{"id":"67273ca7-a000-4d6a-8d7f-7c02d48443ef","arxiv_id":"2501.12608","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A Fisher-matrix forecast says LISA can constrain Λ to 0.7% using z=500 PBH binaries, but the forecast uses an assumed Λ=10^-45, not the standard constant value.","lead":"This paper claims that the LISA space telescope could measure the cosmological constant using gravitational waves from merging primordial black holes at a redshift of 500. The claim rests on an assumption that the cosmological constant was seven orders of magnitude larger in the early universe, an assumption the paper does not justify and that conflicts with the standard constant-Λ model.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (14) turns Λ into a time-dependent quantity, Λ∝1/(ct), and the Fisher forecast uses Λ=10^-45 at z=500; under standard ΛCDM Λ is constant and subdominant, so the claimed 0.7% constraint is not about the cosmological constant.","rationale":"The central claim is a Fisher-matrix forecast that LISA can constrain the cosmological constant to 0.7% using a 500+500 Msun PBH binary at z=500. For that claim to hold, the waveform must contain a physically meaningful, standard-model cosmological-constant effect at that redshift. The paper's Eq. (14) asserts Λ∝1/(ct), which would make Λ grow toward early times; this is not a property of the cosmological constant in ΛCDM, where Λ is constant and its contribution to the Friedmann equation is subdominant during matter and radiation domination. The dimensional inconsistency of Eq. (14) reinforces that this is not a small approximation error but an incorrect identification. The Fisher calculation then adopts Λ=10^-45 as a free fiducial value, seven orders of magnitude above the observed Λ≈10^-52, so the quoted precision is a sensitivity to that chosen model rather than an inference about the actual cosmological constant. The authors do not present an alternative dark-energy model, a derivation from a covariant theory, or a justification for transplanting a de Sitter waveform into an early-time FLRW background. The paper could be reframed as a sensitivity study for an exotic time-varying dark-energy component, but as written the strongest claim—that the cosmological constant can be effectively constrained—is not supported. This matches the reader's identified weakest assumption, so the REJECT verdict stands without modification.","tokens_in":8029,"tokens_out":3098,"duration_ms":32067,"concrete_test":"Recompute the match (Eqs. 15-17) and the Fisher matrix (Eqs. 19-20) for the same Mtot=1000Msun, z=500 system, replacing the fiducial Λ=10^-45 with the standard cosmological constant Λ≈1.1e-52, keeping the waveform model fixed and without invoking Eq. (14). If the resulting ΔΛ/Λ is no longer of order 10^-3 (or the inverse Fisher element is not positive), the claimed 0.7% constraint is an artifact of the nonstandard fiducial value.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Eq. (14), used to promote Λ from 10^-52 to 10^-45 at z=500. It is obtained from H∝1/t in the radiation/matter eras and Λ∝H^2/c^2, but in standard ΛCDM the Friedmann equation is H^2=(8πG/3)ρ+Λc^2/3; during matter/radiation domination the constant Λ term is negligible and does not track H^2(t). Moreover, Eq. (14) is dimensionally inconsistent: H^2/c^2 has units 1/length^2, while 1/(ct) has units 1/length. Using Λ=10^-45 as the fiducial in the Fisher matrix is therefore an assumption of a time-varying dark-energy component, not a prediction about the cosmological constant. The waveform (8)-(9) from Ref. [31] is derived in a pure de Sitter background; applying it to a matter/radiation-dominated FLRW universe at z=500 is unjustified. Consequently the reported ΔΛ/Λ=7.11e-3 measures sensitivity to a large, fiducial, nonstandard Λ and does not establish that LISA can constrain the cosmological constant.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1189,"tokens_out":1365,"duration_ms":35489,"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":[{"comment":"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":"Section III, Eq. (14)"},{"comment":"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":"Section III, Eqs. (8)-(9)"},{"comment":"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.","section":"Section III, Table I"}],"minor_comments":[{"comment":"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).","section":"Section II, text after Eq. (2)"},{"comment":"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.","section":"Eq. (8)"},{"comment":"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).","section":"General"}],"recommendation":"reject","confidential_remarks":"The central claim fails because Eq. (14) is dimensionally inconsistent and not a consequence of standard Friedmann cosmology, and because the waveform model is derived in a de Sitter background that does not describe the early Universe. The Fisher-matrix result is an illustration of sensitivity to an ad hoc fiducial value rather than a physical prediction. These are not local errors that a revision could fix within the current scope; they undermine the paper's main conclusion."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this paper forecasts a 0.7% measurement of Lambda from LISA, but the forecast is built on an assumption that Lambda grows like 1/(ct) in the early universe. That assumption is not part of standard LCDM, and the paper's derivation of it is dimensionally inconsistent. So the headline result is not about the cosmological constant.