Joint reconstruction of $H(z)$ and $f\sigma_8(z)$ with physics informed neural networks

This paper demonstrates a proof of concept for jointly reconstructing the Hubble parameter $H(z)$ and growth rate $f\sigma_8(z)$ using physics-informed neural networks coupled via the linear growth equation, revealing a systematic tension with $\Lambda$CDM predictions while confirming the method's robustness to different local $H_0$ priors.

The Gist

Imagine the universe as a giant, expanding balloon. For decades, scientists have been trying to measure two specific things about this balloon:

1. How fast it is expanding right now and at different times in the past (called $H(z)$).
2. How fast the "clumps" of matter (like galaxies) are growing together due to gravity (called $f\sigma_8(z)$).

For a long time, scientists measured these two things separately, like two different teams working in different rooms. They would measure the expansion, measure the clumping, and then meet up afterward to see if their results made sense together.

This paper introduces a new way of doing things using a type of artificial intelligence called a Physics-Informed Neural Network (PINN). Think of this AI not just as a pattern-matching machine, but as a student who is being taught a strict rulebook (the laws of physics) while they study.

Here is a simple breakdown of what the authors did and found:

1. The Problem: Two Teams, One Universe

In the past, the "Expansion Team" and the "Clumping Team" worked independently.

• The Expansion Team looked at how fast galaxies are moving away.
• The Clumping Team looked at how gravity pulls matter together.
• The Issue: According to Einstein's theory of gravity (General Relativity), these two things are mathematically linked. If you know how fast the universe is expanding, you should be able to predict how fast matter is clumping. If you measure them separately, you might miss a subtle clue that something is wrong with our current understanding of the universe.

2. The Solution: The "Tethered" AI

The authors built a special AI with two heads (one for expansion, one for clumping) but a single brain (a shared backbone).

• The Tether: They tied the two heads together with a "physics rope." This rope is the mathematical equation that links expansion and clumping.
• How it works: As the AI tries to learn from real data, it gets punished (via a "loss function") if the two heads give answers that break the physics rope. It's like training a dog to fetch a ball; if the dog drops the ball before reaching the owner, it gets a gentle correction. The AI learns to fit the data while obeying the laws of physics simultaneously.

3. The Data: A Mix of Measurements

The AI was fed two types of data:

• 50 measurements of how fast the universe is expanding (from "Cosmic Chronometers" and sound waves from the early universe).
• 63 measurements of how fast matter is clumping (from how galaxies move in space).

4. The Big Discovery: The "Hubble Tension" and "Sigma-8 Tension"

There are two famous mysteries in cosmology right now:

• The Hubble Tension: Local measurements say the universe is expanding faster (about 73 km/s) than the early universe models predict (about 67 km/s).
• The Sigma-8 Tension: Measurements of galaxy clumping suggest matter is clumping less than the standard model predicts.

What the AI found:

• Anchoring the Speed: The authors forced the AI to accept the "local" speed of expansion (around 73 km/s) as a starting point.
• The Result: Even with this faster expansion speed, the AI's prediction for how matter clumps stayed consistently lower than what the standard model ($\Lambda$CDM) expects.
• The Analogy: Imagine you are driving a car. The speedometer says you are going 70 mph (the local measurement). The standard map says you should be seeing certain scenery (standard model). But when you look out the window, the scenery is actually different (less clumping). The AI confirmed that even if you trust the speedometer, the scenery still doesn't match the map.

5. Why This Matters

• It Works: The authors proved that tying the two measurements together with physics equations during the learning process is possible and helpful. It creates a smoother, more consistent picture of the universe.
• Robustness: They tried two different "local speed" numbers (73.04 and 73.50). The AI gave almost the exact same answer for how matter clumps in both cases. This suggests the result isn't just a fluke of one specific number.
• The "Null Test": They ran a diagnostic test (called $Om(z)$) to see if the universe behaves like a flat, standard model. The test showed a clear "departure" from the standard model, especially at lower redshifts (closer to us), reinforcing the idea that our current model might be missing something.

What They Did NOT Do

• They did not claim to have solved the mystery of why the universe is behaving this way (e.g., they didn't prove a new type of dark energy exists).
• They did not use supernova data (a different type of measurement) in this specific study, though they mentioned it could be added later.
• They did not claim the results are perfect; they noted that some assumptions (like the amount of matter in the universe) were fixed and could introduce bias.

The Bottom Line

This paper is a "proof of concept." It shows that using AI to learn the expansion and growth of the universe simultaneously, while forcing it to obey the laws of gravity, is a powerful tool. It confirms that the tension between how fast the universe is expanding and how matter is clumping is real and persistent, even when we use the most advanced, physics-aware AI tools available.

Technical Summary

Technical Summary: Joint Reconstruction of $H(z)$ and $f\sigma_8(z)$ with Physics-Informed Neural Networks

Problem Statement
Current cosmological observations present two persistent tensions: the $H_0$ tension (discrepancy between local distance scale measurements $\approx 73$ km s$^{-1}$ Mpc$^{-1}$ and CMB-derived values $\approx 67$ km s$^{-1}$ Mpc$^{-1}$) and the $\sigma_8$ (or $S_8$) tension (galaxy weak lensing surveys indicating less matter fluctuation growth than $\Lambda$CDM predicts). While previous works have reconstructed the Hubble parameter $H(z)$ and the growth rate $f\sigma_8(z)$ independently using Gaussian processes or standard Artificial Neural Networks (ANNs), these methods treat the two observables as statistically independent. This approach discards the known physical coupling between expansion and growth dictated by General Relativity (GR). Specifically, the linear growth equation links $H(z)$ and $f\sigma_8(z)$; ignoring this link during reconstruction prevents the data from constraining the growth history through the expansion history and vice versa.

