Sterile Neutrinos with Secret Interactions - Lasting Friendship with Cosmology

Sterile neutrinos with mass ~1 eV and order 10% mixing with active neutrinos have been proposed as a solution to anomalies in neutrino oscillation data, but are tightly constrained by cosmological limits. It was recently shown that these constraints are avoided if sterile neutrinos couple to a new MeV-scale gauge boson A'. However, even this scenario is restricted by structure formation constraints when A'-mediated collisional processes lead to efficient active-to-sterile neutrino conversion after neutrinos have decoupled. In view of this, we reevaluate in this paper the viability of sterile neutrinos with such"secret"interactions. We carefully dissect their evolution in the early Universe, including the various production channels and the expected modifications to large scale structure formation. We argue that there are two regions in parameter space - one at very small A' coupling, one at relatively large A' coupling - where all constraints from big bang nucleosynthesis (BBN), cosmic microwave background (CMB), and large scale structure (LSS) data are satisfied. Interestingly, the large A' coupling region is precisely the region that was previously shown to have potentially important consequences for the small scale structure of dark matter halos if the A' boson couples also to the dark matter in the Universe.


I. INTRODUCTION
The possible existence of extra, "sterile", neutrino species with masses at the eV scale and O(10%) mixing with the Standard Model (SM) neutrinos is one of the most debated topics in neutrino physics today. Several anomalies in neutrino oscillation experiments [1][2][3][4][5][6] seem to point towards the existence of such particles, but null results from other experiments that did not observe a signal cast doubt on this hypothesis [7][8][9][10][11]. A multifaceted experimental program is under way to clarify the issue and either detect, or conclusively rule out, eV-scale sterile neutrinos with large mixing angle.
If the SM is indeed augmented with one or several such sterile neutrinos, but nothing else, some of the tightest constraints come from cosmological observations. In particular, measurements of the effective number of relativistic particle species in the primordial plasma, N eff [12,13], disfavor the existence of an abundance of light or massless particles beyond the SM neutrinos and the photon in the early Universe. If sterile neutrinos are at the eV scale or above, they are also constrained by the distribution of large scale structure (LSS) in the Universe [14] which would be washed out due to efficient energy transport over large distances by free-streaming neutrinos.
Cosmology, however, only constrains particle species that are abundantly produced in the early Universe. In two recent letters [15,16], it was demonstrated that the production of sterile neutrinos can be suppressed until relatively late times if they are charged under a * xchu@ictp.it † bdasgupta@theory.tifr.res.in ‡ jkopp@uni-mainz.de new interaction. This idea has elicited interest in detailed model building and has interesting phenomenological consequences [17][18][19][20][21][22][23][24][25]. However, Mirizzi et al. [22] have recently pointed out that collisions mediated by the new interaction can result in significant late time production of sterile neutrinos and lead to tensions with CMB and LSS data on structure formation. There are, however, several important caveats to this statement. In particular, the bounds from Ref. [22] can be evaded if the sterile neutrinos either never recouple with active neutrinos or remain collisional until matter-radiation equality.
(We communicated on these caveats with the authors of Ref. [22], who were aware of the first possibility but did not mention it as they found it less interesting in the context of previous literature. They mention the second possibility in the final version of their paper.) Our aim in the present paper is to understand in detail the role and impact of sterile neutrino collisions, and reevaluate if secretly interacting sterile neutrinos remain cosmologically viable. We begin in Sec. II by reviewing the main features of self-interacting sterile neutrino scenarios. Then, in Sec. III, we calculate the additional contribution to N eff at the BBN and CMB epochs. In Sec. IV we consider the impact on the large scale structure in the Universe, focusing in particular on the sensitivity of Sloan Digital Sky Survey (SDSS) and Lyman-α data. We find that there are two regions of parameter space where sterile neutrinos with secret interactions are only weakly constrained. In Sec. V, we discuss our conclusions and summarize the results.

