Calibrating redshift distributions at $z>2$ with Lyman-$α$ forest cross-correlations

TL;DR

This work demonstrates the feasibility of calibrating the high-redshift tail (2<z<3) of photometric galaxy redshift distributions using cross-correlations with the Lyman-α forest. It develops a comprehensive framework that incorporates Lyman-α redshift-space distortions and continuum-fitting distortions, and compares two continuum methods (Picca and LyCAN) within realistic DESI Y5 and LSST Y10 mocks. Baseline results with LyCAN yield a 24σ detection and constrain the mean redshift via a Gaussian Process model to σ_z/(1+z̄) ≈ 0.006, approaching Stage IV requirements, while continuum fitting and contaminants can degrade performance. The study finds that small angular scales maximize SNR, a Δz of 0.1 suffices to map bn(z) in the tail, and SNR scales with the survey area; future improvements include incorporating smaller scales, additional tracers, and bias-mitigation strategies to further tighten redshift calibration for high-redshift cosmology.

Abstract

We explore the feasibility of using Lyman-$α$ (Ly$α$) forests to calibrate the ensemble redshift distribution of the high-redshift tail ($2<z<3$) of photometric galaxies. We use \texttt{CoLoRe} simulations to create mock DESI 5-year Ly$α$ forests and Rubin Observatory LSST 10-year photometric galaxies up to $z=3$, and measure the galaxy redshift distribution via their angular cross-correlations. Due to large redshift-space distortions in the Ly$α$ forest, the conventional $n(z)$ estimator for clustering redshifts does not apply, and we develope a theoretical framework to model the angular cross-correlation directly. Using the simulations, we explore effects of instrumental noise, continuum fitting, and contamination in the Ly$α$ forest, cross-correlation angular scales ($θ$), and redshift bin size ($Δz$) on the signal-to-noise (SNR) of the measurements. We find that continuum fitting methods strongly impact the SNR of the measurements. With our baseline continuum fitting method, \texttt{LyCAN}, at angular scales $θ\sim10$ arcmin and $Δz=0.1$, we measure the cross-correlation signal at $24σ$. If the shape of the redshift distribution and galaxy bias evolution are known well for $z<2$, the cross-correlation can constrain the mean redshift of the galaxy sample to $σ_z/(1+\bar{z}) = 0.006$ at a mean redshift of $\bar{z}=2$. This demonstrates that Ly$α$ cross-correlation is a reliable and promising method to calibrate the high-redshift tails of photometric Stage IV galaxy surveys.

Figures (14)

• Figure 1: Redshift dependent quantities adopted in validation. The left axis shows redshift distributions of the galaxies, $n_g(z)$, and of the Ly$\alpha$ forest using the raw mocks $n_{F, {\rm raw}}^I(z)$, and the LyCAN mocks $n_{F, {\rm LyCAN}}^I(z)$, with $\Delta z=0.1$ averaged over 10 mocks. The shape of $n_g(z)$ is adopted from the DESC SRD for the LSST 10-year source sample in the last tomographic bin. The $n_F^I(z)$ has been scaled by a factor of $0.1$ for visual clarity. The right ordinate shows the tracer biases, $b_{g}(z),b_F(z)$, and RSD parameters, $\beta_{g}(z),\beta_{F}(z)$, for galaxies and Ly$\alpha$ forest, respectively. We have scaled $b_F(z)$ by a factor of $-10$ for better readability.
• Figure 2: Upper panel: An example Ly$\alpha$ forest flux transmission fluctuation field, $\delta_F^q(\lambda)$, between $3997.6 \text{\AA} <\lambda<4612.6\text{\AA}$ in the CoLoRe simulation for a quasar at $z_Q=2.85$. The top panel shows the 'raw' flux fluctuation, which is the ground truth. The range of the wavelength shown corresponds to the minimum and maximum cuts in the quasar rest-frame, and the vertical dashed lines indicate the redshift bin edges for $\Delta z=0.1$. The black dots show the value of the binned fluctuations, $\Delta_F$ in each redshift bin. The figure below shows a zoom-in of the difference between noisy $\delta_F$ and the raw analysis in one of the redshift bins. We over-plot the 'true continuum' case (black dot-dashed line), the picca method without contamination (blue thick line), and the LyCAN method without contamination (orange dashed line). Lower panel: An example of the binned flux transmission field $\Delta_F$ on a cutout of the footprint in one redshift slice $2.4<z<2.5$ for raw, true continuum, Picca, and LyCAN with and without contamination. The $\Delta_F$ values are binned spatially to reduce noise for visual clarity.
• Figure 3: Results of the baseline setup: we use the LyCAN mocks, adopt a redshift bin size of $\Delta z=0.1$, and measure the angular cross-correlation function on the scale $\theta=[10.5, 13.6]$ arcmin. Left panel: the data vector from the stack of 10 CoLoRe mocks (blue data points). The solid error bars correspond to the standard deviation of the 10 mocks, while the faint error bars correspond to the average of the Jackknife covariance. Both of these error bar estimates are quite noisy. The theory prediction is shown as the black solid line. Middle panel: the posterior of the shift parameter $\delta_z$ in the shift mean model using synthetic data vector. The best-fit value and the 68% confidence interval is shown in the box. Right panel: the 1$\sigma$ contour (blue shaded region) on $b(z)n(z)$ from 5000 posterior samples of the Gaussian Process (GP) model. The true $b(z)n(z)$ is shown by the black solid line. The pink shaded regions with cross and dots mark the 68% and 95% prior range, where the GP model is allowed to vary. The constraints on the mean redshift $\delta_z$ is shown in the box.
• Figure 4: Left panel: The Ly$\alpha$-galaxy cross-correlations measured from 10 CoLoRe mocks with different forest implementations: the noiseless truth ('raw', grey squares), the noisy case with known continuum ('true continuum', orange triangles), the noisy case with LyCAN continuum subtraction ('LyCAN (baseline)', blue circles), the noisy case with Picca continuum subtraction ('Picca', red stars), and the contaminated case with LyCAN continuum subtraction ('LyCAN (contaminated)', navy triangles). The theory prediction is shown as the black solid line. For the contaminated case, the data points can be described with the same theory, with an increased effective bias by 25%, shown by the black dashed line. Middle panel: increase in the error bars on the shift mean parameter as more realism is added. Right panel: increase in the constraints on the mean redshift derived from the Gaussian Process (GP) model posterior.
• Figure 5: Signal-to-noise ratio (SNR) of the Ly$\alpha$-galaxy angular cross-correlation for the LyCAN continuum fitting mocks, as a function of redshift and angular scale. The shaded region marks the minimum scales we do not use, below which the simulations deviate significantly from the theory due to lack of power at small scales. The three dashed lines show constant comoving scales of $15,30,45h^{-1}$Mpc for reference.