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In search of an interaction in the dark sector through Gaussian Process and ANN approaches

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that current Hubble and supernova data reveal dark-sector interaction whenever the dark-energy equation of state is fixed away from -1, while at -1 there is no prominent signal.

desk verdict Workmanlike ANN/GP reconstruction of the dark interaction, but the headline w_DE-dependence is built into the setup rather than discovered in the data. read the letter →

arxiv 2505.04336 v2 pith:J72YL53R submitted 2025-05-07 astro-ph.CO gr-qc

Mazaharul Abedin , Guo-Jian Wang , Yin-Zhe Ma , Supriya Pan This is my paper · ORCID
classification astro-ph.COgr-qc PACS 95.36.+x98.80.-k
keywords darkenergymatterdark-sectorinteractionGaussianprocessartificialneuralnetworkcosmicchronometersPantheon+equationofstate
topics Dark EnergyDark Matter
open problems Dark MatterDark Energy
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

This paper tries to establish that the expansion history measured by Cosmic Chronometers and Pantheon+ supernovae carries an imprint of energy exchange between dark matter and dark energy, as long as the dark-energy equation of state $w_{\rm DE}$ is not exactly $-1$. The interaction function is reconstructed without choosing a model for the coupling, using two non-parametric tools: Gaussian-process regression and, for the first time in this context, artificial neural networks. For $w_{\rm DE}=-1$ the reconstructed interaction is compatible with zero in most datasets, with only a mild GP hint in the combined sample; for $w_{\rm DE}\neq -1$, both methods find the interaction moving away from zero as $w_{\rm DE}$ deviates in either the quintessence or the phantom direction. The paper also claims that the ANN reconstructions are more stringent than the GP ones. If true, the result would connect the long-standing question of whether the dark sector is coupled to a single parameter, the equation of state of dark energy.

What carries the argument

The load-bearing object is Eq. (8), an algebraic identity obtained from the Friedmann equations and the two conservation equations for the interacting fluids, which reads $$\frac{w_{\rm DE}\$kappa^{2}$ Q}{$H_0^{3}$} = -($2E^{2}$E''+2E E'^2)(1+z)^2 - $9E^{3}$(1+w_{\rm DE}) + ($4E^{2}$E' + $6E^{2}$E'(1+w_{\rm DE}))(1+z),$$ where $E(z)=H(z)/H_0$ and primes are derivatives with respect to redshift. This identity defines the interaction function $Q$ for any chosen constant equation of state $w_{\rm DE}$ once the dimensionless Hubble rate and its derivatives are known; it is not a physical model of the coupling. The non-parametric methods supply the reconstruction: Gaussian-process regression with the squared-exponential kernel provides $E$, $E'$, $E''$ and their covariances from the data, while the neural network with a Softplus activation builds $H(z)$ from 1000 data realizations, computes its derivatives by finite differences, and propagates the covariance. Eq. (8) then converts either smooth reconstruction into the interaction function and its uncertainty band, which is the quantity compared against zero.

What would settle it

Generate a mock expansion history from a non-interacting model with $Q=0$ but with the true $w_{\rm DE}$ varying with redshift, run the same GP and ANN pipelines with a fixed $w_{\rm DE}\neq -1$, and check whether the reconstructed $\widetilde Q(z)$ departs from zero with significance comparable to the paper's figures; if it does, the reported evidence is an artifact of the fixed-$w_{\rm DE}$ assumption. Alternatively, fit the actual CC+Pantheon+ data to a non-interacting model with a free constant $w_{\rm DE}$ and verify whether that model already matches the reconstructed $E(z)$ within $1\sigma$.