II. SECRET INTERACTIONS AND STERILE NEUTRINO PRODUCTION
We assume that the Standard Model is augmented by a sterile neutrino ν s with mass m s 1 and with order 10% mixing with the SM neutrinos. We moreover assume the existence of a new secret U (1) s gauge interaction, mediated by a vector boson A of mass M at the MeV scale and coupling to sterile neutrinos through an interaction of the form Here, e s is th U (1) s coupling constant and P L = (1 − γ 5 )/2 is the left-handed chirality projection operator. We define the secret fine structure constant as α s ≡ e 2 s /(4π). This new interaction generates a large temperaturedependent potential [16] V eff Cosmology is sensitive to the presence of relativistic degrees of freedom through their contribution to the overall energy density. At early times the sterile sector was presumably in equilibrium with the SM plasma, so that ν s and A were thermally populated. We assume that the sterile sector decouples from the SM sector well above the QCD scale and that oscillations remain suppressed until active neutrinos also decouple. Since the temperature T γ of the SM sector drops more slowly than the sterile sector temperature T s when extra entropy is produced during the QCD phase transition, T s at BBN is significantly smaller than T γ .
It is useful to track the ratio Here, the factor (10.75/106.7) 1/3 gives the ratio of the sterile sector temperature to the active sector temperature before A decay, assuming that the two sectors have decoupled above the electroweak scale. It is based on counting the SM degrees of freedom that freeze out between the electroweak and BBN epochs.
A is present in the Universe at the BBN epoch if M 3T s | BBN 0.5 MeV, and has decayed away if heavier. The factor (2 · 7/8 + 3)/(2 · 7/8) in the first row of Eq. (7) corresponds to the ratio of sterile sector degrees of freedom 4 before and after the decay of A at T s M/3.
The extra radiation in the Universe is parameterized as N eff ≡ (ρ ν a +ρ ν s ,A )/ρ SM ν , i.e., the energy density of all non-photon relativistic species, measured in units of the energy density of a SM neutrino species. The primordial population of ν s and A leads to N eff marginally larger 4 Note that in complete models, for instance in scenarios including a dark Higgs sector to break the U (1) s symmetry, more degrees of freedom may need to be taken into account in the above equations.
EW scale: at the BBN epoch. The first term, N ν a , on the right hand side accounts for the active neutrinos and is equal to 3.045. The second term includes the relativistic sterile sector particles, i.e., only ν s if M 0.5 MeV. If the A bosons are lighter, i.e., M 0.5 MeV, they are present during BBN and contribute g A = 3 degrees of freedom in the sterile sector, in addition to the g ν s = 2 × 7/8 degrees of freedom of a sterile neutrino. Using the fact that also each active neutrino species has g ν a = 2 × 7/8 degrees of freedom, we find  [12]. After BBN, the next important event is a possible secret recoupling of ν a and ν s . If the sterile neutrino production rate Γ s > H, a new hotter population of ν s can be collisionally produced from ν a , and they achieve kinetic equilibrium with the primordially produced colder population of ν s . Also, the A can decay and heat up the sterile neutrinos. Both processes change the number and energy density of neutrinos, and N eff at CMB depends on the order in which they occur.
In Fig. 2, we show the collisional ν a → ν s production rate Γ s , normalized to the Hubble expansion rate H, as a function of the photon temperature T γ : Γ s /H is suppressed at high temperatures (say, above GeV), where sin 2 2θ m is small due to the large V eff . Since Γ s ∝ T −3 γ in this regime (see Eqs. (2), (3) and (5)) and H ∝ T 2 γ , Γ s /H increases with T −5 γ as the temperature decreases. We define the recoupling temperature T re as the temperature where Γ s /H > 1 for the first time since the primordial decoupling of the active and sterile sectors above the QCD phase transition. When T s ∼ M , the energy and temperature dependence of Γ s changes (see Eq. (5)), and when also V eff drops below ∆m 2 /(2T s ) at T s < M , Γ s begins to drop again. The asymptotic behavior is Γ s /H ∝ T 3 γ at T s M and θ m θ 0 . There are then three possible sequences of events: 1. No recoupling: For a sufficiently small interaction strength α s , the scattering rate Γ s always stays below the Hubble rate and there is no recoupling (solid black curve in Fig. 2).
If the interaction is stronger, a recoupling of ν a and ν s can happen either after or before A decay: 2. Recoupling after A decay: If M > few×10 −2 MeV, the recoupling happens after A have decayed (dotted blue curve in Fig. 2).
