arXiv:astro-ph/0608330v1 15 Aug 2006
“The Perfect Slope”: A new robust low-scatter X-ray mass indicator
for clusters of galaxies
Alexey Vikhlinin, Andrey V. Kravtsov, Daisuke Nagai
Harvard-Smithsonian Center for Astrophysics
The University of Chicago
This presentation is a Moriond version of our recent paper (Kravtsov, Vikhlinin & Nagai1)
where we discussed X-ray proxies for the total cluster mass, including the spectral temperature
(Tx), gas mass measured within r500 (Mg), and the new proxy, Yx, which is a simple product
of Tx and Mg. We use mock Chandra images constructed for a sample of clusters simulated
with high resolution in the concordance ΛCDM cosmology. The simulated clusters exhibit
tight correlations between the considered observables and total mass. The normalizations of
the M500− Tx, Mg− Tx, and M500− Yx relations agree to better than ≈ 10 − 15% with the
current observational measurements of these relations. Our results show that Yx is the best
mass proxy with a remarkably low scatter of only ≈ 5−7% in M500 for a fixed Yx, at both low
and high redshifts and regardless of whether clusters are relaxed or not. In addition, we show
that redshift evolution of the Yx− M500 relation is close to the self-similar prediction, which
makes Yx a very attractive mass indicator for measurements of the cluster mass function from
X-ray selected samples.
The evolution of the cluster abundance is one of the most sensitive probes of cosmology. The
potential and importance of this method have motivated efforts to construct several large surveys
of high-redshift clusters during the next several years. However, in order to realize the full
statistical power of the upcoming cluster surveys, it is paramount that the relation between
cluster mass and observables and any potential biases are well known.
Several cluster observables based on the galaxy velocities, optical light, X-ray observables
such as luminosity, temperature, mass of the intracluster medium (ICM), and Sunyaev-Zel’dovich
flux have been proposed as proxies of the total cluster mass (see a recent review by Voit2). In
this study we focus on the mass indicators derived from cluster X-ray observables.
X-ray luminosity is the most straightforward mass indicator to measure observationally.
However, Lxis also the least accurate (internally) of all proposed X-ray proxies for Mtotwith
a large scatter3,4and deviations of the slope of the Lx− M relation from the self-similar
prediction5. The most common choice of mass proxy in the cluster cosmological studies has
been the X-ray temperature of the intracluster gas6,7,8,9. The scatter in the M −Txrelation is
smaller compared to that in the Lx−M relation (the upper limit from observations is ≈ 15% in
M for fixed T for relaxed clusters10). In general, existence of a tight relation such as M − Tx
indicates that clusters are regular population of objects with their global properties tightly
related to total mass, and scatter caused by secondary effects such as substructure in the ICM,
non-gravitational processes, and mergers11. More recently, gas mass was used as a proxy for
Mtot12,13. The practical advantage of Mgover Txis that it can be measured from the X-ray
imaging alone. Also, Mgcan be expected to be less sensitive to mergers which should translate
into smaller scatter in the Mg− M relation. The caveat is that trend of gas mass with cluster
mass and evolution with redshift are not yet fully understood.
The use of clusters as efficient probes for precision cosmology puts stringent requirements on
observable cluster mass proxies: 1) tight, low-scatter correlation between the proxy and mass,
with the scatter insensitive to mergers etc., and 2) simple power-law relation and evolution which
can be described by a small number of parameters and be as close as possible to the prediction
of the self-similar model. The last point is crucial to ensure that the self-calibration strategies
for analyses of large cluster surveys14,15,16,17,18are successful. This is because self-calibration
is powerful when cluster scaling relations and their evolution have a simple form which can be
parameterized with a small number of parameters.
In general, a mass proxy does not have to be a single cluster property, such as Lx, Txor Mg.
Any physically-motivated combination of these variables that is expected to be tightly related
to cluster mass can be used to construct a valid mass indicator. A hint for a better X-ray mass
proxy is provided by recent studies based on cosmological simulations of cluster formation19,20,
which show that integrated SZ flux, YSZis a good, robust mass indicator with low scatter in
the YSZ−M relation, regardless of the dynamical state of the cluster. In addition, the YSZ−M
relation exhibits a simple, nearly self-similar evolution with redshift21,20. The physical reason
for the robustness of the SZ flux is straightforward: YSZis directly related to the total thermal
energy of the ICM and thus to the depth of the cluster potential well.
