A model for updating cosmology’s uncertainty estimates without new simulations

Why uncertainty needs its own model

The matter power spectrum describes the strength of density fluctuations on different spatial scales. Cosmologists compare its predicted and measured values to investigate quantities such as matter density and the properties of dark energy.

Those comparisons require a covariance matrix: a table describing both the uncertainty at each scale and how fluctuations in different measurements are related. Its diagonal entries give variances; its off-diagonal entries describe how measurements fluctuate together. Gravitational evolution couples structures across scales, making these relationships important.

The matrix also depends on the cosmological parameters being tested. Estimating it directly at every candidate cosmology would require many expensive simulations. Holding it fixed avoids that expense, but can misrepresent uncertainties or shift inferred parameters, depending on the analysis.

In their arXiv preprint, Steele and Smith develop a method intended to retain this cosmology dependence without requiring new simulations after the initial calibration.

Correcting a theoretical starting point

The method starts with an approximate theoretical covariance matrix calculated using standard perturbation theory, which describes gravitational growth through an expansion around small density fluctuations. The authors combine its inverse—the precision matrix—with a covariance matrix estimated from simulations.

This produces a correction matrix describing the mismatch between theory and the simulation estimate. If the theoretical model were exact, that correction would be the identity matrix, which leaves other matrices unchanged.

Rather than expanding the entire covariance directly, the researchers make a first-order Taylor expansion of this correction around a reference cosmology. Response matrices describe how the correction changes when cosmological parameters move away from their reference values. Combining the updated correction with the theoretical covariance then gives the reconstructed covariance.

The researchers compare three versions: one retaining the full response, one neglecting changes in the correlation matrix, and one setting the correction’s response to zero. The last version still allows cosmology dependence through the theoretical model; it does not hold the whole covariance fixed.

Testing against Quijote simulations

The study uses the publicly available Quijote simulations. Its reference cosmology has 15,000 realisations, while each of 12 nearby cosmologies has 500. Each nearby case changes one parameter upward or downward, leaving the others fixed.

The six varied parameters describe total matter density, baryon density, the tilt of primordial fluctuations, present-day clustering amplitude, the expansion-rate parameter and the dark-energy equation of state. Each gravity-only simulation follows 512 cubed particles in a periodic cube 1,000 inverse-h megaparsecs across, where h specifies the Hubble constant in conventional dimensionless units.

The authors also test versions with and without super-sample covariance. This accounts for the influence of fluctuations larger than the sampled volume. Here, an approximate theoretical contribution is added to both the theoretical predictions and the simulation-based covariance estimates, rather than being independently measured from the periodic boxes.

What the reconstruction recovered

For the full-response and constant-correlation reconstructions, the authors report diagonal covariance residuals of approximately 6% or less over the tested range up to a wavenumber of about 0.3 h per megaparsec. At wavenumbers below about 0.15 h per megaparsec, the residuals are generally approximately 1% or less, with somewhat larger deviations for variations in h. Lower wavenumbers correspond to larger physical scales.

These percentages describe the diagonal variances, not a uniform accuracy guarantee for every matrix element. Comparisons also show that the reconstructions recover off-diagonal structure more faithfully than the uncorrected perturbation-theory model.

The full response generally performs best. Neglecting the correction’s response becomes less reliable when super-sample covariance is included. Although holding correlations constant can reduce potential noise, the authors find that retaining their changes improves accuracy without substantial additional noise in these tests.

Where the evidence stops

The central limitation is that the method reconstructs the same cosmologies used to establish it. The authors could not afford independent simulation ensembles at additional parameter points. Its accuracy across a broader parameter space, including simultaneous parameter changes, therefore remains untested.

Truncating the expansion can also break matrix symmetry and positive semidefiniteness—the requirement that every combination of measurements have non-negative variance. The method explicitly restores symmetry, but positive semidefiniteness is not automatically guaranteed.

Finally, this is a collisionless-matter calculation, not a complete galaxy-survey analysis. Galaxy bias, redshift-space distortions, survey masks and selection effects still need incorporating. Extending the approach to smaller scales relevant to weak lensing also requires further work. The supplied source is a preprint; it does not establish peer-reviewed publication or demonstrate improved cosmological constraints from real survey data.