Constraining the Self-Interacting Neutrino Interpretation of the Hubble Tension

# Constraining the Self-Interacting Neutrino Interpretation of the Hubble Tension

Nikita Blinov    Kevin J. Kelly    Gordan Krnjaic    Samuel D. McDermott Fermi National Accelerator Laboratory, Batavia, IL, USA
July 1, 2019
###### Abstract

Large, non-standard neutrino self-interactions have been shown to resolve the 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 () 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 -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.

preprint: FERMILAB-PUB-19-175-A-Tthanks: ORCID: http://orcid.org/0000-0002-2845-961Xthanks: ORCID: http://orcid.org/0000-0002-4892-2093thanks: ORCID: http://orcid.org/0000-0001-7420-9577thanks: ORCID: http://orcid.org/0000-0001-5513-1938

## I Introduction

The discrepancy between low-redshift and Cosmic Microwave Background (CMB) determinations of the present-day Hubble parameter, , has grown in significance to over the last several years Riess et al. (2016); Shanks et al. (2019); Riess et al. (2018); Aghanim et al. (2018); Riess et al. (2019). The standard cosmological model, CDM, may need to be augmented if this “ tension” is not resolved by observational systematics. Intriguingly, this tension cannot be addressed by modifying CDM at low redshift Vonlanthen et al. (2010); Verde et al. (2017); Evslin et al. (2018); Aylor et al. (2019), but adding new physics before recombination seems more promising Lesgourgues et al. (2016); Di Valentino et al. (2018); Poulin et al. (2018a); D’Eramo et al. (2018); Poulin et al. (2018b); Pandey et al. (2019); Escudero et al. (2019). Furthermore, low redshift measurements of the matter density fluctuation amplitude on 8 Mpc scales, , also appear to be lower than predicted by CDM from the CMB. This milder tension is not ameliorated in the models of Di Valentino et al. (2018); Poulin et al. (2018a); D’Eramo et al. (2018); Poulin et al. (2018b); Pandey et al. (2019); Escudero et al. (2019).

A particularly interesting resolution to these issues is a non-standard neutrino self-interaction of the form Cyr-Racine and Sigurdson (2014); Lancaster et al. (2017); Oldengott et al. (2017); Kreisch et al. (2019)

 Leff=Geff(¯νν)(¯νν), (1)

where is a dimensionful coupling and flavor indices have been suppressed. If is much larger than the Standard Model (SM) Fermi constant, , neutrinos remain tightly coupled to each other until relatively late times. This inhibits their free-streaming and results in enhanced power on small scales and a shift in the acoustic peaks of the CMB spectrum relative to CDM Bashinsky and Seljak (2004). The effect of neutrino self-interactions is degenerate with other parameters in the CMB fit, such as the angular scale of the sound horizon, the spectral index and amplitude of primordial fluctuations, and extra free-streaming radiation (as parametrized by ).

These approximate degeneracies have been shown to prefer in cosmological data Cyr-Racine and Sigurdson (2014); Lancaster et al. (2017); Oldengott et al. (2017); Kreisch et al. (2019) and relax the tension Oldengott et al. (2017); Lancaster et al. (2017); Kreisch et al. (2019). The latest results of Ref. Kreisch et al. (2019) extended previous analyses by allowing for finite neutrino masses and extra free-streaming radiation at the time of the CMB. They found that in the “strongly interacting” (SI) or “moderately interacting” (MI) regimes

 Geff={(4.6±0.5MeV)−2   (SIν)(90+170−60MeV)−2      (MIν) (2)

could simultaneously reduce the and tensions. Interestingly, the SI cosmology prefers a value of compatible with local measurements at the 1 level, even before including local data in the fit.

The favored range of in Eq. (2) vastly exceeds the strength of weak interactions, whose corresponding coupling is . As we discuss below, such an interaction can only arise from the virtual exchange of a force carrier (“mediator”) with mass and appreciable couplings to neutrinos. This mass range indicates that this scenario faces a variety of stringent cosmological and laboratory constraints.

We find that if the interaction in Eq. (1) resolves the tension, then:

• SI is excluded: The SI-range of Eq. (2) cannot be realized in any of the consistent, low-energy, weakly coupled models that we consider.

• Vector forces are excluded: Constraints from the epoch of Big Bang Nucleosynthesis (BBN) exclude all self-consistent vector mediators.

