The sum of cosmic neutrino masses can be measured cosmologically as the subeV particles behave as hot dark matter whose main effect is to suppress the clustering of matter compared to a universe with the same amount of purely cold dark matter. Current astronomical data provide an upper limit on Sigma mnu between 0.07  0.12 eV at 95 confidence depending on the choice of data. This bound assumes that the cosmological model is LambdaCDM where dark energy is a cosmological constant the spatial geometry is flat and the primordial fluctuations follow a pure powerlaw. Here we update studies on how the mass limit degrades if we relax these assumptions. To existing data from the Planck satellite we add new gravitational lensing data from the Atacama Cosmology Telescope the new Type Ia Supernova sample from the Pantheon survey and baryonic acoustic oscillation BAO measurements from the Sloan Digital Sky Survey and the Dark Energy Spectrosopic Instrument. We find the neutrino mass limit is stable to most model extensions with such extensions degrading the limit by less than 10. We find a broadest bound of Sigma mnu lt 0.19 rmeV at 95 confidence for a model with dynamical dark energy although this scenario is not statistically preferred over the simpler LambdaCDM model.
['Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Phenomenology']
Cosmological limits on the neutrino mass sum for beyondLambdaCDM models
{'Cosmological limits on the neutrino mass sum for beyondΛ CDM models': "Helen Shao, 1, ∗ Jahmour J. Givans, 2, 1 Jo Dunkley, 3, 1 Mathew Madhavacheril, 4 Frank Qu, 5, 6 Gerrit Farren, 5, 6 and Blake Sherwin 5, 6  \n1 Department of Astrophysical Sciences, Peyton Hall,  \nPrinceton University, Princeton, NJ 08544, USA  \n2 Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA  \n3 Department of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA  \n4 Department of Physics and Astronomy, University of Pennsylvania,  \n209 South 33rd Street, Philadelphia, PA, USA 19104  \n5 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK 6 Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, UK (Dated: September 5, 2024)  \nThe sum of cosmic neutrino masses can be measured cosmologically, as the sub-eV particles behave as 'hot' dark matter whose main effect is to suppress the clustering of matter compared to a universe with the same amount of purely cold dark matter. Current astronomical data provide an upper limit on ∑ m ν between 0.07 - 0.12 eV at 95% confidence, depending on the choice of data. This bound assumes that the cosmological model is ΛCDM, where dark energy is a cosmological constant, the spatial geometry is flat, and the primordial fluctuations follow a pure power-law. Here, we update studies on how the mass limit degrades if we relax these assumptions. To existing data from the Planck satellite we add new gravitational lensing data from the Atacama Cosmology Telescope, the new Type Ia Supernova sample from the Pantheon+ survey, and baryonic acoustic oscillation (BAO) measurements from the Sloan Digital Sky Survey and the Dark Energy Spectrosopic Instrument. We find the neutrino mass limit is stable to most model extensions, with such extensions degrading the limit by less than 10%. We find a broadest bound of ∑ m ν < 0 . 19 eV at 95% confidence for a model with dynamical dark energy, although this scenario is not statistically preferred over the simpler ΛCDM model.  \nIntroduction .- Neutrinos are electrically uncharged spin-1 / 2 fermions which exist in one of three active flavor states: electron, muon and tau neutrinos ( ν e , ν µ , ν τ ). The discovery of neutrino flavor oscillations [1-3] showed that neutrinos have mass, with each neutrino flavor occupying a superposition of three mass eigenstates, m i ( i = 1 , 2 , 3). Oscillation experiments measure the masssquared splittings between these states, ∆ m 2 ij = m 2 i -m 2 j , with the mass states either in an inverted hierarchy (IH), where m 3 ≪ m 1 < m 2 , or the normal hierarchy (NH), where m 1 < m 2 ≪ m 3 . In the normal hierarchy the sum of the masses, ∑ m i , has a lower limit of 0 . 06 eV; in the inverted hierarchy it is 0 . 1 eV [4].  \nDirect-detection β -decay experiments give limits of m ≤ 0 . 8 eV (90% confidence level) for the electron neutrino by the KATRIN Collaboration [5]. Future experiments are expected to reach a sensitivity of 0 . 