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Holes in the Microwave Sky
J.G. Bartlett 1 , A. Blanchard 2 , D. Barbosa 3
Observatoire de Strasbourg 4 , 11, rue de l'Universit'e, 67000 Strasbourg, FRANCE
J. Oukbir 5
Danish Space Research Institute, DSKI Juliane Maries Vej 30, DK­2700, Copenhagen 0, DENMARK
We discuss the implications of two possible, recent Sunyaev--Zel'dovich (SZ) detections for
which no optical or X­ray counterparts have been found. This suggests that the objects
reside at high redshift, which is difficult to reconcile with a critical cosmology. We develop
this argument and find that an open model
with\Omega 0 = 0:2 remains consistent with what
we currently know about the two objects. The reasoning also demonstrates the utility of
SZ cluster searches.
1 Introduction
By holes in the microwave sky, we hope to invoke an image of the Sunyaev--Zel'dovich (SZ) effect
at low frequencies: a region of lower than average sky brightness, or a hole in the cosmic microwave
background (CMB). The image presented by B. Partridge at these proceedings is an excellent example
and, along with a similar detection by the Ryle Telescope, is the motivation for this presentation;
because if the two radio decrements are indeed due to SZ effect, this can have powerful implications
for the value
of\Omega 0 .
In what has yielded the deepest radio map to date, the VLA discovered a radio decrement ---
characteristic of the SZ effect below 1.4 mm --- during an observation of one of the HST Medium
Deep Survey fields (Richards et al. (1996)). The object is just resolved, extending over an area of
about 30 \Theta 60 arcsec 2 . The other object (Jones et al. (1996)) was found by the RYLE Telescope
(RT) during an ongoing program of double quasar observations (Saunders (1997)). They find a radio
decrement covering an area of about 120 \Theta 180 arcsecs 2 . In both cases, subsequent follow­up in the
optical and in the X­ray band has not revealed the supposed clusters (Richards et al. (1996), Jones et
al. (1996), Saunders et al. (1997)). Definite confirmation of the SZ nature of the two decrements will
thus come from efforts to measure the effect at different frequencies, to see if the spectra are consistent
with the SZ effect. If they are indeed clusters, then the lack of optical or X­ray counterparts may
be interpreted as evidence that they lie at large redshift. It is in this way that we may obtain very
strong constraints
on\Omega 0 : The number of massive, high­redshift clusters depends sensitively
on\Omega 0 , so
much so that the observation of even a small number of such clusters can eliminate the critical model
(Oukbir & Blanchard (1992), Barbosa et al. (1996), Eke et al. (1996), Oukbir & Blanchard (1997)).
2 The Mass Function
and\Omega 0
The easiest way to understand this dependence is by considering the mass function, the number density
of collapsed objects as a function of mass and redshift (e.g. Bartlett (1997)). In `standard' models,
based on the growth of initially small density perturbations with Gaussian statistics, this takes the
general form of a power law times a Gaussian (Press & Schechter (1974)):
dn
dlnM
=
r
2
ú
! ae ?
M š(M;z)
`
\Gamma
dlnoe(M;z)
dlnM
'
e \Gammaš 2 =2 : (1)
1 mailto:bartlett@astro.u­strasbg.fr
2 mailto:blanchard@astro.u­strasbg.fr
3 mailto:barbosa@astro.u­strasbg.fr
4 http://astro.u­strasbg.fr/Obs.html
5 mailto:jamila@dsri.dk
1

