Negative Neutrino Mass in Cosmology

• Negative neutrino mass is a phenomenological artifact in cosmological analyses, reflecting statistical inferences near the physical parameter boundary.
• Analyses combining CMB, BAO, and structure growth data show a formal best-fit neutrino mass below zero, driven mainly by lensing anomalies and dataset outliers.
• Systematic modeling adjustments in reionization, dark energy dynamics, and statistical methods help resolve the negative mass artifact and guide future neutrino research.

Negative Neutrino Mass

Negative neutrino mass refers to both a phenomenological artifact observed in cosmological parameter inference and, in some theoretical contexts, the introduction of unusual terms or mechanisms leading to negative mass eigenvalues in neutrino mass matrices. The ‘negative mass’ terminology arises most prominently in the context of precision cosmological measurements combining cosmic microwave background (CMB) data, baryon acoustic oscillations (BAO), and structure growth, where the best-fit value for the sum of neutrino masses can formally fall below zero, contrary to physical expectations. This phenomenon has prompted an extensive analysis of its statistical, observational, and theoretical underpinnings.

1. Cosmological Origin of Negative Neutrino Mass

In cosmological analyses, the sum of neutrino masses, $\sum m_\nu$, is constrained by its effect on suppressing the growth of structure via free-streaming—resulting in a small (few percent) reduction of power in the matter power spectrum. Joint analyses with Planck CMB and DESI BAO data provide precise limits, with current constraints reaching $\sum m_\nu < 70$ meV at 95% confidence (Craig et al., 1 May 2024), marginally above the minimal value ($\sim$58 meV) required by oscillation experiments.

In recent data analyses, allowing the total mass parameter to extend to negative values reveals a best-fit $\sum m_\nu = -160 \pm 90$ meV (Craig et al., 1 May 2024), i.e., a formal preference for negative values. This parameterization (here denoted $\tilde{m}_\nu$ or $\sum m_{\nu,\mathrm{eff}}$) is not a literal statement about negative particle mass, but rather reflects an excess in clustering amplitude or lensing compared to predictions with physical (positive) neutrino mass. The excess clustering is inferred primarily through the CMB lensing power spectrum and is thus most sensitive to any anomalies or systematics in lensing reconstruction (Green et al., 10 Jul 2024). The negative mass artifact is not universal; it is dataset- and analysis-dependent, often appearing when specific likelihood chains (e.g., Planck Plik or CamSpec) or particular BAO redshift bins (notably at $z=0.7$ in DESI-Y1) are included (Naredo-Tuero et al., 18 Jul 2024).

2. Lensing Anomaly and Statistical Treatment

The dominant driver behind cosmological negative neutrino mass inference is the lensing anomaly in CMB datasets. The so-called lensing amplitude parameter, $A_\text{lens}$, empirically rescales the amplitude of the CMB lensing power spectrum. Several Planck likelihoods (e.g., Plik 2018) report $A_\text{lens} > 1$ at 2–3$\sigma$, suggesting enhanced lensing relative to $\Lambda$CDM predictions with physical neutrino mass (Green et al., 10 Jul 2024, Naredo-Tuero et al., 18 Jul 2024). The four-point lensing potential reconstruction (via $C_L^{\phi\phi}$) is particularly influential: it directly constrains clustering and is statistically the main source of the negative mass preference (Green et al., 10 Jul 2024). The two-point lensing (smoothing of acoustic peaks) exhibits the effect to a lesser degree.

Statistical methodology near the physical boundary $\sum m_\nu \geq 0$ is relevant. Both Bayesian posteriors and frequentist profile likelihoods (including Feldman–Cousins corrections) show similar behavior: when lensing anomalies are present, the best-fit mass parameter can be negative, but the physical upper limit is only modestly affected (10–20% variation between methods) (Naredo-Tuero et al., 18 Jul 2024). When conservative likelihoods (such as HiLLiPoP) or BAO outlier bins are excluded, the negative mass signal disappears and the bound relaxes upward.

3. Influence of Reionization and Optical Depth

The calibration of the CMB lensing amplitude is intimately tied to the measurement of the reionization optical depth, $\tau$, extracted from large-scale $EE$ and $TE$ CMB polarization. Standard instantaneous reionization models, enforced by low-$\ell$ Planck data, yield low $\tau$ (∼0.054). However, both theoretical degeneracies and dataset tensions allow for higher values ($\tau \sim 0.09$) if additional freedom in the reionization history is introduced (e.g., via principal component analysis such as RELIKE or improved SRoll2 maps) (Jhaveri et al., 30 Apr 2025). The degeneracy between $\sum m_\nu$, $A_\text{lens}$ and $\tau$ is quantified by

$$\Delta M_\nu \simeq 5\Delta\tau \simeq 2.5\Delta\ln(A_L)$$

(Jhaveri et al., 30 Apr 2025). Raising $\tau$ increases the inferred amplitude of lensing and permits a higher (physical) sum of neutrino masses, eliminating the preference for negative values.

Furthermore, cosmic birefringence—a rotation of CMB polarization planes possibly induced by axion-like particles (ALPs)—has been proposed as a mechanism to suppress the reionization “bump,” thereby allowing larger $\tau$ consistent with polarization data. The observed birefringence angle is subject to an $n\pi$ phase ambiguity: $$\beta = \beta_0 + 180n~\text{deg},\ n \in \mathbb{Z}$$ (Namikawa, 28 Jun 2025). Large variance in $\beta$ during reionization phases can suppress the $E$-mode signal at low $\ell$ (reionization bump), permitting higher $\tau$ without discordance with data. This alleviates the tension in inferred neutrino mass (Namikawa, 28 Jun 2025).

