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I am interested in calculating the size of the anisotropies of the CMB to make the comparison with the observed angle through the WMAP satellite. I found an article on

the physics of microwave background anisotropies and which explains all the different physical processes that occur on the size of the primary fluctuations (those that are dominant in determining the cosmological parameters): Acoustic oscillations, Baryon Drag, Doppler Effect, Driving Effect ...

Currently, I am interested only in the projection effect. They say in this article:

Increasing the distance to the last scattering surface also decreases the angular extent of the features. The distance to last scattering [tex]c\,(\eta_{0}-\eta_{*})[/tex] depends mainly on the expansion rate and hence on [tex]H_{0}, \Omega_{0}, \Omega_{k}[/tex] and very weakly on [tex]\Omega_{\Lambda}[/tex]. Putting these quantities together gives us the angular diameter distance d. The angualr extent [tex]\theta \propto l^{-1}[/tex] of a physical feature in the CMB is given by [tex]l_{feature}=k_{feature}\,d[/tex]

My main problem is that I do not know how to determine the angle of arrival from the definition of angular diameter distance. The Dyer-Roder equation allows to calculate the angle [tex]\theta[/tex] as a function of redshift, but for a fixed size of the object that emitted the photons. However, the size of fluctuations changes during their spread to us and the angle [tex]\theta[/tex] between two beams of light changes when they are approaching the observer. I would like to obtain an equation expressing the angle [tex]\theta[/tex] according to the spherical variable[tex]\,\,r\,\,[/tex] for the 3 geometry types (k =- 1,0,1) (what i want to get is the 3 curves on figure in attachment where the size object is the fluctuation size). FLWR metric definition could give this equation but there are 3 variables ([tex]t, r, \theta[/tex]) for one single equation. I have solved numerically the scale factor with time but now I'm stuck for calculating the observed angle.

Do I misunderstood the definition of the Dyer-Roder equation ?

If you have some idea on the subject.

Thanks.

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# Calculation of CMB anisotropies size

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