Finding the One Degree Peak in a Power Spectrum

[gluon]

Can someone tell me how i can determine the one degree peak from power spectrum ?

 

[Chronos]

You plot the angular separation between CMB temperature differences.

 

[Chalnoth]

Can someone tell me how i can determine the one degree peak from power spectrum ?

This post is useful for the plots, and has a good description of how the power spectrum is derived:
http://backreaction.blogspot.com/2007/12/cmb-power-spectrum.html

As for how you can see the 1-degree peak, the power spectrum of the CMB is usually plotted with $\ell$ along the x-axis. The approximate wavelength of a fluctuation is $180 / \ell$ degrees. So the peak that appears close to $\ell = 180$ represents waves with a wavelength of approximately one degree. The plot at the above blog post also lists the angular scale along the top.

 

[gluon]

the temperature difference at 1 degree why is so big?it measures the difference between two points that the sound wave have reached at the time of recombination or from the initial overdense which became underdense and the point that the wave have reached at the time of recombination?

 

[Chalnoth]

the temperature difference at 1 degree why is so big?it measures the difference between two points that the sound wave have reached at the time of recombination or from the initial overdense which became underdense and the point that the wave have reached at the time of recombination?

One degree is the "sound horizon", which is the distance that sound waves in the CMB were capable of traveling since the big bang.

 

[gluon]

and why the temprature difference is big?the temprature difference of two points of sound horizon isn t zero ?

 

[Chalnoth]

and why the temprature difference is big?the temprature difference of two points of sound horizon isn t zero ?

Why would it be zero?

One degree represents matter that fell into a potential well. The next (shorter) peak come from matter that fell into a potential well, then bounced back out.

 

[gluon]

because the two points at the sound horizon have the same temprature or am i wrong?

 

[Chalnoth]

No, definitely not. Where are you getting this idea from?

 

[gluon]

The y-axis is temprature difference between two points and the horizontal is the angle separation.the first peak giving as the fundamental frequency of sound wave right?
if we suppose that we have a overdense region at the time of inflation and until recombination it gets underdense and the wave have propagated at known distance which is given by d=c*t ,c is the speed of sound wave 0,6c and t is the time at recombination 375.000,which giving a distance d=225.000 ly and we know that the distance from cmb surface is 41billion ly away so we get a angle which is 0,5 degree .But if i take that the wave went spherical away from overdense point we get a diameter of 716.000 ly which is giving 1 degree angle.

which difference in temprature we measure? 1) from the initial overdence ,which became underdense,until the region which sound wave reach?or 2) from two positions that sound wave have reached at the time of recombintaion which is separated by distance 716.000 ly ? if we take the 1) case i understand why the temprature difference is so big,but the angle is 0,5 degree not 1.In case 2) the degree is 1 but i can't see why we have temprature difference between this points.should it have zero temprature difference or not?

Sorry for my English!

 

[Chalnoth]

I think the mistake you're making is that it's not just one wave. The CMB is made out of a large, random assortment of waves all traveling in different directions. A peak shows up at the same wavelength as the sound horizon because that's the longest wavelength where constructive interference becomes possible.

 

[gluon]

yes i know its the overtone modes which are at smaller angles.but in the fundumental mode if i measure the temprature difference between two points which is diametrically opposite in the sound horizon, this spherical shell ,with the center in the initial overdense region ,isn't have the same density and thus the same temprature?

 

[Chalnoth]

yes i know its the overtone modes which are at smaller angles.but in the fundumental mode if i measure the temprature difference between two points which is diametrically opposite in the sound horizon, this spherical shell ,with the center in the initial overdense region ,isn't have the same density and thus the same temprature?

Not just overtones. Different waves at the same wavelength. At $\ell = 180$, there are 361 different possible orientations for the waves. Each of those orientations will have its own randomized amplitude.

 

[gluon]

do you mean 361 different waves with the same wavenumber?and this waves can cancel each other or amplifying?

 

[Chalnoth]

Yes. Same wavenumber, different directions. If you want to get some detail on how this works, you can try reading up on spherical harmonics.

 

[gluon]

Thanks Chalnoth!