On Estimating Lyman-alpha Forest Correlations between Multiple Sightlines

TL;DR

This work develops a practical framework for extracting 3D Lyα forest correlations from dense background-sightline samples by introducing a single sensitivity metric, the noise-weighted sightline density $\bar{n}_{\rm eff}$, and deriving optimal sightline weights that maximize the signal-to-noise of the flux power spectrum $P_{\rm F}(\boldsymbol{k})$. It connects these weights to a minimum-variance quadratic estimator, showing that a simple weighting scheme closely approximates the full estimator and can be implemented with iterative solvers for the covariance. The paper analyzes survey designs (quasars and galaxies), cross-correlation opportunities, and provides Fisher-matrix forecasts for cosmological parameters such as $D_A(z)$, $H(z)$, and $\Omega_k$, as well as constraints on ionizing-background and temperature fluctuations. It also discusses systematic concerns, including continuum subtraction and damping wings, arguing that a 3D approach mitigates many pitfalls of 1D analyses and enables robust 3D constraints on cosmology and reionization history.

Abstract

The next frontier of Lyman-alpha forest studies is the reconstruction of 3D correlations from a dense sample of background sources. The measurement of 3D correlations has the potential to improve constraints on fundamental cosmological parameters, ionizing background models, and the reionization history. This study addresses the sensitivity of spectroscopic surveys to 3D correlations in the Lyman-alpha forest. We show that the sensitivity of a survey to this signal can be quantified by just a single number, a noise-weighted number density of sources on the sky. We investigate how the sensitivity of a spectroscopic quasar (or galaxy) survey scales as a function of its depth, area, and redshift. We propose a simple method for weighting sightlines with varying S/N levels to estimate the correlation function, and we show that this estimator generally performs nearly as well as the minimum variance quadratic estimator. In addition, we show that the sensitivity of a quasar survey to the flux correlation function is generally maximized if it observes each field just long enough to achieve S/N ~ 2 in a 1 A pixel on an L_* quasar while acquiring spectra for all quasars with L > L_*: Little is gained by integrating longer on the same targets or by including fainter quasars. We quantify how these considerations relate to constraints on the angular diameter distance, the curvature of space-time, and the reionization history.

Figures (9)

• Figure 1: Solid curves show the effective number density of quasars contributing Ly$\alpha$ forest spectra at redshift $z$ (eqn. \ref{['eqn:neff']}) as a function of the B-band AB magnitude that has $[S/N]_{1A}=1$, $m_{\rm AB}^{1A}$. We have also assumed that $m_{\rm AB}^{1A}$ is equal to the limiting magnitude of the survey. The dashed curves are the actual number of quasars brighter than $m_{\rm AB}$. These curves are calculated for $k_{\parallel} = 0.1~$Mpc$^{-1}$ using the hopkins06 luminosity function and assuming that $S/N \propto\,$flux.
• Figure 2: Impact of varying the limiting magnitude of the survey, $m^{\rm lim}_{\rm AB}$, along with the magnitude at which $[S/N]_{1A} = 1$, $m_{\rm AB}^{1A}$. The solid curves are contours of constant $\bar{n}_{\rm eff}$ evaluated at $k_\parallel = 0.1~$Mpc$^{-1}$ and $z=2.5$, with the labels in units of $10^{-3}~$Mpc$^{-2}$. The corresponding dashed curves are the same but evaluated at $k_\parallel = 0.5~$Mpc$^{-1}$.
• Figure 3: $3$D Ly$\alpha$ forest effective volume divided by survey volume at $|{\boldsymbol{k}}|$, $z=3$, and for $\bar{n}_{\rm eff} = 10^{-1}, ~10^{-2}, ~10^{-3}$, and $10^{-4}~$Mpc$^{-2}$ in order of increasing dash length (decreasing amplitude). The plotted quantity is equal to $(2 \, P_{\rm F, \mu}/\delta P_{\rm F, \mu})^{2}$ and is essentially independent of redshift at the plotted scales.
• Figure 4: Top panel: Fractional increase in variance for the $\widehat{P}_{\rm F}^{(0)}$ estimator relative to the minimum variance quadratic estimator, $\widehat{P}_{\rm F}^{\rm QE}$, as a function of $k_{\perp}$. In order of increasing thickness, the curves are for $\bar{n} = 10^{-3}, \; 3 \times 10^{-3}$, and $10^{-2}~$Mpc$^{-2}$. The black solid curves are the noiseless case, and the red dashed curves include noise as described in the text. These calculations assume $k_{\parallel} = 0.1~$Mpc$^{-1}$, are performed in a $200 {\rm \; Mpc} \times200~$Mpc region, and do not include sample variance. Bottom Panel: The blue dot-dashed curves are the same as the red dashed curves in the top panel but with $k_\parallel = k_\perp$.
• Figure 5: $\bar{n}_{\rm eff}$ as a function of $m_{\rm AB}^{1A}$ for a Ly$\alpha$ forest survey that uses either galaxies or quasars. These curves assume $m^{\rm lim}_{\rm AB} = m_{\rm AB}^{1A}$, where $m^{\rm lim}_{\rm AB}$ is the survey limiting magnitude. The solid curves show $\bar{n}_{\rm eff}$ for quasars at $z=2, 3$, and $4$ (from top to bottom). The dashed curves show this quantity for galaxies at $z=2, 3$, and $4$ (from left to right). Here, $\bar{n}_{\rm eff}$ is calculated at $k_{\parallel} = 0.1~$Mpc$^{-1}$, but its $k_{\parallel}$ dependence is extremely weak.