In the second and third cases, there is also a secret decoupling when Γ s /H again drops below one. If e 2 s /M 2 ≤ O(10 MeV −2 ), this decoupling happens while ν s are still relativistic, i.e. T s m s /3. 5 In the following, we discuss the three aforementioned cases in detail.

No Recoupling
In the no recoupling cases, labeled as A1 and B1 in Fig. 1, the cosmological evolution after BBN is very straightforward. Vacuum oscillations convert a small fraction 1 2 sin 2 2θ 0 0.01 of active neutrinos into sterile neutrinos (and vice versa), but this has negligible impact on the cosmological observables. Therefore, the temperature ratio ξ at CMB can be derived from the separate conservation of entropy in the active neutrino sector and in the sterile neutrino sector. It is independent of 5 In this paper, we will always assume this to be the case since we will find that the parameter region with e 2 s /M 2 ≥ O(10 MeV −2 ) is already disfavored by the requirement that active neutrinos should free stream sufficiently early [29] (see Secs. IV and V). If ν s and ν a are still coupled when the ν s become nonrelativistic, the mostly sterile mass eigenstate ν 4 will undergo a non-relativistic freeze-out and partly annihilate to pairs of mostly active neutrinos. Similarly, there is the possibility that the A decay after the decoupling, but this does not happen for the range of parameters we will discuss here. when the A decay (provided that it happens before the CMB epoch and approximately in chemical equilibrium.) That is, ξ CMBA/B ξ BBN(A) = 0.649. N eff,CMB can be estimated in analogy to Eq. (8A). For the assumed sterile neutrino mass 1 eV, the ν s contribution to the relativistic energy density has to be weighted by an extra factor because they are already semi-relativistic at the CMB epoch, where the photon temperature is T γ 0.30 eV and the kinetic temperature of the sterile sector is T s = ξ CMB · (4/11) 1/3 T γ 0.14 eV. As in [22], we assume that the extra weight factor is characterized by the pressure P . (See Appendix A for the definition and calculations of the kinetic temperature and the pressure P used here.) We thus obtain It is worth noting that the CMB temperature spectrum does not exactly measure the value of N eff,CMB . Instead, the observed spectrum depends on the evolution of the energy density in relativistic degrees of freedom between the epoch of matter-radiation equality (T γ,eq ∼ 0.7 eV) and recombination (T γ,CMB 0.30 eV) [13]. Therefore, the value of N eff,CMB measured from the CMB temperature power spectrum lies between the values of N eff at T γ,CMB and T γ,eq . The latter value, which we will denote by N eff,eq , is given by Both of these values agree with the bound from the 2015 Planck data release, N eff = 3.15 ± 0.23 (68% C.L.) [13].

Recoupling after A decay
The cases of recoupling after A decay are labeled as A2 and B2 in Fig. 1. In both cases, entropy conservation in the sterile sector before recoupling leads to a temperature ratio just after A decay of ξ M ξ BBN(A) = 0.649, which in turn implies Here, we have assumed that during A decay chemical equilibrium holds in the sterile sector. After recoupling, efficient neutrino oscillations and collisions lead to equilibration of the number densities and energy densities of all active and sterile neutrino species. Nevertheless, since number-changing interactions are strongly suppressed at T s M , this recoupling cannot change the total (active + sterile) neutrino number density and energy density beyond what is necessitated by cosmological expansion. The kinetic temperature T ν,re shared by all neutrinos after recoupling is then given by where T SM ν is the active neutrino temperature just prior to recoupling (which is at its SM value) and T s,re,0 denotes the sterile sector temperature just prior to recoupling.
Eventually, the mostly sterile eV-scale mass eigenstate decouples from the light mass eigenstates and becomes semi-relativistic at the CMB time. Its kinetic temperature at this epoch is T s,CMB 0.13 eV. The effective number of relativistic species at the CMB epoch is given by Note that this is smaller than the SM value 3.045. This happens because part of the energy of active neutrinos has been transferred to the mostly sterile mass eigenstate ν 4 , whose kinetic energy gets redshifted away more efficiently after it becomes non-relativistic. Ref. [22] also found N eff < 3 for this scenario. Similarly, we obtain for the time of matter-radiation equality: Both values are in agreement with the Planck bound [13].

