Cosmology-friendly time-varying neutrino masses via the sterile neutrino portal

Cosmology-friendly time-varying neutrino masses via the sterile neutrino portal Guo-yuan Huang,1,* Manfred Lindner ,1,†Pablo Martínez-Mirav´e, 2,‡and Manibrata Sen 1,§ 1Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany 2Departament de Física Teórica, Universitat de Val`encia, and Instituto de Física Corpuscular, CSIC-Universitat de Val`encia, 46980 Paterna, Spain (Received 2 June 2022; accepted 19 July 2022; published 9 August 2022) We investigate a consistent scenario of time-varying neutrino masses, and discuss its impact on cosmology, beta decay, and neutrino oscillation experiments. Such time-varying masses are assumed to be generated by the coupling between a sterile neutrino and an ultralight scalar field, which in turn affects the light neutrinos by mixing. We demonstrate how various cosmological bounds, such as those coming from big bang nucleosynthesis, the cosmic microwave background, as well as large scale structures, can be evaded in this model. This scenario can be further constrained using multiple terrestrial experiments. In particular, for beta-decay experiments like KATRIN, nontrivial distortions to the electron spectrum can be induced, even when time-variation is fast and it gets averaged. Furthermore, the presence of time-varying masses of sterile neutrinos will alter the interpretation of light sterile neutrino parameter space in the context of the reactor and gallium anomalies. In addition, we also study the impact of such time-varying neutrino masses on results from the BEST collaboration, which have recently strengthened the gallium anomaly. If confirmed, we find that the time-varying neutrino mass hypothesis could give a better fit to the recent BEST data. DOI: 10.1103/PhysRevD.106.033004 I. INTRODUCTION Ever since its first detection in 1956 [1,2], the elusive neutrinos remain to be the least known fermion in the Standard Model (SM) [3,4]. Though the early beta-decay data suggest neutrino mass to be either vanishing or extremely small compared to the electron mass [5–12],the discovery of neutrino oscillation phenomenon has firmly established the fact that neutrinos are massive. To weigh those massive neutrinos in a model-independent manner is the major task of modern beta-decay experiments [13–19]. It remains a mystery as to why the absolute scale of neutrino masses is more than six orders of magnitude smaller than its charged lepton partner. This is often ascribed to some new physics at very high energy scales in the spirit of the seesaw mechanism [20–36], whose experimental test is extremely challenging. Alternatively, an interesting possibility is to attribute the origin of neutrino masses to the dark sectors, for example, dark energy (DE) [37,38] and dark matter (DM) [39]. As the persistent direct detection searches of weakly interacting massive particles (WIMPs) come out with null signals so far, there has been increasing attention to other DM candidates, such as the ultralight DM [40,41]. The ultralight DM candidate, due to its tiny mass m<OðeVÞ, acts as a delocalized classical-number field. Such a scenario could be responsible for addressing generic studies of varying physical constants. One particular class of ultralight DM, the fuzzy dark matter with a mass mϕ≳10−22 eV, can also help to alleviate the small-scale structure issues existing between cosmological observations and simulations [42–45]. Such an ultralight scalar field, coupled to neutrinos, can give rise to intriguing phenomenological consequences. Neutrinos coupled to the scalar will naturally get a contribution to their masses, in analogy to the Higgs mechanism. As the classical DM field oscillates, the generated mass term for the neutrinos becomes timevarying. There have been various attempts of studying the phenomenology of a coupling between neutrinos and the ultralight field in the literature [46–71]. A simple realization of this possibility is to couple a classical scalar field (Φ) to neutrinos via the operator ¯ ννΦ, which generates a time-varying neutrino mass term. In a UV-complete model, this can be achieved by directly coupling Φto the lepton *[email protected].de †[email protected] ‡[email protected].es §[email protected]e Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 106, 033004 (2022) 2470-0010=2022=106(3)=033004(18) 033004-1 Published by the American Physical Society doublet lL≡ðν;eÞL, as in the type-II seesaw [32–35]. However, stringent constraints arise because of the accurate determination of electron properties. A more feasible way to couple light neutrinos to the scalar field is by mixing with gauge singlets, such as a right-handed sterile neutrino N. It is also natural to have a considerable Yukawa coupling between the sterile component and the scalar field. Motivated by this, our working Lagrangian consistent with gauge symmetries is described as, −L⊃yDlL ˜ hNþ1 2ðmNþgΦÞNcNþ1 2κlL ˜ h ˜ hTlc L þ1 2 y Λ Φ2NcNþH:c:þ;ð1Þ where Φrepresents the ultralight DM field, his the SM Higgs, the active Majorana neutrino mass term mν≡κv2=2 and the Dirac mass mD¼yDv= ffiffiffi 2 pwill be