4.2 Transparency and Etherington Relation

The Etherington relation [352] implies that, in a cosmology based on a metric theory of gravity, distance measures are unique: the luminosity distance is (1 + z)2 times the angular diameter distance. This is valid in any cosmological background where photons travel on null geodesics and where, crucially, photon number is conserved. There are several scenarios in which the Etherington relation would be violated: for instance we can have deviations from a metric theory of gravity, photons not travelling along unique null geodesics, variations of fundamental constants, etc. We follow here the approach of [65Jump To The Next Citation Point].

4.2.1 Violation of photon conservation

A change in the photon flux during propagation towards the Earth will affect the supernovae (SNe) luminosity distance measures DL (z) but not the determinations of the angular diameter distance. BAO will not be affected so D (z) A and H (z) measurements from BAO could be combined with supernovae measurements of DL (z) to constrain deviations from photon number conservation. Photon conservation can be violated by simple astrophysical effects or by exotic physics. Amongst the former we find, for instance, attenuation due to interstellar dust, gas and/or plasmas. Most known sources of attenuation are expected to be clustered and can be typically constrained down to the 0.1% level [656, 663]. Unclustered sources of attenuation are however much more difficult to constrain. For example, grey dust [14] has been invoked to explain the observed dimming of Type Ia supernovae without resorting to cosmic acceleration. More exotic sources of photon conservation violation involve a coupling of photons to particles beyond the standard model of particle physics. Such couplings would mean that, while passing through the intergalactic medium, a photon could disappear or even (re)appear! Interacting with such exotic particles, modifying the apparent luminosity of sources. Recently, [65Jump To The Next Citation Point] considered the mixing of photons with scalars, known as axion-like particles, chameleons, and the possibility of mini-charged particles which have a tiny, and unquantized electric charge. In particular, the implications of these particles on the SN luminosity have been described in a number of publications [270, 665, 190Jump To The Next Citation Point, 16Jump To The Next Citation Point] and a detailed discussion of the proposed approach can be found in [100, 101, 66Jump To The Next Citation Point, 65Jump To The Next Citation Point].

Any systematic violations in photon conservation can then be interpreted as an opacity effect in the observed luminosity distance, parametrized through a generic opacity parameter, τ(z), as:

2 2 D L,obs = D L,trueexp[τ(z)].
Note that a negative τ(z) allows for apparent brightening of light sources, as would be the case, for example, if exotic particles were also emitted from the source and converted into photons along the line of sight [190]. Following [66] generic deviations from the Etherington relation can be parametrized as:
DL (z) = DA (z)(1 + z)2+𝜖.
Forecast Euclid constraints are shown in Figure 48View Image, taken from [65Jump To The Next Citation Point]. This assumes that Euclid is accompanied by a supernova sample with the characteristic of a Dark Energy Task Force stage IV survey
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Figure 48: Constraints on possible violation of the Etherington relation in the form of deviations from a perfectly transparent universe (𝜖 = 0). Blue regions represent current constraints while orange are forecast Euclid constraints assuming it is accompanied by a Dark Energy Task Force stage IV supernovae sample.

For particular models of exotic matter-photon coupling, namely axion-like particles (ALPs), chameleons, and mini-charged particles (MCPs), the appropriate parameterization parametrization of τ(z) is used instead.

4.2.2 Axion-like particles

Axion-like particles (ALP) can arise from field theoretic extensions of the standard model as Goldstone bosons when a global shift symmetry, present in the high energy sector, is spontaneously broken. Interestingly, these fields also arise naturally in string theory (for a review see [871]). Chameleon scalar fields are another very interesting type of ALPs [169]. They were originally invoked to explain the current accelerated expansion of the universe with a quintessence field which can couple to matter without giving rise to large fifth forces or unacceptable violations of the weak equivalence principle. A chameleon model with only matter couplings will induce a coupling to photons.

The presence of ALPs will have an impact on observations of SNe if their observed light passes through (intergalactic) magnetic fields. The net effect depends on the ratio of the transition probability to the length travelled through a magnetic field, and a parameter A describing the degree of thermalization of the initial flux (A = 1 means thermalized flux where the photon to ALP transition is compensated by the inverse ALP to photon, making the photon number constant). For the simplest ALP model A = 2∕3, the present and forecast constraints are shown in Figure 49View Image taken from [65Jump To The Next Citation Point].

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Figure 49: Constraints on the simplest Axion-like particles models. Blue regions represent current constraints while orange are forecast Euclid constraints assuming it is accompanied by a Dark Energy Task Force stage IV supernovae sample. Here P ∕L is the conversion probability per unit length and is the relevant parameter for τ (z ) (see [65Jump To The Next Citation Point]).

4.2.3 Mini-charged particles

New particles with a small unquantized charge have been investigated in several extensions of the standard model [443Jump To The Next Citation Point, 105Jump To The Next Citation Point]. In particular, they arise naturally in extensions of the standard model which contain at least one additional U(1) hidden sector gauge group [443, 181]. The gauge boson of this additional U(1) is known as a hidden photon, and hidden sector particles, charged under the hidden U(1), get an induced electric charge proportional to the small mixing angle between the kinetic terms of the two photons. In string theory, such hidden U(1)s and the required kinetic mixing are a generic feature [5, 4, 311, 6, 402]. Hidden photons are not necessary however to explain mini-charged particles, and explicit brane-world scenarios have been constructed [105] where MCPs arise without the need for hidden photons.

More interestingly, [16, 395, 17] pointed out that photons propagating in a background magnetic field can actually pair-produce MCPs without the need for a second photon in the initial state. The opacity in this case is parametrized by κy (z) where y is the comoving distance to the source and κ encloses information on the MCP electric charge and the intervening magnetic field strength. Figure 50View Image shows current and forecast Euclid’s constraints, taken from [65] assuming Euclid is accompanied by a supernova sample with the characteristic of a Dark Energy Task Force stage IV survey.

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Figure 50: Constraints on MCP models. Blue regions represent current constraints while orange are forecast Euclid constraints assuming it is accompanied by a Dark Energy Task Force stage IV supernovae sample.

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