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Physics Letters B
www.elsevier.com/locate/physletb
More is different: Reconciling eV sterile neutrinos with cosmological mass bounds ^
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Yong Tang
School of Physics, Korea Institute for Advanced Study, Seoul, South Korea
A R T I C L E I N F 0
Article history:
Received 6 July 2015
Received in revised form 18 August 2015
Accepted 8 September 2015
Available online 10 September 2015
Editor: J. Hisano
A B S T R A C T
It is generally expected that adding light sterile species would increase the effective number of neutrinos, Neff. In this paper we discuss a scenario that Neff can actually decrease due to the neutrino oscillation effect if sterile neutrinos have self-interactions. We specifically focus on the eV mass range, as suggested by the neutrino anomalies. With large self-interactions, sterile neutrinos are not fully thermalized in the early Universe because of the suppressed effective mixing angle or matter effect. As the Universe cools down, flavor equilibrium between active and sterile species can be reached after big bang nucleosynthesis (BBN) epoch, but leading to a decrease of Neff. In such a scenario, we also show that the conflict with cosmological mass bounds on the additional sterile neutrinos can be relaxed further when more light species are introduced. To be consistent with the latest Planck results, at least 3 sterile species are needed.
© 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.
1. Introduction
Although most neutrino experiments can be well described by the standard three light species paradigm, there have been several anomalies that indicate a new sterile state [1-6]. The new state should have an eV scale mass and a mixing angle with sin2 280 ~ 0.01 [7,8]. With such parameters, sterile neutrinos can be copiously produced from oscillation with each species increasing the effective number of neutrinos Neff almost by one unit. This would be in tension with the cosmological bounds from cosmic microwave background (CMB) data [9], Neff < 3.91 and mf < 0.59 eV.
The conflict can be resolved if the sterile species is only partially thermalized with Neff ^ 1. Partial thermalization can be realized in particle physics models where there are secret self-interactions1 in the sterile neutrino sector [10-14]. These self-interactions can induce large matter potentials, effectively suppress the mixing angle, and block sterile neutrino's production from oscillation efficiently [10,11].
However, the situation changed recently when authors in [15] argued that even if sterile neutrinos' production are blocked at BBN time, as the Universe cools down, the new self-interaction will eventually equilibrate sterile neutrinos with the active ones. Then, it can be easily shown that in 3 + 1 scenario, 1/4 of the cosmic background neutrinos would be the heavy sterile ones, which is still in conflict with the above cosmological bounds.
The novelty in this paper is that, we numerically solve the quantum kinetic equations for neutrino mixing, show and confirm that flavor equilibrium is indeed reached after BBN, see Fig. 2. We also propose that the above mentioned conflict can be reconciled easily in an extension that more than one sterile states are introduced. Of the n introduced self-interacting sterile species, only one has eV scale mass and mixes with the active neutrinos. In the early Universe, all of them are out of equilibrium, but can approach flavor equilibrium with active neutrinos after BBN era. So the current relic neutrinos are composed of 3 + n species with equal flavors. As we shall show, Neff in the late Universe can actually decrease and the cosmological bounds can be evaded. To be consistent with the latest Planck results, we found that n = 3 is the minimal number of the introduced sterile species.
E-mail address: ytang@kias.re.kr. 1 In this paper, we do not discuss the case that large lepton asymmetry exists in the active neutrino sector.
http://dx.doi.Org/10.1016/j.physIetb.2015.09.018
0370-2693/© 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/Iicenses/by/4.0/). Funded by SCOAP3.
This paper is organized as follows. In Section 2, we briefly review the effective number of neutrinos, Neff. In Section 3, we discuss the scenario that how Neff in BBN epoch can be different from its value in CMB time when there are secret self-interactions among sterile neutrinos. In Section 4, we show how the cosmological mass bounds on sterile neutrino can be relaxed if more than one light species are introduced. Finally, we give our conclusion.
2. Overview of Neff
In this section, we shall review Neff briefly and establish the related conventions and definitions.
