Recovery of the Power Spectrum of Mass Fluctuations from Observations of the Lyman-alpha Forest

TL;DR

The authors present a framework to recover the linear mass power spectrum $P(k)$ from Ly$\alpha$ forest observations by mapping the flux to a Gaussian density field and inverting the resulting 1-D power to the 3-D spectrum. They then normalize the amplitude with inexpensive pseudo-hydro PM simulations constrained by the observed mean opacity $D_A$, rendering the amplitude largely independent of the detailed cosmology or UV background. Through extensive tests on hydrodynamic simulations, the authors show accurate shape recovery on scales about $1$–$10\,h^{-1}{\rm Mpc}$, even with moderate noise and resolution, and an illustrative application to Q1422+231 yields a CDM-like $P(k)$ with relatively low amplitude, compatible with a low-$\Omega$ CDM model within errors. The method offers a path to measuring the small-scale mass power spectrum at $z\sim 2-4$ from large QSO samples, complementing CMB and galaxy surveys and enabling tests of inflationary scenarios.

Abstract

We present a method to recover the shape and amplitude of the power spectrum of mass fluctuations, P(k), from observations of the high redshift \lya forest. The method is motivated by the physical picture of the \lya forest that has emerged from hydrodynamic cosmological simulations and related semi-analytic models, which predicts a tight correlation between the \lya optical depth and the underlying matter density. We monotonically map the QSO spectrum to a Gaussian density field, measure its 3-d P(k), and normalize by evolving cosmological simulations with this P(k) until they reproduce the observed power spectrum of the QSO flux. Imposing the observed mean \lya opacity as a constraint makes the derived P(k) normalization insensitive to the choice of cosmological parameters, ionizing background spectrum, or reionization history. Thus, in contrast to estimates of P(k) from galaxy clustering, there are no uncertain "bias parameters" in the recovery of the mass power spectrum. We test the full procedure on SPH simulations of 3 cosmological models and show that it recovers their true mass power spectra on comoving scales ~1-10/h Mpc, the upper scale being set by the size of the simulation boxes. The procedure works even for noisy (S/N ~ 10), moderate resolution (~40 km/s pixels) spectra. We present an illustrative application to Q1422+231; the recovered P(k) is consistent with an Ω=1, σ_8=0.5 CDM model. Application to large QSO samples should yield the power spectrum of mass fluctuations on small scales at z ~ 2-4. (Compressed)

Figures (16)

• Figure 1: Recovery of a line-of-sight initial density field from a QSO Ly$\alpha$ spectrum by Gaussianization. (a) An absorption spectrum taken from a TreeSPH simulation of the SCDM model. Simulated photon noise (S/N=50 in the continuum) has been added. (b) The PDF of pixel flux values in a set of 100 simulated spectra from the same model. (c) The PDF of the inferred initial density contrasts $\delta$. The pixels in the simulated spectra are rank ordered according to their flux values, and each pixel is assigned a density contrast such that the original rank order is retained and $P(\delta)$ is Gaussian. The width of the Gaussian, proportional to the amplitude of the density fluctuations, is arbitrary at this point and will be determined later by the normalization process described in § 2.3. (d) The inferred initial density contrast field along the same line of sight as the spectrum in (a).
• Figure 2: Recovery of the power spectrum shape from simulated spectra extracted from the SCDM simulation at $z=3$, with $k$ in comoving $\;h\;{\rm Mpc}^{-1}$. Filled circles show $P(k)$ derived from the Gaussianized flux, while open circles show the flux power spectrum itself. Error bars represent the $1\sigma$ dispersion among five sets of 20 lines of sight through the simulation, each set roughly equal in redshift path length to an observed QSO spectrum. The derived power spectra are normalized on large scales to match the linear theory SCDM mass power spectrum, shown by the solid line. The dashed line shows the linear theory power spectrum multiplied by $\exp(-\frac{1}{2}k^2 r_s^2)$, with $r_s=1.5\;h^{-1}{\rm Mpc}$.
• Figure 3: Recovery of the power spectrum shape for three different CDM models. Points show $P(k)$ derived from Gaussianized simulated spectra, normalized on large scales to match the corresponding linear theory power spectra shown by the solid (SCDM), dashed (CCDM), and dot-dashed (OCDM) curves. See Fig. \ref{['pkscdm']} caption for further details. In all three cases, the shape of the initial power spectrum is recovered quite accurately for $0.5 \leq k \leq 4\;\;h\;{\rm Mpc}^{-1}$.
• Figure 4: The power spectrum of the Gaussianized flux for two non-CDM models. Stars show results for a model with an $n=-1$ power law initial spectrum (the linear theory $P(k)$ is shown as a solid line). Open circles show results from a model where spectra are built from a random superposition of discrete, unclustered lines with Voigt profiles. The linear theory SCDM power spectrum is shown for comparison.
• Figure 5: Dependence of the flux power spectrum $\Delta^{2}_{\rm F}(k)$ on the amplitude of underlying mass fluctuations. Results are shown at $z=3$ for the (a) SCDM, (b) CCDM, and (c) OCDM models. In each panel, points show $\Delta^{2}_{\rm F}(k)$ for 100 spectra from the TreeSPH simulation, with error bars computed from the $1\sigma$ scatter among five groups of 20 spectra each. Lines show $\Delta^{2}_{\rm F}(k)$ from 1000 spectra extracted from PM simulations with the same initial fluctuations as the TreeSPH simulation but different linear theory amplitudes, which are indicated in the legends. In each case, the heavy solid line shows the PM results for the linear theory amplitude used in the corresponding TreeSPH simulation.