Legendre polynomials in boosted temperature approximation

[jouvelot]

Hi all,

In S. Weinberg's book "Cosmology", there is a derivation of the slightly modified temperature of the cosmic microwave background as seen from the Earth moving w.r.t. a frame at rest in the CMB. On Page 131 (1st printing), an approximation (Formula 2.4.7) is given in terms of Legendre polynomials, but I have a hard time seeing how these polynomials get into the picture. Any hints?

Thanks in advance.

Bye,

Pierre

 

[mathman]

(Mathematical explanation) Most expressions can be expanded in terms of a system of orthogonal functions with coefficients. If the coefficients tend to 0 rapidly enough, then a few terms of the expansion can be used as an approximation to the expression.

I have no knowledge of the particulars you are referring to.

 

[jouvelot]

Hello mathman,

Good point. Since the discussion here in the book is about temperature distribution, it makes sense to use Legendre polynomials to look at possible n-polar terms. A usual limited development factorised using Legendre polynomials yields the formula in the book (well, almost, but I get the idea).

Thanks a lot for your very helpful comment.

Bye,

Pierre

 

[jouvelot]

Good point. Since the discussion here in the book is about temperature distribution, it makes sense to use Legendre polynomials to look at possible n-polar terms. A usual limited development factorised using Legendre polynomials yields the formula in the book (well, almost, but I get the idea).

About this "almost" part, the first term in the book is -β²/6 while I find 5β²/6 (there is no mention of a typo here in the Corrections sheet available online). If anyone has an idea... ;)

 

[George Jones]

About this "almost" part, the first term in the book is -β²/6 while I find 5β²/6 (there is no mention of a typo here in the Corrections sheet available online). If anyone has an idea... ;)

Did you subtract $T$ from (2.4.6)? I.e., (2.4.7) = (2.4.6) - $T$.

 

[jouvelot]

Hello George,

I sure did. Moreover, this gives an additional factor term equals to 1, and not something proportional to β² that would help explain the discrepancy.

Thanks for having looked into this :)

Pierre

 

[George Jones]

About this "almost" part, the first term in the book is -β²/6 while I find 5β²/6 (there is no mention of a typo here in the Corrections sheet available online). If anyone has an idea... ;)

Okay, now I have done more of a calculation.

The first time I did the calculation, I got $5 \beta^2/6$.

The second time I got $- \beta^2/6$.

The first time that I did the calculation, I mistakenly used the expansion $\gamma = \left( 1 - \beta^2 \right)^{-1/2} = 1 +\beta^2/2 + \ldots$ instead of $\gamma^{-1} = \left( 1 - \beta^2 \right)^{1/2} = 1 -\beta^2/2 + \ldots$.

 

[jouvelot]

Okay, now I have done more of a calculation.

The first time I did the calculation, I got $5 \beta^2/6$.

The second time I got $- \beta^2/6$.

The first time that I did the calculation, I mistakenly used the expansion $\gamma = \left( 1 - \beta^2 \right)^{-1/2} = 1 +\beta^2/2 + \ldots$ instead of $\gamma^{-1} = \left( 1 - \beta^2 \right)^{1/2} = 1 -\beta^2/2 + \ldots$.

George,

Oh! Great :) I see: one has to be pretty careful about the order with which expansions are performed. I'll sleep better tonight : )

Thanks a lot for your very useful help, and Merry Christmas.

Bye,

Pierre

 

[jouvelot]

Well, ultimately, it was not even a matter of Taylor expansion ordering, but just an oversight on my part. Thanks a lot for spotting it for me, George :)