Methodology
The authors propose a proof-of-concept framework using Physics-Informed Neural Networks (PINNs) to jointly reconstruct $H(z)$ and $f\sigma_8(z)$ without assuming a specific dark energy equation of state.

• Architecture: The model employs a dual-head network with a shared backbone. The input is redshift $z$ (rescaled to $[-1, 1]$), processed through three hidden layers of width 512 using ELU activation functions (chosen for infinite differentiability). The network branches into two independent output heads: one for $H(z)$ and one for $f\sigma_8(z)$. Output activations enforce physical bounds (positivity for $H(z)$ via softplus; a range of $(0.1, 0.9)$ for $f\sigma_8(z)$ via sigmoid).
• Physics Loss: The core innovation is the inclusion of a physics loss term, $\mathcal{L}_{physics}$, derived from the linear growth ODE of GR:
  $$\frac{df}{d \ln a} + f^2 + \left[ 2 + \frac{1}{2} \frac{d \ln H^2}{d \ln a} \right] f = \frac{3}{2} \Omega_m(a)$$
  where $f = f\sigma_8 / \sigma_{8,0}$ and $\Omega_m(a) = \Omega_{m,0}(1+z)^3 H_0^2 / H^2(z)$. The derivatives required for this equation are computed exactly via automatic differentiation through the network's computation graph, avoiding numerical discretization errors.
• Total Loss: The training objective minimizes a weighted sum: $\mathcal{L}_{total} = \mathcal{L}_{data,H} + \mathcal{L}_{data,f\sigma_8} + \lambda \mathcal{L}_{physics}$. The scalar $\lambda$ controls the trade-off between fitting the data and satisfying the physical equation.
• Data and Training: The network is trained on 50 $H(z)$ measurements (32 Cosmic Chronometers, 18 BAO) and 63 $f\sigma_8(z)$ measurements (Redshift Space Distortions). An ensemble of 100 independently seeded networks is trained to quantify uncertainty. The study tests four values of $\lambda \in \{0, 0.01, 0.1, 1.0\}$ and anchors the $H_0$ normalization using two local priors: SH0ES ($73.04 \pm 1.04$) and Local Distance Network ($73.50 \pm 0.81$).

Key Contributions

1. First Joint PINN Reconstruction: This work represents, to the authors' knowledge, the first use of a physics-informed multi-head architecture to jointly model the expansion history and linear growth of structure under a shared physical constraint (the GR growth ODE) during training, rather than as a post-hoc consistency check.
2. Automatic Differentiation for Cosmology: The method demonstrates the utility of automatic differentiation in cosmological reconstruction to compute exact derivatives of the network outputs for the ODE residual, eliminating the need for spatial meshes or finite difference approximations.
3. Coupling Efficacy: The study establishes that coupling the two observables via the growth equation is feasible and beneficial, reducing ensemble spread and ensuring dynamical consistency.

Results

• Optimal Coupling: A coupling weight of $\lambda = 0.1$ is identified as the optimal balance between data fit and physics consistency. At this value, the physics loss and data losses converge to comparable magnitudes, preventing any single term from dominating the gradient.
• $H_0$ Anchoring: Without an explicit $H_0$ prior, the network recovers a biased low $H_0$ ($46\text{--}52$ km s$^{-1}$ Mpc$^{-1}$). When anchored to either the SH0ES or Local Distance Network prior, the network recovers $H_0$ exactly at the prior value. The two different priors yield indistinguishable $f\sigma_8(z)$ reconstructions, demonstrating robustness to the specific $H_0$ tension value adopted.
• Growth Tension: The reconstructed $f\sigma_8(z)$ sits systematically below the $\Lambda$CDM+GR prediction at all redshifts, with the largest deviation around $z \sim 0.4\text{--}1.0$. This is consistent with the $\sigma_8$ tension. The coupling does not remove this signal; rather, it prevents the growth head from absorbing the tension as noise.
• Null Tests:
  • $Om(z)$: The null test shows a marked departure from the flat $\Lambda$CDM expectation (a non-flat profile) at low redshifts, including a negative dip at $z \lesssim 0.2$ (attributed to sparse Cosmic Chronometer data and reconstruction artifacts) and an excess at intermediate redshifts.
  • $Om_{f\sigma_8}(z)$: This diagnostic confirms suppressed growth relative to $\Lambda$CDM, consistent with independent ANN reconstructions found in prior literature (Ref. [9]).
• Systematics: The authors note that the fiducial values $\Omega_{m,0}=0.3$ and $\sigma_{8,0}=0.8$ are hardcoded. The systematic uncertainty from these choices dominates the statistical ensemble spread. Additionally, 18 BAO data points inherit the Planck 2018 sound horizon, introducing a model dependence in the expansion sector.

Significance and Claims
The paper claims to establish a proof of concept that coupling $H(z)$ and $f\sigma_8(z)$ through the GR growth equation during training is both feasible and beneficial. The primary significance lies in the demonstration that:

1. The reconstruction is robust to the choice between two competing local $H_0$ determinations.
2. The physics coupling acts as a physically motivated regularizer, yielding smoother reconstructions and reducing ensemble uncertainty without imposing a specific dark energy model.
3. The method successfully recovers the $\sigma_8$ tension signal (suppressed growth) even when constrained by local $H_0$ values, suggesting the tension is a robust feature of the data rather than an artifact of the reconstruction method.

The authors explicitly state modesty regarding their claims: they do not claim a statistically significant detection of new physics, as the ensemble spread is driven by initialization sensitivity rather than data noise resampling, and full posterior uncertainties are not propagated. They identify the fixed fiducial parameters ($\Omega_{m,0}, \sigma_{8,0}$) and the lack of Alcock-Paczynski corrections for individual surveys as limitations to be addressed in future work.