Viable secret neutrino interactions with ultralight dark matter

TL;DR

This paper explores a model where ultralight dark matter interacts with sterile neutrinos to suppress their early universe production, potentially reconciling laboratory anomalies with cosmological constraints.

Contribution

It introduces a new mechanism coupling sterile neutrinos to ultralight dark matter, analyzing allowed parameter regions consistent with cosmological bounds.

Findings

Parameter regions compatible with cosmological constraints identified.

Suppression of sterile neutrino production in the early universe demonstrated.

Implications for laboratory neutrino oscillation experiments discussed.

Abstract

Several anomalies in neutrino oscillation experiments point to the existence of a $\sim 1\,$eV sterile neutrino $\nu_s$ mixing with $\nu_e$ at the level of $U_{e4}\cong 0.1$, but such a neutrino is strongly disfavored by constraints on additional light degrees of freedom ($\delta N_{\rm eff}$) and total neutrino mass ($\sum_\nu m_\nu$) from cosmology. "Secret neutrino interactions" that have been invoked to suppress the cosmological production of $\nu_s$ typically falter, but recently it was pointed out that $\nu_s$ could get a large mass in the early universe by coupling to ultralight dark matter $\phi$, which can robustly suppress its production. The model has essentially two free parameters: $m_\phi$, and $m_{s,0}$, the mass of the sterile neutrino at early times, enhanced by its coupling to $\phi$. I determine the parameter regions allowed by limits on $\delta N_{\rm eff}$ and $\sum_\nu m_\nu$ from the cosmic microwave background and big bang nucleosynthesis, using a simplified yet accurate treatment of neutrino oscillations in the early universe. This mechanism could have an important impact on laboratory experiments that suggest oscillations with sterile neutrinos.

Equations

$$\left(\genfrac{}{}{0ex}{}{m_{ee}}{m_{es}}\genfrac{}{}{0ex}{}{m_{es}}{m_{ss}}\right)$$

$$m_{es}\cong U_{e4}{m}_4$$

$$\frac{1}{2}\lambda {\bar{\nu }}_s\varphi {\nu }_s$$

$$\varphi (t)\cong 1.08{\varphi }_0\frac{J_{1/4}(m_{\varphi }t)}{(m_{\varphi }t{)}^{1/4}}\equiv {\varphi }_0\hat{\varphi }(t)$$

$$m_{s,0}=\lambda {\varphi }_0$$

$${\varphi }_0=1.0\times 10^{15}\mathrm{GeV}{\left(\frac{10^{-15}\mathrm{eV}}{m_{\varphi }}\right)}^{1/4}$$

$$R\equiv \frac{n_{{\nu }_s}}{n_{{\nu }_e}}=\frac{1}{2}\left(1-\exp \left[-2{\int }_T^{T_i}\left(\frac{\Gamma {\sin }^2{\theta }_m}{H{T}^{\prime }}\right)d{T}^{\prime }\right]\right).$$

$$\Gamma \cong \left(8+5e^{-m_e/T}\right)\frac{7\pi }{216}{G}_F^2{T}^{4}p$$

$${\sin }^{2}2{\theta }_m\cong \frac{4m_{es}^2}{4m_{es}^2+(m_{\mathrm{eff}}+2V_{e}p/m_{\mathrm{eff}}{)}^2}$$

$$V_e$$

$$\delta N_{\mathrm{eff}}(T)=\frac{\int d^{3}pf(p)R(T,p)}{\int d^{3}pf(p)}$$

$$\delta N_{\mathrm{eff}}\cong \frac{1}{2}\left[1-\exp \left(-\frac{65\sqrt{7}{\alpha }^{1/2}{G}_F{M}_p{m}_{es}^2}{576s_W(2+c_W^2{)}^{1/2}{g}_{\ast }^{1/2}{m}_{s,0}}\right)\right]$$

$$\delta N_{\mathrm{eff}}^{BBN}=\frac{4}{7}\left(\frac{4g_{\ast }+7\delta N_{\mathrm{eff}}}{(1+Y_{{\nu }_e}{)}^2}-g_{\ast }\right)$$

