# How can you show the CMB shift parameter?

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1. Mar 24, 2015

Dealing with CMB, people sometimes refer to the shift parameter;
R = \sqrt{\frac{\Omega^0_m}{\Omega^0_k}}sinh(\sqrt{\Omega^0_k}\int^{z_{dec}}_{0}{\frac{dz}{E(z)}})

I know that it is related to the position of the first acoustic peak, however, the amount is around 1.7.
What does this mean?!!!!!!
does it mean without Dark Energy, the peak would be shifted 1.7 in the x axis (the multipole moment) of the CMB anisotropy spectrum?!
To the right or left?!
I think it should be around l=100 or something, so does this mean without Dark Energy, the theory would show a CMB shift parameter shifted about 1.7? like 100\pm1.7?!

Last edited: Mar 24, 2015
2. Mar 24, 2015

### Chalnoth

3. Mar 24, 2015

sure, sorry for the last one!
$$R = \sqrt{\frac{\Omega^0_m}{\Omega^0_k}}sinh(\sqrt{\Omega^0_k}\int^{z_{dec}}_{0}{\frac{dz}{E(z)}})$$

4. Mar 24, 2015

### Chalnoth

I did find this paper:
http://arxiv.org/abs/astro-ph/0702343

They explain that this "shift parameter" is used as one of two parameters determining the location of the first acoustic peak, the other being the baryon density. This seems pretty reasonable if you consider that the sound horizon is set largely by the baryon density, and the rest of the factors that go into the first peak are an estimate of the geometry of the universe between us and the CMB, which is determined by the expansion history between us and the CMB.

As for the parameter itself, it is very similar to a distance. In fact, it looks like it is exactly the comoving distance times $\sqrt{\Omega_m^0}/D_H$, where $D_H = c / H$

5. Mar 25, 2015

Thanks for the explanations.
However, I still don't get the 1.7 number!

I agree that it somehow shows some kind of distance, however, if you see
" L. Amendola and S. Tsujikawa. Dark Energy: Theory and Observations",
they claim that the shift parameter is about 1.7.

What I want to know is :
- Does this number mean that if one does not choose a suitable model (like Lambda CDM), theoretically the first peak would move and of course this is not in agreement with observations?
If yes, in which direction would it move? to the left or right, to the top or bottom?

- In the CMB power spectrum, we have the anisotropy power on one axis and the multipole moment on the other one.
The first peak looks to be of order 10^3 in the anisotropy power axis and 10^2 on the multipole moment axis.
Is this 1.7 related to the horizontal axis or the vertical one?

6. Mar 25, 2015

I think I found it!
In the paper that you mentioned above, they say the location of the first peak is given by
$$l_a \approx \pi \frac{d_A(z_r)}{r_s(a_r)}$$
where
$$d_A = R/\sqrt{\Omega_m}$$
So this way the shift parameter would change the position of the first peak on the horizontal axis (the multipole moment).
Am I right?

Last edited: Mar 25, 2015
7. Mar 25, 2015

### George Jones

Staff Emeritus

This is also given in the book "Dark Energy: Theory and Observations" by Amendola and Tsujikawa that you already referenced; see equation (5.31) and (5.33). See also equation (5.32), which expresses the angular diameter distance of the "surface" of photon decoupling as a constant multiplied by the shift parameter. Thus, the positions of peaks depend on the shift parameter. Equation (5.40) expresses results in terms of the shift parameter and the sound horizon, as noted by Chalnoth.

At first glance, the discussion on pages of 97 - 101 of Amendola and Tsujikawa appears to be quite good.

8. Mar 27, 2015