Probing Long-Range Forces Between Neutrinos with Cosmic Structures

David E. Kaplan; Surjeet Rajendran; Xuheng Luo

REVIEW 3 major objections 5 minor 2 cited by

Probing Long-Range Forces Between Neutrinos with Cosmic Structures

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A long-range force acting only on neutrinos would make the cosmic neutrino background collapse into bound states, and existing matter-power-spectrum and reionization data already rule out a band of such forces with ranges from about 1 kpc…

desk verdict Solid phenomenological study with a credible linear instability mechanism, but the headline constraints hinge on an unvalidated O(1) collapse assumption that the authors openly flag. read the letter →

arxiv 2412.20766 v2 pith:QCLDR7WJ submitted 2024-12-30 hep-ph astro-ph.CO

David E. Kaplan , Xuheng Luo , Surjeet Rajendran This is my paper · ORCID

classification hep-phastro-ph.CO PACS 98.80.-k14.60.Pq

keywords long-rangeneutrinoforcescosmicbackgroundultralightscalarstructureformationmatterpowerspectrumreionizationJeansinstabilityfifth

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that a new long-range force coupling only to neutrinos, with a strength a few orders of magnitude above gravity, would not stay hidden: as the cosmic neutrino background turns non-relativistic in the late universe, the force triggers a rapid Jeans-like instability and makes the neutrinos clump into bound states. These neutrino bound states would act as extra gravitational seeds, boosting the matter power spectrum and, when massive enough, trapping baryons and igniting star formation early enough to reshape the reionization history. Comparing these predicted fingerprints with existing galaxy-clustering, Lyman-$\alpha$ forest, and CMB reionization measurements, the paper derives new 95% confidence limits on the coupling $g$ and scalar mass $m_\phi$ for interaction ranges between about 1 kpc and 10 Mpc. A curious reader should care because this converts the elusive cosmic neutrino background into a practical laboratory for probing forces that ordinary fifth-force experiments cannot reach.

What carries the argument

The carrier of the argument is the Yukawa interaction between the non-relativistic neutrino fluid and an ultralight scalar. The scalar's background value $\phi_0$ is sourced by the C$\nu$B, giving neutrinos a time-dependent effective mass and delaying the non-relativistic transition; the two wavenumbers that govern the perturbation dynamics are the free-streaming scale $k_{\rm fs}\simeq 0.04\,h\,\mathrm{Mpc}^{-1}$ and the Yukawa scale $k_\phi = a m_\phi$. The bound-state mass $M_{\rm bound}$ and the Poisson-noise power spectrum $P_{\rm iso}$ translate the microscopic parameters $(g, m_\phi)$ into observable signatures, while the point-particle condition $a_{\rm obs}\gtrsim 6.5\,a_{\rm NL}$ sets where the power-spectrum and star-formation arguments are applied.

What would settle it

A high-resolution simulation of the non-relativistic cosmic neutrino background with a Yukawa self-interaction at $g\sim 10^{-26}$ and $m_\phi\sim 10^{-29}\,\mathrm{eV}$ that measures the collapsed mass fraction once $\delta_\nu(k_\phi)>1$ and the two-point statistics of the resulting bound states; if the collapsed fraction is far below order unity, or if the objects are not point-like at wavenumbers below $k_{\rm cut}$, the predicted Poisson-noise power spectrum and reionization signatures would not occur.

Watch

Extended reading notes

Core claim

The central claim is that the late-time cosmic neutrino background is a sensitive probe of new neutrino-only fifth forces. In the model, an ultralight scalar $\phi$ couples through $-g\phi\bar\nu\nu$; the background neutrinos source $\phi_0$, which suppresses the effective neutrino mass and delays the non-relativistic transition from $z\approx 120$ down to a redshift $z_{\rm nr}$ that can be as low as about 40 for $g\sim 10^{-26}$ and $m_\phi\sim 10^{-29}\,\mathrm{eV}$. Once the neutrinos become non-relativistic, the Yukawa force in the fluid equations drives a growing mode with index $\gamma(k)\simeq 47\,(g/10^{-26})\,(k^2/(k^2+k_\phi^2))^{1/2}$, so density perturbations grow by orders of magnitude within a Hubble time. When $\delta_\nu(k_\phi)$ crosses unity, the paper assumes the C$\nu$B collapses into bound states of mass $M_{\rm bound}\sim 4\pi m_\nu n_{\nu,0}(a_{\rm NL})/(3 m_\phi^3)$ and radius $\sim m_\phi^{-1}$. These objects add a Poisson-noise term $P_{\rm iso}=f_\nu^2 D_+^2/\bar n_{\rm bound}$ to the linear matter power spectrum and, if massive enough, can capture baryons and trigger star formation at $z\gtrsim 7$. Using the reconstructed matter power spectrum from Lyman-$\alpha$ and galaxy clustering, together with the CMB reionization history, the paper excludes a band of couplings for ranges $1\,\mathrm{kpc}\lesssim m_\phi^{-1}\lesssim 10\,\mathrm{Mpc}$.