Here we show that a similar robust, low-scatter mass indicator can be constructed using X-
ray observables. The indicator, which is simply the product of the X-ray derived gas mass and
average temperature, Yx= MgTx, correlates strongly with cluster mass with only ≈ 5 − 8%
intrinsic scatter. The Yx−Mtotrelation is robust to mergers, in the sense that even for disturbed
unrelaxed systems it gives unbiased estimates of mass with the statistical uncertainty similar to
that for relaxed systems. In addition, we show that evolution of the slope and normalization
of the YX− M relation is nearly self-similar. These properties make YXparticularly useful for
measurements of cluster mass function using X-ray surveys.
2 Mass Proxies
Physical properties of virialized systems, such as clusters, are expected to correlate with their
total mass. For example, in the self-similar model22the cluster gas mass is expected to be
simply proportional to the total mass, M∆c= CMgMg,∆c, where masses are determined within
a radius enclosing a certain overdensity ∆cwith respect to the critical density of the universe
at the epoch of observation, ρcrit(z), and CMgis a constant independent of cluster mass and
redshift. The self-similar relation between cluster mass and temperature is E(z)M∆c= CTT3/2.
Here the function E(z) ≡ H(z)/H0for a flat cosmology with the cosmological constant assumed
throughout this study is given by E(z) =?ΩM(1 + z)3+ ΩΛ
The SZ flux integrated within a certain radius, YSZ, is proportional to the total thermal
energy of the ICM gas and thus to the overall cluster potential, which makes it relatively
insensitive to the details of the ICM physics and merging. YSZis proportional to the ICM mass
and gas mass-weighted mean temperature, YSZ∝ Mg,∆cTm. The self-similar prediction for the
YSZ− M relation is
Cosmological simulations show that YSZ is a good, low-scatter cluster mass proxy and that
YSZ− M relation form and evolution are close to the self-similar prediction21,23,20. Given
the good qualities of YSZas a mass proxy, it is interesting whether a similar indicator can be
Figure 1: Examples of mock Chandra images of our simulated clusters. Small ellipses show substructures detected
by our automated software on the mock 100 ksec images. Large circles show the radii r = r500 and 0.15r500.
constructed from the X-ray observables, which could be used in studies of the X-ray cluster
abundances. The simplest X-ray analog of YSZis
Yx= Mg,∆cTx, (2)
where Mg,∆cis the gas mass derived from the X-ray imaging data (it is measured within a
radius enclosing overdensity ∆c), and Txis the mean X-ray spectral temperature. Txis measured
excluding the central cluster region, which can be achieved with moderate angular resolution
X-ray telescopes (≤ 15′′FWHM). To excise the central regions is desirable because the observed
cluster temperature profiles show a greater degree of similarity outside the core10, and also
because this makes the spectral temperature closer to the gas mass averaged Tm.
3 Mock Chandra Images and Analyses of Simulated Clusters
To test the quality of TX, Mg, and YXas the total mass proxies, we take the following approach.
We use the output of the high-resolution cosmological simulations of clusters in a wide mass
range to predict their X-ray emission maps. These maps are convolved with the response of the
Chandra telescope to create realistic mock “observations” of these clusters. The mock data are
used as an input to the actual X-ray data analysis pipeline and measure the proxies (TX, Mg,
and YX) as they would be derived by observers.
The simulations and analysis procedure is fully described in Kravtsov et al.1The cosmo-
logical simulations accurately follow the cluster growth from inital conditions using Adaptive
Refinement Tree (ART) N-body+gasdynamics code24, a Eulerian code that reaches the high
dynamic range required to resolve cores of mass halos. The peak formal resolution achieved
in these simulations, ≈ 3.66h−1kpc, is sufficient to resolve halos of individual galaxies and to
follow not only dissipationless dynamics of dark matter and gasdynamics of the ICM, but also
star formation, metal enrichment and thermal feedback due to the supernovae type II and type
Ia, self-consistent advection of metals, metallicity dependent radiative cooling and UV heating
due to cosmological ionizing background.
