Flavor-specific Interaction Favors Strong Neutrino Self-coupling in the Early Universe
PPrepared for submission to JCAP
SLAC-PUB-17547
Flavor-specific Interaction Favors Strong
Neutrino Self-coupling
Anirban Das a and Subhajit Ghosh b,c a SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA. b Department of Physics, University of Notre Dame, South Bend, IN 46556, USA. c Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India.E-mail: [email protected], [email protected]
Abstract.
Flavor universal neutrino self-interaction have been shown to ease the tension in themeasurements of Hubble constant between the early and late Universe data. We introduce aself-interaction structure that is flavor-specific in the three active neutrino framework. This ismotivated by stringent constraints on new secret interactions among electron and muon neutrinosfrom several laboratory experiments. Our study indicates the presence of a strongly interactionmode which implies a late-decoupling of the neutrinos just prior to matter radiation equality.Using the degeneracy of the coupling strength with other cosmological parameters, we explainthe origin of this new mode as a result of better fit to certain features in the CMB data. We findthat if only one or two of the three active neutrino flavors are interacting, then the statisticalsignificance of the strongly-interacting neutrino mode increases substantially relative to theflavor universal case. However, the central value of the coupling strength for this interactionmode does not change by a appreciable amount in flavor-specific cases. a r X i v : . [ h e p - ph ] N ov ontents + ) 124.2 Flavor-specific scenario: Two-coupled ( + ) & one-coupled states ( + ) 194.3 Effect on H and σ Neutrino remains to be the most elusive particle in the Standard Model (SM) even after sixty-four years of its discovery. Even though in the SM, neutrinos are predicted to be massless,the observed phenomena of neutrino oscillation predicts at least one state of the neutrinos tohave a mass (cid:38) .
03 eV [1, 2]. Neutrino oscillation requires mixing between different flavors,which implies that their mass matrix is non-diagonal in the flavor basis. Several neutrinooscillation experiments have now measured the neutrino mixing angles to few percent accuracylevel. Terrestrial beta decay experiments probing the absolute masses of neutrinos, however, isyet to reach the sensitivity of that of the oscillation experiments [3–5]. In future, beta decayexperiment KATRIN is expected to reach a sensitivity of 0 . .
12 eV through its dependence on the matter power spectrum [7].CMB and Big Bang Nucleosynthesis (BBN) measurements of the effective number degrees offreedom in neutrinos in the vanilla ΛCDM cosmology agrees with the theoretical prediction ofthe SM. Interestingly, the constraints of neutrino properties from the cosmological data are alsovery sensitive to the underlying cosmological model. In other words, precision cosmologicalmeasurements are sensitive probes for beyond Standard Model (BSM) interactions of neutrinos.– 1 – − − − − [ G eff / MeV − ] P18-TTTEEE+lowE+Lensing + + + Figure 1 : The 1D posteriors for log ( G eff / MeV − ) in the three scenarios- + , + ,and + with three, two, and one self-interacting neutrinos, respectively. The two-modefeature is present in all cases. The relative significance of the larger G eff mode increases withless number of interacting neutrinos, hence, is largest in the + scenario. See section 4 formore details.New physics such as a secret interaction among the neutrinos is difficult to probe in terrestrialoscillation experiments because of their very feeble interaction strength. Whereas, the denseenvironment in the early Universe can amplify the effects of any secret interaction in the neutrinosector due to large number density.Several BSM interaction scenarios have been invoked in the neutrino sector to explainseveral experimental measurements and observations, e.g., anomalous neutrino signal in short-baseline experiments [8, 9], discrepancy between the CMB and local measurement of the Hub-ble parameter [7, 10] etc. One of the proposals to address these observations/tensions is self-interacting neutrino (SINU) [11–18]. The deviation of the neutrino sector from its vanilla free-streaming nature have been shown to affect various cosmological observables. In the earlyuniverse, the dense neutrino environment gives us an opportunity to test these SINU mod-els [19–21]. Several studies have been done on the effects of self-interaction among neutrinoson CMB and large scale structure [22–30]. Most of these studies are based on a parametriza-tion of the self-interacting neutrino sector using c eff and c vis . However, with the availabilityof more precise experimental data, it is now important to model the SINU using a realisticparticle physics-based approach. Previous studies have also shown that the neutrino flux froma supernova will be sensitive to new interactions among the neutrinos [31, 32]. However, thestrongest constraints on the parameter space of the SINU models come from K -meson decay,double beta decay, invisible width of the Z -boson, and τ -decay experiments [33–36]. Therefore,modelling a BSM SINU scenario that can address the above anomalies, while respecting thephenomenological constraints, is a challenging task [37, 38].As alluded earlier, self-interacting neutrinos have been suggested to ease a crisis of moderncosmology - the so-called Hubble tension. Since the early days of the Planck experiment, a– 2 –iscrepancy in the value of the Hubble parameter from CMB observations and local (low redshift)measurements have become a topic of active research [39]. This discrepancy has been exacerbatedwith the latest data from the Planck experiment and the local H -measurement using cepheidsand supernovae [7, 10, 40, 41]. The Planck-measured value of the Hubble constant using thetemperature, polarization and lensing CMB data is H = 67 . ± .
54 km s − Mpc − , whereasthe SH0ES collaboration found H = 74 . ± .
42 km s − Mpc − . There is now a 4 . σ tensionbetween the two values of H . However, an independent local measurement of H was done inRef.[42] which found a value H = 69 . ± . − Mpc − . Apart from unknown experimentalsystematics, this discrepancy could point towards a new physics signature beyond the ΛCDMparadigm. Several such scenarios have been envisaged to address this tension [15, 16, 18, 43–67]. Among them, Ref. [15, 16] introduced new secret interaction between Standard Model activeneutrinos mediated by a new heavy scalar φ . A flavor-universal four-Fermi interaction with acoupling strength G eff was assumed, L ⊃ G eff ¯ νν ¯ νν, G eff ≡ g φ M φ , (1.1)where g φ is the coupling between ν and the mediator φ , and M φ is the mass of φ . A Bayesiananalysis of this model with the CMB data prefers two distinct regions of G eff values: stronglyinteracting (SI) mode with a large value log ( G eff / MeV − ) (cid:39) − . +0 . − . (68% confidencelimit), and the moderately interacting (MI) mode with an upper bound log ( G eff / MeV − ) < − .
57 at 95% confidence level using the Planck 2015 and BAO data. Note that the SI modecoupling strength is about 10 times stronger than the weak interaction Fermi constant G F =1 . × − MeV − .However, SINU scenario meets strong constraints from the laboratory experiments [34, 36].Ref. [34] showed that a simple model of flavor-universal SINU scenario is ruled out by laboratoryconstraints from meson decay, τ decay, and double beta decay. Whereas, Ref. [36] demonstratedthat Standard Model Effective Field Theory (SMEFT) operators, giving rise to flavor universalSINU scenario, are also highly constrained by meson decays, Z decays and electroweak precisionmeasurements etc. These studies also derive flavor-dependent SINU constraints where only oneof the neutrinos is interacting. The constraints in those cases are very stringent for ν e , whereasfor ν µ and ν τ the constrains are comparatively weaker.In this paper, we study flavor-specific neutrino self-interaction scenario using latest cosmo-logical data. We perform a detailed Bayesian analysis of SINU with flavor-specific interactionsfor the first time using the latest 2018 likelihood from the Planck collaboration. As such, weallow the coupling strengths for different neutrino flavors to be different from each other. Ourgoal is to complement the flavor-specific SINU studies from laboratory experiments using thecosmological data. However, there is a notable distinction between the framework of SINUstudies in cosmology vs laboratory searches. In laboratory experiments, the flavor of neutrinocan be identified via electroweak interaction. Whereas, cosmological observables such as CMBare only sensitive to the temperature, free-streaming properties and the total mass of the neu-trinos. Therefore, in cosmology, we cannot distinguish any particular flavor of neutrino and canonly study their effects collectively. All three generation of neutrinos ν e , ν µ and ν τ are on equalfooting for our analysis. – 3 –n this work, we considered only three massless SM neutrinos, and therefore, fixed N eff =3 .
