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The Sunyaev--Zel'dovich effect and Hubble's constant
K. Grainge
MRAO, The Cavendish Laboratory, Madingley Road, Cambridge, England.
Combining measurements of the thermal Sunyaev--Zel'dovich effect with X­ray data allows
calculation of a direct, physical estimate of Hubble's Constant (H 0 ) without the need for
distance­ladder­type arguments. I discuss the random and systematic errors inherent in
this method of estimating H 0 , and present the preliminary results obtained for A2218 of
H 0 = 38 +17
\Gamma12 km \Gamma1 Mpc \Gamma1 and for A1413 of H 0 = 47 +18
\Gamma12 km \Gamma1 Mpc \Gamma1 . These two clusters of
galaxies are part of an orientation unbiased, X­ray selected sample.
1 Introduction
The Sunyaev--Zel'dovich (S--Z) effect (Sunyaev & Zel'dovich (1980)) is a secondary anisotropy in the
Cosmic Microwave Background (CMB), due to inverse­Compton scattering of CMB photons by the
hot electron atmosphere which exists in the potential well of clusters of galaxies. In the Rayleigh­Jeans
region of the CMB spectrum, this results in a decrement in the CMB. This signal is small, only 1 mK
even for the richest clusters, and is also diffuse.
2 Observations of the S--Z Effect with the Ryle Telescope
The Ryle Telescope (RT) (Jones (1991)) is an eight­element interferometer operating at 15 GHz
situated at Lord's Bridge, just outside Cambridge. For S­Z work we use only five of these 13­m
diameter aerials in a compact array, giving access to baselines as short as 18 m; it is these baselines
which give the required sensitivity to detect the extended, low­surface­brightness decrement. The
longer baselines almost entirely resolve out the S­Z effect and are only sensitive to any point sources in
the field (see Figure 1). We are therefore able to use the long baseline observations to remove the effect
of these confusing sources from our short baseline data. We have now successfully detected the S--Z
effect in twelve clusters (Figure 2)(Jones et al. (1993), Grainge et al. (1993), Saunders (1995), Grainge
et al. (1996)). Where X­ray maps exist, they agree well with our S--Z images; for example Figure 3
shows our results for the cluster A1914. The centre of the S--Z decrement does not coincide with the
peak of the X­ray emission, but it does agree well with the X­ray centroid. The off centre X­ray peak
indicates the presence of an over­dense region of gas which will contribute little to the S--Z signal. It is
the bulk of the gas which determines the shape and position of both the S--Z effect and the extended
X­ray emission.
3 Calculating Hubble's Constant
The temperature decrement of the CMB due to the S--Z effect in the low­frequency limit is given by
`
\DeltaT
T
'
RJ
= \Gamma2k B oe T
m e c 2
Z
n e T e dl; (1)
and so is proportional to the line integral of pressure along the line of sight through the cluster. The
electrons in the hot cluster atmosphere also emit thermal bremsstrahlung radiation, which will have
a broad­band X­ray luminosity LX
LX /
Z
n 2
e T
1
2
e dl: (2)
The cluster temperature can be determined from the X­ray spectrum. If S--Z and X­ray data are
combined, then it is possible to solve for both the electron density and the physical size of the emitting
1

Figure 1: Plot of the predicted S--Z flux that would be detected by the RT against observing baseline
for a rich cluster (n 0 = 7:5 \Theta 10 3 m \Gamma3 , fi = 0:65, ` c = 60 00 , z = 0:171)
gas cloud. By measuring its angular extent, and by assuming that the line of sight depth through the
cluster is equal to the width in the plane of the sky, it is possible to directly calculate the distance to
the cluster and hence estimate H 0 (Silk & White (1978), Birkinshaw (1979), Cavaliere et al. (1979)).
In the simplified case of a uniform, isothermal cube of gas, subtending an angle `, which has an X­ray
surface brightness XSB , and assuming that q 0 = 0:5 then (Jones (1995)):
H 0 = 8 (T 0 kB oe T ) 2
m 2
e c 3 K
`
T e
\DeltaT RJ
' 2
`XSB
i
(1 + z) 3 \Gamma (1 + z) 5=2
j
: (3)
where K is an emissivity constant which depends on the gas temperature, the energy response of the
X­ray telescope (corrected for the redshift of the cluster) and the absorbing column to the cluster. In
practise, when combining X­ray and S--Z data we assume that the cluster atmosphere is in hydrostatic
equilibrium and can be described by an isothermal King model (Cavaliere & Fusco­Femiano (1976)),
ae g (r) = ae g0
/
1+
`
r
r c
' 2
! \Gamma3fi=2
: (4)
4 Errors In Determination of H 0
We now considered sources of error which may arise in our determination of H 0 through inaccuracies
in our model.