First, let us recall the thermal history of neutrinos in the standard model (SM) at the early Universe. When the temperature of the thermal bath, Tr, is much higher than MeV, active neutrinos, va (a = e, t ), are in thermal equilibrium with other SM particles through electroweak processes and have the same temperature as TY. Around 2 MeV, active neutrinos are decoupled because electroweak interaction is not strong enough to keep them in equilibrium. Later, electrons/positrons annihilate but heat only the photons. Using the conservation of entropy density, one can obtain the temperature ratio after e± annihilation,
Afterwards, TVa /TY is constant in the standard cosmology. Neff is defined by the energy density ratio,
Pi = Py + Ea Pva = 1 + _( Tv Py Py 8 v t
where pR stands for the total energy density of radiations. In the standard model, we have 3 species of active neutrinos, ve, v¡¡, and vT, so Neff = 3, or 3'046 precisely if instantaneous neutrino decoupling is relaxed. If there is some new physics, it might modify TVa/Ty and/or contribute to pR as extra radiation. For example, dark matter may affect Tva [16,17] or Tr [18], and new particles can decay to or contribute as equivalent neutrinos [19-21]. In this paper, we only consider the extended models with sterile neutrinos.
In the following discussions, we shall use va (a = e, t) to denote the 3 active neutrinos, vs for the sterile species, and or v for all of them if not otherwise specified. And vj will be referred as the i-th mass eigenstate with a mass mvj. For the parameter space we focus on, v12 3 are mainly mixed states of active neutrinos va and vjs (i > 3) are mainly mixed sterile species. We shall also neglect the masses of v1,2,3 when considering the mass constraints on sterile neutrinos.
In principle, Neff can be a function of time or photon temperature. We define 8Neff as the deviation from the standard value, with explicit time/temperature dependence,
PR - Py Py
[3 + 5Neff (t)j,
5 Neff (t) = £
where T0a stands for the neutrino temperature in the standard cosmology without new physics,
T 0a = T Y
a _ before e± annihilation and T°a = (4/11)1/3 Tr afterwards, and f$ runs through all active/sterile neutrinos.
We shall keep in mind that 3 in Eqs. (2.3) and (2.4) is actually 3'046 precisely. But this little difference would not affect our later discussions and we shall use 3 throughout the paper.
3 Nbbn vs Ncmb 3. Neff vs Neff
In this section, we discuss how self-interaction can affect the thermalization of sterile neutrinos. The essential picture is described in Fig. 1 where sterile neutrinos are only partially thermalized at/before BBN time, but flavor equilibrium, pvs = pva, is reached at later time. To be as general as possible, we introduce n sterile species and do not discuss the specific particle physics models, but emphasize that new interaction for the sterile neutrinos is required.
It should be pointed out that in the present discussion for illustrating our points, the minimal setup we need is that one of the n sterile species is mixed with active neutrinos and has eV-scale mass, and the rest may be massless and have negligible mixing to affect current neutrino experimental results. Moreover, sterile neutrinos do not have to mix with each other since all sterile neutrinos can be in flavor equilibrium through new interactions. Therefore, let us just simply assume there is only mixing between a 4-th neutrino and the active species, namely the mixing matrix has the following form,
( Ve \ (Uel Ue2 Ue3 Ue4 V1
Vß U „1 U ß2 U ß3 U ß4 V2
VT U t 1 U t 2 U t 3 U t 4 V3
Vs Us1 Us2 Us3 Us4 V4
/ ■■ 1 ■■
partially thermalized vs
V TVa + TVs MeV
flavor equilibrium
T„. = Tv. = Tv
Fig. 1. Thermal history of active/sterile neutrinos. When the temperature is high, vss are not in thermal equilibrium with vas because of the suppression from a large matter potential. As the Universe cools down, equilibrium between active and sterile neutrinos could be reached.
