- #1
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In Weinberg's cosmology book, section 2.4 we have
[tex]
T' = \frac{T}{\gamma(1+\beta\cos{\theta})}
[/tex]
He then claims, "Expanding in powers of beta, the temperature shift can be expressed as a sum of Legendre polynomials"
[tex]
\Delta T = T' -T = T\left(-\frac{\beta^2}{6} - \beta P_1(\cos{\theta}) + \frac{2\beta^2}{3}P_2(\cos{\theta}) + ...\right)
[/tex]
Can someone help me fill in the algebra here? I really am having a hard time seeing where this is coming from.
[tex]
T' = \frac{T}{\gamma(1+\beta\cos{\theta})}
[/tex]
He then claims, "Expanding in powers of beta, the temperature shift can be expressed as a sum of Legendre polynomials"
[tex]
\Delta T = T' -T = T\left(-\frac{\beta^2}{6} - \beta P_1(\cos{\theta}) + \frac{2\beta^2}{3}P_2(\cos{\theta}) + ...\right)
[/tex]
Can someone help me fill in the algebra here? I really am having a hard time seeing where this is coming from.