Neutrino oscillation experiments measure only differences of squared masses, leaving the absolute neutrino mass scale and the ordering of the mass eigenstates undetermined. Cosmological observations — the cosmic microwave background and the large-scale distribution of galaxies — are sensitive to the sum of neutrino masses Σmν\Sigma m_\nu because relic neutrinos free-stream out of small-scale gravitational potential wells and suppress the growth of structure below their free-streaming scale. The open problem is to extract Σmν\Sigma m_\nu — and through it the lightest mass and the mass ordering — from cosmological data, overcoming parameter degeneracies, cosmic variance in two-point statistics, prior and approximation sensitivity near the physical lower boundary, model dependence in extended cosmologies, and the need for percent-level nonlinear modeling of structure formation with massive neutrinos. Current data deliver only upper limits: Σmν<0.064\Sigma m_\nu<0.064 eV (95% CL) in flat Λ\LambdaCDM from DESI DR2 and the CMB, weakening to 0.10–0.16 eV under alternative likelihoods or evolving dark energy, with fitted central values pushed to or below the physical floor by a CMB lensing anomaly rather than by neutrino mass. No positive cosmological measurement of Σmν\Sigma m_\nu exists and the mass ordering remains undetermined; the problem is open.

Background

Neutrinos are the only fermions of the Standard Model of particle physics whose absolute masses are unknown. Neutrino oscillation experiments (using solar, atmospheric, reactor, and accelerator neutrinos) measure only differences of squared masses, Δm212≈7.4×10−5 eV2\Delta m^2_{21}\approx 7.4\times10^{-5}\,\mathrm{eV}^2 and ∣Δm312∣≈2.5×10−3 eV2|\Delta m^2_{31}|\approx 2.5\times10^{-3}\,\mathrm{eV}^2, proving that at least two of the three mass eigenstates (the states of definite mass that mix to form the observed flavor states) are massive, while leaving the absolute scale and the arrangement of the masses undetermined (Capozzi et al. 2017; 2020). Two orderings are consistent with the oscillation data: the normal ordering (NO), m1<m2<m3m_1<m_2<m_3, and the inverted ordering (IO), m3≪m1<m2m_3\ll m_1<m_2. Because the splittings are fixed, the sum of masses Σmν\Sigma m_\nu is bounded below by about 0.059 eV in NO and about 0.10 eV in IO, so a sufficiently precise measurement of Σmν\Sigma m_\nu would simultaneously fix the lightest mass and determine the ordering — one of the key unknowns of particle physics.

Cosmology is sensitive to Σmν\Sigma m_\nu because the universe is filled with a relic cosmic neutrino background left over from the hot Big Bang (about 112 neutrinos and antineutrinos of each species per cubic centimeter today). Neutrinos with sub-eV masses become non-relativistic after recombination — the epoch at which the cosmic microwave background (CMB), the primordial radiation field released when the first atoms formed, was emitted — and then free-stream, that is, stream nearly unimpeded, out of the gravitational potential wells of the developing cosmic web. This damps the growth of density fluctuations on scales smaller than the neutrino free-streaming length, imprinting a scale- and redshift-dependent suppression of the late-time matter power spectrum (the variance of density fluctuations as a function of wavenumber) whose magnitude is set mainly by Σmν\Sigma m_\nu. Galaxy clustering surveys, weak gravitational lensing (the statistical distortion of distant galaxy shapes by intervening matter), and baryon acoustic oscillations (BAO, fossil sound waves in the galaxy distribution used as standard rulers) measure this suppression, while the CMB constrains Σmν\Sigma m_\nu through the acoustic peak scale, CMB lensing, and geometric degeneracies. The mechanism and its observational consequences are reviewed by Lesgourgues & Pastor (2006).