046 ( N eff ≈ .
015 for each flavor). Massless neutrinos are a fairly good approximation as thebound on the total mass of the neutrinos are an order of smaller compared to the temperatureof the plasma at the last scattering surface. The effect of massive neutrinos is reasonablywell understood and can be speculated from our results. We consider three possible scenariosdepending on the interactions of the three neutrino species – 3-coupled ( + ), 2-coupled + 1-free-streaming ( + ), and 1-coupled + 2-free-streaming ( + ) respectively. We assumedsame coupling strengths for the coupled species for the first two cases. This is justified becausethe coupled neutrino species are completely equivalent due to the massless approximation. Notethat, the + case is identical to the universal flavor coupling scenario which has been studiedpreviously.Our main result is shown in figure 1. We find that the flavor-universal case yields a lowsignificance for the SI mode with the Planck 2018 data. However, the SI mode significance isdrastically increased once the flavor-universality of the coupling strength is relaxed, becomingmaximum when only one neutrino state is self-interacting. We also find that the SI modesignificance even surpasses the MI mode in certain cases. We explain the origin of the SI modeas a results of a better fit of certain features of the CMB data, compared to ΛCDM, usingdegeneracy of G eff with other cosmological parameters. When only one or two neutrino statesare self-interacting, the resulting changes are smaller compared to the scenario when all threeneutrinos are interacting and hence, can be compensated by the other correlated parametersrelatively easily. This results in substantial enhancement of the significance of SI mode in + and + cases. We also find that the SI mode best-fit value of G eff has a milddependence on the number of interacting species.The outline of the paper is as follows. In section 2, we describe our cosmology model andexplain the rationale behind it. In section 3 we detail our methodology and the experimentallikelihoods used in this work. We show and interpret our results in section 4, and conclude witha discussion in section 5. We consider scalar interactions between massless ν and φ below the electroweak scale as L ⊃ g ij φ ¯ ν i ν j , (2.1)where g ij is the coupling between φ and the neutrino flavors i and j . When the temperature ofthe neutrino bath ( T ν ) cools down below the mass of φ , i.e., T ν (cid:28) M φ , a four-Fermi interactionsamong the neutrinos, similar to Eq.(1.1), is generated as follows L ⊃ G ( ijkl )eff ¯ ν i ν j ¯ ν k ν l , G ( ijkl )eff ≡ g ij g kl M φ . (2.2) Note that the mediator could also be a vector particle which would change the details of the interaction, butthe phenomenological aspect of the model remains the same. – 4 –ere we note that for a general interaction of φ with any two flavors of neutrino, the four-Fermiinteraction strength G ( ijkl )eff in Eq.(2.2) has four indices for a process like ν i + ν j → ν k + ν l .Therefore, the most general scenario would involve many different couplings for different flavorcombinations. In such a case, the energy and momentum of individual neutrino species will notbe conserved and one will need to incorporate energy and momentum transfer between differentspecies in the perturbation equations [17]. However, in this work we limit ourselves to a simplerscenario as described below. We consider only diagonal interactions in the flavor space withdifferent coupling strengths: L ⊃ G ( i )eff ¯ ν i ν i ¯ ν i ν i . (2.3)This assumption introduces only three new SINU parameters: G (1)eff , G (2)eff and G (3)eff . As mentionedearlier, in this work we fix the number of neutrino flavors to three with all of them having asame temperature T ν . However, because of the complete equivalence of the interacting states inthe context of CMB, we need to consider only one common coupling parameter G eff for all theinteracting states for a given scenario.We also assume M φ > T ∼ T ν = M φ when φ cannot be produced from scattering of the neutrinos. However for M φ > T <
100 eV) relevant to the present analysis. The annihilationand decay of φ into neutrino increases the temperature of the neutrino bath, but this is model-dependent and we do not take this extra heating into account in this work. Using dimensional analysis, the thermally averaged scattering cross section between the i -th neutrino state goes as (cid:104) σv (cid:105) ∼ ( G eff ) T ν . Therefore the interaction rate scales as Γ ν ≡ n ν (cid:104) σv (cid:105)∼ ( G eff ) T ν because of the T ν -scaling of the neutrino number density. We absorb all other model-dependent prefactors of the interaction rate into G eff . The comoving neutrino self-interactionopacity ˙ τ ν is defined as ˙ τ ν = − a ( G eff ) T ν , (2.4)where a is the scale factor of the Universe. Note that the T ν -scaling of the opacity is a char-acteristic of the four-Fermi interaction. The neutrinos are self-interacting when the interactionrate ˙ τ ν is greater than the comoving Hubble expansion rate aH , i.e., ˙ τ ν > aH . The interactionfreezes-out when ˙ τ ν drops below the Hubble expansion rate at a redshift z dec which is given by1 + z dec (cid:39) . × (cid:18) G eff − MeV − (cid:19) − . (2.5)In the above equation, we have set other background cosmological parameters to their ΛCDMbest-fit values and assumed the decoupling to take place in radiation domination era. We showthe relative strength of the neutrino self-interaction to the Hubble expansion rate in figure 2.From figure 2, we see that the neutrinos are interacting with each other until z (cid:39) G eff = 10 − . MeV − which is much later than SM neutrino decoupling ( z (cid:39) ). For the MI Although, we note that in a concrete model, the heating due to φ leads to a larger N eff which could helpincrease the Hubble parameter to some extent. – 5 – z − − | ˙ τ ν | aH S I : G e ff = − . M e V − M I : G e ff = − . M e V − G e ff = G F ν - d ec o up li n g i n Λ C D M M a tt e r - R a d i a t i o n E q u a li t y ‘ H Figure 2 : Variation of the neutrino self-interaction opacity ˙ τ ν relative to the Hubble expansionrate H with redshift for G eff = 10 − . MeV − (dark blue, solid), 10 − . MeV − (light blue,dashed), and G F (light brown, dot-dashed). The chosen values of G eff are the best-fit values forSI and MI mode in + scenario with Planck temperature and polarization data. The topaxis shows the angular multipole (cid:96) H corresponding to the modes that enter horizon at redshift z . The epochs of neutrino decoupling in ΛCDM and the matter-radiation equality are shown asgray-shaded region.modes the decoupling happens at z > a few × . The neutrino decoupling for the SI modehappens very close to the matter-radiation equality which has interesting implications that willbe discussed later. The Boltzmann hierarchy of the perturbation equations for massless, self-interacting neutrinosin the Newtonian gauge is shown below following Refs. [15, 70].˙ δ ν + 43 θ ν − φ = 0 , ˙ θ ν + 12 k (cid:18) F ν, − δ ν (cid:19) − k φ = 0 , ˙ F ν,(cid:96) + k (cid:96) + 1 (( (cid:96) + 1) F ν,(cid:96) +1 − (cid:96)F ν,(cid:96) − ) = α (cid:96) ˙ τ ν F ν,(cid:96) , (cid:96) ≥ . (2.6)– 6 –ere α (cid:96) are (cid:96) -dependent O (1) angular coefficients that depend on the details of the neutrinointeraction model, and the anisotropic stress σ ν is related to F ν, as σ ν = F ν, /
