A representation learning approach to probe for dynamical dark energy in matter power spectra

TL;DR

The paper introduces DE-VAE, a beta-VAE framework that learns a compact, disentangled latent representation of dynamical dark energy effects on matter power spectrum boosts $\oldsymbol{\mathcal{B}}(k,z)$. By training on a $w_0$-$w_a$CDM model across $k\in(0.01,2.5)\ h/\mathrm{Mpc}$ and $z\in\{0.1,0.48,0.78,1.5\}$, the authors show that a single latent variable suffices to predict DE power spectra within $1\sigma$ (95%) and $2\sigma$ (99%) of the observational error for a Stage IV-like survey, with a second latent offering marginal gains and a third providing no further improvement. Mutual information analysis reveals a strong link between the main latent and the DE parameters $w_0$ and $w_a$, and symbolic regression yields an explicit equation $A_{d=1}(w_0,w_a)$ that connects the latent to the DE parameters. The framework provides a powerful, interpretable approach to compress and interpret beyond-$\Lambda$CDM models, with potential extensions to other cosmological probes, multiple models, and integration into Bayesian inference pipelines to guide survey design and theoretical model development.

Abstract

We present DE-VAE, a variational autoencoder (VAE) architecture to search for a compressed representation of dynamical dark energy (DE) models in observational studies of the cosmic large-scale structure. DE-VAE is trained on matter power spectra boosts generated at wavenumbers $k\in(0.01-2.5) \ h/\rm{Mpc}$ and at four redshift values $z\in(0.1,0.48,0.78,1.5)$ for the most typical dynamical DE parametrization with two extra parameters describing an evolving DE equation of state. The boosts are compressed to a lower-dimensional representation, which is concatenated with standard cold dark matter (CDM) parameters and then mapped back to reconstructed boosts; both the compression and the reconstruction components are parametrized as neural networks. Remarkably, we find that a single latent parameter is sufficient to predict 95% (99%) of DE power spectra generated over a broad range of cosmological parameters within $1σ$ ($2σ$) of a Gaussian error which includes cosmic variance, shot noise and systematic effects for a Stage IV-like survey. This single parameter shows a high mutual information with the two DE parameters, and these three variables can be linked together with an explicit equation through symbolic regression. Considering a model with two latent variables only marginally improves the accuracy of the predictions, and adding a third latent variable has no significant impact on the model's performance. We discuss how the DE-VAE architecture can be extended from a proof of concept to a general framework to be employed in the search for a common lower-dimensional parametrization of a wide range of beyond-$Λ$CDM models and for different cosmological datasets. Such a framework could then both inform the development of cosmological surveys by targeting optimal probes, and provide theoretical insight into the common phenomenological aspects of beyond-$Λ$CDM models.

Figures (5)

• Figure 1: Schematic view of the DE-VAE model employed in this work. The input data consist of power spectrum boosts, as defined in Eq. (\ref{['eq:boost']}), produced at four different redshift bins and for wavenumbers $k \in (0.01 - 2.5) \ h/$Mpc. These boosts are compressed by an encoder, parametrized with convolutional neural networks, to a set of disentangled latent variables (in red). These variables are then concatenated with CDM parameters (in blue) and transformed back to power spectrum boosts through the decoder. All model details are reported in Sect. \ref{['sec:model']}.
• Figure 2: Percentile accuracy for the DE-VAE model trained with different numbers of latent variables $d$. These results are obtained for wavenumbers $k \in (0.01 - 2.5) \ h/$Mpc and redshift values $z \in (0.1, 0.48, 0.78, 1.5)$. The significance is computed using Eq. (\ref{['eq:rel_diff']}), which compares the test power spectra with the predictions from the model, assuming both the predicted and the ground truth power spectra have the same error as in Eq. (\ref{['eq:error']}). The plots show that 99% of the model predictions fall well within 1$\sigma$ (2$\sigma$) of the observational error for most (all) cases. The dashed horizontal lines in each panel correspond to 3$\sigma$.
• Figure 3: Mutual information (MI) between dark energy (DE) parameters $w_0$ and $w_a$ and latent variables. We consider the DE-VAE model trained with different numbers of latent variables $d$, and indicated with the letters A, B or C. When $d=1$, the only latent variable A$_{d=1}$ shows a high MI (i.e. significantly higher than 0) with the DE parameters. Adding a second disentangled variable (A$_{d=2}$ and B$_{d=2}$) slightly improves the predictions of the power spectrum (as shown in Fig. \ref{['fig:percentiles']}), and still shows a significant MI with the DE parameters. When $d=3$, the third latent variable (C$_{d=3}$) is essentially unused, showing no MI with either the DE parameters or any other latent variables. All values lower than 0.01 nat are indicated with a zero, while we omit the MI values in the gray squares. All MI uncertainties are negligible in this instance.
• Figure 4: Scatter plot of $w_0$, $w_a$ and the $d=1$ latent variable for the test set points (blue), together with the graph of the symbolic regression equation reported in Eq. (\ref{['eq:sr']}) (orange). While only accurate within the 2$\sigma$ level for 95% of the predictions on a restricted dataset (see Fig. \ref{['fig:sr_acc']}), the learned analytic expression can be used to bypass the encoder and predict the value of the latent variable given $w_0$ and $w_a$. The red line shows the intersection between the symbolic regression and $A_{d=1}=0$ planes; the latter plane describes a degeneracy with $\Lambda$CDM, which we further discuss, together with broader applications of our framework, in Sect. \ref{['sec:discuss']}.
• Figure 5: Same as Fig. \ref{['fig:percentiles']}(a), but with the latent variable predicted using Eq. (\ref{['eq:sr']}) from $w_0$ and $w_a$, and tested on 1000 boosts generated within three standard deviations of the fiducial Planck values (instead of the five standard deviations considered in the rest of this work).