Table 1: Model Parameters normalised to the local X­ray temperature function (Henry & Arnaud
(1991))
\Omega 0 h oe 8 n
0.2 0.5 1.37 \Gamma1:10
1.0 0.5 0.61 \Gamma1:85
In this equation, ! ae ? is the mean, comoving density of the Universe and oe(M;z) is the power
spectrum as a function of the mass scale, M . The quantity
š(M;z) = ffi c (z)
oe o (M )
D g (z = 0)
D g (z) (2)
is a function of the linear growth factor, D g
(z;\Omega 0 ; \Lambda), which depends
on\Omega 0 and \Lambda, of the critical
density needed for collapse, ffi c , which has only a weak dependence
on\Omega 0 and \Lambda, and of oe o (M ), the
present­day power spectrum, a function of mass only. The appearance of D g in the exponential of the
mass function indicates that
the\Omega 0 dependence can be quite strong; hence, the comment that even a
small number of clusters at large z can severely constrain the density parameter. The key point is that
the shape of the redshift distribution of clusters of a given mass is only determined by the cosmological
parameters (the power spectrum cannot be changed to alter this fact) (Oukbir & Blanchard (1997)).
The problem is that we do not measure mass directly; we need some other, more readily observable
quantity which correlates well with cluster mass. Because we believe that the hot cluster gas is heated
by infall during cluster formation, we expect that the X­ray temperature should represent the depth
of the cluster potential well and, therefore, its mass. This has in fact been well established by various
hydrodynamical simulations (in `standard' scenarios) (Evrard et al. (1996)), which also provide the
exact form of the temperature­mass relation. The X­ray luminosity, on the other hand, is a more
complicated animal, depending not only on the temperature of the gas, but also on its abundance and
spatial distribution. As we discuss below, observing clusters via the Sunyaev--Zel'dovich effect avoids
these problems associated with the X­ray flux, while preserving the simplicity of a straightforward
flux measurement (plus other advantages). This is important because X­ray spectra demand time­
consuming, space­based observations.
For our discussion here, we take a phenomenological point of view and adopt a power­law approx­
imation to the power spectrum: oe o = (1=b)(M=M 8 ) \Gammaff , where b is the bias parameter and M 8 is the
mass contained in a sphere of 8 h \Gamma1 Mpc. We will focus on the comparison of two extreme models,
a critical model and an open model
with\Omega = 0:2 (h = H 0 =100 km=s=Mpc = 1=2 in both cases). The
parameters b and ff for each model are constrained by fitting to the local X­ray temperature function
of galaxy clusters (Henry & Arnaud (1991)), the results of which are given in Table 1 (Oukbir et al.
(1997), Oukbir & Blanchard (1997)).
3 The Sunyaev--Zel'dovich Effect
The Sunyaev--Zel'dovich effect offers unique advantages for finding high redshift clusters and quantify­
ing their abundance. The surface brightness of a cluster relative to the unperturbed CMB is expressed
as a product of a spectral function, j š , and the Compton y­parameter, which is an integral of the elec­
tron pressure along the line­of­sight: y /
R
dl n e T . Integrating the surface brightness over solid angle
yields the following functional form for the total flux density of a cluster with angular size `:
S sz / ` 2 n e T l /M gas TD \Gamma2
a (z)
/ f gas M 5=3
tot (1 + z)D \Gamma2
a (z); (3)
where M tot is the total, virial mass of the cluster, f gas is the hot gas mass fraction of clusters and
D a (z) is the angular­size distance. In the last line, we have used the fact that there exists a tight
relation between X­ray temperature and virial mass: T / M 2=3 (1 + z) (Evrard et al. (1996)). Let's
compare this with the corresponding expression for the X­ray flux of a cluster:
f x = ` 2 n 2
e T 1=2 l(1 + z) \Gamma4 / n e M gas T 1=2 D \Gamma2
a (z)(1 + z) \Gamma4
2

Figure 1: Redshift distribution for clusters of given SZ flux density in the two cosmological models
-- critical (solid lines) and open
with\Omega = 0:2 (dashed lines); see Table 1 for the model parameters.
The curves are drawn for S sz = 4 mJy and S sz = 20 mJy, corresponding to the VLA and RT objects,
respectively, when translated to – = 0:75 mm. For clarity, the two curves for the critical model are
not labelled explicitly. We adopt h = 1=2.
3

Figure 2: SZ source counts with observational constraints, as a function of SZ flux density expressed
at – = 0:75 mm. The two hatched boxes show the 95% one­sided confidence limits from the VLA and
the RT; due to the uncertain redshift of the clusters, there is a range of possible total SZ flux density,
which has for a minimum the value observed in each beam and a maximum chosen here to correspond
to z ? 1. From the SuZIE blank fields, one can deduce the 95% upper limit shown as the triangle
pointing downwards (Church et al. (1997)). We also plot the predictions of our fiducial open model
(\Omega = 0:2) for all clusters (dashed line) and for those clusters with z ? 4. The critical model has great
difficulty explaining the observed objects even with a lower redshift cutoff of only z ? 1; the actual
limit from the X­ray data is stronger, but this would fall well off to the lower left of the plot. We
assume h = 1=2.
/ n e f gas M tot T 1=2 D \Gamma2
l
(z); (4)
with D l denoting the luminosity distance. By comparing these two expressions we see that, in contrast
to the SZ flux density, the X­ray flux suffers cosmological surface brightness dimming, represented by
the extra factors of (1 + z) in the denominator of Eq. (4) which convert the angular­size distance to
the luminosity distance. Besides this well­known difference, which tells us that the SZ effect is the
more efficient way to find high­redshift clusters, we note that the X­ray emission depends on the gas
density in addition to the hot gas mass fraction and temperature. This is unfortunate, because it
means that the X­ray flux from a cluster depends on the core radius and profile of the intracluster
medium (ICM) -- two quantities which are poorly, if at all, understood from the theoretical point of
view. The SZ flux density presents the important advantage that it depends only on the total gas
mass and the temperature, and not on the ICM's distribution. It is also true that the temperature
which appears in the expression for the SZ flux density is a simpler quantity than the X­ray measured
temperature: it is the mean, particle­weighted energy of the gas particles instead of, as in the case of
X­rays, the emission­weighted gas temperature. This SZ temperature is a quantity which should be
all the more closely related to the virial mass of a cluster than even the X­ray temperature, and less
affected by any temperature structure in the cluster.
Now the game is clear: with Eq. (3) we may convert the mass function into a distribution of
clusters in SZ flux density and redshift (the quantitative relation for S sz can be found in Barbosa et
al. (1996)). The redshift distribution of clusters and the total source counts are then easily calculable
4