generated after Higgs takes the vacuum expectation value v= ffiffiffi 2 p≡h ˜ hi¼ 174 GeV, and mNis the Majorana mass term of the sterile neutrino. On top of that, the Yukawa interaction gΦNcN with a coupling constant ggenerates a time-varying mass term to the sterile neutrino. In addition, there might be a dimension-five effective interaction with a coupling constant yand the cutoff scale Λ[49,72,73], similar to the Weinberg operator generating light neutrino masses. In general, the Majorana mass terms mνand mN can be vanishing by imposing additional symmetries such as lepton number conservation (then the remaining Lagrangian will look similar to the singlet Majoron model [74–80]). In the most economical case, one can even generate small neutrino masses, with a minimal interaction form yDlL ˜ hNþgΦNcN, but the number of sterile neutrinos should be extended to at least two in order to explain the oscillation data. Note that even though we assume Majorana neutrinos in this work, the generalization to Dirac neutrinos is not difficult. An alternative scenario is the one in which right-handed neutrinos have no vacuum mass (mN¼0) but get a tiny Majorana mass from their coupling to the scalar field. This can give rise to ultralight scalarinduced pseudo-Dirac neutrinos [71]. The framework explored in this work is different from the earlier discussions about the mass-varying neutrinos, where the tiny neutrino mass is hypothetically generated from the coupling to the dark energy (e.g., acceleron [38], quintessence [81], etc.). In that case, neutrino masses are constant over any observable laboratory timescales and one cannot distinguish the induced mass from the vacuum mass by looking for the time-varying signals. Late-time neutrino mass generation can also arise in models having a late-time cosmic phase transition [82], or models having an anomaly due to a gravitational—θterm [83]. In fact, Ref. [82] analyzed redshift dependent neutrino masses in the light of recent Planck data, and reported a preference for models of late-time neutrino masses. This was further analyzed in a follow-up work [84], where additional data from type-Ia supernovae, as well as structure formation was used. More recently, it was demonstrated that a detection of the diffuse supernova neutrino background could shed light on whether neutrino masses turned on at later redshifts [85]. With the renormalizable terms in Eq. (1), the equation of motion of the scalar in our local galaxy is ð∂2þm2 ϕÞΦ¼g 2ðNcNþ¯ NNcÞ:ð2Þ where the addition of Hubble dilution term, 3H_ Φwith H being the Hubble expansion rate, is necessary if we consider the scalar evolution over cosmological timescales in the expanding Universe. In the absence of the source term in the right-hand side, the scalar field evolves freely in the Universe after production. Because the scalar is assumed to be produced coherently1and its occupation number is very high, it is more appropriate to consider Φas a classical field instead of a quantum state. Neglecting possible spatial variations (arising from structure formation), which are usually suppressed by the DM velocity vϕ∼10−3in our Milky Way, the field evolution simply follows, ΦðtÞ¼ϕsin mϕt: ð3Þ Here the time tis calibrated such that Φð0Þ¼0, and ϕ¼ffiffiffiffiffi 2ρ p=mϕdenotes the field strength, which is approximately ϕ⊙¼2.15×1015 eV·ð10−18 eV=mϕÞwith the dark matter energy density ρ≈0.3GeV · cm−3in our local galaxy. It is worthwhile to setup the magnitude of the coupling gby noting that gϕ⊙≈2.15 eV · ðg=10−15Þ· ð10−18 eV=mϕÞ. Unless otherwise specified, we use the capital Φto denote the complete time-varying field and ϕ for its amplitude. Through the Yukawa coupling to Φ, the sterile neutrino develops an effective time-varying mass term. One may worry about energy-momentum conservation within such a setup. Let us consider a closed system formed only by the scalar and the sterile neutrino. The total Lagrangian explicitly possesses a temporal translation symmetry, so the energy of the entire system must be conserved according to Noether’s theorem. Furthermore, because the Lagrangian with only the neutrino part preserves spatial translation symmetry, the neutrino momentum must be invariant under the time-varying scalar potential. But the neutrino energy might be perturbed by the scalar field. When the neutrino 1The misalignment mechanism serves as such an example, where a complex scalar field Φpicks up the expectation value associated with a broken global symmetry, resulting in a massless pseudo Nambu-Goldstone boson. The boson mass, which tilts the Mexican hat, can be generated by some phase transition. HUANG, LINDNER, MARTÍNEZ-MIRAV´ E, and SEN PHYS. REV. D 106, 033004 (2022) 033004-2 energy evolves with EN¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 Nþðgϕsin mϕtÞ2 q, the energy of the scalar system Eϕwill also change accordingly due to the feedback effect from Nsource term as in Eq. (2), such that ENþEϕ¼const. In the absence of the scalar potential, diagonalization of Eq. (1) leads to light mass eigenstates νi(for i¼1,2,3) and a heavy one ν4. If the vacuum mass m4is too large compared to the