1. CMB Anisotropy
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2. Cosmic Microwave Background
As you know, this is picture of CMB, the Cosmological Microwave background. This picture shows us that our universe is inhomogeneous. What CMB experimentalists do is take a power spectrum of the temperature maps, mush as you would if you wanted to measure background noise.
WMAP, 2012

3. The Big Bang Theory Hans Kristian Eriksen, 2011
According to the theory of the Big Bang, the universe started hot and dense and then expanded and cooled. In the hot dense conditions of the early universe, photons were tightly glued to matter. When the universe was about 300,000 tears old the temperature dropped below 3,000K allowing atomic hydrogen to form and releasing the photons. These photons are make up the Cosmic microwave background we see today.

4. Inflation Wayne Hu, Chicago 2001
In the early universe, because of gravitational instability, noise of CMB was produced.
Wayne Hu, Chicago 2001

5. CMB Power Spectrum Basis!! Hans Kristian Eriksen, 2011
So if what we are dealing with is actually noise in the technical sense, what do we do with it? For “noise-like” phenomena, we are only interested the amplitude of the fluctuations as a function of scale. And the required basis wave functions in a given space is found by solving Laplace’ equation. Since the CMB field is defined on a sphere, on has to solve the spherical Bessel equation. Therefore, the spherical harmonics are linear combination of spherical Bessel function. So, l determines the wave length of the mode. So we can get the size of a spot as 180/l
Basis!!
Hans Kristian Eriksen, 2011

6. CMB Power Spectrum Hans Kristian Eriksen, 2011
So if what we are dealing with is actually noise in the technical sense, what do we do with it? For “noise-like” phenomena, we are only interested the amplitude of the fluctuations as a function of scale. And the required basis wave functions in a given space is found by solving Laplace’ equation. Since the CMB field is defined on a sphere, on has to solve the spherical Bessel equation. Therefore, the spherical harmonics are linear combination of spherical Bessel function. So, l determines the wave length of the mode. So we can get the size of a spot as 180/l
Hans Kristian Eriksen, 2011

7. CMB Power Spectrum Hans Kristian Eriksen, 2011
Mathematically, the power spectrum is actually equal to the square of amplitude of each spherical harmonics. So we can do many things with this graph.
Hans Kristian Eriksen, 2011

8. Overview Wayne Hu, Chicago 2001
To better understand the formation of angular anisotropies form inhomogeneities, Photon pressure in a plane wave gravitational potential fluctuations causes a plane wave temperature variation across space that oscillates in time.
And the photon-baryon fluid stops oscillating at recombination. The modes that freeze in at extrema of their oscillation have enhanced temperature fluctuations across the distance spanned by their wavelength. And we can decompose arbitrary gravitational potential into many discrete “small” gravitational potential. Also we can determine temperature of the peak by decomposing arbitrary gravitational potential. Since 180/l is wave length of gravitational potential wave function. We can know the l-position of our power spectrum peak. We can define sound-horizon as half of the wavelength of longest soundwave.
Sound-horizon means that the maximum distance that sound wave can reach in the recombination term.
Wayne Hu, Chicago 2001

9. Photon-Baryon Fluid Wayne Hu, Chicago 2001
To better understand the formation of angular anisotropies form inhomogeneities, Photon pressure in a plane wave gravitational potential fluctuations causes a plane wave temperature variation across space that oscillates in time.
And the photon-baryon fluid stops oscillating at recombination. The modes that freeze in at extrema of their oscillation have enhanced temperature fluctuations across the distance spanned by their wavelength. And we can decompose arbitrary gravitational potential into many discrete “small” gravitational potential. Also we can determine temperature of the peak by decomposing arbitrary gravitational potential. Since 180/l is wave length of gravitational potential wave function. We can know the l-position of our power spectrum peak. We can define sound-horizon as half of the wavelength of longest soundwave.
Sound-horizon means that the maximum distance that sound wave can reach in the recombination term.
Wayne Hu, Chicago 2001

10. Photon-Baryon Fluid Wayne Hu, Chicago 2001
To better understand the formation of angular anisotropies form inhomogeneities, Photon pressure in a plane wave gravitational potential fluctuations causes a plane wave temperature variation across space that oscillates in time.
And the photon-baryon fluid stops oscillating at recombination. The modes that freeze in at extrema of their oscillation have enhanced temperature fluctuations across the distance spanned by their wavelength. And we can decompose arbitrary gravitational potential into many discrete “small” gravitational potential. Also we can determine temperature of the peak by decomposing arbitrary gravitational potential. Since 180/l is wave length of gravitational potential wave function. We can know the l-position of our power spectrum peak. We can define sound-horizon as half of the wavelength of longest soundwave.
Sound-horizon means that the maximum distance that sound wave can reach in the recombination term.
Wayne Hu, Chicago 2001

