Constraining the Self-Interacting Neutrino Interpretation of the Hubble Tension

FERMILAB-PUB-19-175-A-T

Nikita Blinov,∗ Kevin J. Kelly,† Gordan Krnjaic,‡ and Samuel D. McDermott§ Fermi National Accelerator Laboratory, Batavia, IL, USA (Dated: November 19, 2019) Large, non-standard neutrino self-interactions have been shown to resolve the ∼ 4σ tension in Hubble constant measurements and a milder tension in the amplitude of matter fluctuations. We demonstrate that interactions of the necessary size imply the existence of a force-carrier with a large neutrino coupling (> 10−4) and mass in the keV – 100 MeV range. This mediator is subject to stringent cosmological and laboratory bounds, and we find that nearly all realizations of such a particle are excluded by existing data unless it carries spin 0 and couples almost exclusively to τ-flavored neutrinos. Furthermore, we find that the light neutrinos must be Majorana, and that a UV-complete model requires a non-minimal mechanism to simultaneously generate neutrino masses and appreciable self-interactions.

I. INTRODUCTION They found that Geﬀ in the “strongly interacting” (SIν) or “moderately interacting” (MIν) regimes

The discrepancy between low-redshift and Cosmic ( +0.4 −2 Microwave Background (CMB) determinations of the (4.7−0.6 MeV) (SIν) Geff = (2) present-day Hubble parameter, H , has grown in signif- +171 −2 0 (89−61 MeV) (MIν) icance to 4σ over several years [1–5]. The standard ∼ 1 cosmological model, ΛCDM, may need to be augmented could simultaneously reduce the H0 and σ8 tensions. if this “H0 tension” is not resolved by observational sys- Interestingly, the SIν cosmology prefers a value of H0 tematics. This tension cannot be addressed by modifying compatible with local measurements at the 1σ level, even ΛCDM at low redshift [6–9]; adding new physics before before including local data in the fit. recombination seems more promising [10–17]. The solu- The range of Geff in Eq. (2) vastly exceeds the strength tions in Refs. [10–17] operate at temperatures & 1 eV to of weak interactions, whose coupling is GF (2.9 5 −2 ' × modify the sound horizon and the inferred value of H0. 10 MeV) . We show that this interaction can only arise Low-redshift measurements of the matter density fluc- from the virtual exchange of a force carrier (“mediator”) tuation amplitude on 8 Mpc scales, σ8, also appear to with (MeV) mass and appreciable couplings to neu- be lower than predicted by ΛCDM from the CMB. This trinos.O For this mass scale, the effective interaction in milder “σ8 tension” is not ameliorated in [11–16]. Eq. (1) is valid at energies of order . 100 eV, which pre- One resolution to both issues is non-standard neutrino vail during the CMB era. However, at higher energies, self-interactions [18–22] this mediator is easy to produce on shell, and is subject to stringent cosmological and laboratory bounds. eff = Geff (¯νν)(¯νν), (1) L We find that if strong neutrino self-interactions resolve the H0 tension, then: where Geff is a dimensionful coupling with flavor indices suppressed. If Geff is much larger than the Stan- Flavor-universal Geff excluded: If Geff is neu- dard Model (SM) Fermi constant, GF, neutrinos remain • trino flavor-universal, both SIν and MIν regimes in tightly coupled to each other until relatively late times. Eq. (2) are excluded by laboratory searches for rare This inhibits their free-streaming, resulting in enhanced K decays and neutrinoless double-beta decay. power on small scales and a shift in the acoustic peaks of the CMB spectrum relative to ΛCDM [23]. MIν interactions with ντ favored: Couplings • The effect of self-interactions is degenerate with other to νe, νµ with Geff in the range of Eq. (2) are parameters in the CMB fit, including the angular scale of also excluded, except for a small island for νµ cou-

arXiv:1905.02727v2 [astro-ph.CO] 18 Nov 2019 the sound horizon, the spectral index and amplitude of pling. The only viable scenario involves neutrinos primordial fluctuations, and extra radiation. These de- interacting through their ντ components in the MIν generacies enable a preference for Geff GF in cosmo- regime. logical data [18, 20–22] while relaxing the H0 tension [20– 22]. Ref. [22] extended previous analyses, allowing for fi- Vector forces excluded: Constraints from Big • nite neutrino masses and extra radiation at CMB times. Bang Nucleosynthesis (BBN) exclude most self- consistent vector mediators.

∗ ORCID: http://orcid.org/0000-0002-2845-961X † 1 ORCID: http://orcid.org/0000-0002-4892-2093 These regions correspond to the Planck TT+lens+BAO+H0 ‡ ORCID: http://orcid.org/0000-0001-7420-9577 datasets. Other dataset combinations considered in Ref. [22] § ORCID: http://orcid.org/0000-0001-5513-1938 prefer similar values of Geﬀ . 2

