A change in the photon flux during propagation towards the Earth will affect the supernovae (SNe) luminosity distance measures but not the determinations of the angular diameter distance. BAO will not be affected so and measurements from BAO could be combined with supernovae measurements of 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  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,  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, 190, 16] and a detailed discussion of the proposed approach can be found in [100, 101, 66, 65].
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, , as:
For particular models of exotic matter-photon coupling, namely axion-like particles (ALPs), chameleons, and mini-charged particles (MCPs), the appropriate parameterization parametrization of is used instead.
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 ). Chameleon scalar fields are another very interesting type of ALPs . 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 describing the degree of thermalization of the initial flux ( 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 , the present and forecast constraints are shown in Figure 49 taken from .
New particles with a small unquantized charge have been investigated in several extensions of the standard model [443, 105]. 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  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 where is the comoving distance to the source and encloses information on the MCP electric charge and the intervening magnetic field strength. Figure 50 shows current and forecast Euclid’s constraints, taken from  assuming Euclid is accompanied by a supernova sample with the characteristic of a Dark Energy Task Force stage IV survey.
Living Rev. Relativity 16, (2013), 6
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