Further assume only 1-4 mixing and CP conservation, then the complete 4 x 4 mixing matrix is
U 4x4 =
i C13C12 C13 s12 s13 0\
— C23s12 — s13s23C12 C23C12 — s13s23s12 C13s23 0
s23s12 — s13C23C12 —s23C12 — s13C23s12 C13C23 0
0 0 0 1
i C13C12C14
—c23s12c14 — s13s23c12c14
c13s12 c23c12 — s13s23s12
s23s12C14 — s13C23C12C14 — s23C12 — s13C23s12Ê
—s14 0
( c14 0 0 s14 \
0 10 0
0 0 10 y —s14 0 0 C14 y
s13 C13C12 s14 \
C13s23 —C23s12s14 — s13s23C12s14
C13C23 s23s12 s14 — s13C23C12 s14
0 C14 /
where e.g. Cjj = cos ûjj and sjj = sin etc. Complete investigations with the above multiple-flavor mixing are quite involved numerically when solving the full quantum kinetic equations (QKEs) [22-25] and we refer to Refs. [26-28] for multiple-flavor analysis with non-interacting sterile neutrinos. For simplicity and without loss of generality, we work with only two neutrino states, ve-vs mixing with mass difference Sm2 and mixing angle 80 = 6\4, and shall pay our attention to eV sterile neutrinos with the parameter space suggested by neutrino anomalies, Sm2 ~ 1 eV2 and sin2 2d0 ~ 0.01. We shall keep in mind that vs produced from v/x and vT's oscillation could be equally important since |U/x4 \ ~ |UT4| ~ 0.46|Ue41.
In most of the previous discussions in the Iiterature, steriIe neutrinos are assumed to have no interaction, so they can onIy be produced by osciIIations from the active ones. And the production rate and totaI amount depend on the mass difference, mixing angIe and Iepton asymmetry [29-40].
In case of sterile species having a secret self-interaction, parametrized by GX = g2X/M2X (similar to the Fermi's constant GF in SM),
Gx vs T vs i;sr j vs, Ti, j are products of y^, Y5,...,
the production will also depend on the strength of GX. Models that can give rise to the above types of self-interactions can be found in Refs. [10-14,41,42]. To calculate the Nfn, we use the modified version of LASAGNA [10,43,44] with gX = 0.1.
For 1 + 1 scenario, we can parametrize the system with 2 x 2 Hermitian density matrix in terms of Pauli matrices,
P = ■
/<>( P 0
+ P-à),
where f0 = 1/ (eE/T + 1) and P = (Px, Py, Pz). And the densities of active and sterile neutrino are given by Pa = P0 + Pz = 2 PV1, Ps - P0 - Pz = 2 .
f 0 J0
The kinetic equations governing Pi 's time evolution are
P a = VxPy + Ta P s = —VxPy + Ts
2 feq,a P 2 c 1 a f0
feq, s
2^ — Ps f0
P x = —VzPy — DPx, P y = VZPX — 1 Vx( Pa — Ps) — DPy,
7|—i-1—i-1—|-1—i-1—i-1—i-1—i-1—|-1—i-1—i-1—i-1—i-1—
_Gx=[0Gf 5m2 = leV2, sin220o = O . 01.
Ty[MeV]
Fig. 2. Evolution of 8Neff as temperature Ty decreases. 8N^jf" depends on the self-interaction strength GX. Black curve shows the non-interacting case, GX = 0. The self-interaction can suppress the production of sterile neutrino at high temperature, but lead to flavor equilibrium at later time. Increasing the strength of self-interacting would delay the equilibrium time.
where Vj and ^ are the potentials and scattering kernels [10], respectively,
8m2 8m2 14n2 GF 4 14n2 GX 4
Vx =—r sin 200, Vy = 0, Vz —--- cos 290 --f ~tET4 +-= ~JtET4 ,
x 2E 0, y , z 2E 0 45^2 M2Z va 45V2 M2X vs
Ta - GfETVa
Ts - G2xET4s, D - 2 (Ta + Ts).