2. Energy andmomentum conservation dictates α = α = 0. The values of α (cid:96) for (cid:96) ≥ α (cid:96) = 1 for (cid:96) ≥ CLASS and solve them numerically [71, 72]. For very large value of the coupling G eff when | ˙ τ ν | (cid:29) aH , this system of equations may become difficult to solve as the equations becomestiff, and tight-coupling approximation may be necessary. However, we checked that the default ndf15 integrator in CLASS is able to solve the equations for log ( G eff / MeV − ) ≤ − . k -modein CLASS from a very high redshift ( z ∼ ). Because at z ∼ , | ˙ τ ν | (cid:29) ( aH ) for all SINUmodes including the MI mode, we can safely set initial anisotropic stress to zero for all k values.We also modify other initial conditions accordingly. However, we found that the resulting SINUspectrum with these modified initial conditions differ very slightly ( (cid:46) . k mode, that starts with a non-zero value, vanishes veryquickly due to the strong self-interaction. Since the modifications due to the initial conditions arevery small compared to the precision of the Planck data and also makes the code comparativelymuch slower due to the additional integration time, we chose not to incorporate those for themcmc analysis. In this section, we shall discuss the changes in the CMB angular power spectra due to neutrinoself-interaction. The self-interaction stops the neutrinos from free-streaming before decoupling.As can be seen from Eq.(2.6), the new interaction plays the role of damping in the perturbationsfor (cid:96) ≥
2. Therefore, it impedes the growth of the anisotropic stress σ while the neutrinos arestrongly-coupled. We show the evolution of neutrino anisotropic stress σ ν in the top left panelof figure 3 in both SINU and ΛCDM cosmology. The important difference between them is theinitial suppression of F ν, in the SINU scenario caused by the new interaction. The suppressionis maximum for + where all three neutrinos are interacting, and gradually decreases for + and + where the number of interacting neutrino flavor is two and one respectively.The anisotropic stress is related to the gravitational potentials φ and ψ via the following equation, k ( φ − ψ ) = 12 πGa (cid:88) i = γ,ν ( ρ i + P i ) σ i (cid:39) πGa ρ tot R ν σ ν , (2.7) The modified
CLASS code is available at https://github.com/anirbandas89/class SInu. – 7 – − − − − a − . − . . . . σ ν ( k , a ) Free-streaming Not Free-streaming − − − − a − ψ ( k , a ) ΛCDM : k = 10 − c + 0 f : k = 10 − c + 1 f : k = 10 − c + 2 f : k = 10 − − − − − a − φ ( k , a ) − − − − a − − δ γ ( k , a ) − − − − − . . . ∆ δ γ ( k , a ) − − − − − . . . ∆ φ ( k , a ) − − − − . . . ∆ ψ ( k , a ) log [G eff / MeV − ] = − Figure 3 : Evolution of the neutrino anisotropic stress σ ν (top left), the gravitational potentials ψ (top right), φ (bottom left), and the photon over-density δ γ (bottom right) for the mode k = 0 . − for + (red, solid), + (blue, dotted), + (green, dashed), andΛCDM (black, solid). The white region in each plot, which is determined by | ˙ τ ν | / ( aH ) ≥
10, is approximately the region in scale factor a upto which neutrinos are tighly-coupled for G eff = 0 . − . From the top left plot we see that the σ ν is suppressed in the non-freestreaming region due to the self-interaction. The suppression is largest for + where all threeneutrinos are tightly coupled, and gradually decreases for + and + , respectively. Thesuppression of σ ν results in the enhancement of φ and ψ which in turn enhances δ γ . The insetsshow the absolute changes in the corresponding cases compared to ΛCDM.where ρ i , P i , σ i are individual energy density, pressure and anisotropic stress, respectively, forthe i -th species and ρ tot is the total energy density. In the last step of the above equation, wehave ignored small anisotropic stress of photon σ γ . Also, R ν is the fractional energy density of free streaming neutrinos which, in radiation domination, is given by R ν = ρ ν ρ ν + ρ γ . (2.8)In the radiation domination era R ν ∼ .
41 in ΛCDM. Therefore, the suppression of neutrinoanisotropic stress due to SINU during this period plays an important role in enhancing thegravitational potentials φ and ψ , as can be seen from figure 3. The gravitational potentials inturn affect the evolution of the photon perturbations as can be seen in the bottom-right panelof figure 3.The changes in the CMB power spectrum in figure. 4 can be understood as a result ofchange in the propagation speed for perturbation of the neutrinos as explained below following– 8 – D TT ‘ log [G eff / MeV − ] = − c + 0 f c + 1 f c + 2 f Planck-20180 500 1000 1500 2000 2500 ‘ . . . ∆ D TT ‘ / D TT , Λ C D M ‘ − D EE ‘ ‘ . . . . ∆ D EE ‘ / D EE , Λ C D M ‘ Figure 4 : CMB
T T and EE angular power spectra for + (red, solid), + (blue,dotted), and + (green, dashed) scenarios are shown in the top panels. The ΛCDM powerspectra are also shown for comparison in solid black. The parameters for the ΛCDM spectracorrespond to the best-fit points for the TT,TE,EE+lowE dataset. The bottom panels show therelative changes from the ΛCDM spectra. For SINU plots, we have set log ( G eff / MeV − ) = − c s (cid:39) / √ phase shift φ ν in the acoustic oscillations of the photon [73] , which in the radiation domination,is given by φ ν (cid:39) . πR ν . (2.9)The amplitude of the oscillation is also changed by a factor of (1 + ∆ ν ) where∆ ν (cid:39) − . R ν . (2.10)This results in a suppression of the CMB power spectrum in ΛCDM due to free streamingneutrinos for the modes that enter the horizon before matter-radiation equality. Therefore,these changes in the photon acoustic oscillations affect the CMB angular power spectra for (cid:96) (cid:38) (cid:96) , and suppresses the power spectra [73].This story is changed in the presence of self-interacting neutrinos. The self-interactionstops the neutrinos from free-streaming and delays the neutrino decoupling from the thermalbath until a later time ( z dec ) depending on the strength of the interaction. As a result, the– 9 –ree-streaming neutrino fraction R ν is decreased relative to its ΛCDM value depending on thenumber of neutrino species which are coupled at a certain time: R ν = R ΛCDM ν × , for + / , for + / , for + (2.11)Decreasing R ν in Eq.(2.9) and (2.10) readily implies a phase shift of the CMB spectrum towardslarger (cid:96) , and an enhancement of power relative to ΛCDM. The resulting changes in the CMB TTand EE spectra can be seen in figure 4. In the upper panels, we show D XX(cid:96) = (cid:96) ( (cid:96) + 1) C XX(cid:96) / (2 π )where XX = T T and EE in three SINU scenarios and ΛCDM. In the lower panels, we show thefractional changes of the spectrum relative to ΛCDM. We see that there is an enhancement inthe SINU spectra, and also a phase shift which shows up as wiggles in the fractional differenceplot. In compliance with the explanation above, both of these effects are maximal in + and gradually decreases as the number of self interacting states is decreased. Taking everythinginto account, we can see that the overall changes in the spectrum are milder when less numberof neutrinos are interacting, which allow these changes to be compensated relatively easily bychanging other parameters.A higher Hubble parameter can undo the change in the neutrino induced phase shift φ ν in SINU. This can be understood in terms of the photon transfer function cos( kr ∗ s + φ ν ) whichsources the acoustic peaks in the CMB spectrum. The CMB multipole corresponding to a mode k is given by, (cid:96) ≈ kD ∗ A = ( mπ − φ ν ) D ∗ A r ∗ s (2.12)where D ∗ A is the angular diameter distance, and r ∗ s is the sound horizon at recombination definedas, D ∗ A = (cid:90) z ∗ H ( z ) dz, r ∗ s = (cid:90) ∞ z ∗ c s ( z ) H ( z ) dz (2.13)where H ( z ) is the Hubble rate, and c s ( z ) ≈ / √ φ ν from its ΛCDM value moves the spectrum towards a higher (cid:96) value as can be seen from Eq.