Load-bearing premise

The constraints rest on an unverified step: when neutrino overdensities reach order one, an order-one fraction of the cosmic neutrino background is assumed to collapse quickly into compact, point-like bound states of a specific mass, and if the collapse is inefficient or the lumps are diffuse, the resulting bounds weaken substantially.

Editorial extensions

If this is right

Existing matter-power-spectrum measurements (Lyman-alpha and LRG) already exclude a band of neutrino-only fifth forces with ranges of about 1 kpc to 10 Mpc, a region that laboratory fifth-force and equivalence-principle searches cannot reach because the force is neutrino-specific.
If the mechanism operates, the cosmic neutrino background is not smoothly free-streaming at late times; part of it resides in bound states that act as additional gravitational seeds for structure growth.
Neutrino bound states with $M_{\rm bound}\gtrsim 10^8\,M_\odot$ forming at $z_{\rm NL}\gtrsim 60$ can trap baryons and ignite star formation, producing a reionization history that CMB optical-depth measurements constrain.
The projected DESI galaxy survey will sharpen these limits, and future 21-cm observations of the cosmic dark ages can test the early-star-formation channel.
Smaller bound states from shorter interaction ranges could show up in dark-matter substructure searches or as time-dependent signals in direct cosmic-neutrino detectors.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An N-body or hybrid simulation of the Yukawa-coupled neutrino fluid would settle the paper's main unsimulated step, the Press-Schechter-style extrapolation to non-gravitational collapse; if the collapsed fraction at $\delta_\nu(k_\phi)>1$ is significantly below order unity, the derived bounds would shift to smaller $g$ or a narrower mass range.
The same Poisson-noise route could translate other late-time non-gravitational instabilities, such as long-range forces acting on dark matter, into matter-power-spectrum constraints, so the strategy generalizes beyond neutrinos.
Because the excluded band tracks a roughly constant $g$ at fixed $m_\phi$, the sharpest future improvements will come from better measurements of the matter power spectrum at $k\sim 0.1$ to $2\,h\,\mathrm{Mpc}^{-1}$ and from pinning down the neutrino mass sum, which sets $f_\nu$ and the free-streaming scale.
The reionization channel requires early, massive bound states; high-redshift galaxy surveys such as JWST could either find the predicted star-forming halos or push the star-formation limits tighter.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper does something genuinely useful: it takes a neutrino-only long-range force that was previously weakly constrained and maps it onto existing matter power spectrum and reionization data, producing a specific exclusion band in the (g, mφ) plane. That mapping is new, and the linear perturbation backbone is trustworthy. The authors implement the modified Boltzmann equations in CLASS, match the numerical growth with an analytic fluid approximation, and use external data with no fitted parameters inside the central derivation. The background evolution, the delayed non-relativistic transition, and the Jeans instability criterion are all laid out cleanly.

The soft spot is exactly where the reader and the stress-test note point: the step from linear growth to nonlinear bound states. Section II C explicitly says Press-Schechter applied to the neutrino density field 'has not been rigorously checked' for non-gravitational collapses, then assumes an O(1) collapse fraction when δν > 1 and treats the resulting objects as point masses with a shot-noise power spectrum. At formation those objects are volume-filling, so point-like Poisson noise is not obviously justified. If the collapse fraction is a few percent instead of order one, or if the objects stay diffuse for a while, the Fig. 5 exclusions shrink substantially. The reionization constraint also depends on these objects trapping baryons and cooling efficiently, which is plausible but not demonstrated. I don't think this is fatal: the authors flag the assumption, they choose conservative radii, and they don't oversell the nonlinear physics. But the headline constraint is conditional on a step that needs simulations or at least a parametric treatment of the collapse fraction.

