The L_X-M relation of Clusters of Galaxies Page: 4 of 6
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4 E. S. Rykoff ct al.
- a N200 bins
- O L200 bins -
- - 06
M9200 (1014 h-1 M.).
Figure 1. Points show maxBCG-RASS (algebraic) mean Lx and M200 values found by binning on N200 (solid circles) or L200 (empty
circles). The dark gray band represents the +1Q contours on the best fit relation using the N200 bins. The dot-dashed line shows the
S06 relation, while the dashed line shows the S06 fit to the HIFLUGCS clusters from RB02 based on hydrostatic masses. Both relations
have been scaled assuming self-similar evolution (see 3). The error bar in the legend shows the typical lc systematic error in the SDSS
lensing masses, representing an overall shift in normalization that is possible in the maxBCG-RASS relation. The inset plot indicates
the effects on the slope due to covariance between Lx and N200 at fixed mass (r), as described in 4. If Lx and N200 at fixed mass are
correlated (r = 0.7), this will bias the slope steeper, and if they are anti-correlated, this will bias the slope shallower.
The good level of agreement displayed in Fig. 1 among three
independent approaches to determining the Lx M relation
indicates that optical and X-ray selection methods are find-
ing similar populations of massive halos. Furthermore, the
maxBCG RASS result extends to a lower mass scale than
is probed by RB02 and S06. In this section, we point out
effects that could lead to differences among the three mea-
surements. The discussion is aimed at raising issues to be
addressed by more detailed analysis in future work.
Non-zero bias in hydrostatic mass estimates, displayed
in early, low-resolution gas simulations (Evrard 1990), is
a possible source of systematic error that would shift the
RB02 result relative to the true relation. Recent studies us-
ing mock X-ray exposures of numerical simulations predict
a systematic underestimate of binding mass at the level of
-0.25 in lnM (Rasia et al. 2005; Nagai et al. 2007). Cor-
recting the RB02 result by this amount would more closely
align it with the maxBCG RASS relation. Assuming the
latter is an unbiased estimate of the underlying halo rela-
tion, the luminosity offset between the two relations mea-
sures the Malmquist bias arising from the X-ray flux limit
of the HIFLUGCS sample used by RB02. Good agreement
would signal a small bias, meaning small intrinsic scatter
(< 10%) between luminosity and mass. Such small scatter
is considered unlikely by the analysis of S06.
A separate argument can be made based on slope esti-
mates. If hydrostatic mass estimates scale with true mass as
(Mest) o Mi;, then one would expect the RB02 slope to
differ by 1.5e from the maxBCG RASS value. The measured
slope difference, 0.15 +0.15, implies e - 0.1+0.1. Strongly
mass dependent hydrostatic biases are therefore ruled out.
One could shift the RB02 result to higher masses with-
out requiring a major reduction in scatter, QInI-L, as con-
strained in S06. This would require shifting the S06 result
by modifying the assumed cosmology. The luminosity nor-
malization is sensitive to power spectrum normalization,
Lx ~8 0-4, so raising a8 to 0.95 would shift the S06 result to
lower Lx and preserve the current level of Malmquist-bias
for the RB02 result. However, this adjustment would offset
the S06 and maxBCG RASS relations at the 2a level.
Since it is binned by richness, the maxBCG RASS
result is sensitive to covariance among Lx and N200 at
fixed M200. Simulations suggest mild anti-correlation, as
at fixed mass high concentration halos have higher Lx
but fewer galaxies (Wechsler et al. 2006). As an illustra-
tion of the effect of covariance, consider the case of a bi-
variate, log-normal distribution for Lx and N200 with con-
stant covariance. The off-diagonal term can be character-
ized by the correlation coefficient, r - (SLL SnN), where
61rx = (lnX-lnX)/qinx are the normalized deviations from
the mean relation.
Consider a mass function that is a local power-law,
dn/dlnM ~ M- - e", where p ln M. Convolving
this function with the bivariate log-normal, and using Bayes'
theorem, allows one to write the conditional likelihood
P(, p v), where f - ln Lx and v - ln N200. The result is a
bivariate Gaussian with mean mass p(v) - po(v) - I,,
with Po (v) the inverse of the input mean richness mass re-
lation and ao-,, the scatter in mass at fixed richness. The
X-ray luminosity at fixed optical richness is distributed in a
log-normal manner with mean
f(v) = p' (p(v) + ar a al),
2 2 2 2
at,, p( +=o , 2rQIvQ,,e),
where p is the slope of the halo Lx - M200 relation.
When Lx and N200 are independent (r = 0), the mean
luminosity reflects that of the mean mass selected by the
richness cut. When r - 0, the mean is shifted by an amount
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Rykoff, E.S.; Evrard, A.E.; McKay, T.A.; Becker, M.R.; Johnston, D.E.; Koester, B.P. et al. The L_X-M relation of Clusters of Galaxies, report, May 16, 2008; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc899887/m1/4/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.