Cosmological Constraints on Invisible Neutrino Decays Revisited

Neutrinos could decay. Invisible neutrino decay modes are difficult to target at laboratory experiments, and current bounds on such decays from solar neutrino and neutrino oscillation experiments are somewhat weak. It has been known for some time that Cosmology can serve as a powerful probe of invisible neutrino decays. In this work, we show that in order for Big Bang Nucleosynthesis to be successful, the invisible neutrino decay lifetime should be τν > 10 −3 s. We revisit Cosmic Microwave Background constraints on invisible neutrino decays, and by using the latest Planck observations we find that neutrino lifetimes τν < (1.2− 0.3)× 10 s (mν/0.05 eV) are excluded at 95% CL. We show that this bound is robust to modifications of the cosmological model, in particular that it is independent of the presence of dark radiation. We find that typical invisible neutrino decay modes with rates τν < 10 5 s (mν/0.05 eV) 3 are disfavoured at more than 5σ with respect to ΛCDM given the latest Planck CMB observations. Finally, we show that when including high-` Planck polarization data, neutrino lifetimes τν = (2− 14)× 10 s (mν/0.05 eV) are mildly preferred – with a 1-2 σ significance – over neutrinos being stable.

Neutrinos could decay. Invisible neutrino decay modes are difficult to target at laboratory experiments, and current bounds on such decays from solar neutrino and neutrino oscillation experiments are somewhat weak. It has been known for some time that Cosmology can serve as a powerful probe of invisible neutrino decays. In this work, we show that in order for Big Bang Nucleosynthesis to be successful, the invisible neutrino decay lifetime should be τν > 10 −3 s. We revisit Cosmic Microwave Background constraints on invisible neutrino decays, and by using the latest Planck observations we find that neutrino lifetimes τν < (1.2 − 0.3) × 10 9 s (mν /0.05 eV) 3 are excluded at 95% CL. We show that this bound is robust to modifications of the cosmological model, in particular that it is independent of the presence of dark radiation. We find that typical invisible neutrino decay modes with rates τν < 10 5 s (mν /0.05 eV) 3 are disfavoured at more than 5 σ with respect to ΛCDM given the latest Planck CMB observations. Finally, we show that when including high-Planck polarization data, neutrino lifetimes τν = (2 − 14) × 10 9 s (mν /0.05 eV) 3 are mildly preferred -with a 1-2 σ significance -over neutrinos being stable.
The constraints on the neutrino lifetime are very much dependent upon the neutrino decay products. Radiative neutrino decays are strongly constrained by the nonobservation of neutrino magnetic moments in laboratory experiments τ ν 10 18 yr [19,20], by cosmic microwave background (CMB) spectral distortions τ ν 10 12 yr [21,22], by 21 cm cosmology [23], and by astrophysical considerations τ ν 10 20 yr [24][25][26]. In contrast, the constraints on invisible neutrino decays, namely those that do not involve photons in the final state, are considerably looser. This is a result of the difficulty in detecting the decay products from such a process and due to fact that light active neutrinos are usually highly boosted.
In this work, in light of these somewhat weak constraints on invisible neutrino decays, we study the impact of invisible neutrino decays upon Big Bang Nucleosynthesis (BBN), and also revisit the constraints on invisible neutrino decays derived from the CMB observations made by the Planck satellite.
In the first part of this paper, we exploit the fact that in order for neutrinos to decay invisibly, they should decay into massless or at least to very light species. Because of this, the same interactions that trigger the decay may produce a thermal population of such light species prior to BBN and thereby augment the number of relativistic neutrino species in the early Universe, N eff . We show that, independently of the neutrino decay process and the neutrino type, neutrino lifetimes τ ν < 10 −3 s are ruled out by the current measured primordial nuclei abundances. In this way we improve upon current constraints from accelerator and long-baseline experiments by 8 orders of magnitude, and by 2 orders of magnitude over current constraints from solar neutrino experiments. We note that similar phenomenology has been studied in the past within the context of a considerably heavy τneutrino (m ντ < 23 MeV), see e.g. [40][41][42][43][44][45].
