Optimal Transport Reconstruction of Biased Tracers in Primordial Non-Gaussian Fields

Technical Summary: Optimal Transport Reconstruction of Biased Tracers in Primordial Non-Gaussian Fields

Problem Statement
Optimal Transport (OT) has established itself as an efficient framework for reconstructing the cosmic displacement field, mapping initial positions to present-day locations (or vice versa). Previous work demonstrated that OT accurately recovers displacements for biased tracers under the assumption of Gaussian initial conditions. However, a critical open question remains: does this framework remain valid when initial conditions are non-Gaussian? This is a timely challenge given the increasing efforts to constrain primordial non-Gaussianity (PNG) using next-generation cosmological surveys. A specific difficulty arises because OT implementations often assume a uniform initial spatial distribution. While this holds for the full dark matter field, it does not hold for biased subsets (e.g., halos). In Gaussian scenarios, the unobserved mass ("dust") can be modeled effectively; however, in non-Gaussian scenarios, the dust itself acquires specific scale-dependent biases that must be accounted for to ensure accurate reconstruction.

Methodology
The authors investigate the applicability of OT reconstruction to "local" primordial non-Gaussian fields, where the gravitational potential $\phi_{NG}$ is modeled as a power series in a Gaussian field $\phi_G$ (truncated at second order, $f_{NL} \neq 0$). In this model, the density contrast $\delta_{NG}$ contains a term proportional to the product of Gaussian fields, leading to a distinctive scale-dependent bias in the clustering of biased tracers at large scales (small $k$).

The study utilizes the Quijote-PNG simulation set, consisting of 20 periodic boxes ($L = 1 h^{-1}$ Gpc) with $512^3$ particles, evolving from $z=127$ to $z=0$ under a flat $\Lambda$CDM cosmology. Biased tracers are identified as halos with mass $> 20 m_p$.

The core methodological innovation involves modeling the unobserved "dust" (the mass fraction $1-f_b$ not in the halo catalog) to satisfy the constraint that the full matter field must exhibit no scale-dependent bias. The authors derive that if biased tracers exhibit a scale-dependent bias characterized by parameters $b_\delta$ and $b_\phi$ (where the bias scales as $b(k) \approx b_\delta + 2f_{NL}b_\phi \alpha(k)$), the dust must possess a compensating scale-dependent bias with opposite sign.

Two reconstruction approaches are tested:

1. OT-K: Uses a Wiener-filter model to estimate the dust distribution based on observed tracers. This requires guessing the combination $2f_{NL}b_\phi$ (which can be constrained by survey data) and the mass fraction $f_b$.
2. OT-W: Uses the true dust particles from the simulation (a random sample) combined with the mass-weighted halos to perform the reconstruction, serving as an upper bound on performance.

The reconstruction quality is evaluated by comparing the reconstructed positions to the actual protohalo positions and by analyzing the scale dependence of the divergence of the displacement field ($\theta_{OT}$), which serves as a proxy for the initial density field.

Key Contributions and Results

• Dust Characterization: The paper demonstrates that in local PNG models, the "dust" is not a passive background but exhibits its own characteristic scale-dependent bias. Specifically, the dust's bias amplitude is determined by the requirement that the sum of the biased tracer and dust fields yields a scale-independent total field.
• Extension of OT to Non-Gaussianity: The authors show that the OT methodology extends successfully to non-Gaussian initial conditions. By incorporating a realistic model for the dust that accounts for the $f_{NL}$-dependent bias, the reconstruction of biased tracers remains accurate.
• Reconstruction Accuracy:
  • Tracer Positions: The OT-K reconstruction (using the Wiener-filter dust model) correctly recovers the scale-dependent bias of the protohalos. The OT-W reconstruction (using true dust) shows even better agreement, indicating that the method is robust but sensitive to the fidelity of the dust model.
  • Full Field Proxy: The divergence of the OT displacement field ($\theta_{OT}$) shows almost no scale dependence at small $k$, mirroring the behavior of the true initial field. The OT-W model extends this scale-independence to larger $k$ values compared to OT-K, achieving a factor of 2 improvement in the range of $k$ where $b(k) \approx 1$.
• Robustness to $f_{NL}$: The mean square value of the reconstructed field $\langle \theta_{OT}^2 \rangle$ remains stable across different values of $f_{NL}$ ($+100, 0, -100$), with variations within the standard deviation, suggesting the method is not biased by the presence of non-Gaussianity itself.

Significance and Claims
The paper claims that Optimal Transport provides a useful framework for reconstructing cosmic structures even when initial conditions are non-Gaussian, provided that the model for the unobserved "dust" is sufficiently realistic. The authors emphasize that the success of the method relies on the fact that the distinctive scale-dependent signature of PNG is present in both the biased tracers and the dust, but with opposite signs, allowing the full field to remain unbiased.

The authors modestly state that they do not demonstrate a direct improvement in constraints on $f_{NL}$ from this work alone. Instead, they provide a foundational framework that allows reconstruction-based analyses to be consistently applied in PNG cosmologies. They suggest that this work is a necessary step toward incorporating reconstruction techniques into future high-precision PNG analyses, potentially helping to connect observed biased tracers more closely to their primordial Lagrangian counterparts and mitigating subtle nonlinear or stochastic contributions relevant for next-generation surveys. The paper concludes that improved modeling of the dust (e.g., via iterative methods or machine learning) could yield further gains.