Imprints of a primordial preferred direction on the microwave background

Rotational invariance is a well-established feature of low-energy physics. Violations of this symmetry must be extremely small today, but could have been larger in earlier epochs. In this paper we examine the consequences of a small breaking of rotational invariance during the inflationary era when the primordial density fluctuations were generated. Assuming that a fixed-norm vector picked out a preferred-direction during the inflationary era, we explore the imprint it would leave on the cosmic microwave background anisotropy, and provide explicit formulas for the expected amplitudes $⟨a_{lm}{a}_{l^{\prime }{m}^{\prime }}^{*}⟩$ of the spherical-harmonic coefficients. We suggest that it is natural to expect that the imprint on the primordial power spectrum of a preferred spatial direction is approximately scale-invariant, and examine a simple model in which this is true.