Reconstruction of the neutrino mass as a function of redshift
Christiane S. Lorenz, Lena Funcke, Matthias Löffler, Erminia Calabrese
RReconstruction of the neutrino mass as a function of redshift
Christiane S. Lorenz, ∗ Lena Funcke, † Matthias L¨offler, and Erminia Calabrese Institute for Particle Physics and Astrophysics, ETH Z¨urich,Wolfgang-Pauli-Strasse 27, CH-8093 Z¨urich, Switzerland Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada Seminar for Statistics, Department of Mathematics,ETH Z¨urich, R¨amistrasse 101, CH-8092 Switzerland School of Physics and Astronomy, Cardiff University, The Parade, Cardiff, CF24 3AA, UK (Dated: Received March 1, 2021; published – 00, 0000)We reconstruct the neutrino mass as a function of redshift, z , from current cosmological data usingboth standard binned priors and linear spline priors with variable knots. Using cosmic microwavebackground temperature, polarization and lensing data, in combination with distance measurementsfrom baryonic acoustic oscillations and supernovae, we find that the neutrino mass is consistent with (cid:80) m ν ( z ) = const. We obtain a larger bound on the neutrino mass at low redshifts coinciding withthe onset of dark energy domination, (cid:80) m ν ( z = 0) < .
41 eV (95% CL). This result can be explainedeither by the well-known degeneracy between (cid:80) m ν and Ω Λ at low redshifts, or by models in whichneutrino masses are generated very late in the Universe. We convert our results into cosmologicallimits for models with post-recombination neutrino decay and find (cid:80) m ν < .
19 eV (95% CL),which is below the sensitivity of the KATRIN experiment. Thus, a neutrino mass discovery byKATRIN would hint towards models predicting both post-recombination neutrino mass generationand subsequent relic neutrino annihilation.
I. INTRODUCTION
Cosmological surveys and particle physics experimentsare independent and complementary probes of neutrinoproperties. Neutrino oscillation experiments have mea-sured the squared mass differences between neutrinomass eigenstates, giving a lower bound of 59 meV forthe total sum of the neutrino masses, (cid:80) m ν [1]. Inaddition, the Karlsruhe Tritium Neutrino Experiment(KATRIN) has constrained the electron neutrino massto be lower than m ν,e < . Planck satellite mission, combined with distance measurementsfrom baryonic acoustic oscillations (BAO) from the SloanDigital Sky Survey (SDSS) and 6dF have provided atight upper bound on the total sum of neutrino masses, (cid:80) m ν <
120 meV at 95% CL [4].Contrary to neutrino mass direct detection limits,the cosmological neutrino mass bound assumes a spe-cific cosmological model, usually the ΛCold Dark Mat-ter (ΛCDM) model or its single-parameter extensions.In more extended cosmological models, the cosmologi-cal neutrino mass limits can become less stringent dueto opening wider parameter spaces and/or covering morecomplex physics scenarios.On the one hand, the lower bound of (cid:80) m ν = 59 meVimposed by neutrino oscillation experiments can be re-laxed in cosmological analyses to (cid:80) m ν = 0 meV, forexample, if cosmological neutrinos disappear in the lateUniverse. Although the original “neutrinoless Universe” ∗ [email protected] † [email protected] proposal [5] has been ruled out , other scenarios like mod-els predicting post-recombination neutrino mass genera-tion and subsequent relic neutrino annihilation [9] arestill possible. In such models, the neutrino mass param-eter cannot be captured using cosmological data and canonly be measured using terrestrial and astrophysical ex-periments such as KATRIN. The possible cosmologicaldisappearance of the neutrino mass parameter has alsobeen proposed in the context of modified gravity theo-ries [10–12]; these should soon be tested with surveys likeEuclid [12]. If we allow for strong fine tuning, anotherpossibility to eliminate the cosmological neutrino massbounds would be to postulate a new light scalar particlethat couples to neutrinos, with a coupling constant thatneeds to be smaller than g ∼ − to avoid laboratoryconstraints [13].On the other hand, the upper cosmological neutrinomass bound is sensitive to a number of model assump-tions and can be slightly relaxed, for example whenthe dark energy equation of state is allowed to vary intime [14–20], when the curvature of the Universe is notfixed [20], when considering additional relativistic de-grees of freedom [4], or when assuming non-standard mo-mentum distributions of the cosmic neutrinos [21]. More-over, it has been shown that the cosmological neutrinomass bound can be substantially weakened when neutri-nos are unstable and thus their lifetime is smaller thanthe age of the Universe [22–26] or when neutrino massesare varying in time [9, 27–29]. Note, however, that the By free-streaming of the cosmic neutrino background before pho-ton decoupling [6, 7], by the resulting phase shift in the CMBpeaks [8], and by precise CMB measurements of the effective num-ber of species in the early Universe [4]. a r X i v : . [ a s t r o - ph . C O ] F e b strongly relaxed neutrino mass bounds of (cid:80) m ν < . (cid:80) m ν < . Planck
Planck (cid:80) m ν =0 .
