Probing dark energy using baryonic oscillations in the galaxy power spectrum as a cosmological ruler

TL;DR

This paper proposes using baryon acoustic oscillations in the galaxy power spectrum as a standard cosmological ruler to constrain the dark energy equation of state $w$. By measuring the oscillation wave number $k_A = 2\pi / s$, where the sound horizon $s$ is set by early-Universe physics, the method yields a largely model-independent probe of cosmic geometry; distortions from incorrect $w$ translate into shifts in the measured $k_A$, enabling constraints on $w$. Across back-of-the-envelope estimates and mock simulations for surveys at $z\sim1$ and $z\sim3$, the authors show that with realistic volumes and galaxy densities one can attain $\Delta w \approx 0.1$ (and as tight as $\sim0.04$ for favorable high-redshift cases), complementing Type Ia supernovae. The approach offers an independent, geometry-driven check on the late-time acceleration of the Universe and motivates next-generation spectroscopic redshift surveys to map baryonic oscillations on large scales.

Abstract

We show that the baryonic oscillations expected in the galaxy power spectrum may be used as a "standard cosmological ruler'' to facilitate accurate measurement of the cosmological equation of state. Our approach involves a straight-forward measurement of the oscillation "wavelength'' in Fourier space, which is fixed by fundamental linear physics in the early Universe and hence is highly model-independent. We quantify the ability of future large-scale galaxy redshift surveys with mean redshifts z~1 and z~3 to delineate the baryonic peaks in the power spectrum, and derive corresponding constraints on the parameter w describing the equation of state of the dark energy. For example, a survey of three times the Sloan volume at z ~ 1 can produce a measurement with accuracy dw ~ 0.1. We suggest that this method of measuring the dark energy powerfully complements other probes such as Type Ia supernovae, and suffers from a different (and arguably less serious) set of systematic uncertainties.

Figures (8)

• Figure 1: The model power spectrum of Eisenstein & Hu (1998) for cosmological parameters $\Omega_m = 0.3$, $\Omega_b/\Omega_m = 0.15$, $h = 0.7$. For this choice of parameters, the sound horizon $s \approx 105 \, h^{-1}$ Mpc and the wiggle wavescale $k_A \approx 0.0601 \, h$ Mpc$^{-1}$. In the upper panel we divide the model $P(k)$ by the corresponding zero-baryon model, $\Omega_m = 0.3$, $\Omega_b = 0$, $h = 0.7$. Replacing cold dark matter with baryons produces acoustic oscillations and an overall suppression of power. In the lower panel we divide the model $P(k)$ by a smooth fit to the overall shape of the spectrum. The arrows indicate the approximate position of the linear/non-linear transition at different redshifts, estimated in the following way. At $z=0$, we conservatively defined the linear regime by $k < k_{\rm nl} = 0.1 \, h$ Mpc$^{-1}$. From the model $P(k)$ we computed the variance of mass fluctuations $\sigma^2 (R)$ inside a sphere of radius $R$, where $R = \pi/2k_{\rm nl}$ (i.e. half a fluctuation wavelength, or whole wavecrest, coincides with the diameter $2R$). For $k_{\rm nl} = 0.1 \, h$ Mpc$^{-1}$ we found $\sigma^2 (\pi/2k_{\rm nl}) = 0.35$, and then applied this criterion to fix the linear/non-linear transition at other redshifts. At higher $z$, the amplitude of $P(k)$ is reduced by the growth factor, $P(k) \rightarrow P(k) \, D_1(z)^2$. At $z = 1$, for example, $\sigma^2 (\pi/2k) = 0.35$ for $k = 0.19 \, h$ Mpc$^{-1}$$= k_{\rm nl} (z = 1)$. We fixed the overall amplitude of $P(k)$ such that $\sigma^2 (8 \, h^{-1} \, \rm{Mpc}) = 1$.
• Figure 2: Power spectrum measurement for a simulated survey of $N = 2 \times 10^6$ galaxies over a volume $V = 6 \, V_{\rm Sloan}$ at redshift $z \sim 1$. The power spectrum is divided by the zero-baryon model in the upper panel and the smooth reference spectrum in the lower panel. The solid line is the input (unconvolved) model power spectrum (i.e. Figure \ref{['figpkmod']}). The dashed line in the lower panel is the best fit to the data of a simple decaying sinusoidal function (equation \ref{['eqfit']}). Points are plotted at intervals of $\Delta k = 0.01 \, h$ Mpc$^{-1}$, which are approximately uncorrelated (see Meiksin, White & Peacock 1999).
• Figure 3: Fractional accuracy $\Delta k_A/k_A$ with which the wavescale of the baryonic oscillations in $k$-space can be measured at redshift $z \sim 1$, as a function of the number of galaxies $N$ (as a fraction of $10^6$) and the survey volume $V$ (as a fraction of the Sloan volume $V_{\rm Sloan} = 2 \times 10^8 \, h^{-3}$ Mpc$^3$). Contours are shown corresponding to (beginning in the bottom left-hand corner) $\Delta k_A/k_A = 10\%$, $5\%$, $3\%$ and $2\%$. The positions of the 2dF and Sloan surveys are marked on the plot for comparison. This does not accurately represent their precision in measuring $k_A$, because the linear regime at redshift zero extends only to $k \approx 0.1 \, h$ Mpc$^{-1}$ (whereas the simulations assume $k_{\rm nl} = 0.2 \, h$ Mpc$^{-1}$). Also, the Sloan (LRG) sample will possess a higher linear bias factor $b \approx 1.6$ (whereas the simulations assume $b = 1$), which enhances the "effective" value of $N$ by a factor $b^2$. Hence the Sloan LRG sample may measure the position of the acoustic peak at $k \approx 0.075 \, h$ Mpc$^{-1}$ to an accuracy of $\sim 5\%$ (Eisenstein, Hu & Tegmark 1998). The diagonal dashed line indicates the most efficient observational strategies: fewer galaxies will result in shot noise domination, and more galaxies will be "wasted". The dashed line corresponds to a surface density $\approx 2400$ galaxies deg$^{-2}$ in this case.
• Figure 4: The same plot as Figure \ref{['figdeltalamz1']} for simulated surveys at redshift $z \sim 3$. Note the different scale on the $N-$axis. Contours are shown corresponding to (beginning in the bottom left-hand corner) $\Delta k_A/k_A = 10\%$, $5\%$, $3\%$, $2\%$ and $1.5\%$. The diagonal dashed line corresponds to a surface density $\approx 3400$ galaxies deg$^{-2}$.
• Figure 5: The length distortion of a rod as a function of redshift, supposing the true cosmology is $\Omega_m = 0.3$, $w_{true} = -1$ and the assumed cosmology is $\Omega_m ' = 0.3$, $w_{ass} = -0.9$. The dashed and solid lines illustrate respectively the distortion if the rod is oriented radially (i.e. $dx'/dx$) and tangentially (i.e. $x' \, d\theta/x \, d\theta = x'/x$). Thus it can be seen that the primary effect of assuming an incorrect value of $w$ is a re-scaling of distances away from their true values.