4.5 Physical Properties of Clusters and Groups
4.5.2 Compton Profiles of tSZ Halos
4.5 Physical Properties of Clusters and Groups 93
100 in total, in comparison to the wide survey, which has approximately 40 in total. The ex- tended survey, which contains much lower pixel noise, has significantly more cluster detections, over 700 in total, than both the wide and deep surveys. The majority of these clusters, roughly 550, are at masses below2×1014M¯with few objects detected at the highest masses, due to the smaller area of sky covered in comparison to the wide survey.
Fig. 4.9 also shows the completeness of the adiabatic filtered tSZ maps for the different sur- veys. We expect that the adiabatic model map should contain a larger number of detected clus- ters in comparison to the standard model map as there is neither star formation in the adiabatic model, which increases the average entropy in the cluster centre, nor AGN and supernovae feed- back, which pushes gas into the outskirts of the halo. Both these effects, which are present in the standard model map, reduce the central Compton distortion, making these halos harder to detect.
The differences between the number of detections in the adiabatic and standard model maps are in fact quite small for the different surveys, with the completeness of the adiabatic model map not significantly higher than that of the standard model map. This could be due to the fact that the effects of star formation and feedback only slightly change the integrated Compton distortion i.e., the tSZ flux, in clusters as has been demonstrated in Motl et al. (2005); Nagai (2006); Reid and Spergel (2006) using numerical simulations.
As we will see in §4.5.2 the effects of star formation and feedback are more pronounced on the Compton profile, which suggests that this observable is more sensitive to physical cluster pa- rameters, such as the feedback energy and the amount of star formation. For the halos detected in these maps we will use their radial Compton profiles to probe the underlying physical parameters that determine the distribution of hot gas in each cluster. In the next section we describe how the Compton profiles for each of the detected tSZ halos were extracted from the maps.
4.5 Physical Properties of Clusters and Groups 94
Figure 4.10: Abundance of detected tSZ halos in the standard model maps for the wide (dot- dashed), deep (dashed) and extended (solid) surveys.
has been deblended. As demonstrated in §4.4 the deblending process produces a more accurate radial Compton profile particularly in the outer regions of the cluster. We used angular bins corresponding to size of the ACT beam at the lowest frequency, 148 GHz, so that measurements of the Compton profile in each annulus would be uncorrelated by the beam. The error in the Compton profile,σ(r),in each bin was estimated using the bootstrapping technique described in
§4.4, and includes detector noise, residual contamination from Wiener filtering of the CMB, kSZ and dust, as well as fluctuations in the diffuse tSZ background.
In Fig. 4.11 we show the tSZ profile,∆T(r),of a cluster extracted from the filtered adiabatic model tSZ maps as dotted points, with a set of three error bars that correspond to the tSZ profile errors for the wide, deep and extended survey maps. We observe that the profile error bars are the smallest in the extended survey, because of its lower noise level. However, the profile errors are only slightly larger in the deep and wide surveys, which is a consequence of the inverse noise weighting that results from using the Wiener filter. We have checked that if a uniform weighting of the noise was used instead, then the profile error bars in the wide survey are at least
4.5 Physical Properties of Clusters and Groups 95
Figure 4.11: Radial profiles for a typical cluster in the simulation. The solid and dotted lines label the standard and adiabatic models respectively. Furthermore, the green, red and blue error bars specify the profile errors on the extended, deep and wide surveys respectively.
a factor of two larger. Also shown in Fig. 4.11 is the tSZ profile measured from the input map which in the case of the adiabatic model (dotted line) matches closely with the profile measured in the filtered maps. This indicates that any bias effect of the Wiener filter on the Compton profile is small. Finally to compare the differences of the underlying model on the tSZ profile, we extracted the Compton profile of the same cluster from the input standard model map which is shown as the solid line in Fig. 4.11. We note that the difference between the adiabatic and standard model profiles is larger than the error bars on the adiabatic model which indicates that we can distinguish these models for all three surveys.