$$\frac{{\Gamma }_n}{c_2}$$

$$\frac{{\Gamma }_p}{c_2}$$

$$\sum m_{\nu }\cong [0.06\mathrm{eV}+m_4\delta N_{\mathrm{eff}}]$$

Full text

Introduction. Short baseline (SBL) neutrino oscillation experiments at nuclear reactors suggest at $3\sigma$ an eV-scale sterile neutrino $\nu_{s}$ that mixes with $\nu_{e}$ Giunti:2012tn ; Giunti:2012bc ; Kopp:2013vaa ; Dentler:2018sju ; Diaz:2019fwt . A persistent deficit of low-energy solar $\nu_{e}$ flux in gallium experiments lends support to this interpretation. The NEOS Ko:2016owz and DANSS Alekseev:2018efk experiments that also search for $\nu_{e}$-$\nu_{s}$ oscillations observe features that could be consistent with the SBL anomalies, though are not yet conclusive. Recent fits to the data favor a mass $m_{4}=1.1\,$eV and mixing matrix element $U_{e4}=0.11$ Kostensalo:2019vmv . Moreover there are hints from other experiments, LSND Aguilar:2001ty and MiniBooNE Aguilar-Arevalo:2018gpe , of $\nu_{\mu}\to\nu_{e}$ oscillations via a sterile neutrino with similar mass and mixing parameters. The sterile neutrino intepretation of $\nu_{\mu}\to\nu_{e}$ is clouded by constraints on $\nu_{\mu}$-$\nu_{s}$ oscillations from MINOS Adamson:2017uda and IceCube Aartsen:2017bap ; Jones:2019nix . In this work I focus on the simpler $\nu_{e}$-$\nu_{s}$ scenario that could explain the SBL deficits. The KATRIN experiment will provide an independent probe in the near future Esmaili:2012vg .

A generic challenge to the existence of sterile neutrinos in the indicated mass and mixing range are their oscillations in the early universe that would fully equilibrate the sterile species Enqvist:1991qj ; Dolgov:2003sg ; Gariazzo:2019gyi . This is strongly excluded by big bang nucleosynthesis (BBN) and cosmic microwave background (CMB) constraints on additional effective neutrino species, $\delta N_{\rm eff}$, as well as the sum of neutrino masses $\sum m_{\nu}$. Some means of suppressing oscillations in the early universe while allowing them at the present time is needed.

The use of sterile neutrino interactions to inhibit oscillations has a long history Babu:1991at ; Enqvist:1992ux ; Cline:1991zb . With respect to the current anomalies, refs. Hannestad:2013ana ; Dasgupta:2013zpn suggested that self-interactions of the sterile neutrino could impede the oscillations and thereby satisfy the cosmological constraints. This mechanism is referred to as “secret neutrino interactions,” despite the efforts of PRL to censor the name. Subsequent investigation showed that although the self-interactions in this context could prevent $\nu_{4}$ production until freezeout of the active neutrinos, in accordance with bounds on $N_{\rm eff}$, at lower temperatures their self-scattering combines with oscillations to convert active neutrinos to $\nu_{4}$ and violate the CMB bound on $\sum m_{\nu}$. Saviano:2014esa ; Cherry:2016jol ; Forastieri:2017oma ; Chu:2018gxk ; Song:2018zyl . (An exception is found for self-interactions mediated by a light gauge boson of mass $\lesssim 10\,$MeV Mirizzi:2014ama .)

It was recently pointed out that an effective realization of secret interactions is to couple $\nu_{s}$ to ultralight bosonic dark matter $\phi$ Farzan:2019yvo . In that case the scalar behaves as a coherent condensate, that has not yet started oscillating at early times. It can easily give a large mass to $\nu_{s}$ during this epoch, inhibiting the oscillations. Once the Hubble rate drops below $m_{\phi}$, the field oscillates and redshifts with scale factor as $a^{-3/2}$ as the universe expands. Its contribution to $m_{s}$ quickly disappears, leaving only the bare Lagrangian mass of $\sim 1$  eV. The “secret interaction” moniker is especially appropriate in this case, since the required coupling of $\nu_{s}$ to $\phi$ was shown to be exceedingly weak, $\lambda\sim 10^{-23}$. Similar interactions of light dark matter to standard model neutrinos were considered with respect to their effects on laboratory neutrino oscillations in refs. Berlin:2016woy ; Krnjaic:2017zlz ; Brdar:2017kbt ; Liao:2018byh .

