Blackness of the cosmic microwave background spectrum as a probe of the distance-duality relation

Abstract

A violation of the reciprocity relation, which induces a violation of the distance duality relation, reflects itself in a change in the normalisation of the cosmic microwave spectrum in such a way that its spectrum is grey. We show that existing observational constraints imply that the reciprocity relation cannot be violated by more than 0.01% between decoupling and today. We compare this effect to other sources of violation of the distance duality relations which induce spectral distortion of the cosmic microwave background spectrum.PACS numbers: 98.80. Cq, 04.80.Cc In the standard cosmological model [1,2], in which the universe is described by a spatially homogeneous and isotropic geometry of the Friedmann-Lemaître family, the luminosity distance D L and angular diameter distance D A are related by the distance duality relation,where z is the redshift. This relation is actually far more general [3,4]. It can be shown (see Ref.[5] for a demonstration) that it holds in any spacetime in which (1) the reciprocity relation holds and (2) the number of photons is conserved. The reciprocity relation connects the area distances up and down the past lightcone and is a relation between the source angular distance, r s , and the observer area distance, r o . The former is defined by considering a bundle of null geodesics diverging from the source and which subtends a solid angle dΩ 2 s at the source. This bundle has a cross section d 2 S o at the observer and the source angular distance is defined by the relationSimilarly, the observer area distance is defined by considering a reciprocal null geodesic bundle converging at the observer byAs long as photons propagate along null geodesics and the geodesic deviation equation holds then these two distances are related by the reciprocity relation [5] regardless of the metric and matter content of the spacetime. While r o is related to the angular distance, the solid angle dΩ 2 s cannot be measured so that r s is not an observable quantity. If one further assumes that the number of photons is conserved, the source angular distance is related to the luminosity distance, by D L = (1 + z)r s , which leads to the distance duality relation (1), whereVarying the distance-duality relation: Violations of the distance duality relation (1) can arise from a violation of the reciprocity relation, which can occur in the case where photons do not follow (unique) null geodesics (e.g. in theories involving torsion and/or non-metricity or birefringence [6]) or (2) from the non-conservation of photons, which occur e.g. when photons are coupled to axions [7] or to gravitons in an external magnetic field [8], to Kaluza-Klein modes associated with extra-dimensions [9], or to a chameleon field [10][11][12]. The fact that such a violation of the distance duality relation can account for a dimming of the supernovae luminosity [13,14], since e.g. in the case of photon-axion mixing the luminosity distance has to be rescaled as D L / 1 − P γ−a (r) with P γ−a being the probability for a photon to oscill...

Cosmological tests of the disformal coupling to radiation

“…With this we find that 17) where the round brackets denote symetrization over the indices. Changing back to the true conformal time coordinate η, whose directions we label with 0 and using the fact that dη dη = c p we find that…”

“…The measured spectrum will be related to the intensity I(k, η) of equation (5.5) by a geometrical factor which depends on the way the reciprocity relation is modified by the variation of the speed of light [17] …”

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“…Perhaps owing to a lensing magnification bias, they found a 2σ violation caused by an excess in brightening of SNIa at z > 0.5. Ellis et al (2013) propose an interesting test of Eq. (1) using the cosmic microwave background spectrum.…”

Constraints on cosmic distance duality relation from cosmological observations

“…see refs. [4][5][6][7][8][9][10][11][12][13][14][15][16] and references * Electronic address: xiajq@bnu.edu.cn therein). If η obtained from the d L and d A datasets is different from the unity, the CDDR relation is violated.…”