Generation and analysis of the mock X-ray data reproduces all the effects associated with
the Chandra response and reconstruction of the 3D ICM properties from the observed projected
X-ray spectra. The only exception is that we ignore complications present in reduction of the
real Chandra data related to background subtraction and spatial variations of the effective area
(i.e., we assume that accurate corrections for these effects can be applied to the real data and
any associated uncertainties are included in the reported measurement errors).
Figure 2: Relation between the X-ray spectral tem-
perature, Tx, and total mass, M500. Separate symbols
indicate relaxed and unrelaxed clusters, and also z = 0
and z = 0.6 samples. The dashed line shows the power
law relation with the self-similar slope fit to the entire
sample, and the dotted lines indicate 20% scatter.
Figure 3: Correlation between gas mass and total mass
of the clusters. Both masses are measured within r500.
The meaning of the symbols and lines is the same as
in Fig. 2. The dotted lines indicate 15% scatter.
Our simulations include clusters in a wide range of mass and dynamical state. We use
simulation outputs at redshifts z = 0 and z = 0.6 to test the evolution in the mass vs. proxy
relations. Examples of the mock Chandra images are shown in Fig.1.
4 Comparison of Mass Indicators
Figure 2 shows that the slope and evolution of the M500− Txrelationaare quite close to the
self-similar model. There is a ∼ 20% scatter in M500around the mean relation and much of
the scatter is due to unrelaxed clusters. Note also that the normalizations of the M500− Tx
relation for relaxed and unrelaxed systems are somewhat different: unrelaxed clusters have
lower temperatures for a given mass. This may seem counter-intuitive at first, given that one
can expect that shocks can boost the ICM temperature during mergers. However, in practice
the effect of shocks is relatively small11. The main source of the bias is that during advanced
mergers the mass of the system already increased but only a fraction of the kinetic energy of
merging systems is converted into the thermal energy of the ICM25.
The M500− Mgrelation (Fig. 3) has a somewhat smaller scatter (≈ 10 − 12%) around the
best fit power law relation than the M500− Tx, but its slope is significantly different from the
self-similar prediction — we find M500∝ M0.88÷0.92
is due to the trend of gas fraction with cluster mass, fgas≡ Mg/M500∝ M0.1÷0.2
both the simulated clusters in our sample26and for the observed clusters10. The normalization
of the M500− Mgrelation evolves only weakly between z = 0.6 and z = 0 (yet, the evolution is
statistically significant and it reflects slow evolution of the gas fraction with time26).
The M500− Yxrelation (Fig. 4) has the smallest scatter of only ≈ 5 − 7%. Note that this
value of scatter includes clusters at both low and high-redshifts and both relaxed and unrelaxed
systems. In fact, the scatter in M500−Yxfor relaxed and unrelaxed systems is indistinguishable
within the errors. Note also that the figures include points corresponding to the three projections
compared to the expected M500∝ Mg. This
aHereafter, M500 is the total mass within the sphere corresponding to the mean overdensity of 500 relative to
the critical density at the cluster redshift.
Figure 4: Yx− M500 correlation. The meaning of the
symbols and lines is the same as in Fig. 2. The dotted
lines indicate 8% scatter.
Figure 5: Fractional deviations in TX and Mg for fixed
Mtot from their self-similar relations. Solid and open
circles show clusters at z = 0 and z = 0.6, respec-
tively. The deviations for Mg and TX are generally
anti-correlated. A similar anti-correlation exists in the
trend with redshift.
of each cluster. Figure 4 shows that the dispersion in the projected values of Yxfor each given
cluster is very small, which means that Yxis not very sensitive to the asphericity of clusters.
Remarkably, the scatter of the M500−Yxrelation, which involves direct X-ray observables, is as
small as that in the M500− YSZrelation (≈ 7% for our sample).