Dirac neutrinos disfavored: Mediator-neutrino regime was investigated in Refs. [26, 27], which found no • 2 interactions thermalize the right-handed compo- improvement in the H0 tension. Thus, we focus on mod- 2 2 nents of Dirac neutrinos, significantly increasing els with a new particle φ for which mφ q at energy the number of neutrino species at BBN. This ex- scales relevant to the CMB. | | cludes nearly all scenarios except the MIν regime Throughout this epoch, neutrinos are relativistic, so 2 2 with couplings to ντ . the typical momentum transfer is q T . Eq. (4) is | | ∼ ν valid if mφ Tν . Comparing the values in Eq. (2) to Minimal seesaw models disfavored: Achieving G in Eq. (4), • the necessary interaction strength from a gauge- eff invariant, UV-complete model, while simultane- mφ (4 200) gφ MeV . (5) ' − × | | ously accounting for neutrino masses is challenging Since perturbativity requires gφ . 4π, a new sub-GeV in minimal seesaw models. state is required to realize this self-interacting-neutrino This work is organized as follows: Sec.II demonstrates solution. Since Tν < 100 eV at horizon entry of the high- that a light new particle is required to generate the in- est observed CMB multipoles, the validity of Eq. (1) in teraction in Eq. (1) with appropriate strength; Sec.III the analyses of [18, 20–22, 24] requires mφ & keV (as presents cosmological bounds on this scenario; Sec.IV noted in [22]). From Eq. (5), this translates to −4 discusses corresponding laboratory constraints; Sec.V mφ & keV = gφ & 10 . (6) shows how Eq. (1) can arise in UV complete models; fi- ⇒ | | This bounds the allowed ranges of m and g . Note that nally, Sec.VI offers some concluding remarks. φ φ Eq. (5) precludes the new self-interactions from being described within Standard Model effective theory with no light states below the weak scale [28]. II. THE NECESSITY OF A LIGHT MEDIATOR Finally, we note that Eq. (3) is not gauge-invariant at energies above the scale of electroweak symmetry break- Refs. [20–22, 24] assume that all left-handed (LH) ing (EWSB). We explore UV completions in Sec.V. neutrinos undergo 2 2 flavor-universal scattering described by the interaction→ in Eq. (1). The largest detected CMB multipoles correspond to modes that entered the III. COSMOLOGICAL BOUNDS horizon when the neutrino temperature was < 100 eV. This sets the characteristic energy scale of scattering re- Successful predictions of BBN provide a powerful actions during this epoch: it is important that the form probe of additional light species. New particles in ther- of the Lagrangian in Eq. (1) is valid at this tempera- mal equilibrium with neutrinos increase the expansion ture. At higher energies, however, this description breaks rate during BBN as extra relativistic degrees of freedom down. As previously noted in [18, 20–22], the operator or by heating neutrinos relative to photons. Away from in Eq. (1) is non-renormalizable, and thus is necessarily mass thresholds, both effects are captured by a constant replaced by a different interaction with new degree(s) of shift in Neff , the effective number of neutrinos. We find freedom at a scale higher than the (100 eV) energies that the observed light element abundances constrain probed by the CMB (see Ref. [25] for∼ O a review). ∆Neff < 0.5 (0.7) at 95% CL for the SIν- (MIν-) pre- The interaction in Eq. (1) can be mediated by a par- ferred values of the baryon density, as detailed in App.A. ticle φ with mass mφ and coupling to neutrinos gφ: We emphasize that large ∆Neff 1 at CMB times is crucial for the MIν and SIν results' [22]. Since BBN 1 2 2 1 αβ m φ + (g νανβφ + h.c.), (3) L ⊃ −2 φ 2 φ does not prefer large Neff , the self-interacting neutrino framework requires an injection of energy between nu- where να are two-component left-handed neutrinos, and αβ cleosynthesis and recombination, e.g., via late equilibra- we allow for generic flavor structure gφ of the interaction of a dark sector [36]. Such scenarios may face ad- tion. In Eq. (3) we have assumed that φ is a real scalar; ditional constraints. To remain model-independent, we our conclusions are unchanged if φ is CP-odd or complex. only consider the implications of BBN for the mediator Vector forces face stronger constraints than scalars, as (and right-handed neutrinos if they are Dirac particles) discussed below. needed to implement strong neutrino self-interactions. Using Eq. (3), we see that the νν νν scattering amplitude is g2 /(m2 q2). If the→ momentum transfer M ∝ φ φ − 2 2 2 2 2 Unlike Ref. [22], Refs. [26, 27] fixed N and P m , but we note q satisfies q m , then Geff 1 + q /m + , eff ν | | φ M ∝ φ ··· that a light mediator would affect multipoles between the first where acoustic peak and the diffusion scale. This should be contrasted with the massive mediator case where the self-interaction effects g2 2 2 φ −2 gφ MeV are larger at higher multipoles, allowing for non-standard values Geff 2 = (10 MeV) −1 . (4) P ≡ mφ 10 mφ of Neff and mν to compensate. A strongly-interacting mode could exist here, but is unlikely to result in a larger value of H0 2 2 2 2 P If mφ q , gφ/q , leading to qualitatively differ- after accounting for Neff and mν effects, since these impact ent energy | dependence| M ∝ for neutrino self-interactions; this higher-` modes of the CMB spectrum. 3

Universal coupling νe coupling 1 1

1 1 10− 10−

ν ν SI Lab SI φ

2 Constraints eα 2 BBN φ BBN K eνφ g 10− 10− (Dirac ν) g (Dirac ν) → B (real BBN (complex (real BBN (complex

3 3 10− 10−

ν ν decay

MI φ MI φ φ φ ) ) ) )

