Sunyaev Zel'dovich effect - redshift independence

[v0lD]

So, I have read that the SZ effect is virtually independent of redshift. I follow the argument that the factor of (1+z)^-4 in the surface brightness cancels the (1+z)^4 factor in the photon energy density at the cluster (three factors from space being smaller in each dimension, one from the photon frequency being higher).

However, surely the clusters at higher redshifts have higher temperatures and electron number densities (probably considerably higher, when z is large enough)? I don't understand why this doesn't correspond to a higher contribution to dT/T_{CMB} from these high-z clusters.

Any help would be appreciated.

Thanks.

 

[Chalnoth]

So, I have read that the SZ effect is virtually independent of redshift. I follow the argument that the factor of (1+z)^-4 in the surface brightness cancels the (1+z)^4 factor in the photon energy density at the cluster (three factors from space being smaller in each dimension, one from the photon frequency being higher).

However, surely the clusters at higher redshifts have higher temperatures and electron number densities (probably considerably higher, when z is large enough)? I don't understand why this doesn't correspond to a higher contribution to dT/T_{CMB} from these high-z clusters.

The temperature of the cluster is almost purely dependent upon the mass of the cluster, as it's the temperature that gas gets heated to when it falls into the potential well. I do expect the number density will have an impact, though again that will depend more upon the mass of the cluster than upon its redshift. So in the end, the SZ effect should be highly independent of redshift, with the only significant dependence being on the mass of the cluster and its baryon fraction.

 

[Tomboland]

Sorry to steal your thread but a question on a similar point. If the SZE is independent of redshift how is the distacnce to each cluster worked out when mapping the sky?

If it is close by standard methods fine, but I thought one of the good thing about the SZE was the ability to detect galaxies which are further away and cannot be detected by other means.

 

[Chalnoth]

Sorry to steal your thread but a question on a similar point. If the SZE is independent of e redshift how is the distacnce to each cluster worked out when mapping the sky?

I believe they use images of the same part of the sky from optical or infrared telescopes.

But so far as I know this study is still in its infancy, so we'll have to see where it goes. Perhaps they'll be able to come up with some interesting and robust methods of determining distance just from the SZE observations through combinations of brightness and angular size of the clusters. The errors will obviously be vastly larger than if you can get an optical or infrared redshift, but they may be good enough for large SZE surveys, provided the statistics of the errors are well-understood.

Edit: Btw, for a recent paper that used the SZE to detect some clusters, take a look at this:
http://arxiv.org/abs/0810.1578

The basic gist of it, as far as your question is concerned, is that it's not so much about detecting galaxy clusters that weren't visible in the optical, but instead ones that weren't identified as being clusters.

 

[matt.o]

I believe they use images of the same part of the sky from optical or infrared telescopes.

No, to measure the distance to a cluster using the SZ effect, you need to combine X-ray and radio observations. The X-ray surface brightness of a cluster's hot gas halo is proportional to the integral of the square of the gas density along the line of sight, whereas the SZ decrement is proportional to the density integrated along the line of sight. Combining these observations it is possible to derive a measure of the depth of the cluster along the line of sight. Comparing depth this to the angular size of the cluster, along with some assumptions about the shape of the gas distribution (spherically symmetric etc.), you can get a measure of the distance to the cluster.

See Mark Birkinshaw's excellent description of this http://nedwww.ipac.caltech.edu/level5/Birkinshaw/Birk11_1.html" for more information.

But so far as I know this study is still in its infancy, so we'll have to see where it goes. Perhaps they'll be able to come up with some interesting and robust methods of determining distance just from the SZE observations through combinations of brightness and angular size of the clusters. The errors will obviously be vastly larger than if you can get an optical or infrared redshift, but they may be good enough for large SZE surveys, provided the statistics of the errors are well-understood.

Edit: Btw, for a recent paper that used the SZE to detect some clusters, take a look at this:
http://arxiv.org/abs/0810.1578

The basic gist of it, as far as your question is concerned, is that it's not so much about detecting galaxy clusters that weren't visible in the optical, but instead ones that weren't identified as being clusters.