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Extended reading notes

Core claim

The central claim is that the dimensionless interaction function $\widetilde Q(z)=\kappa^2 Q(z)/[(1+z)^6 H_0^3]$, obtained from Eq. (8) rather than from an assumed coupling, is statistically compatible with zero whenever $w_{\rm DE}=-1$, but is pushed away from zero once $w_{\rm DE}$ is fixed to other constant values in the range considered, roughly $-0.7$ down to $-1.3$. For quintessence-like values $w_{\rm DE}>-1$ the present-day interaction is reconstructed as positive, meaning energy flows from dark matter to dark energy; for strongly phantom values $w_{\rm DE}<-1$ the reconstructed flow can become negative. The two non-parametric approaches agree on the emergence of interaction for $w_{\rm DE}\neq -1$, but differ at $w_{\rm DE}=-1$: the neural network finds no interaction in any dataset, while the Gaussian process finds a mild signal (below $2\sigma$ at present, around $2\sigma$ at intermediate redshifts) for the combined CC+Pantheon+ sample. A secondary claim is that in this reconstruction problem the artificial neural network produces tighter constraints than the Gaussian process.

Load-bearing premise

The whole argument rests on treating every nonzero reconstructed interaction term $\widetilde Q(z)$ as evidence of physical energy exchange, but the reconstruction formula also produces a nonzero $\widetilde Q(z)$ whenever the true dark-energy equation of state differs from the constant value assumed in the analysis.

Editorial extensions

If this is right

  • If $w_{\rm DE}=-1$, no robust dark interaction is required by the current datasets; the only hint is the GP reconstruction from CC+Pantheon+, and it stays below about $2\sigma$.
  • If $w_{\rm DE}$ deviates from $-1$ by as little as $0.1$, both reconstruction methods report a nonzero interaction whose significance grows with the deviation, with the energy-flow direction depending on whether the deviation is quintessence-like or phantom-like.
  • The effective equations of state $w^{\rm eff}_{\rm DM}$ and $w^{\rm eff}_{\rm DE}$ deviate from $0$ and $-1$ in the same redshift regions where $\widetilde Q(z)\neq 0$, so an interacting sector with constant $w_{\rm DE}$ would look like a non-interacting sector with dynamical dark equations of state.
  • Because the pipeline requires no assumed coupling model, applying it to future datasets such as DESI baryon acoustic oscillations would either strengthen or erase the reported signal without changing the method.
  • The claim that ANN gives tighter constraints than GP, if confirmed, gives future model-independent reconstruction studies a concrete reason to prefer neural-network methods for derivative-based quantities.

Reading between the lines

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

  • Eq. (8) is an identity, so a nonzero $\widetilde Q(z)$ obtained with a fixed $w_{\rm DE}\neq -1$ can simply mean that the true expansion history is inconsistent with a constant equation of state equal to that chosen value; a natural check is to fit the same data to a non-interacting model with free constant $w_{\rm DE}$ and test whether the reconstructed $E(z)$ is already matched.
  • Both methods require up to third derivatives of noisy data, so the reported significance levels are likely sensitive to the GP kernel, the ANN architecture, and the number of realizations; a mock test with a known $Q=0$ expansion history would settle whether the pipeline is biased toward finding interaction whenever $w_{\rm DE}\neq -1$.
  • The paper's qualitative agreement between two structurally different reconstruction tools is its strongest empirical hint; if it survives mock calibration, it would make model-independent dark-interaction searches a standard probe of the dark sector.
  • Recent DESI evidence for a dynamical dark-energy equation of state could be re-read through this paper's logic: an interacting dark sector with constant $w_{\rm DE}$ produces effective equations of state that vary with redshift, so the dynamical-$w_{\rm DE}$ signal and a dark interaction may be alternative descriptions of the same data.
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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

3 major / 5 minor

Summary. The paper reconstructs the dark-sector interaction function Q(z) from background data (CC, Pantheon+, and their combination) using two non-parametric methods, Gaussian Process (GP) and Artificial Neural Networks (ANN). The interaction function is derived from the Friedmann and conservation equations for a constant dark-energy equation of state w_DE, and the dependence of the reconstructed Q on w_DE is explored. The authors report that for w_DE = -1 the interaction is not prominent, whereas for w_DE deviating from -1 an interaction emerges, and that ANN provides tighter constraints than GP.