φββ 4 4 10− 3 2 1 2 3 10− 3 2 1 2 3 10− 10− 10− 1 10 10 10 10− 10− 10− 1 10 10 10 mφ [MeV] mφ [MeV]

νµ coupling ντ coupling 1 1 τ `ννφ → 1 1 10− 10−

ν ν K µνφ SI SI 2 → 2 τα µα φ

φ BBN 10− 10− g g BBN (Dirac ν) (Dirac ν) B (real BBN (complex (real BBN (complex

3 3 10− 10− ν ν

MI φ MI φ φ φ ) ) ) ) 4 4 10− 3 2 1 2 3 10− 3 2 1 2 3 10− 10− 10− 1 10 10 10 10− 10− 10− 1 10 10 10 mφ [MeV] mφ [MeV]

FIG. 1. Bounds (shaded regions) on light neutrino-coupled mediators with flavor-universal couplings (top-left), and flavor- specific couplings to νe (top-right), νµ (bottom-left), and ντ (bottom-right). The bands labeled MIν and SIν are the preferred regions from Eq. (2)[22] translated into the gφ-mφ plane. Also shown are constraints from τ and rare meson decays [29– 32], double-beta decay experiments [33–35] (purple), and BBN (red). We combine the τ/meson decay and double-beta decay constraints as “Lab Constraints” in the upper-left panel. BBN yields depend on the baryon density ηb; thick (thin) lines correspond to the SIν (MIν) preferred values of ηb. Nucleosynthesis constraints are stronger for complex scalar mediators (dashed red) than for real scalars (solid red). If neutrinos are Dirac, their right-handed components equilibrate before BBN above the dashed black line.