potential gϕ, only the effective coupling by mixing gij νΦ¯ νiνjwill be relevant at low energies, with iand jbeing the indices of light neutrino mass eigenstates. However, as has been realized, such a scenario faces inevitable constraints from the observation of cosmic microwave background (CMB) [46–48,68]. The growth of ϕwith the redshift will render cosmic neutrinos nonrelativistic in the early Universe, which is in contradiction with the free-streaming property of relativistic neutrinos from the CMB observation. Sensitivities of most of the laboratory searches are not even comparable to the CMB constraint in the order of magnitude. The situation is different in the other regime, when the sterile neutrino mass m4is smaller than gϕ. In this case, as we go back to the dense early Universe (gϕ≫m4), the potential induced for light neutrinos νiwill be suppressed by m2 D=ðgϕÞ. This is very similar to the seesaw mechanism which naturally generates the small neutrino masses with the suppression of heavy degrees of freedom. The details for this observation are given in Appendix A. The sterile neutrino can be as light as OðeVÞ, which is relevant for short baseline anomalies [86–91]. A general issue with the light sterile neutrino is the possibility of its thermalization in the early Universe, leading to an increase in ΔNeff [92–95]. In fact, the parameter space suggested by the short-baseline anomalies is almost ruled out by the ΔNeff constraints from Big Bang nucleosynthesis (BBN) and CMB [96,97]. As has also been noticed previously [49,56], the framework with a time-varying DM field coupled to sterile neutrino can provide a bonus to suppress the production of light sterile neutrinos in the early Universe. In this work, we systematically explore the consequences of a time-varying mass of sterile neutrinos in the early Universe, outlining the impact on BBN and CMB. Furthermore, we discuss the possibility of current generation beta decay experiments like KATRIN to probe the time-varying sterile neutrino hypothesis. Finally, we also highlight how short-baseline neutrino experiments fare in light of such a hypothesis. The mass range of the scalar relevant for various probes is illustrated in Fig. 1, including the neutrino oscillation experiment DUNE, the beta-decay experiment KATRIN, the short-baseline experiment BEST, as well as the equivalence timescales in the early Universe. The corresponding timescales are indicated on the top axis. It is worth mentioning that black hole superradiance constraints exists in several mass ranges for the ultralight scalar mass [98,99]. The structure of the rest of the work is as follows. In Sec. II, we separate the time-varying masses via the sterile neutrino portal into two different scenarios. In Sec. III,we investigate the cosmological consequences if the ultralight DM interacts with the sterile neutrino, including both heavy and light sterile neutrino scenarios. In Sec. IV, we explore the distortion effect induced by the time-varying potential in beta-decay spectrum, taking KATRIN experiment as an example. We proceed in Sec. Vto discuss the impact of a time-varying light sterile neutrino on the short-baseline experiments. Finally, we discuss our results, and conclude in Sec. VI. II. GENERATING TIME-VARYING ACTIVE NEUTRINO MASSES FROM STERILE NEUTRINOS In the limit that the sterile neutrino is very heavy compared to the scalar potential, i.e., m4≫gϕ, the light neutrinos will receive an effective mass in addition to the original vacuum one (see Appendix A), ˜ miðtÞ≈miþsin2θgϕsinmϕt: ð4Þ FIG. 1. The scalar mass scale relevant for various probes including DUNE-like, KATRIN-like, and BEST-like experiments. The corresponding timescales are indicated on the top axis. The mass ranges for which mϕ>Hat Tγ¼1MeV (for BBN) and Tγ¼0.3eV (for CMB) are also given for comparison. COSMOLOGY-FRIENDLY TIME-VARYING NEUTRINO MASSES …PHYS. REV. D 106, 033004 (2022) 033004-3 The active-sterile mixing angle θcan simply be absorbed by redefining gν≡sin2θ·gsuch that the neutrino mass correction reads as gνϕsin mϕt. In such a case, it is technically indistinguishable whether active neutrinos couple directly to the scalar field or by mixing with the sterile neutrino, and the discussion will be reduced to the usual scenario explored in the literature [46–48,50–55,57,58,60–70]. This potential is highly testable in neutrino oscillation experiments, provided the DM oscillation cycle, whose period is given by 2π=mϕ, is at least of the same order as the neutrino time of flight (i.e., the DM field is not rapidly oscillating). If the DM field is oscillating with a period of the order of the experimental running time, i.e., few days to years, the neutrino mass eigenstates can develop oscillation phases continuously with constant effective masses during the flight [46,48,62]. This would manifest as a time modulation of the signal. For shorter DM cycles, the modulation effect will be averaged, but not vanishing. This averaged effect imprints nontrivial distortions to the original neutrino oscillation probability as a function of the neutrino energy. Moreover, it was recently pointed out that even