An improved model for the effect of correlated Si-III absorption on the one-dimensional Lyman-$α$ forest power spectrum

Technical Summary: An Improved Model for the Effect of Correlated Si III Absorption on the One-Dimensional Lyman-𝛼 Forest Power Spectrum

Problem Statement
The one-dimensional Lyman-𝛼 (Ly𝛼) forest power spectrum is a primary tool for constraining cosmological parameters and dark matter models. However, accurate modeling is complicated by contamination from metal absorption lines, specifically those correlated with the Ly𝛼 forest. While uncorrelated metals can be estimated and subtracted using sidebands, correlated metals like triply ionized silicon (Si III, $\lambda1207$) are blended within the forest.

Previous studies, notably McDonald et al. (2006, M06), modeled Si III contamination by assuming the Si III transmitted flux contrast is a linearly rescaled version of the Ly𝛼 flux contrast. This ansatz successfully describes the oscillatory features in the power spectrum induced by the velocity offset between Ly𝛼 and Si III at large scales ($k \lesssim 0.02$ s km$^{-1}$). However, this paper identifies that the M06 model breaks down at smaller scales ($k > 0.02$ s km$^{-1}$). Specifically, the M06 approach fails to account for:

1. Distinct Line Profiles: Si III lines are thermally narrower than Ly𝛼 lines due to the higher mass of silicon, leading to an enhancement of small-scale power that the linear rescaling model misses.
2. Variable Optical Depth Ratios: The ratio of coeval Si III to Ly𝛼 optical depths is not constant but depends on gas density, temperature, and silicon abundance.
3. Redshift Evolution: The amplitude of Si III contamination may not follow the previously assumed scaling with mean Ly𝛼 transmission, particularly due to the hardening of the metagalactic UV background at $z \lesssim 3.5$.

Methodology
The authors utilize the Sherwood-Relics suite of high-resolution cosmological hydrodynamical simulations to construct mock quasar spectra.

• Simulation Setup: The simulations use a $40 h^{-1}$ cMpc box with $2 \times 2048^3$ particles. They employ a non-equilibrium thermo-chemistry solver and a spatially uniform UV background.
• Si III Modeling: Unlike previous simulations that may lack metal production models, the authors assume a fiducial silicon abundance model dependent on gas overdensity ($\Delta$) and redshift ($z$), calibrated against observational constraints from Schaye et al. (2003) and Aguirre et al. (2004). The Si III ionization fraction ($x_{\text{Si III}}$) is calculated using the photo-ionization code Cloudy combined with the UV background spectrum.
• Spectrum Construction: The authors extract 5,000 periodic sightlines at redshift intervals of $\Delta z = 0.1$. They compute Ly𝛼 and Si III optical depths, applying Voigt profile convolutions (with Si III lines resampled onto a finer grid to resolve their narrower thermal widths).
• Power Spectrum Analysis: They compute the 1D power spectrum ($P(k)$) for the total flux, Ly𝛼-only, and Si III-only components. They compare these direct simulation results against the analytical M06 model and their proposed new fitting function.

Key Contributions and Model Development
The paper proposes a new analytical fitting function (Eq. 18) to replace the M06 ansatz for modeling Si III contamination. The new model introduces four free parameters to capture two critical physical effects neglected in previous work:

1. Scale-Dependent Enhancement: The model includes exponential terms ($\exp(k/k_{\text{auto}})$ and $\exp(k/k_{\text{cross}})$) to account for the narrower Si III line profiles and the pixel-dependent ratio of optical depths.
2. Decorrelation: The model allows for the decorrelation between Si III and Ly𝛼 absorption at small scales. This occurs because strong Si III absorbers often correspond to saturated Ly𝛼 lines, breaking the linear relationship assumed in M06.

The proposed total power spectrum is given by:
$$P_{\text{tot}}(k) = P_{\text{Ly}\alpha}(k) + P_{\text{Si III}}(k) + P_{\text{Ly}\alpha-\text{Si III}}(k)$$
Where:

• $P_{\text{Si III}}(k) = a_{\text{auto}} \exp(k/k_{\text{auto}}) P_{\text{Ly}\alpha}(k)$
• $P_{\text{Ly}\alpha-\text{Si III}}(k) = a_{\text{cross}} \cos(\Delta v k) \exp(k/k_{\text{cross}}) P_{\text{Ly}\alpha}(k)$

Results

• Validation: The new model was validated against the Sherwood-Relics simulations across redshifts $2.2 \le z \le 5.0$ and wavenumbers $k < 0.2$ s km$^{-1}$.
• Performance: At large scales ($k < 0.02$ s km$^{-1}$), both the M06 model and the new model fit the data well. However, at smaller scales, the M06 model significantly underestimates the power, whereas the new model reproduces the simulation data with a fractional deviation of less than 1% for the fiducial case.
• Parameter Sensitivity:
  • The scale parameters ($k_{\text{auto}}$ and $k_{\text{cross}}$) are primarily sensitive to the gas temperature ($T_0$). Hotter gas leads to broader lines and different decorrelation scales.
  • The amplitude parameters ($a_{\text{auto}}$ and $a_{\text{cross}}$) are primarily sensitive to the silicon abundance ($[\text{Si/H}]$).
• Redshift Evolution: Contrary to the M06 assumption that Si III power scales inversely with mean transmission ($1 - \bar{F}$), the simulations show that the Si III power spectrum amplitude increases toward lower redshifts ($z \lesssim 3.5$). This is driven by the hardening of the UV background, which reduces the Si III ionization fraction ($x_{\text{Si III}}$) at lower redshifts, but is counteracted by the evolution of the silicon abundance model used.

Significance and Claims
The authors state that their improved model is an essential consideration for future, high-precision Ly𝛼 forest power spectrum measurements, particularly those utilizing high-resolution spectrographs (e.g., Keck/HIRES, VLT/UVES) that probe scales beyond $k \sim 0.02$ s km$^{-1}$.

Regarding cosmological constraints, the paper claims that adopting a physically motivated prior on silicon abundance means the new model has little effect on existing warm dark matter (WDM) constraints derived from the Ly𝛼 forest. However, the authors emphasize that ignoring the small-scale power enhancement from Si III could lead to systematic errors in future analyses as statistical uncertainties decrease. The paper does not propose new observational campaigns but provides the necessary theoretical framework to interpret upcoming high-resolution data accurately.