[astro-ph/0305341] Self Calibration in Cluster Studies of Dark Energy: Combining the Cluster Redshift Distribution, the Power Spectrum and Mass Measurements

Authors:  Subhabrata Majumdar (U Illinois), Joseph J. Mohr (U Illinois)
Abstract:  We examine the prospects for measuring the dark energy equation of state parameter w within the context of any uncertain redshift evolution of galaxy cluster structure (building on Majumdar and Mohr, 2003) and show that including the redshift averaged cluster power spectrum, P_cl(k), and direct mass measurements of 100 clusters helps tremendously in reducing cosmological parameter uncertainties. Specifically, we show that when combining the redshift distribution and the power spectrum information for a particular X-ray survey (DUET) and two SZE surveys (SPT & Planck), the constraints on the dark energy equation of state w can be improved by roughly a factor of 4. Because surveys designed to study the redshift distribution of clusters will have all the information necessary to construct P_cl(k), the benefit of adding P_cl(k) in reducing uncertainties comes at no additional observational cost. Combining detailed mass studies of 100 clusters with the redshift distribution improves the parameter uncertainties by a factor of 3-5. The data required for these detailed mass measurements-- assumed to have 1sigma uncertainties of 30-- are accumulating in the the XMM-Newton and Chandra archives. The best constraints are obtained when one combines both the power spectrum constraints and mass measurements with the cluster redshift distribution; when using the survey to extract the parameters and evolution of the mass--observable relations, we estimate the uncertainties on w of ~4% to 6%. These parameter constraints are obtained from self-calibrating cluster surveys alone. In combination with CMB or distance measurements that have different parameter degeneracies, cluster studies of dark energy will provide enhanced constraints and allow for cross--checks of systematics.
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Sarah Bridle
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[astro-ph/0305341] Self Calibration in Cluster Studies of Da

Post by Sarah Bridle » April 26 2005

This paper calculates how uncertainty in the mass-temperature relation (X-ray) or mass-SZ flux relation can be circumvented when using cluster counts for cosmology by adding info from (1) the clustering of clusters and/or (2) additional mass measurements.

This is a very topical and interesting paper because of the current planned cluster surveys, e.g. SPT which they discuss.

* The authors considered uncertainty in the M-T relation normalisation, power law index and a redshift evolution parameter. In practice presumably for a given mass there is also a scatter in e.g. SZ flux due e.g. to different merger states. I would like to know how much the constraints on cosmology are changed if an unknown scatter in the M-T relation is added and marginalised over.

* I suppose this scatter might not be well described by a Gaussian, since there will be outliers, possibly double peaks due to mergers cf relaxed? I guess more parameters could be used to describe this (any ideas how many?). Might the scatter be a function of mass (and redshift)? How might this affect the number of additional cluster mass measurements required?

* How likely is it that in practice the M-T relation is not a power law (e.g. I thought there was an issue with an entropy floor in groups?)? How might this be parameterised? How much might this affect the constraints?

Has anyone else looked into any of the above? Did I overlook something in this paper? Does anyone have any intuition/guesses?

Gil Holder
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[astro-ph/0305341] Self Calibration in Cluster Studies of Da

Post by Gil Holder » April 26 2005

The authors considered uncertainty in the M-T relation normalisation, power law index and a redshift evolution parameter. In practice presumably for a given mass there is also a scatter in e.g. SZ flux due e.g. to different merger states. I would like to know how much the constraints on cosmology are changed if an unknown scatter in the M-T relation is added and marginalised over.
I know that Lima & Hu looked at the question of scatter recently (astro-ph/0503363), but I think a lot remains to be done in allowing full flexibility in self-calibrating and in using all of the available information to its full potential. They found that the amount of scatter needs to be known at something like the +-10% level, I recall. That is not a crazy number, since the total SZ flux-mass scatter is expected to be around that level, so it is not too crazy to think we can constrain the scatter to within a factor of two.

