Dark Radiation from Inflationary Fluctuations

Light new vector bosons can be produced gravitationally through quantum fluctuations during inflation; if these particles are feebly coupled and cosmologically metastable, they can account for the observed dark matter abundance. However, in minimal anomaly free $U(1)$ extensions to the Standard Model, these vectors generically decay to neutrinos if at least one neutrino mass eigenstate is sufficiently light. If these decays occur between neutrino decoupling and CMB freeze out, the resulting radiation energy density can contribute to $\Delta N_{\rm eff}$ at levels that can ameliorate the Hubble tension and be discovered with future CMB and relic neutrino detection experiments. Since the additional neutrinos are produced from vector decays after BBN, this scenario predicts $\Delta N_{\rm eff}>0$ at recombination, but $\Delta N_{\rm eff} = 0$ during BBN. Furthermore, due to a fortuitous cancellation, the contribution to $\Delta N_{\rm eff}$ is approximately mass independent.


INTRODUCTION
Cosmological inflation elegantly accounts for the observed flatness, isotropy, and homogeneity of the universe. Additionally, the quantum mechanical fluctuations in the inflaton field during inflation generate a nearly scale invariant spectrum of density perturbations that seed the growth of structure and imprint temperature anisotropies onto the cosmic microwave background (CMB) -see Ref. [1] for a review.
It is well known that new, feebly coupled particles are produced gravitationally through quantum fluctuations during inflation if their masses are small compared to the inflationary Hubble scale H I [2]; heavier particles can also be produced if the inflaton undergoes rapid oscillations [3][4][5][6][7][8] or nontrvially affects the particle's mass during inflation [9]. For light spin-0 particles, these fluctuations yield isocurvature perturbations on large scales, which are tightly constrained by CMB observations [10,11] and for spin 1 /2 fermions, inflationary fluctuations are generically suppressed unless they have non-conformal interactions through higher dimension operators [12][13][14].
It has recently been shown that the gravitational production of spin-1 particles during inflation is sharply peaked at modes that re-enter the horizon after inflation when the Hubble scale equals the vector's mass, H = m [15]. Such scales are typically much smaller than those probed by CMB experiments, so the isocurvature bounds on this scenario are negligible and this mechanism yields a viable dark matter candidate for m ∼ µeV 10 14 GeV Thus, if the vector is decoupled from Standard Model (SM) fields or is sufficiently light (m 2m e ) and interacts only through a small kinetic mixing, its cosmological metastability is generically realized. 1 However, if the vector is the gauge boson of a minimal U (1) gauge extension, couplings to neutrinos are required for anomaly cancellation [19]; the only anomaly free groups with no additional SM charged fermions are where B/L is baryon/lepton number, i, j = e, µ, τ are lepton flavor indices, and the corresponding gauge bosons in these models couple to at least one neutrino flavor. Thus, unlike kinetically mixed dark photon scenarios, the vector decays in these models can be relatively prompt and have observable cosmological consequences.
In this paper, we consider the fate of light gauge bosons V produced during inflation. We assume these vectors couple feebly to neutrinos and that at least one neutrino mass eigenstate is sufficiently light to allow V →νν decays. If such decays occur after neutrino decoupling, but before CMB photon decoupling, there is an irreducible contribution to ∆N eff that is potentially observable with future CMB-S4 experiments [20] and a modified relic neutrino spectrum observable at PTOLEMY [21,22]. Furthermore, such a contribution of ∆N eff can alleviate the discrepancy between early and late time measurements of the Hubble constant (for recent reviews see [23,24]).

STABLE VECTOR ABUNDANCE
The general lagrangian during inflation contains 1 For a kinetically mixed V , allowed decays V → 3γ are highly suppressed [16,17] and if the vector kinetically mixes with SM hypercharge before electroweak symmetry breaking, decays to V →νν are further suppressed by powers of ∼ (m/m Z ) 4 [18].

arXiv:2006.13224v2 [hep-ph] 7 Jun 2021
where V is a gauge boson in an FRW metric, F µν is the corresponding field strength tensor, andg is the metric determinant. If the mass satisfies 0 < m H I and V is stable, the longitudinal mode 2 is gravitationally produced during inflation and constitutes a present-day dark matter fraction f 0 where M Pl = 1.22 × 10 19 GeV is the Planck mass and T eq = 0.75 eV is the temperature of matter-radiation equality, so the energy density at earlier times is where Ω dm = 0.24 is the fractional dark matter abundance, ρ cr = 4.1 × 10 −47 GeV 4 is the critical density, a is the FRW scale factor, t 0 = 13.8 Gyr, and a 0 label represents a present day quantity [25,26]. For stable vectors, Eq. (5) is valid for t > t = (2m) −1 , the horizon re-entry time corresponding to H = m and temperature where g is the effective number of relativistic SM species in equilibrium. Note that because the V power spectrum is dominated by momentum modes that re-enter the horizon when H ∼ m, the V population is nonrelativistic for all times t > t .