(Korolyov et al. (1986), Markevitch et al. (1994), Bartlett & Silk (1994), Barbosa et al. (1996), Eke
et al. (1996), Colafrancesco et al. (1997)). In Fig.1, we show the redshift distribution for clusters of
two given SZ flux densities and for two representative cosmologies -- a critical model and a model with
\Omega = 0:2 and \Lambda = 0. For this calculation, we have used a constant gas mass fraction f gas = 0:06 h \Gamma3=2
(Evrard (1997)). The two chosen fluxes are our estimates of the flux density of the VLA and RT
objects, when translated to a wavelength of – = 0:75 mm by using the SZ spectral function, j š ; this is
our fiducial working frequency and corresponds to the peak of the SZ distortion.
The key aspect of this figure is that, at a given flux density, there is an enormous difference between
the number of high­redshift clusters in a critical and open universe. It is for this reason that even the
detection of only two SZ decrements warrants the present discussion, because they appear to be at
large redshift. Let us now quantify this by comparing the predicted number counts of clusters with
redshifts greater than some minimum value with the counts implied by the detection of these two
objects. This is done in Fig.2. The observed counts have been estimated using Poisson statistics and
the amount of sky coverage in each case -- ¸ 0:018 deg 2 for the VLA (two fields) and ¸ 0:034 deg 2
for the RT (three fields). These constraints are given as two boxes because there is actually a range
of possible total SZ flux density from each object, due to the unknown redshifts: at low redshift, the
objects would be resolved and their flux density has to be corrected upwards. The minimum flux
density is clearly the value observed, while for the maximum, we give the values for z ¸ 1 assuming
an isothermal fi \Gamma model. The limits on the counts (i.e., in the vertical direction) are generous in that
they represent the 95% one­sided Poisson confidence limits. We also show an upper limit on the counts
obtained by the SuZIE instrument (Church et al. (1997)), which found no objects in a survey area
of ¸ 0:06 deg 2 down to the limiting flux shown; the symbol represents the resulting 95% confidence
upper limit. Predictions for the number of clusters on the sky for the two cosmological models and
with varying minimum redshifts are shown by the labelled curves.
4 Conclusions
The basic result from Fig.2 is clear: if the two radio decrements are indeed due to the thermal SZ
effect in two clusters, then the critical model is in very serious trouble. On the other hand, an open
model is capable of explaining the objects. While our modelling has been rather simple in that we
have used power­law power spectra and assumed that the cluster gas fraction is constant over mass
and epoch (see, e.g. Colafrancesco et al. (1997) for more detailed treatment of cluster evolution), in
the present circumstance any reasonable evolution in the gas mass fraction would lead to a decrease
in the SZ flux density of objects with small mass and/or at large redshift; hence, it would only make
things more difficult for the critical model.
What about other possible caveats? Besides the fact that we still await definite confirmation of the
true nature of the radio decrements (e.g., detections at other frequencies), the most important thing
to be wary of is the possible bias associated with the fact that the RT was pointed at a known double
quasar, not an a priori blank field. This seems less likely to be true of the VLA detection, in as much
as there was no previously known quasar pair in that field (although one was subsequently found). In
any case, we should have a better idea in the near future from other experiments as the amount of
sky covered by SZ searches appears to be rapidly increasing. The present discussion brings to light
the importance of such SZ searches (we note in particular that a ¸ 1 square­degree search with the
BIMA telescope is now feasible [Holzapfel, private communication]). A satellite mission covering the
full sky, such as the Planck Surveyor, will be the culmination of such efforts.
Finally, we remark that the possible existence of clusters at redshifts much greater than unity
should not be seen as exotic; quite the contrary, in open models, they are expected. If they are indeed
out there, they would not have been detectable up until now by either optical or X­ray observations.
One would imagine that they would first be seen by SZ searches, and these are just now beginning to
provide some very interesting and tantalizing hints.
5

Acknowledgements
We thank the organisers for a very interesting and enjoyable meeting. We would also like to thank
B. Partridge for some helpful and pleasant discussions.
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