Gravitational Weak Lensing of Neutrinos

• Gravitational weak lensing of neutrinos is defined by the deflection and magnification of neutrino trajectories and quantum states due to intervening gravitational potentials.
• It quantifies effects such as neutrino clustering around cold dark matter halos and a ~1% perturbation in weak lensing shear profiles detectable through large-scale stacking techniques.
• Challenges include precise modeling to overcome baryonic feedback and systematic degeneracies, while quantum interference effects offer potential insights into neutrino mass hierarchy and oscillation properties.

Gravitational weak lensing of neutrinos encompasses the deflection, magnification, and phase modification of neutrino trajectories and quantum states due to intervening gravitational potentials, with observable consequences for large-scale structure, cosmological inference, the non-linear mass distribution, and even neutrino flavor transitions. Unlike photons, neutrinos have non-zero rest mass and high thermal velocities, leading to distinctive lensing signatures, both at the classical level (affecting their contribution to the mass profile and lensing observables) and at the quantum level (affecting flavor oscillation probabilities through path interference and decoherence). This article details the theoretical framework, simulation approaches, observational strategies, and the implications for cosmology and fundamental physics.

1. Gravitational Clustering of Massive Neutrinos and Their Halos

Massive relic neutrinos, after becoming non-relativistic at late times, experience non-linear gravitational clustering around existing cold dark matter (CDM) halos. The process leads to the formation of diffuse neutrino halos, which are more extended and less concentrated than CDM or baryonic structures due to the high thermal velocities of neutrinos. The density profile of such halos is determined by solving the collisionless Boltzmann equation for relic massive neutrinos in the gravitational potential of a galaxy cluster, typically modeled with an inner Navarro–Frenk–White (NFW) profile and an outer profile given by the spherical collapse model without shell-crossing.

The clustering of neutrinos is sensitive to both their mass and the depth of the halo potential:

• For higher mass neutrinos (e.g., $m_\nu = 0.3\,{\rm eV}$), the overdensity and spatial extension of the neutrino halo is more pronounced; the phase-space density constraint leads to a smaller core radius for more massive species.
• In more massive clusters, the ratio of the halo's velocity dispersion to the neutrino thermal velocities is larger, enhancing the clustering (Villaescusa-Navarro et al., 2011).

This non-linear neutrino perturbation constitutes a small but spatially extended modification to the total mass distribution, with observable consequences at large radii outside the virialized region.

2. Observable Weak Lensing Signatures and Measurement Strategies

Weak gravitational lensing by clusters or the cosmic web is sensitive to all gravitating matter, including subtle neutrino contributions. For relic neutrinos inhabiting halos, the induced lensing perturbation is manifested as a subtle extension or "puffing up" of the projected mass profile at large radii. The key observables are the tangential shear $\gamma(R)$ and (cylindrically averaged) convergence $\bar{\kappa}(R)$; the presence of neutrino halos induces a fractional change $\Delta C_f/C_f$ in lensing quantities defined as

$$C_f(R) \equiv \bar{\kappa}(R) - \bar{\kappa}(fR) \,,$$

where $f$ is an outer scale factor (e.g., $f=2$).

The predicted neutrino-induced lensing signal, for optimistic neutrino mass scenarios (e.g., all three species at $0.3\,{\rm eV}$), is at the $\sim 1\%$ level in the stacked shear profile of massive clusters. Detection is only feasible statistically by stacking large numbers of clusters (order $10^4$) in all-sky surveys like EUCLID, exploiting the high number density of background source galaxies ($n \sim 30 \,{\rm arcmin}^{-2}$) (Villaescusa-Navarro et al., 2011).

Mathematical Formulations

• NFW dark matter halo profile: $$\rho_{\rm NFW}(r) = \frac{\rho_s}{(r/r_s)(1 + r/r_s)^2}$$
• Shear-convergence relation for a spherical system: $$\gamma(R) = \bar{\kappa}(R) - \kappa(R), \quad \bar{\kappa}(R)= \frac{2}{R^2} \int_0^R R' \kappa(R') \, dR'$$
• Critical surface density: $$\Sigma_{\rm crit} = \frac{c^2 D_s}{4\pi G D_l D_{ls}}$$

Massive neutrinos also suppress the amplitude of the convergence power spectrum $C_\kappa(\ell)$ on non-linear scales, observable in both galaxy lensing and CMB lensing measurements.

3. Systematic Challenges and Degeneracies: Baryons, Feedback, and Modeling

The detection of neutrino-induced lensing effects is limited by the need for sub-percent accuracy in the baseline mass distribution modeling:

• Baryonic physics (e.g., AGN feedback, galactic winds, radiative cooling) alter the inner and outer mass profiles and can both mimic and mask neutrino effects (Natarajan et al., 2014, Harnois-Déraps et al., 2014).
• CDM-only models are inadequate: suppression of power on small scales due to both baryonic feedback and neutrino free streaming are degenerate in weak lensing observables. This necessitates precise simulation-based modeling incorporating both components.

State-of-the-art cosmological hydrodynamical simulations such as MillenniumTNG and OWLS enable direct comparison between full-physics and dark-matter-only runs, allowing the quantification and calibration of baryonic and neutrino impacts on weak lensing statistics (Ferlito et al., 2023).

