Generation of Lognormal Synthetic Lyman-$α$ Forest Spectra for $P_{1D}$ Analysis

Technical Summary: Generation of Lognormal Synthetic Lyman-α Forest Spectra for P1D Analysis

Problem Statement
The one-dimensional flux power spectrum ($P_{1D}$) of the Lyman-$\alpha$ (Ly$\alpha$) forest is a critical probe for small-scale structure in the intergalactic medium (IGM), offering sensitivity to cosmological parameters (e.g., neutrino masses, dark matter nature) and astrophysical processes (e.g., IGM thermal history). However, extracting these parameters requires rigorous validation against systematic errors, including instrumental effects, continuum fitting uncertainties, and astrophysical contaminants. While hydrodynamical simulations provide high-fidelity realizations of the IGM, they are computationally prohibitive for generating the large ensembles of spectra required for robust error estimation and pipeline validation in high-volume surveys like the Dark Energy Spectroscopic Instrument (DESI). Previous lognormal mock generation methods, while faster, often relied on static, analytic forms for the underlying Gaussian power spectrum that lacked the flexibility to accurately reproduce the scale-dependent shape and redshift evolution of the observed $P_{1D}$ across the broad parameter space of modern surveys.

Methodology
The authors present an improved lognormal mock framework designed to generate synthetic 1D Ly$\alpha$ forest spectra tailored for $P_{1D}$ analysis. The method builds upon the formalism established by McDonald et al. (2006) and Karaçaylı et al. (2020) but introduces two key innovations to enhance fidelity:

1. Redshift-Dependent Gaussian Correlation Inversion: Unlike previous approaches that utilized a fixed analytic form for the input Gaussian power spectrum ($P_G$), this method solves directly for the optimal Gaussian correlation function ($\xi_G$) in configuration space. By iteratively adjusting $\xi_G$ such that the resulting flux correlation function ($\xi_F$)—after applying the full sequence of lognormal transformations—matches a target $P_{1D}$, the authors derive a unique, redshift-dependent Gaussian power spectrum $P_G(k, z)$. This inversion is performed point-wise using a non-linear least-squares solver to minimize residuals between the transformed mock statistics and the target model (specifically the DESI Early Data Release measurements).
2. Analytic Mean Flux Fitting: The framework separately optimizes the evolution of the mean transmitted flux. The authors adopt an effective optical depth model ($\tau(z) = \tau_0(1+z)^\gamma$) and fit four transformation parameters ($\tau_0, \tau_1, \nu, \sigma^2_F$) to reproduce the mean flux evolution observed in DESI Data Release 1 (DR1). This is achieved by minimizing a weighted least-squares cost function comparing the analytic mean flux derived from the lognormal transformation to the target observational data.

The generation pipeline involves creating a 1D random Gaussian velocity grid, applying the derived redshift-dependent $P_G(k, z)$, evolving the field with a redshift factor $a(z)$, applying a squared lognormal transformation to approximate non-Gaussianity, and finally converting to optical depth and transmitted flux.

Key Results
The authors validated the framework by generating 1,000 independent line-of-sight realizations across the DESI EDR redshift range ($2.0 \le z \le 3.8$). The results demonstrate:

• Mean Flux Accuracy: The mocks recover the mean transmitted flux evolution with sub-percent accuracy. Across the full DESI EDR redshift range, the root-mean-square (RMS) fractional error is 0.002, with a maximum deviation of 0.5% at the redshift boundaries.
• Power Spectrum Recovery: The method reproduces the shape and amplitude of the $P_{1D}$ with high fidelity. Over the DESI EDR $k$-range ($0.00075 \le k \lesssim 0.035 \text{ s km}^{-1}$), the average RMS fractional error is 0.02. This represents a significant improvement over the previous method (Karaçaylı et al. 2020), which yielded an error of 0.10 in the same range.
• Scalability: The framework maintains robust performance over extended scales ($10^{-5} \le k \lesssim 0.1 \text{ s km}^{-1}$), where the RMS error remains at 0.11, a dramatic improvement over the previous method's error of 18.15. The maximum deviation for the new method in this wide range is below 40%, compared to 300% for the prior approach.
• Convergence: A convergence analysis indicates that 1,000 realizations are sufficient for the RMS fractional error to asymptotically approach a constant value, ensuring statistical stability for validation tests.

Significance and Claims
The paper claims that this improved lognormal framework provides a computationally efficient, analytically tractable alternative to hydrodynamical simulations for generating large volumes of synthetic spectra. By decoupling the modeling of the mean flux amplitude from the detailed flux correlations and solving for the input Gaussian power spectrum directly, the method offers superior control over target observables.

The authors emphasize that while these mocks are not intended for full cosmological inference (as they do not model underlying physics from first principles), they are well-suited for:

• Validating analysis pipelines and quantifying systematic effects (e.g., noise, resolution, continuum fitting).
• Supporting $P_{1D}$ measurements using FFT and likelihood-based estimators in current and future DESI data releases.
• Enabling controlled studies of astrophysical contaminants (e.g., Damped Ly$\alpha$ systems, Broad Absorption Lines) when combined with tools like quickquasars.

The work establishes a foundation for end-to-end analyses where emulator or simulation outputs can serve as direct inputs for lognormal mock generation, bridging theoretical models and observational inference with accurate statistical properties. The authors note that future refinements, such as joint fitting of the mean flux and $P_{1D}$ to better handle variance dependencies, could further enhance the framework's utility for precision cosmology.