Calibrating the baryon oscillation ruler for matter and halos

We characterize the nonlinear evolution of the baryon acoustic feature as traced by the dark matter and halos, using a combination of perturbation theory and $N$-body simulations. We confirm that the acoustic peak traced by the dark matter is both broadened and shifted as structure forms, and that this shift is well described by second-order perturbation theory. These shifts persist for dark matter halos, and are a simple function of halo bias, with the shift (mostly) increasing with increasing bias. Extending our perturbation theory results to halos with simple two parameter bias models (both in Lagrangian and Eulerian space) quantitatively explains the observed shifts. In particular, we demonstrate that there are additional terms that contribute to the shift that are absent for the matter. At $z=0$ for currently favored cosmologies, the matter shows shifts of $\sim 0.5\%$, $b=1$ halos shift the acoustic scale by $\sim 0.2\%$, while $b=2$ halos shift it by $\sim 0.5\%$; these shifts decrease by the square of the growth factor $D(z)$ at higher redshifts. These results are easily generalized to galaxies within the halo model, where we show that simple galaxy models show marginally larger shifts than the correspondingly biased halos, due to the contribution of satellites in high mass halos. While our focus here is on real space, our results make specific predictions for redshift space. For currently favored cosmological models, we find that the shifts for halos at $z=0$ increase by $\sim 0.3\%$; at high $z$, they increase by $\sim 0.5\%D^2$. Our results demonstrate that these theoretical systematics are smaller than the statistical precision of upcoming surveys, even if one ignored the corrections discussed here. Simple modeling, along the lines discussed here, has the potential to reduce these systematics to below the levels of cosmic variance limited surveys.

Figure 1

The mass functions at $z=0$ for our $c\mathrm{CDM}$ (triangles) and $\Lambda \mathrm{CDM}$ (squares) cosmologies, along with two commonly used fitting functions due to Press & Schechter [28] (dotted, blue) and Sheth & Tormen [25] (dashed, red). Since we are using the sum of the particles in a FoF group for our definition of mass, we do not expect perfect agreement with either fitting function. The simulations for the $\Lambda \mathrm{CDM}$ cosmology are discussed in Sec. 5.

Figure 2

The nonlinear matter power spectrum at $z=0$ (red, upper) and $z=1$ (blue, middle), compared to the linear theory (black, lower) for our $c\mathrm{CDM}$ and $\Lambda \mathrm{CDM}$ models. The $z=1$ power spectra have been scaled by $1/D^2$ to match the other power spectra on large scales and the $\Lambda \mathrm{CDM}$ spectra have been offset (vertically) for clarity. Note the strong damping of the oscillations and the large excess power on small scales in the evolved fields.

Figure 3

The shift in the acoustic scale, $\alpha -1$ vs redshift for the mass (black triangles) and $\nu =1.9$ halos (red squares) using Eq. (3) as the template. Also shown are the best fit power-laws (dashed) and the expectations of perturbation theory [$\alpha -1\propto D^2(z)$] (dotted). The points and curves for the halos are shifted by $\delta z=0.05$ for clarity.

Figure 4

The out-of-phase contribution predicted by perturbation theory well approximates the derivative of the acoustic signal. The points plot $P_{22}$, while the line is a scaled version of $d{P}_L/d\ln k$ with the scaling given in the inset. The smooth components of both curves have been subtracted by fitting a cubic spline to the data. All curves are for $z=0$.

Figure 5

The $\Delta {\chi }^2$ for fits of our $z=0$ mass power spectrum to the functional form of Eq. (3) (dashed red), to the form including $P_{22}$ (solid black) and marginalizing over the amplitude of the $P_{22}$ term (dashed blue). There are 60 degrees of freedom in the fit, and in each case the best fit is a reasonable fit. Note that Eq. (3) gives a biased acoustic scale, including the $P_{22}$ term eliminates the bias, and allowing the amplitude of the $P_{22}$ term to float results in very weak constraints.