Recoupling before A decay
The last possibility, labeled as case B3 in Fig. 1, is that recoupling happens before A decay. In this case, all neutrinos, together with A , reach a common chemical equilibrium, which lasts until most of the A particles have decayed. During the formation of chemical equilibrium, the total energy is conserved while entropy increases. Energy conservation allows us to calculate the temperature T ν,re of the active + sterile neutrino sector immediately after recoupling: where T SM ν,re is again the active neutrino temperature just prior to recoupling. Plugging in numbers for the effective numbers of degrees of freedom g ν a , g ν s , g A and using ξ BBN(B) = 0.465, we obtain Later, the A decay and the thermal bath of neutrinos gets reheated by a factor [(3g ν a + g ν s + g A )/(3g ν a + g ν s )] 1/3 1.125. The effective number of relativistic species after A decay is then The next steps are the decoupling of sterile neutrinos and active neutrinos, and then the freeze-out of sterile neutrino self-interactions. Since the number densities and energy densities of the different species do not change during these decouplings, the total effective number of relativistic degrees of freedom at the CMB epoch is given by where again the pressure characterizes the contribution of the semi-relativistic ν 4 to the radiation density in the Universe. Similarly, at matter-radiation equality we have This number is still within the 2σ error of the Planck bound [13].

IV. STRUCTURE FORMATION
Besides the constraints on extra radiation measured by N eff , CMB data also prefers that most of the active (massless) neutrinos start to free-stream before redshift z ∼ 10 5 [29,30]. On the other hand, matter power spectrum observations forbid these free-streaming degrees of freedom from carrying so much energy as to suppress small scale structures [31]. Therefore measurements of the matter power spectrum put the most stringent upper bound on the mass of all fully thermalized neutrino species: m ν 0.2-0.7 eV (95% C.L.) [13]. This concern [22] excludes a large proportion of the parameter region for self-interacting sterile neutrinos considered in [16]. However, like the constraint on N eff discussed in Sec. III, it is avoided if Γ s never exceeds H after the epoch when V eff drops below the oscillation frequency (cases A1 and B1 in Fig. 1).
Interestingly, structure formation constraints are also significantly relaxed when the U (1) s gauge coupling e s is large and/or the gauge boson mass M is small. In this case, sterile neutrinos, although produced abundantly through collisional production (see Sec. II), cannot freestream until late times, long after matter-radiation equality. Thus, their influence on structure formation is significantly reduced. We will now discuss this observation in more detail.
After the active and sterile neutrinos have equilibrated through A -mediated collisions, they should be treated as an incoherent mixture of the four mass eigenstates ν i (i = 1 . . . 4). The reason is that for m 4 ∼ 1 eV, their oscillation time scales are much smaller than both the Hubble time and the time interval between scatterings. For simplicity, assume that only the mostly sterile mass eigenstate ν 4 is massive with mass m 4 1 eV, and that it only mixes appreciably with one of the mostly active mass eigenstates, say ν 1 : We take the vacuum mixing angle to be θ 0 0.1 and we take into account that matter effects are negligible at temperatures relevant for structure formation (after matter-radiation equality). Since it is the flavor eigenstate ν s that is charged under U (1) s , the mass eigenstates ν 1 and ν 4 interact with relative rates sin 2 θ 0 and cos 2 θ 0 , respectively, while ν 2 and ν 3 essentially free-stream.
To study the influence of the secret interaction on structure formation, we estimate the mean comoving distance λ s that each ν 4 can travel in the early Universe. Since neutrinos can transport energy efficiently over scales smaller than λ s , the matter power spectrum will be suppressed on these scales. As long as neutrinos are collisional, they do not free stream, but diffuse over scales of order [32] (λ coll where a(t) is the scale factor of the Universe, t dec s is the time at which sterile neutrino self-interactions decouple, is the thermally averaged interaction cross section of the mostly sterile mass eigenstate ν 4 , estimated here by naïve dimensional analysis, and n s is the number density of sterile neutrinos. For simplicity, we take the kinetic temperature T s of the sterile sector equal to the active neutrino temperature in this section, i.e. T s = T SM ν = (4/11) 1/3 T γ ∝ a −1 (t), as long as ν 4 are relativistic. If sterile neutrinos become non-relativistic (T s < m s ) while they are still strongly self-coupled, the kinetic temperature of the sterile sector scales as T s ∝ a −2 (t) until T s drops below T s,dec . After that, the sterile neutrino momenta are simply redshifted proportional to a −1 (t). This implies in particular that, at T s m s , we have n s T 3 s , while after ν 4 become non-relativistic, but are still strongly coupled, this changes to n s ∝ T 3/2 s . The computation of the average velocity v s of ν 4 entering eq. (22) is discussed in Appendix A.
