Reconstructing the Hubble parameter with future Gravitational Wave missions using Machine Learning

TL;DR

This work assesses the feasibility of reconstructing the Hubble parameter $H(z)$ from future gravitational wave standard siren data using Gaussian process regression (GPR). By generating 500 mock catalogs for two missions (eLISA and ET) across six cosmological models and two fiducial priors (early-time CSB and late-time RSH), the authors perform a non-parametric reconstruction of luminosity distance $d_L(z)$ and its derivative to obtain $H(z)$ via $H(z)=\frac{c(1+z)^2}{d_L'(z)(1+z)-d_L(z)}$, along with $H_0$ uncertainties projected through Monte Carlo propagation. The results show GP reconstructions are robust within the missions’ redshift windows, with $1\sigma$ constraints on $H_0$ improving as the number of detected events increases; notably, a ~10-year eLISA run can achieve precision competitive with a 3-year ET run, and longer durations further tighten constraints. Model- and fiducial-dependent nuances emerge (e.g., varying shifts in $H_0$ and crossing behaviors in $H(z)$ for VM/VM-VEV), offering insight into the Hubble tension at intermediate redshifts. Overall, the study demonstrates the potential of next-generation GW data to constrain cosmic expansion non-parametrically and informs future multi-mission analyses and methodological developments in cosmology with gravitational waves.

Abstract

We study the prospects of Gaussian processes (GP), a machine learning (ML) algorithm, as a tool to reconstruct the Hubble parameter $H(z)$ with two upcoming gravitational wave missions, namely the evolved Laser Interferometer Space Antenna (eLISA) and the Einstein Telescope (ET). Assuming various background cosmological models, the Hubble parameter has been reconstructed in a non-parametric manner with the help of GP using realistically generated catalogs for each mission. The effects of early-time and late-time priors on the reconstruction of $H(z)$, and hence on the Hubble constant ($H_0$), have also been focused on separately. Our analysis reveals that GP is quite robust in reconstructing the expansion history of the Universe within the observational window of the specific missions under consideration. We further confirm that both eLISA and ET would be able to provide constraints on $H(z)$ and $H_0$ which would be competitive to those inferred from current datasets. In particular, we observe that an eLISA run of $\sim10$-year duration with $\sim80$ detected bright siren events would be able to constrain $H_0$ as good as a $\sim3$-year ET run assuming $\sim 1000$ bright siren event detections. Further improvement in precision is expected for longer eLISA mission durations such as a $\sim15$-year time-frame having $\sim120$ events. Lastly, we discuss the possible role of these future gravitational wave missions in addressing the Hubble tension, for each model, on a case-by-case basis.

Figures (8)

• Figure 1: Plot for the posterior covariance matrices (a) $\text{cov}[d_L(Z^\star),d_L(Z^\star)]$, (b) $\text{cov}[d_L'(Z^\star),d_L'(Z^\star)]$, (c) $\text{cov}[d_L(Z^\star),d_L'(Z^\star)]$, defined in Eq. \ref{['eq:posterior_cov_dldl']}, \ref{['eq:posterior_cov_dlpdlp']} and \ref{['eq:posterior_cov_dldlp']} respectively.
• Figure 2: Plots showing the reconstructed functions $d_L$ (left panel) and $d_L'$ (middle panel) in units of Gpc with respect to the test redshifts. The right panel shows the reconstructed posterior distribution of $H_0$, $P(H_0\vert\sigma_f,l)$, for the best-fit values of the hyperparameters.
• Figure 3: Plots for the reconstructed $H(z)$ vs redshift $z$ from $\sim1000$ GW events detectable by ET for a $\sim3$-year mission duration in the redshift range $0<z<2$. Here 'CSB' & 'RSH' denotes the fiducial Hubble function obtained using respective dataset combinations. Set-I & Set-II represent the reconstructed functions with ET mock catalogs generated from the 'CSB' and 'RSH' constraints as fiducials.
• Figure 4: Plots for the reconstructed $H(z)$ vs redshift $z$ using $\sim80$ GW events from the "No Delay" source population detectable by eLISA for a $\sim10$-year mission duration in the redshift range $0<z<5$.
• Figure 5: Comparison of the errors in the reconstruction of $H(z)$ between ET and eLISA in the redshift range $0<z<2$ for mission particulars as outlined in Figs. \ref{['fig:Hz_ET_plot']} and \ref{['fig:Hz_eLISA_plot']}.