Optimal marked statistics from the Effective Field Theory of Large Scale Structure

Technical Summary: Optimal Marked Statistics from the Effective Field Theory of Large Scale Structure

Problem Statement
The next generation of Large Scale Structure (LSS) surveys (e.g., Euclid, DESI, Rubin Observatory) aims to extract maximal cosmological information from the matter distribution, particularly on mildly nonlinear scales where signal richness competes with theoretical modeling complexity. While the matter power spectrum $P(k)$ is the standard workhorse, gravitational evolution generates non-Gaussianities that lock information into higher-order correlators (bispectrum, trispectrum). Recovering this information directly is computationally expensive and theoretically challenging.

Marked statistics offer an alternative: constructing a nonlinearly weighted version of the density field, $\delta_M$, and computing its power spectrum ($M(k)$). This approach compresses higher-order information into a two-point function. Previous studies utilized specific mark functions (e.g., weighting voids more heavily) and found improvements in constraining parameters like $\sigma_8$ and neutrino masses. However, a critical open question remains: Is the standard mark function optimal, or can alternative functional forms capture more information? Furthermore, existing optimization efforts often rely on simulation-based Gaussian processes, which lack the analytical control and physical interpretability of perturbation theory.

Methodology
The authors propose a framework combining the Effective Field Theory of Large Scale Structure (EFTofLSS) with a flexible optimization of the mark function.

1. Theoretical Model:

   • The mark function $m(\delta_R)$, where $\delta_R$ is the density field smoothed on scale $R$, is expanded as a polynomial in powers of $\delta_R$:
     $$m(\delta_R) = c_0 + c_1 \delta_R + c_2 \delta_R^2 + c_3 \delta_R^3 + \mathcal{O}(\delta_R^4)$$
   • The coefficients $\{c_0, c_1, c_2, c_3\}$ are treated as free parameters to be optimized.
   • The authors derive the one-loop expressions for the standard power spectrum $P(k)$, the cross power spectrum $C(k)$, and the marked power spectrum $M(k)$ within the EFTofLSS framework. This includes the necessary counterterms ($c_s^2$) to renormalize UV dependence.

2. Perturbativity Prior:

   • A central novelty is the imposition of a "perturbativity prior" to ensure the marked field remains within the regime of validity of perturbation theory.
   • The authors derive conditions on the coefficients $c_i$ such that higher-order terms in the expansion do not dominate lower-order terms. Specifically, they require $|c_{n+1}|\delta_{R,\text{max}} < |c_n|$, where $\delta_{R,\text{max}}$ is the maximum smoothed overdensity observed in simulations. This restricts the parameter space to physically viable marks.

3. Optimization Procedure:

   • The Fisher information matrix is used as the cost function to maximize constraining power on cosmological parameters $\{\Omega_m, \sigma_8, h\}$.
   • The optimization accounts for affine invariance: the Fisher matrix is invariant under transformations $m \to Am + B$. Consequently, the absolute values of coefficients are less important than their relative ratios (e.g., $c_2/c_1$). The authors fix $c_0=1$ to remove this degeneracy.
   • The optimization scans a grid of $\{c_1, c_2, c_3\}$ values subject to the perturbativity prior.
   • Covariance matrices are initially estimated using Gaussian approximations for speed, then validated and refined using the full non-Gaussian covariance from 1,000 Quijote N-body simulations.

Key Contributions

• Analytical Optimization: The paper provides a fully analytical, simulation-free method to optimize mark functions within the EFTofLSS framework, contrasting with previous Gaussian process approaches that rely heavily on N-body simulations.
• Perturbativity Constraints: It establishes rigorous physical constraints on mark functions to ensure they do not violate the assumptions of perturbation theory, defining a "safe" regime for marked statistics.
• Affine Invariance Clarification: The authors explicitly demonstrate that the information gain from marked statistics arises from the environmental contrast modulated by the mark, rather than a preferential weighting of voids or overdensities specifically. This is a consequence of the affine invariance of the Fisher matrix when the full data vector $\{P, C, M\}$ is used.
• Validation: The analytical predictions for the optimal marks are validated against the Quijote simulation suite, showing agreement at the percent level for power spectra up to $k \approx 0.3 \, h \, \text{Mpc}^{-1}$.

Results

• Optimal Mark Forms: The optimal mark functions depend on the smoothing scale $R$.
  • For small scales ($R=10 \, h^{-1}\text{Mpc}$), the perturbativity prior forces the marks to be simple, monotonically decreasing functions resembling linear expansions.
  • For larger scales ($R=15, 30 \, h^{-1}\text{Mpc}$), the optimal marks exhibit richer, parabolic structures that enhance the differential weighting between high- and low-density environments.
• Parameter Improvements: After marginalizing over the EFT counterterm $c_s^2$, the optimized marked statistics yield significant improvements in parameter error bars compared to the standard power spectrum alone:
  • $\Omega_m$: Improvement factor of $\approx 1.30–2.3$.
  • $\sigma_8$: Improvement factor of $\approx 1.3–8.3$.
  • $h$: Improvement factor of $\approx 1.2–2.5$.
  • These gains vary with redshift and smoothing scale and are consistent with previous simulation-based findings (e.g., [22]).
• Covariance Structure: The authors find that while the Gaussian covariance approximation is sufficient for small smoothing scales ($R=10$), larger scales ($R=15, 30$) exhibit significant off-diagonal correlations in the covariance matrix, necessitating the use of full non-Gaussian covariance for rigorous likelihood analysis.

Significance and Claims
The paper claims that marked statistics, when optimized via an analytical EFTofLSS framework, serve as a powerful tool to extract additional cosmological information beyond standard two-point analyses without the computational cost of full higher-order correlator calculations (like the bispectrum).

The authors emphasize that their approach validates the EFTofLSS description of marked statistics and confirms that a polynomial expansion is a sufficient functional basis for mark optimization in the perturbative regime. They position this method as a fast, fully analytic pipeline suitable for future MCMC analyses of actual survey data. Furthermore, they suggest this framework opens a pathway for exploring optimized marks for primordial non-Gaussianity, where simulation-based approaches are currently too expensive. The work underscores that the "driver" of information gain is the modulation of environmental clustering, providing a physically motivated interpretation for the utility of marked fields.