arxiv: 2604.22731 · v1 · submitted 2026-04-24 · 🌌 astro-ph.CO · gr-qc· hep-ph

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Precision Analysis for boldsymbol{H₀} Using Upcoming Multi-band Gravitational Wave Observations

Setabuddin , Md Riajul Haque , Ratna Koley , Supratik Pal

Authors on Pith no claims yet

Pith reviewed 2026-05-08 09:50 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-ph

keywords primordial black holesHubble constantgravitational wavesmulti-band observationsFisher matrixscalar-induced GWHubble tension

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The pith

Multi-band gravitational wave observations from primordial black holes can constrain the Hubble constant to within 2 km/s/Mpc.

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

The paper examines how signals from primordial black holes observed across multiple frequency bands can tighten constraints on the Hubble parameter H0. It combines scalar-induced gravitational waves accessible to the Square Kilometre Array with merger-induced waves detectable by the Einstein Telescope, then uses signal-to-noise estimates and Fisher matrix analysis on the black hole mass and abundance to forecast errors. These parameter uncertainties are propagated to derive corresponding limits on H0. A sympathetic reader would care because the approach supplies an independent cosmological probe that avoids traditional distance-ladder systematics and could address the Hubble tension with entirely new observations.

Core claim

The central finding is that relative uncertainties of 10 percent or less on the primordial black hole parameters M_PBH and f_PBH translate into an uncertainty on H0 of at most 2 km s^{-1} Mpc^{-1} in a conservative multi-band analysis, improving to order 0.1 km s^{-1} Mpc^{-1} in an optimistic precision case, with the results largely insensitive to the fiducial H0 value and only moderately dependent on collapse efficiency.

What carries the argument

Fisher matrix propagation of uncertainties from the primordial black hole parameters M_PBH and f_PBH, obtained from the joint analysis of scalar-induced and merger-induced gravitational wave signals in multi-band data from SKA and ET.

If this is right

Multi-band gravitational wave observations supply an independent and complementary approach to constraining uncertainties in H0.
The resulting H0 uncertainties remain largely insensitive to the exact fiducial value chosen for H0.
The constraints show only moderate dependence on the assumed efficiency of primordial black hole collapse.
This framework offers a novel cosmic distance ladder-independent route to measuring the Hubble parameter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If such primordial black hole signals are actually detected, the method could serve as a cross-check against other gravitational-wave determinations of H0 such as those from binary neutron star mergers.
The same multi-band framework could be extended to additional future detectors to achieve even tighter joint constraints on both black hole properties and cosmological parameters.
Absence of the predicted signals in real data would directly translate into stronger upper bounds on primordial black hole abundance in the relevant mass window.
Real observations might eventually test whether unmodeled systematics in the multi-band combination alter the propagated errors on H0 beyond the Fisher matrix estimates.

Load-bearing premise

Primordial black holes must exist in sufficient abundance to produce detectable scalar-induced and merger-induced gravitational wave signals with signal-to-noise ratio of at least 1 in the SKA and Einstein Telescope bands.

What would settle it

Observation of the expected multi-band gravitational wave signals from primordial black holes but with measured uncertainties in H0 that substantially exceed the forecasted values of 2 km/s/Mpc or smaller.

read the original abstract

We investigate how multi-band gravitational wave (GW) observations can constrain the uncertainties in the Hubble parameter ($H_0$) using primordial black holes (PBHs) as possible sources. Our framework combines scalar-induced and merger-induced GWs from PBHs, and forecasts on a combination of two future detectors Square Kilometre Array (SKA) and the Einstein Telescope (ET), enabling a multi-band analysis. We perform a statistical forecast of the PBH parameters, $M_{\rm PBH}$ and $f_{\rm PBH}$, using signal-to-noise ratio (SNR) estimates and Fisher matrix analysis. Imposing $\mathrm{SNR} \geq 1$, we identify the accessible PBH parameter space and propagate these uncertainties to estimate the corresponding uncertainties in $H_0$. For $\delta \theta_i/\theta_i \leq 0.1$, with $\theta_i \equiv M_{\rm PBH}(f_{\rm PBH})$, we find $\delta H_0 \lesssim 2~{\rm km\,s^{-1}\,Mpc^{-1}}$ in a conservative approach, improving to $\delta H_0 \lesssim \mathcal{O}(0.1)~{\rm km\,s^{-1}\,Mpc^{-1}}$ for $\delta \theta_i/\theta_i \leq 0.01$ for an optimistic approach of precision measurement. The results are further found to be largely insensitive to the fiducial choice of the $H_0$, with only moderate dependence on the PBH collapse efficiency. These findings demonstrate that multi-band GW observations provide an independent and complementary approach to constraining the uncertainties in $H_0$, with the potential to provide a novel, cosmic distance ladder-independent measure of the Hubble parameter.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a multi-band GW forecast for H0 from PBH signals with SKA+ET, but the headline delta-H0 numbers rest on externally imposed relative precisions for the PBH parameters rather than the direct Fisher outputs.

