Cosmological Effects of Scalar-Photon Couplings: Dark Energy and Varying-alpha Models

We study cosmological models involving scalar fields coupled to radiation and discuss their effect on the redshift evolution of the cosmic microwave background temperature, focusing on links with varying fundamental constants and dynamical dark energy. We quantify how allowing for the coupling of scalar fields to photons, and its important effect on luminosity distances, weakens current and future constraints on cosmological parameters. In particular, for evolving dark energy models, joint constraints on the dark energy equation of state combining BAO radial distance and SN luminosity distance determinations, will be strongly dominated by BAO. Thus, to fully exploit future SN data one must also independently constrain photon number non-conservation arising from the possible coupling of SN photons to the dark energy scalar field. We discuss how observational determinations of the background temperature at different redshifts can, in combination with distance measures data, set tight constraints on interactions between scalar fields and photons, thus breaking this degeneracy. We also discuss prospects for future improvements, particularly in the context of Euclid and the E-ELT and show that Euclid can, even on its own, provide useful dark energy constraints while allowing for photon number non-conservation.

We study cosmological models involving scalar fields coupled to radiation and discuss their effect on the redshift evolution of the cosmic microwave background temperature, focusing on links with varying fundamental constants and dynamical dark energy. We quantify how allowing for the coupling of scalar fields to photons, and its important effect on luminosity distances, weakens current and future constraints on cosmological parameters. In particular, for evolving dark energy models, joint constraints on the dark energy equation of state combining BAO radial distance and SN luminosity distance determinations, will be strongly dominated by BAO. Thus, to fully exploit future SN data one must also independently constrain photon number non-conservation arising from the possible coupling of SN photons to the dark energy scalar field. We show how observational determinations of the background temperature at different redshifts can, in combination with distance measures data, set tight constraints on physical processes in the early universe involving the interaction of scalar fields with photons, thus breaking this degeneracy. We discuss prospects for future improvements, particularly in the context of Euclid and the E-ELT and show that Euclid can, even on its own, provide useful dark energy constraints while allowing for photon number non-conservation.

I. INTRODUCTION
The observational evidence for the acceleration of the universe demonstrates that our canonical theories of gravitation and particle physics are incomplete, if not incorrect. A new generation of ground and space-based astronomical facilities (most notably the E-ELT and Euclid) will shortly be able to carry out precision consistency tests of the standard cosmological model and search for evidence of new physics beyond it.
After a quest of several decades, the recent LHC detection of a Higgs-like particle [1,2] finally provides strong evidence in favour of the notion that fundamental scalar fields are part of Nature's building blocks. A pressing follow-up question is whether the associated field has a cosmological role, or indeed if there is another cosmological counterpart.
If there is indeed a cosmologically relevant scalar field, the natural expectation is for it to couple to the rest of the degrees of freedom in the model, unless there are symmetry principles suppressing these couplings. Therefore, * tavgoust@gmail.com † Carlos.Martins@astro.up.pt ‡ up090322024@alunos.fc.up.pt § up110370652@alunos.fc.up.pt ¶ gluzzi@lal.in2p3.fr not allowing for such couplings may significantly bias the analysis of current and future cosmological datasets. In this paper, which is a sequel to [3], we focus on potentially observable signatures of the interaction of cosmological scalar fields with the electromagnetic sector, specifically changes to the standard evolution of the Cosmic Microwave Background temperature with redshift, The two canonical ways to reconstruct the evolution of the CMB temperature with redshift rely on spectroscopy of molecular/ionic transitions triggered by CMB photons [4][5][6] and on the thermal Sunyaev-Zel'dovich (SZ) effect towards clusters [7][8][9][10]. New sources will soon become available for measurements by the above methods, and entirely new methods for measuring T (z) are also being developed, so it is timely to consider their impact on cosmology and on searches for new physics.
While in [3] deviations from the standard evolution were constrained in a purely phenomenological (but nevertheless model-independent) way, here we discuss in considerably more detail the possible links between T (z) and two other astrophysical observables: measurements of nature's fundamental dimensionless couplings and the equation of state of dark energy. As emphasised in [11], such joint measurements will be crucial for the next generation of cosmological experiments, which will carry out precision consistency tests of the underlying scenarios. We will base our discussion on two specific examples, ESA's Euclid 1 [12] and ESO's European Extremely Large Telescope (E-ELT) [13],

