Sterile neutrino dark matter via GeV-scale leptogenesis?
J. Ghiglieri, M. Laine
TL;DR
This paper investigates a scenario where keV to GeV right-handed neutrinos explain neutrino masses, dark matter, and baryon asymmetry, by numerically solving cosmological evolution equations at temperatures up to 5 GeV.
Contribution
It provides a detailed numerical analysis of sterile neutrino production in a 1+2 flavor framework, including late entropy and lepton asymmetry effects, for the first time at these temperatures.
Findings
Approximately 10% of dark matter abundance can be produced with favorable parameters.
Late entropy production influences dark matter abundance.
Potential methods to increase dark matter production are discussed.
Abstract
It has been proposed that in a part of the parameter space of the Standard Model completed by three generations of keV...GeV right-handed neutrinos, neutrino masses, dark matter, and baryon asymmetry can be accounted for simultaneously. Here we numerically solve the evolution equations describing the cosmology of this scenario in a 1+2 flavour situation at temperatures GeV, taking as initial conditions maximal lepton asymmetries produced dynamically at higher temperatures, and accounting for late entropy and lepton asymmetry production as the heavy flavours fall out of equilibrium and decay. For 7 keV dark matter mass and other parameters tuned favourably, of the observed abundance can be generated. Possibilities for increasing the abundance are enumerated.
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CERN-TH-2019-062
July 2019
aainstitutetext: Theoretical Physics Department, CERN,
CH-1211 Geneva 23, Switzerlandbbinstitutetext: AEC, Institute for Theoretical Physics, University of Bern,
Sidlerstrasse 5, CH-3012 Bern, Switzerland
Sterile neutrino dark matter via GeV-scale leptogenesis?
J. Ghiglieri b
M. Laine
Abstract
It has been proposed that in a part of the parameter space of the Standard Model completed by three generations of keV…GeV right-handed neutrinos, neutrino masses, dark matter, and baryon asymmetry can be accounted for simultaneously. Here we numerically solve the evolution equations describing the cosmology of this scenario in a 1+2 flavour situation at temperatures GeV, taking as initial conditions maximal lepton asymmetries produced dynamically at higher temperatures, and accounting for late entropy and lepton asymmetry production as the heavy flavours fall out of equilibrium and decay. For 7 keV dark matter mass and other parameters tuned favourably, % of the observed abundance can be generated. Possibilities for increasing the abundance are enumerated.
Keywords:
Thermal Field Theory, CP violation, Neutrino Physics, Resummation
1 Introduction
The idea of accounting for dark matter through keV-scale sterile neutrinos dw ; sf is strongly constrained by now (for a review see, e.g., ref. review ). The non-observation of -rays from putative sterile neutrino decays restricts their Yukawa couplings to be very small, |h_{{\mbox{\tiny\rm{I}}}a}|<10^{-12}. With such small couplings a sufficient number of sterile neutrinos can be produced in the Early Universe only if the production is enhanced through a resonant mechanism sf , requiring the presence of large lepton asymmetries. Some time ago, it was pointed out singlet that this scenario could be embedded in a framework in which two generations of GeV-scale right-handed neutrinos first generate a baryon asymmetry ars ; as , and then continue to generate lepton asymmetries, which subsequently boost dark matter production shifuller ; aba ; dmpheno .
In the most detailed dark matter computation carried out so far dmpheno , it was assumed that lepton asymmetries are produced first, at T\mathop{\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}5 GeV, whereas dark matter production is only active at T\mathop{\raise 1.29167pt\hbox{<\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}5 GeV. However, if the mass scale of the heavier sterile neutrinos is M\;\mathop{\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}2 GeV, they decay at T\ll M/\pi\;\mathop{\raise 1.29167pt\hbox{<\kern-7.5pt\raise-4.73611pt\hbox{\sim}}} GeV, and these decays may produce further lepton asymmetries singlet ; late . Then lepton asymmetry generation and dark matter production may proceed simultaneously, and need to be accounted for within a unified framework.