• Dirac neutrinos are excluded: If neutrinos are Dirac particles, mediator-neutrino interactions efficiently thermalize the right-handed component of neutrinos, thereby significantly increasing the number of neutrino species at BBN.

• Minimal seesaw models are excluded: Achieving the necessary interaction strength in Eq. (2) from a gauge-invariant, UV-complete model is challenging with minimal field content in a Type-I or -II seesaw model; a non-minimal mechanism is required to generate neutrino masses.

• Interactions with are favored: Couplings to , with in the range of Eq. (2) are largely excluded by laboratory searches for rare decays and for neutrinoless double-beta decay, except for a small island for the coupling. This means that the mass-eigenstate neutrinos only interact via their components.

This work is organized as follows: Sec. II demonstrates that a light new particle is required to generate the interaction in Eq. (1) with appropriate coupling strength; Sec. III presents the cosmological bounds on this scenario; Sec. IV discusses the corresponding laboratory constraints; Sec. V shows how Eq. (1) can arise in UV complete models; finally, Sec. VI offers some concluding remarks.

## Ii The Necessity of a Light Mediator

The Boltzmann equations used in Refs. Oldengott et al. (2017); Lancaster et al. (2017); Kreisch et al. (2019); Barenboim et al. (2019) assume that left-handed (LH), mass-eigenstate neutrinos participate in elastic scattering processes. They also assume that the interactions in Eq. (1) involve constant and flavor-universal values of during all epochs relevant for the CMB. The largest CMB multipoles observed by Planck correspond to modes that entered the horizon when the universe had a temperature of .This temperature sets the characteristic energy scale of scattering reactions during this epoch, and it is important that the form of the Lagrangian shown in Eq. (1) is valid in this regime. At higher energies, however, this description can break down. In this section, we therefore emphasize the need to introduce new particle content to study laboratory and early universe processes that occur at energies .

As noted in Refs. Oldengott et al. (2017); Cyr-Racine and Sigurdson (2014); Lancaster et al. (2017); Kreisch et al. (2019), the operator in Eq. (1) is non-renormalizable, and thus is necessarily replaced by a different interaction with new degree(s) of freedom at some energy scale higher than the energies probed by the CMB (see Ref. Cohen (2019) for a review). Since in Eq. (1) is momentum-independent, we will assume this interaction arises from “integrating out” a particle with mass and a perturbative coupling to neutrinos :

 Lphen⊃−12m2ϕϕ2+12(gαβϕνανβϕ+h.c.), (3)

where are two-component left-handed neutrinos, and are flavor indices and the subscript “phen” indicates that we will use this Lagrangian for our phenomenological analysis. In Eq. (3) we have assumed that is a real scalar without loss of essential generality. In particular, our conclusions remain unaffected if is a CP-odd or complex scalar. We focus on a scalar mediator here for clarity; introducing a new vector force instead, for example, follows the same reasoning but comes with additional, stronger constraints discussed in more detail below. We also explicitly allow for generic couplings between different neutrino species, and discuss implications of different choices of below.

Using Eq. (3), we see that the scattering amplitude is always proportional to two powers of multiplying the propagator, as shown schematically in Fig. 1. If the momentum transfer satisfies , we have

 M∝g2ϕm2ϕ−q2→Geff(1+q2m2ϕ+⋯), (4)

where we have suppressed flavor indices and defined

 Geff≃g2ϕm2ϕ=(10 MeV)−2(gϕ10−1)2(MeVmϕ)2 . (5)

In the opposite limit, , then , leading to a qualitatively different energy and temperature dependence of neutrino self-interactions; this regime was investigated in Refs. Forastieri et al. (2015, 2019), which found no improvement in the tension. Thus, for the remainder of this work we focus on models in which at energy scales relevant to the CMB. The intermediate regime, where the mediator mass is negligible for some CMB wave-numbers and not for others, is beyond the scope of this work. We therefore require a new degree of freedom with .