This model is quite economical, depending only upon $m_{\phi}$ and the $\nu_{s}$-$\phi$ coupling $\lambda$, assuming $\phi$ constitutes all of the dark matter (DM) so that its initial amplitude is determined by its relic density. Equivalently, one can trade $\lambda$ for the new contribution $m_{s,0}$ to the $\nu_{s}$ mass at early times, before $\phi$ has started to oscillate. The purpose of this note is to determine the allowed parameter space, more quantitatively than was done in ref. Farzan:2019yvo .

Theoretical framework. Considering mixing between $\nu_{s}$ and $\nu_{e}$ only, the neutrino mass matrix is

[TABLE]

It is assumed that $m_{ss}\gg m_{ee}$. Then for small mixing one can show that $m_{es}$ is related to the mass eigenvalue $m_{4}\sim 1\,$eV by

[TABLE]

Fits to the SBL data favor $m_{4}=1.13\pm 0.04$, $U_{e4}\in[0.04,\,0.13]$ Dentler:2018sju ; for definiteness I adopt the central value $m_{4}=1.1\,$eV and $U_{e4}=0.11$ of ref. Kostensalo:2019vmv , giving $m_{es}=0.12$ eV and $m_{ss}\cong m_{4}$.

The sterile neutrino, taken to be Majorana, couples to bosonic DM via

[TABLE]

leading to the effective mass $m_{\rm eff}=m_{ss}+\lambda\phi$ when DM has a VEV. For ultralight DM, such a VEV is presumed to exist Hu:2000ke ; Hui:2016ltb , assuming some initial value in the early universe, that persists to account for the present relic density. If $\phi$ is sufficiently weakly coupled, it never thermalizes and remains coherent, behaving like a classical field. Its time dependence in the expanding cosmological background is 111The normalization is such that $\hat{\phi}(0)=1$

[TABLE]

during radiation domination (when $a(t)\sim t^{1/2}$). The relevant combination of parameters affecting neutrino oscillations is

[TABLE]

so that $m_{\rm eff}=m_{ss}+m_{s,0}\,\hat{\phi}(t)$.

For $t\gg m_{\phi}$ (but before matter-radiation equality) it can be shown that $\rho_{\phi}\cong 0.37\,m_{\phi}^{2}\phi_{0}^{2}(m_{\phi}t)^{-3/2}$. Matching to the present DM density, one finds

[TABLE]

Such a large VEV could arise if $\phi$ is an axion-like particle, the phase of a complex field $\Phi=|\Phi|e^{i\phi/f_{\phi}}$, with decay constant $f_{\phi}>\phi_{0}$. At early times $\rho_{\phi}\sim m_{\phi}^{2}\phi_{0}^{2}$ would be negligible compared to the energy density of radiation, and $\phi_{0}$ could take random values in the interval $[0,\,2\pi f_{\phi}]$.

Production of $\nu_{s}$. Although a rigorous study of $\nu_{s}$-$\nu_{e}$ oscillations in the early universe requires solving the Boltzmann equation for the density matrix Stodolsky:1986dx ; Enqvist:1990ad ; Sigl:1992fn , a good approximation can be obtained in a simpler approach, described in refs. Kainulainen:1990ds ; Cline:1991zb , which in some regimes leads to analytic results.222The quantitative agreement of the two formalisms was recently demonstrated in ref. Bringmann:2018sbs   The method is based upon solving the Schrödinger equation for the two-state system, including an imaginary term $-i\Gamma/2$ in the Hamiltonian representing scattering of $\nu_{e}$ in the plasma, that causes decoherence.

The solution yields the probability for a $\nu_{e}$ to oscillate into $\nu_{s}$ between an arbitrary initial time and a later time $t$. From this, a rate of $\nu_{s}$ production is derived, and the associated Boltzmann equation can be solved for the ratio of $\nu_{s}$ occupation number relative to that of $\nu_{e}$, as a function of temperature and neutrino momentum,333The factors of 2, missing in Cline:1991zb , account for the back-reaction from $\nu_{s}\to\nu_{e}$ Bringmann:2018sbs

[TABLE]

Here $\theta_{m}$ is the mixing angle including matter effects, and the initial temperature $T_{i}$ can be taken to infinity. The total interaction rate, including elastic scattering, is

[TABLE]

for a $\nu_{e}$ of momentum $p$ Cline:1991zb ; Notzold:1987ik . The exponential factor approximates the change at low temperatures when electrons have decoupled from the plasma. For relativistic neutrinos,

[TABLE]

(recall that $m_{\rm eff}(t)$ is the total $\nu_{s}$ mass) and

[TABLE]

is the thermal self-energy for $\nu_{e}$ or $\bar{\nu}_{e}$. The second line incorporates effects of the electron and baryon asymmetries, where $c_{1}\cong 0.95$, $\eta_{b}$ is the baryon-to-photon ratio, and $y_{n}$ is the neutron to proton ratio as a function of temperature, which I take to be the standard result as shown in ref. Kolb:1990vq . Numerically it turns out to have a negligible effect ($<0.1\%$) on the following results; hence we can treat $\nu_{e}$ and $\bar{\nu}_{e}$ on the same footing.