Significance. If the central claim were established, the paper would be a useful addition to the interacting-dark-energy literature, extending the GP-based reconstruction of Yang et al. (2015) and related works to ANN, and it carefully checks sensitivity to the H0 prior. The use of a consistent pipeline on two independent non-parametric tools, with public codes and explicit treatment of data covariance, is a strength. However, the evidence statement about w_DE != -1 is not currently supported because the reconstruction is not validated against mock non-interacting universes and no non-interacting baseline model is fitted. The paper's main conclusion therefore rests on an unproven identification of algebraic reconstruction with empirical detection.

major comments (3)
  1. [Eq. (8); Secs. V A2 and V B2] The reconstructed interaction function is defined, not measured, by Eq. (8) once w_DE is fixed: substituting any smooth E(z) into Eq. (8) yields a nonzero Q when w_DE != -1, even if the true expansion history is a non-interacting wCDM model with a different constant w_DE. The paper never fits the non-interacting constant-w_DE model (E(z) = [Omega_m(1+z)^3 + (1-Omega_m)(1+z)^{3(1+w_DE)}]^{1/2}) to the same datasets, nor reports a likelihood comparison between the interacting and non-interacting cases for each w_DE. Consequently, the abstract's statement that for w_DE != -1 'evidence of interaction is found' confuses model misspecification with a physical energy transfer; the w_DE-dependence of Q is largely algebraic. This is the load-bearing claim of the paper and needs to be re-framed or supported by a baseline test.
  2. [Sec. III B, Eq. (30); Secs. V A2 and V B2] The ANN pipeline obtains H'(z) and higher derivatives by central finite differences of 1000 reconstructed H(z) realizations, and the GP pipeline uses analytic derivative kernels; neither is validated on mock data from a known non-interacting cosmology. Since the significance claims (e.g., Figs. 4 and 9) rest on the ratio of reconstructed Q to its uncertainty, a mock-based calibration of the bias and false-detection rate is required to interpret the 68% CL crossings as evidence of interaction. At minimum, the authors should demonstrate on simulated CC-like and Pantheon+-like data from a non-interacting wCDM model with w_DE = -0.7, -1, and -1.3 that the pipeline recovers Q = 0 within the quoted uncertainties.
  3. [Secs. V A1, V B1, and VI] The two methods disagree for the same w_DE = -1 case: GP with CC+Pantheon+ shows a mild interaction (less than 2 sigma at z = 0 and about 2 sigma in the intermediate regime, Fig. 2), while ANN with the same data shows no interaction (Fig. 7). The paper acknowledges this in Sec. VI, but the headline claim for w_DE != -1 assumes that both methods provide reliable reconstructions. The discrepancy at w_DE = -1 suggests that method-specific systematics, most likely in derivative reconstruction, dominate the difference, and therefore the joint claim that 'an emergence of dark interaction is observed from both GP and ANN reconstructions' for w_DE != -1 is not yet robust.
minor comments (5)
  1. [Abstract] The phrase 'In particularly' should be 'In particular'.
  2. [Sec. V A1] There is a typo: 'the possibility of sign changeable interaction' appears as 'he possibility of sign changeable interaction'.
  3. [Sec. V A1 and figure captions] The units 'km s^{-1} Mpc^{-1}c' appear with a stray 'c' in the text near the H0 = 73.04 value, and also in the lower-panel description of Fig. 2.
  4. [Sec. III B] The ANN architecture description gives the number of nodes (1024) and optimizer, but it omits the number of hidden layers and the learning rate; adding these details would improve reproducibility.
  5. [Figs. 4 and 9] The figures show deviation of Q from zero normalized by its uncertainty, but the shading or band conventions are not explicitly described in the captions, making it ambiguous which confidence levels correspond to the plotted quantities.

Circularity Check

1 steps flagged · score 6.0 of 10

For w_DE≠−1, nonzero Q is partly forced by Eq. (8)'s definition; the w_DE=−1 analysis remains data-driven.