Combined Modeling Requirements

Effect	Regime	Impact on WL Signal	Degeneracy Direction
Baryonic Feedback	Small scales ($k > 1\,h/{\rm Mpc}$)	Suppression/"spoon" shape	Suppression degenerate with neutrino mass
Massive Neutrinos	Intermediate/large scales	Smoothed profile, power suppression	Overlaps baryonic feedback effect at high $\ell$

4. Quantum Effects: Lensing of Neutrino Flavor and Oscillation States

Beyond classical deflection, gravitational lensing can induce quantum interference effects in neutrino flavor oscillations. Neutrinos propagating along distinct geodesics (corresponding to different impact parameters) accrue different phases: $$\Phi_i^{(p)} = \frac{m_i^2}{2E} (L_p + \delta_{\rm grav}(b_p, M, \ldots))$$ where $L_p$ is the geometric path length, $b_p$ the impact parameter, $M$ the lens mass, and possibly additional metric deformation parameters (as in Rezzolla–Zhidenko or Kaluza–Klein metrics) (Alloqulov et al., 23 Aug 2024, Chakrabarty et al., 2023).

The flavor transition probability in the lensing scenario depends not only on the mass-squared splittings ($\Delta m_{ij}^2$), but also on the absolute neutrino masses and the sum $m_i^2 + m_j^2$, due to path difference-dependent interference. Thus, gravitational lensing introduces sensitivity (in principle) to the neutrino mass hierarchy and absolute scale, breaking the usual degeneracy of vacuum oscillations (Swami et al., 2020).

Decoherence arises when wave packet separation due to distinct group velocities exceeds the coherence length, suppressing interference terms. In the Schwarzschild or deformed metrics, the decoherence length depends on the proper time (geodesic interval) between emission and detection, with curved spacetimes generally elongating this scale compared to flat spacetime. However, metric deformations (e.g., in the RZ class) tend to have only subleading impact on the decoherence length—all principal features remain set by neutrino masses and source/detector separation (Swami et al., 2021, Alloqulov et al., 23 Aug 2024).

5. Simulation Frameworks and Statistical Analyses

N-body and hydrodynamical simulations provide the theoretical infrastructure for modeling lensing signatures from neutrino-induced perturbations. Massive neutrinos are implemented as either additional particles (with proper thermal velocities and transfer functions) or via grid-based methods. The matter power suppression appears in the three-dimensional spectrum $P_m(k)$ and propagates into the convergence power spectrum $C_\kappa(\ell)$ through line-of-sight integration.

Weak lensing observables are then computed in both single- and multi-redshift (tomographic) bins. Non-Gaussian information from higher-order lensing statistics (peak counts, PDFs of $\kappa$) further enhances sensitivity to the non-linear regime influenced by massive neutrinos (Li et al., 2018, Ferlito et al., 2023). Bayesian statistical inference, often employing Gaussian process interpolation and Fisher matrix formalism, enables parameter estimation and systematic error propagation.

"Fixed & paired" initial conditions across simulation ensembles reduce cosmic variance and enhance statistical robustness for lightcone and full-sky lensing predictions (Ferlito et al., 2023).

6. Cosmological and Particle Physics Implications

The lensing effects of massive neutrinos are directly utilized for:

• Probing the total neutrino mass sum, with projected sensitivities (if model systematics are under control) approaching the minimal normal-hierarchy scale ($\sigma(m_\nu) \sim 0.01\ {\rm eV}$) for Stage IV surveys (Natarajan et al., 2014, Ferlito et al., 2023).
• Constraining scenarios of non-standard neutrino cosmology, such as long-range interactions that leave imprints in the CMB lensing anomaly or alter the neutrino equation of state, thus resolving apparent discrepancies in $A_{\rm lens}$ without invoking unphysical parameters (Esteban et al., 2022).
• Providing a "beyond-linear" probe of relic neutrino clustering: the potential identification of neutrino halos via CMB lensing (matched filtering or stacking around clusters) would serve as direct, non-linear confirmation of relic neutrino clustering properties, offering an independent avenue to constrain their masses and velocity distributions (Hotinli et al., 2023).

Additionally, quantum and semi-classical lensing effects offer a route to paper:

• Gravitationally-induced phase shifts in neutrino oscillations, permitting, in principle, experimental discrimination between standard and alternative gravity models (e.g., Kaluza–Klein, non-commutative black holes) through the detailed structure of oscillation probabilities (Chakrabarty et al., 2023, Alloqulov et al., 23 Aug 2024, Filho et al., 11 Dec 2024).
• Tests of the absolute mass scale, mass hierarchy, and mixing parameters from the properties (frequency, amplitude, decoherence) of gravitationally lensed neutrino interference patterns.

7. Future Prospects and Observational Strategies

Practical detection of neutrino lensing signatures is challenging, due to the small amplitude of the effect and severe systematic requirements. Key strategies and outlook:

• Continuation of wide and deep weak lensing surveys (e.g., EUCLID, LSST, Roman, CMB-S4) providing both high source densities and precise multi-redshift shear/convergence maps.
• Aggressive modeling efforts to account for baryonic feedback, with joint analyses combining galaxy and CMB lensing to break degeneracies and minimize systematics.
• Tomographic and non-Gaussian lensing analyses to efficiently extract the small-scale and temporal information sensitive to massive neutrinos.
• Indirect searches for gravitationally magnified high-energy neutrino sources (e.g., lensed GRBs by supermassive black holes) as a means to probe cosmic distances, star formation epochs, and relic neutrino backgrounds (Vieyro et al., 2013).
• The theoretical and phenomenological development of techniques to extract lensing signatures encoded by neutrino halos and quantum interference from stacking, matched-filtering, and advanced statistics.

If realized, a robust detection or constraint of gravitational lensing signatures unique to neutrinos would provide a profound insight into both the cosmic neutrino background and the intersection of gravity and quantum field theory.