11. Photon-Baryon Fluid Wayne Hu, Chicago 2001
To better understand the formation of angular anisotropies form inhomogeneities, Photon pressure in a plane wave gravitational potential fluctuations causes a plane wave temperature variation across space that oscillates in time.
And the photon-baryon fluid stops oscillating at recombination. The modes that freeze in at extrema of their oscillation have enhanced temperature fluctuations across the distance spanned by their wavelength. And we can decompose arbitrary gravitational potential into many discrete “small” gravitational potential. Also we can determine temperature of the peak by decomposing arbitrary gravitational potential. Since 180/l is wave length of gravitational potential wave function. We can know the l-position of our power spectrum peak. We can define sound-horizon as half of the wavelength of longest soundwave.
Sound-horizon means that the maximum distance that sound wave can reach in the recombination term.
Wayne Hu, Chicago 2001

12. 1st Peak
1st peak
The l correspond to the 1st peak can measure the sound horizon. Actually higher order peaks affected by the curvature of the space, but the 1st peak most related to the sound-horizon.
By calculating size of horizon, we can know the curvature of our universe. And we can know our space is almost flat.
WMAP, 2012

13. 1st Peak Hans Kristian Eriksen, 2011 1st peak
The l correspond to the 1st peak can measure the sound horizon. Actually higher order peaks affected by the curvature of the space, but the 1st peak most related to the sound-horizon.
By calculating size of horizon, we can know the curvature of our universe. And we can know our space is almost flat.
Hans Kristian Eriksen, 2011

14. 2nd Peak Wayne Hu, Chicago 2001 2nd peak
Imagine two condition, one is low baryons and the other is high baryons. Gravitational potential and initial condition are same. In this case we assumed they have the same maximum rarefaction. Than we can assume both system oscillate. With the same initial condition, high baryon system have higher maximal compression than the low baryon system. But they have same maximal rarefaction. So, only the odd peaks will be affected by baryon density. We can see these effects in this diagram.
So by comparing the odd and even peaks, we can know baryon density of the universe. Surprisingly, there are more baryons than those we can observe. Therefore, we can know there must be some baryonic dark matter.
Wayne Hu, Chicago 2001

15. 2nd Peak Wayne Hu, Chicago 2001 2nd peak
Imagine two condition, one is low baryons and the other is high baryons. Gravitational potential and initial condition are same. In this case we assumed they have the same maximum rarefaction. Than we can assume both system oscillate. With the same initial condition, high baryon system have higher maximal compression than the low baryon system. But they have same maximal rarefaction. So, only the odd peaks will be affected by baryon density. We can see these effects in this diagram.
So by comparing the odd and even peaks, we can know baryon density of the universe. Surprisingly, there are more baryons than those we can observe. Therefore, we can know there must be some baryonic dark matter.
Wayne Hu, Chicago 2001

16. 3rd Peak Wayne Hu, Chicago 2001 3rd peak
The series of higher acoustic peaks is sensitive to the energy density ratio of dark matter to radiation in the universe. Because the amount of radiation is know from the measured temperature of the CMB and the thermal history, under normal assumptions the higher acoustic peaks are sensitive to the dark matter density in the universe.
What happens is that if the energy density of the radiation dominates the matter density, we can no longer consider the photon-baryon fluid to be oscillating in a fixed gravitational potential well. In fact, the potential decays away at just the right time to drive the amplitude of the oscillations up.
Wayne Hu, Chicago 2001

17. 3rd Peak Wayne Hu, Chicago 2001 3rd peak
The series of higher acoustic peaks is sensitive to the energy density ratio of dark matter to radiation in the universe. Because the amount of radiation is know from the measured temperature of the CMB and the thermal history, under normal assumptions the higher acoustic peaks are sensitive to the dark matter density in the universe.
What happens is that if the energy density of the radiation dominates the matter density, we can no longer consider the photon-baryon fluid to be oscillating in a fixed gravitational potential well. In fact, the potential decays away at just the right time to drive the amplitude of the oscillations up.
Wayne Hu, Chicago 2001

18. Photon Diffusion Damping
In the higher l values, we see the damping signal, and this damping affected by many cosmological constants. This is called the photon diffusion damping. Photon diffusion damping is effected by various cosmological parameters. Therefore, we can do a consistency check with the matter density, curvature, dark matter density values we calculated from the lower l peaks.
Wayne Hu, Chicago 2001

19. Parameter Estimation
By using CMB anisotropy, WMAP calibrate many cosmological constants.

20. Reference
Intermediate Guide to the Acoustic Peaks and Polarization Wayne Hu, Chicago (
Hans Kristian Eriksen, (
Wilkinson Microwave Anisotropy Probe (