in the rapidly oscillating (dynamical and beyond) regime, the neutrino flavor transition induced by the scalar field is still possible but with a decreased impact [62]. Even though the time-varying analysis of neutrino oscillation experiments itself is interesting and very rich in phenomenology, the available model parameters are severely constrained from cosmology. In fact, a reasonable cosmological scenario, without fine-tuning the DM evolution, might rule out all the parameter space to which neutrino oscillations are sensitive. Irrespective of whether the scalar field accommodates all the DM abundance or not, its field strength averaged over space in the early Universe as a function of redshift zwill read as ϕðzÞ2¼ϕð0Þ2·ð1þzÞ3;ð5Þ up to the time when the Hubble expansion rate becomes comparable to the scalar mass, i.e., H≈mϕ. To see how an experimentally testable coupling is disfavored by cosmology, we take gνϕ⊙¼ffiffiffiffiffiffiffiffiffiffiffi Δm2 21 p≈8.7×10−3eV, where the local DM overdensity is approximately ϕ2 ⊙≈105ϕð0Þ2 [48,49]. During the era of matter-radiation equality ðzeq ≈3000Þ, we obtain gνϕðzeqÞ¼4.5eV, which exceeds the neutrino temperature at that time Tν≈0.5eV by almost one order of magnitude. As a result, neutrinos become nonrelativistic, and do not free-stream at the speed of light before recombination, thereby spoiling CMB observations. The above conclusion is actually rather conservative and requires DM particles to be populated just before the matter-radiation equality. On the other hand, if the DM production is not fine-tuned and takes place early before BBN at Tν≈1MeV (corresponding to zBBN ≈6×109) such as the misalignment production of QCD axions, a severe limit gνϕ⊙<7×10−7eV can be obtained by conservatively requiring gνϕðzBBNÞ<1MeV. The allowed tiny local effective time-varying mass clearly rules out all the possibility of realistic laboratory searches. The above picture changes if m4<gϕðzÞduring the photon decoupling and/or neutrino decoupling era. In Fig. 2, we show the lighter effective neutrino mass ˜ mL≡ Minf˜ m1;˜ m4gwithin a DM oscillation period for different gϕ. It is important to note that when the gϕsin mϕt¼ −m4−m1is satisfied, it is possible for ˜ mLto swap from ˜ m1 to ˜ m4due to the resonance encountered. We choose the lighter eigenstate mL, because the lighter neutrino is defined to have a dominant overlap with active neutrino, providing the active-sterile mixing is small. For the case of gϕ¼100m4, the lighter mass ˜ mLis given by ˜ m1for gΦ> −m1−m4and ˜ m4for gΦ<−m1−m4. Previous analyses simply assume an ad hoc sinusoidal variation, e.g., Eq. (4), in the active neutrino mass or mixing parameters, which need not necessarily to be the case. As shown in Fig. 2and is clear from Eq. (A2), the variation of light neutrino parameters can have a more complicated form, especially when gϕis large compared to the sterile neutrino mass. In the limit of 0eV <m 4≪gΦ, the mass-variation is instead given by (see Appendix Afor a derivation) ˜ m1≃m1þm4−ðm4−m1Þcos2θ 2−ðm4−m1Þ2sin22θ 4gΦ; ð6Þ ˜ m4≃m1þm4þðm4−m1Þcos2θ 2þgΦ:ð7Þ In the extreme case of m4≪gΦand sin θ≪1, a constant shift ∼m4sin2θwith a time variation with magnitude ∼m2 4sin2θ=ðgΦÞin the neutrino mass will be induced. In this case, we do not expect a large effective neutrino FIG. 2. The time-varying neutrino mass ˜ mLvia the sterile neutrino portal as a function of time t. The vacuum parameters have been chosen as m1¼0.1eV, m4¼3eV and sin22θ¼0.25. The value of the potential, gϕ, has been taken to be gϕ¼0.1m4(black curve), 0.2m4(dotted red curve), and 100m4(dashed blue curve), respectively. HUANG, LINDNER, MARTÍNEZ-MIRAV´ E, and SEN PHYS. REV. D 106, 033004 (2022) 033004-4 mass in the very early Universe as long as m4sin2θis chosen to be small. III. COSMOLOGICAL CONSEQUENCES OF TIME-VARYING STERILE NEUTRINO MASSES A. Neutrino decoupling and Big Bang nucleosynthesis The measurements of primordial helium and deuterium abundances agree very well with the theoretical predictions, assuming standard electroweak interactions with massless neutrinos. The weak interaction rates are expected to be altered if neutrinos are very massive at the redshift of decoupling, e.g., if ˜ miðzBBNÞ∼Tν. A severe constraint can be imposed on gϕ⊙if ultralight DM particles are assumed to be populated before the BBN era. This can translate into a constraint on the effective neutrino mass for increasing values of gϕas described by Eq. (5). Figure 3shows the average of the lighter neutrino mass h˜ mLias a function of the potential gϕ. The vacuum mass and mixing angle have been taken as m1¼0eV, m4¼3eV and θ¼π=12. We observe that starting from very small field strength with gϕ<m 4, the average mass h˜ mLiincreases linearly with gϕ. At the point around gϕ¼m4, the average mass h˜ mListops growing as expected from Eq. (A11). The turning point around gϕ¼m4is the key to alleviate the tension between testable time-varying signals and BBN. The maximum of h˜ mLi, which is given by h˜ mLimax ≈m4sin2θ, is under control no matter how gϕ changes in the early Universe. Hence, in order not to spoil the BBN observations with a large potential, the