While the large scale SZE effect surveys planned may use the method I outlined above for measuring distances and H(z), I think the main purpose of them is to determine cosmological parameters by measuring the mass function of clusters as a function of redshift, since the growth of clusters is sensitive to these parameters. The SZE surveys are good for this because the SZE is independent of redshift, thus you can theoretically detect more clusters at high redshift compared to X-ray, optical etc. cluster searches.

 

[Chalnoth]

No, to measure the distance to a cluster using the SZ effect, you need to combine X-ray and radio observations. The X-ray surface brightness of a cluster's hot gas halo is proportional to the integral of the square of the gas density along the line of sight, whereas the SZ decrement is proportional to the density integrated along the line of sight. Combining these observations it is possible to derive a measure of the depth of the cluster along the line of sight. Comparing depth this to the angular size of the cluster, along with some assumptions about the shape of the gas distribution (spherically symmetric etc.), you can get a measure of the distance to the cluster.

See Mark Birkinshaw's excellent description of this http://nedwww.ipac.caltech.edu/level5/Birkinshaw/Birk11_1.html" for more information.

Well, that would certainly be one way. It's nice in that it's a separate measurement of the same physical stuff, the cluster gas. But I somehow doubt it's going to be nearly as accurate as optical/infra-red redshifts. So I would naively expect that as long as they have access to optical and/or infrared as well as x-ray data, that they would only use the x-ray data to cross-check and make sure the galaxies they think are in the cluster actually are.

For really distant clusters, I have to wonder if the uncertainty in angular size due to beam (aka PSF) considerations will become a significant confounding factor in comparing the x-ray and SZE observations.

 

[matt.o]

Well, that would certainly be one way. It's nice in that it's a separate measurement of the same physical stuff, the cluster gas. But I somehow doubt it's going to be nearly as accurate as optical/infra-red redshifts.

I'm not sure what you mean here. This is a direct measurement of distance to the cluster, independent of redshift. Sure, you need the redshift for measuring H(z), but my understanding of Tomboland's question was that he/she wanted to know how to derive the distance to the cluster using SZE.

In any case, the iron K line emission present in the X-ray data for clusters can generally be used to get an estimate of the redshift, although it is not quite as accurate as optical redshifts, depending on the optical data available.

http://adsabs.harvard.edu/abs/2006ApJ...647...25B" et al. 2006 put fairly tight constraints on the Hubble parameter using this method with only 38 clusters.

So I would naively expect that as long as they have access to optical and/or infrared as well as x-ray data, that they would only use the x-ray data to cross-check and make sure the galaxies they think are in the cluster actually are.

I'm not sure what you are getting at here, the X-ray data is an integral part of this distance measurement. Furthermore, in the method where the cluster mass function is used, the X-ray data can be used for cluster mass determination.

For really distant clusters, I have to wonder if the uncertainty in angular size due to beam (aka PSF) considerations will become a significant confounding factor in comparing the x-ray and SZE observations.

Not a problem for Chandra -- 85% of the light at ~0.3 keV is dispersed within a 1 arcsecond radius for an on-axis source. I'm not sure if spatially resolved studies of the SZ decrement are needed for these measurements. The SZ data is certainly not needed to measure the angular size of the cluster.

Edit to add: I am not sure if Bonamente et al. even measure the angular diameter of the clusters -- they just assume spherical symmetry so the line-of-sight length of the cluster (determined from the X-ray and SZ measurements) is equivalent to the angular size. If you want details, see their paper.

 

[Chalnoth]

I'm not sure what you mean here. This is a direct measurement of distance to the cluster, independent of redshift. Sure, you need the redshift for measuring H(z), but my understanding of Tomboland's question was that he/she wanted to know how to derive the distance to the cluster using SZE.

In any case, the iron K line emission present in the X-ray data for clusters can generally be used to get an estimate of the redshift, although it is not quite as accurate as optical redshifts, depending on the optical data available.

http://adsabs.harvard.edu/abs/2006ApJ...647...25B" et al. 2006 put fairly tight constraints on the Hubble parameter using this method with only 38 clusters.

