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Lorentz Violation with Gravitational Waves: Constraints from NANOGrav and IPTA Data

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In a modified-gravity model with extrinsic-curvature derivative terms, pulsar-timing observations of the nanohertz gravitational-wave background imply a Lorentz-violating scale $M_{LV} > 10^{-19}$ GeV at 68% confidence.

desk verdict New PTA constraint on Lorentz-violating GW damping, but the central bound is unreproducible without the reported horizon-entry scale factor and has a sign inconsistency. read the letter →

arxiv 2505.22736 v1 pith:HXDOW5GF submitted 2025-05-28 astro-ph.CO astro-ph.HEgr-qc

Alireza Allahyari , Mohammadreza Davari , David F. Mota This is my paper
classification astro-ph.COastro-ph.HEgr-qc
keywords Lorentzviolationgravitationalwavespulsartimingarraysstochasticgravitational-wavebackgroundmodifiedgravityextrinsiccurvaturespectralenergydensity
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's central claim is that pulsar-timing observations of the nanohertz gravitational-wave background can directly constrain Lorentz violation in gravity. It adds extrinsic-curvature derivative terms to the gravitational action, making gravitational-wave propagation frequency-dependent through an energy scale $M_{LV}$, and derives the resulting spectral energy density. Fitting this spectrum to two recent pulsar-timing datasets yields $M_{LV} > 10^{-19}$ GeV at 68% confidence, excluding Lorentz-violating gravitational modifications with energy scales below that value and improving on the earlier binary-merger bound by about two orders of magnitude.

What carries the argument

The central object is the modified tensor-mode transfer function and the spectral energy density built from it. The added extrinsic-curvature term changes the gravitational-wave mode equation to $h_A'' + (2+\bar\nu)\mathcal{H}h_A' + k^2 h_A = 0$, where $\bar\nu\mathcal{H} = [\ln(1 + c_1 k^2/a^2)]'$. The load-bearing piece is the closed-form approximation $T = e^{D}T_{\mathrm{GR}}$ with $D$ as above; it converts the single parameter $M_{LV}$ into a frequency-dependent deformation of $\Omega_{GW}(f)$, and the fits map the data onto a lower bound for $M_{LV}$.

What would settle it

Compute $a_e$ from the standard horizon-entry condition $k = a_e H(a_e)$ and evaluate Eq. (3.8) at $f = 10^{-9}$ Hz for $M_{LV} = 10^{-19}$ GeV. If the exponential factor is not a suppression of the GR spectrum, the reported bound is not a damping bound. A direct numerical integration of the mode equation (2.6) at that frequency would settle which sign is physical.

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Extended reading notes

Core claim

The central claim is that adding the term $c_1(\nabla_k K^{ij}\nabla^k K_{ij} - R_{ij}R^{ij})$ to the gravitational action, with $c_1 = \alpha_{\bar\nu}/M_{LV}^2$, changes the stochastic gravitational-wave background at pulsar-timing frequencies. The transfer function picks up an exponential factor $e^{2D}$ with $D = -\frac{1}{2}[\alpha_{\bar\nu}(2\pi f/(a c M_{LV}))^2]_{a_e}^{a}$, producing the present-day spectral energy density in Eq. (3.8). Fitting that spectrum to two pulsar-timing datasets gives $\log_{10}(M_{LV}/\mathrm{GeV}) > -19$ at 68% confidence for both, with best-fit strain amplitudes $\log_{10}A \approx -14.13$ and $-14.34$, and spectral indices $\gamma \approx 3.22$ and $4.08$. The paper reads this as evidence that pulsar timing arrays probe Lorentz violation more strongly than binary-merger observations.

Load-bearing premise

The bound depends on the sign and size of the exponential factor in Eq. (3.8), which in turn depends on the horizon-entry scale factor $a_e$ and the sign convention adopted; if either is wrong, the reported $M_{LV}$ limit changes.