Throughout this cosmological epoch the neutrinos are relativistic, so the typical momentum transfer is . Therefore, the expansion indicated by the arrow in Eq. (4) is valid (and we may neglect the momentum- and, hence, temperature-dependence in ) only if . Comparing the values in Eq. (2) to the expression for in Eq. (5), we see that

 mϕ≃(4−200)×|gϕ|MeV , (6)

so a new sub-GeV state is generically required to realize the self-interacting-neutrino solution to the tension. Since at horizon entry of the highest momentum modes relevant for CMB anisotropies, the validity of the effective interaction in Eq. (1) in the analyses of Refs. Cyr-Racine and Sigurdson (2014); Lancaster et al. (2017); Oldengott et al. (2017); Kreisch et al. (2019); Barenboim et al. (2019) requires (as already pointed out in Ref. Kreisch et al. (2019)). From Eq. (6), this condition translates to

 mϕ≳keV  ⟹  |gϕ|≳10−4 . (7)

Eqs. (6) and (7) bound the range of .

Finally, we note that the interaction in Eq. (3) is not gauge-invariant at energies above the scale of electroweak symmetry breaking (EWSB). This is because is a component of the doublet of the group, whose gauge transformations rotate into the corresponding charged lepton . Thus, some particle with non-trivial electroweak quantum numbers must replace the interaction of Eq. (3) at energies above the Higgs vacuum expectation value . We defer a complete discussion of this issue to Sec V; for now we assume that the interaction of Eq. (1) is valid well below the electroweak scale.

## Iii Cosmological Bounds

The successful predictions of BBN in standard cosmology (with only SM particle content) provide a powerful probe of additional light species. New particles in thermal equilibrium with neutrinos increase the Hubble expansion rate during BBN as extra relativistic degrees of freedom or by heating neutrinos relative to photons. Away from mass thresholds, both effects are captured by a constant shift in , the effective number of neutrinos. We find that the observed light element abundances constrain () at 95% CL for the SI- (MI-) preferred values of the baryon density, as explained in more detail in App. A.

Note that we do not use the Planck limit on  Aghanim et al. (2018), since that result assumes free-streaming neutrinos. In fact, a large at the time of the formation of the CMB is crucial for reconciling the MI and SI cosmologies with the observed CMB power spectrum Kreisch et al. (2019). The concordance of the CMB and BBN measurements of within the self-interacting neutrino framework thus seems to require an injection of energy between nucleosynthesis and recombination; we remain agnostic on this point. Given this discussion, we conservatively only apply a constraint on from considerations of BBN physics alone.

### iii.1 Mediators and ΔNeff

The interaction in Eq. (3) induces decays and inverse decays; this process can bring into into thermal equilibrium with the neutrino bath before BBN.111Annihilation and scattering processes (e.g. and ) also contribute to equilibration, but the corresponding rates are suppressed by additional powers of the small coupling , so their contribution is always subdominant. From Eqs. (6) and (7), the mediator that generates sufficiently large neutrino self-interactions must have mass in the range. Consequently, this particle is at least semi-relativistic when neutrinos decouple from photons at MeV. We can estimate the resulting contribution to the expansion rate as parametrized by assuming is relativistic:

 ΔNeff≃87(TγTν)4ρϕργ, (8)

where is the energy density of photons and is their temperature. If becomes non-relativistic before the end of BBN, is increased further as decays raise the neutrino temperature (see, e.g., Ref. Boehm et al. (2012)). Here, we show that a mediator whose interactions realize the necessary in Eq. (2) necessarily equilibrates with neutrinos before .

Vector Mediators: If Eq. (1) arises from integrating out a vector particle with mass , the Lagrangian at energies above is

 L⊃12m2ϕϕμϕμ+(gϕϕμν†¯σμν+h.c.), (9)

where is the dimensionless gauge coupling. At low energies, the interaction in Eq. (9) yields

 Geff≃g2ϕm2ϕ≃(10 MeV)−2(gϕ10−4)2(keVmϕ)2. (10)

The vector equilibrates before via if the corresponding thermally averaged rate exceeds Hubble when :

 Γνν→ϕH∼g2ϕm2ϕMPlmax(Tdec,mϕ)3>108Geff(10MeV)−2, (11)

where GeV and we have used Eqs. (10) and (7). Thus, this reaction is in equilibrium for all values of couplings and masses of interest. As a result, has a thermal number density at in both the MI and SI scenarios. Using Eq. (8), we find assuming remains relativistic throughout BBN. If, instead, becomes non-relativistic between and keV at the end of BBN, then , so must become non-relativistic well before . In Ref. Escudero et al. (2019) it was found that the Boltzmann suppression for massive vector particles is effective for at 95% CL. Using Eq. (10), this requires , which is excluded in all theoretically-consistent (or anomaly-free) vector models with neutrino couplings Bauer et al. (2018). Anomaly-free vectors, such as those coupled to currents, where is a lepton family number, would additionally introduce large interactions which would likely spoil the CMB fit.

