Cosmology with galaxy clusters: Systematic effects in the halo mass function

We investigate potential systematic effects in constraining the amplitude of primordial fluctuations ${\sigma }_8$ arising from the choice of halo mass function in the likelihood analysis of current and upcoming galaxy cluster surveys. We study the widely used $N$-body simulation fit of Tinker et al. (T08) and, as an alternative, the recently proposed analytical model of excursion set peaks (ESP). We first assess the relative bias between these prescriptions when constraining ${\sigma }_8$ by sampling the ESP mass function to generate mock catalogs and using the T08 fit to analyze them, for various choices of survey selection threshold, mass definition and statistical priors. To assess the level of absolute bias in each prescription, we then repeat the analysis on dark matter halo catalogs in $N$-body simulations designed to mimic the mass distribution in the current data release of Planck Sunyaev-Zel’dovich clusters. This $N$-body analysis shows that using the T08 fit without accounting for the scatter introduced when converting between mass definitions (alternatively, the scatter induced by errors on the parameters of the fit) can systematically overestimate the value of ${\sigma }_8$ by as much as $2\sigma$ for current data, while analyses that account for this scatter should be close to unbiased in ${\sigma }_8$. With an increased number of objects as expected in upcoming data releases, regardless of accounting for scatter, the T08 fit could overestimate the value of ${\sigma }_8$ by $\sim 1.5\sigma$. The ESP mass function leads to systematically more biased but comparable results. A strength of the ESP model is its natural prediction of a weak nonuniversality in the mass function which closely tracks the one measured in simulations and described by the T08 fit. We suggest that it might now be prudent to build new unbiased ESP-based fitting functions for use with the larger data sets of the near future.

Figure 1

Halo mass function at different redshifts. (Top panel): Smooth curves show the mass function from the three prescriptions discussed in the text—ESP [37, solid], the $m_{200\mathrm{b}}$ fit of T08 [22, dashed] and the ST99 fit [20, short dashed]—at three redshifts, from top to bottom $z=0.1$ (red), $z=0.3$ (blue), $z=0.7$ (black). The data points show the average mass function measured in 9 realizations of an $N$-body simulation (see Sec. 4a) with the error bars representing the standard deviation of the 9 runs. The red circles, blue triangles and black squares respectively show the measurements at $z=0.1$, 0.3, 0.7. (Bottom panel): Ratios of the mass function at each redshift with the corresponding ESP mass function, with line styles and color code as in the top panel. Dotted vertical lines in each panel show the limiting mass computed using Eq. (1) for $z=0.1$, 0.3, 0.7 from left to right.

Figure 2

The Fisher information density for constraints on the single parameter ${\sigma }_8$, defined as $F_{ij,{\sigma }_8{\sigma }_8}/(\mathrm{\Delta }\log m\mathrm{\Delta }z)$ for bin widths $\mathrm{\Delta }\log m$ and $\mathrm{\Delta }z$ in the log-mass and redshift directions, respectively, where $F_{ij,{\sigma }_8{\sigma }_8}$ was defined in Eq. (19) and computed using the T08 $m_{200\mathrm{b}}$ mass function for the fiducial cosmology. The left panel shows the density for a Planck-like survey with selection threshold (1) (seen as the sharp lower boundary of the colored region), while the right panel shows the result for an SPT-like survey with limiting mass (2). Note the different redshift ranges in the two panels. The constraints on ${\sigma }_8$ are driven by the approximate redshift range $0.2\lesssim z\lesssim 0.5$ for the Planck-like survey and $0.5\lesssim z\lesssim 1.0$ for the SPT-like survey. For a fixed redshift, the constraints are driven by the smallest mass bins allowed by the selection threshold.

Figure 3

Randomly chosen posterior distributions $p({\sigma }_8)$ from the likelihood analysis of mock catalogs generated using the ESP mass function with the fiducial cosmology (vertical dotted line shows the input value of ${\sigma }_8$) and analyzed using the T08 mass function for the $m_{200\mathrm{b}}$ mass definition, when ${\mathrm{\Omega }}_{\mathrm{m}}$ is fixed at its true value (solid curves) and marginalized over a broad, flat prior (dashed curves). Each curve is approximately symmetric around its peak value, a feature that is shared by all the posterior distributions we analyze (not shown).