The purpose of the present paper is to assume that the initial lepton asymmetries have been dynamically produced by two generations of GeV-scale right-handed neutrinos. In a recent work degenerate , we showed that in this case lepton asymmetries \mathop{\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}10^{3} times larger than the baryon asymmetry can arise. Furthermore, the lepton asymmetries have an intriguing structure, being evenly distributed amongst all flavours and settling into a stationary state (see also ref. eijima ). We now follow that state down to lower temperatures, at which the GeV-scale right-handed neutrinos freeze out and decay. This non-equilibrium dynamics modifies the expansion rate of the universe and may also produce new lepton asymmetries. The question is whether this could help to boost the asymmetries that, according to ref. degenerate , were too small to have a substantial effect in the dark matter context. For the dark matter sector itself we fix the mass to the prototypical 7 keV scenario, with the corresponding Yukawa couplings pushed to the maximal allowed range as suggested by supposed observations observe1 ; observe2 .111At the time of writing these observations continue to be controversially discussed.
The presentation is organized as follows. The rate equations applying to the 1+2 sterile neutrino system are summarized in sec. 2. In sec. 3 we explain how the falling out of equilibrium of the “heavy” GeV-scale flavours modifies the expansion of the universe, and transcribe the rate equations to this situation. Subsequently, the heavy part of the rate equations can be simplified as explained in sec. 4, whereas the “light” keV-scale part may experience resonant enhancement, cf. sec. 5, which prohibits any substantial simplification. Parameter choices are justified in sec. 6, and numerical results are presented in sec. 7. A brief summary and outlook conclude this investigation in sec. 8.
2 Review of rate equations for the 1+2 flavour situation
The theory we work with is described by the Lagrangian
[TABLE]
where M_{\mbox{\tiny\rm{I}}}\geq 0 are Majorana masses; is a Higgs doublet; are chiral projectors; is a left-handed lepton doublet of generation ; h_{{\mbox{\tiny\rm{I}}}a} are the components of the neutrino Yukawa matrix; and summations over indices are left implicit.
We consider the situation , and assume the 2nd and 3rd generations to be almost degenerate in mass. The average mass is denoted by M_{\mbox{\tiny\rm{H}}}\equiv(M_{2}+M_{3})/2. The heavy flavours and the associated Yukawa couplings |h_{{\mbox{\tiny\rm{I}}}a}|\mathop{\raise 1.29167pt\hbox{<\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}10^{-7} are chosen to reproduce the active neutrino mass differences and mixing angles, whereas the first generation has much smaller Yukawa couplings, , as is suitable for playing a role in dark matter physics.
The density matrix of the hierarchical 1+2 flavour system is expressed as
[TABLE]
and similarly for other objects. Here
[TABLE]
denotes a symmetrization/antisymmetrization with respect to helicity , and off-diagonal heavy-light components of have averaged out up to effects suppressed by 1/M_{{\mbox{\tiny\rm{H}}}}. The lepton asymmetry in flavour is denoted by , whereas n_{\mbox{\tiny\rm{B}}} is the baryon asymmetry.
The evolution equation for lepton asymmetries can be split into the contributions of the light and heavy flavours,222In order to derive the evolution equations (2)–(2.6), we have generalized the considerations in refs. sr ; dmpheno ; selfE ; cptheory to apply to three flavours of sterile neutrinos possessing an arbitrary mass spectrum, and at the end simplified the setup by specializing to a hierarchical 1+2-flavour system.