For the second part of the work, we calculate the effect of neutrino decays in the density perturbations of the neutrino fluid and use this to test the neutrino decay hypothesis against the 2015 temperature and polarization CMB power spectra as measured by the Planck satellite [46,47]. For previous CMB analysis see [35-39, 48, 49], particularly Ref. [38]. In this study, we are maximally conservative and perform analyses assuming various types of neutrino decay modes. We consider invisible neutrino modes in which an active neutrino decays into another active neutrino plus a massless scalar field and obtain a lower limit on the lifetime using the 1. Annihilation (left) and scattering (right) neutrino-φ diagrams as induced by the same interactions that trigger invisible neutrino decays (middle).
Planck 2015 data of τ ν > 1.2 × 10 9 s (m ν /0.05 eV) 3 at 95% CL. This bound is the same as the previous limit obtained in Ref. [38] that used Planck 2013 data, but unlike Ref. [38] here we consider that only the two neutrinos that participate in the decay process are interacting. In addition, we explore the possible degeneracies between a finite neutrino lifetime and a variation in N eff , and show that contrary to previous expectations [39,50,51] even if only one neutrino species decays and a non-interacting N eff is allowed to vary, neutrino lifetimes of τ ν < 0.9 × 10 9 s (m ν /0.05 eV) 3 are still excluded at 95% CL by Planck CMB observations. Finally, we find that when including Planck 2015 high-polarization data in the analysis, neutrino lifetimes in the range τ ν = (2 − 14) × 10 9 s (m ν /0.05 eV) 3 are preferred over neutrinos being purely stable with a ∼ 1-2 σ significance.
This paper is organized as follows. In Section II, we consider a simple and generic model for invisible neutrino decays. In Section III, we consider the production in the early Universe of beyond the Standard Model light neutrino decay products and set constraints on such production using BBN. We also include a discussion of the applicability of the derived BBN constraints. In Section IV, we outline how we model the impact of neutrino decays upon cosmological perturbations and test the neutrino decay hypothesis with Planck 2015 data to set constraints on invisible neutrino decays. We summarize and discuss the main results of this work in Section V. Finally, in Section VI, we comment on how invisible neutrino decays are expected to be constrained in the future.

II. INVISIBLE NEUTRINO DECAYS
Fast and invisible neutrino decays are a typical prediction of models in which global lepton number is spontaneously broken so as to generate light Majorana neutrino masses. In such models, as a result, a massless Goldstone boson appears in the spectrum, the majoron [7][8][9].
Here, we shall consider the following effective interaction between neutrinos and a massless scalar φ: where the ν i correspond to the massive neutrino eigenstates, i, j = 1, 2, 3 and we shall assume neutrinos are Majorana particles 1 . λ ij are coupling constants, of which the off-diagonal elements with i = j induce neutrino decay. Given the interactions above, the rate of neutrino decay ν i → ν j + φ is: where in the last step we have assumed that m νi m νj .

III. BIG BANG NUCLEOSYNTHESIS CONSTRAINTS
We place early Universe constraints on the invisible neutrino lifetime by exploiting the fact that the same interactions that allow for fast invisible neutrino decays also mean that processes of the typeνν → φφ will be active in the early Universe (see Figure 1 for an illustration of these processes). These processes can potentially lead to a thermal population of massless or very light φ species in the early Universe. This would thereby impact the primordial nuclei abundances and the number of effective neutrino species as inferred from CMB observations.