4.1 Hot/Cold Gas Halos
We know from X­ray observations that, on large scales, the intra­cluster gas is typically very close
to being isothermal out to radii of 750 kpc (Mushotsky (1996)). However, X­ray observations are
insensitive to the low density gas far from the cluster centre (see equation 2), and so do not give us
information on the temperature distribution of ``halo'' gas surrounding the central region.
2

A611 IPOL 15206.116 MHZ A6110­1.ICLN.1
PLot file version 1 created 04­APR­1997 15:26:36
Peak flux ­6.0032E­04 JY/BEAM
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6.000, 7.000, 8.000, 9.000, 10.00)
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RIGHT ASCENSION (B1950)
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A697 IPOL 15355.000 MHZ A697 20D S5.SUM.1
PLot file version 1 created 20­JAN­1995 18:44:55
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DECLINATION
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RIGHT ASCENSION (B1950)
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PLot file version created 31­JAN­1995 11:30:57
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­1.00, 1.000, 2.000, 3.000, 4.000, 5.000,
6.000, 7.000, 8.000, 9.000, 10.00)
DECLINATION
(B1950)
RIGHT ASCENSION (B1950)
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00
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54
52
50
48
A990 IPOL 15355.000 MHZ 25D S2 SH.ICLN.3
PLot file version created 11­OCT­1995 15:11:18
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DECLINATION
(B1950)
RIGHT ASCENSION (B1950)
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26
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22
20
18
A1413 IPOL 15355.000 MHZ S6 0­1.ICLN.1
PLot file version 1 created 04­APR­1997 18:53:01
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DECLINATION
(B1950)
RIGHT ASCENSION (B1950)
11 53 00 52 45 30
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44
42
40
38
36
A1914 IPOL 15355.000 MHZ A1914 S.HGEOM.2
PLot file version 2 created 27­JUN­1997 10:48:59
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Levs = 1.0000E­04 * ­7.00, ­6.00, ­5.00,
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3.000)
DECLINATION
(B1950)
RIGHT ASCENSION (B1950)
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04
02
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PLot file version created 11­DEC­1995 17:11:57
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Levs = 1.5000E­04 * ­10.0, ­9.00, ­8.00,
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­1.00, 1.000, 2.000, 3.000, 4.000, 5.000,
6.000, 7.000, 8.000, 9.000, 10.00)
DECLINATION
(B1950)
RIGHT ASCENSION (B1950)
14 52 30 15 00 51 45 30 15 00 50 45
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A2111 IPOL 15355.000 MHZ A2111 S S1.APCLN.1
PLot file version created 04­APR­1997 17:07:37
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Levs = 1.0000E­04 ­10.0, ­9.00, ­8.00,
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­1.00, 1.000, 2.000, 3.000, 4.000, 5.000,
6.000, 7.000, 8.000, 9.000, 10.00)
DECLINATION
(J2000)
RIGHT ASCENSION (J2000)
15 40 00 39 45 30 15
34 30
28
26
24
22
20
A2218 IPOL 15360.000 MHZ A2218C RM5.ICLN.2
PLot file version created 07­FEB­1995 13:33:30
Peak flux = ­4.8373E­04 JY/BEAM
Levs = 1.0000E­04 ­10.0, ­9.00, ­8.00,
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­1.00, 1.000, 2.000, 3.000, 4.000, 5.000,
6.000, 7.000, 8.000, 9.000, 10.00)
DECLINATION
(B1950)
RIGHT ASCENSION (B1950)
16 36 45 30 15 00 35 45 30 15 00 34 45
66 24
22
20
18
16
14
0016+16 IPOL 15355.000 MHZ 0016 0­2.ICLN.1
PLot file version created 30­JAN­1995 21:55:34
Peak flux = ­4.8499E­04 JY/BEAM
Levs = 1.0000E­04 * ­10.0, ­9.00, ­8.00,
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­1.00, 1.000, 2.000, 3.000, 4.000, 5.000,
6.000, 7.000, 8.000, 9.000, 10.00)
DECLINATION
(B1950)
RIGHT ASCENSION (B1950)
00 16 05 00 15 55 50
16 12
11
10
09
08
07
1643+465 IPOL 15355.000 MHZ S3 0­1 LTSS.ICLN.1
PLot file version 1 created 05­JAN­1996 14:20:01
Peak flux 8.4487E+05 RATIO
Levs 6.5000E­05 * ­10.0, ­9.00, ­8.00,
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­1.00, 1.000, 2.000, 3.000, 4.000, 5.000,
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DECLINATION
(B1950)
RIGHT ASCENSION (B1950)
16 44 15 00 43 45 30 15
46 36
34
32
30
28
26
Figure 2: S--Z detections with the RT
3

Figure 3: Map of A1914. The colour plot is the ROSAT PSPC image, and the contours show the
Cleaned map of the naturally weighted RT 0--1 k– visibilities, after subtraction of 1 source.