12 eV (95% CL,
Planck √ ρ Λ ∼ m ν ∼ meV. If cosmological data permitlarger neutrino masses during dark energy domination,there could be an intriguing theoretical connection be-tween these two phenomena (see, e.g., Refs. [9, 27, 36–39]and references therein).The tight neutrino mass limit from cosmology is mak-ing a direct detection from Tritium β -decay experimentslike KATRIN very challenging. Indeed, KATRIN’s tar-get sensitivity of 200 meV at 90% CL for the ν e mass [40]will unlikely hit the cosmological limit obtained in eitherΛCDM or its simple extensions [4, 20]. Assuming all ob-servations and analysis assumptions are correct and freeof systematic effects, we are entering a regime where a di-rect detection of neutrino masses with particle detectorssuch as KATRIN could become a strong hint for non-standard neutrino physics. At the same time, the nextgeneration of cosmological surveys aim to improve theirreach in neutrino mass sensitivity and to make the firstdetection [41, 42].To work towards these future goals, we present herea model-independent approach to investigate the pos-sibility that neutrino masses change on cosmologicaltimescales, reconstructing the neutrino mass as a func-tion of redshift.A similar methodology has been applied to reconstructpossible variations in other cosmological parameters, inparticular dark energy parameters [35, 43–55] and theHubble parameter [35, 56–61] as a function of redshift,the shape of the primordial power spectrum as a function of wavenumber [62–66], and parameters describing devia-tions from general relativity as a function of redshift andwavenumber [67]. Exploring variations of cosmologicalparameters in time also gains in importance in light oftensions between parameters inferred from current high-and low-redshift data (see, e.g., Ref. [68]).In general, reconstructions fully accounting for degen-eracies and the interplay between cosmological param-eters, such as the dark matter density and dark energyparameters, are especially hard to achieve [69–71]. In thecase of the neutrino mass, a model-independent recon-struction is particularly challenging. A comprehensivemodel for time-varying neutrino masses would require aninteraction with other energy sectors, such as dark radi-ation or dark energy, to satisfy energy conservation laws.Additionally, including such interactions permits to fullycapture and exploit the physics signatures of the model.While in the case of dark radiation this interaction isnegligible for many cosmological scenarios (e.g., the darkradiation resulting from neutrino decays has negligiblecosmological impact [23]), in the case of dark energy thisinteraction could alter w de ( z ) and thus generate multiplesignatures that allow us to place stronger limits on themodel.In this paper, we take the conservative approach to ig-nore additional constraining power coming from the in-clusion of the coupling between the neutrino and the darksector, and perform a model-independent analysis thatonly focuses on neutrinos without assuming any specificinteractions with other dark sectors. The method usedhere expands on the work done in Ref. [29], which as-sumed a specific time-varying neutrino mass model, andalso spans models with neutrino decays for which we willset new limits.The paper is structured as follows. In Sec. II we ex-plain the theoretical background of cosmological neutrinomass constraints and neutrino mass models. In Sec. IIIwe present our methodology, in particular the differentdatasets and reconstruction methods. We present ourresults in Sec. IV and summarize and discuss them inSec. V. II. THEORETICAL BACKGROUNDA. Neutrino mass constraints from cosmologicalprobes
At the beginning of their cosmic journey, neutrinosare relativistic particles and behave as a radiationcomponent in the early Universe. When their kineticenergy term drops below the mass term due to coolingin an expanding Universe, neutrinos start to behaveas non-relativistic, massive particles. The redshift ofthis transition is inversely proportional to the neutrinomass [72]: neutrinos with a smaller mass become non-relativistic at a later time compared to neutrinos with ahigher mass. Neutrino masses thus affect cosmologicalobservations during different cosmic epochs, which inturn allows us to probe (cid:80) m ν at different redshifts. The cosmic microwave background–
Constraining neu-trino properties with the CMB has been a rich researcharea with extensive literature (see, e.g., Refs. [73–75]).To summarize, the key effects we look for in CMB probesare:(i) Effects on the background evolution of the Universevia changes in the angular diameter distance at recombi-nation, D A ( z rec ) and the Hubble parameter [73, 74]. Ingeneral, there is a strong degeneracy between neutrinoparameters and the Hubble constant [73, 76–79].(ii) Effects on the evolution of perturbations. In partic-ular, the integrated Sachs-Wolfe effect is affected, both atearly times during radiation domination (eISW) [73, 74],as well at late times when dark energy starts to dominatethe evolution of the Universe ( (cid:96) ISW) [80]. The eISW de-pends on the relativistic degrees of freedom, N eff , as wellon neutrino masses, but in a different way. Whereas N eff mostly changes the amplitude of the first peak in theCMB anisotropy power spectrum, neutrino masses affectthe amplitude of the eISW on a large range of multipolesdepending on the neutrino mass and the correspondingfree-streaming scale [73, 74, 81]. The (cid:96) ISW effect dom-inates at (cid:96) <