Figure 6

The six combinations of $Q_n(k)$, including the exponential damping (at $z=0$) that appear in the nonlinear halo power spectrum for $c\mathrm{CDM}$. For each of these, we subtract a smooth component (by fitting a five point cubic spline), and compare to the “no-wiggle” power spectrum of [4]. The dashed [red] line shows the same procedure applied to the linear power spectrum (divided by 5), while the dotted [blue] line is the $P_{22}$ correction to the nonlinear matter power (upper left panel). In each panel we have scaled $P_{22}$ by the multiplicative factor shown to better match each combination of $Q_n$.

Figure 7

(Upper) The best fit values of ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ for the cross-power spectra of halos and the evolved matter density; triangles, squares, circles, and crosses are the $z=0$, 0.3, 0.7 and 1 data, respectively. Note that increasing ${\mathcal{B}}_1$ corresponds to increasing $\nu$. The solid lines are Sheth-Tormen [25] predictions, while the dashed lines are for Press-Schecter [28]; the thick [red] lines are for Lagrangian theory, while the thin [blue] lines are for Eulerian theory. The dot-dashed lines are based on a quadratic fit to $b_2(b_1)$, again both for Eulerian and Lagrangian bias models. The theoretical scatter in these relations is $\sim 20\%$. (Lower) As above but for the halo-halo autospectrum.

Figure 8

Measurements of $b_2$ vs $b_1$ for our $c\mathrm{CDM}$ simulations. The symbols are as in Fig. 7. The solid [red] and dashed [blue] lines are the peak-background split predictions for the Lagrangian and Eulerian bias, assuming a Sheth-Tormen [25] mass function. The dotted line is a simple quadratic fit to the data.

Figure 9

As in Fig. 4, except for $\Lambda \mathrm{CDM}$.

Figure 10

An example of the out-of-phase components, as in Fig. 6, except for a $\Lambda \mathrm{CDM}$ cosmology. Note the reduced amplitude of the $P_{mn}$ terms, indicating smaller shifts than for our toy cosmology. We also note that the factors that scale $P_{22}$ to the other $Q_n$ combinations appear to be independent of the underlying cosmology.

Figure 11

As in Fig. 8, except for $\Lambda \mathrm{CDM}$. These measurements were based on an additional set of simulations employing ${1200}^3$ particles in cubic boxes of side $1250h^{-1}\mathrm{Mpc}$. The symbols—triangles, squares, circles, stars and crosses correspond to $z=0$, 0.3, 0.5, 0.7 and 1.0, respectively. The dotted line shows the fit from Fig. 8 while the dot-dashed line is the quadratic fit in the inset.

Figure 12

Estimates of the shift (if not corrected) as a function of halo bias and redshift for our $\Lambda \mathrm{CDM}$ cosmology. The width of the shaded regions denotes the error estimated from the difference between the Press-Schechter and Sheth-Tormen forms in the conversion between ${\mathcal{B}}_1$ and ${\mathcal{B}}_2/{\mathcal{B}}_1$ and demonstrates approximately how errors in the theory propagate into a residual shift. We highlight two example populations of halos: a $b(z=0)=1$ sample [blue crosses] to represent emission line galaxies, and a $b(z=0)=1.5$ sample [red circles] to represent an elliptical sample. In both cases, the clustering of the sample is assumed to be constant with redshift. The error bars correspond to a 10% measurement of the bias. Also shown are nominal distance accuracies for Stage III (currently underway) and Stage IV (future) experiments.

Figure 13

The shifts for three example galaxy samples at $z=0$ for $\Lambda \mathrm{CDM}$, as a function of the galaxy bias. The dotted line shows the shifts for halos, while the dashed [red] and solid [blue] lines are for threshold and satellite samples, respectively, (see the text for more details). We assume the Eulerian Sheth-Tormen peaks-bias model for definiteness.