sterile neutrino parameter should stay within the range m4sin2θ≪1MeV. Light sterile neutrinos (lower than the neutrino decoupling temperature, i.e., m4<1MeV) have an additional risk of thermalization and increasing the amount of extra radiation in the early Universe, measured by ΔNeff [100,101]. It has been noticed that such a risk can be evaded by introducing secret interactions among sterile neutrinos. An effective potential suppressing the activesterile mixing will hence be induced in the presence of just a small background of sterile neutrinos [102–106]. The spirit is similar in our scenario. In the presence of a DM potential, sterile neutrino production will be strongly suppressed. In the following, we numerically investigate this possibility by directly solving for ΔNeff. In a simplified two-neutrino setup, the evolution equation of the sterile neutrino phase-space distribution fNis given by [107] dfN dTν¼− Γ 4HTν sin22 ˜ θmðfνa−fNÞ:ð8Þ where Tνis the neutrino temperature, fνais the active neutrino phase-space distribution, and Γ∝T5 ν=m4 Wencodes the net neutrino interaction rate. Here, ˜ θmis the effective mixing angle in matter, sin22 ˜ θm ¼h˜ Δ2ihsin22 ˜ θ14i h˜ Δ2ihsin22 ˜ θ14iþΓ2=4þðh˜ Δihcos2 ˜ θ14i−VTÞ2;ð9Þ where ˜ Δ¼ð˜ m2 4−˜ m2 1Þ=ð2EÞgives the vacuum oscillation frequency arising from the mass difference between active and sterile species, and hidenotes the time-averaging operation. Here VT∝ðT5 ν=m4 WÞis a measure of the forward scattering thermal potential experienced by the neutrinos. The scattering term in the denominator encodes the Quantum Zeno effect, where the flavor conversion is suppressed for a large scattering rate. Furthermore, the selfscattering process 2N→2Nmay freeze-in and become relevant post BBN, leading to a scattering-induced decoherent population of sterile neutrinos [104]. However, note that this effect is proportional to the coupling g, and is subdominant in our case with an extremely small coupling. In the limit where fνacan be approximated by a FermiDirac function (or any generic function of p=Tν), Eq. (8) can be solved approximately to obtain the contribution of sterile neutrinos around the time of BBN [108], ΔNBBN eff ¼fN fνa ≃1−exp− 2×103 4ffiffiffiffiffi g phsin22 ˜ θ14ih˜ m4i eV ; ð10Þ where gis the effective number of relativistic degrees of freedom. In Fig. 4, we illustrate the dependence of ΔNeff on the local DM potential gϕ⊙for a given sterile neutrino parameter choice Δm2 41 ¼9eV2and sin22θ¼0.25. Notably, the presence of just a tiny FIG. 3. The lighter neutrino mass average h˜ mLias a function of the scalar potential, gϕ. The vacuum parameters have been chosen as m1¼0eV, m4¼3eV and sin22θ¼0.25. The turning point around gϕ¼m4is marked as the dashed vertical line. COSMOLOGY-FRIENDLY TIME-VARYING NEUTRINO MASSES …PHYS. REV. D 106, 033004 (2022) 033004-5 potential, e.g., gϕ⊙≳10−7eV, is able to reduce ΔNeff to a negligible level, i.e., ΔNeff ≲0.01. Increasing the potential gϕ⊙will further reduce ΔNeff, thereby making this scenario safe from primordial abundance constraints. One might worry if the presence of the light scalar Φitself can act as extra radiation, and run into trouble with BBN predictions. This is again prevented in our scenario due to tiny coupling gconsidered, i.e., according to the relation gϕ⊙≈2.15 ×10−7eV · ðg=10−22Þ·ð10−18 eV=mϕÞ. Sterile neutrinos with masses in the keV range and produced through such a freeze-in mechanism can also act as possible dark matter candidates, as was pointed out by Dodelson and Widrow [107]. However, such a mechanism is in tension with the nonobservation of x-rays originating from the decay of sterile neutrino dark matter [109]. Nonetheless, in a scenario such as ours, the sterile neutrino need not be a DM candidate, or can only be a tiny fraction of the DM density of the Universe. As a result, all the x-ray bounds will be rescaled by the fractional density of sterile neutrinos in the DM energy budget, and hence, not be relevant for our scenario [110]. Interestingly, introducing new interactions among the sterile neutrinos, as well as the active neutrinos have been a popular way to relax these x-ray bounds on these models [111,112]. B. Cosmic microwave background and large scale structure After decoupling around Tν¼1MeV, neutrinos evolve freely without scattering in our scenario. However, the neutrino masses will keep varying with the background scalar field, and can be possibly large during the recombination epoch. Meanwhile, the CMB and large scale structure (LSS) observations are very sensitive to the absolute scale of neutrino masses. In fact, a world-leading constraint on the sum of neutrino masses has been set, i.e., Σimi<0.12 eV using the dataset Planck TT, TE, EE þlowE þlensing þBAO [113]. Recently, stronger limits, Pmν<0.09 eV at 95% C.L. were derived [114,115] One of the key effects of large neutrino masses during recombination is to reduce the free-streaming length λfs compared to the massless neutrino case. Free-streaming neutrinos can damp all the perturbation modes for distances less than λfs, which is well consistent with the current cosmological data. Hence, any