4. Model-Dependence and Effects of Dark Energy Dynamics

Allowing extensions to $\Lambda$CDM, such as time-dependent dark energy (parameterized by $w(a) = w_0 + w_a (1 - a)$), significantly affects the negative neutrino mass artifact. Evolving dark energy models (including “mirage” models with $w$ crossing $-1$ near $z \sim 0.4$) increase the inferred sum of neutrino masses such that the cosmological posteriors become consistent with oscillation experiment lower bounds: e.g., $\sum m_{\nu,\mathrm{eff}} = 0.06^{+0.15}_{-0.10}$ eV for $w_0w_a$CDM (Elbers et al., 15 Jul 2024). This reflects an underlying degeneracy—some excess in clustering can equally be attributed to a reduction in neutrino mass or a modification of late-time cosmic acceleration (dark energy). However, even large departures from $w(z) = -1$ are not favored by DESI+Planck data unless CMB lensing and $\tau$ constraints are modified (Jhaveri et al., 30 Apr 2025, Green et al., 10 Jul 2024).

The table below summarizes the effect of modeling choices:

Analysis Choice	Neutrino Mass Bound	Negative Mass Preference
$\Lambda$CDM + Planck (Plik 2018)	$\sum m_\nu < 0.072$ eV (95%)	Present ($\sim 2-3\sigma$)
+ HiLLiPoP Likelihood	$\sum m_\nu < 0.11$ eV (95%)	Absent
wCDM extensions (DESI+Planck)	$\sum m_{\nu,\mathrm{eff}} \sim 0.04-0.06$ eV	Absent/Reduced

A plausible implication is that negative neutrino mass artifacts are highly sensitive to both CMB lensing amplitude and the expansion history encoded by dark energy parameterization.

5. Systematics, Outliers, and Robustness

Multiple papers have identified that the statistical preference for “negative” neutrino mass is strongly connected to known systematics or outlier data. Notably, the $z=0.7$ DESI BAO bin exhibits a $\sim 3\sigma$ tension with Planck expectations in $\Lambda$CDM; removing this bin weakens the cosmological neutrino mass bounds and eliminates the negative mass artifact (Naredo-Tuero et al., 18 Jul 2024). The same applies for CMB analysis choices—switching to likelihoods less affected by the lensing anomaly (HiLLiPoP) relaxes constraints (Naredo-Tuero et al., 18 Jul 2024).

Frequentist and Bayesian inference methods produce nearly identical results for cosmological neutrino mass bounds near the physical parameter space boundary, with differences below 20% (Naredo-Tuero et al., 18 Jul 2024).

6. Theoretical Models: Negative Mass Mechanisms

In particle physics contexts distinct from cosmological inference, negative mass eigenvalues can arise from theoretical model-building. For example, the introduction of a Higgs singlet with a “negative kinetic term” in the Lagrangian: $$\mathcal{L}_\text{kinetic} = - [D_\mu S]^\dagger [D^\mu S]$$ can, after symmetry breaking and diagonalization, produce a mass matrix for neutrinos with negative eigenvalues (Lu, 2020). These negative values are unphysical for mass observables but can be rendered positive via field redefinitions (e.g., phase rotations). Their real significance in this context is to enable a cancellation mechanism protecting the photon from acquiring a mass when both the charged and neutral components of an extended Higgs sector acquire vacuum expectation values.

More generally, negative effective mass parameters in cosmology are a proxy for enhanced late-time structure formation, not an indication of particles with negative physical mass.

7. Implications and Future Prospects

The apparent negative neutrino mass in cosmological inference exposes deeper tensions in the standard cosmological model’s internal consistency—especially between low-redshift geometric distance measures (BAO) and early-universe physics (CMB) (Jhaveri et al., 30 Apr 2025). This artifact signals either the presence of subtle systematics (lensing anomaly, BAO outliers, reionization model dependencies), underestimated parameter degeneracies, or, conceivably, the need for new physics in the neutrino, dark energy, or dark matter sectors.

Sterile new physics scenarios—such as neutrino decay, environmental mass variation (“chameleon” models), or new long-range dark sector forces—have been shown to mimic the effects attributed to negative $\sum m_\nu$, without requiring actual negative masses (Craig et al., 1 May 2024). Additionally, primordial trispectra from inflation could mimic lensing signatures in CMB four-point functions, thus biasing inferred clustering, though current constraints limit these possibilities (Craig et al., 1 May 2024).

Future observational efforts, including CMB polarization missions (e.g., CMB-S4, LiteBIRD), high-resolution lensing experiments, and extended BAO and supernova surveys, alongside improved reionization and birefringence modeling, will play a critical role in clarifying whether the negative neutrino mass signals persist, abate, or are replaced with consistent, physically plausible parameter estimates.

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In summary, negative neutrino mass in cosmology is a proxy for excess clustering due to anomalies in lensing statistics, outlier datasets, or modeling assumptions, and highlights the interconnectedness and calibration sensitivity of cosmological probes. In field theory contexts, negative mass terms following diagonalization or field redefinition reflect artifact bases that can always be made physically meaningful by suitable transformations. The ongoing dialogue between data, theoretical modeling, and systematics underpins the interpretation of these results and guides future directions in neutrino physics and cosmology.