In order to make a precise statement about the constraint on the coupling constant λ ij , and therefore (via equation (2)) upon the lifetime of the neutrino, we need to calculate the abundance of massless φ particles in the early Universe. The presence of a thermal abundance of φ particles will only influence N eff or the primordial element abundances if the φ population is generated prior to neutrino decoupling, at T 2 MeV [52], corresponding to an era in the Universe in which neutrinos can be efficiently produced via e + e − →νν annihilations. If there is a thermal population of φ particles prior to neutrino [53,54]. Note that ∆N eff > 0.4 is excluded at more than 95% CL from current measurements of the primordial nuclei abundances [55], see also [56]. We will assume that all relevant species can be described by thermal distribution functions with negligible chemical potentials, and proceed as in [57,58], to find the following temperature evolution equations: Where ρ i , p i correspond to the energy density and pressure of a given species and their respective antiparticle. H = 8πρ total /(3m 2 Pl ) is the Hubble parameter with m Pl = 1.22 × 10 19 GeV. P int and its derivatives take into account finite temperature corrections to the electromagnetic pressure and energy density, and δρ/δt are the energy density transfer rates, see [57,58] for details. The SM neutrino↔electron energy transfer rate, neglecting the electron mass, reads [58]: The neutrino-φ energy transfer rate takes into account the energy transfer resulting from the following processes  (1). The horizontal dashed lines correspond to the N eff + 2σ measurements from Planck [60] (within ΛCDM) and as inferred from the observed primordial nuclei abundances at the time of BBN [55]. We also show the expected sensitivity from Stage-IV CMB [61] experiments.
νν ↔ φφ, νφ ↔ νφ, and ν i ↔ ν j + φ. We have disregarded the scattering interactions since they are subdominant for massless species as compared to annihilations. We also neglect the contribution from neutrino decays since the rate of neutrino decay Γ ∼ λ 2 m 2 ν /T is not relevant for T > 1 MeV because it is tiny when compared to 2 ↔ 2 processes, Γ ∼ λ 4 T because of neutrinos being highly boosted. Thus, the relevant energy transfer rate is given by annihilation processes, and reads (see Appendix A for the derivation): We evolve the system of equations (3) . The temperature evolution for some values of the neutrino-φ coupling constant is displayed in Figure 2. Notice that if λ ij > 8 × 10 −6 a thermal population of φ particles will be produced at T 3 MeV which will yield ∆N eff 0.57.
In order to constrain the φ-neutrino coupling, we shall use the latest constraints on N eff as inferred from the measured primordial Helium and Deuterium abundances taken from the recent comprehensive analysis of Ref. [55] (see also [56]). This analysis used Y P = 0.2449±0.0040 [62] and D/H = (2.527±0.030)×10 −5 [63]. At 95.4% CL, the N eff constraint from BBN reads [55]: Note that within the neutrino decay scenario, N eff is the same at the time of CMB formation and during BBN. This is because the φ population can only lead to a change in N eff provided that it is generated before neutrino decoupling at T > 2 MeV. Since the protonto-neutron interactions freeze-out at T p→n ∼ 0.7 MeV, BBN occurs at T BBN ∼ 0.07 MeV [64][65][66], and recombination happens at T CMB ∼ 0.26 eV, then this is clearly the case. We show the resulting N eff as a function of the value of the φ-neutrino Yukawa coupling in Figure 3. The comparison between N eff as a function of λ ij and that required for successful BBN results in the following constraint on λ ij : Finally, to translate the bound on the coupling into the neutrino decay lifetime, we need to specify the mass of one of the neutrinos in the decay process since only mass differences are known [1][2][3]. Therefore, our bound on τ ν depends upon the mass of one of the neutrinos in the process, and we choose this mass to be that of the final state neutrino m lightest .
We therefore have shown that in order for a successful BBN, invisible neutrino decay modes of the type ν i → ν j + φ (where i, j represent massive neutrino states, and φ is a massless scalar) should have a lifetime This bound applies to any neutrino mass eigenstate (provided that the decay is kinematically accessible) and for both normal and inverted ordering. Supernova cooling can also be used to set constraints on the neutrino-φ coupling, and thereby on the neutrino decay lifetime. The agreement of SN1987A observations with supernova models excludes couplings in the range 3 × 10 −7 λ ij 2 × 10 −5 or λ ij 3 × 10 −4 [67,68]. This bound is shown in Figure 4 in grey.