effect which reduces the neutrino free-streaming length λfs during recombination can be constrained from CMB data. This scenario is different from the BBN epoch, where neutrinos scatter very rapidly before decoupling. In this case, it is important to figure out how a free neutrino mass or flavor eigenstate propagates within the oscillating scalar field. To demonstrate this idea, we plot in Fig. 5the two mass eigenvalues ˜ m1and ˜ m4described by Eqs. (A2) and (A3), within one DM oscillation cycle. Other parameters are taken as m1¼0.1eV, m4¼3eV and θ¼π=12. The solid curves stand for the mass ˜ m1, which coincides with the vacuum value when ϕis vanishing. On the other hand, the dotted curves are for the ˜ m4.AsΦðtÞoscillates, we note a nontrivial evolution pattern for gϕ¼20 eV FIG. 4. Contribution of sterile neutrinos to the effective number of neutrinos, ΔNeff, around the time of BBN as a function of gϕ. The mass splitting and mixing for the sterile neutrino are chosen to be Δm2 41 ¼10 eV2and sin22θ¼0.25. FIG. 5. The evolution of ˜ m1(solid curves) and ˜ m4(dotted curves) as functions of time, within one DM cycle. The DM potential is taken to be gϕ¼20 eV (blue curves) or yϕ2=Λ¼20 eV (red curves) for the left panel. Vacuum neutrino parameters are fixed as m1¼0.1eV, m4¼3eV and θ¼π=12. Conventions are the same for the right panel except that we take gϕ¼2eV or yϕ2=Λ¼2eV. HUANG, LINDNER, MARTÍNEZ-MIRAV´ E, and SEN PHYS. REV. D 106, 033004 (2022) 033004-6 (left panel, in blue). At t¼π=ð2mϕÞ, the neutrino masses reach their local maximal value. The lighter neutrino mass is just around ˜ m1≈0.3eV. After t>π=mϕ,ΦðtÞbecomes negative, and ˜ m1continuously grows to ∼10 eV, which is around the original ˜ m4value. Note that the mixing sin2θ(in gray, unitless) also reaches values around one, implying that the corresponding relations of flavor and mass eigenstates are swapped. The other eigenvalue ˜ m4follows an opposite behavior. During recombination, if these two mass eigenstates evolve adiabatically as ΦðtÞoscillates, we would expect the neutrino flavors, being a linear combination of the mass eigenstates, to be relativistic half of the time, and nonrelativistic the other half. This is equivalent to setting neutrino velocity to Oð0.5cÞ, which will hence reduce the free-streaminglengthλfs by a factor of aroundtwo. However, we notice that two eigenvalues also critically hit each other at the point of gΦðtÞ¼−m4, enforcing a nonadiabatic transition. As a consequence, each time a neutrino state, say j˜ ν1i, reaches this point, there will be a certain probability for it to transit to j˜ ν4i, and vice versa. The net effect is to reduce the neutrino free-streaminglength,byatmostafactoroftwo.We do not investigate numerically the details about the energymomentum conservation in the scenario where the scalar, active neutrinos and sterile neutrinos form a highlyentangled system,butinsteadgiveour remarks.Asinthecasewithonly sterile neutrino and scalar, the momentum of neutrinos must bealwaysconservedbecauseofspatialtranslationsymmetry. When j˜ ν1itransits to j˜ ν4i, the energy of each neutrino state is increased by approximately gϕif p≪gϕ.However,we notice that j˜ ν4iat t≳π=mϕhas more overlap with the sterile state jNi, hence increasing the contribution of the source term in Eq. (2). This feedback effect should balance the energy between the neutrino and scalar systems. Even though such a scenario, to our knowledge, has never been strictly investigated with cosmological simulations, one might get an intuition from the neutrino mass limit. The neutrino mass information one can extract depends on the redshift. Firstly, the quoted limit from Planck Pimi< 0.12 eV is derived from the data of all available redshifts (from z¼0to z¼3000). Because the scalar field strength isheavierinthe denseearlyUniverse,weare moreinterested in the consequence at higher redshifts. There are fits using the CMB and LSS data by allowing neutrino masses to vary with the redshift, which find the mass limit derived from the data at z>1100 can only be Σimi<0.40 eV at 95% CL [84]. This can be translated it into a constraint on neutrino velocity, hvνi>0.97c, by using the temperature Tνðz¼1100Þ≈0.18 eV and the relation hpνi≈3Tν.Even without a dedicated analysis, it indicates that the scenario with hvνi>0.5cis in tension with CMB observations, if gϕ is too large in the early Universe. However, the above estimate is too stringent, because the impact on CMB perturbation due to a smaller freestreaming is only one of the effects offinite neutrino masses. The inclusion of more effects, such as background evolution effect, should lead to a more conservative limit on the neutrino velocity, which can only be obtained with a more detailed analysis. Furthermore, it is important to note that while the scalar field keeps oscillating with the frequency mϕ=ð2πÞ, the overall field strength decreases with Eq. (5) as the Universe expands. At certain point when gϕ<m 4the two neutrino mass eigenvalues start to separate. This is demonstrated in the right panel of Fig. 5, where we show the eigenstates for a smaller value of gϕ¼2eV (blue curve). The neutrino velocity