Markov Chains and the Metropolis Algorithm

Markov Chain Monte Carlo (MCMC) simulations (Chib & Greenberg 1995; Gamerman 1997; Neal 1993) are used to draw samples from a probability distribution. The statistical properties of the distribution, such as its mean and variance can then be estimated from the sample. Usually, one wishes to sample the posterior distribution of the (cosmological) parameters, _P(p_|d), but the

technique can equally well be used to sample any other probability distribution. MCMCs are especially well-suited for high-dimensional problems, because the computational effort increases only linearly with the number of parameters.

The samples are drawn by running a Markov Chain, which is defined as a sequence of random variables (in our case points in parameter space) chosen by a random process such that a given element of the sequence, pi, depends solely on the previous element, pi−1. The aim is to choose the next point in the chain based on the previous point such that the distribution of the points becomes stationary, with_P(p_|d) being the stationary distribution, in the limit of the number of points going

to infinity.

One possibility of implementing such a process is the Metropolis-Hastings (M-H) algorithm (Metropolis et al. 1953), which we briefly introduce in the following. For a given point pi in the chain, the M-H algorithm draws a point ˜p from a proposal distribution q( ˜p_|pi). The proposed point

2.4 Parameter sampling 47

is accepted, i.e. pi+1 ≡ ˜p, with the transition probability

a(pi, ˜p)=min (_q(p i| ˜p)P( ˜p|d) q( ˜p_|pi)P(pi|d) ,1 ) . (2.31)

In practise, this is implemented by drawing a uniformly distributed random variable u from [0,1], accepting the proposed point if

q(pi| ˜p)P( ˜p|d)

q( ˜p_|pi)P(pi|d)

>u, (2.32)

and rejecting it otherwise. If ˜p is rejected, we retain the old point and set pi+1 ≡ pi. If the proposal distribution is symmetric, q( ˜p_|p_i) = q(p_i| ˜p), the algorithm is called the Metropolis algorithm (Metropolis et al. 1953). The Metropolis algorithm is used in the MCMC driver of CMBEASY

Chapter 3

Optimal methods for detecting the

integrated Sachs-Wolfe e_ffect

Note: Sections 3.2-3.4 and section 3.7 of this chapter, as well as appendix B.1 and A.1, have been published in Frommert et al. (2008). The bulk of section 3.5 and appendix B.2 have been published in Frommert&Enßlin (2009a). Section 3.6 has been added.

3.1

Introduction

As we have seen in section 1.4.3, the integrated Sachs-Wolfe effect (Sachs & Wolfe 1967) is an important probe of the existence and nature of dark energy (see also Crittenden & Turok 1996) and the nature of gravity (see Lue et al. 2004; Zhang 2006b). However, the detection of the ISW signal is a challenging task, for it is much smaller than the primordial temperature fluctuations in the CMB, which originate at the time of last scattering. We can try to detect the ISW effect via its cross-correlation with the large-scale structure (LSS). Such a correlation exists, since the ISW effect is created by the interaction of CMB photons with the gravitational potential of the LSS. The primordial temperature fluctuations of the CMB, on the other hand, should be uncorrelated with the LSS distribution. In recent years, substantial effort has been made to detect the ISW effect via cross-correlation of the CMB temperature fluctuations with LSS surveys, such as optical galaxy and quasar surveys1, radio surveys2, and X-ray surveys3.4

The standard method for detecting the cross-correlation between the LSS and the CMB, which has been used by the studies mentioned above, involves comparing the observed cross-correlation function with its theoretical prediction for a given fiducial cosmological model. The theoretical prediction is by construction an ensemble average over all possible realisations of the universe given the fiducial parameters, i.e. over all possible realisations of the primordial CMB, which originates at the surface of last scattering, and all realisations of the local matter distribution. Assuming ergodicity, this second ensemble average can also be thought of as an average over all possible