A. Mediators and ∆Neff gauge coupling. φµ equilibrates before Tdec via νν φ ↔ if the corresponding thermally-averaged rate Γνν→φ ex- Eq. (3) induces φ νν decays and inverse decays, ceeds Hubble when T = max(Tdec, mφ): which can equilibrate↔φ with neutrinos before neutrino- 2 2 Γ g m MPl G photon decoupling at T 1 2 MeV. Here we show νν→φ φ φ > 108 eff , (7) dec H max(T , m )3 (10 MeV)−2 that this necessarily happens∼ for− mediators that realize ∼ dec φ 19 Geff in Eq. (2). Annihilation and scattering processes also where MPl = 1.22 10 GeV and we have used Eqs. (4) contribute, but the corresponding rates are suppressed by and (6). This reaction× is in equilibrium for all values additional powers of gφ. of couplings and masses of interest. As a result, φµ has a thermal number density at Tdec in both MIν and Vector Mediators: If Eq. (1) arises from a vector SIν scenarios. Counting degrees of freedom, we find particle φµ with mass mφ, then at energies above mφ ∆Neff = (8/7)(3/2) 1.7 assuming φµ remains rela- 1 2 µ † µ ' m φ φµ + gφφµν σ¯ ν + h.c. , where gφ is the tivistic throughout BBN; if φµ becomes non-relativistic L −→ 2 φ 4 between Tdec and the end of BBN, then ∆Neff 2.5. We therefore assume that neutrinos are Majorana parti- ≈ Thus, φµ must become non-relativistic well before Tdec. cles for the remainder of this work. Ref. [16] found that Boltzmann suppression for massive vectors is effective for mφ > 10 MeV (95% CL). Using Eq. (4), this requires gφ & (0.1), which is excluded C. Secret Neutrino Interactions in all theoretically-consistentO (or anomaly-free) vector models with neutrino couplings [37, 38]. Anomaly-free The Neff bounds considered here can, in principle, be vectors, such as those coupled to lepton-family-number evaded by “secret” interactions which are communicated currents, would introduce largeνν ¯ ee¯ interactions which to active neutrinos via mixing with a light sterile neu- would likely spoil the CMB fit. trino, which couples directly to a mediator. In these scenarios the active-sterile mixing angle is suppressed Scalar Mediators: Similarly, any scalar mediator φ at early times by plasma effects, but can become large −4 that realizes Geff from Eq. (3) with gφ & 10 (re- at later times when the universe is cooler [43–46]. The quired by Eq. (6)) also has a thermal abundance at mixing angle may be smaller than 10−9 for T 50 ∼ & Tdec. Relativistic scalars in equilibrium with neutrinos keV when BBN ends (to avoid thermalization) and sub- contribute ∆Neff = 0.57 (1.1) for a real (complex) φ, sequently grow to (1) by T 100 eV (to enable which has one (two) degree(s) of freedom. The φ den- a large active neutrino∼ O self-interaction∼ during the CMB sity must become Boltzmann-suppressed before neutrino- era, thereby resolving the H0 tension). This sharp transi- photon decoupling, leading to a lower limit on mφ. We tion over a narrow temperature range requires significant use AlterBBN 2.1 [39, 40] as described in App.A to ob- fine tuning of the active-sterile mass-splitting and a large tain lower bounds (95% CL) lepton asymmetry. See App.D for a discussion. ( 1.3 MeV (real scalar) mφ > , (8) 5.2 MeV (complex scalar) IV. LABORATORY BOUNDS for the SIν preferred values of the baryon density (cor- Because terrestrial experiments routinely reach ener- responding MIν bounds are somewhat weaker – see gies above the MeV scale, the model of Eq. (3) is well App.A). SI ν and MIν BBN bounds are presented in constrained. We focus on scalar mediators, commenting Fig.1 as thick and thin red vertical lines, respectively. on pseudoscalars in Sec.V. Laboratory constraints arise from: B. Constraining Dirac Neutrinos ee Double Beta Decay: If gφ = 0 and φ is lighter than the Q-value of a double-beta-decaying6 nucleus, the pro- If neutrinos are Dirac all neutrino masses arise from cess (Z,A) (Z + 2,A) e−e−φ may occur, contribut- → the interaction Dirac yν HLνR mν ννR, where ing to measured 2νββ rates. Measurements constrain L ⊃ → T ee −4 mν yν v/√2,H is the Higgs doublet, L = (ν, `) is a gφ . 10 if mφ . 2 MeV [33–35], shown in the top ≡ | | lepton doublet, νR is a right-handed neutrino (RHN), and row of Fig.1. flavor indices have been suppressed. The Weyl fermions αβ Meson Decays: Nonzero gφ can allow for meson de- ν and νR become Dirac partners after EWSB and acquire ± ± cays m `α νβφ if mφ < mm m`α [29–32, 47, 48]. identical masses. In the SM alone, the Yukawa coupling + → + + + − −12 Br(K e νe)/Br(K µ νµ) = (2.416 0.043) yν 10 (mν /0.1 eV) is insufficient to thermalize right- ∼ −5 → eβ → ± × handed states, so relic neutrinos consist of left-handed 10 constrains gφ as shown in the top row of Fig.1[49, + + −6 µα neutrinos and right-handed antineutrinos [41]. 50]. Br(K µ νµνν¯) < 2.4 10 [51] constrains gφ , The interactions in Eq. (2) are much stronger than the shown by the→ purple region in bottom-left× panel of Fig.1. weak force at late times, so φ and νR can both thermalize. − ττ τ Decays: The decay τ `βνβντ φ constrains gφ . Approximating the RHN production rate as Γφ→ννR ττ → (m /m )2Γ , for m = 0.1 eV we have ' Ref. [49] found gφ . 