30 and is therefore elusive because of cos-mic variance in CMB data alone. However, when cross-correlated with galaxy number counts, this effect is apromising avenue for measuring neutrino masses [82].(iii) Effects on the matter distribution deflecting theCMB photons by gravitational lensing [83]. This willleave both an imprint on the CMB temperature and po-larisation anisotropies (in the high- (cid:96) region of the spec-tra), as well as generate a CMB lensing convergencesignal, C φφ(cid:96) . The latter probe will capture the small-scale suppression of the matter power spectrum due tolarge neutrino thermal velocities and corresponding free-streaming out of density fluctuations [74, 84]. Dependingon whether the neutrino wavelength is above or below thefree-streaming wavelength, neutrinos cluster as cold darkmatter and baryons, or free-stream out of gravitationalwells. This slows down the clustering of matter, leadingto a suppression of the matter power spectrum on thecorresponding scales which is more pronounced for largerneutrino masses [85]. The CMB damping tail is measuredwith high precision with current data [86–88] and CMBlensing is now in a high signal-to-noise regime [89, 90].(iv) The optical depth to reionization τ is degeneratewith the amplitude of scalar primordial fluctuations A s .Since the amount of clustering of cosmic structures istightly linked to (cid:80) m ν , this in turn becomes a strongdegeneracy between τ and (cid:80) m ν . CMB polarizationmeasurements of τ enable us to obtain a tighter con-straint on (cid:80) m ν [16, 91]. Baryonic acoustic oscillations–
Before CMB decou-pling, baryons and photons are tightly coupled to eachother. At the time of recombination, the CMB photons decouple from the baryons, and the oscillations of thebaryon-photon fluid are frozen in the CMB anisotropies.In addition to the CMB, these oscillations also leavea characteristic imprint in the large scale structureof the Universe, both transverse as well as along theline of sight [92, 93]. In particular, the size of theBAO is known, and therefore BAO can be used asstandard rulers to infer either H ( z ) r s ( z ∗ ) or D A /r s ( z ∗ ),where D A ( z ) is the angular diameter distance (see, e.g.,Refs. [94, 95] for reviews) and r s the sound horizon atdecoupling z ∗ . These quantities are in particular sensi-tive to the matter density Ω m h . BAO measurementsallow to constrain the energy contribution of massiveneutrinos to the matter density. The BAO feature hasbeen detected in the clustering of galaxies [92, 93, 96], aswell as in the clustering of low-redshift quasars [97–99],and in the correlations of Lyman- α systems [100, 101](see below). This latter probe provides an additionalmeasurement to standard BAO data and extends BAOobservations towards high redshifts. Lyman- α forest– The absorption lines of neutralhydrogen in the intergalactic medium (IGM) in quasarspectra are sensitive to cosmological parameters, andprobe cosmic structure formation at redshifts between z ∼ −
6. In particular, Lyman- α forest measurementscharacterize small structures on the scale of sub-Mpcto Mpc. This anchors the level of the matter powerspectrum on scales between k ∼ . − α forest measurements to constrain neutrinomasses. The current upper bound from Lyman- α measurements alone is Σ m ν < .
71 eV (95% CL) [106].Probing cosmological parameters with the Lyman-alphaforest is also challenging because of observational andastrophysical systematics, which need to be modeled inthe analyses [110, 111].
Supernovae–
Measurements of the luminosity distance D L ( z ) from supernovae explosions are not directlysensitive to neutrino properties. However, they can beused to constrain dark energy parameters, such as thedark energy density Ω Λ and the dark energy equationof state w de ( z ) (see, e.g., Ref. [112] for a review), whichhelps significantly to break degeneracies with neutrinoparameters.Combining the different cosmological probes describedabove enhances significantly the constraining power com-ing from only one of the datasets (see, e.g., Refs. [4, 41,73, 75, 113]). This is due to the fact that (i) differentprobes are capturing the physics of the Universe at differ-ent redshifts, and (ii) different probes depend on differentparameter combinations, which follow different degener-acy directions. For example, Ref. [73] describes in detailhow the combination of CMB and BAO helps to breakthe H − (cid:80) m ν degeneracy. B. Neutrino mass models
Most of the possible neutrino mass models, in par-ticular the ones arising from the famous seesaw mecha-nism [114–119], cannot be tested using cosmological data.From a cosmological perspective, more focus is then natu-rally placed on studying neutrino mass mechanisms thatmake cosmologically testable predictions, such as mod-els providing neutrino mass variations on cosmologicallyinteresting timescales.As mentioned earlier, when allowing for the neutrinomass to vary as a function of redshift, an interactionwith the dark energy or dark radiation sector is requiredto satisfy energy conservation laws. The most popularmodels that couple the neutrino and dark energy sectorsare mass varying neutrino (MaVaN) scenarios (see, e.g.,Ref. [27]). Here, the neutrinos couple to a light scalarfield, which slowly rolls in a flat potential. Originally,this direct link between neutrino masses and dark en-ergy was proposed to explain the similarity between theenergy scales of neutrino masses and quintessence-likedark energy ( E ∼ − eV). However, the coupling toa light scalar mediates an attractive force between theneutrinos and leads to bound state formation [120], suchthat the light scalar field can only explain dark energyunder rather special circumstances (see, e.g., Refs. [36–38]). For example, the Growing Neutrino Quintessencemodel studied in Ref. [37] yields time-dependent neutrinomasses, which