Editorial extensions

If this is right

  • Lorentz-violating gravitational modifications with an energy scale below $10^{-19}$ GeV are excluded by current pulsar-timing data, within this model.
  • The same datasets imply a blue-tilted background spectrum ($\gamma \approx 3.2$ and $4.1$), so the observed signal can be accommodated as an inflationary relic with modified propagation.
  • Because the exponential factor grows with frequency, the bound strengthens toward higher reference frequencies; the paper reports roughly an order-of-magnitude improvement at $f_{yr}$ compared with $10^{-9}$ Hz.
  • Pulsar timing arrays become a competitive probe of spacetime symmetries, reaching about two orders of magnitude below the binary-merger energy scale.

Reading between the lines

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

  • A natural check is to fix $a_e$ from the horizon-entry condition $k = a_e H(a_e)$ and recompute Eq. (3.8); the reported lower bound would shift if this changes the sign or size of the exponential factor.
  • The same exponential transfer function predicts a departure from a pure power-law background at higher frequencies, so future pulsar-timing data with sensitivity above roughly $10^{-8}$ Hz could confirm or rule out the model independently of the current fit.
  • The derivation is not tied to nanohertz frequencies; recasting it at millihertz band would extend the bound by several orders of magnitude if the model holds, though the paper does not perform that extrapolation.
  • If the model is correct, power-law spectral-index templates partially absorb the Lorentz-violating curvature, so joint fits that include the exponential shape should be used when comparing models.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a Lorentz-violating modification to the gravitational action by adding extrinsic-curvature derivative terms, derives a modified stochastic gravitational-wave background spectrum, and fits this model to NANOGrav 15-year and IPTA second data release data using PTArcade. It reports a 68% lower bound M_LV > 10^-19 GeV, along with best-fit strain amplitudes and spectral indices for both datasets.

Significance. If the derivation and statistical treatment are correct, the result would strengthen constraints on this particular Lorentz-violating gravity model by about two orders of magnitude over the LIGO/Virgo bound quoted in the paper, and it would demonstrate that pulsar timing arrays can probe the Lorentz-violating scale in this class of theories. A clear strength is that the paper takes a concrete, falsifiable model and confronts it with public PTA likelihoods through a standard tool (PTArcade). However, the central spectral formula is not derived in the manuscript, the sign of the claimed 'damping' effect is inconsistent with the equations as written, and the horizon-entry scale a_e is left unspecified; these issues currently prevent the headline constraint from being reproduced or verified.