The paper is honest, well-cited, and technically sound in its linear regime. It is not a finished map of the parameter space; it is a first, clearly motivated probe. I would send it to a serious referee. The referee should push for a prominent caveat and ideally for the CLASS code to be released, but the core result—that this instability can be constrained by existing large-scale structure and reionization data—deserves to be in the literature.

Referee Report

3 major / 5 minor

Summary. This paper considers an ultralight scalar field coupled only to a Dirac neutrino, generating long-range forces in the cosmic neutrino background (CνB). The authors compute the background evolution, showing that the scalar is sourced by the CνB and delays the non-relativistic transition to a redshift z_nr given approximately by Eq. (5). They then study linear perturbations with a modified Boltzmann code (CLASS) and a non-relativistic fluid approximation, identifying a Jeans-like instability with growth rate γ(k) in Eq. (16). For parameter regions where δν(kφ) exceeds unity, they assume that an O(1) fraction of the CνB collapses into bound states of mass M_bound (Eq. 17) and radius ~ mφ^{-1} (Eq. 19), forming at redshift zNL. These bound states contribute a Poisson-noise term P_iso to the matter power spectrum (Eq. 22), which they compare with Lyα forest, LRG galaxy clustering, and a DESI forecast, and they also argue that the bound states can trigger early star formation constrained by reionization. The headline result is new constraints on the coupling g and mass mφ for ranges 1 kpc ≲ mφ^{-1} ≲ 10 Mpc, for a default mν = 60 meV.

Significance. The linear perturbation analysis is a solid contribution: it is based on a modified Boltzmann system implemented in a public code, it is matched by an analytic fluid growth calculation, and the growth rate follows from the model Lagrangian without any fitted parameters. If the nonlinear step were validated, the resulting constraints would close a previously open band in the neutrino fifth-force parameter space and establish the late-time CνB as a sensitive probe of new neutrino interactions. The paper is also honest about the limitations of its nonlinear treatment. However, the headline constraints are conditional on an unvalidated Press-Schechter-like collapse fraction and on a shot-noise model for objects that at formation are volume-filling; the significance of the numerical exclusions is therefore not yet established at the level claimed in the abstract.

major comments (3)

[Sec. II C, Eqs. (17) and (22)] The exclusions in Fig. 5 rest on the assumption that when the linear density perturbation δν(kφ) exceeds unity, an O(1) fraction of the CνB collapses into bound states of mass M_bound ≈ 4π mν nν,0(aNL)/(3 mφ³). The paper itself states in Sec. II C that it 'has not been rigorously checked whether the Press-Schechter theory is applicable to non-gravitational collapses.' A collapse fraction f_coll significantly below unity changes n̄bound and hence P_iso in Eq. (22), and therefore directly modifies every constraint shown in Fig. 5. Since this is the load-bearing step for the central claim, the authors should provide physical justification or supporting simulation for f_coll ≈ O(1), or show how the excluded region shrinks as f_coll is varied.
[Sec. II C and Sec. III A, Eqs. (19)–(23)] The Poisson term P_iso = fν² D_+² / n̄bound treats the bound states as discrete point particles. At formation, however, these objects are volume-filling: from Eq. (17), their physical number density is ~ (3/4π) mφ³, so their mean separation is of order their assumed radius mφ^{-1} from Eq. (19). The paper's cutoff in Eq. (24) only ensures that 2R_bound < r_ta at the observation redshift; it does not justify treating the still-virializing, extended density distribution as a Poisson sample of point masses. Given that the Lyα and LRG constraints in Fig. 5 are computed with this shot-noise model, the authors should test the sensitivity of their constraints to the assumed bound-state radius and density profile, or restrict claims to regimes where the point-mass approximation is demonstrably valid.
[Sec. III A, Eq. (25) and Fig. 5] The mapping from the phenomenological parameters (zNL, Mbound) to the model parameters (g, mφ) uses the analytic growth rate γ(k) of Eq. (16) and the criterion δν(kφ) ≳ 1. The agreement between this analytic estimate and the CLASS calculation is shown for a single benchmark (Fig. 2), and for mφ ≳ 10^{-28} eV the numerical code breaks down, leaving only the analytic estimate. Because the final exclusion region in the bottom panel of Fig. 5 is the main result, the authors should validate the mapping over the full shown parameter range, or quantify the systematic uncertainty in zNL and Mbound that propagates to the boundary of the excluded region.