(2.12), which can be compensated by an increase in θ ∗ ≡ r ∗ s /D ∗ A .This shift in θ ∗ can be accommodated by changing D ∗ A via modifying H and Ω Λ which changesthe Hubble evolution at late times [18]. The increase in Hubble constant H is only relevant forSI mode where the self interaction strength is significantly strong inducing a large phase shift.In other words, higher values G eff is positively correlated with H . Note that, the increase of H for SI mode is maximum in + case where all three neutrinos interact producing largestphase-shift compare to ΛCDM. We performed a Bayesian analysis of this model using the latest version of the Markov ChainMonte Carlo (MCMC) sampler
MontePython3.3 [75, 76]. To analyze the MCMC chains and plot– 10 – able 1 : The nested sampling settings used in this work.Parameter ValueSampling efficiency 0 . . GetDist 1.1.2 software package [77]. As the typical pos-teriors in SINU are multimodal, we use
MultiNest interfaced with
MontePython to sample theparameter space and analyze the multimodal posteriors to separate the modes. The operationalsettings for
Multinest that we used in our analysis are shown in table 1 [78–80].To constrain the model, we used the Planck 2018 likelihoods for the temperature andpolarization power spectra [81]. Here‘TT+lowE’ denotes the combination of low- (cid:96)
TT ( (cid:96) < (cid:96)
EE and high- (cid:96) TT plik-lite ( (cid:96) >
30) likelihood, and ‘TTTEEE+lowE’ denotes thecombination of low- (cid:96)
TT, low- (cid:96)
EE and high- (cid:96)
TTTEEE plik-lite likelihood. In addition wealso use Planck 2018 lensing likelihood. For BAO, we used the 6DF Galaxy survey, SDSS-DR7MGS data, and the BOSS measurement of BAO scale and f σ from DR12 galaxy sample [82–84] . Finally for H data, we used the latest measurement of local Hubble parameter from SH0EScollaboration [41]. We use the following combinations of likelihoods in our analysis: ‘TT+lowE’,‘TTTEEE+lowE’, ‘TTTEEE+lowE+lensing’, ‘TTTEEE+lowE+lensing+BAO’, and ‘TTTEEE+lowE+lensing+BAO+ H ’. In table 2, we show the prior ranges used. We use a log-prior forthe extra parameter G eff because the expected features in its posterior distribution span overmany orders of magnitude. Also, we do not vary the relativistic degrees of freedom N eff in thisanalysis to disentangle the effect of only neutrino self-coupling on the CMB power spectra, fixit to N eff = 3 . In this work, we used the latest corrected version of the BAO likelihood implemented in
MontePython3.3 .See Ref. [85] for more details.
Table 2 : Prior ranges used in this work.Parameter PriorΩ b h [1 . , . c h [0 . , . θ s [2 . , . τ reio [0 . , . A s ) [2 . , . n s [0 . , . ( G eff / MeV − ) [ − . , − . Results
Table 3 : Parameter values and 68% confidence limits in + .Parameters TT+lowE TTTEEE+lowESI MI SI MIΩ b h . ± . . ± . . ± . . ± . c h . ± . . ± . . ± . . ± . θ s . ± . . ± . . ± . . ± . A s ) 2 . ± . . ± . . ± . . ± . n s . ± . . ± . . ± .
004 0 . ± . τ reio . ± . . ± . . ± . . ± . ( G eff / MeV − ) − . ± . − . ± . − . ± . − . ± . H ( km s − Mpc − ) 68 . ± .
78 67 . ± .
79 69 . ± .
52 67 . ± . σ . ± . . ± . . ± . . ± . + scenario, weassume that all of three species self-interact and we use the same value of G eff for all of them.This is because all of them are massless, and hence are equivalent. We have explicitly checkedthat even if we let the couplings for each species vary independently, their final posteriors areidentical. Therefore, we use the same value of coupling without loss of generality. In case of + , because of the shape of the posteriors, MultiNest was not able to separate MI and SImodes , and as a result, we are not able to quote the inferred values of the parameters alongwiththeir 68% confidence limits. However, we show the posteriors and the best-fit χ for both modesfor all datasets. Below we present the results in flavor universal and flavor-specific scenario.In figure 5 we show the posteriors for log ( G eff / MeV − ) and other relevant parameters for allthree SINU scenarios and ΛCDM. +
0f )
This is the only scenario which has been analysed before in Refs. [16, 66, 86] albeit with 2015and older Planck likelihoods. We show the results here using the Planck 2018 likelihood for thefirst time. The inferred parameter values and their 68% confidence limits for the TT+lowE andTTTEEE+lowE datasets are given in table 3.Our results for the posteriors for log ( G eff / MeV − ) and other relevant parameters areshown in figure 6a. We find a multimodal posterior for log ( G eff / MeV − ) in agreement withthe previous analyses. The SI mode corresponds to a larger value of log ( G eff / MeV − ) = − . ± .
11, whereas the MI mode is characterized by log ( G eff / MeV − ) < − . ± .
40 for In this case, the relatively larger value of χ the intermediate region between the two peaks in the 1-d posteriorof G eff , as shown in figure 1, makes it difficult for Multinest to distinguish one mode from the other. – 12 – − − [ G eff / MeV − ]0 . . . . n s . . A s e − τ r e i o H ( k m / s / M p c )
67 68 69 70 H (km / s / Mpc) 1 . . A s e − τ reio .
94 0 . n s P18-TTTEEE+lowE+Lensing
ΛCDM + + + Figure 5 : The 68% and 95% confidence limits of log ( G eff / MeV − ) , H , A s e − τ reio , and n s forneutrino interaction scenarios + (green) , + (blue) and + (red) for Planck 2018TTTEEE+lowE+lensing dataset. For reference we also show the posteriors for ΛCDM model(black) for the same dataset. We see that both A s e − τ reio and n s are negatively correlated withlog ( G eff / MeV − ), whereas H is positively correlated.TTTEEE+lowE dataset. These values do not change appreciably for other datasets, except forTT+lowE, the reason of which we discuss later. The most important effect of including thePlanck 2018 polarization data is that it suppresses the significance of the SI mode substantiallycompared to TT+lowE. Inclusion of the polarization data also shifts the log ( G eff / MeV − )posterior towards smaller value as can be seen in the marginalized posteriors in figure 6a. In therest of this subsection, we explain different aspects of these results.First, let us try to understand the origin of the SI mode which can be explained using thedegeneracy of G eff with parameters. To this end, we first show the changes in the residuals of the– 13 – − − − − log [G eff / MeV − ]
66 68 70 H (km / s / Mpc) .
75 1 .
80 1 .
85 1 . A s e − τ reio .
92 0 .
94 0 .
96 0 . n s P18-TT+lowE P18-TTTEEE+lowE P18-TTTEEE+lowE+lensing+BAO+H0 (a) + − − − − − log [G eff / MeV − ]
66 68 70 H (km / s / Mpc) .
