In Weinberg's cosmology book, section 2.4 we have(adsbygoogle = window.adsbygoogle || []).push({});

[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.

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# Dipole anisotropy expansion

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