II. CMB TEMPERATURE EVOLUTION
We start with a brief review on the redshift evolution of the CMB temperature. This is not meant to be exhaustive, but simply to introduce the formalism that will be used. Further details can be found in the original analysis of [14], as well as in [3]. We will be assuming the presence of a canonical scalar field in an FRW background, with a Lagrangian with Introducing a coupling C φ between the scalar field and the radiation fluid, the evolution equations for the radiation energy and number densities reaḋ where Ψ depends on the coupling C φ . This will in general distort the behaviour of the radiation fluid, and in particular the photon temperature-redshift relation, away from its standard evolution. Restricting our attention to the observationally relevant case of adiabatic evolution, the adiabaticity condition implies [3,14] and one obtains the following evolution equation for the CMB temperatureṪ For future use, let us define a correction to the standard behaviour, y(z), such that and clearly y(z) = 1 corresponds to the standard cosmological model; we can then write 1 http://www.euclid-ec.org The simplest ansatz for the source term Ψ in Eqs. (5) and (7) is Ψ = 3βHn γ , which yields the relation this has been fairly widely used in the past, with the available measurements of T (z) providing a constraint on the parameter β [3,6,10]. The corresponding evolution of the radiation density is A generalisation has been suggested by Bassett and Kunz [15] 2 , with Ψ = 3βH(1 + z) λ−1 n γ and again assuming adiabaticity. The previous case is recovered for λ = 1, while for λ = 1 we get Naturally, if we linearise in β and then in redshift we recover the usual linear modification to the standard temperature-redshift relation, For this case we have for the scalar field energy densitẏ which might possibly be related to the properties of dark energy (with a suitable parametrisation, e.g. for its equation of state w φ ), as will be further discussed below.