Well, yes, as a measure of expansion history I'm sure that the X-ray measurements are fantastic. I was just thinking in terms of getting down the distribution of clusters in 3D space, where distance measures and redshift measures are more or less equivalent estimates of the same thing.

I'm not sure what you are getting at here, the X-ray data is an integral part of this distance measurement. Furthermore, in the method where the cluster mass function is used, the X-ray data can be used for cluster mass determination.

I would have thought that if you have redshift information, the brightness and angular size information of the cluster would give you this, though you'd have to make use of an independent dataset for the expansion history, which is certainly not ideal.

Not a problem for Chandra -- 85% of the light at ~0.3 keV is dispersed within a 1 arcsecond radius for an on-axis source. I'm not sure if spatially resolved studies of the SZ decrement are needed for these measurements. The SZ data is certainly not needed to measure the angular size of the cluster.

Ah, right, should have considered that. I do most of my stuff on the CMB end, so am more used to arcminute+ beams.

Edit to add: I am not sure if Bonamente et al. even measure the angular diameter of the clusters -- they just assume spherical symmetry so the line-of-sight length of the cluster (determined from the X-ray and SZ measurements) is equivalent to the angular size. If you want details, see their paper.

Unfortunately that's probably not a very good assumption to make...

 

[matt.o]

Well, yes, as a measure of expansion history I'm sure that the X-ray measurements are fantastic. I was just thinking in terms of getting down the distribution of clusters in 3D space, where distance measures and redshift measures are more or less equivalent estimates of the same thing.

Yes, they are equivalent assuming the cosmology is correct. However, my interpretation of Tomboland's question was that he/she was asking about measures of distance using the SZE, not redshift.

I would have thought that if you have redshift information, the brightness and angular size information of the cluster would give you this, though you'd have to make use of an independent dataset for the expansion history, which is certainly not ideal.

I'm not sure which part of my quote you are answering here. What do you expect you would get given the redshift, brightness and angular size information?

Unfortunately that's probably not a very good assumption to make...

It does introduce large scatter (in combination with other uncertainties like substructure in the gas distribution) into individual distance measurements. However, if you choose your sample carefully, these effects can be minimised and I'd expect they'd average out reasonably well over a large enough sample.

 

[Chalnoth]

Yes, they are equivalent assuming the cosmology is correct. However, my interpretation of Tomboland's question was that he/she was asking about measures of distance using the SZE, not redshift.

Right, and ideally we want as many independent measures of the cosmology as possible, so I guess this sort of measure makes a lot of sense.

I'm not sure which part of my quote you are answering here. What do you expect you would get given the redshift, brightness and angular size information?

I meant the cluster mass, sorry.

It does introduce large scatter (in combination with other uncertainties like substructure in the gas distribution) into individual distance measurements. However, if you choose your sample carefully, these effects can be minimised and I'd expect they'd average out reasonably well over a large enough sample.

Well, the worry would be that you'd have to have an understanding as to the probability distribution of the cluster shapes to estimate the errors from this effect. A large sample will only help here if you can also measure the cluster shapes within that sample, which I suppose should be possible with X-ray data.

 

[matt.o]

I meant the cluster mass, sorry.

Yes, but ideally you need the temperature as a function of radius, too.

Well, the worry would be that you'd have to have an understanding as to the probability distribution of the cluster shapes to estimate the errors from this effect. A large sample will only help here if you can also measure the cluster shapes within that sample, which I suppose should be possible with X-ray data.

True, you can get the projected shapes of the clusters from the X-ray. This is what i meant when I said "if you choose your sample carefully", ie., those with fairly round X-ray morphologies. This becomes increasingly difficult at high redshifts, since clusters have only formed recently. Like I said, you'd have to get the details from the Bonamente et al. 2006 paper to determine the effect of cluster shape distribution. I think they talk about it, but if not there have been studies aimed at determining how these parameters affect the measurements.

As a side note, I have also seen attempts to measure the oblateness/prolateness of clusters using the combination of SZ and X-ray data (http://adsabs.harvard.edu/abs/2006ApJ...645..170S"), although they assume the cosmology is correct to do this.