REVIEW 3 major objections 3 minor 35 references
Inferring the cosmological constant in early Universe only by gravitational waves
T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read Gravitational waves from early-universe black-hole mergers could constrain the cosmological constant to under one percent.
desk verdict The 0.7% Lambda forecast rests on an invalid time-dependent-Lambda assumption, so the paper is a sensitivity study for exotic dark energy, not a measurement of the cosmological constant. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the $\Lambda$-modified, frequency-domain waveform of a circular binary on a de Sitter background (Eqs. 8-9), whose amplitude carries the factor $\left(1 + \Lambda R_f^3 f/(36c)\right)^{-1/16}$ and whose phase carries a corresponding $\Lambda$-term; physically, the cosmological constant acts as a frequency-dependent correction that grows toward low frequencies and early times. Coupled with the Friedmann-derived scaling $\Lambda \propto 1/(ct)$, this turns high-redshift sources into amplifiers of the $\Lambda$ effect. The Fisher information matrix $\Gamma_{ij} = \langle \partial h/\partial\lambda_i \,|\, \partial h/\partial\lambda_j\rangle$ then converts the waveform's $\Lambda$-derivative into a forecast uncertainty, which is the quantitative engine of the claimed constraint.
What would settle it
The claim would be settled by matched-filtering real LISA data for a confirmed high-redshift 1000 $M_\odot$ PBH merger: if the waveform with $\Lambda=10^{-45}$ is not preferred over the $\Lambda=0$ waveform, or if the best-fit $\Lambda$ is consistent with the constant local value $10^{-52}$, the $\Lambda\propto 1/(ct)$ scaling and the resulting detectability forecast are refuted.
Extended reading notes
Core claim
The paper's central claim is that a single coalescing 500+500 $M_\odot$ primordial black hole binary at redshift $z\approx 500$ emits a signal whose de Sitter-corrected waveform differs from the standard waveform by a factor that LISA can detect. The $\Lambda$-dependent amplitude factor $(\Lambda R_f^3 f/(36c)+1)^{-1/16}$ and its matching phase term shift the waveform enough that the Fisher matrix gives $\Delta\Lambda/\Lambda \approx 7.1\times 10^{-3}$ for LISA and $\approx 4.4\times 10^{-3}$ for a LISA+Taiji network. The key to making the effect visible is not the detector but the background: at $z=500$ the effective $\Lambda$, taken from $\Lambda \propto H^2/c^2$ and hence $\Lambda \propto 1/(ct)$, is $\sim 10^{-45}$ instead of the local $\sim 10^{-52}$, several orders of magnitude larger. The paper then verifies the choice of source by computing the merger rate and signal-to-noise ratio, and reports that all waveform parameters, including masses and spins, are measurable simultaneously with sub-percent precision.
Load-bearing premise
The load-bearing premise is that the cosmological constant obeys $\Lambda \propto 1/(ct)$ at early times (so that $\Lambda \approx 10^{-45}$ at $z=500$), because all of the detectability hinges on $\Lambda$ being orders of magnitude larger in the early universe than its present-day value.
Editorial extensions
If this is right
- LISA could measure the cosmological constant to about 0.7% from a single 1000 $M_\odot$ binary at $z\approx 500$; combining LISA and Taiji improves this to about 0.44%.
- High-redshift PBH mergers become a self-contained cosmological probe: with no electromagnetic counterpart required, the gravitational waveform alone carries the $\Lambda$ information.
- The mismatch between $\Lambda$ and no-$\Lambda$ waveforms increases with binary mass, so heavier PBH binaries—up to where the merger rate drops—are more sensitive to $\Lambda$; the 1000 $M_\odot$ total mass is the chosen balance point.
- All intrinsic parameters (masses, spins, time, phase) are simultaneously measurable with sub-percent precision, meaning the $\Lambda$ measurement does not degrade the rest of the source parameter estimation.
Reading between the lines
- In the standard $\Lambda$CDM picture, $\Lambda$ is constant and the relation $\Lambda = H^2/c^2$ holds only in a pure de Sitter phase, so the paper's high-redshift effective $\Lambda$ is really an inference about the background geometry; if taken literally, this measurement would probe an evolving dark-energy component or a modified gravity rather than the constant $\Lambda$ of the concordance mod
- The same de Sitter waveform correction could be used as a test of any long-wavelength modification of the wave zone; a measured $\Lambda$ consistent with zero at $z\approx 500$ would place direct bounds on the effective cosmological constant in the early universe.
