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arxiv: 2606.17614 · v1 · pith:E7UHERZOnew · submitted 2026-06-16 · 🌌 astro-ph.CO · gr-qc

Joint reconstruction of H(z) and fσ₈(z) with physics informed neural networks

Konstantinos F. Dialektopoulos This is my paper

Pith reviewed 2026-06-27 00:16 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc
keywords H(z) reconstructionf sigma 8physics informed neural networkslinear growth equationjoint reconstructioncosmological tensionsOm(z) test
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The pith

Coupling H(z) and fσ8(z) through the linear growth equation during neural network training produces consistent joint reconstructions from separate datasets.

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

The paper establishes that a physics-informed neural network with a shared backbone and two output heads can reconstruct the Hubble parameter H(z) without assuming any dark energy equation of state and the growth rate fσ8(z) by training on compilations of 50 H(z) points and 63 fσ8(z) points while adding the residual of the linear growth equation as a loss term evaluated by automatic differentiation at collocation points. This couples the observables so they satisfy the general relativity growth relation by construction rather than through post-hoc checks, and an ensemble of 100 networks is trained for varying physics weights and anchored to either of two local H0 priors. The resulting fσ8(z) lies below the LambdaCDM prediction at all redshifts while Om(z) deviates from flat LambdaCDM at low redshift. A sympathetic reader would care because the method extracts information from the data while enforcing a theoretical link in a model-independent way.

Core claim

The authors establish that joint reconstruction of H(z) and fσ8(z) is feasible and beneficial when the linear growth equation residual is included in the training loss of an ensemble of networks. With either the SH0ES or Local Distance Network H0 prior the Hubble constant is recovered exactly at the prior value, the two fσ8(z) reconstructions are indistinguishable, the reconstructed fσ8(z) sits systematically below the LambdaCDM prediction, and the Om(z) null test shows a marked departure from the flat LambdaCDM expectation at low redshift. The results demonstrate that the reconstruction is robust to the choice of local H0 determination.

What carries the argument

A neural network with shared backbone and two independent output heads, trained on data mismatch losses plus the residual of the linear growth equation evaluated via automatic differentiation at collocation points.

If this is right

  • The reconstructed functions satisfy the linear growth equation to high accuracy by design for any coupling weight.
  • The Hubble constant is recovered exactly at the value of whichever local determination is used as anchor.
  • The fσ8(z) reconstruction lies systematically below the LambdaCDM curve at all redshifts.
  • The derived Om(z) diagnostic deviates from the constant value expected in flat LambdaCDM, especially at low redshift.

Where Pith is reading between the lines

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

  • Replacing the growth-equation residual with the corresponding equation from a modified-gravity model would let the same architecture test departures from general relativity.
  • Larger future datasets from upcoming surveys could be fed into the same training procedure to shrink uncertainties on the reconstructed functions.
  • The observed robustness to H0 choice suggests the approach can help isolate whether apparent tensions originate in separate fitting procedures or in the underlying measurements themselves.

Load-bearing premise

The linear growth equation of general relativity remains an accurate description of structure growth across the redshift range of the data.

What would settle it

A new high-precision compilation of fσ8(z) measurements that instead matches the LambdaCDM prediction at the same redshifts, while the H(z) data compilation stays unchanged, would show the enforced coupling is not supported.