If the temperature and density of the halo change in a manner such that the gas remains in
hydrostatic equilibrium with the central region, then the line of sight pressure integral will be identical
to that predicted by a purely isothermal King model. Thus the S--Z effect will be exactly that predicted
by our simulations, and we will correctly estimate H 0 .
We have performed simulations to quantify the effect of hydrostatic equilibrium not being preserved
in a gas halo. In one extreme example we assume that the gas density is unchanged from that predicted
by the King model (equation 4) but allow the temperature to increase steadily to 3 times its central
value beyond a radius of 750 kpc. Although this increases the temperature decrement which would be
observed by a single dish telescope by 20%, the S--Z flux that we would measure with the RT shortest
baselines would change by only 4%. The reason for this is that the RT is an interferometer and so
resolves out any structure much larger than its synthesised beam, such as the changes in the line of
sight pressure integral due to gas halos.
We therefore conclude that the exact nature of the outer regions of the intracluster gas does not
significantly affect our H 0 determinations.
4.2 Cooling Flows
In many clusters the gas density is sufficiently high at the centre that the radiative cooling time is less
than the age of the cluster (Fabian et al. (1991)). To maintain pressure balance, this cool gas collapses
inwards and becomes denser. These cooling flows have now been detected in most rich clusters through
the greatly increased X­ray emission of the cool, dense, central gas.
The cooling flow radius is typically of the order of 100 kpc; outside this region the gas remains
isothermal. We therefore blank the pixels at the centre of the X­ray map and fit an isothermal model
to only the regions where there has not been any significant cooling. If quasi­hydrostatic equilibrium
is maintained in the central regions where cooling flows are formed, then through similar arguments
to those used in section 4.1, the magnitude of the S­Z effect in such a cluster is identical to that
predicted by a purely isothermal model. Thus we will be able to accurately estimate H 0 in clusters
with quasi­hydrostatic cooling flows. Further, we have performed simulations with worst­case, non­
hydrostatic cooling flow models which indicate that even in these clusters the effect on the expected
flux measured by the RT would be negligible.
4

4.3 Small Scale Clumping
It is possible that although clusters appear to be isothermal on large scales, there may be temperature
structure on small scales, below the resolution of X­ray telescopes; the intracluster mediummay consist
of a mixture of cold, dense and hot, diffuse clumps, still in hydrostatic equilibrium, and with thermal
conduction suppressed by magnetic fields. Modelling such a cluster assuming that it was isothermal
would incorrectly estimate both the gas temperature and the X­ray surface brightness. However,
these two errors tend to cancel each other when calculating H 0 . Using a simple two­phase model of a
clumped cluster where a volume fraction Ÿ has density j and the remaining (1 \Gamma Ÿ) has density 1, we
find that the fractional error in H 0 resulting from modelling the cluster as being isothermal is given
by
\Gamma
Ÿ
\Gamma
j 3=2 \Gamma 1
\Delta
+1
\Delta 1=2
\Gamma
Ÿ
\Gamma
j 1=2 \Gamma 1 \Delta
+1 \Delta 3=2
:
(5)
Thus for a cluster where half the gas volume has double the temperature of the other half (Ÿ =
0:5; j = 2), the error in estimating H 0 is 4%. An extreme case where a tenth of the gas mass occupies
only a hundredth of the volume (Ÿ = 0:01; j = 10) leads to a 11% error; higher levels of clumping
than given in this model could easily be detected from the cluster's X­ray spectrum (Edge, private
communication).