is reduced in the large gϕcase due to the conversion between the lower and upper mass eigenstates when gΦðtÞ¼−m4. However, such an issue does not arise if the mass-variation is due to the higher dimensional term, governed by yϕ2=Λcase with y>0. This is demonstrated via the red curves in Fig. 5. In this case, no matter how large the potential becomes, the eigenstate with ˜ m1, which has dominant mixing with active neutrinos, always stays below m4sin2θ. The threats from cosmological observations can thus be removed by this higher-dimensional operator. In such a case, the local potential yϕ2=Λcan be large without spoiling BBN, CMB and LSS to have observable time-varying effect for active neutrinos at laboratories. Note that our only requirement here is m4sin2θ<0.40 eV, where the sterile neutrino mass can actually be heavy, e.g., m4¼1TeV with θ¼10−7. This requirement is well compatible with the current collider searches of heavy right-handed neutrinos. For instance, the ATLAS and CMS collaborations at LHC have set constraints jVμNj2;jVeNj2≲10−3[116–118] at m4¼100 GeV, corresponding to a very loose result m4sin2θ<0.1GeV. In this section, we explored the effects of either gϕor yϕ2=Λterm in the early Universe, by keeping the other subdominant. In principle, we can simultaneously have these two effects, where yϕ2=Λis suppressed by some cutoff scale. However, as we go to the early Universe, the effective term, yϕ2=Λ, growing faster might dominate over the gϕterm, and save the model from cosmological bounds. In the remaining part of the work, we will explore the consequences of a time-varying neutrino mass on beta decays and light sterile neutrino phenomenology. For these analyses, we shall ignore the higher dimensional operator, and focus on the renormalizable interaction gΦfor simplicity. IV. TRITIUM BETA DECAYS A time-varying mass of the sterile neutrino can also leave potentially observable imprints in beta-decay experiments, depending on the mass of the sterile neutrino, as well as the mass and amplitude of the scalar field Φ. When the sterile neutrino is heavy (e.g., m4>MeV), larger than gϕand decoupled from the energy scale of beta-decay experiments, the scalar potential can only affect the beta-decay COSMOLOGY-FRIENDLY TIME-VARYING NEUTRINO MASSES …PHYS. REV. D 106, 033004 (2022) 033004-7 spectrum by mixing with active ones [67]. In the standard 3νscenario, beta-decay experiments, such as Mainz [119], Troisk [120,121], and KATRIN [15,17], measure the effective neutrino mass mβ≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P3 i¼1jUeij2m2 i q, which receives incoherent contributions from three generations of neutrinos. By analyzing the electron spectrum from tritium beta decays, stringent limits have been set, e.g., mβ<0.8eV from the combination of the first and second campaign of KATRIN (KNM1þKNM2)[17]. In order to show the effect of a time-varying scalar on beta-decay spectrum, we consider a simplified modification to the effective neutrino mass, in the limit gϕ<m 4,as ˜ mβðϕÞ≈mβþgνϕsin mϕt; ð11Þ where gν≡sin2θ14 ·grepresents the effective neutrino coupling suppressed by the active-sterile mixing angle. For Eq. (11), a uniform mixing of the sterile neutrino to three generations of neutrinos has been assumed, such that ˜ UeiðϕÞ¼Uei and ˜ miðϕÞ¼miþgνΦ(for i¼1,2,3) hold, thereby leading to the simplified relation in Eq. (11). In general, the diagonalization of the sum of vacuum mass matrix and DM-induced mass matrix will result in complex dependence on the scalar field. For the accuracy of current beta-decay experiments, the major observable is ascribed to a single mass parameter ˜ mβ. Different coupling patterns are assumed to not affect much the overall magnitude of modifications. The beta spectrum with the effective neutrino mass ˜ mβðϕÞcan be parametrized as RβðEe;˜ mβÞ¼G2 F 2π3jVudj2ðg2 Vþ3g2 AÞm3He m3HFðZ; EeÞ ×Eeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2 e−m2 e qHðEe;˜ mβÞ;ð12Þ with the spectral function HðEe;˜ mβÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðKend;0−KeÞ2−˜ m2 β q×ðKend;0−KeÞ:ð13Þ Here, GFis the Fermi coupling constant, Vud is the weak mixing matrix element, gV¼1and gA¼1.247 are the vector and axial-vector weak coupling constants of Tritium and the function FðZ; EeÞis the ordinary Fermi function describing the spectral distortion in the atomic Coulomb potential. Throughout this work, we use Eeand Keto distinguish the total and kinematic electron energies and Kend;0is the electron endpoint energy in the massless neutrino limit. The step function, which is necessary to make the square root real, is not explicitly shown. The ultralight scalar may manifest itself as a modulation effect to the beta spectrum, if the DM cycle can be covered by KATRIN runs, and also be resolvable for the duration of KATRIN spectrum scans. If the above condition is not satisfied, one can also look for the distortion effect by averaging over the DM oscillations. In this circumstance, we investigate analytically how the averaged scalar field modifies the beta spectrum. For the current KATRIN sensitivity, it is a good approximation to keep up to the first order