- A natural next step is to replace the Fisher approximation with a full Bayesian posterior for $\Lambda$ and the binary parameters using the same waveform and LISA sensitivity; the Fisher numbers suggest the posterior would be informative, but the true error bars may differ due to non-Gaussianities and parameter degeneracies.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims that gravitational waves from high-redshift primordial black hole binaries (M_tot = 1000 solar masses, z > 500) can be used with LISA to constrain the cosmological constant. The authors compute a PBH merger rate, a waveform model with a cosmological-constant correction, signal-to-noise ratios, and a Fisher information matrix. They report a relative uncertainty on Lambda of about 0.7% for LISA and conclude that the cosmological constant can be effectively constrained in the early Universe.
Significance. If the central claim were correct, it would open a new observational window on the cosmological constant using high-redshift GW sources that lack electromagnetic counterparts. The paper does use a standard PBH merger-rate formalism and provides concrete SNR and Fisher-matrix estimates for LISA and Taiji, which is a useful exercise in assessing GW detector capabilities. However, the core result depends on an invalid scaling of Lambda with cosmic time and on a waveform model whose background spacetime does not describe the early Universe. The claimed constraint is therefore not a constraint on the cosmological constant of standard cosmology.
major comments (3)
- [Section III, Eq. (14)] Equation (14) asserts Lambda proportional to 1/(ct). This is dimensionally inconsistent: the left-hand side has units of inverse length squared (in the usual convention where Lambda multiplies g_mu_nu in Einstein's equations), while 1/(ct) has units of inverse length. The text derives this relation from H(t) = a_dot/a and Lambda proportional to H^2/c^2, but the Friedmann equation is H^2 = (8 pi G / 3) rho + Lambda c^2 / 3, so during radiation and matter domination the constant Lambda term is negligible and does not scale as H^2. In standard Lambda-CDM, Lambda is constant; the value used in the Fisher matrix, Lambda = 10^-45 m^-2, is not the cosmological constant at z = 500 but an ad hoc, nonstandard time-varying dark-energy parameter.
- [Section III, Eqs. (8)-(9)] The waveform model (8)-(9) is taken from Ref. [31], which derives gravitational-wave solutions in a pure de Sitter background (empty Universe with a cosmological constant). The authors then apply this model to a matter/radiation-dominated FLRW Universe at z = 500. This is unjustified: the background expansion history, including the matter and radiation content that dominates at z = 500, is absent from the waveform model. Consequently, the match calculations in Fig. 2 and the Fisher matrix results in Table I are based on a waveform that does not describe the stated early-Universe scenario.
- [Section III, Table I] The Fisher matrix is computed with the fiducial value Lambda = 10^-45 m^-2, which is about seven orders of magnitude larger than the observed cosmological constant (about 10^-52 m^-2). The authors themselves note that a GW with Lambda = 10^-52 cannot be distinguished by LIGO. Since the waveform's dependence on Lambda is monotonic and the Fisher error scales with the fiducial value, the reported Delta-Lambda / Lambda = 7.11 x 10^-3 for LISA is an artifact of the chosen fiducial, not a prediction of measurability for the actual cosmological constant. The conclusion that LISA can constrain Lambda therefore does not follow.
minor comments (3)
- [Section II, text after Eq. (2)] The phrase 'the condition for this pair to potentially form a binary system is given by x' is incomplete; it should specify the condition, e.g., that the separation is less than some scale (presumably the mean separation).
- [Eq. (8)] The quantity R_f is introduced as 'the observation time' but its role in the waveform amplitude and phase is not explained; please define it clearly and state how it is fixed in the calculations.
- [General] Some notation is inconsistent, e.g., the cosmological constant is written as both Lambda and Lambda in different places, and the chirp mass is defined but not clearly used in Eqs. (8) and (9).
Circularity Check
The claimed constraint on the cosmological constant is computed around an injected fiducial Λ=10^-45 justified by a nonstandard, dimensionally inconsistent relation Λ∝1/(ct); the measurability result is therefore built into the input.
