21cmFirstCLASS II. Early linear fluctuations of the 21cm signal

In a companion paper we introduce 21cmFirstCLASS, a new code for computing the 21-cm anisotropies, assembled from the merger of the two popular codes 21cmFAST and CLASS. Unlike the standard 21cmFAST, which begins at $z=35$ with homogeneous temperature and ionization boxes, our code begins its calculations from recombination, evolves the signal through the dark ages, and naturally yields an inhomogeneous box at $z=35$. In this paper, we validate the output of 21cmFirstCLASS by developing a new theoretical framework which is simple and intuitive on the one hand, but is robust and precise on the other hand. As has been recently claimed, using consistent inhomogeneous initial conditions mitigates inaccuracies, which according to our analysis can otherwise reach the $\mathcal O\left(20\%\right)$ level. On top of that, we also show for the first time that 21cmFAST over-predicts the 21-cm power spectrum at $z\gtrsim20$ by another $\mathcal O\left(20\%\right)$, due to the underlying assumption that $\delta_b=\delta_c$, namely that the density fluctuations in baryons and cold dark matter are indistinguishable. We propose an elegant solution to this discrepancy by introducing an appropriate scale-dependent growth factor into the evolution equations. Our analysis shows that this modification will ensure sub-percent differences between 21cmFirstCLASS and the Boltzmann solver CAMB at $z\leq50$ for all scales between the horizon and the Jeans scale. This will enable 21cmFirstCLASS to consistently and reliably simulate the 21-cm anisotropies both in the dark ages and cosmic dawn, for any cosmology. The code is publicly available at https://github.com/jordanflitter/21cmFirstCLASS.