Finally, we note that Eq. (9) can arise if active neutrinos mix with heavy sterile-neutrinos, which couple directly to the vector  Schmaltz and Weiner (2019); Bertuzzo et al. (2018); Ballett et al. (2019). The mediators in these models are subject to the same BBN constraint above, but because their coupling to active neutrinos is derived from the active-sterile mixing angle, the resulting faces additional bounds from laboratory searches de Gouvêa and Kobach (2016) that exclude the interaction strengths in Eq. (2) required to resolve the tension.

Scalar Mediators: By a similar argument, any scalar mediator that realizes from the interaction in Eq. (3) with as required by Eq. (7) also has a thermal abundance at . Relativistic scalars in equilibrium with neutrinos contribute for a real (complex) , which has one (two) degree(s) of freedom. As for the vector case, the density must become Boltzmann-suppressed before neutrino-photon decoupling, leading to a lower limit on . We use AlterBBN 2.1 Arbey (2012); Arbey et al. (2018) as described in App. A to obtain the CL lower bounds

 mϕ>{1.3MeV  (real % scalar)5.2MeV  (complex scalar), (12)

for the SI preferred values of the baryon density (the corresponding MI bounds are somewhat weaker – see App. A). These bounds are are shown as red vertical lines labeled BBN in Fig. 2.

### iii.2 Excluding Dirac Neutrinos

If neutrinos are Dirac particles, then all neutrino masses arise from the Higgs-neutrino Yukawa interaction

 LDirac⊃yνHLνR→mνννR  ,  mν≡yνv/√2  , (13)

where is the Higgs doublet, is a lepton doublet, is a right-handed neutrino (RHN), and flavor indices have been suppressed. The Weyl fermions and become Dirac partners after EWSB and acquire identical masses. In the absence of additional interactions, the Yukawa coupling in Eq. (13) is insufficient to thermalize right handed helicity states, so the relic neutrinos consist exclusively of left-handed neutrinos and right-handed antineutrinos Dolgov et al. (1995).

The light mediator interactions in Eq. (2) are much stronger than the weak force at late times, so and can both thermalize. Approximating the RHN production rate as , for eV we have

 Γϕ→ννRH≃g2ϕm2νMPlm3ϕ=106Geff(10MeV)−2MeVmϕ, (14)

where is the temperature at which RHN production is maximized relative to Hubble expansion. For a more detailed discussion, see App. B.

Observations of neutrino oscillations require that at least two of the light neutrinos are massive, with one heavier than eV and one heavier than eV Esteban et al. (2019). Thus, for all values of we consider in the SI range, at least one RHN will thermalize before BBN. For this reason, we assume that neutrinos are Majorana particles for the remainder of this work.

### iii.3 Supernova 1987A

A light new particle with appreciable couplings to neutrinos can alter the qualitative behavior of neutrino emission that was observed in the aftermath of Supernova 1987A. We find that for the parameter space shown in Fig. 2 the particle is trapped and does not drain energy from the supernova. See App. C for further discussion.

## Iv Laboratory Bounds

Because terrestrial experiments routinely reach energies above the MeV scale, there are many possible probes of the models we consider. Here we specialize to CP-even scalar mediators without loss of essential generality; we comment on CP-odd scalars in Sec. V. The main constraints are as follows:

Double Beta Decay: If in Eq. (3) and is lighter than the -value of a double-beta-decaying nucleus, the process may occur, contributing to measured rates. Measurements of this process constrain if MeV Agostini et al. (2015); Blum et al. (2018); Brune and Ps (2018). This constraint is shown in the top row of Fig. 2.