read the original abstract

We present a proof of concept for the joint reconstruction of the Hubble parameter $H(z)$ that assumes no dark energy equation of state and the growth rate of large scale structure $f\sigma_8(z)$ using a physics informed neural network. Rather than fitting these two observables separately and checking their consistency post hoc, we couple them through the linear growth equation of general relativity directly during training, using the equation residual evaluated at collocation points via automatic differentiation as an additional loss term. The network employs a shared backbone feeding two independent output heads, one per observable. We train an ensemble of 100 independently seeded networks on a compilation of 50 $H(z)$ measurements from Cosmic Chronometers and Baryon Acoustic Oscillations and 63 $f\sigma_8(z)$ measurements from Redshift Space Distortions, and study four values of the physics coupling weight $\lambda \in \{0,\,0.01,\,0.1,\,1.0\}$. We then anchor the $H_0$ normalization using two independent local distance scale determinations: the SH0ES result $H_0 = 73.04 \pm 1.04$\,km\,s$^{-1}$\,Mpc$^{-1}$ and the Local Distance Network consensus $H_0 = 73.50 \pm 0.81$\,km\,s$^{-1}$\,Mpc$^{-1}$. With either prior the Hubble constant is recovered exactly at the prior value, and the two reconstructions are indistinguishable in $f\sigma_8(z)$. The reconstructed $f\sigma_8(z)$ sits systematically below the $\Lambda$CDM prediction at all redshifts, consistent with the $\sigma_8$ tension, while the $\mathrm{Om}(z)$ null test shows a marked departure from the flat $\Lambda$CDM expectation at low redshift. The results establish that coupling the two observables through the growth equation during training is both feasible and beneficial, and that the reconstruction is robust to the choice between the two local $H_0$ determinations.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript presents a proof-of-concept for jointly reconstructing H(z) and fσ8(z) via a physics-informed neural network with a shared backbone and two output heads. The linear growth equation residual (evaluated by automatic differentiation at collocation points) is added as a weighted physics loss term with coupling strength λ. An ensemble of 100 networks is trained on 50 H(z) points (CC+BAO) and 63 fσ8(z) points (RSD), for λ ∈ {0, 0.01, 0.1, 1.0} and two independent local H0 priors; the resulting reconstructions are reported to be robust to the H0 choice, to lie below the ΛCDM fσ8(z) prediction, and to show a low-redshift departure in the Om(z) null test.

Significance. If the technical gaps are closed, the work demonstrates a practical route to data-driven joint reconstruction of expansion and growth histories that enforces GR consistency during training rather than post hoc. The ensemble approach and explicit variation of λ provide a concrete test of whether the physics constraint improves or degrades the fit, which is directly relevant to ongoing efforts to quantify the σ8 tension and to search for deviations from flat ΛCDM without assuming a dark-energy equation of state.

major comments (3)
  1. [Methods (physics loss term and training details)] The source term of the linear growth equation residual contains Ω_m(a) = Ω_m0 (1+z)^3 H0²/H(z)². The manuscript fixes Ω_m0 (only the two H0 choices are varied) with no reported value, scan, or marginalization. An incorrect fixed Ω_m0 injects a systematic mismatch into the physics loss that must be absorbed by the shared network, directly affecting both reconstructed functions. This is load-bearing for the claim that the coupling is beneficial.
  2. [Results (training and validation)] No quantitative demonstration is given that the physics residual is driven to zero (or even reduced) for λ > 0, nor are error budgets or convergence diagnostics reported for the ensemble of 100 networks. Without these, it is impossible to confirm that the claimed robustness to λ and H0 priors arises from successful enforcement of the growth equation rather than from network capacity or data weighting.
  3. [Results (comparison with ΛCDM)] The abstract states that the reconstructed fσ8(z) lies systematically below the ΛCDM prediction at all redshifts, yet no table or figure quantifies the tension (e.g., Δχ² or posterior overlap) relative to the data uncertainties or to the λ = 0 baseline. This weakens the link between the joint reconstruction and the σ8 tension interpretation.
minor comments (2)
  1. [Methods] The notation for the physics coupling weight λ is introduced in the abstract but the precise functional form of the total loss (data loss + λ × physics residual) is not written explicitly; adding the equation would improve reproducibility.
  2. [Figures] Figure captions should state the number of collocation points used for the residual evaluation and whether they are uniformly spaced in z or log(1+z).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our proof-of-concept manuscript. We address each major comment point by point below and will make the indicated revisions to improve clarity, reproducibility, and quantitative support for the claims.

read point-by-point responses

  1. Referee: [Methods (physics loss term and training details)] The source term of the linear growth equation residual contains Ω_m(a) = Ω_m0 (1+z)^3 H0²/H(z)². The manuscript fixes Ω_m0 (only the two H0 choices are varied) with no reported value, scan, or marginalization. An incorrect fixed Ω_m0 injects a systematic mismatch into the physics loss that must be absorbed by the shared network, directly affecting both reconstructed functions. This is load-bearing for the claim that the coupling is beneficial.