4.4 Cluster Ellipticity
In calculating H 0 we have assumed that the line of sight depth, l ? , through the cluster is equal to
the width in the plane of the sky, l k . If this is not the case then the calculated value of Hubble's
constant, H 0;calc will be related to its real value by
H 0;calc =H 0
l ?
l k
(6)
X­ray maps show that clusters are elliptical in the plane of the sky, with ellipticities of 1:2 : 1 being
common. Therefore to obtain a robust estimate of H 0 we must observe an orientation­unbiased sample
of clusters. This sample can be compiled from a X­ray catalogue by selecting clusters above a certain
luminosity limit (rather than a surface brightness limit). At present we are working on such a sample
derived from the ROSAT All Sky Survey. The true value of H 0 is then the geometric mean of the
individual estimates.
4.5 Cosmological Model
Combining X­ray and S--Z data depends on the cosmological deceleration parameter q 0 as well as
H 0 (equation 3 assumes that q 0 = 0:5). In theory, observing two clusters will yield both of these
parameters. We have estimated the possible error in calculating H 0 from assuming that q 0 = 0:5
by simulating the response of the RT to a rich cluster (n 0 = 10 4 m \Gamma3 ; T e = 6:7 keV; r C = 250 kpc;
fi = 0:65) at redshifts between 0.1 and 10 for q 0 = 0 and 0.5. The results are plotted in Figure 4. It
can be seen that the value of q 0 adopted makes little difference to the predicted flux observed by the
RT, especially between redshifts of 0.1 and 0.5. It is also interesting to note that Figure 4 implies that
if rich clusters exist in the early universe, then we should be able to detect them with the RT out to
redshifts of 10. q 0 will, however, affect our fit to the X­ray emission from the cluster. We calculate
that at z = 0:2 the change in our estimate of H 0 between assuming q 0 = 0 and 0:5 is only 5%. The
error rises to 13% for a cluster at z = 0:5 and to 70% at z = 1:0.
5 Error Balance Sheet and Conclusions
Combining these and some other sources of error together, we obtain the preliminary results for the
clusters A2218 and A1413 shown in Table 1. Both of these clusters are at z ! 0:2, so we neglect the
possible error that could arise from using a value of q 0 different to 0:5.
5

0 1 10
Redshift
300.0
400.0
500.0
600.0
700.0
Flux
Density
at
900
l
(µJy)
0 1 10
Redshift
300.0
400.0
500.0
600.0
700.0
Flux
Density
at
900
l
(µJy)
a) q 0 = 0:5 b) q 0 = 0
Figure 4: Predicted S--Z flux density on RT shortest baseline when observing a rich cluster with
n 0 = 10 4 m \Gamma3 ; T e = 6:7 keV; r C = 250 kpc; fi = 0:65 for different values of the cosmological deceleration
parameter, q 0 .
Table 1: Preliminary values of H 0 for the clusters A2218 and A1413.
A2218 A1413
Random Error combining S--Z and X­ray \Sigma25% \Sigma27%
Primary Flux Calibration \Sigma10% \Sigma10%
Source Subtraction \Sigma10% \Sigma10%
Cluster Halo Emission \Gamma20% ---
Electron Temperature \Sigma12% \Sigma6%
Clumping +10% +10%
Kinetic S--Z Effect \Sigma4% \Sigma4%
Estimated Value of H 0 (with 1oe error estimates) 38 +17
\Gamma12 47 +18
\Gamma12
6

We note that although the gas properties are different in these two clusters, the estimates of H 0
are consistent. These two clusters comprise part of an X­ray luminosity limited sample, which will be
independent of cluster orientation. This sample will allow us to calculate an unbiased estimate of H 0
from a direct physical method, with a robust error budget.
Acknowledgements
This work was carried out in collaboration with M. Jones, G. Pooley, R. Saunders, J. Baker, T. Haynes
and A. Edge.
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