In order to interpret the radial tSZ profiles in terms of cluster parameters, we have constructed models for the distribution of hot gas in clusters. Moreover, we consider two models studied in Bode et al. (2009), their ‘zero’ model which we refer to as the ‘adiabatic’ model, which contains no star formation or feedback, and the ‘standard’ model, which includes star formation and feedback. The gas models are described in detail in Appendix B but we summarise their main
4.5 Physical Properties of Clusters and Groups 96
features here. The dark matter profile of the halo is assumed to be a generalised, spherically symmetric, NFW profile with concentration,c, and inner slope, α,with its normalisation fixed by requiring that the integrated mass out to the virial radius equals the virial mass, Mvir. The gas is assumed to be in hydrostatic equilibrium with the dark matter halo with polytropic index γ = 1.2, and to have mass which equals the baryonic mass less the stellar mass. Here the baryonic mass and stellar mass are, respectively, the baryon fraction,fb,and stellar fraction,fs, times the virial mass of the halo.
Removing gas that cools to form stars changes the average entropy of the halo. Other pro- cesses that increase the entropy of the halo include feedback from AGN and supernovae which we model as a term proportional to the mass in formed stars, with coefficient²F. The model also allows for a dynamical energy to be transferred from the dark matter halo to the gas, which we assume has a coefficient²D = 0.05and a non-thermal component of the gas pressure with frac- tional contribution,δrel,which we assume to be zero following Bode et al. (2009). These terms are incorporated into the energy conservation equation, together with a term that allows the halo to expand or contract adiabatically. The gas density and pressure are obtained by solving the energy conservation equation together with a conservation equation for surface pressure at the cluster boundary. Once the gas pressure profile is known we integrate it along the line of sight to obtain the projected Compton profile which can be compared to the simulated profile.
To test the accuracy of our models we compared the radial tSZ profile for each halo (with Mvir > 1× 1014M¯) in the standard model simulation, to the radial profile computed from our model for that halo. We avoided the contamination due to the projection of tSZ flux from other halos in the simulation along the line of sight by comparing to catalogue profiles, which are integrated over the cluster path length, rather than profiles extracted from the tSZ map. To calculate the cluster profile we needed to specify the parameters of the cluster model. The cluster mass and redshift were obtained directly from the halo catalogue. In the case of the concentration and the inner slope of the dark matter density profile we fitted to the dark matter mass and density profiles respectively. The feedback coefficient and stellar fraction were specified by the choice of model, in this case the standard model, as described in Appendix B. The baryon
4.5 Physical Properties of Clusters and Groups 97
fraction for each halo was calculated through the use of a scaling relation (see Eq. (4.5.13) for further explanation) – which was formulated by using the catalogue. We found that our models reproduced the simulated profiles with an accuracy better than 20% over a large range of radii,halo mass and redshift. The small discrepancy between the model and simulated profiles is likely due to the slight differences between the underlying parameters of the simulated halo and the model parameters, such as the concentration, inner slope of the dark matter density profile and the stellar and baryon fractions, that were fitted using the information available in the catalogue. Moreover, the simulation used the full three dimensional gravitational potential for the halo, which is in general aspherical, whereas our model uses a spherically symmetric gravitational potential that could in general have small differences with the spherically averaged simulated profile.
The fact that our implementation of the model reproduces the radial tSZ profiles with reason- able accuracy for clusters spanning a wide range in mass and redshift, indicates that the model provides a good description of the underlying physics of the simulated halos, in particular the change in cluster properties due to variations of the underlying cluster parameters. This suggests that our implementation of the cluster model can be used to study the constraints that radial tSZ profiles will set on the underlying cluster parameters, such as the stellar fraction, baryon fraction and feedback energy coefficient. We note that this model has also been used by Vanderlinde et al. (2010) to interpret the results from their tSZ cluster survey and has the advantage, over models like the β model, that have been used because of their simple parametric form for the Compton profile, that it makes a direct connection with the underlying gas physics. We turn to the objective of constraining cluster parameters in the next section.
4.5 Physical Properties of Clusters and Groups 98