vanish in the early Universe and raise from m ν = 0 eV at a (cid:46) . m ν ∼ . a = 1.Coupling the mass-varying neutrino sector to dark ra-diation is less trivial and usually occurs with simultane-ously coupling both sectors to the dark energy sector.For example, it has been proposed that a simultaneousvariation of neutrino masses, a light sterile neutrino frac-tion, and dark energy can be used to test certain aspectsof the Weak Gravity Conjecture [39].Another gravitational avenue to couple neutrinos todark energy and dark radiation is the gravitational neu-trino mass model proposed in Ref. [9], which servedas a motivation for the previous cosmological study inRef. [29]. The key cosmological prediction of this model isthat the cosmological neutrino mass parameter vanishes.Indeed, the model predicts massless neutrinos in the earlyUniverse and the generation of neutrino masses in a late-time cosmological phase transition at T (cid:46) m ν . In thephase transition, neutrino masses are generated, followedby neutrino decay into the lightest mass eigenstate and rapid annihilation into massless Goldstone bosons (seeRefs. [9, 121, 122] for more details). Thus, the model pre-dicts massless neutrinos in the early Universe and mass-less dark radiation in the late Universe. The intermediateregime with massive neutrinos, which exists directly af-ter the phase transition, exists only for cosmologicallynegligible timescales. This implies that the cosmologi-cal neutrino mass parameter would be zero, as currentlypreferred by cosmological data [4]. This also implies thatexperiments aiming at a direct detection of the relic neu-trino background, such as the proposed PTOLEMY ex-periment [123], would not be able to detect this back-ground, unless we allow for substantial neutrino asym-metries (see below).If the model in Ref. [9] is extended by allowing forneutrino asymmetries (i.e., more neutrinos than antineu-trinos or vice versa), not all neutrinos would find an an-tineutrino partner to annihilate, leaving behind a frac-tion of relic neutrinos with a non-zero cosmological massparameter. As Ref. [29] demonstrated, even in this case,the cosmological neutrino mass bound would be substan-tially weakened to (cid:80) m ν < . Planck
Planck (cid:80) m ν strengthened bya factor of O (2) from (cid:80) m ν = 0 .
21 eV (95% CL,
Planck (cid:80) m ν = 0 .
12 eV (95% CL,
Planck
Planck (cid:80) m ν (cid:46) . Planck At first sight, one might expect that PTOLEMY could detect themassless Goldstone bosons, which are neutrino-composite bosons(similar to the light quark-composite mesons in QCD). However,the boson’s energy is E = T ν < m ν , while the neutrino capturewould release the other neutrino of the bound state, requiring anenergy of at least m ν . This would violate energy conservation,unless one of the neutrinos is almost massless. We thank PedroMachado for bringing up this argument. These and other theoretical motivations to studyneutrino mass variations on cosmologically interestingtimescales, which typically yield non-trivial interactionswith other cosmological sectors such as dark energy anddark radiation, are the main rational for the work thatwe present here.
III. METHODOLOGYA. Data
Motivated by the expected contribution highligthed inSec. II A, we include the following datasets in our analy-sis:1.)
CMB and CMB lensing:
We use CMB tempera-ture, polarization and lensing data from the
Planck
BAO:
We use BAO from 6dF [125], the Sloan Dig-ital Sky Survey (SDSS) DR7 Main Galaxy Sam-ple (MGS) [126] and the SDSS Baryon OscillationSpectroscopic Survey (BOSS) twelfth data release(DR12) [127]. The mild discrepancies seen between
Planck and Lyman-alpha BAO data have decreasedin recent releases of the Lyman-alpha BAO fromSDSS DR14 eBOSS [128, 130]. Therefore we in-clude also this additional BAO dataset which pro-vide two data points at z ∼ . z ∼ .
3. Wealso include the BAO measurement from quasarsfrom the 14th data release of the extended BOSS(eBOSS) quasar sample [98], giving us another dat-apoint at z ∼ . Supernovae:
We additionally add type IA su-pernova data from from the Pantheon SupernovaeSample [129] in order to break the degeneracies be-tween dark energy and neutrino masses.We demonstrate the impact of these specific datasetslater in Sec. IV, and summarize their respective redshiftrange in Tab. I.
B. Reconstruction method
We modify the publicly available Einstein-Boltzmanncode
CAMB [131] and the corresponding Monte-CarloMarkov chain package
CosmoMC [132] in order to im-plement different reconstruction methods for redshift-dependent neutrino masses.
1. Binned reconstruction
As a first attempt, we parameterize the neutrino masswith a step function for which it is assumed that theneutrino mass has a constant positive amplitude in each redshift bin. In the statistics literature this is called aregressogram [133]. To maximally exploit our datasetswe use a parameterization with six redshift intervals: (cid:88) m ν ( z ) = (cid:80) m ν, (0 ≤ z < z ) (cid:80) m ν, ( z ≤ z < z ) (cid:80) m ν, ( z ≤ z < z ) (cid:80) m ν, ( z ≤ z < z ) (cid:80) m ν, ( z ≤ z < z ) (cid:80) m ν, ( z ≥ z ) . (1)We choose the edges of the redshift bins to pinpoint spe-cific transitions in the composition of the Universe andto highlight the impact of using different cosmologicalprobes. We set z = 0 . z = 3, z = 10, z = 100 and z = 1100. The first bin explores (cid:80) m ν during dark en-ergy domination, the second bin spans the BAO interval,the third and fourth bin include most of the remaininginformation expected in CMB lensing, and the second tothe last bin stretches out to the time of CMB decoupling.The last bin for z > w de ( z ) from gamma-ray bursts [134].