Figure 4

Cumulative distributions of the significance $s$ [Eq. (20)] of the bias in ${\sigma }_8$ for the likelihood analysis of $N$ mock catalogs generated using the ESP mass function with the fiducial cosmology and analyzed using the T08 mass function for various choices of survey selection threshold, mass definition and prior on ${\mathrm{\Omega }}_{\mathrm{m}}$. Unless specified, we set $N=300$. The thin dashed black line in each panel shows the Gaussian distribution for comparison. (Left panel): Mass definition $m_{200\mathrm{b}}$, Planck-like selection threshold (1). The bias on ${\sigma }_8$ moves to more negative values as the prior on ${\mathrm{\Omega }}_{\mathrm{m}}$ is relaxed from a fixed value (dotted red) to a tight 2% Gaussian (thick long-dashed brown) to a broad flat distribution (solid red). The result is independent of the fiducial value of ${\sigma }_8$, as shown by the distribution for ${\sigma }_{8,\text{fid}}=0.80$ (thick long-dashed blue). (Right panel): Mass definition $m_{500\mathrm{c}}$, with catalogs generated and analysed using the mass calibration scheme $C_1$ of Appendix pp1-s1. This time, the bias for a Planck-like survey when using a tight prior on ${\mathrm{\Omega }}_{\mathrm{m}}$ (thick long-dashed green) is more negative than for a flat prior (solid blue). As before, the result is independent of the fiducial value of ${\sigma }_8$, as shown by the distribution for ${\sigma }_{8,\text{fid}}=0.80$ (thick long-dashed yellow). The bias for an SPT-like survey [selection threshold (2)] with a flat ${\mathrm{\Omega }}_{\mathrm{m}}$ prior (short-dashed purple) is considerably less significant than the corresponding bias for a Planck-like survey. As a sanity check, the likelihood analysis using the ESP mass function for the Planck-like survey leads to unbiased results (dotted blue, $N=125$).

Figure 5

Joint distributions of the number of mock clusters $n_{\text{clusters}}$ with the standard deviation ${\mathrm{\Sigma }}_{{\sigma }_8}$ of the posterior $p({\sigma }_8)$ (left panel) and with the absolute bias ${\overline{\sigma }}_8-{\sigma }_{8,\text{fid}}$ (right panel) as measured in the Monte Carlo mock catalogs discussed in Fig. 4. Results are displayed for different survey selection thresholds (Planck/SPT), mass definitions ($m_{200\mathrm{b}}/m_{500\mathrm{c}}$) and ${\mathrm{\Omega }}_{\mathrm{m}}$ priors (flat/tight/fixed) as marked. The insets zoom in on the $m_{200\mathrm{b}}$ results using a linear scale on both axes. The dotted curve in the left panel shows the behavior ${⟨n_{\text{clusters}}⟩}^{-1/2}$ normalized to the Planck-$m_{500\mathrm{c}}$ flat-${\mathrm{\Omega }}_{\mathrm{m}}$ cloud. The trends in these panels help understand the behavior of the cumulative distributions of Fig. 4 (see text for a discussion).

Figure 6

The two Planck-like selection thresholds used in the text, Eq. (1) shown by the solid red curve and Eq. (24) shown by the dashed blue curve. The axes are chosen for ease of comparison with Fig. 3 of [12]; the dashed blue curve in particular is in quite good agreement with the green curve for the “shallow zone” in Fig. 3 of [12].

Figure 7

Joint distributions of the number of mock clusters $n_{\text{clusters}}$ with the standard deviation ${\mathrm{\Sigma }}_{{\sigma }_8}$ of the posterior $p({\sigma }_8)$ (left panels) and with the absolute bias ${\overline{\sigma }}_8-{\sigma }_{8,\text{fid}}$ (right panels) as measured in the $N$-body based mock catalogs when using the T08 (top panels) and ESP (bottom panels) mass functions for the likelihood analysis. The format of the left and right panels is similar to the corresponding panels of Fig. 5, and the dotted line in the left panels is the same as that in the left panel of Fig. 5. Results are shown for the various combinations of mass definition and prior on ${\mathrm{\Omega }}_{\mathrm{m}}$ as displayed in Table 2, for the survey selection threshold (1). Note in particular that the values of absolute bias ${\overline{\sigma }}_8-{\sigma }_{8,\text{fid}}$ for the case when the intrinsic scatter of $p(\ln m_{500\mathrm{c}}|\ln m_{200\mathrm{b}})$ is ignored (labeled “no scat”) are systematically more positive than all other cases (see also Table 2).