From the QKEs, we can recover neutrino oscillation in vacuum if we take the non-interacting limits, GF ^ 0, GX ^ 0,
P 0 = 0, P = Vx P,
which describe the precesses of P around V. Non-zero ^'s effects are to repopulate different momentum modes to reach thermal distribution and D-terms would damp and shrink P. As we can see in the above equations, the introduced new self-interaction leads to rs = 0 and contributes to Vz and D in the above QKEs.
The introduced self-interactions for sterile neutrinos have two effects. One is to block the thermalization at high temperature. The other effect is the collisions that lead a scattering-induced decoherent production at later time. In Fig. 2, we show the non-interacting case with a black curve and compare it with two interacting cases, GX = 10GF and GX = 104GF. If GX is large, for instance GX = 104GF, vs can only be partially thermalized with Tvs < Tva when Tr > 2 MeV. However, if GX is not large enough, for instance GX = 10GF, it will block the thermalization first at high temperature but enhance the production of sterile neutrino at a later time even before BBN epoch, Ty — 5 MeV. The GX = 104GF case, however, has shown that the equilibrium time is later than the GX = 10GF case, less than 1 MeV. Generally, increasing GX would delay the equilibrium time.
A simplified picture to understand these two effects is to use the effective mixing angle,
sin2 29eff =
sin2 290
(cos 20o - m Veff)2 + sin2 29o
(3.10)
where E is the energy of oscillating neutrino, -cos290 is usually called the vacuum oscillation term [45], and Veff = VSM — Vm is
the matter potential. Simple analysis would give
VSM ^ z^FT4 Vnew ~ ZXFT
' m . ^ ' va ' " m h
' Vs '
(3.11)
which highly depend on the temperature. A familiar case, MSW effect [46-48], happens when Veff = cos2908m2/2E, leading to a maximal mixing angle 0eff = n/4. However, when the matter potential is much larger than the vacuum term, the mixing angle is effectively suppressed,
sin2 29eff < sin2 290, when | Veff| »
cos 290
(3.12)
which can efficiently block the production of sterile neutrinos. The value of TVs to block production at BBN time can be roughly estimated as follows:
/ T \ 1/4 Gx -T4 Sm2 Tvs ( Sm2 M2/
Veff~ —tET4 >-^ —— > —T—-
ett MX Vs 2E MeV I 2E2 Gx
Take GX ~ 104 GF and MX ~ 1 GeV, we get Tv BBN time from the dot-dashed line in Fig. 2.
10 3 MeV around BBN time. This is what wee see the smallness of TVs all the way to
As the Universe cools down, the matter potential Veff gets smaller very quickly. When |Vf <
cos 2d0
, matter effect can be
neglected and 0f ~ 80. Again, we discuss in the simplified framework of two-flavor case, va-vs or v1-v2 in the mass eigenstates,
va = cos 9eff v\ — sin 9eff v2, vs = sin 9eff v\ + cos 9eff v2.
Before BBN, because of the highly suppressed 0eff, there are mostly va or v1 neutrinos in the thermal bath, and a small amount of vs or v2 states. After 6eff is not suppressed any more, v1 has the scattering process to produce v2 through v1 + v2 ^ v2 + v2 with rate
G2XT5s sin2200, and v1 + v1 ^ v2 + v2 with rate r
G2XT5a sin4200. If r is larger than Hubble parameter H =
-pR, then v1
and 2, or a and s, will reach equal numbers quickly. Since the sterile neutrinos' self-interaction will induce rapidly elastic scattering, vs vs ^ vs vs, it can redistribute momenta among sterile neutrinos. As long as the scattering rate rX ~ GX T5s is larger than the Hubble parameter H (G is the Newton's constant), sterile neutrinos will soon reach the Fermi-Dirac distribution, leading to the flavor equilibrium, nVs = nVa. For GX = 108Gf and MX = 1.2 MeV it was estimated that the equilibrium would be approached around Tr ~ 40 KeV [15], although numerically being a formidable challenge for such a large interaction.