[TABLE]
where denotes the Fermi distribution, , and \omega_{{\mbox{\tiny\rm{I}}}}\equiv\sqrt{k^{2}+M_{\mbox{\tiny\rm{I}}}^{2}}. The light and heavy components of the density matrix evolve as
[TABLE]
The coefficients associated with the light flavours read
[TABLE]
where (...)_{{\mbox{\tiny\rm{L}}}} indicates the use of a “light” mass , whereas the heavy coefficients read
[TABLE]
where (...)_{{\mbox{\tiny\rm{H}}}} stands for a “heavy” mass M_{{\mbox{\tiny\rm{H}}}}. We have denoted leptonic chemical potentials by , and expressed the dependence on neutrino Yukawa couplings through
[TABLE]
whereas {Q}^{\pm}_{\mbox{\tiny\rm{(a)}}}=[{Q}_{\mbox{\tiny\rm{(a+)}}}\pm{Q}_{\mbox{\tiny\rm{(a-)}}}]/2 denote symmetrization and antisymmetrization with respect to helicity. The coefficients and parametrize the C-even and C-odd parts, respectively, of “absorptive” reactions (i.e. real processes), whereas and parametrize “dispersive” corrections. Specifically,
[TABLE]
where is a retarded correlator associated with the operator to which the sterile neutrinos couple; denotes helicity; refers to the flavour; and and can be extracted by symmetrizing and antisymmetrizing in chemical potentials, respectively.
For a practical determination of , we have generalized the computations of refs. numsm ; broken ; degenerate to arbitrary kinematics (i.e. not only the ultrarelativistic regime but also or ), restricting however still to the approximation M\ll m_{\mbox{\tiny\rm{W}}} in the treatment of scatterings below the electroweak crossover.
In order to close the system, the chemical potentials appearing in eqs. (2.7)–(2.15) need to be re-expressed in terms of the number densities appearing on the left-hand side of eq. (2). This requires the determination of “susceptibilities”. We follow the approach in appendix A of ref. degenerate , simplifying the formulae by restricting to , where GeV, but adding charged lepton and light quark masses according to ref. dmpheno . Hadronic contributions are smoothly switched off at low by a replacement , as proposed in ref. numsm .
3 Non-equilibrium expansion
The GeV-scale flavours that are responsible for leptogenesis at GeV, freeze out and subsequently decay when \pi T\ll M_{{\mbox{\tiny\rm{H}}}}. These non-equilibrium decays release entropy st , an effect which has been argued to be substantial for M_{{\mbox{\tiny\rm{H}}}}\simeq 1...10 GeV entropy , and which therefore needs to be included in dark matter and baryogenesis computations. (When \pi T\gg M_{{\mbox{\tiny\rm{H}}}}, the GeV-scale flavours already have a small effect on the energy and entropy densities, however this is on the percent level and thus insignificant on our resolution.)
Denoting by the cosmological scale factor and by GeV the Planck mass, and assuming a flat universe, Friedmann equations can be expressed as
[TABLE]
where is the energy density, is the pressure, and is the Hubble rate. We write
[TABLE]
where e_{\mbox{\tiny\rm{T}}} and p_{\mbox{\tiny\rm{T}}} are the Standard Model energy density and pressure at a temperature , whereas e_{{\mbox{\tiny\rm{H}}}} and p_{{\mbox{\tiny\rm{H}}}} represent the contribution of the heavy right-handed neutrinos. If we denote by
[TABLE]
a co-moving momentum mode and by the corresponding phase space integral, the energy density and pressure carried by the heavy flavours can be expressed as
[TABLE]
We now insert eq. (3.3) into eq. (3.2), and move the thermal terms to the left-hand side. Making use of {\rm d}e_{\mbox{\tiny\rm{T}}}=T{\rm d}s_{\mbox{\tiny\rm{T}}} and e_{\mbox{\tiny\rm{T}}}+p_{\mbox{\tiny\rm{T}}}=Ts_{\mbox{\tiny\rm{T}}}, where s_{\mbox{\tiny\rm{T}}} is the Standard Model entropy density, we find
[TABLE]
In order to proceed, it is helpful to express the phase-space integrals in eq. (3.5) in terms of the time-independent variable (cf. eq. (3.4)), because then \rho^{+}_{{\mbox{\tiny\rm{I}}}{\mbox{\tiny\rm{I}}}} appears in a form for which a time-evolution equation is available. For a combination appearing in eq. (3.6) this implies
[TABLE]
A derivative with respect to now operates on two terms, \rho^{+}_{{\mbox{\tiny\rm{I}}}{\mbox{\tiny\rm{I}}}} as well as the last piece,
[TABLE]
Once p_{{\mbox{\tiny\rm{H}}}} from eq. (3.5) is inserted, the contribution from eq. (3.8) cancels against the contribution from p_{{\mbox{\tiny\rm{H}}}} in eq. (3.6). In total, then,
[TABLE]
At this point we make use of the equation of motion of \rho^{+}_{{\mbox{\tiny\rm{I}}}{\mbox{\tiny\rm{I}}}}. It follows from eq. (2.6) that, to a good approximation,
[TABLE]
Inserting eq. (3.10) into eq. (3.9) and going subsequently back to co-moving momenta as integration variables, we get
[TABLE]
We see that entropy is generated only if \rho^{+}_{{\mbox{\tiny\rm{I}}}{\mbox{\tiny\rm{I}}}} falls out of equilibrium.