0.3 for light φ, depicted as a purple ν φ φ→νν ν band in the bottom-right panel of Fig.1. g2 m2 M Γφ→ννR φ ν Pl 6 Geff MeV 3 = 10 −2 , (9) H ' mφ (10 MeV) mφ Fig.1 summarizes our findings: values of Geff from where T = mφ & Tdec is the temperature at which RHN Eq. (2) favored by the H0 tension are excluded if φ cou- production is maximized relative to H. See App.B for ples universally to all neutrinos (top-left), which was ex- more details. plicitly considered in Refs. [18, 20–22, 24], or (in the Neutrino oscillation results require that at least two SIν solution) if φ couples predominantly to νe or νµ of the light neutrinos are massive, with one heavier than (top-right/bottom-left panels, respectively). Similarly, 10−2 eV and one heavier than 10−1 eV [42]. For all we can exclude the possibility that φ couples to any ∼ ∼ values of mφ we consider in the SIν range, at least one single mass-eigenstate neutrino, since the νe- and νµ- RHN will thermalize before BBN, leading to ∆Neff & 1. composition of each mass eigenstate is similar. Moreover, 5 in this case, the collisional Boltzmann equations would neutrino masses and the operator in Eq. (1) with the be much more complicated to solve (different eigenstates magnitude in Eq. (2) are not possible. will start to free-stream at different times), and the re- These arguments also apply to the Majoron, the sults of Refs. [18, 20–22, 24] may not apply. Nambu-Goldstone boson of lepton-number breaking [56– However, a flavor-restricted coupling leads to approx- 58]. In these models, φ is a pseudoscalar particle, but its imately the same neutrino mass-eigenstate interactions coupling to neutrinos is still suppressed by mν /f. How- as in Refs. [18, 20–22, 24], since the flavor eigenstates ever, the bounds we considered still apply, because all are well-mixed in the mass basis. A ντ -only coupling, in limits derive from relativistic neutrinos, for which there which the matrix gαβ is zero except for gττ , is potentially is no distinction between scalar and pseudoscalar. viable since τ decays are less constraining than meson de- Finally, we note that large Geff can be obtained us- cays. Thus, we are unable to fully exclude an interaction ing separate seesaw mechanisms to generate the neutrino τ Geff ν¯τ ντ ν¯τ ντ . masses and the φνν interaction – we can use the type-I τ In this case, Geff = A Geff for Geff defined in Eq. (4) Seesaw for the light neutrino masses and the type-II see- × ττ and A (1) is a constant that accounts for the re- saw mechanism can produce large gφ (as long as it does duced scattering∼ O probability of each mass eigenstate. Be- ττ not contribute to mν ). The size of gφ decouples from cause mixed mass-eigenstate vertices are possible in this the neutrino masses. scenario, there are additional diagrams compared to the mass-diagonal case. For this reason, we caution that the effect on the CMB anisotropies of flavor-specific neutrino VI. CONCLUDING REMARKS self-interactions can be mildly different than that considered in Refs. [18, 20–22, 24]. Nonetheless, we expect that the preferred coupling range should shift slightly up We have shown that the self-interacting neutrino ex- relative to the flavor-universal case; a complete study is planation of the H0 tension requires the existence of a light MeV-scale mediator, subject to stringent cosmo- necessary to know how this affects the full SIν range. ∼ The MIν range is still allowed in a τ-flavor-only scenario, logical and laboratory bounds. Consequently, for both though a dedicated study is needed. the SIν and MIν regimes in Eq. (2), the flavor-universal interactions considered in Refs. [18, 20–22] are robustly excluded by BBN-only bounds on ∆Neff and by labora- V. ULTRAVIOLET COMPLETIONS tory searches for rare K decays and neutrinoless double- beta decay; the SIν regime is excluded for all flavor struc- tures. In this section we consider models of Majorana neutri- Intriguingly, we find that flavor-dependent variations nos with an additional particle φ, specifically the type-I of the MIν regime may viably resolve the H0 tension if and II seesaw mechanisms. In both, we find the resulting a 10 MeV scalar mediator with large coupling inter- φνν coupling is suppressed by factors of the light neutrino acts∼ almost exclusively with ν or ν (though there is mass. In these minimal models, it is therefore impossible τ µ little parameter space for νµ coupling). A dedicated ex- to simultaneously generate neutrino masses and a large ploration of the τ-only scenario is necessary to determine enough Geff to address the H0 tension. if the preferred region to resolve the H0 tension persists We note that the coupling of φ to LH neutrinos in Eq.3 without running afoul of laboratory measurements. Our violates lepton number in analogy to neutrino masses, results also motivate exploration of the “intermediate” so it is a compelling possibility to relate these phenom- mediator-mass regime, where neutrino scattering is rele- ena. The SM Lagrangian preserves lepton number, so vant for a partial range of redshifts explored by the CMB. the the scale f of lepton-number violation must arise However, realizing such strong, flavor specific interac- from new interactions. In type-I models, f is related tions in UV-complete, gauge-invariant models is challeng- to the RH neutrino mass, while in type-II it is propor- ing. We find that sufficiently strong interactions cannot tional to the Higgs-triplet mixing parameter [52]. The arise in models that generate neutrino masses via a single interaction of φ with the neutrino sector occurs through type-I or -II seesaw mechanism: the resulting neutrino- the combination f + λφ, where λ is a coupling con- scalar coupling is suppressed by factors of m /f where stant. While the relation of λ to the neutrino-masses ν f mν is the appropriate seesaw scale. A compelling is model-dependent, the interaction with φ takes on the 2 2 2 2 and viable model remains to be found. universal form gφ λmν /f Geff λ mν /(mφf ) in both type-I and≈ type-II seesaw⇒ scenarios.∼ Realizing −2 3 Geff (4 300 MeV) requires f 10 mν 10 eV. In the≈ type-I− model, this scale sets the∼ mass of the∼ RHNs, ACKNOWLEDGMENTS which thermalize before BBN and contribute to ∆Neff as in the Dirac case discussed in Sec.IIIB. In type- We thank André de Gouvêa, Francis-Yan Cyr- II models this scale is bounded by non-observation of Racine, Joshua Isaacson, Martina Gerbino, Stefan Höche, rare lepton-number-violating processes [52–55]. There- Matheus Hostert, Joachim Kopp, Massimiliano Lattanzi, fore, minimal scenarios where the same seesaw generates Shirley Li, Kohta Murase, Jessica Turner, and Yue Zhang 6

