We analyze the spectrum of cosmic microwave background (CMB) anisotropies in the timescape cosmology: a potentially viable alternative to homogeneous isotropic cosmologies without dark energy. We exploit the fact that the timescape cosmology is extremely close to the standard cosmology at early epochs to adapt existing numerical codes to produce CMB anisotropy spectra, and to match these as closely as possible to the timescape expansion history. A variety of matching methods are studied and compared. We perform Markov Chain Monte Carlo analyses on the parameter space, and fit CMB multipoles 50 <= l <= 2500 to the Planck satellite data. Parameter fits include a dressed Hubble constant, H_0 = 61.0 km/s/Mpc (+/- 1.3% stat) (+/- 8% sys), and a present void volume fraction f_{v0} = 0.627$ (+/- 2.3% stat) (+/- 13% sys). We find best fit likelihoods which are comparable to that of the best fit Lambda CDM cosmology in the same multipole range. In contrast to earlier results, the parameter constraints afforded by this analysis no longer admit the possibility of a solution to the primordial lithium abundance anomaly. This issue is related to a strong constraint between the ratio of baryonic to nonbaryonic dark matter and the ratio of heights of the second and third acoustic peaks, which cannot be changed as long as the standard cosmology is assumed up to the surface of last scattering. These conclusions may change if backreaction terms are also included in the radiation-dominated primordial plasma.
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