The effective number of extra neutrino species produced by the oscillations requires integrating over momentum, weighted by the massless Fermi-Dirac distribution function $f(p)$ for $\nu_{e}$ ,

[TABLE]

Before numerically evaluating $\delta N_{\rm eff}$, an analytic result can be found, in the regime where $m_{\phi}\lesssim 10^{-14}\,$eV, sufficiently small that $\phi$ does not start oscillating until the integral in eq. (7) has converged. In that case $m_{\rm eff}\cong m_{s,0}$ can be treated as constant, and $m_{es}^{2}$ can be ignored in the denominator. The integral can be evaluated analytically (ignoring the weak $T$-dependence of $g_{*}$ in the Hubble rate $H=1.66\sqrt{g_{*}}\,T^{2}/M_{p}$), to obtain

[TABLE]

where $W$ denotes the Weinberg angle, $M_{p}$ is the unreduced Planck mass, and $g_{*}\cong 10.75$ for the parameters of interest. The dependence on $T$ and $p$ is negligible for $T\lesssim 1\,$MeV, making it unnecessary to integrate over momenta.

BBN constraints. For larger values of $m_{\phi}$, the DM starts oscillating before nucleosynthesis, which tends to activate the neutrino oscillations. This can be compensated by also increasing $m_{s,0}$, but an analytic treatment is no longer possible. One should numerically integrate over $T^{\prime}$ and $p$ in eqs. (7,11).

Additionally for BBN, we should distinguish between oscillations that produced a real excess in $N_{\rm eff}$, occuring before the freezeout temperature $T_{f}=3.2\,$MeV of $\nu_{e}$, versus the subsequent oscillations that conserve total neutrino number but convert some $\nu_{e}$ into $\nu_{s}$. The reduction in $\nu_{e}$ density impacts BBN by changing the $p\leftrightarrow n$ equilibrium. One can account for this by defining an effective $\delta N_{\rm eff}^{BBN}$ Dolgov:2003sg ,

[TABLE]

where $g_{*}=10.75$ and $Y_{\nu_{e}}\lesssim 1$ is the relative abundance of $\nu_{e}$, reduced by oscillations between $T_{f}$ and nucleosynthesis, $T_{n}\cong 0.1\,$MeV. We estimate $Y_{\nu_{e}}$ by computing the change in $\delta N_{\rm eff}$ from the temperature interval $[T_{f},T_{n}]$, using eqs. (7,11).

The treatment (13) is valid when the effect of the oscillations is to deplete the density of $\nu_{e}$ without changing its energy spectrum too dramatically. Such spectral distortions can change the neutron-to-proton ratio and subsequent production of 4He in a way that cannot be simply modeled by a reduction in $\nu_{e}$ density Kirilova:1997sv .

To check whether it is justified to neglect the spectral distortion effect, I computed the collision terms of the Boltzmann equations for $n$ and $p$ in the region of parameter space, relevant to the BBN constraint, where $R(T,p)$ has the strongest momentum dependence. This occurs along the BBN exclusion contour at its upper right-most extreme, at the lowest temperature ($T=0.1\,$MeV), where $R(T,p)|_{p=xT}\cong 0.1\sqrt{x}$. The thermally averaged value is $\bar{R}=0.177$. The collision terms in the Boltzmann equation multiplying the neutron and proton densities are respectively

[TABLE]

where $c_{2}=G_{F}^{2}(g_{\scriptscriptstyle V}^{2}+3g_{\scriptscriptstyle A}^{2})T^{5}$, $\rho=Re^{-x}$, $f(x,y)=x^{2}(x+y)\sqrt{(x+y)^{2}+\zeta^{2}y^{2}}$, $y=(\Delta+m_{e})/T$, $\Delta=m_{n}-m_{p}$ and $\zeta=m_{e}/\Delta$. In the absence of spectral distortions, the rates are given by $\bar{\Gamma}_{n,p}$ evaluated as in (LABEL:nprates) but with $\bar{\rho}=\bar{R}e^{-x}$ in place of $\rho$. I find that the approximation $\Gamma_{n,p}=\bar{\Gamma}_{n,p}$ is good to $(2-3)\%$, justifying the use of eq. (13) for determining the BBN constraint.