for helpful conversations. This manuscript has been au- Model Neff 95% upper limit thored by Fermi Research Alliance, LLC under Contract P ΛCDM+Neff + mν 3.19 ± 0.135 3.47 No. DE-AC02-07CH11359 with the U.S. Department of SIν 3.27 ± 0.14 3.56 Energy, Office of Science, Office of High Energy Physics. MIν 3.43 ± 0.13 3.72

TABLE I. Preferred values and upper limits on the effective Appendix A: Calculating ∆Neff number of neutrino species, Neff , from primordial nucleosynthesis. The cosmological models differ through their values The Hubble expansion rate at the time of BBN is of the baryon density parameter ηb (and its uncertainty) de- sensitive to the energy density in neutrinos and other termined from the CMB power spectrum as described in the relativistic species when photon temperatures are be- text. low an MeV [59–61]. New relativistic particles or an injection of energy into the Standard Model neutrino bath (via the decays or annihilations of a ν-coupled ues and uncertainties for Ωb in the SIν and MIν cos- species) can increase the Hubble expansion rate at this mologies [22]. For MIν, other data sets prefer slightly time. A larger expansion rate at the time of BBN mod- lower values of Ωb, which would increase the D/H yield ifies the neutron-to-proton ratio and the freeze-out of and lead to stronger constraints on Neff (and light ther- deuterium-burning reactions, leading to larger yields of mal particles as discussed below). This makes our choice Helium-4 and Deuterium. The primordial abundances of of TT+lens+BAO+H0 conservative. In contrast, the SIν these elements are measured in pristine gas clouds to be fit that uses TTTEEE + lens + H0 has a larger Ωb; this 5 would weaken the constraints on extra relativistic species. Yp = 0.2449 0.004 [62] and 10 D/H = 2.527 0.03 [63], respectively.± These observations are in good± agreement However, even this fit has a smaller value of Ωb than the with predictions assuming only Standard Model particle MIν result discussed above (and shown in Tab.I); thus content at the time of BBN [64]. we expect the BBN constraints for TTTEEE + lens + H0 to lie between the SIν and MIν results given in Tab.I. We use the measurements of Yp and D/H to constrain P modifications to the expansion rate using the BBN Boltz- We also note that our ΛCDM+Neff + mν value of Ωb mann code AlterBBN 2.1 [39, 40]. We follow the Monte does not include the local H0 measurement in the fit, Carlo procedure outlined in Ref. [65] to estimate theo- since the CMB-only H0 is incompatible with the local retical uncertainties from nuclear reaction rates. In de- measurement. Combining these (inconsistent) data sets riving limits we marginalize over a Gaussian prior on the gives a preferred value of Ωb that is close to that of the MIν cosmology [22], and would therefore result in BBN baryon density ηb corresponding to the best-fit points of P bounds similar to the MIν results. ΛCDM+Neff + mν [4], SIν and MIν [22]. Using the re- 2 sults of Refs. [64, 66] to convert Ωbh to ηb, we find that The upper bounds in Tab.I apply to light particles 10 10 ηb = 6.086 0.066, 6.146 0.082, and 6.248 0.082 that are fully relativistic at the time of nucleosynthe- ± P ± ± in ΛCDM+Neff + mν (TT+lowl+lowE+BAO data com- sis. If these particles have masses at the MeV scale, bination), SIν and MIν (TT+lens+BAO+H0), respectively. then their decays or annihilations can heat the neutri- Note that the best fit values of ηb in these cosmologies nos relative to photons; if they are much heavier, they 10 are all compatible with the range 5.8 < 10 ηb < 6.6 (95 transfer their entropy to neutrinos while those are still in % CL) extracted only from BBN data [51]. equilibrium with photons. The resulting change in neu- If the new particles remain relativistic throughout nu- trino temperature and the corresponding ∆Neff can be cleosynthesis, their modification of the Hubble rate is estimated assuming instantaneous neutrino-photon de- specified by a constant shift of the number of relativis- coupling at Tdec 1 2 MeV and by using entropy ≈ − tic degrees of freedom, Neff . We find that the observed conservation [67]. This crude estimate along with Tab.I values of Yp and D/H favor the values of Neff shown suggests that neutrino-coupled scalars in the SIν cosmol- in Tab.I for the different cosmologies. Note that the ogy with mφ 2 7 MeV should be incompatible with . − ΛCDM result is higher than reported in, e.g., Ref. [64], the observed light element abundances, where the range and it has a smaller uncertainty. This is because of up- corresponds to varying Tdec and the number of scalar dated D burning rates and observed abundances used in degrees of freedom from 1 (real scalar) to 2 (complex our analysis; for further discussion, see Ref. [65]. These scalar). The results of a full calculation using AlterBBN improvements actually weaken the BBN constraint com- (which does not make these approximations), shown in pared to Ref. [64] because of the slight preference for Tab.II, are compatible with this estimate. Neff > 3, which is driven by the slight underprediction We note that we do not use the Planck limit on of D/H compared to the observed value. The upper lim- ∆Neff [4], since that result assumes free-streaming neu- its in SIν and MIν are weaker still, due to their larger trinos. In fact, a large ∆Neff 1 at the time of the central values and uncertainties of the baryon density: a formation of the CMB is crucial' for the successful fits of larger baryon density reduces the Deuterium yield, which MIν and SIν cosmologies to the observed CMB power can be compensated by increasing Neff [64]. Different spectrum [22]. The concordance of the CMB and BBN choices of datasets feature slightly different best-fit val- measurements of ∆Neff within the self-interacting neu- 7

