To obtain constraints on an X-ray luminosity for a gas halo, possibly seen in the SZ effect, we take the measured CMB decrement as the observation of hot cluster gas, with a spherical density profile, given by an isothermal halo with .
In an analytic approximation we find the flux limit to be given by
A is a normalisation factor accounting for the structural parameters of the assumed profile. It has the value
for the X-ray flux expected within an aperture radius ( here) for the model. is the temperature dependent cooling function, the core radius, and and are the observed quantities, the central Comptonization value from the measured decrement and the X-ray flux limit. Given the parameter dependence in the above relation, a choice of , and , together with the assumed density profile, is conservative.
In Fig. 3 we show the SZ and x-ray limits on the x-ray luminosity of a gas halo in the energy range (observer) 0.5--2.0 keV.
Figure 3: We show limits on the luminosity of a gas halo for different redshifts. The lower limits are set by the detected CMB decrement assuming the SZ effect. The upper limits at 99.7 % and 95 % confidence for exclusion are set by our X-ray observation derived with a Raymond-Smith code and assuming Galactic absorption, cosmic helium and 10 % cosmic metal abundances. For a high assumed temperature of 10 keV, the redshift has to be greater than 2.8 (95 % CL). The solid vertical line marks the redshift of the quasar pair and the dashed line the redshift of a damped Ly absorption feature in the spectrum of quasar A.
For Fig. 3 we used a Raymond-Smith model below 10 keV and a thermal Bremsstrahlungs-model above to convert the emission integral derived from the SZ effect to a luminosity. We relate, also including Galactic absorption of HI column density and the PSPC instrument effective area and response matrix, our limit on the count rate directly to the luminosity at various redshifts. For the Raymond-Smith model we used cosmic helium abundance and 10 % of the cosmic metal abundance and only cosmic helium abundance for the Bremsstrahlungs-model. Not excluded halos lie at high redshift. For 5 keV gas the redshift limit is 4.2, higher than the redshift of the quasar pair , for a hot 10 keV halo a redshift of 2.8 is still excluded at a 95 % confidence level and the redshift of the quasar pair with at least 85 %.
Comparing to a sample of clusters with measured SZ effect and which were found to have a comparable temperature decrement: A2218, A478, A773, A1656, CL0016+16, A2142 and A2163, we find their X-ray luminosities to be , indicating that our luminosity estimate of is conservative.