What are the parameters that are the same in CMB data for multiple models?

[Arman777]

In cosmology, CMB tells a lot about which cosmological model can be acceptable or not. For instance we know that, whatever the cosmological model we use $\theta_*$ parameter will be always the same. Is there any other parameters that is listed in this picture is **model-independent** ?(i.e always stays the same for any given model)

https://arxiv.org/abs/1807.06209

 

[kimbyd]

There's not really any such thing as a model-independent parameter, unfortunately. The parameters make up the model.

I think what you're asking can be more correctly rephrased as, "Which parameters are constrained tightly by the CMB data, and don't change much based upon model assumptions?" I think the main ones there are the matter densities ($\Omega_b h^2$, $\Omega_c h^2$, $\Omega_m h^2$) and the distance to the surface of last scattering ($r_*$).

That's not quite the right chart to understand this, however. The columns are not different models, but different sets of data.

 

[Arman777]

I think what you're asking can be more correctly rephrased as, "Which parameters are constrained tightly by the CMB data, and don't change much based upon model assumptions?"

Yes that was my point.

I think the main ones there are the matter densities (Ωbh2, Ωch2, Ωmh2) and the distance to the surface of last scattering (r∗).

Well that make sense I guess. Do you know more ? or that's is just it

That's not quite the right chart to understand this, however. The columns are not different models, but different sets of data.

Yes indeed.

 

[kimbyd]

Well that make sense I guess. Do you know more ? or that's is just it

The redshift at last scattering for sure. Curvature is only tightly-constrained if the CMB data is combined with nearby data. The Hubble parameter has that unsolved tension which makes its value uncertain, which also means the density fractions aren't very well-constrained. This means that the dark energy density is not well-constrained, even if we assume a cosmological constant.

The optical distance to the surface of last scattering ($\tau$) I'm not sure about. It does rely largely upon polarization data which is notoriously difficult to manage systematic errors for, both because the foregrounds are brighter relative to the CMB and because Planck's polarimeters are an older design that doesn't control systematics as well as more current technologies. So I'm a little suspicious of it, but my suspicions may be unwarranted.

I suppose the power spectrum amplitude $A_s$ must be very tightly-constrained. So is $\theta_{MC}$ (the angular size of the sound horizon).

The spectral index and its tilt are likely much more model-dependent.

 

[Arman777]

The redshift at last scattering for sure

You mean $z_*$ ?

Curvature is only tightly-constrained if the CMB data is combined with nearby data. The Hubble parameter has that unsolved tension which makes its value uncertain, which also means the density fractions aren't very well-constrained. This means that the dark energy density is not well-constrained, even if we assume a cosmological constant.

I understand that from the CMB data we can tightly constrain $\theta_* \equiv \frac{r_*}{D_A(z_*)}$. And every cosmological model must approximatley give the same value.
Meanwhile, when we are trying to obtain $\theta_*$ we need some parameters such as

$$h, \omega_m, \omega_r, \omega_{DE}, \omega_k$$

In the CMB data we are fixing $\omega_m$ and $\omega_r$ as you said in your earlier post. So we are left with

$$h, \omega_{DE}, \omega_k$$

In general we are assuming a flat universe and $\omega_k$ becomes $0$. This is also called geometrical degeneracy where $\omega_k$ is degenerate with $\omega_{DE}$ (See Fig 1. of this article [astro-ph/9807103] Cosmic Confusion: Degeneracies among Cosmological Parameters Derived from Measurements of Microwave Background Anisotropies (arxiv.org) )

But I understand that for small scale perturbation ($l \approx 10$ this degeneracy can be broken ? Could you look at the section 3.3 and explain what does Eq. 12 means ? Does that article shows that the degeneracy can be broken my considering "something" into account ?

In my opinion, in Eqn. 10(a) He is defining those values for a fictitious model. To sum up my problem is something like this; Does Eq. 12 can help to break the geometrical degeneracy for any type of given model ?

So is θMC (the angular size of the sound horizon).

I am doing a cosmological analysis and I am using $\theta_*$ as the model independent parameter. Should I use $\theta_{MC}$ instead of $\theta_*$ ?

 

[Arman777]

I have also another question. I created a python code to calculate the $\theta_*$ by using $r_*$ and $D_A(z_*)$. When I use the Planck values such as TT,TE,EE+lowE+lensing. I am getting weird results.