The comparison of the mass proxies, clearly shows that Yx, the product of gas mass and X-ray
spectral temperature, is more robust and self-similar mass indicator than either of these X-ray
observables. Why is the product better than its parts? The answer is obvious from Figure 5
where we show the residuals of temperature and gas mass from their respective relations with
total mass. Figure 5 shows that the clusters with temperatures lower than the mean temperature
for a given total mass tend to have gas mass higher than the mean, and vice versa. Note also
that there is some redshift evolution between z = 0 and z = 0.6 — more clusters have negative
deviations of temperature and positive deviations of measured gas mass at z = 0.6 compared to
z = 0. This redshift evolution is thus in the opposite direction for the gas mass and temperature
deviation. The measured Mgsystematically increases at higher z for a fixed total mass because
high-z clusters are less relaxed on average. For unrelaxed clusters, the ICM density distribution
is non-uniform which results in overestimation of Mg from the X-ray data27. Some of the
decrease of Mg at lower z may be due to continuing cooling of the ICM which decreases the
mass of hot, X-ray emitting gas. The anti-correlation of residuals and opposite evolution with
redshift for gas mass and temperature is the reason why the behavior of their product, on
average, has smaller scatter and is closer to the self-similar expectation in both the slope and
5 Discussion and Conclusions
We presented comparison of several X-ray proxies for the cluster mass — the spectral tempera-
ture Txand gas mass Mgderived from the X-ray data within r500, and the new proxy, Yx, defined
as a simple product of Txand Mg. Analogously to the integrated Sunyaev-Zel’dovich flux, Yxis
related to the total thermal energy of the ICM. To test these mass proxies, we use mock Chandra
“observations” of a sample of clusters simulated in the concordance ΛCDM cosmology.
The main result of this study is that Yx is a robust mass indicator with remarkably low
scatter of only ≈ 5 − 7% in M500 for fixed Yx, regardless of whether the clusters are relaxed
or not. In addition, the redshift evolution of the Yx− M500relation is close to the self-similar
prediction given by equation 1, which makes this indicator a very attractive observable for
studies of cluster mass function with the X-ray selected samples.
The Tx− M500relation has the largest scatter (≈ 20%), most of which is due to unrelaxed
clusters. The unrelaxed clusters have temperatures biased low for a given mass because a certain
fraction of the kinetic energy of merging systems is still in the form of bulk motions of the ICM.
The Mg− M500relation shows an intermediate level of scatter, ≈ 10 − 12%. This relation does
not appear to be sensitive to mergers. It does, however, exhibit significant deviations from self-
similarity in its slope, which is due to the dependence of gas fraction within r500on the cluster
mass26(a similar dependence exists for the observed clusters10).
Generally, all the observable–mass relations we tested demonstrate a remarkable degree of
regularity of galaxy clusters as a population. Tx, Mg, and Yxall exhibit correlations with M500
which are close to the expectation of the self-similar model, both in their slope and evolution
with time, within the uncertainties provided by our sample. The only exception is the slope of
the Mg− M500relation.
Given that our analysis relies on cosmological simulations, it is reasonable to ask whether
the simulated clusters are realistic. Although simulations certainly do not reproduce all of the
observed properties of clusters, especially in their core regions, the ICM properties outside the
core in simulations and observations agree quite well. We illustrate this in Fig. 6, which shows
that the Mg− Tx relations for simulated and observed clusters10. Clearly, both simulated
and observed clusters exhibit tight correlations between Mgand Txwhich agree remarkably in
their slope (Mg∝ T1.75) and normalization. The normalizations derived from simulated and real
clusters agree to ≈ 10%, while slopes are indistiguishable and both deviate significantly from the
expected self-similar value of 1.5. This is a consequence of significant trends in the gas fraction
with cluster mass, Mg/M500∝ M0.2÷0.25
deviations from the self-similar model also manifest themselves in the absence of any noticeable
evolution with redshiftb. Interestingly, the real clusters show a similarly weak evolution in the
Mg− Txrelation28. Figure 5 shows that the likely explanation is that the clusters at z = 0.6
tend to be colder for the fixed Mtotbut have higher estimated Mgthen their counterparts at
z = 0 because they are less relaxed.
