Large scale bias and the peak background split

TL;DR

Dark matter haloes are biased tracers of the underlying matter field, and bias depends on halo mass. The authors show that, on large scales, the halo bias can be computed from the unconditional mass function via the peak-background split, making the bias largely determined by the mass-function shape rather than detailed merger histories. They test this with GIF simulations across SCDM, OCDM, and ΛCDM, replacing the standard Press-Schechter mass function with a modified GIF form (a=0.707, p=0.3) to predict bias, expressed as $b_{ m Eul} = 1 + b_{ m Lag}$. The predictions agree well with measurements across mass and redshift, including haloes observed after formation, and show that low-mass haloes can be more biased than PS-based estimates would imply. These results provide a practical link between mass-function modeling and large-scale structure predictions, with implications for galaxy formation and reionization studies and guiding future moving-barrier improvements.

Abstract

Dark matter haloes are biased tracers of the underlying dark matter distribution. We use a simple model to provide a relation between the abundance of dark matter haloes and their spatial distribution on large scales. Our model shows that knowledge of the unconditional mass function alone is sufficient to provide an accurate estimate of the large scale bias factor. Then we use the mass function measured in numerical simulations of SCDM, OCDM and LCDM to compute this bias. Comparison with these simulations shows that this simple way of estimating the bias relation and its evolution is accurate for less massive haloes as well as massive ones. In particular, we show that haloes which are less/more massive than typical M* haloes at the time they form are more/less strongly clustered than formulae based on the standard Press-Schechter mass function predict.

Figures (6)

• Figure 1: The square root $b(k)$ of the ratio of the power spectra of dark matter haloes to that of the dark matter, computed at the time the haloes were identified: $z=0$, $z=2$ and $z=4$ for the top, middle and bottom rows, respectively. The different line types in each panel correspond to haloes of different mass ranges (mass bins increase in size by factors of two). The more massive haloes at a given redshift typically have larger values of $b(k)$.
• Figure 2: The unconditional mass functions from five different output times (filled triangles, open triangles, open squares, filled circles, open circles show results for $z=0$, $z=0.5$, $z=1$, $z=2$ and $z=4$) in the GIF simulations plotted as a function of the scaled variable $\nu$. Dotted curve shows the Press--Schechter prediction, dot-dashed curve shows the mass function associated with the Zeldovich approximation, and solid curve shows our modified fitting function.
• Figure 3: The large scale Eulerian bias relation at $z_{\rm obs}=z_{\rm form}$ between haloes which are identified at $z_{\rm form}$, and the mass. Solid curves show the relation we predict using the GIF mass function, dotted curves show the relations which follow from the Press--Schechter mass function, and dot-dashed curves show the relations associated with the Zeldovich mass function.
• Figure 4: The large scale Eulerian bias relation of the previous figure, rescaled as indicated by the axis labels. Solid curves show the relation we predict using the GIF mass function, dot-dashed curves show the corresponding relation predicted using the Zeldovich mass function, and dotted curves show the relation which follows from the Press--Schechter mass function.
• Figure 5: The square root of the ratio of the halo and matter power spectra for haloes that formed at $z_{\rm form}=4$ (top panels) and $z_{\rm form}=2$ (bottom panels) but were observed at $z_{\rm obs}=0$.