  1. self definitional [Eq. (8), Section II; interpreted in Sections V A2/V B2 and Summary Section VI.]
    "wDE κ2Q H3 0 =− [2E2E′′ + 2EE′2](1 +z)2− 9E3(1 +wDE) + [4E2E′ + 6E2E′(1 +wDE)](1 +z), (8) ... if wDE̸=−1, then both GP and ANN predict an evidence of interaction is found when wDE deviates from −1; in fact, this evidence is pronounced much for ANN."

    Eq. (8) algebraically determines Q once E, E′, E″ and w_DE are specified. For any reconstructed E there is a unique w_DE = w₀ that makes Q=0; for ΛCDM-like data this is w₀=−1. The paper fixes w_DE to other values, so the resulting Q≠0 is a mathematical consequence of the input and the definition, not a measurement of energy transfer. The w_DE-dependent terms (1+w_DE) appear explicitly in Eq. (8), so moving w_DE away from −1 shifts Q by construction. No non-interacting constant-w_DE model is fitted to the same data, no likelihood ratio is computed, and no mock test calibrates the false-detection rate; hence the conclusion 'evidence of interaction is found when w_DE deviates from −1' reduces to the input choice.

full rationale

Most of the pipeline is self-contained: the CC and Pantheon+ data are public, the GP and ANN implementations are standard public packages, and the w_DE = −1 reconstruction of Q is a genuine data-driven exercise, with GP finding a mild CC+Pantheon+ signal and ANN finding none. No load-bearing self-citation or imported uniqueness theorem was found; the self-citations for H0 priors and ANN methodology are standard and not decisive for the central claim. The circularity enters specifically when w_DE is treated as a freely hand-specified input and the resulting Q from Eq. (8) is then reported as empirical 'evidence of interaction.' Because Eq. (8) defines Q for every E(z) and w_DE, and because a nonzero Q for w_DE ≠ −1 is the algebraic consequence of choosing an input away from the value that would give Q=0, the headline claim for w_DE ≠ −1 is partly built into the formula rather than discovered. The paper does not compare to the non-interacting constant-w_DE model with the same w_DE, nor run mock validations, so the statistical force of that particular claim is missing. This is partial circularity confined to the w_DE ≠ −1 interpretation; the w_DE = −1 results and the GP-versus-ANN constraint comparison retain independent content.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The reconstruction rests on flat-GR background equations, a hand-fixed constant DE equation of state, external H0 priors, and smoothness assumptions for the machine-learning reconstructions. No new particles, forces, or physical entities are introduced.

free parameters (4)
  • w_DE scanned constant values = -0.7, -0.8, -0.9, -1, -1.1, -1.2, -1.3
    The central result is conditional on hand-chosen values of the dark energy equation of state; w_DE is not fitted or marginalized.
  • H0 priors and CC-estimated H0 = 67.36 +/- 0.54 and 73.04 +/- 1.04 km/s/Mpc from Planck and SH0ES; CC-only H0 estimated from the reconstructed H(z) but…
    E(z) = H(z)/H0 enters Eq. (8). Two external priors are tested for Pantheon+ and combined data, while the CC-alone pipeline derives H0 from the same data.
  • GP kernel hyperparameters = Optimized via marginal likelihood, not reported
    The GP derivative reconstruction depends on the trained hyperparameters of the squared-exponential kernel, which are fitted to the data.
  • ANN architecture and training settings = 1024 hidden nodes, 30,000 iterations, L2 = 0.005 or 0.0005, Softplus, beta = 1
    The ANN reconstruction, including its derivative approximation, depends on these hand-chosen training and architecture settings.
assumptions (3)
  • domain assumption Spatially flat FLRW background with GR as the gravitational theory
    The background equations in Section II assume flatness and GR; a different geometry or gravity theory would change the relation between Q and H(z).
  • ad hoc to paper Dark energy has a constant equation of state w_DE over the full redshift range
    The derivation of Eq. (8) assumes constant w_DE. The paper discusses DESI hints of dynamical w_DE but does not model them, leaving the inferred Q dependent on an assumption that may be false.
  • domain assumption GP and ANN reconstructions of H(z), D(z) and their derivatives are unbiased enough for the derived Q
    No mock validation is provided. The ANN uses central finite differences (Eq. 30), which amplify noise, and GP derivative errors inherit the kernel smoothness assumption.