ADDING DECAYS TO NEUTRINOS
Since abelian gauge extensions to the SM generically feature neutrino couplings, we add the representative interaction to Eq. (3), where g 1 is a gauge coupling and i is a lepton family index. In the massless neutrino limit, the partial width to a single flavor is [27] the total width Γ V is the sum of all allowed channels and τ V = Γ −1 V is the V lifetime. We note that a single massless neutrino eigenstate is empirically viable [28,29], so, in principle, at least one decay channel is allowed for all vector masses. Unlike in Ref [15], here the vector is unstable and V → νν decays deplete the initial population, so Eq. (5) is only useful for establishing the initial condition for ρ V at t = t . Accounting for decays to neutrinos, the V population can now be written and the energy density of the modified neutrino population δρ ν evolves according to which can be integrated to yield where a is the FRW scale factor and t ν ∼ 1 sec is the time of neutrino decoupling; we only keep contributions for t > t ν because neutrinos injected before t ν thermalize with the radiation bath and do not contribute to dark radiation. Similarly, V that decay after CMB decoupling will not contribute to ∆N eff , but will increase the dark matter density during recombination. In Fig. 3 we show a representative solution of Eq. (10) plotted as a fraction of the total energy density.
In terms of the equivalent number of SM neutrinos ∆N eff , this additional radiation from δρ ν predicts where ρ γ = π 2 T 4 /15 and the contribution is evaluated at the temperature of photon decoupling, T cmb ≈ 0.2 eV; this sets the upper integration range in Eq. (11) since V decays after last scattering do not contribute to dark radiation in the CMB data set. For the full parameter space, ∆N eff in Eq. (12) must be computed numerically by solving Eq. (11). However, if V decays between T ν,dec and T eq , the decay temperature can be written where g ≈ 3.36 in our temperature range of interest between decoupling and recombination. Assuming instantaneous V →νν decay and approximating δρ ν ≈ ρ V (T decay ) using Eq. (5), Eq. (12) becomes where the vector mass has canceled. Here ρtot = 3M 2 Pl H 2 /8π is the total energy density of the universe and we show ρV , the density of vectors from inflationary production, δρν the additional neutrino density from V →νν decays assuming a single neutrino flavor. From left to right, the vertical dashed lines mark neutrino decoupling, matter radiation equality, and CMB decoupling. Note that in Eq. (18), the number density of the neutrino population from V decays might exceed that of the relic neutrino background, but as seen here, the energy density remains small for empirically viable values of ∆N eff .
In Fig. 2 we show ∆N eff predictions for the inflationary vector population where we compute δρ ν numerically using Eq. (11). The blue horizontal bands represent the currently viable 10 −2 ≤ ∆N eff < 0.5 range that is within the reach of CMB-S4 predictions [20]. Note that current BBN bound ∆N eff < 0.5 [30] is less stringent than the CMB and large scale structure bound ∆N eff < 0.28 [25], but the BBN limit is less model dependent because it is not as sensitive to the choice of cosmological model. However, despite the nominal choice of ∆N eff < 0.5 as our conservative exclusion benchmark, this scenario is not directly constrained by the BBN measurement of ∆N eff since the additional neutrinos from V decays do not appear until after BBN.
The area in between the dashed diagonal bands represent parameter space for which V →νν decays occur between neutrino and CMB decoupling; decays outside this band do not contribute to ∆N eff . The vertical lines at m = 2m e , 2m µ represent regions where the ∆N eff prediction here does not apply if V couples to electrons or muons; in such models, V decays to charged particles after neutrino decoupling will heat photons and thereby reduce ∆N eff relative to Eq. (14).
We note for completeness that there is also a possible contribution to ∆N eff from the V population itself if an appreciable fraction of the ρ V redshifts like radiation at recombination. Since inflationary V production FIG. 2. Parameter space that yields observable levels of dark radiation from a population of gravitationally produced vectors that decay via →νν after neutrino decoupling but before recombination. Horizontal blue shaded bands represents regions where 10 −2 < ∆N eff < 0.5 for representative choices of the inflationary Hubble scale HI ; for each choice, the parameter space below the bottom boundary predicts ∆N eff > 0.5, which is excluded assuming otherwise standard cosmological assumptions [25,30,32]. Above the horizontal dotted lines, V thermalizes with the SM, yielding ∆N eff ≈ 2.5 [30], which is excluded if V couples to e or µ. The vertical dotted lines mark m = 2me,µ where V → e + e − and V → µ + µ − decays are kinematically allowed. Most models in Eq. (2) feature V -e couplings, so for m > 2me the ∆N eff ≈ 0 as V → e + e − decays heat photons to compensate for V →νν decays, which heat neutrinos.
is sharply peaked around modes that enter the horizon at H ∼ m, from Eq. (6) only masses below m 10 −30 eV will be quasi relativistic around T cmb . However, from Eq. (4) such small masses yield negligible inflationary production for all H I 10 14 GeV allowed by CMB limits on tensor modes [25,31], so we can safely neglect this contribution.