The decoupling temperature T s,dec and the corresponding time t dec s are defined by the condition that the sterile neutrino interaction rate is just equal to the Hubble rate: n s σv s t=t dec = H(t dec ) .
After t dec , sterile neutrinos start to free stream. The total comoving distance that a ν 4 travels between the time t dec and the present epoch t 0 is [32] λ fs The overall damping scale is then given by At scales larger than λ s , structure formation is unaffected by the existence of sterile neutrinos, while at smaller scales, structures are washed out.
corresponding to a wave number of k s ≡ 2π/λ s 0.085 h/Mpc .  Figure 3. (a) Three-dimensional matter power spectrum P M (| k|) derived from the SDSS Luminous Red Galaxy (LRG) Sample [33] and (b) one-dimensional flux power spectrum ∆ 2 (k) = k P F (k)/π of Lyman-α photons at various redshifts, compared to the qualitative predictions of sterile neutrino models with (red dashed curves) and without (green dotted curves) selfinteractions. Note that the relation between P M (| k|) and P F (k) is non-linear, see e.g. [34]. The assumed self-interaction parameters are e s = 0.1, M = 0.1 MeV, and the assumed sterile neutrino mass is 1 eV. The data points and the SM prediction (solid green curves) are taken from [33] and and from [35], respectively. The predictions including sterile neutrinos are obtained by multiplying the SM predictions by the k-dependent suppression profile from Fig. 7 of [31], shifted such that the onset of the suppression is at our calculated damping scale k s (Eq. (27); see text for details), and scaled such that the maximum suppression is given by Eq. (29) for panel (a) and by (30) for panel (b). Note that the error bars shown here are statistical only, and large systematic uncertainties, especially at small scales (large k) should be kept in mind.
This should be compared to the free streaming scale of a decoupled sterile neutrino with a mass 1 eV, This factor of ∼ 5 decrease in the free streaming scale compared to a conventional sterile neutrinos without selfinteractions implies that data on large scale structure (LSS) and baryon acoustic oscillations (BAO) will be in much better agreement with our model than with sterile neutrino models that do not feature self-interactions. The strongest constraints will come from data probing very small scales, in particular Lyman-α forests.
Even at scales k > k s , the suppression of the matter power spectrum P M (| k|) does not set in abruptly, but increases gradually. For non-interacting sterile neutrinos, numerical simulations show that the suppression saturates at k 50 k s . At even smaller scales (even larger k), the deviation from the prediction of standard cosmology is [31,36] δP M (| k|) in the linear structure formation regime. Here, f ν = 3m s ζ(3)/(2π 2 ) T 3 s (t 0 )×8πG/(3H 2 (t 0 ))/Ω m 0.07 is the ratio of the sterile neutrino mass density Ω s to the total mass density Ω m 0.3 today. In the regime of nonlinear structure formation, δP M (| k|)/P M (| k|) is somewhat larger [31,36], but N-body simulations show that it decreases again at scales k few h/Mpc [34,37].