2. Spline priors
To perform a smoother fit and to potentially identifyfeatures in the neutrino mass sum that are hidden in thebinned reconstruction, we also consider Bayesian regres-sion splines with variable knot points [135].The knots correspond to the bin margins of the re-gressogram prior function described above, and modelchange points where the trajectory of the neutrino massmight change its slope. Previous literature in cosmol-ogy has often used splines with fixed knot positions (see,e.g. [35, 58, 63]). This, however, requires to choose thepositions of the knots in advance and can therefore signif-icantly influence or bias the result of the reconstruction.Here, we estimate the position of the knots from the dataand include them as free parameters in the analysis. Us-ing variable knot positions yields a more flexible fit thanin the case with fixed knots and allows the fit to adaptto underlying features of the model. This methodologyhas previously successfully been applied several times incosmology [47, 136, 137], most recently in Ref. [61].Compared to binned priors, the resulting reconstruc-tion is smooth and not piecewise constant anymore. Wemodel two knots, z and z , and linearly interpolate be-tween (cid:80) m ν,z , (cid:80) m ν,z , (cid:80) m ν,z and (cid:80) m ν,z . For our Data set Redshift rangePlanck 2018 CMB TTTEEE [4, 87] mostly z = 1100Planck 2018 CMB lensing [4, 124] 0 ≤ z ≤ z = 0 . z = 0 . z = 0 . , . , . z = 1 . α ) [128] z = 2 . α -QSO) [128] z = 2 . . < z < . analysis, we choose z = 0 and z = 1100. In that case,the neutrino mass at redshift z is given by (cid:88) m ν ( z ) (2)= (cid:80) m ν,z + ( (cid:80) m ν,z − (cid:80) m ν,z ) zz ( z ≤ z < z ) (cid:80) m ν,z + ( (cid:80) m ν,z − (cid:80) m ν,z ) z − z z − z ( z ≤ z < z ) (cid:80) m ν,z + ( (cid:80) m ν,z − (cid:80) m ν,z ) z − z z − z ( z ≤ z < z ) (cid:80) m ν,z ( z ≥ z ) . We then choose a uniform prior on the logarithm of thecorresponding redshift with − ≤ log ( z i ) i =1 , ≤ . z has to beequal or larger than z , in accordance to the definitionabove in Eq. (2). In addition, we choose z , > . (cid:80) m ν at z , z and z , and twofor the positions of z and z . In order to obtain point-wise credible bands for (cid:80) m ν ( z ), we compute the neu-trino mass sum for two hundred points in z as derivedparameters with GetDist [138] to sample well the entireredshift range. We then compute pointwise the mean andcredible intervals for each point in z . IV. CONSTRAINTS FROM CURRENT DATA
For our neutrino mass reconstruction, we run MCMCchains with our modified version of
CosmoMC . Since bothour methods only touch the neutrino mass modelling,we vary the standard ΛCDM parameters alongside withthe new parameters of the reconstruction: the cold darkmatter density Ω c h , the baryon density Ω b h , the scalarspectral index n s , the amplitude of primordial fluctu-ations A s and the optical depth to reionization τ . Inaddition, we have five additional parameters for the re-construction with fixed bins ( (cid:80) m ν, , (cid:80) m ν, , (cid:80) m ν, , (cid:80) m ν, and (cid:80) m ν, ), and five for the reconstructionwith linear splines and variable knots ( (cid:80) m ν,z , (cid:80) m ν,z , (cid:80) m ν,z , (cid:80) m ν,z and the position of the knots log ( z ) and log ( z )). We choose a uniform prior between [0:5]for (cid:80) m ν in the individual redshift bins (similarly to the Planck (cid:80) m ν in the differentbins, leaving them to vary independently. In addition,we assume a normal neutrino mass hierarchy, in line withcurrent cosmological constraints [4, 20, 139]. The choiceof the neutrino mass hierarchy should not significantlyaffect our final results presented in Sec. IV, as currentcosmological data cannot (yet) distinguish between thedifferent neutrino mass hierarchies [20, 75, 139–148]. A. Binned reconstruction
We show the results of the binned neutrino mass re-construction in Fig. 1, and in Table II. We plot both the95% and the 68% CL upper limits for each mass param-eter, as well as the 95% limit for (cid:80) m ν ( z ) = const . (i.e.,the standard single parameter extension of ΛCDM formassive neutrinos) obtained from the same data combi-nations. For the full data combination including CMB,CMB lensing, BAO and SN, we find (cid:88) m ν (0 ≤ z < . < .
13 eV (cid:88) m ν (0 . ≤ z < < .
42 eV (cid:88) m ν (3 ≤ z < < .
37 eV (cid:88) m ν (10 ≤ z < < .
19 eV (cid:88) m ν (100 ≤ z < < .
32 eV (cid:88) m ν ( z ≥ < .