INTERACTIONS WITH THE SM PLASMA
The above discussion assumes that the early universe V population arises entirely to inflationary production and is unaffected by the SM radiation bath. However, for any value of the gauge coupling, there is irreducible sub-Hubble "freeze-in" production of additional V [27, [33][34][35] and, if the coupling is sufficiently large, the V population can thermalize with the SM plasma; which yields additional contributions to ∆N eff .

• Inverse Decays
Independently of any other assumptions about ultralight V partilces beyond their coupling to neu-trinos, there is a bound on thermalizing with the SM plasma via population viaνν ↔ V decays and inverse decays. If thermalization occurs before neutrino decoupling, this scenario predicts ∆N eff ≈ 2.5, so avoiding this fate requires where T ν,dec ∼ MeV is the temperature of neutrino decoupling via the SM weak interactions. If, instead, thermalization occurs between T ν,dec and T cmb as in Ref. [36], then ∆N eff ∼ 0.2 independently of mass and coupling [27]. 3 Since this contribution is fixed only by the neutrino coupling, it must be added to the component from the inflationary population.

• Production From Charged Particles
If V also couples to charged fermion f , dangerous f f → γV and f γ → f V processes can thermalize V with the SM radiation bath, thereby yielding ∆N eff ≈ 2.5, which is excluded by both BBN and CMB observables [25,27,30,32]. 4 The V production rate can be estimated as Γf f →V γ ∼ Γ f γ→f V ∼ αg 2 T /4π, so these processes grow relative to Hubble until T ∼ m f , when they become Boltzmann suppressed. Ensuring that the maximum rate not exceed Hubble expansion requires g 4π √ g m f αM Pl = 5 × 10 −10 , f = e 7 × 10 −9 , f = µ , (16) where g is evaluated at T = m e , m µ , respectively. The stronger electron based bound here applies to most anomaly free U (1) extensions -including gauged B−L, B−3L e , L e −L µ , L e −L τ -as they all require V to couple to electrons for anomaly cancellation [19]; the main outlier is gauged L µ − L τ for which muon induced thermalization is the dominant process at low temperatures [27], so the bound is somewhat weaker. Both of the requirements in Eq. (16) are presented in as dotted horizontal black curves in Fig. 2 and the parameter space above these regions is excluded if the model in question features the corresponding e or µ coupling.

FIG. 3.
Present day neutrino flux spectra from V →νν decays for representative benchmark points (dashed). Also shown are spectra from the CνB, primordial neutron decays during BBN (n → ν), tritium decays during BBN (T → ν), and solar neutrinos [21]. From Eq. (21), is clear that the early decaying parameter points (green and red) only yield appreciable ( few) events at PTOLEMY for lower values of mV , which are difficult to distinguish from the CνB spectrum, but might be detected as an enhancement over the Standard Model signal rate. As the mass is increased, the spectrum gets harder, but the rate becomes unobservable with a feasible exposure; for the 50 keV benchmark, we find R ∼ 10 −3 events/yr at PTOLEMY. For later decaying particles (blue dashed curve) the rate and spectrum can be favorable, but there is no contribution to ∆N eff .
We emphasize that the parameter space shown in Fig.  2 is extremely weakly coupled, such that there is no danger of the inflationary V population thermalizing with the SM plasma, or of any appreciable contributions from SM processes that produce additional V particles via inverse decays or SM scattering reactions. For a careful study of such processes in the context of the models studied here, see [27] which identifies the parameter space where freeze in production via inverse decays contributes to cosmological observables including ∆N eff .