It is, however, difficult to directly measure P M (| k|) at these nonlinear scales. The most sensitive data sets are Lyman-α forests, from which the 1-dimensional flux power spectrum P F (k) of Lyman-α photons can be extracted. Translating P F (k) into a measurement of P M (| k|) requires a determination of the bias b(k), which is obtained from numerical simulations of structure formation that include the dynamics of the gas clouds in which Lyman-α photons from distant quasars are absorbed. For SM neutrinos, such simulations have been performed for instance in [34], and we can estimate from Fig. 13 of that paper that the maximal suppression of P F (k) is of order where m ν is the sum of all neutrino masses. This estimate is crude but conservative, and ignores the fact that the maximal suppression is actually smaller at lower redshifts. The suppression of P F (k) described by Eq. (30) is smaller than the suppression of P M (| k|) from Eq. (29) because of the nonlinear k-dependent relation between the matter power spectrum and the flux power spectrum (see for instance [38], especially Fig. 16 in that paper). Since no dedicated simulations are available for our selfinteracting sterile neutrino model, we will assume in the following that δP F (k)/P F (k) saturates at the value given by Eq. (30) even when m ν is dominated by the sterile neutrino mass m s . This amounts to assuming that the impact of secretly interacting sterile neutrinos on these small scales is qualitatively similar to that of active neutrinos. A more detailed treatment requires a dedicated simulation including these secretly interacting sterile neutrinos. Note that neutrino free-streaming after CMB decoupling may lead to less suppression than described in Eqs. (29) and (30) because perturbation modes well within the horizon have already grown significantly by that time. We will not include this effect in the following discussion to remain conservative.
We show the qualitative impact of self-interacting sterile neutrinos on large scale structure in Fig. 3. Panel (a) compares theoretical predictions in models with and without sterile neutrinos to data on the three-dimensional matter power spectrum P M (| k|) from the Sloan Digital Sky Survey (SDSS) Luminous Red Galaxy (LRG) catalog [33]. Panel (b) compares to onedimensional flux power spectra ∆ 2 (k) ≡ k P F (k)/π from Lyman-α forest data [35]. SDSS-LRG data corresponds to a mean redshift of z 0.35, while Lyman-α data is split up according to redshift and reaches up to z 5.4. Note that the data in [35] is presented as a function of the wave number k v in velocity space, measured in units of sec/km. The conversion to the wave number k in coordinate space, measured in units of h/Mpc, is done according to the formula k = k v H(z)/(1+z), where H(z) is the Hubble rate at redshift z. The theoretical predictions for the SM with vanishing neutrino mass (solid green curves in Fig. 3) are taken from [33] and [35], respectively. Our (qualitative) predictions for sterile neutrino models with and without self-interactions are obtained in the following way: we start from the numerical prediction for the neutrino-induced suppression of the matter power spectrum from Ref. [31]. In particular, we use the curve corresponding to f ν = 0.07 from Fig. 7 in that paper. We then shift this curve such that the onset of the suppression coincides with our calculated damping scale k s , and we rescale it such that the maximal suppression is −8f ν in Fig. 3 (a) (linear regime) and 10% in Fig. 3 (b) (nonlinear regime), see Eqs. (29) and (30). We then multiply with the SM prediction to obtain the dotted green curves for sterile neutrinos without self-interactions and the red dashed curves for sterile neutrinos with self-interactions in Fig. 3. We use e s = 0.1, M = 0.1 MeV and m s = 1 eV. Since we neglect a possible upturn of the power spectrum at k 1 h/Mpc [37], our estimates are very conservative. From Fig. 3 (a), we observe that the suppression of the matter power spectrum at scales 0.2 h/Mpc due to self-interacting sterile neutrinos is completely negligible, while a fully thermalized non-interacting sterile neutrino with the same mass leads to a clear suppression already at these scales. This implies that self-interacting sterile neutrinos with the parameters chosen here are not constrained by data on linear structure formation. Going to smaller scales or larger k (Fig. 3 (b)), where nonlinear effects become relevant, we see that both the sterile neutrino model with self-interactions and the one without lead to suppression, but the amount of suppression is reduced in the self-interacting case. It was shown in Ref. [35] that the data disfavors suppression larger than 10% at k = 10 h/Mpc. Self-interacting sterile neutrinos at the benchmark point shown in Fig. 3 appear to be marginally consistent with this constraint. It should be kept in mind, however, that our predictions are only qualitative. Therefore, only a detailed fit using simulations of non-linear structure formation that include sterile neutrino self-interactions could provide a conclusive assessment of the viability of such a scenario.