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Pith. "Pith review of In search of an interaction in the dark sector through Gaussian Process and ANN approaches." pith.science (2026). https://pith.science/paper/J72YL53R

@misc{pith2026250504336,
  author       = {Pith},
  title        = {Pith review of: In search of an interaction in the dark sector through Gaussian Process and ANN approaches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J72YL53R}},
  note         = {Machine review of arXiv:2505.04336}
}
abstract

Whether the current observational data indicate any evidence of interaction between the dark sector is a matter of supreme interest at the present moment. This article searched for an interaction in the dark sector between a pressure-less dark matter and a dark energy fluid with constant equation of state, $w_{\rm DE}$. For this purpose, two non-parametric approaches, namely, the Gaussian Process (GP) and the Artificial Neural Networks (ANN) have been employed and using the Hubble data from Cosmic Chronometers (CC), Pantheon+ from Supernovae Type Ia and their combination we have reconstructed the interaction function. We find that for $w_{\rm DE} =-1$, the interaction in the dark sector is not prominent while for $w_{\rm DE} \neq -1$, evidence of interaction is found depending on the value of $w_{\rm DE}$. In particularly, we find that if we start deviating from $w_{\rm DE} = -1$ either in the quintessence ($w_{\rm DE} > -1$) or phantom ($w_{\rm DE} < -1$) direction, an emergence of dark interaction is observed from both GP and ANN reconstructions. We further note that ANN which is applied for the first time in this context seems to play a very efficient role compared to GP.

Figures

Figures reproduced from arXiv: 2505.04336 by the authors.

Figure 1
Figure 1. The structure of an ANN model with given redshift [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Reconstructed interaction function Qe(z) using the Gaussian process considering 34 CC H(z), Pantheon+, and CC+Pantheon+ data for wDE = −1. The Hubble constant H0 = 67.36 ± 0.54 km s−1 Mpc−1 at 68% CL [103] and H0 = 73.04 ± 1.04 km s−1 Mpc−1 at 68% CL [104] are used in the reconstruction of D(z). The horizontal dashed line (red) in each plot corresponds to Qe(z) = 0, i.e. no-interaction between the dark sectors and t… view at source ↗
Figure 3
Figure 3. Reconstructed effective EoS parameters, w eff DM and w eff DE, using the Gaussian process considering 34 CC H(z), Pantheon+, and CC+Pantheon+ data for wDE = −1. The Hubble constant H0 = 67.36 ± 0.54 km s−1 Mpc−1 at 68% CL [103] and H0 = 73.04±1.04 km s−1 Mpc−1 at 68% CL [104] are used in the reconstruction of D(z). The horizontal dashed line (red) in the plots for w eff DM (w eff DE) corresponds to w eff DM = 0 (w e… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The deviation of Qe from zero, where ∆Qe = (Qe(z)− 0)/σQe(z) is shown for the GP. The figure corresponds to the reconstructions using CC+Pantheon+ and for H0 = 73.04 ± 1.04 km s−1 Mpc−1 [104]. Overall, we notice a mild evidence of interaction to￾gether with its sign ch…
Figure 6
Figure 6. Figure 6: The deviation of w eff DM from 0 where ∆w eff DM = (w eff DM(z) − 0)/σweff DM(z) is shown for CC+Pantheon+ with H0 = 73.04 ± 1.04 km s−1 Mpc−1 [104] considering differ￾ent values of wDE and using GP. w eff DM in Eqs. (10a) or (11a)). Taking CC, Pantheon+ and CC+Pantheo…
Figure 7
Figure 7. Figure 7: Reconstructed interaction function Qe(z) using the ANN approach considering 34 CC H(z), Pantheon+, and CC+Pantheon+ data considering wDE = −1. The Hubble constant H0 = 67.36 ± 0.54 km s−1 Mpc−1 at 68% CL [103] and H0 = 73.04±1.04 km s−1 Mpc−1 at 68% CL [104] are used i…
Figure 8
Figure 8. Figure 8: Reconstructed effective EoS parameters, w eff DM and w eff DE, using ANN considering 34 CC H(z), Pantheon+, and CC+Pantheon+ data. The Hubble constant H0 = 67.36 ± 0.54 km s−1 Mpc−1 at 68% CL [103] and H0 = 73.04 ± 1.04 km s−1 Mpc−1 at 68% CL [104] are used in the reco…
Figure 9
Figure 9. Figure 9: The deviation of Qe from zero, where ∆Qe = (Qe(z)− 0)/σQe(z) is shown for ANN. The figure corresponds to the reconstructions using CC+Pantheon+ and for H0 = 73.04 ± 1.04 km s−1 Mpc−1 [104]. to what we have noticed in GP reconstructions (see [PITH_FULL_IMAGE:figures/fu…