read the letter

The core result is a forecast that multi-band observations of scalar-induced and merger GWs from PBHs could constrain H0 to roughly 2 km/s/Mpc or better if the PBH mass and fraction can be measured to 10% or 1% relative accuracy. They combine SKA and ET data, apply SNR cuts to define the accessible (M_PBH, f_PBH) region, run Fisher matrices on those parameters, and then propagate to H0. The work is incremental but cleanly executed on the forecasting side. They show the results are fairly insensitive to the fiducial H0 value and only moderately dependent on collapse efficiency, which is useful to see. The SNR >=1 threshold and the two-detector setup are standard and applied consistently. The paper does a decent job mapping the detectable PBH parameter space for this channel. The main limitation is in the final step from PBH-parameter uncertainties to delta-H0. The quoted bounds appear to come from separately assuming delta-theta/theta <=0.1 or 0.01 and then translating, rather than using the actual Fisher covariance matrix to compute the propagated error on H0. The abstract does not show the explicit functional dependence H0(M_PBH, f_PBH) or the error-propagation formula, so it is not obvious whether the numbers are the direct statistical forecast or conditional on those extra precision targets. This makes the central claim harder to evaluate without the full derivation. The analysis also assumes PBHs exist at levels that produce SNR >=1 signals, which remains an open question. This paper is mainly for people already working on GW forecasts and the Hubble tension who want to see one more independent channel quantified. A reader who follows similar Fisher studies will find the setup familiar and the numbers easy to compare. It is solid enough on methods and topic to deserve a serious referee, though the propagation details will need clarification in review.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a forecast for constraining the Hubble constant H0 using multi-band gravitational wave observations of primordial black holes (PBHs). It combines scalar-induced gravitational waves and merger-induced signals from PBHs, using the Square Kilometre Array (SKA) and Einstein Telescope (ET) for multi-band analysis. The authors perform signal-to-noise ratio (SNR) estimates to identify the accessible parameter space for PBH mass M_PBH and abundance f_PBH, followed by Fisher matrix analysis to forecast uncertainties in these parameters. These uncertainties are then propagated to estimate the uncertainty in H0, yielding δH0 ≲ 2 km s^{-1} Mpc^{-1} for 10% relative precision on PBH parameters and δH0 ≲ O(0.1) km s^{-1} Mpc^{-1} for 1% precision. The results are reported to be insensitive to the fiducial H0 value and moderately dependent on PBH collapse efficiency.

Significance. If the error propagation from PBH parameters to H0 is rigorously demonstrated and the assumptions about PBH abundance and signal detectability hold, this work could provide a novel, independent probe of H0 that is free from the cosmic distance ladder. The multi-band approach combining SKA and ET is a positive aspect, as is the use of standard Fisher forecasting methods. The finding of insensitivity to fiducial H0 adds robustness. However, the significance is tempered by the reliance on PBHs existing in sufficient numbers to produce SNR ≥1 signals, which remains an open question in the field.

major comments (2)

[§4 (Results and H0 propagation)] §4 (Results and H0 propagation): The central results for δH0 are obtained by imposing δθi/θi ≤ 0.1 and ≤ 0.01 on θi ≡ M_PBH(f_PBH) rather than using the uncertainties directly computed from the Fisher matrix in the SNR ≥1 region. The manuscript does not provide the explicit functional form of H0(M_PBH, f_PBH) or the error propagation formula (e.g., via partial derivatives ∂H0/∂M_PBH and ∂H0/∂f_PBH). This makes the quoted δH0 bounds conditional on external precision assumptions instead of being the direct output of the statistical forecast.
[§3.2 (Fisher matrix analysis)] §3.2 (Fisher matrix analysis): While the Fisher matrix is applied to forecast σ(M_PBH) and σ(f_PBH), there is no demonstration that these computed uncertainties satisfy or relate to the imposed relative error thresholds used for the δH0 estimates. The paper should show the actual Fisher-derived relative errors for the accessible parameter space and then propagate those specifically to H0.