minor comments (5)

[Abstract] The abstract states constraints for ranges '1 kpc ≲ mφ^{-1} ≲ 10 Mpc' without qualifying that these are derived for the default neutrino mass mν = 60 meV; the constraints for other masses are not computed, and this qualification should be added.
[Fig. 3] The caption of Fig. 3 does not define the dashed lines or the meaning of the orange shaded region in terms of the semi-analytical criteria; the text in Sec. II C refers to 'the orange shaded regime' and 'green shaded regime' but the figure legend would benefit from a more explicit description.
[Eq. (16)] The approximate expression in the second line of Eq. (16) would be clearer if the square-root factor were written explicitly, e.g., γ(k) ≈ 47 (k²/(k²+kφ²))^{1/2} (g/10^{-26}), and if the condition γ(k) ≳ 1 were restated next to it.
[Sec. III A, Eq. (22)] Please state explicitly that n̄bound is the comoving number density and that P_iso is the comoving power spectrum; the notation is conventional but would help the reader avoid a unit ambiguity.
[Sec. III A] Reference [139] is used for both the Lyα and LRG power spectrum reconstructions; a sentence describing whether the data points are treated as independent and how the covariance is accounted for (or neglected) in the χ² of Eq. (25) would make the analysis more reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the headline constraints are derived from the model Lagrangian and compared with external data, not fitted to them.

full rationale

The derivation chain is self-contained. The model is defined by Eq. (1); background evolution (Eqs. (3)-(5)) gives znr; linear perturbation theory (Eqs. (9)-(16), with the fluid limit derived in Appendix A) gives the growth rate gamma(k); the nonlinear step uses an explicitly flagged Press-Schechter-style extrapolation (Sec. II C) to assign Mbound (Eq. (17)) and Rbound (Eq. (19)); and the signal is the Poisson contribution Piso = f_nu^2 D_+^2 / nbar_bound (Eq. (22)) added to the LambdaCDM spectrum. The observed Ly-alpha/LRG power spectra and the reionization optical depth enter only at the comparison stage (Sec. III), so the 'prediction' is not obtained by inverting the data. The mapping from (Mbound, zNL) back to (g, m_phi) via Eqs. (5), (16), and (17) is a bijection within the model, not a fit to the constrained observables. Self-citations (e.g., Ref. [64] for a DESI sensitivity estimate) are not load-bearing. The paper's own limitation that the Press-Schechter extrapolation 'has not been rigorously checked' for non-gravitational collapses (Sec. II C) is a correctness risk: if the collapse fraction is not O(1), or if the bound states are not point-like on the relevant scales, the Fig. 5 exclusions weaken. That concern is substantive, but it is not circularity: it concerns the validity of an assumption, not an identity or a fitted input disguised as output.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central claim rests on a small number of chosen parameters and assumptions. The neutrino mass and the O(1) collapse fraction are the most important free parameters. The linear perturbation theory is standard, but the nonlinear collapse step and the Poisson-noise model are ad hoc to this paper and carry the main risk. No genuinely new particles beyond the already-hypothetical scalar are invoked.

free parameters (2)

mν (neutrino mass) = 60 meV
Fiducial neutrino mass set by hand in Sec. II; sets fν ≈ 0.45% and the mapping from (zNL, Mbound) to (g, mφ). The constraints scale with this choice, as noted by the authors for the pink curve in Fig. 4.
Collapse fraction of CνB forming bound states = O(1), assumed
Sec. II C assumes that when δν(kφ) exceeds unity, an order-one fraction of the CνB collapses. The amplitude of Piso scales linearly with this fraction, so it directly controls the strength of the constraints.

