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Constraining the Hubble Constant with a Simulated Full Covariance Matrix Using Neural Networks

Jing Niu, Peng He, Tong-Jie Zhang

TL;DR

This work tackles the problem of constraining the present-day Hubble constant $H_0$ from Cosmic Chronometers by modeling the full covariance between $H(z)$ measurements. It introduces PD-CovNet, a neural network-based method that learns a positive-definite covariance matrix and extends the published $15\times15$ block to a full $33\times33$ matrix, with hyperparameters chosen via leave-one-z-out cross-validation and compared to a Gaussian Process baseline. Constraining $H_0$ is performed with two independent methods, EMCEE and GP, across multiple covariance configurations, and the results show no statistically meaningful shift in the central $H_0$ value, though precision is sensitive to covariance modeling and the constraint method. The study highlights the importance of accurate covariance representation in CC analyses and demonstrates that PD-CovNet provides a more reliable covariance generator than GP in this low-data setting, with potential implications for resolving or understanding the Hubble tension. Overall, the approach offers a principled way to propagate full covariance information into cosmological parameter inferences from CC data.

Abstract

The Hubble parameter, $H(z)$, plays a crucial role in understanding the expansion history of the universe and constraining the Hubble constant, $\mathrm{H}_0$. The Cosmic Chronometers (CC) method provides an independent approach to measuring $H(z)$, but existing studies either neglect off-diagonal elements in the covariance matrix or use an incomplete covariance matrix, limiting the accuracy of $\mathrm{H}_0$ constraints. To address this, we use a Positive-Definite Covariance Network (PD-CovNet) to simulate the full $33 \times 33$ covariance matrix based on a previously published $15 \times 15$ covariance matrix. Hyperparameters are chosen via leave-one-z-out validation, and performance is benchmarked against a Gaussian-process (GP) baseline. Under identical five-fold cross-validation over redshift groups, we prove that PD-CovNet is a reliable generator of the full covariance compared to the GP baseline. Using this full PD-CovNet-simulated covariance alongside three comparators with different covariance specifications, we constrain $\mathrm{H}_0$ with two independent methods (EMCEE and GP). Across all covariance specifications and both constraint methods, standardized differences and two-sided p-values show no statistically meaningful shift in the central value of the constrained $\mathrm{H}_0$. However, the precision of the constrained $\mathrm{H}_0$ depends on both covariance and method: EMCEE is uniformly more precise than GP once covariance is modeled; within a fixed method, incorporating more covariance reduces precision; and PD-CovNet hyperparameters have a modest effect on uncertainty. These results indicate the importance of accurate covariance modeling in CC-based $\mathrm{H}_0$ constraints.

Constraining the Hubble Constant with a Simulated Full Covariance Matrix Using Neural Networks

TL;DR

This work tackles the problem of constraining the present-day Hubble constant from Cosmic Chronometers by modeling the full covariance between measurements. It introduces PD-CovNet, a neural network-based method that learns a positive-definite covariance matrix and extends the published block to a full matrix, with hyperparameters chosen via leave-one-z-out cross-validation and compared to a Gaussian Process baseline. Constraining is performed with two independent methods, EMCEE and GP, across multiple covariance configurations, and the results show no statistically meaningful shift in the central value, though precision is sensitive to covariance modeling and the constraint method. The study highlights the importance of accurate covariance representation in CC analyses and demonstrates that PD-CovNet provides a more reliable covariance generator than GP in this low-data setting, with potential implications for resolving or understanding the Hubble tension. Overall, the approach offers a principled way to propagate full covariance information into cosmological parameter inferences from CC data.