Let us simplify the setup by making use of so-called momentum averaging. Even though not associated with any formally small expansion parameter, this turns out to represent a reasonable approximation in many cases degenerate ; kinetic ; cpnumerics . We integrate eq. (3.10) over and then change variables into , which leads to
[TABLE]
Now introduce the ansatz
[TABLE]
where Y^{+}_{{\mbox{\tiny\rm{I}}}{\mbox{\tiny\rm{I}}}} is a yield parameter, and denote
[TABLE]
Then eqs. (3.11) and (3.12) become
[TABLE]
As a final step, the evolutions of Y^{+}_{{\mbox{\tiny\rm{I}}}{\mbox{\tiny\rm{I}}}} and s_{\mbox{\tiny\rm{T}}}a^{3} can be decoupled from each other, by inserting eq. (3.15) into (3.16). Moreover, introducing
[TABLE]
we can rewrite
[TABLE]
where is the speed of sound squared. From eqs. (3.15) and (3.18) the Jacobian can be solved for,
[TABLE]
Then the basic equations become
[TABLE]
Numerical solutions for s_{\mbox{\tiny\rm{T}}}a^{3}, obtained from eqs. (3.20) and (3.21), are shown in fig. 1(left). In fig. 1(right) we show the corresponding from eq. (3.19), normalized to its Standard Model value. According to fig. 1(left), entropy release substantially reduces any yields that were generated at GeV, if M_{{\mbox{\tiny\rm{H}}}}\;\mathop{\raise 1.29167pt\hbox{<\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}\;2 GeV.333This statement depends somewhat on the values of the Yukawas chosen, cf. sec. 6 and item (ii) in sec. 8.
In contrast the effect from is moderate, if dark matter production is peaked at GeV.
Even if only the evolution of Y^{+}_{{\mbox{\tiny\rm{I}}}{\mbox{\tiny\rm{I}}}} is directly coupled with the evolutions of s_{\mbox{\tiny\rm{T}}}a^{3} and (cf. eqs. (3.20), (3.21)), the results in fig. 1 also influence other evolution equations. The redshift factor from eq. (3.4) can be expressed as
[TABLE]
As we have replaced as an integration variable through with the help of , the co-moving momentum will from now on be denoted by k_{\mbox{\tiny\rm{T}}}. Defining Y\equiv n/s_{\mbox{\tiny\rm{T}}}, evolution equations for particle densities and phase space distributions from sec. 2 are transcribed as
[TABLE]
4 Simplified treatment of heavy flavours
It was indicated in the previous section that for the heavy flavours it is advantageous to resort to momentum averaging. Moreover, it is convenient to go over into an interaction picture. In order to simplify the notation of eqs. (3.23) and (3.24), let us denote
[TABLE]
Then the role of the free Hamiltonian is played by \mathop{\mbox{diag}}\langle\widehat{\omega}_{2},\widehat{\omega}_{3}\rangle_{1}-\bigl{\langle}\widehat{H}^{+}_{\!{\mbox{\tiny\rm{H}}}}\bigr{\rangle}_{1}, where the averaging is defined in eq. (3.14). From here we can subtract the trace part without loss of generality. The remaining upper diagonal appearing in eq. (2.6) is defined as
[TABLE]
After the change of a picture, these diagonals do not appear on the right-hand side of the equations, whereas all non-diagonal coefficient functions get modified, as , , where the phase factor satisfies .