The bound of τ νi→νj +φ > 10 −3 s represents an improvement of 8 orders of magnitude as compared with constraints obtained from accelerator and long-baseline neutrino experiments [32]. Separately, the bound of τ νi→νj +φ > 10 −3 s is still 2 orders of magnitude more stringent that those inferred from solar neutrino experiments [29][30][31]. However, in some regions of parameter space this BBN bound is less constraining than the bound that can be inferred from SN1987A observations [67,68]. Here we comment on how relaxing some of the assumptions that we made in order to obtain the constraint on the neutrino lifetime of τ ν > 10 −3 s (8) from BBN could affect them, and we argue that they cannot be significantly altered.
1. Majorana-Dirac: For a given neutrino decay rate, the annihilation cross section for Dirac neutrinos is 1/2 that of Majorana neutrinos, since the neutrinos are not their antiparticles. Therefore, the constraint on λ should be relaxed by a factor of 2 1/4 1.2 in the Dirac case. And therefore, the constraint on the lifetime should naively be relaxed by a factor √ 2. However, if neutrinos are Dirac, the ν −φ interaction will lead also to a thermal population of massless right handed neutrinos and ∆N eff will greatly exceed 0.57, which will result in an even tighter constraint.

φ mass:
Regardless of what the mass of the φ scalar is, if the φ scalar is light enough to be in the neutrino decay final state, then its mass is negligible in the early Universe (T 1 MeV) and therefore m φ will not impact the annihilation rate. The mass may change the decay width at rest, however, the phase space suppression will be O(1) unless m φ is very fine tuned m φ m νi − m νj . Hence, a non-negligible m φ will not impact our conclusions.

III.1.2. Other scenarios
Here we comment how the BBN constraint of τ ν > 10 −3 s (8) applies to other particle physics scenarios in which the decay is not necessarily ν i → ν j + φ.
1. ν i → ν j + Z . Invisible neutrino decays also generically result from vector mediated neutrino selfinteractions [69][70][71][72][73], provided that m Z < |m ν3 − m ν1 | 0.05 eV. For such types of models, our bounds still apply since the presence of a thermal population of very light Z s prior to neutrino decoupling would render ∆N eff = 1.71, a value which is clearly excluded by CMB observations and successful BBN (6). In addition, as a result of processes of the type + − → γ Z [73], coupling constants of O(10 −8 ) would be ruled out for m Z < |m ν3 − m ν1 |, therefore rendering very strong constraints on the neutrino lifetime in such type of models, τ ν /m ν O(10 4 ) s eV −1 .
2. ν i → ν 4 + φ. If one of the light massive eigenstates decays into a scalar plus a fourth very light neutrino (that has very small mixing with the three active flavours e, µ, τ ), then our cosmological constraint still applies since within this scenario ∆N eff could be as large as ∆N eff = 1.57 at the time of BBN, which is again clearly excluded by current data (6).
3. ν 4 → ν 1 +φ. This scenario will be ruled out for τ ν4 < 10 −3 s since the same interactions that trigger the 4th neutrino decay will render a thermal population of ν 4 and φ particles, thereby rendering ∆N eff = 1.57, which is incompatible with a successful BBN. Note that this bound will apply for m ν4 1 MeV.