Abstract

The Hubble parameter, , plays a crucial role in understanding the expansion history of the universe and constraining the Hubble constant, . The Cosmic Chronometers (CC) method provides an independent approach to measuring , but existing studies either neglect off-diagonal elements in the covariance matrix or use an incomplete covariance matrix, limiting the accuracy of constraints. To address this, we use a Positive-Definite Covariance Network (PD-CovNet) to simulate the full covariance matrix based on a previously published covariance matrix. Hyperparameters are chosen via leave-one-z-out validation, and performance is benchmarked against a Gaussian-process (GP) baseline. Under identical five-fold cross-validation over redshift groups, we prove that PD-CovNet is a reliable generator of the full covariance compared to the GP baseline. Using this full PD-CovNet-simulated covariance alongside three comparators with different covariance specifications, we constrain with two independent methods (EMCEE and GP). Across all covariance specifications and both constraint methods, standardized differences and two-sided p-values show no statistically meaningful shift in the central value of the constrained . However, the precision of the constrained depends on both covariance and method: EMCEE is uniformly more precise than GP once covariance is modeled; within a fixed method, incorporating more covariance reduces precision; and PD-CovNet hyperparameters have a modest effect on uncertainty. These results indicate the importance of accurate covariance modeling in CC-based constraints.

Paper Structure

This paper contains 14 sections, 48 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Data
  3. Simulate Covariance Matrix
  4. PD-CovNet
  5. Model Architecture
  6. selection of hyperparameters.
  7. GP baseline
  8. Five-fold cross-validation
  9. Constraining $\mathrm{H}_0$ Using the CC Dataset
  10. EMCEE
  11. GP
  12. Constraints on $\mathrm{H}_0$
  13. Conclusion
  14. Correction of the Hubble Parameter at $z=0.09$

Figures (3)

  • Figure 1: Architecture of the shared FCNN feature map and PD-CovNet. (a) FCNN feature map $\phi_\theta(z_i)$: a fully connected network with three hidden layers (128 neurons each, employing the ReLU activation function) mapping each standardized $z_i\in\mathbb{R}$ to an $r$-dimensional feature vector $\phi_\theta(z_i)$; the same weights $\theta$ are used for all $i$. (b) PD-CovNet constructs a covariance over the redshift grid by applying the shared FCNN to $N$ inputs and stacking the outputs row-wise to form $\Phi_\theta(z_{1:N})$ in Equation (\ref{['eq:Model Architecture5']}). The covariance $\Sigma_\theta(z_{1:N})$ is designed by Equation (\ref{['eq:Model Architecture6']}). In this paper, parameters $(\theta,\sigma^2)$ are learned from the published $15\times15$ covariance via the multivariate normal log-likelihood and then used to simulate the full $33\times33$ covariance.
  • Figure 2: The published covariance matrix (top) and PD-CovNet-simulated covariance matrices (bottom). The top panel shows the $15\times15$ covariance matrix reported by 2020ApJ...898...82M; the remaining entries for the full set of 33 $H(z)$ CC data points (see Table \ref{['tab:cc_data']}) are set to zero. The bottom panels are PD-CovNet simulations under two hyperparameter sets: H1 (bottom left), specified as epochs $=500$, feature dimension $=12$, hidden width $=32$, hidden depth $=2$, and learning rate $=3\times10^{-3}$; H2 (bottom right), specified as epochs $=3250$, feature dimension $=4$, hidden width $=128$, hidden depth $=3$, and learning rate $=10^{-3}$. The $x$- and $y$-axes are the redshifts $z_1$ and $z_2$, respectively. The top panel has its own colorbar; the two bottom panels share a common colorbar.
  • Figure 3: The published covariance matrix (left panel) and the GP-simulated covariance matrix (right panel). The left panel shows the published $15\times15$ covariance matrix $\boldsymbol{\Sigma}_{\rm pub}$ on the redshift set $\mathbf Z_{15}$. The right panel shows the simulated $33\times33$ covariance matrix $\boldsymbol{\Sigma}_{33}$ on $\mathbf Z_{33}$ obtained by fitting a Gaussian Process with an anisotropic Matérn $(\nu = 5/2)$ kernel to $\boldsymbol{\Sigma}_{\rm pub}$ and evaluating on $\mathbf Z_{33}$. Both panels use the same colormap with a single shared colorbar. The $x$- and $y$-axes are the redshifts $z_1$ and $z_2$, respectively.