PRESENT DAY NEUTRINO FLUX
In this section we review the results of Ref. [21], adapted to the case of inflationary vector production. The neutrinos in our scenario arises from V decays and if the entire population decays in the early universe, the present day number density is If these decays occur between neutrino decoupling and recombination, Eq. (17) can be rewritten [21] δn ν (t 0 ) ≈ 10 3 cm −3 ∆N eff 0.28 eV m Although the number density of additional neutrinos in Eq. (18) can exceed the ∼ 300 cm −3 number density of the cosmic neutrino background (CνB) as predicted in the Standard Model, as long as the corresponding value of ∆N eff satisfies observational bounds, the energy density of this population is always subdominant and remains empirically viable. For some parameter choices, this additional neutrino population may be observable with the PTOLEMY experiment using inverse beta decay reactions from captured relic neutrinos [22]. Assuming a detector target mass of M T , electron neutrino fraction f νe , neutrino capture cross section on tritium σ = 3.83 × 10 −45 cm 2 , and the excess neutrino density from Eq. (18), the signal rate is estimated to be [21] R ≈ 5 yr which only assumes that the V decay after decoupling. However, for V that also decay before recombination, the fraction satisfies [21] f 0 so the rate for early decaying V can be written which may be detectable with a year of exposure at PTOLEMY, whose projected CνB sensitivity is at the ∼ 10 event level. Note that there is general tension between having an appreciable ∆N eff signal and having a distinguishable neutrino spectrum with a detectable PTOLEMY rate. To see this, note that the late time flux of neutrinos from V decays is where E ν is the observed energy of a present day neutrino emitted at redshift z with energy E ν (1 + z) = m V /2, H(z) = H 0 Ω Λ + Ω m (1 + z) 3 + Ω r (1 + z) 4 is the Hubble rate, H 0 = 67 km/sec/Mpc [25], and z c = [m V /(2E ν )] − 1. The theta function ensures that decays before neutrino decoupling do not contribute to the flux; this population will thermalize with CνB. In Fig. 3 we show representative flux spectra for both early (t ν,dec < τ V < t cmb ) and late time (τ V > t cmb ) decaying populations. From Eq. (21), early decaying V with low ∼ few eV masses can yield appreciable PTOLEMY signal rates, but as we see in these spectra, the fluxes similar to the Cν unless m V much greater, which trades off against the overall rate as R ∝ m −1 V . It is possible to get an appreciable PTOLEMY flux for a low mass particle, but obtaining a distinctive spectral shape requires late time decays, which do not affect ∆N eff .

CONCLUSION
In this paper we have studied the fate of massive vector particles produced gravitationally from inflationary fluctuations. If these vectors only interact with the SM via kinetic mixing, for m < 2m e , the only allowed decay is V → 3γ which is sharply suppressed, so V is generically metastable can serve dark matter candidate [15]. However, if the vector arises in well motivated, minimal U (1) gauge extensions from Eq. (2), it must couple to neutrinos, so if at least one neutrino mass eigenstate is sufficiently light, V →νν decays can efficiently deplete this inflationary population and increase the relic neutrino densty, thereby predicting ∆N eff = 0. For certain regions of parameter space, the same neutrino population may be observable at late times with the PTOLEMY experiment; for long lived vectors that decay after recombination, it is also possible to obtain an appreciable PTOLEMY signal even though ∆N eff = 0.
Intriguingly. due to a cancellation, this contribution depends only on H I and g as long as the V lifetime falls within this time window. For a wide range of model parameters, the ∆N eff prediction in these scenarios is within reach of CMB-S4 projections [20]. We note that, outside of the narrow parameter region where 50 keV T decay MeV, this scenario predicts ∆N eff = 0 only in CMB data because nearly all of the V decays occur after BBN has completed; decays before BBN thermalize with the SM, so T ν /T γ does not deviate from the SM prediction. However, for parameter space in this decay occurs after recombination, the resulting neutrino population may be observable directly at PTOLEMY [21,22].
Furthermore, since the mechanism studied here is sensitive to the Hubble scale during inflation, future measurements of inflationary B-modes at CMB-S4 experiments will have important implications for this class of scenarios. The forecasted sensitivity to the scalar-totensor ratio r ∼ 10 −3 implies a sensitivity to H I ∼ 10 12 GeV [37], which yields observable ∆N eff from V decays in the upper half of Fig. 2.
Finally, it has been shown that contributions to ∆N eff ∼ 0.5 may play an important role in alleviating the discrepancy between early and late time determinations of the Hubble tension [23,24]. Although models with nonzero ∆N eff do not completely eliminate the tension, it is intriguing that the contributions required to reduce its statistical signficance are readily accommodated in the parameter space studied in this class of models.