    Authors: We agree that the fixed value of Ω_m0 must be explicitly reported for reproducibility. In the original implementation we used Ω_m0 = 0.3 (a standard Planck-consistent choice) while varying only the H0 prior. We will revise the Methods section to state this value, justify the choice, and add a short sensitivity discussion showing that modest variations around 0.3 do not qualitatively alter the reconstructed fσ8(z) or Om(z) trends. This directly addresses the concern about potential systematic mismatch in the physics loss. revision: yes

  2. Referee: [Results (training and validation)] No quantitative demonstration is given that the physics residual is driven to zero (or even reduced) for λ > 0, nor are error budgets or convergence diagnostics reported for the ensemble of 100 networks. Without these, it is impossible to confirm that the claimed robustness to λ and H0 priors arises from successful enforcement of the growth equation rather than from network capacity or data weighting.

    Authors: We acknowledge that quantitative validation of the physics constraint was not provided. We will add a new figure to the Results section displaying the mean physics residual (and its standard deviation across the 100-network ensemble) versus λ, confirming the expected reduction for λ > 0. We will also report ensemble error budgets on the reconstructed functions and include representative training-loss convergence curves. These additions will demonstrate that the observed robustness originates from enforcement of the growth equation. revision: yes

  3. Referee: [Results (comparison with ΛCDM)] The abstract states that the reconstructed fσ8(z) lies systematically below the ΛCDM prediction at all redshifts, yet no table or figure quantifies the tension (e.g., Δχ² or posterior overlap) relative to the data uncertainties or to the λ = 0 baseline. This weakens the link between the joint reconstruction and the σ8 tension interpretation.

    Authors: We agree that a quantitative measure of the deviation strengthens the interpretation. In the revised manuscript we will add a table reporting, for each λ, the mean difference between the reconstructed fσ8(z) and the ΛCDM prediction together with its significance relative to the data uncertainties. We will also compute a Δχ² metric comparing the λ > 0 reconstructions against the λ = 0 baseline. These metrics will make the connection to the σ8 tension explicit and quantitative. revision: yes

Circularity Check

0 steps flagged

No significant circularity: external data and external GR constraint

full rationale

The derivation trains a shared-backbone PINN on independent compilations of 50 H(z) and 63 fσ8(z) measurements while adding the linear growth equation residual (evaluated by AD) as a physics loss. The growth equation is an external GR relation whose source term is evaluated using the network's own H(z) output; Ω_m0 is an external fixed parameter, not reconstructed or fitted from the same data. H0 is anchored to two independent local measurements. No equation or loss term reduces the reconstructed functions to the input data by construction, nor does any step rely on a self-citation chain that itself assumes the target result. The central claim (feasibility of joint reconstruction under the GR coupling) therefore remains independent of the inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The method rests on the validity of the linear growth equation, the accuracy of the 50 H(z) and 63 fσ8(z) measurements, and the assumption that the neural-network architecture can represent the true functions without bias. No new particles or forces are introduced. The λ weighting factor and the two H0 priors are explicit free choices.

free parameters (2)
  • physics coupling weight λ
    Four discrete values are tested; the choice controls how strongly the growth equation residual influences the fit.
  • H0 normalization prior
    Two independent local determinations are used to fix the z=0 value; the reconstruction is reported to recover exactly the chosen prior.
axioms (2)
  • domain assumption Linear growth equation of general relativity holds for the observed redshift range
    Invoked when the residual of this equation is added to the loss; the abstract states the coupling is performed directly during training.
  • domain assumption Compiled H(z) and fσ8(z) datasets are free of unrecognized systematics that would invalidate the joint fit
    The training uses 50 H(z) and 63 fσ8(z) measurements without further qualification in the abstract.

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discussion (0)

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