Before a detailed discussion on Neff at or after CMB time, we should note that the above investigation only took into account the sterile neutrinos produced from oscillations. However, oscillation is not the only contributing process. Generally, we expect there is a whole dark sector accompanying with self-interacting sterile neutrinos. Once the whole dark sector is decoupled from the SM thermal both, entropy is transfered to sterile neutrinos (and other particles in thermal equilibrium with sterile neutrinos). We may call this part as the "primordial" portion (denoted as SNf which depends on the physical degrees of freedom in the dark sector and decoupling temperature, therefore model-dependent, see, for example, Refs. [12,13] for concrete models. Since at high temperature oscillation is effectively blocked by the large matter potential, the effects of these primordial sterile neutrinos are to change the initial condition for QKEs and lift up the curves at high temperature by SNf in Fig. 2. Hence, if not stated explicitly, we shall treat TVs at BBN time as a free parameter in the rest of our discussion.
Now, we are in a position to discuss the effect on Neff at or after CMB time. Assume there are n sterile species with common temperature T Vs, if sterile and active neutrinos reach the equilibrium when they are still relativistic, they would have the same temperature TV determined by the conservation of neutrino number or entropy density,
3 x (T°)3 + n x T3s = (3 + n) x T3,
(3.13)
where n is the number of sterile species that have self-interactions.2
With the new temperature T V, we can calculate SNeff at CMB time with Eq. (2.4),
S Nf = (3 + n)—1/3
3 + n xl ^
— 3,
in comparison with
eff ■
(3.14)
(3.15)
If sterile neutrinos were fully thermalized at BBN time, we would have TVs = T0a, and Eq. (3.14) gives the same result as Eq. (3.15) does. However, for partially thermalized Vs, TVs < T^a. As shown in Fig. 3, it is evident that SNemb < SNfn, and that the difference can be significant in the low TVs/T*0a region and it increases as n gets bigger. An interesting observation is that SNf can even be negative for small values of T s /T 0a . If future experiment data indicate a deficit of Neff, it would be natural to consider the scenario that active neutrinos are mixing with self-interacting sterile species.
In the right panel of Fig. 3, we show SNf as functions of SNbf for different n. The solid black line is for S Nf = S Nfn in non-interacting case, and from up to down, dashed lines correspond to n = 1 , 2, 3, 6, respectively. Dot-dashed orange lines mark the current 95% CL preferred ranges for Neff from Planck [49],
Neff = 3.11 +0 57 He + Planck TT + low P. Since SNbffn > 0 in our scenario, the available parameter space is inside the region with arrows indicated.
(3.16)
4. Cosmological neutrino mass bounds
In this section, we show how the conflict between eV sterile neutrino and cosmological mass bounds can be relaxed when more than one light sterile species are introduced.
2 We note that n = 1 case has been discussed in the latest version of Ref. [15] whose results agree with ours.
0.0 0.2 0.4 0.6 O.f
0.0 0.2 0.4 0.6 0.8 1.0
J 1 1 1 1 1 1 1 1 1 1 ■ i i i i i J 1 1 L
: sum vs <5B$? n=6y II n=y.
I r / // /. /n=2T
40- n=y-
■---------- — r^ss.
---- 1-1 ....... ....... . . r
0.0 0.2 0.4 0.6 O.f
TV,/??
1.0 .1 i i i | i i i | i i i : vs ■ i i i i i i i i iij.
0.5 1*5 § 0.0 - z' -
■ /// ^ ■
-0.5 Y/ 1________T_____ -/ / ■ / <- -
-1.0 ............. .............
n*(rv/7i)4or«5N3m
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Fig. 3. SNfn vs S(Left panel) We choose several cases for the number of sterile species, as indicated by n. For each case, the solid curve shows SNfn while the dashed one gives SNfb. Sizable differences can arise in the low TVs/T0 region. (Right panel) SN^m15 as function of SNf". The solid black line is for SNf = SNf in non-interacting case, and from up to down, dashed lines correspond to n = 1, 2,3, 6. Dot-dashed orange lines mark the current bounds from Planck [49]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. (Left panel) mf vs n. mf is a model-dependent quantity. We show how it changes as TVs/T0a varies in four cases with different n. mV4 = 1 eV is assumed to be dominant on the mass. (Right panel) mf as functions of SNf and n. Parameter space inside region marked with arrows is still allowed.