Meson Decays: Nonzero can allow for the meson decays if  Blum et al. (2014); Berryman et al. (2018); Kelly and Zhang (2019); Krnjaic et al. (2019). Note that, unlike decays, this process is not helicity suppressed. The strongest constraints come from kaon decay. The measured ratio of constrains as shown in the top row of Fig. 2 Lessa and Peres (2007); Fiorini (2007). The limit on  Tanabashi et al. (2018) constrains as shown by the purple-shaded region in bottom-left panel of Fig. 2. Non-zero still contributes when via off-shell  Kelly and Zhang (2019). Note that, if decays to visible matter via or through some other interaction (e.g. -higgs mixing), the branching ratio must be subdominant to decays to avoid tight coupling below MeV, which would spoil the solution to the tension. Thus, the laboratory bounds presented here are unaffected by other possible couplings (see Ng and Beacom (2014) for a review).

Decays: Similar to meson decays, that couples to with allows for decays . For , this constrains  Lessa and Peres (2007). This is depicted as a purple band in the bottom-right panel of Fig. 2.

In Fig. 2 we summarize our main findings: values of from Eq. (2) favored by the tension are excluded if couples universally to all flavor- and mass-eigenstates (top left panel), which was explicitly considered in Refs. Cyr-Racine and Sigurdson (2014); Lancaster et al. (2017); Oldengott et al. (2017); Kreisch et al. (2019); Barenboim et al. (2019), or if couples predominantly to or (top-right and bottom-left panels, respectively). Similarly, we can rule out the possibility that couples to any single mass-eigenstate neutrino, since the - and -composition of each mass eigenstate is similar. Moreover, in this case, the collisional Boltzmann equations would be much more complicated to solve (because different eigenstates will start to free-stream at different times), and the results of Refs. Cyr-Racine and Sigurdson (2014); Lancaster et al. (2017); Oldengott et al. (2017); Kreisch et al. (2019); Barenboim et al. (2019) may not apply.

However, a flavor-restricted coupling leads to approximately the same neutrino mass-eigenstate interactions as in Refs. Cyr-Racine and Sigurdson (2014); Oldengott et al. (2017); Lancaster et al. (2017); Kreisch et al. (2019); Barenboim et al. (2019), since the flavor eigenstates are well-mixed in the mass basis. In particular, a -only coupling, in which the matrix is zero except for the entry, is potentially viable since decays are less constraining than meson decays. Thus, we are unable to fully exclude an interaction . In this case, for defined in Eq. (5) and is a constant that accounts for the reduced scattering probability of each mass eigenstate. However, because mixed mass-eigenstate vertices are possible in this 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 self-interactions can be mildly different than that considered in Refs. Cyr-Racine and Sigurdson (2014); Lancaster et al. (2017); Oldengott et al. (2017); Kreisch et al. (2019); Barenboim et al. (2019). Nonetheless, we expect that the preferred coupling range should shift slightly up relative to the flavor-universal case. For this reason, the SI range is ruled out regardless of choice of flavor structure in Eq. (3). The MI range is still allowed in a -flavor-only scenario, though a dedicated study is necessary and well motivated.

## V Possible Ultraviolet Completions

In this section we quantify the difficulty of realizing the operator in Eq. (1) from renormalizable, gauge invariant interactions that also accommodate neutrino masses and mixings. In light of the discussion in Sec III, we only consider models of Majorana neutrinos. We focus on the type-I and type-II seesaw mechanisms of Majorana neutrino mass generation with an additional scalar particle . In both cases, we find the resulting Yukawa coupling is suppressed by factors of the light neutrino mass. In these minimal models, it is therefore impossible to simultaneously generate neutrino masses and a large enough in Eq. (2) to address the tension.

### v.1 Type-I Seesaw

We consider the Lagrangian

 LI⊃yNLHN+12(MN+λNϕ)NN+h.c., (15)

where we have suppressed flavor indices. After EWSB, the first term in Eq. (15) and the term generate light neutrino masses after is integrated out:

 mν≈−y2Nv22MN. (16)

These interactions also generate the coupling of to LH neutrinos in Eq. 3 with

 gϕ=λNmMN⇒Geff=λ2Nm2νm2ϕM2N. (17)

Thus, realizing requires . If this is true, the RHNs thermalize before BBN and spoil as discussed in analogy to the Dirac case discussed in Sec. III.2. Therefore, a minimal scenario where the same Type-I Seesaw generates the neutrino mass and the operator in Eq. (1) with the magnitude in Eq. (2) is not possible.

