$Λ$CDM and early dark energy in latent space: a data-driven parametrization of the CMB temperature power spectrum

TL;DR

This paper introduces a data-driven reparameterization of CMB temperature power spectra using a β-VAE to obtain a disentangled latent space that captures the independent degrees of freedom constrained by data. By training separate VAEs on ΛCDM and early dark energy (EDE) spectra, the authors identify that 5 latents suffice for ΛCDM and 8 for EDE to reconstruct spectra within Planck errors, with clear physical interpretations of the latents (amplitude, sound-horizon scale, peak modulation, tilt, and lensing; plus an EDE-specific latent). They demonstrate that latent-posteriors inferred from Planck TT data align with traditional cosmological constraints and show that CMB TT alone cannot decisively distinguish small EDE fractions, while a latent dedicated to EDE isolates its unique signatures. The approach reduces the cosmological parameter space, mitigates prior-volume effects, and offers a generalizable framework for compressing and interpreting cosmological observables beyond ΛCDM.

Abstract

Finding the best parametrization for cosmological models in the absence of first-principle theories is an open question. We propose a data-driven parametrization of cosmological models given by the disentangled 'latent' representation of a variational autoencoder (VAE) trained to compress cosmic microwave background (CMB) temperature power spectra. We consider a broad range of $Λ$CDM and beyond-$Λ$CDM cosmologies with an additional early dark energy (EDE) component. We show that these spectra can be compressed into 5 ($Λ$CDM) or 8 (EDE) independent latent parameters, as expected when using temperature power spectra alone, and which reconstruct spectra at an accuracy well within the Planck errors. These latent parameters have a physical interpretation in terms of well-known features of the CMB temperature spectrum: these include the position, height and even-odd modulation of the acoustic peaks, as well as the gravitational lensing effect. The VAE also discovers one latent parameter which entirely isolates the EDE effects from those related to $Λ$CDM parameters, thus revealing a previously unknown degree of freedom in the CMB temperature power spectrum. We further showcase how to place constraints on the latent parameters using Planck data as typically done for cosmological parameters, obtaining latent values consistent with previous $Λ$CDM and EDE cosmological constraints. Our work demonstrates the potential of a data-driven reformulation of current beyond-$Λ$CDM phenomenological models into the independent degrees of freedom to which the data observables are sensitive.

Figures (12)

• Figure 1: Our method consists of a variational autoencoder (VAE), which compresses the CMB temperature power spectrum into a low-dimensional latent representation (via the encoder); the representation is then sampled to reconstruct CMB spectra (via the decoder). Our goal is to (i) find the minimum number of latents required to reconstruct accurate spectra, (ii) physically interpret the information captured by the latents, and (iii) provide constraints in latent space using Planck data and relate them to latents for different cosmological models.
• Figure 2: Top panels: Examples of $D_{\ell}^\mathrm{TT}$ for two different cosmologies -- a $\Lambda$CDM cosmology with the Planck best-fit cosmological parameters (left panel) and an EDE one selected randomly from the test set (right panel). In black we show the spectra generated with the Einstein-Boltzmann solvers, while the shaded regions cover the range of reconstructed spectra returned by the VAE decoder given 100 samples in latent space. Bottom panels: Ratio between the reconstructed and CLASS or CLASS_EDE spectra. In gray, we show the Planck$1\sigma$ errors for comparison.
• Figure 3: Mean and 99% confidence interval of the residual error for the entire test set cosmological parameter space. Top panel: Residual error of two VAEs, both trained on $\Lambda$CDM TT spectra, with 4D and 5D latent dimensionality respectively. Bottom panel: Same for two VAEs trained on EDE TT spectra with 7D and 8D latent dimensionality respectively. The values of $\beta$ for all these models are tuned such that we obtain disentangled latents. The Planck$1\sigma$ errors are shown in gray.
• Figure 4: 1D and 2D marginalized posterior probability distributions for the eight latent parameters $\bm{z}$ given the two examples of mock data. The mock data are generated by the decoder given a 'ground truth' point in the 8D latent space; these are marked by dashed lines. These points correspond to respectively the most likely latent values of a Planck best-fit $\Lambda$CDM cosmology (orange) and an EDE model with $f_{\rm EDE}=0.15$ (blue). We show unbiased and accurate constraints in latent space, thus validating the robustness and trustworthiness of our pipeline.
• Figure 5: 1D and 2D marginalized posterior probability distributions for the latent parameters $\bm{z}$ given the Planck data; the left panel shows the $\Lambda$CDM latent parameters in orange, and the right panel the EDE ones in blue. We compare the latent posterior constraints to theoretical expectations for the range of latent values allowed by a given set of cosmologies: a $\Lambda$CDM cosmology with best-fit parameters from Ref. Planck:2018vyg (green in both panels) and an EDE cosmology with best-fit parameters from Plik_lite (pink in the right panel). Our constraints in latent space are thus consistent with previous constraints from the literature.