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In a companion paper we introduce 21cmFirstCLASS, a new code for computingthe 21-cm anisotropies, assembled from the merger of the two popular codes21cmFAST and CLASS. Unlike the standard 21cmFAST, which begins at $z=35$ withhomogeneous temperature and ionization boxes, our code begins its calculationsfrom recombination, evolves the signal through the dark ages, and naturallyyields an inhomogeneous box at $z=35$. In this paper, we validate the output of21cmFirstCLASS by developing a new theoretical framework which is simple andintuitive on the one hand, but is robust and precise on the other hand. As hasbeen recently claimed, using consistent inhomogeneous initial conditionsmitigates 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 firsttime that 21cmFAST over-predicts the 21-cm power spectrum at $z\gtrsim20$ byanother $\mathcal O\left(20\%\right)$, due to the underlying assumption that$\delta_b=\delta_c$, namely that the density fluctuations in baryons and colddark matter are indistinguishable. We propose an elegant solution to thisdiscrepancy by introducing an appropriate scale-dependent growth factor intothe evolution equations. Our analysis shows that this modification will ensuresub-percent differences between 21cmFirstCLASS and the Boltzmann solver CAMB at$z\leq50$ for all scales between the horizon and the Jeans scale. This willenable 21cmFirstCLASS to consistently and reliably simulate the 21-cmanisotropies both in the dark ages and cosmic dawn, for any cosmology.

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