We also consider the case in which, instead of violating lepton number explicitly, we replace the term in parentheses in Eq. (15) with a complex scalar . This acquires a vacuum expectation value (VEV) of , dynamically generating a mass for  Chikashige et al. (1981, 1980); Schechter and Valle (1982). The light neutrinos couple to the pseudo-Goldstone boson , the Majoron, of this symmetry breaking. The coupling is of the form , so the Majoron is a pseudoscalar mediator. We note that our results from the previous sections apply to the Majoron, since all processes we have considered involve relativistic neutrinos for which the distinction between scalar and pseudoscalar is irrelevant. As with the above case, we see that the neutrino self-interaction strength will be suppressed by the light neutrino masses.

### v.2 Type-II Seesaw

We augment the Standard Model with a complex triplet . The type-II Seesaw Lagrangian is

 LII ⊃yTLTL+(μT+λTϕ)HT†H+h.c. −M2TTr(T†T), (18)

where the dimensionful coupling is a soft breaking of Lepton number. After integrating out and EWSB, neutrinos acquire masses of order

 mν≈yTμTv2M2T, (19)

and a coupling to of the form of Eq. 3 with

 gϕ=λTmνμT⇒Geff=λ2Tm2νm2ϕμ2T. (20)

The scale is bounded by non-observation of rare lepton-number-violating processes such as or  Akeroyd et al. (2009); Dinh et al. (2012); Dev et al. (2017). Thus, for keV, the suppression is too severe to permit values of in Eq. (2), as required to resolve the tension.

As with the Type-I Seesaw, we may consider the Majoron case where is replaced by a field which acquires a VEV. The neutrinos couple to the pseudo-Goldstone boson of this lepton number symmetry breaking, with coupling of the form . Again, this coupling is highly suppressed.

We conclude that the coupling to active neutrinos cannot arise from interactions with the same field that generates neutrino masses (the RH neutrino in Type-I seesaw and the triplet in Type-II), without being suppressed by neutrino masses. A large enough can therefore be obtained using separate seesaw mechanisms to generate the neutrino masses and the interaction. For instance, we can use the Type-I Seesaw with for the light neutrino masses and add the Type-II Seesaw with to allow for large . In this scenario, the size of is independent of the neutrino masses.

## Vi Concluding Remarks

In this work we have shown that the self-interacting neutrino explanation of the tension requires the existence of a light MeV-scale mediator, which is subject to stringent cosmological and laboratory bounds. Consequently, most realizations of this scenario are robustly excluded by the BBN-only bounds on , accelerator constraints for rare decays, and precise results from neutrinoless double-beta decay searches. If self-interactions resolve the tension, they cannot viably arise in models with vector-mediated forces or with Dirac neutrinos, and they cannot reside in the strongly interacting parameter space in Eq. (2).

Our analysis identifies viable phenomenological models with a 10 MeV scalar mediator and large couplings to -flavored neutrinos only. However, realizing such interactions in UV-complete, gauge-invariant models is challenging. We find that sufficiently strong interactions cannot arise in models that generate neutrino masses via a single Type-I or -II seesaw mechanism: the resulting neutrino-scalar coupling is suppressed by factors of where is the appropriate seesaw scale.

Looking forward, a dedicated exploration of the -only scattering scenario favored by our analysis will be necessary to see if the moderately interacting-scenario is indeed able to resolve the tension while not running afoul of laboratory measurements. Our results also motivate exploration of the “intermediate” mediator-mass regime, where neutrino scattering is relevant for a partial range of redshifts explored by the CMB.

###### Acknowledgements.
We thank André de Gouvêa, Francis-Yan Cyr-Racine, Joshua Isaacson, Martina Gerbino, Stefan Höche, Massimiliano Lattanzi, Kohta Murase, Jessica Turner, and Yue Zhang for helpful conversations. This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

## Appendix A Calculating ΔNeff

The Hubble expansion rate at the time of BBN is sensitive to the energy density in neutrinos and other relativistic species when photon temperatures are below an MeV Serpico and Raffelt (2004); Boehm et al. (2013); Nollett and Steigman (2015). New relativistic particles or an injection of energy into the Standard Model neutrino bath (via the decays or annihilations of a -coupled species) can increase the Hubble expansion rate at this time. A larger expansion rate at the time of BBN modifies the neutron-to-proton ratio and the freeze-out of deuterium-burning reactions, leading to larger yields of Helium-4 and Deuterium. The primordial abundances of these elements are measured in pristine gas clouds to be  Aver et al. (2015) and D/H  Cooke et al. (2018), respectively. These observations are in good agreement with predictions assuming only Standard Model particle content at the time of BBN Cyburt et al. (2016).

