Cosmological limits on the neutrino mass sum for beyond-$Λ$CDM models
Helen Shao Jahmour J. Givans Jo Dunkley Mathew Madhavacheril Frank Qu Gerrit Farren Blake Sherwin




hshao@alumni.princeton.edu
Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA



 Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA

Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA


Department of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA
Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA


Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA, USA 19104

 
DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UKKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, UK


DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UKKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, UK 


DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UKKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, UK





§ ABSTRACT 
The 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  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 < 0.19  eV at 95% confidence for a model with dynamical dark energy, although this scenario is not statistically preferred over the simpler  model.


Cosmological limits on the neutrino mass sum for beyond-ΛCDM models
    Blake Sherwin
    September 9, 2024
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§ INTRODUCTION

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 <cit.> 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 mass-squared splittings between these states, Δ m_ij^2 = m_i^2 - m_j^2, 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 <cit.>. 

Direct-detection β-decay experiments  give limits of m ≤ 0.8 (90% confidence level) for the electron neutrino by the KATRIN Collaboration <cit.>.
Future experiments are  expected to reach a sensitivity of 0.2  eV (90% c.l.) <cit.>. A tighter indirect limit has already been placed on the sum of neutrino masses from cosmological data <cit.>, 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 < 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) <cit.>.
This bound assumes that the correct cosmological model is , 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 <cit.>. In this paper we update bounds using the latest data from the Atacama Cosmology Telescope (ACT) <cit.>, the Pantheon+ supernova survey <cit.>, 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 <cit.>, 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 <cit.>. We note that the PR3 Planck likelihood showed a 2.9σ inconsistency between  predictions of the CMB lensing parameter A_L and its value as measured via its smearing effect on the temperature and polarization power spectra <cit.>. This led to artificially tightened constraints on the neutrino mass. Hence, for our baseline data, as in <cit.>, we choose to use the Planck likelihood constructed from the PR4 NPIPE processed maps <cit.>. We also include BAO measurements which characterize the typical separation of galaxies and provide a measure of the total amount of dark matter. Our `BAO-1' dataset includes data measured by the 6dF <cit.> and Sloan Digital Sky Survey DR7 <cit.>, the latter of which is combined with the BOSS DR12 and DR16 data <cit.>. We also use a composite `BAO-2' dataset combining results from both SDSS and DESI,  described in the appendix.[We use the following  <cit.> likelihoods for primary CMB data: , and for BAO: .]

To this baseline data combination, we add the angular power spectrum of the reconstructed gravitational lensing of the CMB measured by Planck <cit.> and ACT <cit.>; 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 <cit.>, offering a direct way to measure the expansion rate of the universe. Here, we do not include the SH0Es calibration <cit.> and refer to this dataset as `SNe'.[We convert the Pantheon+ likelihood used with CosmoSIS to a version compatible with Cobaya and check we get the same results as using <cit.>.] Consistent constraints on  have also been placed using the full-shape power spectrum of galaxies and redshift space distortions (e.g., <cit.>), though we do not use these in our analyses. 

Our baseline model is , 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[In  there are 3.046 effective neutrino species and a power law spectral index with no running.]. As in <cit.>, we generate theoretical predictions for these models with the CAMB numerical code, using the Parameterized Post-Friedmann (PPF) prescription for dark energy perturbations <cit.>, and the Chevallier-Polarski-Linder (CPL) parameterization of the form <cit.>, 

    w(z) = w_0 + w_a(z/1+z)

when we consider variations of dark energy equation of state as a function of redshift.[To compute the small-scale nonlinear matter power spectrum we use the halofit model <cit.> implemented with HMcode <cit.> and the MEAD fitting method <cit.>.] We estimate the posterior distribution of parameters with Metropolis Hastings methods using the  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  to be positive. Previous works have also employed hierarchy-agnostic priors <cit.> but did not find conclusive results on a preferred hierarchy. 

As described in <cit.>, 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. <ref> shows how a model with =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 3σ significance level. 




§ RESULTS
 We initially obtain similar results to <cit.>, finding < 0.12  eV for the  model, before including SNe data.[<cit.> uses different BAO likelihoods, resulting in slightly different  constraints.] Including the SNe data shrinks the 1σ 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. <ref>. If we incorporate DESI measurements, using the CMB (ACT+Planck) + BAO-2 + SNe dataset, the constraint marginally tightens to <0.12 eV, but we still find a non-zero peak in the  posterior distribution as shown in Fig. <ref>. This is a higher upper bound compared to the limit reported in <cit.>, 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 <cit.>, who provide analogous constraints using the  Planck likelihoods <cit.> and a slightly different combination of DESI+SDSS data. A summary of the 95% upper limits for different datasets is given in Table <ref>.


