Calculation of CMB anisotropies size

[fab13]

hello,

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

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 $$\theta$$ 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 $$\theta$$ between two beams of light changes when they are approaching the observer. I would like to obtain an equation expressing the angle $$\theta$$ according to the spherical variable$$\,\,r\,\,$$ 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 ($$t, r, \theta$$) 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.

 

[AleLucca]

However, the size of fluctuations changes during their spread to us

This is true, but the only quantity that matters is the size of the fluctuation at the instant of the decoupling of the CMB (z = 1100).

and the angle $$\theta$$ between two beams of light changes when they are approaching the observer.

That is not correct. For symmetry reasons (the universe is isotropic at first order), light beams connecting an observer with an event at an angle coordinate $$\theta$$ travel along world-lines of constant $$\theta$$: there is no reason why a light ray should deviate its path in a specific direction or another. This is true regardless of the flatness of the space metric.

You can then calculate the angular scale by dividing the intrinsic size of the source by the angular diameter distance, which you can calculate as a function of redshift for a given cosmological model.

I hope this helps

 

[fab13]

Hello,

i wrote a Matlab program which calculate the apparent angle of an anisotropy with a size equal to "380.000 light-years" , i.e "0.1166 Mpc".

For this, i have used the following equation defining the apparent angle ( in arcminute unit) as a function of the redshift "z" with "D_anisotropy=size of the anisotropy" :

$$\theta= \dfrac{180*60}{\pi}\,D_{anisotropy}\,\big[\dfrac{c}{H_{0}\,(1+z)} \ ,\int_{z_{begin}}^{z} \,(\Omega_{m}\,(1+z)^{3}+\Omega_{\Lambda}+\Omega_{k}\,(1+z)^{2})^{-1/2}\,dz\big]^{-1}$$

For $$\Omega_{m}=0.3$$ and $$\Omega_{\Lambda}=0.7$$, i get the figure in attachment. For "z=1100", I have an angle apparent equal to about 30 arcminutes for D_anisotropy=380.000 ly , i.e 0.5 degrees.

This value is relatively quite far from the first peak in CMB power spectrum which is localized at "l=200", i.e "theta= 1 degree".

Does the relation between multipole "l" and "theta" angle take the form : "theta(°)=200/l" ?

If it does, then my calculation is the value of one of the others peaks which are localized for smaller angles ?

If anyone could explain me this result,

Thanks in advance.

ps: here is the Matlab code :

function dist_diam_ang

c=3*10^(5);
H0=71;

Omega2_m=0.3;
Omega2_red=0.7;
Omega2_k=1-Omega2_m-Omega2_red;

z_begin=0.1;
z_final=1100;

size1=(380000/3.26)*10^(-6); % Megaparsec
inter=1;

j=1;
for z=z_begin:inter:z_final
theta1(j)=(180*60/(pi))*size1*(c/(H0*(1+z))*(quad(@(x)myfunc(x,Omega2_m,Omega2_red,Omega2_k),z_begin,z)))^(-1);
j=j+1;
end
plot(z_begin:inter:z_final,theta1);
ylim([0 60]);
xlabel('Redshift z');
ylabel('Apparent Angle (arcminute)');%-------------------%
function y = myfunc(x,Omega_m,Omega_red,Omega_k)
y=(Omega_m*(1+x).^(3)+Omega_red+Omega_k*(1+x).^(2)).^(-1/2);
end
%-------------------%end