\n\nWhat the paper does well: the merger-rate calculation follows Sasaki et al. and is standard; the SNR and Fisher-matrix machinery is competently applied; and the match between waveforms with and without Lambda is a sensible way to see where the effect could matter. The identification of 500 solar-mass PBHs at z~500 as a possible probe is a legitimate extension of previous work, and the paper is clearly organized.\n\nThe soft spots are not minor. Equation (14) says Lambda proportional to 1/(ct). In LCDM, Lambda is constant; during radiation and matter domination the Friedmann equation has H^2 ~ rho + Lambda c^2/3, and the Lambda term is negligible, not equal to H^2/c^2. The step from H proportional to 1/t to Lambda proportional to 1/(ct) also has a units problem: H^2/c^2 has dimensions 1/length^2, while 1/(ct) is 1/length. The waveform in Eqs. (8)-(9) comes from a de Sitter background (Nef et al. 2009), and using it in a matter/radiation-dominated FLRW universe at z=500 is unjustified. The Fisher forecast then takes Lambda = 10^-45 m^-2 as a free input; the 0.7% uncertainty is a sensitivity to that input, not a measurement of the cosmological constant. Under the standard value, the effect would be negligible.\n\nSo the paper is best read as a sensitivity study for an exotic time-varying dark-energy model, not as a method to infer Lambda in the standard model. It does not establish the claim in the abstract.\n\nI would not send this to peer review in its current form; the central argument has a load-bearing invalid step. A serious referee might help reframe it as a study of early-time dark energy, but as written it overclaims.","headline":"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.","tokens_in":8833,"tokens_out":2722,"would_cite":false,"duration_ms":26712,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C35","83F05"],"pacs":["04.30.-w","98.80.-k"],"model":"deepseek-v4-flash","headline":"Gravitational waves from early-universe black-hole mergers could constrain the cosmological constant to under one percent.","keywords":["cosmological constant","gravitational waves","primordial black holes","LISA","Fisher information matrix","de Sitter background","early universe","parameter estimation"],"falsifier":"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.","tokens_in":7750,"feed_emoji":"🌌","tokens_out":9775,"duration_ms":87859,"temperature":0.7,"pith_summary":"The paper argues that gravitational waves from mergers of primordial black holes in the early universe carry a measurable imprint of the cosmological constant, because the waveform is computed on a de Sitter background rather than flat spacetime. At high redshift, using the Friedmann relation $\\Lambda \\propto 1/(ct)$, the effective value of $\\Lambda$ grows far above its local value, making the effect visible to LISA. The authors identify a 500+500 solar-mass binary at $z\\approx 500$ as the best compromise between event rate and waveform difference, and a Fisher information matrix forecast shows $\\Lambda$ could be constrained to about 0.7% with LISA alone and 0.44% with LISA and Taiji combined. If correct, this would make the cosmological constant measurable from gravitational waves alone, without electromagnetic counterparts.","feed_headline":"Early-universe black holes could reveal the cosmological constant","feed_subtitle":"LISA could measure the cosmological constant to under 1% from one 1000-solar-mass merger at redshift 500.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the de Sitter-background gravitational waveform with $\\Lambda$-dependent amplitude and phase that the analysis uses.","marker":"[31]"},{"why":"Provides the merger-rate formula for PBH binaries used to select the 1000 $M_\\odot$, $z=500$ configuration.","marker":"[30]"},{"why":"Defines the LISA sensitivity curve used for overlap, SNR, and Fisher forecasts.","marker":"[13]"},{"why":"Gives the phenomenological inspiral phase model used to build the $\\Lambda$-modified waveform for parameter estimation.","marker":"[32]"},{"why":"Establishes the Fisher information matrix formalism for gravitational-wave parameter uncertainties.","marker":"[35]"},{"why":"Provides the toolkit used to compute signal-to-noise ratios for the LISA detector.","marker":"[34]"}],"fun_headline_variants":["Ancient black hole merger could reveal the cosmological constant to 0.7%","LISA could measure the cosmological constant to <1% from one z=500 merger","Cosmological constant from a single primordial black hole binary at z=500","Early universe gravitational waves could pin down the cosmological constant"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Ancient black hole merger could reveal the cosmological constant to 0.7%","LISA could measure the cosmological constant to <1% from one z=500 merger","Cosmological constant from a single primordial black hole binary at z=500","Early universe gravitational waves could pin down the cosmological constant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000859,"raw_usage":{"total_tokens":3716,"prompt_tokens":919,"completion_tokens":2797,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":2726}},"tokens_in":535,"tokens_out":2797,"duration_ms":21286,"temperature":1.0,"reasoning_tokens":2726,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:59:34.602291+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"N¨ af, P","cited_arxiv_id":null,"evidence_quote":"Supplies the de Sitter-background gravitational waveform with $\\Lambda$-dependent amplitude and phase that the analysis uses."}],"review_version":1}