major comments (4)
  1. [§2-§3, Eqs. (2.8), (3.4), (3.8)] The sign of the claimed 'damping' effect is inconsistent with the equations as written. Setting ᾱ_ν=1 makes c1 constant, and Eq. (2.8) then gives Hν̄ = (c1 k^2/a^2)' = -2 H c1 k^2/a^2 < 0. Inserting this into Eq. (3.4) yields D = +1/2 (a_e^{-2} - a^{-2})(k/M_LV)^2 > 0 for a > a_e, so the exponential in Eq. (3.8) amplifies the GW spectrum rather than damping it. The text repeatedly describes the effect as 'damping,' and the reported lower bound on M_LV relies on this sign: with a positive friction coefficient the spectrum would be suppressed and the claimed lower bound would not follow. Please state the assumed sign of c1, reconcile the terminology and the equations, and rerun the analysis if the sign changes.
  2. [§3, Eq. (3.4); §4] The scale factor at horizon entry, a_e, is never computed or specified. The text merely says that 'ae refers to the scale factor at horizon entry,' but a_e^{-2} enters exponentially in Eq. (3.8); for f=10^{-9} Hz in standard cosmology a_e is of order 5×10^{-12}, so a_e^{-2} is of order 4×10^{22}, and an error in a_e shifts the inferred M_LV by a factor proportional to a_e^{-1}. Please give the explicit formula for a_e(k) (e.g., the radiation-entry relation a_e = H0 sqrt(Ω_r)/k) and use it consistently across the fitted band. The statement that the result is 'obtained with f=10^{-9} Hz' also needs clarification: if a_e depends on k, the exponent scales faster than f^2 and the posterior should be driven by the highest frequencies, not the lowest; if a_e is instead fixed, that approximation must be justified.
  3. [§2, Eq. (2.6); §3, Eqs. (3.3)-(3.4), (3.7)-(3.8)] The central spectral formulas are asserted without derivation. Eq. (2.6) is stated as the equation of motion of action (2.1), but the variation is not shown; in particular, the claim that the R_ij R^ij term cancels the ∇_k K_ij ∇^k K^ij contribution so that GWs remain luminal is not demonstrated. Likewise, the transfer function decomposition in Eqs. (3.3)-(3.4) and the evaluation of D are taken as given, and the step from Eq. (3.7) to Eq. (3.8) skips the computation of D' and the validity of the approximation T'_GR = k T_GR. Please provide these derivations or give precise references for each step so the modified spectral energy density can be verified.
  4. [§4] The statistical setup is under-specified. The manuscript states only that PTArcade is used with uniform priors, without saying which likelihood is adopted (full Hellings-Downs correlation vs. common-spectrum process), which pulsar noise models are included, and how the one-sided 68% limit is defined. The marginal posterior for log10 M_LV should be shown directly; if it is truncated by the prior boundary at log10 M_LV = -25, the quoted lower limit may be prior-driven. Please report the full posterior or at least the one-dimensional marginalized distribution and convergence diagnostics.
minor comments (4)
  1. [§5, Table 1] There are several typographical errors: 'IPT A2' and 'ML V' appear in Section 5, and the Table 1 caption repeats 'MLV MLV' and 'γ γ'; these should be corrected.
  2. [§3, Eq. (3.8)] The factor (2πf H0/(c M_LV^2) + 1)^2 is numerically very close to unity for the quoted parameter ranges; stating this explicitly would help readers see that the exponential term dominates the constraint.
  3. [Figure 2] The caption does not specify the axes or the exact quantity plotted; please add axis labels and clarify whether the theoretical curves are Ω_GW from Eq. (3.8) evaluated at the best-fit parameters.
  4. [References] Reference [54] (PTArcade) lacks a journal or volume entry, and reference [47] is cited for the construction of the action; please complete the bibliographic details.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the M_LV bound is a standard parameter-estimation result from public PTA data, not a prediction built from the data it claims to constrain.

full rationale

The paper derives a Lorentz-violating gravitational-wave spectrum from a specified action (Eq. 2.1), propagates the modified tensor equation (Eq. 2.7), introduces a damping factor D (Eq. 3.4), and obtains the spectral energy density (Eq. 3.8). It then fits the free parameters A, gamma, and M_LV to the NANOGrav 15-year and IPTA second data release using PTArcade. This is ordinary Bayesian parameter estimation rather than a circular construction: the data are used to infer the parameters, and the reported lower bound M_LV > 10^-19 GeV is a posterior constraint, not a quantity defined in terms of the fitted values. The choices alpha_nu = 1 and the horizon-entry scale factor a_e are model inputs, not fitted outputs, so they cannot make the inference circular, though they may affect the numerical result and reproducibility. The action and transfer-function ingredients are drawn from prior literature, but these citations are not self-referential in a load-bearing way; reference [37] is an external LIGO/VIRGO bound used only for comparison, and the transfer-function ansatz of [50,51] is standard and not used to smuggle in the target conclusion. Concerns about the sign of the exponential factor and the unspecified computation of a_e are correctness or robustness issues, not circularity. The paper makes no claim to predict the PTA signal from first principles without using the PTA data; its central claim is explicitly an observational constraint. Therefore no step reduces by definition or by self-citation to the paper's own inputs, and the circularity score is 0.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central result depends on the model action (2.1), the quadratic expansion and equation of motion (2.3)-(2.6), the transfer function approximation (3.3)-(3.4), and the choice alpha_nu = 1. The fitted parameters are M_LV, A, and gamma. The horizon-entry scale factor a_e is an input that is not specified in the paper, and alpha_nu is set by hand. No new particles or fields are introduced.