assumptions (6)

domain assumption ΛCDM background with Planck fiducial cosmological parameters
Used in Sec. III A to compute the reference matter power spectrum Pad and growth factors.
domain assumption Fluid approximation: shear and higher multipoles negligible, c_s ≈ 3Tν/mν
Invoked in Sec. II B and Appendix A to reduce the Boltzmann hierarchy to the continuity and Euler equations (Eqs. 9 and A18).
domain assumption Background evolution of φ0 follows Ref. [57], with φ0 decaying as a^{-3} after z_nr
Basis for Eq. (5) for a_nr, which sets the start of perturbation growth and the bound-state mass scale.
ad hoc to paper Press-Schechter-like collapse applies to non-gravitational Yukawa collapse, with O(1) fraction collapsing at δν > 1
Stated in Sec. II C as an assumption, not rigorously checked; this produces the bound states that source the signal.
ad hoc to paper The bound states act as discrete point masses for the matter power spectrum on scales k < k_cut
Used in Eq. (22) Piso = fν^2 D_+^2/n̄bound, borrowed from PBH/MACHO literature. The authors' own condition Eq. (24) is an attempt to ensure point-likeness, but the initial volume-filling is a concern.
domain assumption Rees-Ostriker-Silk cooling criterion: Tvir ≥ 10^4 K and tff ≤ 1/H trigger star formation
Adopted from Refs. [151-157] in Sec. III B to map reionization constraints onto the model parameter space.

invented entities (1)

Ultralight scalar φ coupled to neutrinos
purpose: Mediates the long-range force between neutrinos; generates a time-varying neutrino mass and the Yukawa-driven perturbation instability
Introduced as a hypothetical BSM field. No detection is claimed; the paper only excludes parts of its parameter space. The field has no independent falsifiable handle outside the cosmological signals used to set the constraints.

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Pith. "Pith review of Probing Long-Range Forces Between Neutrinos with Cosmic Structures." pith.science (2026). https://pith.science/paper/QCLDR7WJ

@misc{pith2026241220766,
  author       = {Pith},
  title        = {Pith review of: Probing Long-Range Forces Between Neutrinos with Cosmic Structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QCLDR7WJ}},
  note         = {Machine review of arXiv:2412.20766}
}

abstract

We study the consequences of new long-range forces between neutrinos on cosmic scales. If these forces are a few orders of magnitude stronger than gravity, they can induce perturbation instability in the non-relativistic cosmic neutrino background in the late time universe. As a result, the cosmic neutrino background may form nonlinear bound states instead of free-streaming. The implications of the formation of nonlinear neutrino bound states include enhancing matter perturbations and triggering star formation. Based on existing measurements of the matter power spectrum and reionization history, we place new constraints on long-range forces between neutrinos with ranges lying in $1 \text{ kpc}\lesssim m_\phi^{-1} \lesssim 10 \text{ Mpc}$.

Figures

Figures reproduced from arXiv: 2412.20766 by the authors.

**Figure 2.** FIG. 2. The C [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Overview of the parameter space. In the area shaded [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Illustration of the impact of the formation of neutrino [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Constraints on the formation of neutrino bound states. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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Forward citations

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Reference graph

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ad”) and neutrino bound states induced pertur- bations (labeled “iso

After this point, the neutrinos become non-relativistic and ϕ0 rapidly decreases at the rate ∝ a−3, and the effective mass of neutrino returns to its bare mass. The dynamics of ϕ0 result in an evolving neutrino mass, which delays the transition of the C νB from relativistic to non-relativistic until a lower redshift znr, where anr ≈ max   g2nν,0(a0) m2 ...

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(A8) Substituting Eqs

Background evolution We separate the homogeneous and perturbed parts of the fields ϕ and Fν as follows: ϕ = ϕ0(τ ) + δϕ(xµ), (A7) Fν = Fν,0(τ, p)(1 + Θ(xµ, pµ)). (A8) Substituting Eqs. (A7) and (A8) into Eq. (A1) and re- taining only the background part yields the equation of motion for the scalar field: ¨ϕ0 + 3H ˙ϕ0 + m2 ϕϕ0 = −g ⟨¯νν ⟩0 (ϕ0). (A9) At th...

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Perturbation evolution The full equation of motion governing the evolution of Θ and δϕ can be found in Refs. [ 57, 79, 87]. Following 4 Note that the expression Eq. (A4) works in the regime where mν + gϕ <0. The “negative” sign of the effective mass can be absorbed by redefining the fermion state. the conventions in Ref. [ 195], in the synchronous gauge, ...

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