In spite of the near-degeneracy of and , defined in eq. (4.2) becomes large at low temperatures (recall that \widehat{\omega}_{\mbox{\tiny\rm{I}}} are normalized by , which decreases like the Hubble rate, as ). In this situation the fast oscillations between the heavy sterile neutrinos, induced by , can be “integrated out”. Working to leading order in as described in ref. cpnumerics , we find that in this regime
[TABLE]
Here we have complemented the momentum average in eq. (3.14) through
[TABLE]
The helicity-symmetric diagonal components of the density matrix evolve according to eq. (3.20), whereas the other components obey
[TABLE]
5 Resonant contribution in light flavour
The question arises whether momentum averaging could also be adopted for . This is, however, hindered by the possible appearance of a “resonance” in the coefficients {Q}^{\pm}_{(a){\mbox{\tiny\rm{L}}}},\,{\!\bar{Q}}^{\pm}_{(a){\mbox{\tiny\rm{L}}}}, which parametrize the evolution of through eqs. (2.7)–(2.9). The resonance originates through the helicity-conserving indirect contribution, which for M_{1}\ll k_{\mbox{\tiny\rm{T}}} has the form
[TABLE]
The function has a C-even and C-odd part; the latter, which is proportional to chemical potentials, is denoted by . At low temperatures the C-even part is to a good approximation proportional to nr . Therefore we may write
[TABLE]
The function is positive, whereas is odd in the interchange . Therefore, after extracting the C-even and the C-odd from eq. (5.1) by symmetrizing and antisymmetrizing in chemical potentials, respectively, both contain one appearance of
[TABLE]
For small this can be approximated as
[TABLE]
This is qualitatively different from non-resonant contributions, which are proportional to .
We observe from eqs. (5.3) and (5.4) that resonances exists if , and they are located at
[TABLE]
Recalling that and nr , where is the Fermi constant, resonances are important if |\mu_{a}|\mathop{\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}10M_{1}. For keV and GeV, this requires .
6 Parameter values and initial conditions
We start by considering the benchmark point
\begin{picture}(5.0,5.0)(0.0,0.0) \put(0.0,0.0){} \end{picture}
from ref. degenerate , tuned to produce the observed baryon asymmetry as well as maximally large low-temperature lepton asymmetries, within a specific slice of the parameter space. The most important parameters are M_{{\mbox{\tiny\rm{H}}}}\approx 0.7732 GeV, GeV, , where the last one refers to the Casas-Ibarra parameter fixing the absolute value of the neutrino Yukawas ci . The small implies |h_{{\mbox{\tiny\rm{I}}}a}|\simeq 2\times 10^{-8}. The corresponding active-sterile mixings, {|h_{{\mbox{\tiny\rm{I}}}a}|v}/({\sqrt{2}M_{{\mbox{\tiny\rm{I}}}}})\simeq 4.5\times 10^{-6}, are tiny and thus challenging to constrain in (future) experiments.
For the light sector we fix the overall active-sterile mixing angle to a maximal suggested value observe2 , , i.e. for keV. Furthermore we set , which according to the web site associated with ref. dmpheno leads to maximal efficiency in dark matter production. Thus, for all .
The initial conditions for the evolution are set at a temperature GeV, where the system is to a good approximation in a stationary state degenerate . Taking also into account that rate coefficients are dominated by helicity-conserving contributions at low temperatures, eqs. (4.3)–(4.6) imply that
[TABLE]
where and . To be optimistic, we multiply lepton asymmetries obtained in ref. degenerate by a factor two, leading to the initial condition Y_{a}-Y_{\mbox{\tiny\rm{B}}}/3\simeq-6.2\times 10^{-7} for all , which fixes the chemical potential appearing in eq. (6.1) as . The initial baryon asymmetry is set at the observed value Y_{\mbox{\tiny\rm{B}}}=0.87\times 10^{-10}; in view of entropy dilution, it should be taken to be somewhat larger at the beginning, but this has little effect on our considerations here, and is also easy to achieve in practice degenerate .