2 eV (90% c.l.) [6]. A tighter indirect limit has already been placed on the sum of neutrino masses from cosmological data [7-15], with the primary effect on cosmology being the suppression of the clustering of matter compared to a universe with the same amount of purely cold dark matter. Expressed as the sum of neutrino masses, the current upper bound at 95% confidence is in the range ∑ m ν < 0 . 07 -0 . 12 eV depending on the choice of data, with the tightest limit obtained when including new baryon acoustic oscillation (BAO) data from the Dark Energy Spectroscopic Instrument (DESI) [16-20]. This bound assumes that the correct cosmological model is  \nΛCDM, where dark energy is a cosmological constant, the spatial geometry is flat, cold dark matter interacts only via gravity, and the power spectrum of primordial perturbations follows a pure power-law. Previous studies have examined how the mass limit degrades if we loosen these assumptions, since some model extensions can mimic the effect of non-zero neutrino mass on the clustering of matter [e.g, 14, 19-24]. In this paper we update bounds using the latest data from the Atacama Cosmology Telescope (ACT) [18, 25], the Pantheon+ supernova survey [26], and a composite of BAO data from DESI and the Sloan Digital Sky Survey, allowing for variations in, for example, the spatial curvature and the equation of state of dark energy. Methodology .- We follow similar analysis methods as in [14, 18], estimating the posterior distribution for sets of cosmological parameters that include a varying neutrino mass sum, using a mixture of cosmological datasets. This includes the angular power spectra of the CMB intensity and polarization anisotropy as measured by the Planck satellite [17, 27]. We note that the PR3 Planck likelihood showed a 2.9 σ inconsistency between ΛCDM predictions of the CMB lensing parameter A L and its value as measured via its smearing effect on the temperature and polarization power spectra [13]. This led to artificially tightened constraints on the neutrino mass. Hence, for our baseline data, as in [18], we choose to use the Planck likelihood constructed from the PR4 NPIPE processed maps [27]. We also include BAO measurements which characterize the typical separation  \nFIG. 1. A ΛCDM model with a neutrino mass sum of 0 . 19 eV (green), and a CDM model with dark energy equation of state w and neutrino mass sum of 0 . 32 eV (red), are not distinguishable from a zero neutrino mass model using the CMB intensity power spectrum (top). However, neutrino mass sums of this size, within these different scenarios, can both be excluded at 3 σ significance using current data combination. The lower panel shows that their CMB lensing power spectra can be ∼ 6-9% lower than for the zero-mass ΛCDM model, and are disfavored by the indicated ACT DR6 lensing data.  \n<!-- image -->  \nof galaxies and provide a measure of the total amount of dark matter. Our 'BAO-1' dataset includes data measured by the 6dF [28] and Sloan Digital Sky Survey DR7 [29], the latter of which is combined with the BOSS DR12 and DR16 data [30]. We also use a composite 'BAO-2' dataset combining results from both SDSS and DESI, described in the appendix. 1  \nTo this baseline data combination, we add the angular power spectrum of the reconstructed gravitational lensing of the CMB measured by Planck [32] and ACT [25]; this provides a measure of the growth of cosmic structure at later times, peaking at typically half the age of the universe. We label these datasets as: 'CMB (Planck)' and 'CMB (ACT+ Planck )'. For the latter, we emphasize that we are using a variation of the ACT likelihood that includes Planck lensing data. Finally, we include the distances to Type Ia supernova as measured by the Pantheon+ survey [26], offering a direct way to measure the expansion rate of the universe. Here, we do not include the SH0Es calibration [33] and refer to this dataset as 'SNe'. 2 Consistent constraints on ∑ m ν have also been placed using the full-shape power spectrum of galaxies and redshift space distortions (e.g., [35]), though we do not use these in our analyses.  \nOur baseline model is ΛCDM, characterized by two initial conditions (the amplitude and spectral index of power-law scalar fluctuations, A s and n s , respectively), three ingredients (baryon density, cold dark matter density and dark energy fraction), and an optical depth to reionization, τ , to which we add the sum of neutrino masses. The extension model parameters we consider are the curvature, Ω k , the equation of state of dark energy that is either constant with time or varies with time, w ( z ), the effective number of neutrino species, N eff , and a running of the primordial scalar spectral index, n run 3 . As in [18], we generate theoretical predictions for these models with the CAMB numerical code, using the Parameterized Post-Friedmann (PPF) prescription for dark energy perturbations [36], and the Chevallier-PolarskiLinder (CPL) parameterization of the form [37, 38],  \nw ( z ) = w 0 + w a ( z 1 + z ) (1)  \nwhen we consider variations of dark energy equation of state as a function of redshift. 4 We estimate the posterior distribution of parameters with Metropolis Hastings methods using the Cobaya sampling code, quoting marginalised one-dimensional medians or 95% upper limits. To remain independent of neutrino oscillation experiments, we do not impose the lower mass limit for either NH or IH as a prior. However, we do require ∑ m ν to be  \nΛCDM+Σm  \n∑  \nFIG. 2. The 1D marginalized posterior distributions for ∑ m ν in the ΛCDM+ ∑ m ν model, with the lower bounds on neutrino mass sum required for the normal ( ∑ m ν < 0 . 06 eV) and inverted ( ∑ m ν < 0 . 1 eV) hierarchies shaded. The vertical lines indicate the 95% upper bound for each dataset. All three datasets include primary CMB data from Planck and CMB lensing from ACT and Planck . 'BAO-1' refers to SDSS measurements while 'BAO-2' refers to a mixture of SDSS+DESI data, as described in the text. Including SNe data results in a non-zero peak in ∑ m ν .  \n<!-- image -->  \npositive. Previous works have also employed hierarchyagnostic priors [e.g 14, 42] but did not find conclusive results on a preferred hierarchy.  \nAs described in [21], in this baseline model the neutrino mass is not strongly correlated with any of the other six parameters, but it has weak correlations with the primordial amplitude and the optical depth to reionization as they both affect the amplitude of the clustering. Fig. 1 shows how a model with ∑ m ν = 0 . 19 eV cannot be distinguished from a zero neutrino mass model using only primary CMB anisotropy data-neutrinos would have become non-relativistic only after the CMB formed. However, such a model would lower the CMB lensing signal by about 6%, which is sufficient to be disfavored at the 3 σ significance level.  \nResults .- We initially obtain similar results to [18], finding ∑ m ν < 0 . 12 eV for the ΛCDM model, before including SNe data. 5 Including the SNe data shrinks the 1 σ uncertainty by a small amount, but the preferred value is slightly non-zero, so the upper limit increases to 0 . 13 eV, as shown in Fig. 2. If we incorporate DESI measurements, using the CMB (ACT+ Planck ) + BAO2 + SNe dataset, the constraint marginally tightens to ∑ m ν < 0 . 12 eV, but we still find a non-zero peak in  \nν  \nFIG. 3. 2D marginalized distribution for Ω k and ∑ m ν in the ΛCDM+Ω k + ∑ m ν cosmology. A spatially flat universe has Ω k = 0, an open universe has Ω k > 0, and a closed universe has Ω k < 0. Including the new ACT CMB lensing data gives a tighter constraint than with earlier lensing data from Planck. Adding BAO-2 or SNe data does not further constrain ∑ m ν .  \n<!-- image -->  \nthe ∑ m ν posterior distribution as shown in Fig. 2. This is a higher upper bound compared to the limit reported in [20], since our BAO-2 dataset conservatively does not include the new DESI data in the first two redshift bins, using SDSS data instead. With this choice of data the inverted hierarchy is still viable in the minimal extension. This is consistent with the findings of [19, 24], who provide analogous constraints using the HiLLiPop+LoLLiPoP Planck likelihoods [43] and a slightly different combination of DESI+SDSS data. A summary of the 95% upper limits for different datasets is given in Table I. In Fig. 3 we additionally vary the spatial curvature, where Ω k < 0 ( > 0) corresponds to a closed (open) universe. As has been shown in e.g., [14, 17, 21, 22], a slightly larger neutrino mass is allowed in an open universe with Ω k > 0. Due to the geometric degeneracy [44], an open universe needs more dark energy to conserve the peak positions in the primary CMB spectrum.  \nThese models would have a smaller CMB lensing signal than the best-fitting ΛCDM model [45], which can be partly compensated by decreasing the neutrino mass. However, BAO data provides orthogonal constraints on ∑ m ν and curvature, switching the direction of this degeneracy [21]. The new ACT lensing data better measures the clustering, resulting in ∑ m ν < 0 . 13 eV compared to 0.17 eV with only Planck lensing.  \nIncluding the SNe data does not further tighten constraints for this model. If the dark energy is not a constant vacuum energy, its equation of state, w = p/ρ , is anti-correlated with neutrino mass. As discussed in [14, 46], if w < -1, the dark energy density Ω DE ( z )  \nw  \nFIG. 4. The 2D marginalized posterior for w and ∑ m ν in the w CDM+ ∑ m ν cosmology, where w = -1 is a cosmological constant. Adding SNe data reduces the degeneracy between w and ∑ m ν , resulting in ∑ m ν < 0 . 13 eV, similar to the baseline model.  \n<!-- image -->  \n̸  \n̸  \nis decreased, then the total matter density must increase to fit the primary CMB data. This enhances the growth of cosmic structures, whose effects on CMB lensing can be counteracted by increasing the neutrino mass. This correlation increases the neutrino mass bound to ∑ m ν < 0 . 18 eV with CMB(ACT+ Planck )+BAO-1 data. In this case, the SNe data measure the time dependence of the expansion rate of the universe, providing an additional constraint on w . This partly breaks the degeneracy, as shown in Fig. 4, bringing the limit back to ∑ m ν < 0 . 13 eV. Including BAO-2 data further decreases this bound to ∑ m ν < 0 . 11 eV but does not improve the constraint on w . Moreover, allowing both w and Ω k to vary together does not further decrease the neutrino mass uncertainty. Neither w = -1 nor Ω k = 0 are preferred by the data.  \nThe effects of non-zero neutrino mass can only be partially mimicked by a constant equation of state of dark energy. If this component is dynamic, corresponding to a new scalar field for example, the equation of state would have a time dependence. For the time-varying model we consider, the dark energy density scales as [47]  \nΩ DE ( z ) = Ω DE (0)(1 + z ) 3(1+ w 0 + w a ) exp ( -3 w a z 1 + z ) . (2)  \nw  \nw  \nFig. 5 shows how the correlation of the neutrino mass with the equation of state today, w 0 , is reduced, and instead becomes anti-correlated with w a . This happens because w a governs the late time dynamics of dark energy, as seen in Eq. 2 for small z . Consequently, it has a larger effect on H ( z ) and on ∑ m ν than does w 0 [14]. As discussed earlier, decreasing the dark energy equation of state through an increase in w a can result in a good fit to the data (namely, the measured comoving distance to de-  \n<!-- image -->  \nw  \n0  \nw  \nw  \n0  \na  \nw  \nCDM+Σm  \na  \nν  \nCDM+Σm  \n0  \n<!-- image -->  \nw  \nFIG. 5. The 2D marginalized posteriors for the w a w 0 CDM+ ∑ m ν model where the equation of state is a function of z as in Eqn. 1. The indicated w 0 = -1 and w a = 0 correspond to a cosmological constant. Adding SNe data significantly tightens the constraints on w a and w 0 , but has only a modest effect on the ∑ m ν bound. Including BAO-2 measurements, indicated by the green contours, do not affect the dark energy constraints but reduces the tail on ∑ m ν to < 0 . 19 eV (95% c.l), as seen in the second plot. Moreover, additionally varying Ω k does not affect the constraint, as demonstrated by the orange contour in the first plot.  \n<!-- image -->  \nν  \nFIG. 6. Summary of upper limits, at 95% confidence, on the sum of neutrino masses for the extended models indicated on the horizontal axis. The ∑ m ν bound obtained by the blue dataset for the ΛCDM+ ∑ m ν model is indicated at 0 . 13 eV, and the line at 0 . 2 eV denotes the expected future bound from KATRIN on the electron neutrino mass [5]. As expected, the effect of adding SNe data is most apparent for the extensions with dark energy. Including BAO-2 measurements tightens the constraints for all models, achieving an upper bound of ∑ m ν < 0 . 11 eV in the minimally extended cosmology.  \n<!-- image -->  \ncoupling) if ∑ m ν is increased. Meanwhile, the w 0 + w a term in the exponent of Eq. 2 induces anti-correlations between the two parameters, as seen in the third plot of Fig. 5. Thus, w 0 and ∑ m ν are now slightly correlated due to their mutual anti-correlations with w a . Including the SNe data, the bound is ∑ m ν < 0 . 23 eV; without the ACT and SNe data this limit is 25% weaker at 0 . 