CMB constraints. For the CMB constraints, there is an analogous effect from late time $\nu_{e}\to\nu_{s}$ conversions. Even though oscillations occuring after freezeout of $\nu_{e}$ should not change $\delta N_{\rm eff}$, they can increase the sum of neutrino masses by converting some $\nu_{e}$ to $\nu_{s}$. Therefore the extra contribution to $\sum_{\nu}m_{\nu}$ can be estimated as $m_{ss}$ times the asymptotic value of $\delta N_{\rm eff}$ that results at low $T\sim 1\,$eV, neglecting the conservation of neutrino number below $T_{f}$.

The results are shown in fig. 1, which displays three contours for $\delta N_{\rm eff}$ in a region constrained by CMB measurements Aghanim:2018eyx . The exact upper limit determined by the Planck Collaboration depends upon which data sets are combined. At 95% c.l.  $\delta N_{\rm eff}<0.5$ is a typical value (using TT+lowE or TT,TE,EE+lowE+lensing+BAO+R18), although a more stringent bound $\delta N_{\rm eff}<0.23$ is derived from TT,TE,EE+lowE alone. To illustrate the BBN constraint I show the $2\sigma$ limit from ref. Hufnagel:2017dgo , which is somewhat weaker than that obtained in ref. Cyburt:2015mya . The BBN contour at $\delta N_{\rm eff}=0.31$ illustrates the effect of conversions $\nu_{e}\to\nu_{s}$ after $\nu_{e}$ freezeout; for low $m_{\phi}$ it coincides with the corresponding CMB $\delta N_{\rm eff}$ (since no such conversions take place), but at higher $m_{\phi}$, $\delta N_{\rm eff}^{BBN}$ is seen to deviate from its CMB counterpart, as expected.

The strongest constraint is the CMB limit on neutrino masses. Their sum goes as

[TABLE]

taking account of the standard contribution, assuming normal mass hierarchy. Ref. RoyChoudhury:2019hls recently constrained $\sum m_{\nu}<0.145\,$eV for the normal hierarchy, implying $\delta N_{\rm eff}<0.08$. This implies a lower limit on $m_{s,0}>160\,$eV, hence $\lambda\gtrsim 10^{-22}\times(m_{\phi}/10^{-15}$ eV)1/4.

Discussion. For DM with $m_{\phi}\lesssim 10^{-14}\,$eV, we have seen that the cosmological analysis is relatively simple, since $\nu_{e}$ has frozen out before $\phi$ starts to oscillate. A favored value for $m_{\phi}$ from considerations of cosmological structure formation is considerably lower, $m_{\phi}\sim 10^{-22}\,$eV. In this regime, the de Broglie wavelength is so large that structure at galactic scales can be suppressed, providing a possible solution to the cusp/core problem of DM halos Hu:2000ke .

Such light DM has an oscillation frequency of order 1 y, which could have interesting consequences for laboratory oscillation experiments, if $\lambda$ is large enough to signficantly impact the effective mass $m_{\rm eff}$ of $\nu_{s}$ during the timescale of the experiment. For example, if the extra contribution to $m_{\rm eff}$ is as large as the bare mass $m_{4}$, one would need $\lambda\sim 10^{-15}$, which is technically natural since there are no significant loop corrections. In this situation, the usual analysis of oscillation data could lead to ambiguous results, since the $\Delta m^{2}$ being fitted would be varying in time. This effect has already been considered with respect to active neutrinos coupling to $\phi$ in refs. Berlin:2016woy ; Krnjaic:2017zlz ; Brdar:2017kbt . It could be interesting to reconsider the experiments that suggest active-sterile neutrino oscillations in this light. A search for time dependence of the signal has been performed by the Daya Bay collaboration Adey:2018qsd .

Acknowledgment. I thank K. Kainulainen and J. Kopp for very useful comments on the manuscript, and P. Huber for pointing out ref. Adey:2018qsd . This work was supported by NSERC (Natural Sciences and Engineering Research Council, Canada).