Let us finally discuss how the cosmological effects of the three active neutrinos are modified in the selfinteracting sterile neutrino scenario. The dynamics of the mass eigenstates ν 2 and ν 3 , which we assume not to mix with ν 4 , is the same as in standard cosmology: they start to free stream at redshift z 10 5 . ν 1 , however, starts to free stream later than a non-interacting neutrino, but earlier than ν 4 . The free-streaming condition for ν 1 is, in analogy to Eq. (23), As shown in Ref. [29], free-streaming of active neutrinos before redshift z ∼ 10 5 is required to sufficiently suppress the acoustic peaks in the CMB power spectrum. The change from three to two truly free streaming neutrino species in our model will lead to minor modifications of the CMB power spectrum, but the analysis from [29] suggests that these are unlikely to spoil the fit to CMB data, in particular since they may be compensated by changes in the best fit values of other cosmological parameters.
Note that also Planck CMB data alone, without including data on large scale structure observations, imposes an upper limit on the mass of sterile neutrinos, which, for a fully thermalized species is m s 0.5 eV at 95% C.L. [13]. A much weaker bound is expected if self-interactions among sterile neutrinos are so strong that they remain collisional until after the CMB epoch. In this case, the early Integrated Sachs-Wolfe (ISW) effect induced by ν s perturbations at low multipole order (50 ≤ l ≤ 200) will be suppressed [31]. Thus, the main effect on the CMB will come from the shift of matter-radiation equality, to which the sensitivity is, however, much weaker.

V. DISCUSSION AND CONCLUSIONS
As we have seen in the previous section, there are two main scenarios in which self-interacting sterile neutrinos do not run into conflict with cosmological data: (i) The ν s production rate Γ s drops below the Hubble expansion rate H before the effective potential |V eff | drops below the oscillation frequency |∆m 2 /(2E)| and the dynamic suppression of active-sterile mixing due to V eff ends. In this case, sterile neutrinos are not produced in significant numbers in the early Universe and hence cosmology is not sensitive to their existence. An explanation of small scale structure anomalies as advocated in [16] is, however, not possible in this scenario. 6 In particular, even if the new interaction also couples to dark matter as proposed in [16], it is too weak to have phenomenological consequences.
This disadvantage can be avoided if more than one selfinteracting sterile neutrino exists. Consider for example, a model with three mostly sterile neutrino mass eigenstates ν 4 , ν 5 , ν 6 . Let ν 4 and ν 5 have a relatively large mixing θ 0 ∼ 0.1 with active neutrinos, as motivated for instance by the short baseline oscillation anomalies. Let their coupling to the A gauge boson be e (4,5) s 10 −5 , large enough to dynamically suppress their mixing with the mostly active mass eigenstates until after BBN, but small enough to prevent their equilibration afterwards. On the other hand, let ν 6 have a vanishing mixing with ν 1,2,3 , but a larger secret gauge coupling e (6) s 0.1. Due to its small mixing, it is never produced through oscillations. However, its primordial population-the relic density produced before the visible and sterile sectors decoupled in the very early Universe-still acts as a thermal bath to which the dark matter may be strongly coupled, thus potentially solving the missing satellites problem [41][42][43][44][45].
(ii) The self-interactions are so strong that sterile neutrinos remain collisional at least until matter-radiation equality. In this scenario, sterile neutrinos are produced when |V eff | ≤ |∆m 2 /(2E)|. However, as shown in Sec. III, the effective number of relativistic degrees of freedom in the Universe, N eff , remains close to 3 because equilibration between active and sterile neutrinos happens after neutrinos have decoupled from the photon bath. Moreover, as argued in Sec. IV, the impact of self-interacting sterile neutrinos on structure formation is much smaller in this scenario than the impact of conventional noninteracting sterile neutrino because they cannot transport energy efficiently over large distances due to their reduced free-streaming. Structure formation constraints could be further relaxed in models containing, besides an eV-scale mass eigenstate ν 4 , one or more additional mostly sterile states with much lower masses [25]. It is intriguing that the parameter region corresponding to scenario (ii) contains the region where small scale structure anomalies can be explained, as shown in [16].
