CMB , Spherical Harmonics and Rotational Invariance

In summary: That's the assumption of isotropy. As the different coefficients for the same ##\ell## are just rotations of one another, assuming isotropy requires that they all have the same probability distribution (provided you make use of the appropriate normalization for the ##Y_l^m## functions).
  • #1
center o bass
560
2
In Dodelson's "Modern Cosmology" on p.241 he states that the ##a_{lm}##-s -- for a given ##l##-- corresponding to a spherical harmonic expansion of the photon-temperature fluctuations, are drawn from the same probability distribution regardless of the value of ##m##. Dodelson does not explain this any further, but other authors claim that it is due to the fact that ##m## somehow corresponds to an orientation and this should not matter as the universe is (believed to be) statistically rotational invariant.

Question:
What is the precise property of the spherical harmonic ##Y_l^m## for a given ##l## that justifies this claim?
 
Space news on Phys.org
  • #2
That [itex]Y_l^m \sim e^{i m \phi}[/itex]?
 
  • #3
center o bass said:
Question:
What is the precise property of the spherical harmonic ##Y_l^m## for a given ##l## that justifies this claim?
The ##Y_\ell^m## functions for a given ##\ell## can be morphed into one another through rotations in any direction. That is, if you rotate the coordinate system, the resulting ##a_{\ell m}## parameters are a linear combination of the pre-rotated ##a_{\ell m}## parameters. During this rotation, only the ##a_{\ell m}## values with the same ##\ell## are mixed.
 
  • #4
Chalnoth said:
The ##Y_\ell^m## functions for a given ##\ell## can be morphed into one another through rotations in any direction. That is, if you rotate the coordinate system, the resulting ##a_{\ell m}## parameters are a linear combination of the pre-rotated ##a_{\ell m}## parameters. During this rotation, only the ##a_{\ell m}## values with the same ##\ell## are mixed.
Thanks for the reply! From what you've now said, how would one go on to argue (fairly rigorously) that the ##a_{lm}##-s for a given ##l## must be drawn from the same probability distributions?
 
  • #5
center o bass said:
Thanks for the reply! From what you've now said, how would one go on to argue (fairly rigorously) that the ##a_{lm}##-s for a given ##l## must be drawn from the same probability distributions?
That's the assumption of isotropy. As the different coefficients for the same ##\ell## are just rotations of one another, assuming isotropy requires that they all have the same probability distribution (provided you make use of the appropriate normalization for the ##Y_l^m## functions).
 

FAQ: CMB , Spherical Harmonics and Rotational Invariance

What is CMB?

CMB stands for Cosmic Microwave Background. It is the leftover radiation from the Big Bang and is the oldest light in the universe. It is observed as a faint glow of microwave radiation that is uniform in all directions.

What are Spherical Harmonics?

Spherical Harmonics are mathematical functions used to describe the variations in temperature and polarization of the CMB across the sky. They are derived from the spherical coordinate system and are represented as a series of harmonics with different frequencies and amplitudes.

How are Spherical Harmonics used to analyze the CMB?

Spherical Harmonics are used to decompose the CMB map into different angular scales, allowing scientists to study the temperature and polarization fluctuations at different resolutions. They also help in identifying patterns and anisotropies in the CMB, which provide valuable insights into the early universe.

What is rotational invariance in the context of CMB?

Rotational invariance refers to the concept that the CMB is uniform and isotropic in all directions. This means that no matter which direction we observe the CMB from, its properties and characteristics remain the same. This is a fundamental property of the CMB and is a key aspect of the Big Bang theory.

Why is rotational invariance important in the study of the CMB?

Rotational invariance is important because it allows scientists to make accurate predictions and models about the early universe based on the properties of the CMB. It also helps in ruling out alternative theories of the universe's origin that do not fit with the observed isotropy and uniformity of the CMB. Additionally, rotational invariance allows for more precise measurements of the CMB, leading to a better understanding of the universe's evolution.

Back
Top