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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    Matérn: In this case the form of the kernel takes Kν=p+ 1 2 (z, ˜z) =σ2 f exp −√2p + 1 l |z− ˜z| p! (2p)! × pX i=0 (p +i)! i!(p−i)! 2√2p + 1 l |z− ˜z| p−i . (21) For each value ofp, there is a distinct covariance function of Matérn, which are Matérn 9/2 (for p = 4), Matérn 7/2 (p = 3), Matérn 5/2 (p = 2), and Matérn 3/2 (p = 1). Note that the Matérn covar...

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    This (1 +z)6 has been used for the scaling purpose and there is no physics as- sociated with it. Now, using the expression ofeQ(z) in terms ofE and its derivatives, the effective EoS parameters can also be expressed as weff DM =−(2EE′′ + 2E′2)(1 +z)2− 9E2(1 +wDE) 3 3(1 +wDE)E2− 2(1 +z)EE′ + (4E′ + 6E′(1 +wDE))(1 +z) 3 3(1 +wDE)E− 2(1 +z)E′ , (10a) weff DE...

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    (23) In the descriptions of the kernels presented above, σf, α and l are defined as the hyperparameters

    Rational Quadratic: The kernel assumes the fol- lowing form K(z, ˜z) =σ2 f 1 + (z− ˜z)2 2αl2 −α . (23) In the descriptions of the kernels presented above, σf, α and l are defined as the hyperparameters. The posterior probability distribution function, denoted as Eq. (17), is determined by the Bayes theorem, which states that it is dependent on both the li...

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    Now fol- lowing the Gaussian distributionN (H(z),σH(z)), we create 1000 realizations of a data-like sample of n H(z) measurements

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    We recreate 1000 re- constructed H(z) by repeating this process for ev- ery sample using the trained ANN model

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    Pantheon+ sample of Supernovae Type Ia: Given that in Pantheon+ sample of Supernovae Type Ia (SNIa) [98], there are total 1701 light curves in the given range of0.01≤ z≤ 2.26 with SH0ES included. In this paper, we use 1590 SNIa data points with some SH0ES data in this redshift...

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    For wDE =−1 We first start with the reconstructions ofeQ(z) using 34 CC data alone. The initial step is to reconstruct the Hubble function H(z) and its first and second deriva- tives, namely H′(z) and H′′(z) (here prime stands for the derivative ofH(z) with respect to the reds...

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    Fund for Improvement of S&T Infrastructure (FIST)

    at certain redshift regimes, while for ANN we do not find any evidence of non-nullweff DM. For other constant values ofwDE, we notice that whenwDE deviates from −1 either in the quintessence or phantom regime, the ef- fective EoS parameter for DM starts deviating from zero. Th...

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Reviewed August 15, 2026 · model on record in the stance chip above.