Toward a Measurement of the Cosmological Geometry at z~2: Predicting Lyman-alpha Forest Correlation in Three Dimensions, and the Potential of Future Data Sets

TL;DR

This work proposes and tests a framework to measure cosmological geometry at z~2 via the Alcock–Paczyński test using the three-dimensional Lyα forest flux power spectrum. By employing Hydro-PM simulations to compute P_F(k,μ) and deriving Lyα bias parameters, the author builds a robust analytic fit that captures nonlinear, thermal, and velocity effects, enabling predictions across scales. The study assesses numerical requirements (box size, resolution) and demonstrates that accurate AP measurements require precise modeling of redshift-space anisotropy, with forecasts showing Ω_Λ can be constrained to a few percent with future data, including SDSS and follow-up spectroscopy. The results provide a practical pathway to exploit the Lyα forest as a high-redshift geometric probe and outline the data quality and modeling needs to realize this potential.

Abstract

The correlation between Lyman-alpha absorption in the spectra of quasar pairs can be used to measure the transverse distance scale at z~2, which is sensitive to the cosmological constant (Omega_Lambda) or other forms of vacuum energy. Using Hydro-PM simulations, I compute the three-dimensional power spectrum of the Lyman-alpha forest flux, P_F(k,mu), from which the redshift-space anisotropy of the correlation can be obtained. I find that box size ~40 Mpc/h and resolution ~40 Kpc/h are necessary for convergence of the calculations to <5% on all relevant scales, although somewhat poorer resolution can be used for large scales. I compute directly the linear theory bias parameters of the Lyman-alpha forest, potentially allowing simulation results to be extended to arbitrarily large scales. I investigate the dependence of P_F(k,mu) on the primordial power spectrum, the temperature-density relation of the gas, and the mean flux decrement, finding that the redshift-space anisotropy is relatively insensitive to these parameters. A table of results is provided for different parameter variations. I investigate the constraint that can be obtained on Omega_Lambda using quasars from a large survey. Assuming 13 (theta/1')^2 pairs at separation <theta, and including separations <10', a measurement to <5% can be made if simulations can predict the redshift-space anisotropy with <5% accuracy, or to <10% if the anisotropy must be measured from the data. The Sloan Digital Sky Survey (SDSS) will obtain spectra for a factor ~5 fewer pairs than this, so followup observations of fainter pair candidates will be necessary. I discuss the requirements on spectral resolution and signal-to-noise ratio (SDSS-quality spectra are sufficient).

Figures (14)

• Figure 1: Redshift evolution of $f(z)$, relative to an Einstein-de Sitter model. The dashed lines show flat models (with $\omega=-1$), while the dotted lines show open models.
• Figure 2: solid lines: contours of constant $f(z)$ (separated by 0.05), assuming $\omega=-1$. dotted line: $\Omega_m+\Omega_\Lambda=1$.
• Figure 3: Contours of constant $f(z)$ (separated by 0.05), assuming a flat universe ($\Omega_m+\Omega_\Lambda=1$).
• Figure 4: Ly$\alpha$ forest power vs. mass power and galaxy power. The thin solid and thin dotted lines show predictions for $P_F({\mathbf k})$ along and across the line of sight, respectively, for a $\Lambda$CDM model at $z=2$. The thick solid line shows the linear theory, real space power of the mass fluctuations, while the thick dotted line shows the non-linear mass power. Short, vertical lines indicate the central wavenumbers for several recent determinations of the mass power spectrum using the one-dimensional flux power (the vertical positions of these lines are completely arbitrary). Points with error bars show the $z=0$ galaxy power spectrum in the linear regime, from Hamilton & Tegmark (2000) (arbitrarily rescaled in amplitude).
• Figure 5: Fractional changes in power using different pressure approximation methods. All curves are relative to a simple PM simulation, and in all cases the solid lines show power along the line of sight ($0.75<\mu<1.0$) and the dotted lines show power across the line of sight ($0.0<\mu<0.25$). The thick, black lines show the difference between HPM and PM power. The squares connected by thin lines show Gaussian smoothing, $\exp[-(k r)^2/2]$, applied to the mass distribution, with $r=35 \, h^{-1} \, {\rm kpc}$ ( red with filled squares) and $r=49 \, h^{-1} \, {\rm kpc}$ ( green with open squares).