In addition to the benchmark point
\begin{picture}(5.0,5.0)(0.0,0.0) \put(0.0,0.0){} \end{picture}
, we have carried out further scans like in ref. degenerate , and again multiplied the corresponding lepton asymmetries by a factor two. This leads to two further parameter points which serve to illustrate the dependence on M_{{\mbox{\tiny\rm{H}}}}. Initial lepton asymmetries can be kept large by decreasing M_{{\mbox{\tiny\rm{H}}}}, but there is not much room here, given that according to refs. Neff0 ; Neff there is a cosmological lower bound M_{{\mbox{\tiny\rm{H}}}}\mathop{\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}0.1 GeV. We have chosen M_{{\mbox{\tiny\rm{H}}}}=0.2 GeV as a lighter mass; as we will see, this is already problematic (the other parameters are GeV, , Y_{a}-Y_{\mbox{\tiny\rm{B}}}/3=-5.0\times 10^{-7}). As a heavier mass we have settled on M_{{\mbox{\tiny\rm{H}}}}=4.0 GeV ( GeV, , Y_{a}-Y_{\mbox{\tiny\rm{B}}}/3=-5.4\times 10^{-11}), which clearly illustrates how results depend on M_{{\mbox{\tiny\rm{H}}}}. We have also carried out further runs with M_{{\mbox{\tiny\rm{H}}}}=2.0,10.0 GeV and these confirm the overall picture.
7 Numerical solution
Important ingredients characterizing the solution of the rate equations are equilibration rates, which determine how efficiently different components of the density matrix approach their would-be equilibrium values. As an example, consider the dimensionless combination appearing in eq. (4.5) but for simplicity normalized to the thermal Hubble rate rather than ,
[TABLE]
The result is shown in fig. 2(left). We observe that the system can follow equilibrium (i.e. that \tilde{\Gamma}_{{\mbox{\tiny\rm{H}}}}\;\mathop{\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}\;1) when T\;\mathop{\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}\;2 GeV, but at GeV there is a period when this should not happen. At very low temperatures, rates are dominated by vacuum decays, and the system again approaches equilibrium.
For the light flavour, we show an equilibration rate from eqs. (2.5), (2.9), evaluated at a fixed comoving momentum, in fig. 2(right) (\Gamma_{{\mbox{\tiny\rm{L}}}}\equiv 2\sum_{a}\phi^{+}_{(a)11}\,{Q}^{+}_{(a){\mbox{\tiny\rm{L}}}}). Given that in the full range \tilde{\Gamma}_{{\mbox{\tiny\rm{L}}}}\equiv\Gamma_{{\mbox{\tiny\rm{L}}}}/({3c_{s}^{2}H_{{\mbox{\tiny\rm{T}}}}})\ll 10^{-3}, the light flavour never comes near thermal equilibrium.
For benchmark
\begin{picture}(5.0,5.0)(0.0,0.0) \put(0.0,0.0){} \end{picture}
(i.e. M_{{\mbox{\tiny\rm{H}}}}\approx 0.8 GeV), the solutions of the rate equations for lepton asymmetries and the density matrix of the heavy sector are shown in fig. 3. At first the density matrix follows the equilibrium form, but at GeV the equilibrium form starts to decrease as mass effects become important. The actual solution cannot immediately follow this change, given that the equilibration rate has become small.
We note from fig. 3 that even if the density matrix deviates from equilibrium at low temperatures, there is no substantial re-generation of lepton asymmetries taking place in this regime. The reason is that the rate coefficients are so small that the source terms, cf. the last lines of eq. (4.3), remain inefficient.
Making us of planck and eV pdg , where is the current entropy density, the fraction of dark matter carried by the lightest right-handed neutrinos can be expressed as
[TABLE]
We observe from fig. 3(right) that intermittently about 8.5% of the total dark matter abundance could be accounted for, before entropy dilution kicks in at late times.