Figure 8

Mass calibration using scheme $C_1$. (Top panels): Colored region shows the normalized joint distribution of $\log m_{500\mathrm{c}}$ and $\log m_{200\mathrm{b}}$ in our 9 $N$-body realizations at redshift $z=0.1$ (left panel) and $z=0.3$ (right panel). The yellow squares show the median of $\log m_{500\mathrm{c}}$ in bins of $\log m_{200\mathrm{b}}$ (with the bin “center” being defined as the median value of $\log m_{200\mathrm{b}}$ in the bin) and error bars show the standard deviation of $\log m_{500\mathrm{c}}$ in the bin. The solid yellow line shows the analytical calculation for the conditional expectation value $⟨\log m_{500\mathrm{c}}|\log m_{200\mathrm{b}}⟩$ using the procedure described in the text, where we set the mean concentration-mass-redshift relation to be Eq. (a2) with $\alpha =9.0,\beta =0.4$, which we call scheme $C_1$. The diagonal dotted black line shows the one-to-one relation; the mean relation is clearly significantly biased. The horizontal dashed green line shows the limiting value of $m_{500\mathrm{c}}$ at the corresponding redshift as given by the Planck-like selection threshold (1). The vertical dotted black line indicates the value of $m_{200\mathrm{b}}$ at which the horizontal line lies $3{\sigma }_{\log m_{500\mathrm{c}}}$ away from the measured median $\log m_{500\mathrm{c}}$ of the bin, giving a rough indication of the range of $m=m_{200\mathrm{b}}$ values that contribute to the integral over $m$ in Eq. (3). (Bottom panels): Ratio of the measured value of ${10}^{\log m_{500\mathrm{c}}{|}_{\text{median}}}$ to the analytical calculation ${10}^{⟨\log m_{500\mathrm{c}}|\log m_{200\mathrm{b}}⟩}$. The error bars in this case are the standard error computed over all halos that contribute to the bin, and are therefore typically significantly smaller than the corresponding scatter shown in the top panels. Clearly, the scheme $C_1$ is accurate at $\lesssim 5\%$ over the mass range of interest; similar results are true at other redshifts as well.

Figure 9

Mass calibration using scheme $C_2$. Comparing the same data as in Fig. 8 with the analytical calculation using the calibration scheme $C_2$ where we set $\alpha =8.0,\beta =0.375$ in Eq. (a2). The format is identical to Fig. 8 and the results are qualitatively similar, except that the scheme $C_2$ describes the data better at lower masses than $C_1$.

Figure 10

Resolution study. The four panels correspond to the following settings for box size, particle resolution and initial conditions generated using Music. (Top left): One of the primary runs ($r3$) with $L_{\text{box}}=2h^{-1}\mathrm{Gpc}$, $N_{\text{part}}={1024}^3$ and initial conditions (ICs) set at resolution level ${1024}^3$. (Top right): res-test-C with $L_{\text{box}}=1h^{-1}\mathrm{Gpc}$, $N_{\text{part}}={512}^3$ and ICs set at level ${512}^3$. (Bottom left): res-test-$F_1$ with $L_{\text{box}}=1h^{-1}\mathrm{Gpc}$, $N_{\text{part}}={1024}^3$ and ICs set at two levels, ${512}^3$ (with the same seed as res-test-C) and ${1024}^3$. (Bottom right): res-test-$F_2$ with $L_{\text{box}}=1h^{-1}\mathrm{Gpc}$, $N_{\text{part}}={1024}^3$ and ICs set at two levels, ${512}^3$ (with the same seed as res-test-C) and ${1024}^3$ (with seed different from the corresponding number in res-test-$F_1$). The format of the panels is identical to Figs. 8 and 9. The solid yellow line in each main panel is the same, corresponding to the analytical value of $⟨\log m_{500\mathrm{c}}|\log m_{200\mathrm{b}}⟩$ computed using scheme $C_1$, and is used for the ratio comparison in each subpanel. The results are shown at redshift $z=0.3$, with qualitatively similar results holding at all other redshifts.

Figure 11

An illustration of one of the 6 cones that make up each light cone (here, $l_1$). The observer is placed at the center of the box [labeled (0,0,0)]. The opening angle $\theta$ is chosen so that the last bin of interest at $z=1.0$ is entirely contained in the first periodic copy of the box (shown as the shaded cube) along the axis of the cone, and intersects the adjacent faces in this case at $x=r_{\max }=r_{\text{com}}(z_{\max }+\mathrm{\Delta }z/2)$. The yellow shaded rectangles indicate the approximate positions and comoving widths of bins at redshift $z=0.2$, 0.4, 0.9 from left to right. To minimize repetition of halos in the full light cone $l_1$, halos are assigned from realization $R_1$ when the redshift bin is entirely contained in the main box, from $R_2$ for bins straddling the main box and the first copy, and from $R_3$ for bins entirely in the first copy. See text for more details.