1_{Sloan Digital Sky Survey, Adelman-McCarthy et al. (2008), Two-Micron All-Sky Survey, Jarrett et al. (2000)} 2_{NRAO VLA Sky Survey, Condon et al. (1998)}

3_{High Energy Astrophysics Observatory, Boldt (1987)}

4_{Such cross-correlation studies have, for example, been done by Boughn et al. (1998), Boughn & Crittenden (2004),}

Boughn & Crittenden (2005), Afshordi et al. (2004), Rassat et al. (2007), Raccanelli et al. (2008), McEwen et al. (2007), Pietrobon et al. (2006), Fosalba et al. (2003), Fosalba & Gazta˜naga (2004), Vielva et al. (2006), Liu & Zhang (2006), Ho et al. (2008) and Giannantonio et al. (2008), just to name a few of them.

50 Optimal methods for detecting the integrated Sachs-Wolfe effect

positions of the observer in the Universe (’cosmic mean’). The specific realisations of both the LSS and the primordial temperature fluctuations of the CMB in our Universe thus contribute to the error budget of the detection. We estimate the contribution of these two sources of uncertainty to the total variance in the detected signal under the simplifying assumption that there is no shot noise in the galaxy distribution. The contribution of the LSS to the total uncertainty, which we refer to as local

variance, amounts to about 11 per cent in the case of an ideal LSS survey going out to about redshift

2 and covering enough volume to include the large scales relevant for the ISW. We will show that this local variance leads to a biased detection significance in the standard method for ISW detection. In this chapter, we present new methods for the detection of the ISW effect, which reduce both sources of uncertainty mentioned above by working conditional on the LSS distribution and on the measured CMB polarization. The method which only operates conditional on the LSS distribution, without using polarization data, will be referred to as the optimal temperature method. The conditionality on the LSS implies that the signal-to-noise ratio or detection significance in the optimal temperature method depends on the specific realisation of the LSS in our Universe. Note that we use the two expressions signal-to-noise ratio and detection significance as synonyms. On average, the detection significance is about 7 per cent higher than for the standard method, due to the reduction of local variance. Here, we have assumed a perfect galaxy survey covering all of the relevant volume. In addition to reducing local variance, we can reduce the variance coming from the primordial temperature fluctuations of the CMB by inferring information about the latter from CMB polarization data. The resulting method is called the optimal polarization method. Note that, of course, the optimal polarization method uses not only polarization data but also temperature data. The latter reaches a detection significance of up to 8.5, which is about 16 per cent higher than the standard one for shallow LSS surveys such as the SDSS main galaxy sample, and about 23 per cent for a full-sky survey reaching out to a redshift of 2. Again, these estimates hold for ideal (noiseless) data. Unfortunately, for currently available CMB and LSS surveys, the detection significance of our optimal polarization method is not notably above the standard one, which is mainly due to the high contamination of the WMAP polarization data by detector noise and Galactic foregrounds. A very crude estimate for data from the Planck Surveyor mission promises an enhancement of detection significance of at least 10 per cent for the optimal polarization method as compared to the standard method.

Many of the cross-correlation studies mentioned above have attempted to constrain cosmological parameters using a likelihood function for the cosmological parameters p given the observed cross-correlation function between CMB temperature fluctuations and LSS data. Just like the detection significance, these parameter estimates suffer from biasing due to local variance. Furthermore, to our knowledge, there is no straightforward way of combining the likelihood function for the cross-correlation with the likelihoods for CMB and LSS data so far. In this chapter, we derive the correct joint likelihood function _P(T,P, δg|p) for cosmological parameters, given the CMB temperature and polarization maps T and P and the LSS data δ_g, from first principles for the linear LSS formation regime. This joint likelihood consistently includes the coupling between the two data-sets introduced by the ISW effect, which so far has been neglected in analyses deriving cosmological parameter constraints by combining CMB and LSS data (Tegmark et al. 2004; Spergel & et al 2007). For parameter sampling studies using our likelihood, we expect small changes of the dark-energy related parameters with respect to studies neglecting the coupling between the data-sets.