In Fig. <ref> we additionally vary the spatial curvature, where Ω_k<0 (>0) corresponds to a closed (open) universe. As has been shown in e.g., <cit.>,
a slightly larger neutrino mass is allowed in an open universe with Ω_k>0. Due to the geometric degeneracy <cit.>, an open universe needs more dark energy to conserve the peak positions in the primary CMB spectrum.

These models would have a smaller CMB lensing signal than the best-fitting  model <cit.>, which can be partly compensated by decreasing the neutrino mass. However, BAO data provides orthogonal constraints on  and curvature, switching the direction of this degeneracy <cit.>. The new ACT lensing data better measures the clustering, resulting in <0.13 eV compared to 0.17 eV with only Planck lensing.

Including 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 <cit.>, if w<-1, the dark energy density Ω_ DE(z) is 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 <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. <ref>, bringing the limit back to <0.13 eV. Including BAO-2 data further decreases this bound to <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_k0 are preferred by the data.

The 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 <cit.>

    Ω_ DE(z) = Ω_ DE(0) (1+z)^3(1+w_0+w_a)exp(-3w_az/1+z).
 
Fig. <ref> 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. <ref> for small z. Consequently, it has a larger effect on H(z) and on  than does w_0 <cit.>. 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 decoupling) if  is increased. Meanwhile, the w_0 + w_a term in the exponent of Eq. <ref> induces anti-correlations between the two parameters, as seen in the third plot of Fig. <ref>. Thus, w_0 and  are now slightly correlated due to their mutual anti-correlations with w_a. Including the SNe data, the bound is <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 , as shown the green contours of Fig. <ref>, resulting in < 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. <ref>.



§ CONCLUSIONS

With an upper limit from cosmological data on the neutrino mass sum of <0.12 eV, rising to <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  model is still preferred when applying Bayesian model selection. 
A lower bound on  has not yet been seen with cosmological data: we have yet to find evidence for non-zero neutrino mass. Improved data 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 non-zero mass detection <cit.>. In such a scenario, it will be important to distinguish a non-zero mass from a time-varying 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.





§ ACKNOWLEDGMENTS

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 .



§ APPENDIX

Recent BAO measurements by DESI placed tight constraints on the upper bound of neutrino mass sum in various cosmologies <cit.>. Notably, they obtain < 0.072 eV in the + model. Given that this bound appears to exclude the IH at 95% confidence, we investigate the impact of this constraint when using select BAO measurements from SDSS in place of DESI. In particular, we note the ∼ 3σ difference found between the DESI and SDSS LRG results in the redshift interval 0.6 < z < 0.8 <cit.>. 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 <cit.>, 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:

    
  * 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  likelihoods are . This coincides with the selection described in <cit.>, as the SDSS survey contains larger effective volume at these redshifts. 
    
  * We use the SDSS measurements for 0.6 < z <  0.8 in place of the LRG2 results from DESI, corresponding to the   likelihood. This differs from the selection described in <cit.>, which used the DESI measurements to maximize effective survey volume.
    
  * We use the DESI LRG+ELG combination over 0.8 < z < 1.1, and the higher redshift ELGs and QSOs. The corresponding  likelihoods are .
    
  * For the Lyα measurements, we use the combined DESI+SDSS data provided by the  likelihood.

Using this in our CMB (ACT+Planck) + BAO-2 dataset for the minimal cosmological extension: +, we find an upper bound of < 0.104 eV (95% c.l). The marginalized posterior distribution is shown in green in Fig. <ref>. If instead we swap out the SDSS LRG measurement at z=0.6 with the corresponding data point from DESI, we find < 0.081 eV (95% c.l). These results can be compared to the upper bound of < 0.075 eV (95% c.l) obtained using CMB (ACT+Planck)+ BAO(DESI), where BAO measurements are from DESI only (blue curve in Fig. <ref>). Hence, we find an increased (∼ 44%) neutrino mass bound when excluding both DESI LRG measurements. While these upper bounds still place pressure on the IH, they allow for NH as a viable mass paradigm in this model. The constraint is further relaxed when including SNe data, as shown in Table <ref>, in agreement with <cit.>.