75 1 .
80 1 .
85 1 . A s e − τ reio .
92 0 .
94 0 .
96 0 . n s (b) + − − − − − log [G eff / MeV − ]
66 68 70 H (km / s / Mpc) .
80 1 .
85 1 . A s e − τ reio .
94 0 .
96 0 . n s (c) + Figure 6 : Marginalised 1D posteriors for log ( G eff / MeV − ) , H , A s e − τ reio , and n s for threeSINU scenarios - + (top), + (middle) and + (bottom) for three datasets.TT, TE, and EE power spectra in figure 7 as we change G eff and the other correlated parameters,successively, to their SI mode best-fit values for TTTEEE+lowE dataset, starting from theΛCDM best-fit spectrum. These other parameters are H , A s , τ reio , n s , and Ω m , respectively.When we first incorporate G eff (red, solid), we see that the spectrum moves upwards and shiftstowards larger (cid:96) according to our discussion in the previous section. This gives rise to the positiveresiduals and the oscillations. Next, we change the best-fit value of H (blue, dashed) whichprimarily compensates for the phase shift of the spectrum. What remains after this is mostly– 14 –
00 1000 1500 2000 2500 ‘ − D TT ‘ − D TT , Λ C D M ‘ ΛCDM G eff G eff + H G eff + H + A s + τ reio G eff + H + A s + τ reio + n s G eff + H + A s + τ reio + n s + Ω m
250 500 750 1000 1250 1500 1750 2000 ‘ − − D T E ‘ − D T E , Λ C D M ‘
250 500 750 1000 1250 1500 1750 2000 ‘ − − D EE ‘ − D EE , Λ C D M ‘ P18 : TTTEEE + lowE | + Figure 7 : The residuals of the CMB TT (top left), TE (top right), and EE (bottom) powerspectra starting from ΛCDM best-fit, and with successive inclusion of the SI mode best-fitvalues of G eff , H , A s , τ reio , n s , and Ω m ≡ Ω b + Ω c , respectively, in + . Starting from G eff , H corrects for the phase of the spectrum, A s and τ reio reduce the amplitude, and n s red-tiltsthe whole spectrum. See section 4.1 for more details.an overall amplitude offset barring a small amount of residual phase-shift. The amplitude offsetis taken care of by the best-fit values of A s and τ reio (magenta, dotted). A smaller value of A s and a larger τ reio suppress the amplitude bringing down the residuals very close to the ΛCDMvalues. This overall amplitude change over-compensates the modifications at very large scalewhich entered the horizon after neutrino decoupling, resulting in the dip in the (cid:96) ∼
200 region.A smaller value of the n s (orange, dot-dashed) corrects for this and red-tilts the spectrumincreasing power at low (cid:96) while suppressing power at large (cid:96) . Finally, a smaller Ω m (green,solid) reduces the residuals even more, bringing it very close to the ΛCDM spectrum. A similarbehavior is shown by the EE spectrum as well which, however, has much larger error bars athigh (cid:96) than TT, and has less constraining power. This exercise shows that the Planck CMB dataallows for a larger value of G eff using its degeneracy with other parameters giving rise to the SImode. However, this compensation mechanism works for a special range of G eff values whereit is large enough to enhance all the acoustic peaks so that the effects can be compensated byother global parameters, and also the same time does not impart very large modification of thefirst acoustic peak and large phase shift. For very high values of G eff , neutrino remains stronglycoupled till very late times, even after recombination, and behaves almost always like a perfectfluid before recombination, and this scenario is disfavored by CMB data [74]. All of these prefer G eff values for which neutrino decoupling happens slightly prior matter-radiation equality in theSI mode. The origin of the MI mode is rather easier to understand. For smaller value of G eff ,– 15 –he changes in the spectrum are very small compared to the ΛCDM. This results in the plateauat small value in the posterior of log ( G eff / MeV − ). TTTT , C D M c + 0 f : MI 3 c + 0 f : SI 3 c + 0 f : VA500 1000 1500 2000 25000.02.55.07.5 , TT CDM : 85.83 c +0 f : MI : 85.1 3 c +0 f : SI : 79.93 c +0 f : VA : 116.6 T E T E , C D M c + 0 f : MI 3 c + 0 f : SI 3 c + 0 f : VA250 500 750 1000 1250 1500 1750 20000510 , T E CDM : 81.13 c +0 f : MI : 80.5 3 c +0 f : SI : 86.53 c +0 f : VA : 85.6 EEEE , C D M c + 0 f : MI 3 c + 0 f : SI 3 c + 0 f : VA250 500 750 1000 1250 1500 1750 20000510 , EE CDM : 73.93 c +0 f : MI :73.3 3 c +0 f : SI :79.93 c +0 f : VA :79.3 P18 : TT + lowE
TTTT , C D M c + 0 f : MI 3 c + 0 f : SI 3 c + 0 f : VA500 1000 1500 2000 25000.02.55.07.5 , TT CDM : 85.93 c +0 f : MI : 82.7 3 c +0 f : SI : 78.63 c +0 f : VA : 97.2 T E T E , C D M c + 0 f : MI 3 c + 0 f : SI 3 c + 0 f : VA250 500 750 1000 1250 1500 1750 20000510 , T E CDM : 76.63 c +0 f : MI : 77.3 3 c +0 f : SI : 78.63 c +0 f : VA : 80.6 EEEE , C D M c + 0 f : MI 3 c + 0 f : SI 3 c + 0 f : VA250 500 750 1000 1250 1500 1750 20000510 , EE CDM : 72.93 c +0 f : MI :72.9 3 c +0 f : SI :73.33 c +0 f : VA :76.7 P18 : TTTEEE + lowE
Figure 8 : Residual plots relative to the ΛCDM best-fit for high- (cid:96) ( (cid:96) = 29 − (cid:96) = 29 − χ (cid:96) ( see Eq.(4.1))with (cid:96) for ΛCDM and SINU modes. In the legend of each plot, we show the total ˜ χ . Thegray-shaded TE (middle-left)and EE (bottom-left) plots for TT+lowE signifies that high- (cid:96) TEand EE mode data are not included in that analysis. VA denotes the ‘best-fit’ point in the‘valley’ between the two modes in G eff posterior.– 16 –ow we shall explain the significance of the SI mode quantitatively relative to that of theMI mode. From figure 6a, clearly the significance of the former depends on the dataset used. Tobetter understand the origin of this variance in significance, we start by defining the quantity˜ χ (cid:96) as ˜ χ (cid:96) ≡ ( D BF (cid:96) − D Planck (cid:96) ) σ (cid:96) , (4.1)where D Planck (cid:96) is the binned Planck data, and D BF (cid:96) is the power spectrum corresponding to ourbest-fit point, and σ (cid:96) is the error bar of the Planck binned data. Therefore, ˜ χ (cid:96) carries informationabout the goodness-of-fit of the spectrum in different regions of (cid:96) , and gives an idea about whichpart of the spectrum prefers/penalizes the fit. We define a quantity ˜ χ as ˜ χ = (cid:80) (cid:96) ˜ χ (cid:96) whichgives the approximate goodness-of-fit information in the whole spectrum. Here we are using thebinned Planck data for visual clarity in the plots. In ˜ χ , we ignore the bin-by-bin correlation ofthe CMB data. Note that, the ˜ χ is not the same quantity as the χ calculated using the fullPlanck likelihood.In figure 8, we plot the residuals and ˜ χ (cid:96) for TT, TE, and EE spectrum in the high- (cid:96) region for the SI and MI mode best-fit points in + for the TT+lowE (left panels)and TTTEEE+lowE (right panels) datasets . We also show the corresponding plots for thebest-fit point in the ‘valley’ (VA) between the two modes which is defined within the range − . < log ( G eff / MeV − ) < − . We define the VA point to help explain the origin behindthe separate