95% CL lower bound on mφ (MeV) the Standard Model plasma before 1 MeV, since they ∼ Model real φ complex φ would give ∆Neff & 1 during nucleosynthesis, which is clearly incompatible with the results of App.A. In this ΛCDM+N +P m 2.3 6.3 eff ν section we derive the conditions for RH neutrino ther- SIν 1.3 5.2 malization. MIν 0.6 3.7 In the Standard Model, RH neutrinos (neutrinos with the “wrong” helicity) can be created in any interac- TABLE II. Lower limits on mediator mass mφ from primor- tion that produces LH neutrinos, i.e. weak interac- dial nucleosynthesis, assuming φ was in thermal equilibrium tions. The characteristic production rate of RH neutri- before BBN. The cosmological models differ through their val- 2 5 2 nos in scattering reactions is ΓRH GFT (mν /T ) [41]. ues of the baryon density parameter ηb (and its uncertainty) ∼ as determined from the CMB power spectrum. If weak interactions are the only interactions produc- ing neutrinos, then RHNs do not thermalize as long as mν (100 keV), which is comfortably satisfied in the . O trino framework thus seems to require an injection of Standard Model. However, the interaction introduced energy between nucleosynthesis and recombination; we in Eqs. (1) and (2) of the main text is many orders of remain agnostic on this point. Given this discussion, we magnitude stronger than its weak counterpart, and the resulting RH neutrino production rate is much larger. conservatively only apply a constraint on ∆Neff from con- siderations of BBN physics alone. We further point out Thermalization is described by a Boltzmann equation of the form that the contribution of a scalar to ∆Neff at the 95% CL limit from our BBN analysis is also compatible with the ρ˙ + 4Hρ = C, (B1) extra radiation density at the best fit point in Ref. [22] νR νR at only the 2σ level. In order to be in better agreement where ρν is the energy density in “wrong helicity” with ∆Neff 1.02 0.29 for the SIν mode, new (semi) R relativistic degrees' ± of freedom may need to come into neutrinos and C is the collision term encoding the equilibrium with the neutrino bath after BBN is com- processes that produce these neutrinos. Thermaliza- tion/equilibration occurs when C 4Hρeq. The dom- plete [36, 45]. ∼ R In deriving constraints from light element abundances inant process responsible for RH neutrino production is the decay φ νν(λ = +1/2), where λ = +1/2 ( 1/2) we have ignored the possible influence of neutrino self- → − interactions and potentially large neutrino mass. How- is a helicity label corresponding to RH (LH) neutrinos; scattering reactions such as νν νν(λ) are suppressed ever, these effects have a negligible impact on the yields. 2 → For example, large neutrino self-interactions at the time by an additional factor of gφ. We therefore evaluate the collision term for φ νν(λ = +1/2): of nucleosynthesis maintain thermal equilibrium for the → three neutrino species, and affect flavor oscillations. In Z the standard cosmology, the non-instantaneous decou- 2 C 2 dΦ3EνR (λ = +1/2) fφ(1 fν ), (B2) pling of weak interactions leads to a small shift to Neff ≈ |M | − and spectral distortions of the neutrino distributions. The former effect is already taken into account in pub- where dΦ3 is the Lorentz-invariant phase-space (including the momentum conservation delta function) and fφ, lic BBN codes by setting Neff = 3.046, while the latter fν are the phase-space distributions of φ and LH neutri- has a tiny effect on the abundances of Yp and D/H [68]. Thus, if the standard spectral distortions are driven to nos. We have neglected the inverse decay contribution zero by the novel self-interactions considered here, then and the RH neutrino Pauli-blocking factors. These ap- their impact would still be unobservable. Similarly, the proximations are adequate for estimating the onset of equilibrium, assuming the initial abundance of RH neu- shift to Neff from the presence of non-standard interac- trinos is negligible, i.e. fνR 1. We have also multiplied tions is also much smaller than a percent [69]. Another ∗ important parameter in the CMB fit is the neutrino mass, the right-hand side by two to account for φ decays which which, along with interaction-induced matter potential, can also produce wrong helicity neutrinos. can affect flavor oscillations during BBN. However, oscil- We use the Lagrangian in Eq. (3) of the main text (with lations have a negligible impact on the abundances [69]. a complex φ for a Dirac neutrino) to evaluate the matrix We therefore conclude that the constraints from nucle- element squared in the plasma frame, without summing over one of the helicities [70] osynthesis derived above are sensitive to Neff and ηb, but not to other aspects of the strongly-interacting neutrino ( Eν (1 + cos θ)m2 λ = +1/2 model. 2 2 2Eν ν (λ) gφ R , (B3) |M | ≈ m2 λ = 1/2 φ − Appendix B: Dirac Neutrino Thermalization where θ is the angle between the νR and φ direction of motion, and we have kept only the leading terms in If neutrinos are Dirac fermions, their right-handed mν /mφ. Note that when cos θ = 1, the “wrong” helic- (RH) components must not come into equilibrium with ity amplitude vanishes at this order− as a result of angular 8

4 momentum conservation (the next term is mν ). Using We ﬁnd that Lφ given by Eq. (C1) exceeds Lν if gφ is this result in Eq. (B2), we ﬁnd that ∝ roughly in the range