Cosmological bounds on the neutrino masses from the combination of CMB, large scale structure and distance measurements are constraining the following effective quantity [9],
mf ^Zi^m =£ fe) mVi - 94.1 eV x Vvh2, (4.1)
n0a i\T vj
where nVj stand for Vi's number density, nV for the value of active neutrino in standard cosmology, h2 accounts for its energy density fraction in the Universe. After the later flavor equilibrium discussed above, all neutrinos share the same temperature, T Vj = T v . Using the same minimal setup as we did in last section, we assume only one sterile state has eV-scale mass and all the others are almost massless. So we can reduce the above summation only over the heaviest one, i = 4,
In Fig. 4, we plot how mf changes with TVj/T0a in four cases, n = 1, 2, 3, 6. When more light states are added with but fixed SNeff, the individual number density of each species is decreased. Therefore, the total number of the heaviest state 4 is reduced and me ff then gets smaller correspondingly. Other light states are just radiations and red-shifted, contributing only negligibly in late Universe.
Now, we compare with the cosmological bounds. We should note that cosmological bounds on sterile neutrino mass and abundance depend on the cosmological models and the chosen data set [50-58] 3 mf only, and using the Planck+WP+High L data
combination, the latest result from Planck collaboration was able to give bounds with 95% CL [49],
2.53 < Nfb < 3.7, mf < 0.52 eV. (4.3)
As we show in right panel of Fig. 4, if we the face value of the above constraint,4 then n = 3 is on the intersect point with marginal status and is the minimal number of introduced sterile species. This amusing accidental agreement recovers the symmetry between active and sterile neutrinos. If future experimental pushed the upper limit further stringent, from the trend shown in the four cases of Fig. 4, it is easy to introduce more light sterile states in the discussed scenario to relax the cosmological bounds. When putting the lower bounds on Neff, we should be aware of the assumption that no other relativistic particles contribute as radiations. In cases where there are quite a mount of massless particles such as Goldstone or Majaron particles, the lower bounds on Neff then do not apply and n < 3 will be allowed.
5. Conclusion
In this paper, we have discussed a scenario that eV sterile neutrinos are partially thermalized before BBN era but equilibrated with active ones in later time. A mechanism to realize such a scenario is to introduce secret self-interactions for sterile neutrinos. The self-interactions can induce large matter potentials at high temperature, suppress the mixing angle and block the production of sterile neutrinos from oscillations. They can also lead to a rapid scattering-induced decoherent production of sterile neutrinos at later times before CMB. When flavor equilibrium between active and sterile species is approached, it surprisingly leads to a decrease of Neff.
We also discussed how the conflict with cosmological neutrino mass bounds can be relaxed in this scenario. We have found that, the more light sterile species we add, the less constrained they would be. If we take the latest Planck bounds [49], Nf113 < 3.7 and mf < 0.52 eV, at least three sterile species are needed to evade such a constraint.
Acknowledgements
The author is grateful to Rasmus Hansen for helpful discussions and sharing his code. This work is supported in part by National Research Foundation of Korea (NRF) Research Grant 2012R1A2A1A01006053.
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3 Currently, there is a very loose constraint on self-interaction of active neutrinos from CMB data [58], around ~ 108GF. The bound on self-interaction of sterile neutrino can be inferred and should be similar.
4 Note that the above constraint is only intended for non-interacting sterile neutrinos. Self-interactions may change the "free-streaming" scale, see Ref. [59] for example.
Since no analysis with self-interacting sterile neutrino is available, here we only took the face value from Planck [49].
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