SI mode. Firstly, we note that the MI residuals are very small and very close tothe ΛCDM points, i.e., the x-axis. This is expected as the CMB data loses any sensitivity for G eff values below the MI mode limit, and the spectrum becomes practically indistinguishablefrom ΛCDM. Interestingly however, the SI mode best-fit point yields a better fit to the ΛCDMresiduals which is evident in the TT spectrum in the top panels of figure 8. This substantiallyenhances the significance of the SI mode which is reflected in the ˜ χ (cid:96) distribution. In several (cid:96) -regions, e.g. around (cid:96) (cid:39) , , χ (cid:96) is below the ΛCDM orthe MI mode values, which largely compensates for the other (cid:96) regions with relatively worsefit. These are exactly the places where the SI best-fit point fits the ΛCDM residuals. In fact,according to the simplified analysis, the SI mode ˜ χ for the TT+lowE data is better than bothMI mode or ΛCDM.This is, however, not true for the TE and EE spectra though. The peaks in the EEspectrum are sharper than in the TT spectrum. Therefore, the polarization spectrum is moresensitive to phase shift due to SINU than the temperature spectrum, even though the error barsare larger [74]. This is reflected in the bottom panels in figure 8, where we see that the SI mode˜ χ (cid:96) is always equal to or above the MI or ΛCDM plots, yielding a poorer fit especially to thelow and intermediate- (cid:96) polarization data. Similar behaviour is also observed in the TE residuals(middle panels). As a result, when the polarization data is included, the significance of the SImode is reduced.The presence of the ‘valley’ between the two modes can be understood from the lowest (cid:96) mode that is affected by G eff . From the top axis of figure 2, we see that a typical SI mode G eff In the low- (cid:96) region, the residuals are small compared to the errorbar in the data. The best-fit point in the valley region is the sampling point that has the smallest χ in that range. – 17 –
00 1000 1500 2000 2500 ‘ − − D TT ‘ − D TT , Λ C D M ‘ c + 0 f : SI 2 c + 1 f : SI 1 c + 2 f : SI 250 500 750 1000 1250 1500 1750 2000 ‘ − − D T E ‘ − D T E , Λ C D M ‘
250 500 750 1000 1250 1500 1750 2000 ‘ − D EE ‘ − D EE , Λ C D M ‘ P18 : TTTEEE + lowE
Figure 9 : The residuals of the SI mode best-fit spectra of TT (top left), TE (top right), andEE (bottom) relative to ΛCDM in three scenarios. We note that these curves exhibit a betterfit to the ΛCDM residuals which shown in gray. circles.affects all modes (cid:96) (cid:38)
200 which happens to be the approximate position of the first peak inthe CMB spectrum. Therefore, it is easier to compensate for the modifications by changing theother parameters, as explained earlier, yielding a reasonably good fit. On the other hand, MImode values of G eff affects only (cid:96) (cid:38) which are far beyond the range of the present CMBexperiments. Planck measured the TT spectrum only upto (cid:96) = 2500, and the EE spectrumupto (cid:96) = 2000. Therefore, the MI mode is virtually indistinguishable from ΛCDM as far as thePlanck data is concerned. For intermediate values of G eff between the two modes, the CMBspectrum is modified only in the high- (cid:96) part of the Planck (cid:96) -range. In this case the degeneracywith other parameters, which impart changes in whole spectrum, cannot be exploited to get anoverall good fit. Thus, a ‘valley’ appears between the two modes as a result of the poor fit tothe CMB data.The polarization data (TTTEEE+lowE) also shifts of the whole posterior G eff to the leftrelative to TT+lowE which evident from the left panels of figure 6. This can be understoodfrom the relative sizes of the errorbars in the TT, TE, and EE spectra. The TT spectrum hassmaller errorbars at large (cid:96) . Whereas, the TE and EE spectra uncertainties are smaller at small (cid:96) (see figure 8). This essentially means that the TT data can accommodate large deviations atsmall- (cid:96) , whereas TE and EE data have more freedom at larger- (cid:96) . Therefore, inclusion of thepolarization data penalizes any deviation at low (cid:96) . Because, stronger self-interaction implies alater decoupling of the neutrinos affecting relatively smaller (cid:96) , TTTEEE+lowE data prefers aslightly smaller value of G eff compared to TT+lowE.– 18 – .2 Flavor-specific scenario: Two-coupled (2c +
1f ) & one-coupled states (1c +
2f )
In this section, we shall concentrate on the flavor-specific neutrino self-interaction scenarios,namely, + with only two self-interacting neutrino states, and + with only one self-interacting state. For the results in this section, we used the same coupling strength for thetwo interacting states instead of having two independent couplings in the MCMC sampling.We have explicitly verified the validity of this assumption by running chains with independentcouplings and same prior for the states which yielded same posterior distributions. This wasrather expected as all states are massless, and are not distinguishable from one another as far astheir cosmology is concerned. Hence, we chose the same coupling to derive all the results herereducing the number of free parameters without loss of generality. In passing, we also note thatthe one could have chosen different priors for different states inspired by the bounds from BBNand laboratory experiments which would yield distinct posteriors [34, 36]. We want to point outthat + scenario is equivalent to the flavor-specific ν e /ν µ /ν τ coupling cases discussed inRef. [34].The posteriors for log ( G eff / MeV − ) are shown in figure 5, and the inferred parametervalues and their 68% confidence limits for the TT+lowE and TTTEEE+lowE datasets aregiven in table 4 for + . The two modes are present in both of these cases. The SI modeappears at almost at the same value of G eff as in + . As discussed before, this specificvalue of G eff is determined by the (cid:96) -range of the Planck data. The interplay between G eff andother degenerate parameters seeks out this value. At the same time there is a very small shiftin the SI mode G eff values across three SINU scenarios. However, the most striking differencebetween + and + or + is the enhancement of the SI modes for all datasets inthe latter, as can be seen from figure 1 and 5 (see also figure 12 and 13). The enhancement ismore pronounced in + than + . From Using Eq. (2.9), (2.10), (2.11) and we see thatthe phase shift and the amplitude suppression due to neutrino free-streaming are proportionalto the number of free-streaming neutrino states. With less number of self-interacting states,these changes in the CMB spectra are relatively milder compared to + . This is evidentfrom figure 4. As a result, there exists more room to use the degeneracy between G eff andother parameters, especially A s and n s , to achieve a better fit. This is seen in figure 9 where wesee that SI mode the residuals in the flavor-specific scenario are always smaller than the flavoruniversal case.Another new feature here is that the upward rise of the valley between the two modes inthe log ( G eff / MeV − ) posterior. In the previous section, we explained the origin of the valleyas a result of the fact that intermediate values of G eff affect only a part of the CMB spectraobserved by Planck, which cannot be compensated by varying other degenerate parameters.Hence, the intermediate values are disfavored. However in + or + , those changesare comparatively modest, and can be partly undone by varying other parameters. As a result,the valley points yield smaller χ compared to + . H and σ We know that neutrino self-coupling strength G eff has a positive correlation with H throughthe phase-shift as explained in section 2.3. From table 3, we see that the mean value of theHubble shifts to H = 69 . ± .