Things are phrased in terms of M-T, but the SZ observable is SZ flux within a filter (to get rid of primary CMB, for instance), so there could also be scatter coming from clusters with the same mean thermal energy but different amounts of entropy causing more or less of the cluster to be partially cut out by the filter, and this will be a fairly strong function of z. I doubt that is going to be much more than 10% scatter, but I don't think anyone really knows what typical z=1 clusters will look like in SZ. Simulations say these things are not a problem, but things that mess around with the gas profile are exactly the physics in the simulations that is mainly put in by hand as subgrid. Hopefully the SZA will tell us more about this soon, but I don't know where they (or AMI) are in terms of getting images of high z clusters at intermediate masses.
How likely is it that in practice the M-T relation is not a power law (e.g. I thought there was an issue with an entropy floor in groups?)? How might this be parameterised? How much might this affect the constraints?
This could also use a careful look, but you should be careful reading too much into the discussions about entropy floors in groups. In SZ, everything gets confused around 1e14 h^{-1} solar masses, which is always at least several keV. Allmost all of the plots you see showing clusters doing crazy things at low mass are showing deviant behaviour kicking in at masses that SZ surveys have no hope of picking out. I wouldn't expect really weird deviations from power laws in the higher masses accessible to SZ, but there is no reason to expect a strict power law, either. It would be interesting to see what happens if you allow a gently rolling power law and marginalize over it. In general, I think the z dependence is much more of a concern than the M-T mapping, though. The power law in (1+z) is a concern because high z is where the SZ is most likely to not have good X-ray or weak lensing supplementary info, so it is hard to be sure that the evolution in z is what you think it is.

Note that the independent mass measurements are assumed to be fairly coarse in this paper, but that it is essential that they be unbiased (both overall and as a function of redshift). I'm not convinced that weak lensing observations of clusters can give results that can be compared to the simulations at the level of a few percent in an unbiased way out to z=1. There are questions of whether the baryons start to become a problem in the simulations at that level, and whether the observations will be able to control systematics at that level. I like the idea of entirely internal self-calibration much more than trying to graft surveys together.

The data is coming soon, presumably, so it would be great if we theorists were on top of this.

Jochen Weller
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[astro-ph/0305341] Self Calibration in Cluster Studies of Da

Post by Jochen Weller » April 28 2005

Hi Sarah

we looked at this a bit in our paper. The parameterization originally comes from a Verde et al paper:

astro-ph/0106315 [abs, ps, pdf, other] :
Title: Are Clusters Standard Candles? Galaxy Cluster Scaling Relations With the Sunyaev-Zeldovich Effect
Authors: Licia Verde (Princeton, Rutgers), Zoltan Haiman (Princeton), David N. Spergel (Princeton)
Comments: Submitted to ApJ
Journal-ref: Astrophys.J. 581 (2002) 5-19

I find the errorbars they get on the "cluster parameters" hard to believe but of course have not checked myself. I think using Fisher matrix in this context is dangerous. However I heard a talk by Majumdar (here in Chicago) and they switched to MCMC and find different error estimates on the cluster parameters.

Yours
Jochen

Scott Dodelson
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[astro-ph/0305341] Self Calibration in Cluster Studies of Da

Post by Scott Dodelson » April 29 2005

I was playing around with this several months ago and found an odd thing: as the uncertainty in the mass estimate (e.g., via scatter in the M-T relationship) goes up, the error on cosmological parameters can get smaller! Think about it though: as the mass error goes up, you get many more clusters in your sample (much more leakage in from low mass than out). With more clusters, as long as you assume you know the mass function, your constraints get tighter.

Has anyone else found this?

subha majumdar
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Re: [astro-ph/0305341] Self Calibration in Cluster Studies o

Post by subha majumdar » April 29 2005

Think about it though: as the mass error goes up, you get many more clusters in your sample (much more leakage in from low mass than out). With more clusters, as long as you assume you know the mass function, your constraints get tighter.

Has anyone else found this?
I agree with Scott...there are far more clusters of lower mass that are upscattered to be detected in the survey than clusters of higher mass that are downscattered to be not detected. However, I have not really played with increasing the scatter to see what happens at the end. It is easier to do this in MCMC and I've been playing with it.

Also, to reply to Jochen's comment, I must say that its difficult to compare numbers between two codes unless one really compres apples to apples. The cluster parameter errors in this paper was independently confirmed by Zoltan and his students. Moreover, I had the opportunity to compare some of my numbers with Martin Whites code and YT Lin's code.