-
fitted input called prediction
[Section III, after Eq. (10); Table I]
"A study by [31] revealed that a gravitational wave with Λ = 10−52 cannot be distinguished by LIGO. Hence, in this study, we explore the possibility of increasing the value of Λ to achieve a more significant discernible difference. ... Now, we consider the PBH binary with m1 = m2 = 500 M⊙, a1 = a2 = 0.5, and Λ = 10−45 to get the FIM."
The central result (ΔΛ/Λ=7.11e-3) is a Fisher forecast around Λ=10^-45, a value chosen specifically because it is large enough to make the waveform differ visibly from the Λ=0 waveform. No independent physical derivation fixes this fiducial; the abstract's claim that 'the cosmological constant can be effectively constrained' is thus a restatement of the injected amplitude, not an inference about the standard cosmological constant (Λ≈10^-52), for which the paper itself notes the effect is indistinguishable. The prediction reduces by construction to the chosen input.
-
self definitional
[Section III, Eqs. (12)-(14)]
"From the above equations and Λ ∝ H^2/c^2, we can get the relationship between cosmological constant Λ and time t: Λ ∝ 1/(ct) (14)"
In the standard Friedmann equation H^2=(8πG/3)ρ+Λc^2/3, Λ is constant and is not proportional to H^2 except in a pure de Sitter phase. The paper instead defines an early-universe Λ through Λ∝1/(ct) (a relation that is also dimensionally inconsistent: H^2/c^2 has units of inverse length squared, while 1/(ct) has inverse length). This definition is the only justification for Λ=10^-45 at z=500, so the subsequent Fisher 'constraint' is a property of the definition rather than an independent measurement of the cosmological constant.
full rationale
No load-bearing self-citations are present: the merger-rate and SNR calculations use standard external tools and references, and the waveform is taken from Ref. [31], an external work. The circularity lies in the fiducial construction. The paper increases Λ from 10^-52 to 10^-45, uses the dimensionally suspect relation Λ∝1/(ct) to make this seem natural at z=500, and then reports a Fisher-matrix precision around that injected value. A Fisher forecast is a legitimate sensitivity tool, so the calculation is not vacuous, but the headline claim that the cosmological constant can be constrained is conditional on a nonstandard, assumed value; for the standard Λ≈10^-52 the same pipeline would show negligible sensitivity. Hence the central prediction is partially forced by the input (score 6), rather than being an independent first-principles result.
Assumptions & free parameters
free parameters (3)
- Cosmological constant value used in Fisher matrix, Λ=10^-45 m^-2 =
10^-45 m^-2
- PBH formation parameters α and β =
α=β=1
- PBH spin a1=a2 =
0.5
assumptions (3)
- ad hoc to paper Λ ∝ H^2/c^2 holds in the early universe, and hence Λ ∝ 1/(ct)
- domain assumption The de Sitter waveform from Nef et al. (2009) applies at z=500 in a FLRW universe
- domain assumption PBH binary formation model of Sasaki et al. (2016) with uniform spatial distribution
Cite this review
Pith. "Pith review of Inferring the cosmological constant in early Universe only by gravitational waves." pith.science (2026). https://pith.science/paper/NS27IMMT
@misc{pith2026250112608,
author = {Pith},
title = {Pith review of: Inferring the cosmological constant in early Universe only by gravitational waves},
year = {2026},
howpublished = {\url{https://pith.science/paper/NS27IMMT}},
note = {Machine review of arXiv:2501.12608}
}
abstract
The expansion of the Universe is accelerating which can be interpreted as due to the cosmological constant $\Lambda$. In this study, we investigate the behavior of gravitational waves in the presence of a cosmological constant in the early universe. We rigorously analyze the merger rate of binary primordial black holes (PBHs) and the corresponding signal-to-noise ratio within the framework of Laser Interferometer Space Antenna (LISA). We find that binary PBHs with a total mass of $M_{\mathrm{tot}}=1000M_{\odot}$ and a redshift larger than $z=500$ are the ideal system for studying the effect of the cosmological constant through LISA. By computing the fisher information matrix, we establish that the cosmological constant can be effectively constrained.
Figures
Reference graph
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