52 km s − Mpc − for the TTTEEE+lowE dataset for the SI– 19 – able 4 : Parameter values and 68% confidence limits in + .Parameters TT+lowE TTTEEE+lowESI MI SI MIΩ b h . ± . . ± . . ± . . ± . c h . ± . . ± . . ± . . ± . θ s . ± . . ± . . ± . . ± . A s ) 2 . ± . . ± . ± . . ± . n s . ± . . ± . . ± . . ± . τ reio . ± . . ± . . ± . . ± . ( G eff / MeV − ) − . ± . − . ± . − . ± . − . ± . H ( km s − Mpc − ) 68 . ± .
71 67 . ± .
64 68 . ± .
46 67 . ± . σ . ± . . ± . . ± . . ± . ∼ σ . On the otherhand, when compared with the CCHP measurement H = 69 . ± . − Mpc − which usesthe tip of the red giant branch calibration, our value of H is fully consistent with that [42].However, the SI mode value of H slightly decreases in the flavor-specific + and + scenarios because of smaller phase shift, but the significance of the mode is increased. This isevident from figure 10. In table 5, we show H , Ω Λ , 100 θ s , r ∗ s , and D ∗ A for + , + and ΛCDM. The shift in θ s due to the phase shift in SINU decreases D ∗ A that helps increase thevalue of H and Ω Λ . Note that, the value of r ∗ s changes by only ∼ σ from ΛCDM, and plays asub-dominant role in changing the Hubble constant.The significance of the SI mode is enhanced when the local Hubble measurement data H = 74 . ± . − Mpc − from SH0ES is included [87]. At the same time, it suppresses theMI mode significantly as can be seen from figure 6. This is not surprising given the fact thatthe SH0ES likelihood includes only Hubble data, and favors SI mode which yields a larger H .However, one should be cautious about combining the Planck with the SH0ES data becauseof the more than ∼ σ discrepancy between the two. Even though we show the results withPlanck+lensing+BAO+SH0ES here, we do not draw any conclusion from them. Note that,the + scenario yields higher H but its significance is smaller. In + and + scenarios, although the significance of the SI mode is increased, the corresponding value of H decreases due to smaller phase shift. Therefore, both flavor-universal and flavor-specificSINU scenarios fail to yield a large enough value of H to completely solve the Hubble tension(see figure 6). The parameters inferred from the combined TTTEEE+lowE+lensing+BAO+H0dataset are quoted in table 9 and 10.Apart from H , another parameter that shows a small discrepancy between CMB andlow redshift data is the late-time matter clustering amplitude σ . Planck 2018 results found σ = 0 . ± .
006 using TTTEEE+lowE+lensing data [7]. However, the gravitational weaklensing observation by KiDS-1000 found σ = 0 . +0 . − . [88]. There is about ∼ . σ discrepancy– 20 – − − − − [ G eff / MeV − ]6667686970 H ( k m / s / M p c ) P18-TTTEEE+lowE+Lensing + + + Figure 10 : The contours of 68%, 95%, and 99% confidence levels for log ( G eff / MeV − ) and H in + (green), + (blue), and + (red) for the Planck TTTEEE+lowE+lensingdataset. The SI mode contour for + yields the largest value of H because of bigger phaseshift. However, the significance of the mode is much less compared to + and + .between the two values. The value of σ in SINU cosmology is mainly determined by twocompeting effects. The lack of anisotropic stress in the neutrino bath boosts the perturbationmodes which enter horizon before neutrino decoupling. In case of SI mode values of G eff , thesemodes happen to be in range of σ , and thereby boosting its value. On the other hand, the lowervalue of best-fit A s and n s suppress σ . As a result, G eff and σ show a very weak correlationas can be seen in figure 11, 12, and 13. We find σ = 0 . ± . . ± . + and + , respectively for TTTEEE+lowE (see table 3 and 4). Table 5 : Parameter values and 68% confidence limits for SI mode in + and + , andΛCDM in TTTEEE+lowE+lensing dataset.SI: + SI: + ΛCDM H ( km s − Mpc − ) 69 . ± .
46 68 . ± .
39 67 . ± . Λ . ± . . ± . . ± . θ s . ± . . ± . . ± . r ∗ s (Mpc) 144 . ± .
25 144 . ± .
21 144 . ± . D ∗ A (Mpc) 12 . ± .
027 12 . ± .
023 12 . ± . .4 Mode comparison In this subsection, we analyze the relative significance of the two modes in the SINU posterior.First, we compare the maximum likelihoods which corresponds to the best-fit point for eachmodes from the full MCMC analysis. We define the maximum likelihood ratio R SI as R SI ≡ max[ L ( θ SI | d)]max[ L ( θ MI | d)] . (4.2)Here d is the data and θ is the set of parameters in the model and L ( θ | d) is the likelihood. Weshow R SI for all cases in table 6. It is evident from table 6, that R SI increases with less numberof self-interacting neutrinos, especially R SI > + scenario. Thissignifies that fit to the CMB data is better for the SI mode. Inclusion of the H data enhances R SI in all cases, as expected. Also, addition of BAO data lowers R SI for all scenarios. Next, wecompute the Bayesian evidence Z for the two modes as follows Z ≡
Pr(d | M ) = (cid:90) Pr(d | θ, M )Pr( θ | M )d θ . (4.3)Here M is the cosmological model. The Bayesian evidence essentially is the probability of thedata d given a model M . We use the mode separation option in Multinest to compute theBayesian evidence for the two modes. Then we compute the ratio of the probabilities Pr( M | d)of model M for given data d, which is the Bayes factor, to find the relative significance of eachmode assuming equal prior. This gives, B SI ≡ Pr( M SI | d)Pr( M MI | d) = Z SI Z MI Pr( M SI )Pr( M MI ) = Z SI Z MI . (4.4)A Bayes factor B SI >
1, therefore, means that the SI mode is more significant than the MI mode.In Table 7, we show the Bayes factors for + and + for all datasets. As mentionedearlier, for + , the log ( G eff / MeV − ) posterior does not completely vanish in the regionbetween MI and SI modes. As a result, the MultiNest module is not able to distinguish the SIand MI modes.We show the change in the best-fit χ relative to ΛCDM, ∆ χ = χ − χ for allcases and all datasets in table 8. Table 6 : Maximum likelihood ratio R SI .Dataset +
0f 2c +
1f 1c + TT+lowE 0.25 1.08 1.63TTTEEE+lowE 0.05 0.46 1.15TTTEEE+lowE+lens 0.07 0.43 1.08TTTEEE+lowE+lens+BAO 0.01 0.34 1.07TTTEEE+lowE+lens+BAO+ H able 7 : Bayes factor B SI .Dataset +
0f 2c + TT+lowE 0.045 0.223TTTEEE+lowE 0.018 0.100TTTEEE+lowE+lens 0.014 0.074TTTEEE+lowE+lens+BAO 0.002 0.061TTTEEE+lowE+lens+BAO+ H Secret self-interactions among the active neutrinos could leave an observable imprint in the CMBanisotropy power spectra. Such interaction stops the neutrinos from free-streaming much laterthan weak decoupling . As a result, the CMB power spectra experience a phase shift and an enhancement for the modes which entered the Hubble horizon before the decoupling of the neu-trinos. We studied these effects in three cases- + , + , and + with three, two,and one self-interacting massless neutrino species, respectively, and their cosmological implica-tions using the latest CMB data from Planck 2018, BAO data, and local Hubble measurementdata from SH0ES. This study of the flavor-specific scenario is inspired by the recent strongconstraints from several laboratory experiments on the flavor universal scenario. We make thefollowing key observations in this work. Table 8 : Best-fit ∆ χ relative to ΛCDM. +
0f 2c +
1f 1c + TT+lowE MI -0.16 -0.05 -0.07SI 2.6 -0.2 -1.05TTTEEE+lowE MI 0.11 0.11 0.15SI 6.03 1.67 -0.12TTTEEE+lowE+lensing MI 0.16 0.1 0.17SI 5.34 1.8 0TTTEEE+lowE+lensing+BAO MI 0.09 0.13 0.15SI 8.48 15.47 0TTTEEE+lowE+lensing+BAO+ H MI 0.17 0.11 0.15SI 1.68 -1.29 -2– 23 –