40 eV (95% CL) . These results are consistent with neutrino masses con-stant in time. In addition, we observe that the neu-trino mass bound becomes less stringent at low redshifts( z ≤ z ≥ Top row:
In the top row of Fig. 1, we plot the z -dependent constraints on the neutrino mass sum fromCMB anisotropy data alone. Because the binnedreconstruction introduces more free parameters, theconstraints in all bins are consistent but systematicallylarger than the standard neutrino mass bound from thelatest Planck release [1, 4] . The CMB anisotropy dataput a strong constraint on (cid:80) m ν at high redshifts, butonly a weak constraint at low redshifts. As describedearlier, at low redshifts, the CMB anisotropy data wouldmostly be sensitive to neutrino masses via the lateintegrated Sachs-Wolfe effect at low multipoles (cid:96) , whichis cosmic variance limited and subject to significantparameter degeneracies. In the lowest redshift bin(0 < z < . (cid:80) m ν = 2 . (cid:80) m ν . This mightbe due to the fact that at recombination the constrainton (cid:80) m ν comes mostly from the early integrated Sachs-Wolfe effect, the CMB damping tail, and backgroundeffects, while all other effects from neutrino masses onlybecome relevant at intermediate to low redshifts. Second row:
In the second row in Fig. 1, we plotthe constraints on (cid:80) m ν ( z ) from CMB anisotropiesplus the CMB lensing reconstruction. The CMB lensingkernel peaks around z ∼ . ≤ z <
3. Other constraints are also improved byan indirect effect: CMB lensing is also sensitive to thetotal matter density and the dark energy density [151],and therefore helps to constrain (cid:80) m ν (0 < z < . Λ , Ω m , and (cid:80) m ν . This can also be seenin Fig. 2, where we show the results for (cid:80) m ν in thedifferent bins, as well as its correlation with Ω Λ and Ω m . Third row:
The BAO data points used in this workare distributed over the whole redshift range between z = 0 . − .
35. This mostly improves the constraints Note that the (cid:80) m ν ( z ) = const. line should be compared to the95% shaded region. In addition, we have plotted in Figs. 1 and 4the 95% CL and 68% CL upper limits (i.e., we have removed the32% and 5% highest samples for these limits). in the first two neutrino mass bins. In addition, theinclusion of BAO reduces significantly the degeneracybetween Σ m ν and Ω m , and therefore also improves theoverall constraints in all redshift bins. Bottom row:
We include SN data from the Pan-theon 18 data set to constrain further the dark energycomponent. We observe the smallest uncertainty of (cid:80) m ν in the bin between 10 ≤ z ≤ f ( k, z ) at differentscales and redshifts [152]. The scale-dependent featurein this quantity is a source of information for neutrinomasses [153–156]. Including the growth rate measure-ments, we found a decrease of the bound of (cid:80) m ν of38% in the lowest redshift bin and only a marginal im-provement in other bins. The exact modeling of RSD isunder active development and therefore we present ourfinal results without including RSD. B. Reconstruction with splines
Next we reconstruct the neutrino mass sum with linearsplines and variable knots (see Sec. III). We investigate atwhich redshift the sensitivity on (cid:80) m ν starts to weakenby allowing the knots z and z to vary as well. We showour results for this reconstruction on the right-hand sideof Fig. 1. We also present our results in Table ?? . Forthe final data combination we find (cid:88) m ν ( z = 0) < .
41 eV (cid:88) m ν ( z = 1100) < .
54 eV (95% CL) . Furthermore, we find the largest uncertainty of the neu-trino mass constraint, (cid:80) m ν < .
47 eV, at a redshift of z ∼ . (cid:80) m ν at different redshifts are similar. Thereare small differences, in particular for (cid:80) m ν ( z = 0), be-cause in the binned reconstruction (cid:80) m ν is fixed betweenthe bin margins. We also note that, as expected, weobtain smooth curves for the reconstruction with linearsplines. This is due to the fact that the positions ofthe knots z and z are free parameters in the analysis, Dataset CMB CMB + CMBL CMB + CMBL + BAO CMB + CMBL + BAO + SN (cid:80) m ν ( z ≤ .
5) (eV) 4.84 3.07 1.73 1.13 (cid:80) m ν (0 . ≤ z ≤
3) (eV) 3.20 2.85 0.51 0.42 (cid:80) m ν (3 ≤ z ≤
10) (eV) 1.64 1.62 0.40 0.37 (cid:80) m ν (10 ≤ z ≤ (cid:80) m ν (100 ≤ z ≤ (cid:80) m ν ( z ≥ (cid:80) m ν ( z = 0) (eV) 4.79 2.90 1.44 1.41 (cid:80) m ν ( z = 0 .
5) (eV) 3.52 2.75 2.31 0.75 (cid:80) m ν ( z = 3) (eV) 2.09 1.85 0.19 0.18 (cid:80) m ν ( z = 10) (eV) 1.05 1.11 0.16 0.15 (cid:80) m ν ( z = 100) (eV) 0.69 0.47 0.17 0.18 (cid:80) m ν ( z = 1100) (eV) 0.52 0.50 0.51 0.54TABLE III. 95% upper limits on the amplitudes of the neutrino mass at different redshifts reconstructed with linear splinesand variable knots and using different data combinations. smoothing the posterior means and credible bands. Com-pared to the binned reconstruction, we observe that thecredible bands of the reconstructed curve via splines withvariable knots exhibit more features despite the samenumber of free parameters.For constant neutrino masses in time, we would, inprinciple, expect flat posteriors for the two knots z and z , as the position of the knots does not matter whenfitting a constant function. However, even in that case,the posterior for z and z can still vary because of pos-sible degeneracies between the neutrino mass sum andother cosmological parameters and because of the differ-ent amount of datapoints at different redshifts, resultingin local changes of the uncertainty of the neutrino masssum.In Fig. 3, we see that there is no clear preference forthe position of the knots z and z . For the position ofthe first knot (change point) we find z < .