2 2 3 1 gφmν 2ζ(3)T gexcl 5 10−6 6 10−5 . (C2) C +(mφ/T ), (B4) φ 1 + m /keV ≈ 32π π2 C ' × − × × φ where the factor in parentheses is the number density of The sharp change in the shape of the bound at mφ keV is due to the fact that rate of decay and inverse' relativistic φ and the function + is 1 for mφ/T < 1 2 2 C ∼ decay, Γφ↔νν g m /T , becomes subdominant to the and becomes Boltzmann-suppressed for mφ/T > 1. φ φ ∝ 4 annihilation rate, Γφφ↔νν g T , for masses mφ gφTc, We can apply the nucleosynthesis bound as computed ∝ φ . in SectionA if thermalization occurs before T where Tc (30 MeV) is the core temperature. These dec ∼ O 1 MeV. By using the approximate thermalization cri-∼ bounds are approximately compatible with those shown terion below Eq. (B1), we find that RH neutrinos ther- in [35] at masses above 10 keV. From Eq. (C2), we see malize before BBN if that bounds arising from the luminosity of φ particles from Supernova 1987A are generally below the coupling max(m ,T )3/2 0.1 eV range of interest in this work. g 5 10−3 φ dec . (B5) φ & MeV m It is also interesting to understand the constraints on × ν ee gφ arising from deleptonization of the core, which have This bound (evaluated using the full numerical ) is been obtained in the (MeV) mass range in [35] and + ∼ O shown in Figure 1 of the main text as a black dashedC which approximately overlap the range in Eq. (C2). At line. lower masses, this likely has an effect on the early phases of collapse and the collapse progenitor. Such a possibility was suggested in [76] and was studied in the aftermath Appendix C: Supernova 1987A of Supernova 1987A by [77]. This latter study found a ee −4 constraint gφ < 3 10 , neglecting any mφ-dependence. Because these bounds× are determined by physics at the A new weakly coupled particle can change the behavior beginning of the core collapse, when temperatures are of the neutrino emission that was observed from the ex- (few MeV), the bound likely cuts off at 5 MeV, plosion of Supernova 1987A. The proto-neutron star cool- similar∼ O to the 0νββ bounds cited above. A detailed∼ study ing phase that was observed in large water Cherenkov de- is of interest, but beyond the scope of this work. tectors was qualitatively similar to the Standard Model- only expectation [71]. If a new particle species X car- ried away too much energy during the proto-neutron Appendix D: Strong Neutrino Self-Interactions via star cooling phase, the time over which neutrinos ar- Sterile Neutrino Interactions rived would have been unacceptably reduced [72, 73]. A semi-analytic criterion that the luminosity of this parti- 52 Here we extend our exploration into the possibility that cle should obey is LX Lν = 3 10 erg / s at times of order 1 second after the≤ core bounce× [74]. Following the the interactions between the light neutrinos and the me- procedure described in more detail in [75], we have diator φ are generated by mixing with sterile neutrinos, which we refer to as N. This requires interactions be- ! tween N and φ – we refer to this coupling as g – and Z Rν Z d3k Z Rg N prod abs 0 mixing between the light and sterile neutrinos, which Lφ = dV 3 ωΓφ exp Γφ dr , (C1) 0 (2π) − r we refer to as U . This scenario is appealing because the mixing U may| | be temperature-dependent, allowing ~ abs | | where: the φ has four-momentum (ω, k); Γφ is the φ the constraints regarding thermalization of φ (and poten- absorptive width; Γprod is the φ production rate, which tially N) prior to BBN. The mixing U can then become φ | | is related to the absorptive width in equilibrium by large before T 100 eV, allowing for strong interactions prod abs to modify CMB≈ observations as desired by Ref. [22]. This Γφ = exp( ω/T )Γφ ; Rν is the radius of the neu- trinosphere, outside− of which neutrinos free-stream; and could also explain the necessary ∆Neff. 1 at the time of CMB preferred by the fits in Ref. [22]≈ if the increased Rg = 100 km is the radius inside of which neutrinos gain energy on average in elastic scattering events. We cal- mixing causes some particle to thermalize between BBN and CMB times. culate Γφ including φ decay and φ annihilation to neutrino pairs, both of which are important for the masses If such a scenario exists, then the four-neutrino scatter- 4 4 8 of interest. We have not included contributions from ing with σ gφ now becomes σ gN U – we identify ∝ −1 ∝ | | 2 the neutrino effective potentials, which may be signifi- the desired coupling gφ 10 with gN U . Allowing ≈ 2| | −2 cant at small mφ [35]. We have also neglected neutrino gN to be as large as √4π, this dictates U 10 by the | | ≈ Pauli blocking in Γφ, which is important near the proto- time of CMB, T 100 eV. In the main text and in Sec- neutron star core, since this will become unimportant tionA, we discussed≈ the criteria for the thermalization between Rν and Rg. of φ prior to BBN, contributing to Neff . In SectionA, we 9 focused on the values of mφ that are incompatible with constraints of thermalization of φ while still having an BBN observations. Additionally, there is the requirement appreciable mixing during the time of the CMB, this sup- −11 that gφ & 10 in order for φ to thermalize. In order pression must be active when T 100 keV - 1 MeV, but to avoid this constraint, allowing for a time-dependent no longer present by T 1 keV.≈ These two conditions 2 −11 ≈2 mixing with N, we can enforce U . 10 until after place a constraint on ∆m /ηL: BBN. | | 2 In the following, we explore the specifics of one scenario −16 2 ∆m −5 2 10 eV . . 10 eV . (D2) in which mixing can change rapidly between BBN and ηL CMB, and we highlight the difficulties of realizing such a Even with order-one lepton number asymmetry, this in- scenario. Ref. [43] explored the temperature-dependent dicates very small sterile-active neutrino mass-squared suppression of the sterile-active neutrino mixing, partic- splittings. Ref. [43] derives its results assuming there is ularly depending on a lepton-number asymmetry to gen- one “active” neutrino species with which the sterile neu- erate a large neutrino matter potential at early times in trino mixes. If ∆m2 10−5 eV2, the sterile neutrino is the universe. This temperature-dependent potential can . nearly degenerate with the active ones, and this formal- lead to rapid changes in the sterile-active neutrino mix- ism breaks down. ing, where U 2 T −8 for particular epochs, depending We conclude that, even if sterile neutrino interactions on parameters| | associated∝ with the sterile neutrino. The may be responsible for the large neutrino self interac- suppression of U 2 is in effect when | | tions, further work is required to determine whether this mechanism can occur for the temperatures of interest. If, 2 3 ∆m p 2 however, this mechanism may provide appreciable inter- √2GF ηLT 1 U0 , (D1) T − | | actions at CMB times without thermalizing φ prior to BBN, we note that the results of Ref. [22] prefer large where GF is the Fermi constant, ηL is the lepton number Neff at the time of CMB. There is a possibility that the asymmetry, ∆m2 the new sterile-active neutrino mass- mixing U 2 turning on rapidly can cause φ to thermalize squared splitting. Here, we indicate the vacuum mixing between| BBN| and CMB, providing the extra contribu- as U0 for clarity. In order to simultaneously avoid BBN tion to Neff preferred by the fits in Ref. [22]. | |

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