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How to get transfer function for linear matter power spectrum?

Physics Asked by apprentice on August 2, 2021

I understand that the linear matter power spectrum is given by (Dodelson, Modern Cosmology):

$$P_L(k) = P_R(k)dfrac{4}{25}dfrac{k^4}{Omega_m H_0^2}T^2(k)D_+(a),$$

where $P_R(k) = (2pi^2/k^3)A_s(k/k_p)^{n_s-1}$, $D_+(a)$ is the growth rate, and $T(k)$ is the transfer function.

The growth rate $D_+(a)$ I can compute from the Mészáros equation:

$$delta_{cdm}” + dfrac{a’}{a}delta_{cdm}’ – dfrac{3}{2}left(dfrac{a’}{a}right)Omega_{cdm}(a)delta_{cdm} = 0,$$

where the derivative is with respect to the conformal time.

How do I get $T(k)$, particularly for large $k$?

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