  The internal linear combination (ILC) method is a popular approach for
constructing component-separated maps in cosmic microwave background (CMB)
analyses. It optimally combines observed maps at different frequencies to
produce an unbiased minimum-variance map of a component. When performed in
harmonic space, it is straightforward to analytically compute the contributions
of individual sky components to the power spectrum of the resulting ILC map.
ILC can also be performed on a basis of needlets, spherical wavelets that have
compact support in both pixel and harmonic space, capturing both
scale-dependent and spatially varying information. However, an analytic
understanding of the power spectra of needlet ILC (NILC) component-separated
maps, as needed to enable their use in cosmological parameter inference, has
remained an outstanding problem. In this paper, we derive the first analytic
expression for the power spectra of NILC maps, as well as an expression for the
cross-spectrum of a NILC map with an arbitrary second map, in terms of
contributions from individual sky components. We validate our result using
simulations, finding that it is exact. These results contain useful insights:
we explicitly see how NILC power spectra contain information from contaminant
fields beyond the two-point level, and we obtain a formalism with which to
parameterize NILC power spectra. However, because this parameter dependence is
complicated by correlations and higher-point functions of the component maps
and weight maps, we find that it is intractable to perform parameter inference
using these analytic expressions. Instead, numerical techniques are needed to
estimate parameters using NILC maps -- we explore the use of likelihood-free
inference with neural posterior estimation in a companion paper. Our code to
produce the results in this paper is available in
https://github.com/kmsurrao/NILC-PS-Model.