92 (68% CL)for the full data combination. This limit is close to theonset of dark energy domination at z ∼ . z is themost pronounced for the final data combination.The position of the second knot, z , is less constrained,but also moves towards lower redshifts when includingmore data. Whereas the first knot, z , mostly is neededto model the increase of the uncertainty of (cid:80) m ν ( z ) atlow redshifts, the second knot, z , determines the widthof the regime where the uncertainty of (cid:80) m ν ( z ) is thesmallest. C. Limits on neutrino mass decay models
In this section, we note that our a priori model-independent reconstruction can constrain specific modelsthat predict new physics beyond the Standard Model of Particle Physics. In particular, as discussed in Sec. II B,it has been proposed that neutrinos can decay into darkradiation after they become non-relativistic. Due toneutrino free-streaming for z (cid:38) Planck (cid:80) m ν (cid:46) . z ∼
600 [22–25]. The cosmological impact of theresulting dark radiation is negligible in these models, andtherefore we can use convert our limits into a bound forthese models.The constraint is shown in our final result plot, Fig. 4,which will be discussed in more detail in the next section.In this plot, the red dashed line is the neutrino mass sumas a function of the redshift z nr at which the largest neu-trino mass eigenstate m ν becomes non-relativistic [72],1 + z nr ∼ m ν meV , (3)assuming a normal neutrino mass hierarchy. The smallerneutrino mass eigenstates will become non-relativisticlater and thus can be neglected in this curve. Takingthe intersection of the red dashed line and our resultcurve, we can significantly strengthen the current limitof (cid:80) m ν < . (cid:88) m ν < .
19 eV (95% CL) . (4)Thus, we conservatively bound (cid:80) m ν for this case withour result for (cid:80) m ν ( z = 187), where 87 < z nr < (cid:80) m ν = 0 .
19 eV (95% CL) (see Fig. 4). We note thatthe bound in Eq. (4) is tightened due to two different ef-fects. First, we use more recent CMB data from the lat-est 2018
Planck release [4]. Second, Refs. [22–25] assume z012345 m ( e V ) CMB 68% CL95% CL m ( z )=const. z012345 m ( e V ) CMB 68% CL95% CL m ( z )=const. z012345 m ( e V ) CMB + CMBL 68% CL95% CL m ( z )=const. z012345 m ( e V ) CMB + CMBL 68% CL95% CL m ( z )=const. z012345 m ( e V ) CMB + CMBL + BAO 68% CL95% CL m ( z )=const. z012345 m ( e V ) CMB + CMBL + BAO 68% CL95% CL m ( z )=const. z012345 m ( e V ) CMB + CMBL + BAO + SN 68% CL95% CL m ( z )=const. z012345 m ( e V ) CMB + CMBL + BAO + SN 68% CL95% CL m ( z )=const. FIG. 1. Reconstructed upper bounds of the neutrino mass as function of redshift. The panels from top to bottom showincremental addition of data starting from
Planck
Planck
CMB lensing, BAO from BOSS DR12, 6dF and MGS, eBOSS DR14 quasars and Lyman-alpha, and SN from Pantheon18. The left panels show the results obtained with the binned parameterization and the right panels the limits from thereconstruction with linear splines and variable knots. In all cases, we also report with the black solid line the 95% CLconstraint from the same data combination for (cid:80) m ν ( z )=const. (i.e., the standard single parameter extension of ΛCDM formassive neutrinos). m m , m , m , m , m , m ,0 m m ,1 m ,2 m ,3 m ,4 m ,5 CMBCMB + CMBLCMB + CMBL + BAOCMB + CMBL + BAO + SN
FIG. 2. 2-dimentional contours showing the 68% and 95% CL and 1-dimensional posteriors for the neutrino mass parametersin the individual bins, Ω Λ and Ω m . Different colors show different data combinations, as in Fig. 1. that the neutrinos decay after becoming non-relativistic,which results in a lower decay redshift z decay (cid:46) z nr com-pared to the analyses of Refs. [23, 24]. For their upperneutrino mass bound, (cid:80) m ν = 0 . z decay (cid:46) z = 600, our bound would bestrengthened to (cid:80) m ν = 0 .
34 eV (95% CL). However,for a neutrino mass sum of (cid:80) m ν = 0 .
34 eV, the redshift z nr would be much lower, which further strengthens the mass bound.We note that the bound would become weaker whenassuming that the neutrinos decay before they becomenon-relativistic, as predicted by models proposed inRefs. [158, 159]. In this case, we can conservatively bound (cid:80) m ν with our result for (cid:80) m ν ( z = 1100), which resultsin (cid:80) m ν < .