We summarize these results in Fig. 4. The yellow crosshatched region on the right is ruled out because active and sterile neutrinos come into thermal equilibrium before the active neutrino decoupling from the SM plasma. In the lower part of this region, this happens simply because V eff is negligibly small. In the upper part, V eff is large, but also Γ s is large so that collisional production of sterile neutrinos is efficient in spite of the suppressed inmedium mixing θ m . This leads to constraints from N eff and from the light element abundances in BBN [21]. In the blue vertically hatched region, sterile neutrinos are produced after ν a decoupling, so that CMB constraints on N eff remain satisfied. However, sterile neutrinos freestream early on in this region and violate the CMB and structure formation constraints on their mass. This mass constraint can be considerably relaxed if the sterile neutrinos remain collisional until after the CMB epoch at T γ 0.3 eV. This defines the upper edge of the blue hatched region. In the red shaded region in the upper left corner, the secret interaction is too strong and ν 1 free streams too late. CMB data requires that active neutrinos free stream early enough, and thus strongly disfavors this region. Two white regions remain allowed: Scenario (i) with weak self-interactions, corresponds to the wedge-shaped white region in the lower part of the plot. Scenario (ii), with strong self-interactions, is realized in the thin white band between the blue vertically hatched region and the red shaded region. As explained above, whether or not this white band is allowed depends strongly on systematic uncertainties at Lyman-α scales and on the possible existence of additional states with masses 1 eV. There are several important caveats and limitations to the above analysis. First, we have only worked with thermal averages for the parameters characterizing each particle species, such as energy, velocity, pressure, etc. To obtain more accurate predictions, it would be necessary to solve momentum-dependent quantum kinetic equations. This would be in particular interesting in the temperature regions where V eff changes sign and where V eff ∼ ∆m 2 /(2T ν a ) × cos 2θ 0 . We expect that our modeling of flavor conversions in this region as fully nonadiabatic transitions is accurate, but this assumption remains to be checked explicitly. Moreover, the impact of self-interacting sterile neutrinos on non-linear structure formation at the smallest scales probed by Lyman-α data should be calculated more carefully. Improving these issues is left for future work.
In conclusion, we have argued in this paper that selfinteracting sterile neutrinos remain a cosmologically viable extension of the Standard Model. As long as the selfinteraction dynamically suppresses sterile neutrino production until neutrinos decouple from the photon bath, the abundance produced afterwards is not in conflict with Figure 4. Schematic illustration of the parameter space for eV-scale sterile neutrinos coupled to a new "secret" gauge boson with mass M and a secret fine structure constant α s . The vacuum mixing angle between active and sterile neutrinos was taken to be θ 0 = 0.1. The white region in the lower half of the plot is allowed by all constraints, while the narrow white band in the upper left part satisfies all constraints except possibly large scale structure (LSS) limits from Lyman-α data at the smallest scales. The red stars show representative models in scenarios (i) and (ii). The colored regions are excluded, either by LSS observations (blue vertically hatched), by the requirement that active neutrinos should free stream early enough (red shaded), or by a combination of CMB and BBN data (yellow cross-hatched).
constraints on N eff . Moreover, if the self-interaction is either weak enough for scattering to be negligible after the dynamic mixing suppression is lifted, or strong enough to delay free streaming of sterile neutrinos until sufficiently late times, also structure formation constraints can be avoided or significantly relaxed. In the transition region T s ∼ m s , the solution needs to be obtained numerically. The result is plotted in Fig. 5.
Finally, we comment on the calculation of the pressure P and the average velocity v s of sterile neutrinos. The pressure is given by [46] and the average velocity is Here, N ≡ dp 4πp 2 /(2π) 3 × f (p, t) is a normalization factor. Besides the conditions of kinetic equilibrium discussed above, we also need to take into account that sterile neutrino self-interactions freeze out at a time t dec and sterile sector temperature T s = T s,dec , after which kinetic equilibrium is lost and sterile neutrino momenta are simply redshifted as a −1 (t). This implies for the momentum distribution function: for t < t dec 1 exp 1 T s,dec for t > t dec .
Here, µ s (t dec ) is the chemical potential at the time of decoupling. We have checked that the exact decoupling time only slightly changes the evolution of P , so we regard our solution in Fig. 5 as universal for all parameter values of interest. In Sec. IV, we have for simplicity as-sumed a sudden decoupling of self-interactions though. For the value e 2 s /M 2 1 MeV −2 chosen there, this leads to T s,dec ∼ 0.0024 eV, corresponding to a photon temperature of 0.038 eV.