of perturbative expansions on ˜ m2 β(unless gνϕis very large) in the spectral function, namely, HðEe;˜ mβÞ∝ðKend;0−KeÞ− ˜ m2 β 2ðKend;0−KeÞ:ð14Þ The square of effective neutrino mass as in Eq. (11) averaged over one DM cycle reads h˜ m2 βi¼m2 βþðgνϕÞ2 2:ð15Þ Hence, in such a case the presence of the scalar field directly adds a constant term to the square of effective neutrino mass. For the sensitivity of first KATRIN campaign, the averaged scalar effect is degenerate with a usual neutrino mass, but it leads to large neutrino mass cosmology [122]. This degeneracy is nonlinear and obvious from Fig. 6, where we have fitted the parameter space of m2 βand gϕusing the KATRIN data from the first campaign (KMN1). Following the analysis strategy from the KATRIN Collaboration, we allow m2 βto become negative during the fit. From Fig. 6and Eq. (15), one can see that in this scenario, the effective neutrino mass measured, h˜ m2 βiis always larger than the true m2 βand hence, FIG. 6. Allowed regions in the m2 β−gνϕplane at 68% and 95% C.L. using data from KATRIN’s first campaign (KNM1) and in the limit in which m4is very heavy and ˜ m2 βexhibits a time modulation as in Eq. (11). HUANG, LINDNER, MARTÍNEZ-MIRAV´ E, and SEN PHYS. REV. D 106, 033004 (2022) 033004-8 the upper limits derived when assuming gϕ¼0are conservative. Up to this point, we limited the discussion to the case in which the sterile neutrino is heavy. However, when the sterile neutrino is light enough, additional emission channel of beta decays will be open. In the ð3þ1Þνscenario, we can split the 3νand sterile neutrino contributions as [123] Rð3þ1Þν βðEeÞ¼ð1−j˜ Ue4ðϕÞj2ÞRβðEe;˜ mβÞ þj˜ Ue4ðϕÞj2RβðEe;˜ m4Þ;ð16Þ with j˜ Ue4j¼sin ˜ θ. The ϕðtÞ-dependent mixing angle ˜ θ as well as masses ˜ mβand ˜ m4can be calculated from Eqs. (A2)–(A4), respectively. For simplicity, we assume that the ϕðtÞdependence in ˜ mβis approximately that of ˜ m1. This is well justified in light of the current KATRIN sensitivity to the effective neutrino mass. For a general light sterile neutrino, one has to integrate the exact spectral function over DM modulations. A generic numerical treatment can be performed in the analysis without making approximations. Aiming to provide a deeper comprehension of the experimental signatures, we show in the upper panel of Fig. 7the beta-decay spectra for different scenarios, with a configuration similar to the first KATRIN campaign. The lower panel demonstrates the difference of rates of various scenarios with respect to the standard one with mβ¼0eV. The region where KATRIN can obtain most information about neutrino masses is slightly above the beta-decay endpoint. The statistics are very limited close to the endpoint, while far above the endpoint the beta-decay rate is too large and hence, insensitive to very small distortions. A finite neutrino mass will induce not only kink structure, but also provide a constant shift to the beta-decay spectrum away from the endpoint. This is illustrated in Fig. 7for four different values of mβ¼f0;0.4;0.7;1geV, which correspond to the four gray lines, from right to left. However, the kink structure is still unresolvable given the sensitivity of current generation of beta-decay experiments, resulting in the current limit of m2 β<0.9eV from KATRIN’s second campaign [17]. A light sterile neutrino will induce a second kink in the decay spectrum. Such spectral feature, which is expected around Ke−Kend;0≲m4, can be well separated from a normal neutrino mass term providing the kink is away from the endpoint, as shown by black dashed curve. The size of the distortion is related to the size of the mixing jUe4j2. Consequently, beta-decay experiments set limits to the sterile neutrino mass and mixing [124,125]. Red curves in Fig. 7illustrate how adding large scalar potentials to the sterile neutrino will smooth these distortions, mainly because the averaged mixing will be suppressed by the potential. Consequently, this scenario allows to open up the sterile neutrino parameter space in the context of shortbaseline anomalies, as we will discuss later in Fig. 9. The case in which the sterile neutrino is heavy is given by the blue curve. We have shown that the effect of the scalar is degenerate with the mass term (see Fig. 6). V. IMPACT ON GALLIUM AND REACTOR ANOMALIES Until now, we have discussed the impact of a consistent scenario of time-varying neutrino masses on cosmology and beta decay experiments. If the sterile neutrino is light today, it will also have important consequences for the FIG. 7. The beta-decay spectra for various scenarios including: the standard beta decays with mβ¼f0;0.4;0.7;1geV (gray curves from right to left), the heavy sterile neutrino case as in Eq. (11) with an effective potential gνϕ¼1eV (left panel, blue curve), a benchmark choice of light sterile neutrino with m4¼3eV and sin22θ¼0.25 (right panel, black dashed curve), and the addition of scalar potentials gϕ¼10 eV (red dashed curve) and 50 eV (red dotted curve) upon the light sterile case. Note that we assume mβ¼0 for all the cases with nonzero scalar potentials. 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