Erratum: Imprints of a primordial preferred direction on the microwave background [Phys. Rev. D75, 083502 (2007)]

Abstract

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 ′ m ′ 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.

Anisotropy in solid inflation

“…[43] contracted the Einstein equations with a normal vector n µ , to obtain their equations (9) and (10). In the Bianchi-I geometry (29) and in our notation, these equations read, respectively,…”

“…In this expression, K and σ µν are, respectively, the trace and the tracefree part of the extrinsic curvature on surfaces orthogonal to n µ . For us , K = 3α, and σ µν σ µν = 6σ 2 , which are, respectively, the isotropic and anisotropic Hubble rates in (29). As long as the dominant and strong energy condition hold, D, S ≥ 0, the two equations (41) imply [43] …”

“…First, from Eq. (63) we can read the anisotropic amplitude of the power-spectrum g * in the parameterization [29] …”

“…On different scales (and marginalizing over the preferred direction) Large-Scale Structure data analysis constrain −0.41 < g * < 0.38 at 95% C.L. [61] (the amplitude of the anisotropy may in general be scale dependent [29]). Therefore a 10% level anisotropy, |g * | = 0.1 ((1%) level, |g| * = 0.01) would correspond to an anisotropy parameter σ 0.1 (0.01).…”

“…Such models are characterized by (i) a suitable vector potential V A 2 [26], (ii) a specific coupling to the scalar curvature L ⊃ 1 12 RA 2 [8,27,28], or (iii) a lagrange multiplier λ that enforces a fixed norm for the vector, L ⊃ λ A 2 − v 2 2 [29]. Due to the broken gauge invariance, the vector field has also a longitudinal mode.…”