A similar level of agreement between the simulations and latest Chandra measurements ex-
ists also for the total mass vs. temperature relation, M500− Tx. In fact, the normalization for
our simulated sample agrees with the observational results10to ≈ 10%. This is a considerable
improvement over the situation of just several years ago when there was ≈ 30−50% discrepancy
between observational measurements and cosmological simulations29,30. The M − Txnormal-
ization was revised both in simulations and observations due to (1) inclusion of more realistic
physics in cosmological simulations (e.g., radiative cooling and star formation,31,32, (2) im-
proved analyses of observed clusters using more realistic gas density profiles33,10, (3) more
reliable measurements of the cluster temperature profiles34,35,10, and (4) the use of uniform
definition of Txin observations and in simulations analyses36,37,38. The remaining systematic
10% difference observed at present is likely caused by non-thermal pressure support from bulk
gas motions39,40,41, which is unaccounted for by the X-ray hydrostatic mass estimates. In
Figure 7 we compare the Yx− M500 relation for the simulated clusters and for the Chandra
sample10. The observed clusters show a tight correlation with the slope close to the self-similar
value. There is ≈ 15% difference in normalization, likely explained also by neglecting the tur-
bulent pressure support in the Chandra hydrostatic mass estimates. The excellent agreement of
for both simulated26and observed clusters10. The
bNote that Mg in Fig.6 is not multiplied by the E(z) factor unlike the total mass in Fig.2 and 4.
Figure 6: Relation between X-ray spectral temperature
and gas mass for the relaxed subsample of simulated
clusters (circles) and for a sample of relaxed Chandra
clusters (stars). Both gas mass and temperature are
the quanities derived from analysis of real and mock X-
ray data. The error bars in the Chandra measurements
are comparable to the symbol size and are not shown
for clarity. The dashed line shows the best fit power
law relation with the slope 1.75.
Figure 7: Yx − M500 relation for the z = 0 sample
of the simulated clusters (circles) and for a sample of
relaxed Chandra clusters. The gas masses for the sim-
ulated clusters are appropriately rescaled (see caption
to Fig.6). The dot-dashed line shows the best fit power
law relation for the simulated clusters with the slope
fixed to the self-similar value of 3/5. The dashed line
shows the same best fit power law, but with the nor-
malization scaled down by 15%.
simulations and observations in terms of the relation between the two X-ray observables used to
compute Yx(Mg−Tx) and a relatively good agreement in the Tx−M500and Yx−M500relations,
gives us confidence that the results presented here are sufficiently realistic.
Our results show that Yx is clearly most robust and most self-similar X-ray cluster mass
indicator. The biases existing in mass estimates based on Mg and Txanti-correlate both for
a given redshift and in terms of evolutionary trends (see Figure 5). This explains why their
product, Yx, is a better mass indicator than Txand Mgindividually. The quality of Yxcompares
well to that for the actual three-dimension integral of the ICM thermal energy (proportional to
YSZ) in terms of its low scatter and self-similarity. Yxmay prove to be an even better mass proxy
than YSZ, given that we use ideal 3D measurement of the latter while reproducing the actual
data analysis for the former. Note also that YSZis more sensitive to the outskirts of clusters,
because it involves gas mass-weighted temperature (as opposed to the spectral temperature more
sensitive to the inner regions), and thus should be more prone to projection effects.
Note that Yxis also an attractive mass proxy from the data analysis point of view. First, it
reduces observational statistical noise by combining the two independently measured quantities,
Mgand Tx, into a single quantity. For example, a 10% measurement uncertainty in Txtranslates
into a ∼ 15% mass uncertainty through the M − Txrelation and only 6% uncertainty through
the Yx− M relation. Yxis also less sensitive to any errors in the absolute calibration of the
X-ray telescope because the biases in the derived TXand Mgtend to anticorrelate.
The robustness and low scatter make Yxan excellent mass indicator for observational mea-
surements of cluster mass function at both z = 0 and higher redshifts. The necessary data —
an X-ray brightness profile and a wide-beam spectrum excluding the core — are easily obtained
with sufficiently deep observations with Chandra, XMM-Newton, and Suzaku (for low-redshift
clusters). The small scatter and simple, nearly self-similar evolution of the Yx−M relation hold
promise for the self-calibration strategies for future large X-ray cluster surveys.
This project was supported by the NSF under grants No. AST-0206216 and AST-0239759,
and by NASA through grants NAG5-13274 & NAG5-9217 and contract NAS8-39073. D.N. is
supported by a Sherman Fairchild postdoctoral fellowship at Caltech. Last but not least, we
would like to thank the organizers of the XLIst Rencontres du Moriond.
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