We use the measurements of and D/H to constrain modifications to the expansion rate using the BBN Boltzmann code AlterBBN 2.1 Arbey (2012); Arbey et al. (2018). We follow the Monte Carlo procedure outlined in Ref. Berlin et al. (2019) to estimate theoretical uncertainties from nuclear reaction rates. In deriving limits we marginalize over a Gaussian prior on the baryon density corresponding to the best-fit points of CDM Aghanim et al. (2018), SI and MI Kreisch et al. (2019). Using the results of Refs. Steigman (2006); Cyburt et al. (2016) to convert to , we find that , , and in CDM, SI and MI, respectively. Note that the best fit values of in these cosmologies are all compatible with the range (95 % CL) extracted only from BBN data Tanabashi et al. (2018).

If the new particles remain relativistic throughout nucleosynthesis, their modification of the Hubble rate is specified by a constant shift of the number of relativistic degrees of freedom, . We find that the observed values of and D/H favor the values of shown in Tab. 1 for the different cosmologies. Note that the CDM result is higher than reported in, e.g., Ref. Cyburt et al. (2016), and it has a smaller uncertainty. This is because of updated D burning rates and observed abundances used in our analysis; for further discussion, see Ref. Berlin et al. (2019). These improvements actually weaken the BBN constraint compared to Ref. Cyburt et al. (2016) because of the slight preference for . The upper limits in SI and MI are weaker still, due to their larger central values and uncertainties of the baryon density: a larger baryon density reduces the Deuterium yield, which can be compensated by increasing  Cyburt et al. (2016).

The upper bounds in Tab. 1 apply to light particles that are fully relativistic at the time of nucleosynthesis. If these particles have masses at the MeV scale, then their decays or annihilations can heat the neutrinos relative to photons; if they are much heavier, they transfer their entropy to neutrinos while those are still in equilibrium with photons. The resulting change in neutrino temperature and the corresponding can be estimated assuming instantaneous neutrino-photon decoupling at and by using entropy conservation Boehm et al. (2012). This crude estimate along with Tab. 1 suggests that neutrino-coupled scalars in the SI cosmology with MeV should be incompatible with the observed light element abundances, where the range corresponds to varying and the number of scalar degrees of freedom from 1 (real scalar) to 2 (complex scalar). The results of a full calculation using AlterBBN (which does not make these approximations), shown in Tab. 2, are compatible with this estimate.

We point out that the contribution of a scalar to at the 95% CL limit from our BBN analysis is also compatible with the extra radiation density at the best fit point in Ref. Kreisch et al. (2019) at only the level. In order to be in better agreement with for the SI mode, new semi-relativistic degrees of freedom may need to come into equilibrium with the neutrino bath after BBN is complete Berlin et al. (2019).

## Appendix B Dirac Neutrino Thermalization

If neutrinos are Dirac fermions, their right-handed (RH) components must not come into equilibrium with the Standard Model plasma before , since they would give during nucleosynthesis, which is clearly incompatible with the results of App. A. In this section we derive the conditions for RH neutrino thermalization.

In the Standard Model, RH neutrinos (neutrinos with the “wrong” helicity) can be created in any interaction that produces LH neutrinos, i.e. weak interactions. The characteristic production rate of RH neutrinos in scattering reactions is  Dolgov et al. (1995). If weak interactions are the only interactions producing neutrinos, then RHNs do not thermalize as long as , which is comfortably satisfied in the Standard Model. However, the interaction introduced in Eqs. (1) and (2) is many orders of magnitude stronger than its weak counterpart, and the resulting RH neutrino production rate is much larger.