The yields of the helicity asymmetries are illustrated in fig. 4. Helicity asymmetries remain modest (Y^{-}_{{\mbox{\tiny\rm{I}}}{\mbox{\tiny\rm{I}}}}\ll Y^{+}_{{\mbox{\tiny\rm{I}}}{\mbox{\tiny\rm{I}}}}, ), which is a manifestation of the fact that thermal production dominates over resonant production (one helicity state is produced from neutrinos, the other from antineutrinos). The dark matter phase space spectra, which are strongly tilted towards the IR compared with kinetically equilibrated fermions, are shown in fig. 4(right).
Finally we consider the dependence of the final dark matter abundance on the parameters of the heavy sector. Results for M_{{\mbox{\tiny\rm{H}}}}=0.2 GeV are shown in fig. 5, and for M_{{\mbox{\tiny\rm{H}}}}=4.0 GeV in fig. 6. Despite a large variation in the original lepton asymmetries and a re-generation of new ones in the latter case, the only important effect for dark matter abundance are the variations in the expansion history of the universe, as shown in fig. 1.
8 Summary and outlook
The purpose of this paper has been to update our previous sterile neutrino dark matter analysis dmpheno by fixing the initial lepton asymmetries to maximal values that can be produced by the dynamics of GeV-scale right-handed neutrinos degenerate . The parameters of the latter are constrained to be responsible for generating the active neutrino masses and mixing angles ci . We permit for the generation of further lepton asymmetries in the low-temperature decays of the right-handed neutrinos,444However, the parametric fine tunings described in sec. 2.6.2 of ref. late , requiring a specific choice of CP phases, have not been imposed. by including both the light and heavy sterile flavours in the set of rate equations, and track the modification of the universe expansion caused by the energy density carried and entropy released by the heavy flavours. In addition we resolve both helicity states of the sterile neutrinos; this is important for heavy flavours given that initial lepton asymmetries are correlated with helicity asymmetries eijima ; degenerate , and for the light flavour given that resonant production sf affects one helicity state only.
Even though we do find rich dynamics in the heavy sector (cf. figs. 3, 5, 6), the dark matter abundance does not vary greatly between the cases, reaching typically less than 10% of the observed value. The reason for this behaviour can be understood as follows. The dependence of on lepton asymmetries must be quadratic at small , given that energy density is a C-even quantity. A strongly growing dependence only sets in at , cf. fig. 7. This is associated with the dominance of resonant production, which in the language of eq. (5.4) requires . Given that the asymmetries obtained in ref. degenerate are below this level, dark matter production takes place predominantly through normal thermal processes. Therefore, our dark matter results are rather insensitive to the heavy sector, apart from its influence through the expansion of the universe, as depicted in fig. 1. In order to account for 100% of dark matter, initial lepton asymmetries should be a factor larger than those found in ref. degenerate , i.e. of the order , depending on entropy dilution.
Even if we have failed to account for all of dark matter through the dynamics of keV…GeV scale sterile neutrinos, the results could change in the future, for several reasons (in line with the basic minimalistic premise of our study, we keep the Lagrangian of eq. (2.1) intact for this discussion, without any extra non-SM fields, and assume a standard cosmology):
- (i)
As the dark matter production is largely thermal rather than resonant, it is proportional to the rate coefficient , which contains large hadronic uncertainties hadronic . It would be interesting to estimate or constrain through lattice simulations.
- (ii)
We have chosen the Yukawa couplings of the heavy flavours to be as small as possible, in order to diminish lepton number washout and therefore to have maximal initial asymmetries degenerate . However, as the initial asymmetries have little influence in any case, the Yukawas could be made larger, without spoiling baryogenesis (cf., e.g., refs. ph ; inar ; degenerate ). Then the heavy flavours would stay closer to equilibrium and decay faster, producing less entropy. Even though we do not expect substantial variations of the dark matter abundance from here, a comprehensive study of the heavy flavour Yukawas would be welcome. This should also include the search for potential “atypical” CP phases where late-time lepton asymmetries might be anomalously large.