free parameters (5)
  • M_LV = log10 M_LV > -19 GeV (68% CL)
    The Lorentz-violating energy scale, constrained by fitting the modified GW spectrum to NANOGrav and IPTA data.
  • A (strain amplitude at fyr) = log10 A = -14.13 +/- 0.15 (NG15), -14.34 +0.21/-0.14 (IPTA2)
    Amplitude of the power-law strain spectrum, fitted jointly with gamma and M_LV.
  • gamma (spectral index) = 3.22 +/- 0.37 (NG15), 4.08 +/- 0.39 (IPTA2)
    Spectral index of the strain power law, fitted jointly with A and M_LV.
  • alpha_nu = 1 (fixed)
    Time-dependence of the coupling c1; set to unity by hand, which affects the inferred scale of M_LV.
  • a_e = not reported
    Scale factor at horizon entry appearing in the damping exponent; the paper does not state how it is computed for each frequency in the PTArcade likelihood.
assumptions (6)
  • domain assumption The action (2.1) with the extrinsic-curvature derivative term and the R_ij R^ij subtraction is the Lorentz-violating theory under consideration.
    Motivated by spatial covariant gravity and Horava-Lifshitz gravity, cited from refs [37, 47, 48, 49]; the paper does not derive this action.
  • standard math The quadratic expansion (2.3) and the equation of motion (2.6) for tensor perturbations follow from the action (2.1).
    Asserted in Sec. 2 without a detailed derivation; a reader cannot check the cancellation of the dispersion modification.
  • domain assumption For small c1, the relation H*nu_bar approximately (c1 k^2/a^2)' holds and the Fourier equation reduces to Eq. (2.7).
    Used in Eq. (2.8) to build the transfer function; valid only when c1 k^2/a^2 is small.
  • domain assumption The PTA background is described by a power-law strain spectrum h_c(f) = A (f/fyr)^alpha with fitted A and alpha, modified by the Lorentz-violating factor.
    The analysis fits A, gamma, and M_LV to NG15 and IPTA2 data; this assumes the background is a stochastic GW signal of this form.
  • standard math The standard formula for Omega_GW from [50] and the approximation T'_GR = k T_GR apply at PTA frequencies.
    Used in deriving Eqs. (3.7)-(3.8); standard for sub-horizon modes with k >> k_eq.
  • ad hoc to paper The time-dependence coefficient is set to alpha_nu = 1.
    Eq. (2.9) parametrizes c1 = alpha_nu/M_LV^2 and the paper fixes alpha_nu = 1 without justification; the bound on M_LV would scale with this choice.

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Pith. "Pith review of Lorentz Violation with Gravitational Waves: Constraints from NANOGrav and IPTA Data." pith.science (2026). https://pith.science/paper/HXDOW5GF

@misc{pith2026250522736,
  author       = {Pith},
  title        = {Pith review of: Lorentz Violation with Gravitational Waves: Constraints from NANOGrav and IPTA Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HXDOW5GF}},
  note         = {Machine review of arXiv:2505.22736}
}
abstract

We explore a theoretical framework in which Lorentz symmetry is explicitly broken by incorporating derivative terms of the extrinsic curvature into the gravitational action. These modifications introduce a scale-dependent damping effect in the propagation of gravitational waves (GWs), governed by a characteristic energy scale denoted as $M_{{LV}}$ . We derive the modified spectral energy density of GWs within this model and confront it with recent observational data from the NANOGrav 15-year dataset and the second data release of the International Pulsar Timing Array (IPTA). Our analysis yields a lower bound on the Lorentz-violating energy scale, finding $M_{{LV}} > 10^{-19}$ GeV at 68\% confidence level. This result significantly improves upon previous constraints derived from LIGO/VIRGO binary merger observations. Our findings demonstrate the potential of pulsar timing arrays to probe fundamental symmetries of spacetime and offer new insights into possible extensions of general relativity.

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Reviewed August 7, 2026 · model on record in the stance chip above.