Separate universe consistency relation and calibration of halo bias

The linear halo bias is the response of the dark matter halo number density to a long-wavelength fluctuation in the dark matter density. Using abundance matching between separate universe simulations which absorb the latter into a change in the background, we test the consistency relation between the change in a one-point function, the halo mass function, and a two-point function, the halo-matter cross-correlation in the long-wavelength limit. We find excellent agreement between the two at the 1%–2% level for average halo biases between $1\lesssim {\overline{b}}_1\lesssim 4$ and no statistically significant deviations at the 4%–5% level out to ${\overline{b}}_1\approx 8$. The halo bias inferred assuming instead a universal mass function is significantly different and inaccurate at the 10% level or more. The separate universe technique provides a way of calibrating the linear halo bias efficiently for even highly biased rare halos in the $\mathrm{\Lambda }$ cold dark matter model. Observational violation of the consistency relation would indicate new physics, e.g. in the dark matter, dark energy, or primordial non-Gaussianity sectors.

Figure 1

Abundance matching relates the number density weighted bias above threshold mass $M_{\mathrm{th}}$ to the shift of that threshold. The halo abundance above $M_{\mathrm{th}}$ grows in proportion to the bias function when increasing ${\delta }_{\mathrm{b}}$, which we can compensate by moving $M_{\mathrm{th}}$ accordingly. This figure graphically illustrates Eq. (14).

Figure 2

Cumulative (thick solid green) and differential (thin solid blue) mass functions at $z=0$ calibrated by penalized-spline smoothing the cumulative number density of all $(500\mathrm{Mpc}/h{)}^3$ fiducial simulations. Shaded regions show the standard deviation of bootstrap resamples. The T08 fitting mass functions [48] (dashed black) are also shown for reference with the lower panel showing the difference for each case.

Figure 3

Threshold mass shift as a response of varying ${\delta }_{\mathrm{b}}$ at fixed cumulative abundance at $z=0$. The solid blue line and shaded region show the smoothed estimate and the bootstrap error.

Figure 4

Average bias for halos with mass $>M$. The solid blue line and shaded area show the SU response bias with bootstrap errors, whereas the dashed red line and shaded area show the same for the clustering bias. The dotted line shows the bias of T10 [22] integrated over the mass function of T08 [48] for comparison.

Figure 5

Average bias for halos in mass bins. Blue $+$ points show the SU response bias with bootstrap errors centered on average masses (19), and red $\times$ points show the same for the clustering bias. The dotted line shows the fitting formula of the clustering bias from T10 [22].

Figure 6

Robustness of the smoothing procedure verified by comparing smoothed abundance estimates from 1000 mocks drawn from the fitting mass function T08 [48] to the function itself (solid). We generate each mock catalog for halos between $1.4\times {10}^{12}{M}_{\odot }$ and ${10}^{16}{M}_{\odot }$, in a volume of $4{\mathrm{Gpc}}^3/h^3$, the same as that of all fiducial $(500\mathrm{Mpc}/h{)}^3$ simulations combined. Lines and shaded regions show the mean and scatter of the estimated cumulative (thick green) and differential (thin blue) mass functions.

Figure 7

Abundance matching robustness and efficiency. Blue $+$ points show the response bias with bootstrap errors centered on average masses [Eq. (19)] by abundance matching, and grey • points show the same by the direct measurement of abundance changes within fixed mass bins. The two methods give consistent results, while the former has much reduced errors by a factor of 3–5. The dotted line shows the fitting formula of the clustering bias from T10 [22].

Figure 8

Dependence of clustering bias calibration on $k_{\max }$, in the $V_{\mathrm{c}}=(500\mathrm{Mpc}/h{)}^3$ simulations (solid, shaded) and $V_{\mathrm{c}}=(1\mathrm{Gpc}/h{)}^3$ (dashed, hatched) at $z=0$. Shown are the mean and bootstrap errors for the 0.2 dex mass bins centered from $7.9\times {10}^{12}{M}_{\odot }$ to $1.2\times {10}^{15}{M}_{\odot }$. Larger $k_{\max }$ gives more modes and thus smaller variance but also introduces bias due to scale dependence approaching the nonlinear scale. We choose to use modes below the dashed line (see the text for discussion of robustness).

Figure 9

Clustering bias robustness to simulation volume $V_{\mathrm{c}}=(500\mathrm{Mpc}/h{)}^3$ (small) and $(1\mathrm{Gpc}/h{)}^3$ (large). Overlapping points show the level of robustness to the 400 particle criteria for the minimum halo mass in the large volume and fluctuations due to the lack of high mass halos in the small volume.

Figure 10

Robustness of the UMF response bias estimator verified using 1000 mock catalogs of the Sheth-Tormen (ST) mass function compared with the analytic ST bias. The dotted-dashed line shows the mean of the estimated ${\overline{b}}_1$ matches the analytic result (dashed) to $\lesssim 1\%$, well within the scatter of the estimated ${\overline{b}}_1$ (shaded region).

Figure 11

Average bias for halos with mass $>M$. the dotted-dashed green line and shaded area show the UMF response bias with bootstrap errors, whereas the dashed red line and shaded area show the same for the clustering bias. The dotted line shows the clustering bias of T10 [22] integrated over the mass function of T08 [48] for comparison.

Figure 12

Average bias for halos in mass bins. Green dots show the UMF response bias with bootstrap errors centered on average masses (19), and red $\times$ points show the same for the clustering bias. The dotted line shows the fitting formula of clustering bias from T10 [22].