29 eV. Using the BAO-2 data also shrinks the parameter space for larger ∑ m ν , as shown the green contours of Fig. 5, resulting in ∑ m ν < 0 . 19 eV. Finally, we find that allowing curvature to be additionally varied in this model does not further increase the uncertainty, as seen in Fig. 5.  \nConclusions .- With an upper limit from cosmological data on the neutrino mass sum of ∑ m ν < 0 . 12 eV, rising to ∑ m ν < 0 . 19 eV in a time-varying dark energy model, we are approaching the minimum predicted level for either neutrino mass hierarchy (0.06 or 0.1 eV). Our analysis shows that the upper limit does not depend strongly on assumptions about the cosmological model, and that the ΛCDM model is still preferred when applying Bayesian model selection. A lower bound on ∑ m ν has not yet been seen with cosmological data: we have yet to find evidence for non-zero neutrino mass. Improved  \ndata from future DESI releases and the Simons Observatory, as well as measurements of the optical depth from CLASS, TauRUS or LiteBIRD, hold promise for a nonzero mass detection [48]. In such a scenario, it will be important to distinguish a non-zero mass from a timevarying dark energy. We can expect dark energy properties will be better constrained by future supernova, weak lensing and galaxy clustering data from, e.g., the Vera C. Rubin Observatory and the Euclid satellite. Combinations of multiple cosmological datasets will be crucial for solidifying our understanding of neutrino properties beyond the Standard Model.  \nAcknowledgments .- JG acknowledges support from Princeton's Presidential Postdoctoral Research Fellowship. JD acknowledges NSF grant AST2108126. MM acknowledges support from NSF grants AST-2307727 and AST-2153201 and NASA grant 21-ATP21-0145. We use the Cobaya code .  \nAppendix .- Recent BAO measurements by DESI placed tight constraints on the upper bound of neutrino mass sum in various cosmologies [20]. Notably, they obtain ∑ m ν < 0 . 072 eV in the ΛCDM+ ∑ m ν model. Given that this bound appears to exclude the IH at  \nTABLE I. The 95% upper limits on ∑ m ν [eV] in the extended cosmological models. All results are obtained using Planck primary CMB data, and the six cases show the effects of adding ACT lensing data, DESI measurements, and/or Type Ia supernova data from Pantheon+. BAO-1 indicates SDSS-only data while BAO-2 denotes our DESI+SDSS combination. The rightmost column indicates the most constraining combination.  \n95% confidence, we investigate the impact of this constraint when using select BAO measurements from SDSS in place of DESI. In particular, we note the ∼ 3 σ difference found between the DESI and SDSS LRG results in the redshift interval 0 . 6 < z < 0 . 8 [49]. Thus, we choose to use a composite 'BAO-2' dataset containing measurements from both DESI and SDSS, conservatively swapping in the SDSS LRG measurements for these ranges in our data mixture. A similar data combination is tested in [19, 20], where BAO results from each redshift range are selected from the survey that covers the larger effective volume at that redshift. Our 'BAO-2' consists of the following BAO measurements:  \n- 1. We use the SDSS measurements at z = 0 . 15, 0.38, and 0.51 in place of the BGS and LRG1 results from DESI. The corresponding Cobaya likelihoods are bao.sdss dr7 mgs, sdss dr12 consensus bao . This coincides with the selection described in [20], as the SDSS survey contains larger effective volume at these redshifts.\n- 2. We use the SDSS measurements for 0 . 6 < z < 0 . 8 in place of the LRG2 results from DESI, corresponding to the sdss dr12 consensus bao Cobaya likelihood. This differs from the selection described in [20], which used the DESI measurements to maximize effective survey volume.\n- 3. We use the DESI LRG+ELG combination over 0 . 8 < z < 1 . 1, and the higher redshift ELGs and QSOs. The corresponding Cobaya likelihoods are bao.desi 2024 bao lrgpluselg z1, bao.desi 2024 bao elg z2,  \nbao.desi 2024 bao qso z1  \n.  \n- 4. For the Ly α measurements, we use the combined DESI+SDSS data provided by the", 'bao.desi 2024 eboss bao lya likelihood.': "Using this in our CMB (ACT+ Planck ) + BAO-2 dataset for the minimal cosmological extension: ΛCDM+ ∑ m ν , we find an upper bound of ∑ m ν < 0 . 104 eV (95% c.l). The marginalized posterior distribution is shown in green in Fig. 7. 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