Investigations of Faraday Rotation Maps of Extended Radio Sources ...

5.2. MAXIMUM LIKELIHOOD ANALYSIS 101
where the brackets 〈〉 denote the expectation value and, thus, C ij (a p ) describes the
expectation based on the proposed model characterised by a particular set of a p ’s.
Now, the likelihood function L ∆ (a p ) has to be maximised for the parameters a p . Note,
that although the magnetic fields might be non-Gaussian, the RM should be close
to Gaussian due to the central limit theorem. Observationally, RM distributions are
known to be close to Gaussian (e.g. Taylor & Perley 1993; Feretti et al. 1999a,b; Taylor
et al. 2001).
Ideally, the covariance matrix is the sum of a signal and a noise matrix term which
results if the errors are uncorrelated to true values. Writing RM obs = RM true +δRM
results in
C ij (a p ) = 〈RMi
true RMj true 〉 + 〈δRM i δRM j 〉
= C RM (⃗x ⊥i , ⃗x ⊥j ) + 〈δRM i δRM j 〉 (5.3)
where ⃗x ⊥i is the displacement of point i from the z-axis and 〈δRM i δRM j 〉 indicates
the expectation for the uncertainty in the measurement. Unfortunately, while in the
discussion of the power spectrum measurements of CMB experiments the noise term
is extremely carefully studied, for the discussion here this is not the case and goes
beyond the scope of this work. Thus, this term will be neglected throughout the rest of
this chapter. However, Johnson et al. (1995) discuss uncertainties involved in the data
reduction process in order to gain a model for 〈δRM i δRM j 〉.
Since one is interested in the magnetic power spectrum, one has to find an expression
for the covariance matrix C ij (a p ) = C RM (⃗x ⊥i , ⃗x ⊥j ) which can be identified
as the RM autocorrelation 〈RM(⃗x ⊥i ) RM(⃗x ⊥j )〉. This has then to be related to the
magnetic power spectra.
The observable in any Faraday experiment is the rotation measure RM. For a line of
sight parallel to the z-axis and displaced by ⃗x ⊥ from it, the RM arising from polarised
emission passing from the source z s (⃗x ⊥ ) through a magnetised medium to the observer
located at infinity is expressed by
RM(⃗x ⊥ ) = a 0
∫ ∞
z s(⃗x ⊥ )
dz n e (⃗x) B z (⃗x), (5.4)
where a 0 = e 3 /(2πm 2 ec 4 ), ⃗x = (⃗x ⊥ , z), n e (⃗x) is the electron density and B z (⃗x) is the
magnetic field component along the line of sight.
In the following, it is assumed that the magnetic fields in galaxy clusters are
isotropically distributed throughout the Faraday screen. If one samples such a field
distribution over a large enough volume they can be treated as statistically homogeneous
and statistically isotropic. Therefore, any statistical average over a field quantity
will not be influenced by the geometry or the exact location of the volume sampled.
Following Sect. 3.3, one can define the elements of the RM covariance matrix using
the RM autocorrelation function C RM (⃗x ⊥i , ⃗x ⊥j ) = 〈RM(⃗x ⊥i )RM(⃗x ⊥j )〉 and introduce
a window function f(⃗x) which describes the properties of the sampling volume
C RM (⃗x ⊥ , ⃗x ′ ⊥) = ã 0
2
∫ ∞
z s
dz
∫ ∞
z ′ s
dz ′ f(⃗x)f(⃗x ′ )〈B z (⃗x ⊥ , z)B z (⃗x ′ ⊥, z ′ )〉, (5.5)
where ã 0 = a 0 n e0 , the central electron density is n e0 and the window function is
defined by
f(⃗x) = 1 {⃗x⊥ ∈Ω} 1 {z≥zs(⃗x ⊥ )} g(⃗x) n e (⃗x)/n e0 , (5.6)