minor comments (2)

[Abstract] Abstract: The notation 'O(0.1)' should be typeset as mathcal{O}(0.1) for consistency.
[Methods] Ensure that all references to the PBH collapse efficiency parameter are clearly defined in the methods section with its range of values explored.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive comments that help improve the clarity and rigor of our analysis. We address each major comment point by point below, agreeing that revisions are needed to better connect the Fisher matrix results directly to the H0 uncertainty estimates.

read point-by-point responses

Referee: [§4 (Results and H0 propagation)] The central results for δH0 are obtained by imposing δθi/θi ≤ 0.1 and ≤ 0.01 on θi ≡ M_PBH(f_PBH) rather than using the uncertainties directly computed from the Fisher matrix in the SNR ≥1 region. The manuscript does not provide the explicit functional form of H0(M_PBH, f_PBH) or the error propagation formula (e.g., via partial derivatives ∂H0/∂M_PBH and ∂H0/∂f_PBH). This makes the quoted δH0 bounds conditional on external precision assumptions instead of being the direct output of the statistical forecast.

Authors: We thank the referee for this observation. The current presentation uses assumed relative precisions (10% and 1%) on M_PBH and f_PBH to delineate conservative and optimistic cases for δH0, as these thresholds illustrate the potential reach of multi-band observations within the SNR ≥1 region identified by our Fisher analysis. However, we agree that the manuscript would be strengthened by explicitly deriving the functional dependence H0(M_PBH, f_PBH) from the underlying GW signal models (e.g., via frequency or amplitude scalings in the scalar-induced and merger-induced spectra) and by providing the error propagation formula using partial derivatives. In the revised manuscript, we will add this derivation in §4, compute the actual Fisher-derived relative errors across the accessible parameter space, and propagate those specific uncertainties to obtain δH0. This will make the quoted bounds a direct output of the forecast rather than conditional on external assumptions. revision: yes
Referee: [§3.2 (Fisher matrix analysis)] While the Fisher matrix is applied to forecast σ(M_PBH) and σ(f_PBH), there is no demonstration that these computed uncertainties satisfy or relate to the imposed relative error thresholds used for the δH0 estimates. The paper should show the actual Fisher-derived relative errors for the accessible parameter space and then propagate those specifically to H0.

Authors: We acknowledge that the connection between the Fisher-computed uncertainties and the imposed thresholds is not explicitly demonstrated in the current version. The Fisher matrix in §3.2 yields σ(M_PBH) and σ(f_PBH) for the SNR ≥1 region, but these are not directly mapped to the 0.1 and 0.01 relative error levels used for δH0. In the revision, we will include a new figure or table in §3.2 (or §4) displaying the Fisher-derived relative errors δM_PBH/M_PBH and δf_PBH/f_PBH as a function of the PBH parameters within the detectable space. We will then use these computed values, together with the propagation formula, to derive the corresponding δH0, ensuring full consistency between the statistical forecast and the final H0 constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard Fisher forecasting with conditional bounds on assumed precisions.

full rationale

The paper's chain consists of SNR-based identification of PBH parameter space, Fisher matrix estimation of uncertainties on M_PBH and f_PBH, and propagation to H0. The quoted δH0 values are presented as conditional on externally imposed relative precisions (δθi/θi ≤ 0.1 or 0.01) rather than direct outputs of the computed Fisher covariances. No equations reduce to each other by construction, no parameters are fitted and then renamed as predictions, and no load-bearing self-citations or imported uniqueness theorems appear in the abstract or description. The analysis remains self-contained against external benchmarks using standard statistical methods without self-referential loops.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The forecast rests on standard cosmological assumptions plus PBH-specific modeling choices whose independent evidence is not supplied in the abstract.

free parameters (1)

PBH collapse efficiency
Mentioned as having moderate dependence; value chosen to model scalar-induced GW amplitude.

axioms (2)

domain assumption PBHs produce both scalar-induced and merger-induced GW backgrounds that can be jointly observed in radio and ground-based bands.
Invoked to justify the multi-band framework and SNR ≥ 1 selection.
standard math Fisher matrix provides accurate uncertainty propagation from PBH parameters to H0.
Standard assumption in forecast papers; no validation shown.

pith-pipeline@v0.9.0 · 5638 in / 1420 out tokens · 22987 ms · 2026-05-08T09:50:03.796503+00:00 · methodology

discussion (0)

Reference graph

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