On comparing Fisher to MCMC, my answer would be the results are pretty similar for most of the parameters, except a couple (like evolution) which makes fisher estimates pretty useful. Infact for these parameters the fisher errors are more than MCMC . Its not that surprising because the likelihood surface is non-gaussian...see similar conclusion in recent paper by Sehgal, Kosowsky & Holder. The old classic paper by Bond and Efstathiou also discusses this issue of Fisher vs MCMC.


Coming back to Sarah's question on M-T power law: As Gil commented that smaller masses (group scales) would not be detected in these surveys. They would however become important when one looks at the SZ pow spec and there one would need to know the M-T more carefully to see if there is a general rolling. Adding SZ powspec to number counts can help in untangling cluster physics effects from cosmological effects (see paper by Diego & Majumdar (astro-ph/0402449). Btw, recent observations (see Ponman & Sanderson entropy paper) shows conclusively that previously one was thinking of some kind of `entropy floor' only because of less data at low masses. Withe low mass side now being populated , there is really no `entropy floor' but a rather shallower slope in mass-entropy relation spanning over the entire mass range.

The idea of entirely self-calibration is nicer but a direct mass measurment still gives the best handle on M-T evolution.

Tom Crawford
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[astro-ph/0305341] Self Calibration in Cluster Studies of Da

Post by Tom Crawford » May 01 2005

Hi Scott:

I think your result is valid, but only if you know the scatter very well. As Lima & Hu point out, it's the uncertainty on the scatter that kills you. If we make the oversimplification that all we care about is [tex]\delta(N_{cl})/N_{cl}[/tex] (where [tex]N_{cl}[/tex] is the total number of clusters found), then for a steep enough mass function, a perfectly known scatter in the [tex]M-T[/tex] relation gets you more clusters found and beats down the Poisson errors, but any uncertainty in that scatter adds back in a term that may or may not cancel the benefit.

For the toy model of Gaussian scatter in the (linear) [tex]M-T[/tex] relation, then the number of clusters in a single (narrow) temperature bin [tex]T[/tex] to [tex]T+\Delta T[/tex] is

[tex]
N_{cl} = \Delta T \int_0^\infty \frac{dN}{dM} P(T|M) dM
= \Delta T \int_0^\infty \frac{dN}{dM} \; A \exp \left[-\frac{1}{2}\left(\frac{T-f(M)}{\sqrt{2} \sigma_T}\right)^2 \right] dM,
[/tex]

so the total uncertainty on [tex]N_{cl}[/tex] goes like

[tex]\left(\frac{\delta(N_{cl})}{N_{cl}} \right)^2 = \frac{1}{N^2_{cl}} \left[ N_{cl} + \left( \frac{\partial N_{cl}}{\partial \sigma_T} \right)^2 \delta^2_{\sigma_T} \right]
= \frac{1}{N_{cl}} + \left[\frac{\int_0^\infty \frac{dN}{dM} \; \exp \left[-\frac{1}{2}\left(\frac{T-f(M)}{\sqrt{2} \sigma_T}\right)^2 \right] \left(\frac{T-f(M)}{\sigma_T}\right)^2 dM}{\int_0^\infty \frac{dN}{dM} \; \exp \left[-\frac{1}{2}\left(\frac{T-f(M)}{\sqrt{2} \sigma_T}\right)^2 \right] dM} \right]^2 \left(\frac{\delta_{\sigma_T}}{\sigma_T}\right)^2
[/tex]

In this same toy model, if the [tex]T[/tex] bin we're interested in corresponded to [tex]3\sigma[/tex] peaks in a Gaussian mass function, then a 20% scatter in the [tex]M-T[/tex] relation would knock down the error on [tex]N_{cl}[/tex] by ~20%, but then a 10% uncertainty on the scatter would put you back where you started.

Cheers,

Tom

Jochen Weller
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[astro-ph/0305341] Self Calibration in Cluster Studies of Da

Post by Jochen Weller » May 01 2005

Dear Scott

this is exactly what we found as well. It is in the Battye and Weller 2003 paper. But as Tom said if the scatter is too large you wreck a lot.

Yours
Jochen

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