Turning this sensitivity into a determination of the mass scale faces four recognized obstacles. First, parameter degeneracies: Σmν\Sigma m_\nu anticorrelates with the matter density Ωm\Omega_m, the clustering amplitude (σ8\sigma_8, the rms mass fluctuation in spheres of 8 Mpc/h), the reionization optical depth τ\tau, and, in extended cosmologies, the dark-energy equation of state; upper limits weaken substantially outside minimal Λ\LambdaCDM (the standard cold-dark-matter-plus-cosmological-constant model). Second, cosmic variance: the two-point statistics of any single tracer saturate at the sampling-variance ceiling set by the finite observable volume, so extracting the few-percent neutrino signature requires multi-tracer combinations, cross-correlations with CMB lensing, or higher-order statistics; Ballardini & Maartens (2022) forecast that a multi-tracer combination of Square Kilometre Array 21cm intensity mapping (mapping unresolved hydrogen emission rather than individual galaxies), the LSST photometric galaxy survey (the Legacy Survey of Space and Time at the Vera C. Rubin Observatory), and Stage-4 CMB lensing can reach σ(Σmν)≃12\sigma(\Sigma m_\nu)\simeq 12 meV using only linear scales, and controlling theoretical error on mildly nonlinear scales is essential to such forecasts (Chudaykin & Ivanov 2019). Third, the inference is prior- and approximation-sensitive: analyses that approximate the spectrum by three degenerate (equal) masses can produce non-physical bounds, and Bayesian inference near the physical lower boundary depends strongly on the prior (Loureiro et al. 2019; Herold & Kamionkowski 2025). Fourth, nonlinear modeling: upcoming surveys measure the matter distribution deep in the nonlinear regime, where the power spectrum and halo bias in massive-neutrino cosmologies must be predicted at the percent level, motivating dedicated simulation suites such as MassiveNuS (Liu et al. 2018). The target precision of the next generation of surveys — Euclid, Rubin/LSST, and CMB-S4 — is of order 0.01–0.02 eV, precisely what is needed to separate the minimal NO and IO floors. Laboratory probes are complementary but currently far weaker: the KATRIN tritium beta-decay experiment bounds the electron-flavor neutrino mass at 0.45 eV (90% confidence), about an order of magnitude above the cosmologically relevant scale (KATRIN Collaboration 2025).

Problem Statement

Determine, from cosmological observations — the cosmic microwave background together with large-scale-structure data such as galaxy clustering, BAO, weak lensing, and CMB lensing — the absolute neutrino mass scale, expressed as a positive measurement of the sum of neutrino masses Σmν\Sigma m_\nu, with precision sufficient to fix the lightest neutrino mass at quantified significance and to determine, or exclude at quantified significance, the normal versus inverted mass ordering (whose minimal sums differ by about 0.04 eV). The determination must be robust in the sense that its claimed significance survives: (i) exact treatment of the neutrino mass spectrum with the measured oscillation splittings rather than degenerate-mass approximations; (ii) defensible prior choices near the physical lower boundary (Σmν≳0.059\Sigma m_\nu\gtrsim 0.059 eV for the normal ordering, ≳0.10\gtrsim 0.10 eV for the inverted ordering); (iii) modeling of nonlinear structure formation in massive-neutrino cosmologies whose validated error is subdominant at the scales used; and (iv) standard extensions of the cosmological model, in particular evolving dark energy, together with the dataset and likelihood alternatives known to matter (for example, different Planck releases and lensing treatments, and supernova samples). A cosmological upper limit on Σmν\Sigma m_\nu, however tight, does not by itself answer the question, and neither does an ordering preference whose significance is not stable under conditions (i)–(iv).