FIG. 4. Constraints on the lifetime of neutrino decay processes of the type νi → νj + φ, where i, j label active neutrino mass eigenstates and φ is a massless sterile scalar. m lightest corresponds to mν j . In the left panel the bounds are shown for ν2 → ν1 + φ (NO) and ν1 → ν2 + φ (IO) decay processes, while in the right panel we show the constraints for the ν3 → ν1 + φ (NO) and ν1 → ν3 + φ (IO) decay processes. The purple and magenta contours are ruled out by accelerator, long-baseline, and solar neutrino experiments [29][30][31][32]. The grey area is excluded by SN1987A observations [67,68]. The blue contours correspond to the cosmological constraints obtained in this work by the requirement of successful BBN, see Section III. The cyan contours correspond to the bounds obtained from the Planck 2015 CMB analysis, see Section IV. In addition, in red, we highlight the preferred region of parameter space by Planck 2015 CMB observations.

IV. CMB CONSTRAINTS
If neutrinos decay efficiently while still relativistic into other massless species, the decay process will effectively make the neutrino fluid no longer free-streaming [35,36]. In particular, neutrino decays will erase the neutrino anisotropic stress that otherwise arises in the course of expansion in a purely non-interacting massless fluid [74,75]. In this section, we describe how we implement the effect of neutrino decays upon the neutrino cosmological perturbations and use the latest public CMB measurements by the Planck satellite to set constraints on invisible neutrino decays.

IV.1. Modelling neutrino decays
We follow Ref. [36] in order to calculate the effective neutrino decay rate that erases the neutrino anisotropic stress, Γ . Ref. [36] argues that Γ Γ (m ν /E ν ) 3 , which by thermally averaging m/E and m 2 /E 2 separately yields: Written as a function of the scale factor a, the neutrino lifetime and the neutrino mass, reads: where a 0 = 1.
In order to account the effect of neutrino decays in the neutrino cosmological perturbations, we follow the relaxation time approximation for the neutrino collision term [76]. This approximation amounts to modifying the massless neutrino Boltzmann hierarchy for the perturbed neutrino phase space in the following manner: where F ν represents the contribution from the th Legendre polynomial to the perturbed neutrino phase space distribution [75]. The neutrino fluid is regarded as the neutrinos plus the massless species produced in the decay. We implement equation (11) in the cosmological Boltzmann code CLASS [77,78]. For simplicity, we assume that neutrinos are massless since given Planck 2018 constraints [60], m ν < 0.12 at 95% CL, and therefore neutrinos decay while relativistic for the relevant cosmological evolution considered in this study.

IV.2. CMB Analysis
In order to test the neutrino decay hypothesis with CMB observations we use the latest public CMB data from the Planck satellite [46,47]. In particular, we use both the high-Planck 2015 temperature and polarization spectra, the low-temperature and polarization spectra, and also the lensing measurements from the 2015 data release [46]. We consider the following data set combinations Planck 2015 TT+lowP+lensing and Planck 2015 TTTEEE+lowP+lensing. To perform the CMB analysis, since Γ is the quantity that directly enters the Boltzmann hierarchy, we define and we use a logarithmic prior on Γ eff over the range [10 0 , 10 8 ]. Converting a constraint on Γ eff into a constraint on the neutrino lifetime is trivial by using equation (12). For the rest of the cosmological and nuisance parameters we use the same priors as the Planck collaboration in their 2015 base ΛCDM analysis [46,47] with the exception of the reionization width τ reio which for which we require τ reio > 0.04 in order to account for the Gunn-Peterson effect [60].
In order to be maximally conservative, we consider several decay scenarios and also consider to which extent the presence of additional non-interacting massless speciesencoded in terms of ∆N eff -can alter the invisible neutrino decay constraints.
We consider the same decay scenario as in Section II in which one active massive neutrino decays into another one by emitting a massless scalar particle φ; namely, ν i → ν j + φ. Within this scenario, the number of interacting neutrino species is N int = 2, while the other neutrino simply free-streams. We consider another scenario in which an active neutrino decays into a sterile and very light neutrino ν 4 by emitting a massless scalar field φ; namely, ν i → ν 4 + φ. In this scenario the number of interacting neutrino species is N int = 1 while we consider the other two active neutrino species to be noninteracting and therefore purely free-streaming. We contrast both scenarios by varying Γ eff and also ∆N eff , for which we use a linear prior in the range [−2, 10]. We perform a Monte Carlo Markov Chain (MCMC) analysis using MontePython-v3 [79,80] and we quote results of analyses in which the maximum Gelman-Rubin coefficient [81] for any parameter is R − 1 < 0.05. Parameter log 10 (Γ eff ) τν /(10 9 s) · (mν /0.05 eV)  (12). Note that we also account for Planck 2015 lensing measurements.

