might happen in the plasma after the electroweak phase transition can be ultimately suppressed in this model.

As an interesting additional feature, the hidden scalar of the model of this section can be a DM candidate.888Unless the oscillations begin at plasma temperature below $200\,\text{eV}$, which is the latest time when Dark Matter must appear in the Universe [24]. Let us consider the late-time cosmology of the hidden sector, assuming the oscillations starting at the radiation domination, $T_{\text{osc}}>T_{\mathrm{RD/MD}}$. The energy remains in the oscillating scalar all the way until the present time. The scalar relic in the Universe contributes to the present dark matter component an amount of

[TABLE]

where $z_{*}$ is redshift of the oscillation epoch (3.7). The scalar field saturates the dark matter if

[TABLE]

with $\epsilon_{\phi}=1$, otherwise contributes a fraction of $\epsilon_{\phi}$ to the dark matter density. The oscillations onset is given by the condition (3.7), and for $T_{\mathrm{osc}}\lesssim 1\,\text{MeV}$ one obtains999For higher oscillation onset temperature $1\,\text{MeV}<T_{\mathrm{osc}}<T_{\mathrm{max}}$, that is for heavier scalars, the calculation should be straightforwardly corrected for the non-constant number of degrees of freedom in the primordial cosmic plasma.

[TABLE]

with $H_{0}$ and $T_{0}$ referring to the present Hubble parameter and relic photon temperature. Then from eqs. (3.10), (3.9), (3.11) one finds for the initial amplitude

[TABLE]

where the redshift of the oscillation onset is

[TABLE]

With oscillation temperature constrained from above by (1.1), $T_{\mathrm{osc}}\ll T_{\mathrm{max}}\sim 100\,\text{MeV}$, the scalar mass must be below

[TABLE]

Evidently, this forbids scalar perturbative decay into keV scale sterile neutrinos. The production by oscillating scalar at zero neutrino mass crossing is also suppressed in the wide region of parameter space.

Overall, for the mechanism outlined in this section the DM can be explained by the oscillating scalar field condensate. The active-sterile neutrino mixing can be arbitrary large as far as cosmology and astrophysics are concerned.

4 Discussion of cosmological and astrophysical constraints

Sterile neutrino models are constrained by astrophysical observations and by direct searches. We start with summarising the present status of astrophysical constraints and combine them with the generation mechanisms described in the previous sections. Later we compare the results with the current and future potential of the direct laboratory searches.

We consider here three sources of constraints from cosmology and astrophysics.101010Sterile neutrinos of masses and mixing angles in the interesting ranges may contribute to other astrophysical processess, like supernovae explosions, Universe reionization etc, see reviews [4, 1, 2]. However, these processes are quite involved by itself and application of the results found in literature to our models with hidden sector is not straightforward. First constraint comes from the requirement that the sterile neutrino contribution to the energy density does not exceed the DM contribution, $\Omega_{N}\leq\Omega_{\mathrm{DM}}$.

Second, due to the mixing with active neutrino there is a radiative decay mode of the sterile neutrino, $N\to\gamma\nu$, leading to the monochromatic photons of energy $E=M/2$ arriving from the relic sterile neutrino background. Cosmic X-ray observations place upper bounds on the photon flux at a given energy, which limits the sterile-active mixing angle, see [2] for review. Let us denote by $\theta_{\text{X-ray}}(M)$ the limiting value in the assumption that sterile neutrino composes all of the DM. Then, if the sterile neutrino composes only fraction of the DM, the bound is rescaled as

[TABLE]

Third, if the sterile neutrinos compose a significant part of the DM, $\Omega_{N}\lesssim\Omega_{\mathrm{DM}}$, one has to take into account bounds from the structure formation, arriving from the relative abundance of dwarfs and other small satellite galaxies [25], phase space density of galactic dark matter [26], and analyses of Lyman-$\alpha$ forest [27]. These bounds constrain the level of free streaming in the dark matter, so they actually limit the average velocity of the dark matter particles.

With a sketch in figure 2 we illustrate the role of each constraint in different areas in the plane of sterile neutrino mass and mixing angle. The black line corresponds to the mixing angle, $\sin^{2}2\theta_{\text{NRP}}$, leading to the generation in oscillations of DM amount of sterile neutrinos, $\Omega_{N}=\Omega_{\mathrm{DM}}$ (non-resonant production). In the scenarios described in present work this line corresponds to $T_{c}\to\infty$. The blue line is the X-ray constraint for sterile neutrinos $\theta_{\text{X-ray}}$, assuming $\Omega_{N}=\Omega_{\mathrm{DM}}$. Then, in the region 0 the sterile neutrino composes only a part of DM, and is not constrained by X-ray observations, even without introduction of the hidden sector with phase transition and low critical temperature $T_{c}$. The same is true in the region 1, which lies below the line

[TABLE]

Really, the amount of neutrino produced here by the non-resonant oscillations is small, and the X-ray bound (4.1) is satisfied.

In the region 2 it is possible to suppress the amount of produced sterile neutrinos by reducing $T_{c}$ (see eq. (2.6)), and make $\Omega_{N}=\Omega_{\text{DM}}$. However, the DM composed out of sterile neutrino produced in this way strongly violates the structure formation bounds. Therefore, only some part of DM can be generated here. A rough constraint on this part can be obtained from [28], but we do not present a detailed analysis here (in our case this constraint is always satisfied once the sterile neutrino fraction does not exceed roughly one third).