For instance for these parameters

c = 299792.458
N_eff = 3.046
w_b = 0.02237
w_r = 2.469 * 10**(-5) * (1 + (7/8)*(4/11)**(4/3) * N_eff)
w_m = 0.1430
h0 = 0.6736

I am getting these results

z_*:  1091.9068900826396
D_A(z_*):  13870.09921075586
r_*: 150.31680242574282
theta_*: 0.01083747

Here is my code

from numpy import sqrt, infty
from scipy import integrate

########## Our data analaysis for the LCDM model #########
c = 299792.458
N_eff = 3.046
w_b = 0.02237
w_r = 2.469 * 10**(-5) * (1 + (7/8)*(4/11)**(4/3) * N_eff)
w_m = 0.1430
h0 = 0.6736

g1 = (0.0783 * w_b**(-0.238)) / (1 + 39.5 * w_b**(0.763))
g2 = 0.560 / (1 + 21.1 * w_b **(1.81))
z_rec = 1048 * (1 + 0.00124 * w_b**(-0.738)) * (1 + g1 * w_m**g2)

def r_zrec_finder(h):
    def r_zrec(z):
        c_s = c / sqrt(3 + (9/4) * (w_b/w_r) * (1 / (1+z)))
        return (c_s/100) / sqrt(w_m * (1+z)**3 + w_r * (1+z)**4 + (h**2 - w_m - w_r))
    result, error = integrate.quad(r_zrec, z_rec, infty)
    return resultdef D_zrec_finder(h):
    def D_zrec(z):
        return (c/100) / sqrt(w_m * (1+z)**3 + w_r * (1+z)**4 + (h**2 - w_m - w_r))
    result, error = integrate.quad(D_zrec, 0, z_rec)
    return resultD_zrec1 = D_zrec_finder(h0)
r_zrec1 = r_zrec_finder(h0)theta = r_zrec1 / D_zrec1print("z_*: ", z_rec)
print("D_A(z_*): ", D_zrec1)
print("r_*:", r_zrec1)
print("theta_*:", round(theta, 8))

Here the main problem is even I am using the Planck values and I should obtain $r_* = 144.43 Mpc$but I am gettting $r_* = 150 Mpc$ I know that my equations are correct but I cannot see where is the problem.

Can you think of any reason why doing numeric calculations to find $r_*$ gives so wrong results ?

The same is true for $z_*$ the Planck result is $\approx 1089$ but I am getting $\approx 1091$ Which I am using one of the most well known approximations by used in arXiv:astro-ph/9510117v2 19 Apr 1996 Eq. E-1

Well the Planck uses complex data analysis etc but then how can I compare a fictitious model by with LCDM by using the Planck data ? Do I obligated to use complex data analysis "stuff".

 

[kimbyd]

You mean $z_*$ ?

In general we are assuming a flat universe and $\omega_k$ becomes $0$. This is also called geometrical degeneracy where $\omega_k$ is degenerate with $\omega_{DE}$ (See Fig 1. of this article [astro-ph/9807103] Cosmic Confusion: Degeneracies among Cosmological Parameters Derived from Measurements of Microwave Background Anisotropies (arxiv.org) )

It's fairly close to degenerate, but not exactly. And you can break that degeneracy very effectively by combining it with nearby data.

The bigger problem, in my view, is $H_0$, which also interacts with these parameters, and has some currently unknown issues.

But I understand that for small scale perturbation ($l \approx 10$ this degeneracy can be broken ? Could you look at the section 3.3 and explain what does Eq. 12 means ? Does that article shows that the degeneracy can be broken my considering "something" into account ?

Without looking at it, at large angular scales (low=$\ell$) dark energy is detectable via the Integrated Sachs-Wolfe Effect. Small-scale perturbations average out, because there are many more along the line of sight. Large-scale perturbations tend to result in an amplification of the power spectrum.

In my opinion, in Eqn. 10(a) He is defining those values for a fictitious model. To sum up my problem is something like this; Does Eq. 12 can help to break the geometrical degeneracy for any type of given model ?

It's pretty late here, so hopefully I'll remember to come back to this in the morning. But if it's the ISW Effect, then I guess it depends upon what you mean by "model". Certainly the ISW effect depends upon the large-scale distribution of matter, and in particular how large-scale gravitational potentials evolve over time.

I am doing a cosmological analysis and I am using $\theta_*$ as the model independent parameter. Should I use $\theta_{MC}$ instead of $\theta_*$ ?

I don't know that it matters too much.

 

[Arman777]

And you can break that degeneracy very effectively by combining it with nearby data.

But if we don't combine it with another data, can we break the degeneracy ?

The bigger problem, in my view, is H0, which also interacts with these parameters, and has some currently unknown issues.

That is indeed a problem, but for now there is not much to do with it..

But if it's the ISW Effect,

Yes its the ISW effect.

I don't know that it matters too much.

 

[kimbyd]

But if we don't combine it with another data, can we break the degeneracy ?

Not really, no. The problem is that these parameters don't affect the physics that produced the CMB very much. Their primary effect is in impacting how the image of the CMB appears today relative to when it was emitted. The CMB can only tightly-constrain things that are a result of physics before and during the emission of the CMB.