IV.3. Planck 2015 Constraints
In the left panel of Figure 5, we display the marginalized posterior distribution of the parameter Γ eff , which is directly related to the neutrino lifetime (12). In the right panel of Figure 5, we show the two-dimensional marginalized posterior between Γ eff and ∆N eff . It is obvious that the two parameters are not degenerate and from the left panel of Figure 5 we notice that the posterior distributions for both a varying ∆N eff and when it is fixed are fairly similar.
In Table I we quote the best fit, mean 68% CL error bars and 95% CL exclusions for the parameter Γ eff and for the invisible neutrino decay lifetime. The reader is deferred to Table II in Appendix B where we quote the mean and 68% error bars for the standard cosmological parameters too. From Table I we clearly appreciate that the derived limits from the Planck 2015 TTTEEE+lowP+lensing dataset are less stringent than those from the Planck 2015 TT+lowP+lensing dataset. This is essentially because when including high-polarization data there is a 1-2σ preference for a non-infinite invisible neutrino decay lifetime. We therefore choose the Planck 2015 TTTEEE+lowP+lensing to quote both 95% CL upper and lower limits and ± 68 % CL measurements.
We show that Planck 2015 CMB observations bound the lifetime of neutrino decay processes like ν i → ν j + φ to be τ νi→νj +φ > 1.2 × 10 9 s m νi 0.05 eV 3 Planck , (13) at 95% CL. The lower bound on the neutrino lifetime of the decay mode of the type ν i → ν 4 + φ, where ν 4 is a very light and sterile neutrino, at 95% CL reads: Furthermore, we also perform analyses allowing for an additional massless and non-interacting contribution to the energy density of the Universe, encoded in terms of ∆N eff . We find that, when letting ∆N eff vary, the bounds are only slightly relaxed, and at 95% CL read: τ νi→νj +φ > 0.9 × 10 9 s m νi 0.05 eV 3 Planck , (15) τ νi→ν4+φ > 0.3 × 10 9 s m νi 0.05 eV 3 Planck , (16) and hence are barely affected by an additional contribution to N eff from massless non-interacting species.
At 68% CL the neutrino lifetimes are bounded to be Finally, in order to highlight the constraining power of Planck CMB observations on invisible neutrino decays, we study how much the fit to the Planck 2015 data is degraded in a scenario with a considerably short neutrino decay lifetime: τ ν = 1.3 × 10 5 s (m ν /0.05 eV) 3 . We run an MCMC fixing this lifetime and allowing to vary the six standard cosmological parameters, the Planck nuisance parameters, and also ∆N eff . For the Planck 2015 TT+lowP+lensing data set, we find that the bestfit points have a higher minimum χ 2 , as compared to ΛCDM, of: Similarly, for the Planck 2015 TTTEEE+lowP+lensing data set we find: Within Gaussian statistics, these results demonstrate that fast neutrino decays at a rate of τ ν = 1.3 × 10 5 s (m ν /0.05 eV) 3 are clearly disfavoured by Planck CMB observations with a 5.4−7.7 σ and 3.1−3.9 σ significance for ν i → ν j +φ and ν i → ν 4 +φ decays respectively.

V. SUMMARY AND DISCUSSION
In this work we have revisited the cosmological constraints on invisible neutrino decay modes in light of the rather weak constraints from solar, atmospheric and long-baseline neutrino experiments. Collectively, we have exploited cosmological observations to place stringent constraints on invisible neutrino decays. See Ref. [38] for the previous CMB analysis. Figure 4 highlights the main constraints on invisible neutrino decays derived in this work.