In the regions 3 and 4 the overall allowed abundance of the sterile neutrino is determined solely by the X-ray bound (4.1). The respective $T_{c}$ is obtained from combination of (4.1) and (2.6) (or, in case of the model with phase transition, of the sum of (2.6) and (3.3)).

For moderate values of $\theta$, which are interesting for the direct laboratory searches, $T_{c}$ should be quite low, in order to significantly suppress the sterile neutrino production. In the scenario with phase transition (section 3.1) the suppression can not be arbitrary due to the production (3.2) of neutrino above $T_{c}$, thus leading to the maximal allowed mixing angle. For the scenario of section 3.2 this limit is practically absent, as far as $T_{c}\sim T_{\text{osc}}\ll M_{N}$ are allowed.

Figure 3 gives the numerical results obtained using the full numerical solution of (2.2) and (3.2) in the scenario of section 3.1. The plots show the maximal possible sterile neutrino abundance $\Omega_{N}/\Omega_{\text{DM}}$ and maximal possible $T_{c}$ that satisfies the total abundance and X-ray constraints. Below we refer to the particular parts of plots in Figure 3 by making use of the names introduced in figure 2.

In the regions 0 and 1 no $T_{c}$ is required, and $\Omega_{N}$ just corresponds to non-resonant production. In regions 3 and 4 the maximal abundance saturates the bound (4.1). The upper-right white region corresponds to the area, where required suppression can not be achieved in the model of section 3.1.

In the region 2 of the plot the X-ray bound allows for $\Omega_{N}=\Omega_{\text{DM}}$. However, as mentioned above, this would strongly violate the structure formation constraints. Even rather conservative constraints give for the case of non-resonantly produced DM the bounds of $m_{\text{NRP}}>8\,\text{keV}$ from Lyman-$\alpha$ [30] and of $m_{\text{NRP}}>5.7\,\text{keV}$ [31] from phase space density. For the sterile neutrino with distribution (2.5) we can rescale the mass as

[TABLE]

obtaining the constraints of $M>11\,\text{keV}$ and $8\,\text{keV}$, respectively, which exclude the whole region.111111Strictly speaking, the distribution of sterile neutrinos for the model of section 3.1 has also the colder component (3.2) with $\langle\!\langle p\rangle\!\rangle=1.2T$, but it is not sufficient to relax the structure formation constraints significantly enough. It is hard to precisely estimate what fraction of Warm DM is still allowed, so we do not take the structure formation constraint into account in the plot, but dashed the region where they are important (note, that it is slightly larger than the area 2 of the sketch in figure 2, as far as it corresponds to the region where the amount of the produced WDM component is above 30%).

Note, that in the model with the feebly interacting scalar of section 3.2 there may be nonperturbative generation of super-cool sterile neutrinos by the oscillating background. This may lead to sterile neutrino Cold DM, thus evading the structure formation constraints in region 2. We leave the investigation of this possibility for future work.

It is clearly seen from figure 3, that in the model of section 3.1 with allowed $T_{c}>T_{c,\mathrm{min}}$ (3.4) it is possible to reach the area of the direct ground-based constraints (green solid lines in figure 3). In the model of section 3.2 even larger values of mixing angle are compatible with the cosmological constraints.

Note in passing, that in our analysis sterile neutrino mixes with the electron active neutrino only. However, the upper bound of available region (the upper edges of colour areas in figure 3) remains valid for the case of mixing with muon and tau species. The reason is that at low temperatures (below electron decoupling) there is no difference in production rate of sterile neutrinos from mixing with any active species. The X-ray limits are the same for mixing with all the three active neutrinos.

Finally, the model of section 3.1 with phase transition at some moment in the past leads to significant extension of available region in the plane $(M,\theta^{2})$. In the model of section 3.2 with feebly interacting scalar, the sterile neutrino production in the early Universe can be fully suppressed making X-ray limits irrelevant. Then all the region below the direct limits is allowed. Moreover, the presently developing projects, e.g. [29], can directly probe the mixings (2.7) consistent with minimal implication of the seesaw type-I mechanism of generation of active neutrino masses.

5 Conclusions

To summarise the results in one sentence, we show how to suppress the sterile neutrino production in the early Universe to the level which makes the direct laboratory searches competitive to the investigations with the X-ray satellites of the radiative decays of the cosmic sterile neutrino background.

Mechanisms suggested in the paper the sterile neutrino production can be suppressed to the level where even present direct limits become the strongest ones. This shows the importance of the direct searches, in particular in exploring the see-saw type I mechanism of active neutrino mass generation. On the contrary, the astrophysical indirect searches are subject to model-dependent uncertainties.

The possible scenarios are not limited by the two outlined, but the common statement is that the change of the sterile neutrino mass should happen at $T_{c}<T_{\mathrm{max}}$, and that requires hidden sector with new physics at low energy scale.

The authors are indebted to Igor Tkachev for valuable discussions and help with depicting the very recent limits [7] of Troitsk $\nu$-mass experiment. The works is supported by the RSF grant 17-12-01547.