In all three cases and all datasets, the posterior of log ( G eff / MeV − ) has two distinctmodes, namely, the SI mode characterized by a larger log ( G eff / MeV − ), and the MI modewith a smaller value. The origin of the SI mode is explained as a result of degeneraciesbetween G eff and other ΛCDM parameters. The absence of free streaming neutrinos inSINU leads to a phase shift and enhancement of the CMB angular power spectra. Thesechanges are compensated by the other ΛCDM parameters. Specifically, H compensatesfor the phase shift, A s and τ reio correct the overall amplitude, and n s tilts the wholespectrum to achieve a good fit to the data. We showed that this compensation mechanismworks only for those values of G eff for which neutrino decoupling happens close to matterradiation equality. • In intermediate values in the ‘valley’ between the SI and MI modes are not favored by data.The large G eff corresponding to the SI mode globally affects the whole CMB spectrumobserved by Planck, which can be undone by changing other parameters. In contrast,the MI mode spectrum is virtually indistinguishable from the ΛCDM as it only affectsvery high- (cid:96) modes, due to small G eff , which are not observed by Planck. The valley inbetween corresponds to intermediate G eff values which modifies only the high- (cid:96) portion ofthe spectrum. These partial changes are difficult to compensate using other parameters,resulting to a worse fir to the data. • In the + scenario, the significance of the SI mode is greatly diminished if CMBpolarization data is included. This is a consequence of the relatively poorer fit of themodel to the low- l polarization data. Even though the phase shift from the free-streamingneutrinos is the same for both temperature and polarization spectra, the peaks in thepolarization spectrum are relatively sharper. As a result, the EE data is more sensitive toany change in the number of free-streaming neutrinos than the TT data [74]. The inclusionof the polarization data also shifts the whole posteriors towards smaller G eff . • However, the SI mode is rejuvenated in the flavor-specific + and + scenarioswhich are in lesser conflict with laboratory experiments. We showed that even with thepolarization data, the SI mode significance is comparable or sometimes even greater thanthat of the MI mode. In these cases, the number of self-interacting neutrinos are less than + , and as a result, the changes in the CMB spectra are also relatively moderate.This allows for more freedom to use other degenerate parameters to achieve a significantlybetter fit. • Due to the phase shift in CMB spectrum, SINU scenario favors a larger H in the SImode. For example, in the + scenario, we find H = 69 . ± .
41 km s − Mpc − forPlanck temperature, polarization, and lensing data. Although in flavor-specific scenario,the significance of the SI mode increases, the value of H slightly decreases due to lesserphase-shift. We do not find any strong correlation between σ and G eff .In this work, we have confined our discussion within the standard framework of only threeactive neutrino states. However, additional sterile neutrino states are interesting to consider forvarious different reasons. The presence of such an additional neutrino state(s) might change– 24 –ur results significantly. To begin with, the relativistic degrees of freedom N eff is expected toincrease, which would in turn increase the value of H [66]. As hinted by the short-baselineneutrino oscillation experiments, the mass of the additional sterile state is predicted to bearound ∼ + scenario, we can get a sense of whatcould happen in the case where the sterile state is self-interacting but the active states are not.This scenario will evade all laboratory constraints on the active sector. However, for a CMBanalysis, one needs to take into account the mass of the sterile state which is of the order of thetemperature at recombination. The finite neutrino mass will have further implications for ouranalysis. In presence of massive neutrinos, the flavor-structure of the interaction matrix wouldcertainly be non-diagonal which can have interesting signatures. Finite neutrino mass will alsoaffect the matter power spectrum at late times. The strong neutrino self interaction will alsoenhance the primordial CMB B-mode at small scale which can have interesting implications forfuture B-mode experiments [18, 89].In this work, we have kept the helium fraction Y p fixed to its BBN value. However, thereis a well-known degeneracy between Y p and the number of free-streaming neutrinos. We planto investigate it in a future work. Note that, the present and all previous analyses on this topichave assumed a diagonal coupling matrix between different neutrino flavors. However, this isfar from a realistic scenario as off-diagonal couplings will be present even in the most simplemodel of neutrino self-interaction. The evolution of the cosmological perturbations become muchmore complicated when off-diagonal couplings are present, and a more general set of evolutionequations need to be used [17].Future CMB observations like, CMB-S4, will make measurements at smaller scales andwill probe even higher (cid:96) values than Planck. This would probe the neutrino interaction at evenearlier times, and perhaps could shed light on the nature of the mediator particle φ . This workis the first step towards studying a more general and realistic flavor profile of neutrino selfinteraction. A more rigorous analysis including CMB, BBN, and laboratory experiment datawould be worthwhile. Acknowledgment
AD was supported by the U.S. Department of Energy under contract number DE-AC02-76SF00515.SG was supported in part by the National Science Foundation under Grant Number PHY-2014165 and PHY-1820860. SG also acknowledges the support from Department of AtomicEnergy, Government of India during the initial stage of this project. This work used the compu-tational facility of Department of Theoretical Physics, Tata Institute of Fundamental Research.
A Best-fit Parameter Values
Here we show the parameter values and their 68% confidence limits for the TTTEEE+lowE+lensingand TTTEEE+lowE+lensing+BAO+ H datasets.– 25 – able 9 : Parameter values and 68% confidence limits in + .Parameters TTTEEE+lowE+lens TTTEEE+lowE+lens+BAO+ H SI MI SI MIΩ b h . ± . . ± . . ± . . ± . c h . ± . . ± . . ± . . ± . θ s . ± . . ± . . ± . . ± . A s ) 2 . ± .
012 3 . ± . . ± . . ± . n s . ± . . ± . . ± .
003 0 . ± . τ reio . ± . . ± . . ± .
006 0 . ± . ( G eff / MeV − ) − . ± . − . ± . − . ± . − . ± . H ( km s − Mpc − ) 69 . ± .
41 67 . ± .
52 69 . ± .
38 68 . ± . σ . ± . . ± . . ± . . ± . Table 10 : Parameter values and 68% confidence limits in + .Parameters TTTEEE+lowE+lens TTTEEE+lowE+lens+BAO+ H SI MI SI MIΩ b h . ± . . ± . . ± . . ± . c h . ± . . ± . . ± . . ± . θ s . ± . . ± . . ± . . ± . A s ) 3 ± . . ± .
011 3 ± . . ± . n s . ± . . ± . . ± .
003 0 . ± . τ reio . ± . . ± . . ± .
005 0 . ± . ( G eff / MeV − ) − . ± . − . ± . − . ± . − . ± . H ( km s − Mpc − ) 68 . ± .
36 67 . ± .
42 69 . ± .
31 68 . ± . σ . ± .
004 0 . ± . . ± . . ± . B Posterior distributions of all parametersReferences [1] I. Esteban, M. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni, and T. Schwetz,
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