References

  1. Arthur Loureiro, Andrei Cuceu, Filipe B. Abdalla, Bruno Moraes, Lorne Whiteway, Michael McLeod, Sreekumar T. Balan, Ofer Lahav, Aurélien Benoit-Lévy, Marc Manera, Richard P. Rollins, Henrique S. Xavier, Upper Bound of Neutrino Masses from Combined Cosmological Observations and Particle Physics Experiments, Physical Review Letters 123, 081301 (2019), DOI 10.1103/PhysRevLett.123.081301, https://doi.org/10.1103/PhysRevLett.123.081301 (arXiv:1811.02578)
  2. Mario Ballardini, Roy Maartens, Constraining the neutrino mass using a multitracer combination of two galaxy surveys and cosmic microwave background lensing, Monthly Notices of the Royal Astronomical Society 510, 4295–4301 (2022), DOI 10.1093/mnras/stab3480, https://doi.org/10.1093/mnras/stab3480 (arXiv:2109.03763)
  3. Jia Liu, Simeon Bird, José Manuel Zorrilla Matilla, J. Colin Hill, Zoltán Haiman, Mathew S. Madhavacheril, Andrea Petri, David N. Spergel, MassiveNuS: cosmological massive neutrino simulations, Journal of Cosmology and Astroparticle Physics 2018(03), 049 (2018), DOI 10.1088/1475-7516/2018/03/049, https://doi.org/10.1088/1475-7516/2018/03/049 (arXiv:1711.10524)
  4. J. Lesgourgues, S. Pastor, Massive neutrinos and cosmology, Physics Reports 429, 307–379 (2006), DOI 10.1016/j.physrep.2006.04.001, https://doi.org/10.1016/j.physrep.2006.04.001
  5. Francesco Capozzi, Eleonora Di Valentino, Eligio Lisi, Antonio Marrone, Alessandro Melchiorri, Antonio Palazzo, Global constraints on absolute neutrino masses and their ordering, Physical Review D 95, 096014 (2017), DOI 10.1103/PhysRevD.95.096014, https://doi.org/10.1103/PhysRevD.95.096014
  6. Francesco Capozzi, Eleonora Di Valentino, Eligio Lisi, Antonio Marrone, Alessandro Melchiorri, Antonio Palazzo, Addendum to 'Global constraints on absolute neutrino masses and their ordering', Physical Review D 101, 116013 (2020), DOI 10.1103/PhysRevD.101.116013, https://doi.org/10.1103/PhysRevD.101.116013
  7. Planck Collaboration: N. Aghanim, Y. Akrami, M. Ashdown, et al., Planck 2018 results. VI. Cosmological parameters, Astronomy & Astrophysics 641, A6 (2020), DOI 10.1051/0004-6361/201833910, https://doi.org/10.1051/0004-6361/201833910
  8. DESI Collaboration: M. Abdul-Karim, J. Aguilar, S. Ahlen, et al., DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints, Physical Review D 112, 083515 (2025), DOI 10.1103/tr6y-kpc6, https://doi.org/10.1103/tr6y-kpc6 (arXiv:2503.14738, https://arxiv.org/abs/2503.14738)
  9. Daniel Green, Joel Meyers, Cosmological preference for a negative neutrino mass, Physical Review D 111, 083507 (2025), DOI 10.1103/PhysRevD.111.083507, https://doi.org/10.1103/PhysRevD.111.083507 (arXiv:2407.07878)
  10. Itamar J. Allali, Alessio Notari, Neutrino mass bounds from DESI 2024 are relaxed by Planck PR4 and cosmological supernovae, Journal of Cosmology and Astroparticle Physics 2024(12), 020 (2024), DOI 10.1088/1475-7516/2024/12/020, https://doi.org/10.1088/1475-7516/2024/12/020
  11. Laura Herold, Marc Kamionkowski, Revisiting the impact of neutrino mass hierarchies on neutrino mass constraints in light of recent DESI data, Physical Review D 111, 083518 (2025), DOI 10.1103/PhysRevD.111.083518, https://doi.org/10.1103/PhysRevD.111.083518 (arXiv:2412.03546)
  12. Peter W. Graham, Daniel Green, Joel Meyers, New interpretations of the cosmological preference for a negative neutrino mass, Physical Review D 113, 043514 (2026), DOI 10.1103/1bqb-qlrj, https://doi.org/10.1103/1bqb-qlrj
  13. KATRIN Collaboration: M. Aker, D. Batzler, A. Beglarian, et al., Direct neutrino-mass measurement based on 259 days of KATRIN data, Science 388, 180–185 (2025), DOI 10.1126/science.adq9592, https://doi.org/10.1126/science.adq9592
  14. Anton Chudaykin, Mikhail M. Ivanov, Measuring neutrino masses with large-scale structure: Euclid forecast with controlled theoretical error, Journal of Cosmology and Astroparticle Physics 2019(11), 034 (2019), DOI 10.1088/1475-7516/2019/11/034, https://doi.org/10.1088/1475-7516/2019/11/034