Thermalization is described by a Boltzmann equation of the form

 ˙ρνR+4HρνR=C, (21)

where is the energy density in “wrong helicity” neutrinos and is the collision term encoding the processes that produce these neutrinos. Thermalization/equilibration occurs when . The dominant process responsible for RH neutrino production is the decay , where () is a helicity label corresponding to RH (LH) neutrinos; scattering reactions such as are suppressed by an additional factor of . We therefore evaluate the collision term for :

 C≈2∫dΦ3EνR|M(λ=+1/2)|2fϕ(1−fν), (22)

where is the Lorentz-invariant phase-space (including the momentum conservation delta function) and , are the phase-space distributions of and LH neutrinos. We have neglected the inverse decay contribution and the RH neutrino Pauli-blocking factors. These approximations are adequate for estimating the onset of equilibrium, assuming the initial abundance of RH neutrinos is negligible, i.e. . We have also multiplied the right-hand side by two to account for decays which can also produce wrong helicity neutrinos.

We use the Lagrangian in Eq. (3) (with a complex for a Dirac neutrino) to evaluate the matrix element squared in the plasma frame, without summing over one of the helicities Dreiner et al. (2010)

 |M(λ)|2 ≈g2ϕ⎧⎨⎩Eν2EνR(1+cosθ)m2νλ=+1/2m2ϕλ=−1/2, (23)

where is the angle between the and direction of motion, and we have kept only the leading terms in . Note that when , the “wrong” helicity amplitude vanishes at this order as a result of angular momentum conservation (the next term is ). Using this result in Eq. (22), we find that

 C≈g2ϕm2ν32π(2ζ(3)T3π2)C+(mϕ/T), (24)

where the factor in parentheses is the number density of relativistic and the function is for and becomes Boltzmann-suppressed for .

We can apply the nucleosynthesis bound as computed in Appendix A if thermalization occurs before . By using the approximate thermalization criterion below Eq. (21), we find that RH neutrinos thermalize before BBN if

 gϕ≳5×10−3(max(mϕ,Tdec)MeV)3/2(0.1eVmν). (25)

This bound (evaluated using the full numerical ) is shown in Fig. 2 as a black dashed line.

## Appendix C Supernova 1987A

A new weakly coupled particle can change the behavior of the neutrino emission that was observed from the explosion of Supernova 1987A. The proto-neutron star cooling phase that was observed in large water Cherenkov detectors was qualitatively similar to the Standard Model-only expectation Burrows and Lattimer (1987). If a new particle species carried away too much energy during the proto-neutron star cooling phase, the time over which neutrinos arrived would have been unacceptably reduced Burrows et al. (1989, 1990). A semi-analytic criterion that the luminosity of this particle should obey is at times of order 1 second after the core bounce Raffelt (1996). Following the procedure described in more detail in Chang et al. (2017), we have

 Lϕ=∫Rν0dV∫d3k(2π)3ωΓprodϕexp(−∫RgrΓabsϕdr′), (26)

where: the has four-momentum ; is the absorptive width; is the production rate, which is related to the absorptive width in equilibrium by ; is the radius of the neutrinosphere, outside of which neutrinos free-stream; and is the radius inside of which neutrinos gain energy on average in elastic scattering events. We calculate including decay and annihilation to neutrino pairs, both of which are important for the masses of interest. We have not included contributions from the neutrino effective potentials, which may be significant at small  Brune and Ps (2018). We have also neglected neutrino Pauli blocking in , which is important near the proto-neutron star core, since this will become unimportant between and .

We find that given by Eq. (26) exceeds if is roughly in the range

 gexclϕ≃(5×10−6−6×10−5)×11+mϕ/keV. (27)

The sharp change in the shape of the bound at is due to the fact that rate of decay and inverse decay, , becomes subdominant to the annihilation rate, , for masses , where is the core temperature. These bounds are approximately compatible with those shown in Brune and Ps (2018) at masses above 10 keV. From Eq. (27), we see that bounds arising from the luminosity of particles from Supernova 1987A are generally below the coupling range of interest in this work.

It is also interesting to understand the constraints on arising from deleptonization of the core, which have been obtained in the mass range in Brune and Ps (2018) and which approximately overlap the range in Eq. (27). At lower masses, this likely has an effect on the early phases of collapse and the collapse progenitor. Such a possibility was suggested in Kolb et al. (1982) and was studied in the aftermath of Supernova 1987A by Fuller et al. (1988). This latter study found a constraint , neglecting any -dependence. Because these bounds are determined by physics at the beginning of the core collapse, when temperatures are , the bound likely cuts off at , similar to the bounds cited above. A detailed study is of interest, but beyond the scope of this work.

## References

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