- (iii)
We have restricted ourselves to the SHiP window, M_{{\mbox{\tiny\rm{H}}}}<M_{B}\simeq 5 GeV ship , but nature may have chosen otherwise. Increasing M_{{\mbox{\tiny\rm{H}}}} in the analysis of ref. degenerate , we find that initial lepton asymmetries would be smaller then. However, as anticipated in refs. singlet ; late , novel lepton asymmetries are produced later on (cf. fig. 6). Therefore it seems promising to explore what happens with larger values of M_{{\mbox{\tiny\rm{H}}}}. Then, however, scatterings entering the rate coefficients need to be addressed without resorting to the approximation M_{{\mbox{\tiny\rm{H}}}}\ll m_{\mbox{\tiny\rm{W}}}, which poses a significant technical challenge. The initial temperature should be chosen in the regime T\gg M_{{\mbox{\tiny\rm{H}}}}, i.e. larger than here.
- (iv)
Additional semi-conserved quantities such as chiral charges or helical magnetic fields have long been speculated to play a role in cosmology (cf., e.g., refs. eR1 ; eR2 ; Bext1 ; Bext2 ; Bext3 ; Bext4 ), and they could conceivably interfere with late-time lepton asymmetries as well.
- (v)
Last but not least, the observational status of the dark matter sterile neutrino remains unclear. Here we have relied on the indications in refs. observe1 ; observe2 , however other parameter values could be studied within our framework, and might change the conclusions.
To summarize, we have established a framework which permits to study sterile neutrino dark matter production in a non-degenerate 1+2 flavour situation, with ongoing lepton number violation and with the heavy flavours falling out of equilibrium and gradually decaying. As a proof of concept, we have excluded several SHiP-like benchmarks as an explanation for all of dark matter. Broader parameter scans may help to bridge the gap.
Acknowledgements
We thank M. Shaposhnikov for encouraging us to verify, several years ago, that sterile neutrino dark matter production is most efficient when all are equally large; these tests are documented on the web page associated with ref. dmpheno . We are grateful to S. Eijima, M. Shaposhnikov and I. Timiryasov for helpful suggestions. M.L. was partly supported by the Swiss National Science Foundation (SNF) under grant 200020-168988.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1(1) S. Dodelson and L.M. Widrow, Sterile-neutrinos as dark matter, Phys. Rev. Lett. 72 (1994) 17 [hep-ph/9303287].
- 2(2) X.-D. Shi and G.M. Fuller, A New dark matter candidate: Nonthermal sterile neutrinos, Phys. Rev. Lett. 82 (1999) 2832 [astro-ph/9810076].
- 3(3) M. Drewes et al. , A White Paper on ke V Sterile Neutrino Dark Matter, JCAP 01 (2017) 025 [1602.04816].
- 4(4) M. Shaposhnikov, The ν 𝜈 \nu MSM, leptonic asymmetries, and properties of singlet fermions, JHEP 08 (2008) 008 [0804.4542].
- 5(5) E.K. Akhmedov, V.A. Rubakov and A.Y. Smirnov, Baryogenesis via neutrino oscillations, Phys. Rev. Lett. 81 (1998) 1359 [hep-ph/9803255].
- 6(6) T. Asaka and M. Shaposhnikov, The ν 𝜈 \nu MSM, dark matter and baryon asymmetry of the universe, Phys. Lett. B 620 (2005) 17 [hep-ph/0505013].
- 7(7) M. Laine and M. Shaposhnikov, Sterile neutrino dark matter as a consequence of ν 𝜈 \nu MSM-induced lepton asymmetry, JCAP 06 (2008) 031 [0804.4543].
- 8(8) T. Venumadhav, F.Y. Cyr-Racine, K.N. Abazajian and C.M. Hirata, Sterile neutrino dark matter: Weak interactions in the strong coupling epoch, Phys. Rev. D 94 (2016) 043515 [1507.06655].