54 eV (95% CL). All of our bounds shouldget further tightened when taking into account the al-tered neutrino perturbations (see, e.g., Ref. [22, 23]).1 log ( z ) log ( z ) l o g ( z ) CMBCMB + CMBLCMB + CMBL + BAOCMB + CMBL + BAO + SN
FIG. 3. The results for the change points z and z from thereconstruction with linear splines and variable knots. V. DISCUSSION AND CONCLUSION
In this paper, we reconstructed the cosmologicalneutrino mass sum as a function of redshift. Thisreconstruction was model independent, such that nospecific mass model or interaction with the dark energyor dark radiation sector was assumed. Our final resultis shown in Fig. 4. The figure shows the reconstructedneutrino mass sum for the full data combination ofCMB temperature, polarization and lensing, combinedwith BAO and SN data. We highlight the redshiftwhen dark energy starts to dominate. We also highlightthe redshift z nr at which the largest neutrino masseigenstate becomes non-relativistic, in order to constrainmodels [22–25] that predict neutrino decay after thisnon-relativistic transition (see Sec. IV and below).Our result is consistent with neutrino masses that areconstant in time, as predicted by the Standard Model ofCosmology. We observe a small increase of the boundon (cid:80) m ν at high redshifts ( z (cid:38) Λ and (cid:80) m ν at low redshifts, whichis shown in Fig. 2. We have explicitly tested this scenarioby imposing a strong prior on Ω Λ = 0 . ± .
01. Inthat case, the constraint on Σ m ν ( z = 0) decreased by44% to (cid:80) m ν ( z = 0) = 0 .
79 eV, which shows an impactfrom correlations with dark energy but is still larger thanstandard neutrino mass bounds. This latter effect mightbe due to the large number of additional free parametersneeded for the reconstruction. On the other hand, the large mass bound at lowredshifts could be explained by new physics. It has beena long-standing puzzle why the energy scales of neutrinomasses and dark energy are close ( m ν ∼ √ ρ Λ ∼ meV)but far away from all other known fundamental en-ergy scales, such as the Higgs or Planck scales. Ifcosmological data allow for larger neutrino masses inthe late Universe, 0 ≤ z ≤ Λ was found.The analysis presented in this paper could be extendedin different ways. On the theory side, it would be interest-ing to reconstruct (cid:80) m ν ( z ) fully implementing potentialinteractions with the dark energy or the dark radiationsector (see below). In particular, a coupling between thedark energy and the neutrino sectors would affect thedark energy perturbations, leading to more features inthe model and additional ways to decrease the degener-acy between dark energy and massive neutrinos.On the methodology side, one could model the num-ber of knots or bins with a hyperparameter [135] in orderto allow the data to decide on the number of knots. Tospeed up computations, this can be fitted with empiri-cal Bayes, where the hyperparameters are chosen as themaximizers of the marginalised likelihood [57, 161]. Theexact number of knots or bins can impact the results byunder- or overfitting the reconstructed function.On the data side, it would be interesting to re-construct (cid:80) m ν including more large scale structuredata, in particular galaxy clustering, cosmic shear, andcross-correlations of either the (cid:96) ISW effect and galaxyclustering or of CMB lensing and tracers of the largescale structure. These additional measurements of thesuppression of the matter power spectrum would also bean additional way to break the degeneracy with darkenergy (see, e.g. [42, 82, 113, 162–169]). To fully exploitthese data will however require careful modeling of thenon-linear scales (see, e.g. [170–178]), and of potentialsystematics, such as the scale-dependent galaxy bias inthe presence of massive neutrinos [146, 179–182].Finally, we comment on the implications of our resultsfor particle physics models and experiments. In partic-ular, our study considerably strengthens the previouslyreported cosmological mass bound (cid:80) m ν < . (cid:80) m ν < .
19 eV (95% CL). We note that theconstraints are still weaker than for constant neutrinomasses in time, because decay scenarios are insensitive2 z0.000.250.500.751.001.251.501.752.00 m ( e V ) CMB + CMBL + BAO + SN 68% CL95% CLNon-relativistic transitionDE domination m ( z )=const. (95% CL) FIG. 4. Neutrino mass limits as function of redshift obtained from
Planck (cid:80) m ν in the case when the neutrino with the largest neutrino mass becomes non-relativistic at redshift z , assuming a normal neutrino mass hierarchy. to late-time cosmological data from BAO and SN.However, our study pushes the neutrino mass boundsof such models below the sensitivity of the KATRINexperiment. This implies that a neutrino mass discoveryat KATRIN would hint towards models predicting post-recombination neutrino mass generation and subsequentrelic neutrino annihilation, such as proposed in Ref. [9].The cosmological disappearance of the neutrino massparameter might also be explained in the context ofmodified gravity [10–12], but this degeneracy betweenmassive neutrino and modify gravity effects will bebroken by future surveys, such as Euclid [12]. ACKNOWLEDGEMENTS
We thank Alexandre R´efr´egier, Thejs Brinckmann,Sunny Vagnozzi, Pedro Machado, Roni Harnik, Rapha¨elSgier and Neal Dalal for helpful discussions and / or com-ments on the draft. We also acknowledge helpful discus-sions at the Cosmology from Home Conference in 2020.CSL acknowledges support by Alexandre R´efr´egier’s Cos-mology research group at ETH Z¨urich. Research atPerimeter Institute is supported in part by the Govern-ment of Canada through the Department of Innovation,Science and Industry Canada and by the Province ofOntario through the Ministry of Colleges and Univer-sities. EC acknowledges support from the STFC ErnestRutherford Fellowship ST/M004856/2 and STFC Con-solidated Grant ST/S00033X/1, and from the EuropeanResearch Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (Grantagreement No. 849169). ML has been funded in part byETH Foundations of Data Science (ETH-FDS). 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