Cosmic variance or galaxy bias? Disentangling finite-volume and galaxy formation effects in cosmological analysis

Technical Summary: Disentangling Finite-Volume and Galaxy Formation Effects in Cosmological Analysis

Problem Statement
Current and forthcoming large-scale galaxy redshift surveys (e.g., DESI, Euclid) aim to determine cosmological parameters with unprecedented precision. However, achieving unbiased results requires rigorous control over theoretical systematics. A primary challenge is distinguishing between two strongly coupled uncertainty sources: cosmic variance, arising from the intrinsic variance of the initial Gaussian random field in a finite volume, and galaxy bias, describing the complex non-linear processes of galaxy formation.

In standard perturbative frameworks, bias parameters are estimated by fitting observations to the ensemble-averaged matter power spectrum. This approach implicitly assumes the survey volume represents the ensemble average, thereby neglecting the specific cosmic variance of the observed realization. Consequently, the estimated bias parameters become degenerate, conflating true galaxy formation physics with the statistical fluctuations of the specific cosmic volume surveyed. This degeneracy complicates applications such as Baryon Acoustic Oscillation (BAO) reconstruction and the inference of cosmological parameters, where assumptions about bias values are critical.

Methodology
The authors develop a novel formalism based on Eulerian Perturbation Theory (EPT) to mathematically disentangle cosmic variance from galaxy bias.

1. Reframing Cosmic Variance as Bias: The authors propose treating the density field of an arbitrary finite-volume realization ($\delta'$) as a "biased" tracer of the ensemble-averaged density field ($\delta$). By introducing a new set of bias parameters ($b_{1,dm}, b_{2,dm}$), they express the finite-volume field as:
   $$\delta'(x) = b_{0,dm} + b_{1,dm}\delta(x) + \frac{1}{2}b_{2,dm}\delta^2(x) + \dots$$
   In this framework, as the volume approaches infinity, $b_{1,dm} \to 1$ and $b_{2,dm} \to 0$. This formulation allows the separation of the "cosmic variance bias" from the physical "galaxy bias" ($b_{1,g}, b_{2,g}$).

2. Application to BAO Shift: The authors apply this formalism to the non-linear shift of the BAO scale. Using the analytical expressions derived by [22], they decompose the total BAO shift ($\Delta \alpha$) into two distinct components:

   • A component driven by the intrinsic cosmic variance of the dark matter field (controlled by $b_{1,dm}, b_{2,dm}$).
   • A component driven by galaxy biasing (controlled by $b_{1,g}, b_{2,g}$).
     This decomposition reveals that cosmic variance alone can induce a non-negligible BAO shift, particularly in small volumes, which is degenerate with the shift caused by non-linear gravitational growth.

3. Integral Constraint Handling: The paper addresses the integral constraint (the bias introduced by forcing the mean density of a survey volume to zero). The authors demonstrate that this effect can be reabsorbed into the constant term of the bias expansion ($b'_0$), ensuring it does not introduce additional bias parameters or complicate the separation of cosmic variance.

4. Numerical Validation: To validate the analytical arguments, the authors utilize the UNIT simulation suite. They analyze four distinct sets of $N$-body simulations ($2 \times 10^3$ particles in $(1 h^{-1}\text{Gpc})^3$ volumes):

   • 100 standard simulations.
   • 100 fixed-amplitude simulations (suppressing amplitude variance).
   • 2 $\times$ 50 paired simulations (suppressing phase variance).
   • 2 $\times$ 50 fixed-and-paired simulations (suppressing both).
     They measure the BAO peak position in a model-independent manner (fitting a Gaussian to the residual correlation function) to quantify the variance reduction.

Key Results

• Analytical Disentanglement: The new formalism successfully separates the contributions of cosmic variance and galaxy bias to the non-linear BAO shift. The analysis confirms that cosmic variance can induce a shift comparable to that of galaxy bias, necessitating extra care when using perturbative approaches on finite volumes.
• Variance Suppression: Numerical results from the UNIT simulations demonstrate that controlling initial conditions significantly reduces uncertainty in the BAO peak position:
  • Fixing (controlling amplitude) reduces uncertainty by a factor of $\sim 1.5 - 2.1$.
  • Pairing (controlling phase) reduces uncertainty by a factor of $\sim 1.5 - 1.7$.
  • Combined (Fixed-and-Paired): The combination of both techniques yields a variance suppression factor of $\sim 3.4 - 5.4$ compared to standard ensembles.
• Convergence: The BAO peak position is found to be well-converged at the sub-percent level across all simulation types, with no apparent systematics introduced by the fixing or pairing techniques relative to the standard ensemble.

Significance and Proposed Strategies
The paper concludes that current methods for estimating bias parameters inadvertently include the degenerate effects of cosmic variance. To address this, the authors propose two practical strategies for future cosmological analyses:

1. Optimal Multi-Tracer Combination: Instead of solely maximizing statistical precision, future analyses could combine complementary tracers (e.g., galaxies and cosmic voids) specifically to cancel out cosmic variance. This could involve redefining pre-reconstruction samples to trace complementary density regimes or optimizing weights in the combined correlation function to mimic the effect of paired realizations.
2. Variance-Suppressed Reconstruction: The authors suggest applying variance suppression techniques (fixing and pairing) to both the observed data (via reconstruction methods like backward inversion or Bayesian forward reconstruction) and the mock catalogs used for covariance estimation. This would allow for the analysis of a "variance-suppressed" version of the observed Universe and its paired anti-Universe. They note that this requires careful implementation to avoid biasing the BAO shift through the cosmology adopted in the reconstruction process.

The authors emphasize that while these strategies show promise, they require further testing on detailed mock catalogs (e.g., DESI-like) and integration with state-of-the-art analysis techniques such as BAO reconstruction and Effective Field Theory.