In summary, the main results obtained in this paper are: 1. The invisible neutrino decay lifetime should be τ ν > 10 −3 s at 95% CL in order for the primordial elements to be synthesised successfully. In addition, we have discussed other neutrino decay scenarios for which it applies beyond ν i → ν j + φ decays.
3. The CMB constraints on invisible neutrino decays are robust upon modifications of the cosmological model. In particular, we have shown that the bounds are barely affected by possible contributions to N eff from non-interacting dark radiation.
4. Invisible neutrino lifetimes τν < 10 5 s (mν /0.05 eV) 3 are highly disfavoured by Planck CMB observations. Neutrino decays νi → νj + φ occurring at such a rate are excluded with a 7.7 − 5.4 σ significance, where the two numbers correspond to the full Planck 2015 data set, with and without including high-polarization data respectively.

VI. OUTLOOK
Cosmological constraints on invisible neutrino decays are typically orders of magnitude more stringent than those derived from laboratory and solar neutrino experiments. However, it must be noted, that in order to set FIG. 6. TT power spectrum for various values of the neutrino decay lifetime as compared to ΛCDM, assuming the same cosmological parameters. We consider the neutrino decay process νi → νj + φ and we fix mν i = 0.05 eV for concreteness. The grey band indicates cosmic variance. cosmological constraints we have implicitly assumed that the neutrino interactions that trigger neutrino decays are time independent. This is not the case in some models in which neutrinos do decay today, but would not have done so in the early Universe [18]. Hence, all terrestrial, astrophysical and cosmological bounds are meaningful.
Sensitivity to invisible neutrino decays is generically expected to improve in the future. Bounds from current and upcoming laboratory experiments and neutrino telescopes have been a subject of intense study, see e.g. [50,51,[82][83][84][85][86][87][88]. From the cosmological side, the positive detection of the neutrino energy density would represent a very strong constraint on the neutrino lifetime [89]. In addition, since baryon acoustic oscillations have now been shown [90] to require the presence of free-streaming neutrino species, we think they could also be used to set constraints on the invisible neutrino lifetime, and could potentially reach τ ν ∼ O(10 15 ) s.
In this work we have focused on CMB constraints upon invisible neutrino decays. One may naively think that future CMB observations could help to tighten the constraints on invisible neutrino decays. However, we do not expect this to be the case. In Figure 6, we show the TT power spectrum for some neutrino decay scenarios as compared to ΛCDM. One can clearly appreciate that, for neutrino lifetimes that are not already excluded by Planck 2015 observations, the only modification to the power spectrum occurs for < 1000, which corresponds to angular scales that have been measured already with cosmic variance error bars by the Planck satellite. This means that we expect constraints to improve only slightly, and particularly from future polarization measurements. Hence, in the very near future, it would be interesting to analyze invisible neutrino decays with the final Planck data release [60] -once it becomes publicly available -since the high-polarization likelihood has changed with respect to the 2015 Planck data release [60].
To conclude, in this work we have shown that when the full Planck 2015 data is considered, neutrino decay lifetimes of τ ν = (2 − 14) × 10 9 s (m ν /0.05 eV) 3 are preferred over neutrinos being stable with a 1-2 σ significance. From the particle physics perspective, although beyond the scope of this work, it would be very interesting to work out a UV complete model that is capable of generating such neutrino lifetimes while being consistent with all other laboratory constraints, in particular those arising from the null searches of charged lepton-flavour violation processes. TABLE II. Marginalized posteriors for the standard cosmological parameters plus the neutrino decay lifetime and ∆N eff from the the analysis to Planck 2015 data. i, j label active neutrino mass eigenstates. The rows correspond to the mean and ± 1σ errors but for the case of the neutrino decay parameters in which we quote a 95% CL bound.