III. LINKS TO VARYING ALPHA
In the Bekenstein-Sandvik-Barrow-Magueijo (BSBM) class of models [16] there is a scalar field ψ coupling to the Maxwell F 2 term in the matter Lagrangian, thus yielding a varying fine-structure constant, α. In this case we havė with and we immediately have, assuming adiabaticity, thaṫ ln leading to Regardless of the BSBM specific case, we can take this as a phenomenological relation that can be tested observationally.
Recently, Webb et al. [17] found a significant indication (at the 4.2-σ level,) for a spatial dipole in the fine structure constant, α. If this is not a hidden systematic effect, and under the assumption that this class of models is correct, there should also be an additional CMB temperature dipole (that is, in addition to the standard one) in the same direction of the α dipole, and with µKelvin amplitude. Although this is beyond the scope of the current analysis, it should be possible to disentangle this from the 'usual' CMB dipole. In particular, a signal of this magnitude may be of some relevance for the recently released Planck results.
Note that above we did not need an explicit expression for the redshift behaviour of α or T . In this class of models, if one neglects the recent dark energy domination one can in fact find an analytic solution (i.e., a matter era one) or in other words where k is a dimensionless parameter to be constrained by data. This is also useful more generally, as a one-parameter toy model alternative to the standard β parametrisation. Naturally the two coincide at low redshifts (i.e., if linearised), but at higher redshifts they will differ. The parametrisation described by Eq. (20) must satisfy the atomic clock bounds at z = 0. With the above redshift dependence for α we have and consequently today we have converting years into seconds (or vice-versa) one finds that k should not exceed This is smaller than the value obtained by fitting to the pre-dipole data by Webb and collaborators [19,20], but consistent with the dipole one: if one ignores the direction of the sources on the sky and naively fits the entire dataset to the above function, there is no strong evidence for a non-zero k. The spatial dependence of these measurements will be addressed in subsequent work. As discussed in [3] the typical precision expected for the temperature measurements will significantly increase with the next generation of facilities. Specifically we consider ESPRESSO [21], under construction for the VLT, and the planned HIRES for the E-ELT (for which the CODEX Phase A study [22] provides a realistic benchmark); their typical expected precisions in the temperature measurements are respectively and ∆T Hires ∼ 0.07 K , which are about three orders of magnitude larger than what one would expect the temperature variation to be in the BSBM model at z ∼ 4, on the assumption that the Webb detection is correct. To get an intuitive picture of the sensitivity of T (z) measurements within this class of models, one can determine what would be the smallest value of k detectable by a single measurement by those two future spectrographs. This result is shown in Fig. 1, giving then a detection limit around k = 0.004 for HIRES and k = 0.02 for ESPRESSO. Note that this obviously depends on the redshift at which the measurements are made: the higher the redshift, the stronger the constraints that can be achieved. Clearly a detection of a Webb-level value of k would require a very large number of sources, which are not currently known. However, this may be possible for clusters (whose expected sensitivity in the case of Planck is also depicted): even though they are at much lower redshifts (when deviations from the standard behaviour are correspondingly smaller), samples of thousands of clusters are expected to become available very soon. This is further discussed in [3,23].
Recently Muller et al. [24] have provided the very tight measurement T = (5.08 ± 0.10) K at z = 0.89, using radio-mm molecular absorption measurements. With the ALMA array [25] gradually becoming available, the number and quality of these measurements will steadily increase, although they will be time-consuming and there will be strong competition for ALMA observing time. Nevertheless this method offers the exciting prospect of a new tool to map T (z) over a very wide redshift range, potentially up to z = 6.5 (J. Black, private communication).
One can also check that this agrees with the Oklo natural nuclear reactor bound (corresponding to an effective redshift z = 0.14), which is according to Petrov et al. [26], or according to Gould et al. [27]; this does happen, but in any case there are several caveats with interpreting these Oklo analyses. The most obvious one is that the nuclear reactions being considered are mostly sensitive to the strong nuclear coupling, so assuming that only the fine-structure constant α varies while the rest of the physics is unchanged is a naive assumption. At higher redshifts, an additional consistency test will be provided by the redshift drift measurements carried out by highresolution spectrographs like HIRES [11,28].
In [3] we have shown that if the temperature-redshift relation is changed to in models where photon number is not conserved, then the distance duality relation is correspondingly affected where d L and d A are the luminosity and angular diameter distance measures respectively. Therefore for this class of varying-alpha models we predict that Again this relation can be tested for both time and/or spatial variations of α; even though the effect is small, there are hundreds (and in the future there will be thousands) of type Ia supernova measurements. Recently [29] has claimed that there is also a 'supernova dipole' aligned with the α dipole, but that analysis ignores the effects of varying α on the supernova brightness (see [30] for a succinct discussion); such effects must be included in a fully consistent analysis. There are therefore a number of consistency tests for this class of models, involving on the one hand astrophysical measurements of T and α, and on the other hand distance measurements such as d L .

IV. LINKS TO DARK ENERGY
Now let us instead consider that the scalar field coupled to radiation is the one responsible for dark energy, and specifically thaṫ thus we assume that some fraction of the energy in the scalar field is lost to radiation. For our present purpose we need to assume a specific dark energy parametrisation, so we shall take the simple and well-known Chevallier- in other words the coupling produces an effective correction to the CPL equation of state It follows that within the context of this class of models we can use T (z) measurements to impose constraints on the dark energy equation of state parameters w 0 and w a , as well as on the coupling k. This has already been done in two papers by Jetzer & Tortora [32,33], although there this was done for a specific and somewhat unrealistic decaying-lambda model.
For the radiation component we havė ρ γ + 4Hρ γ = 3kHρ φ ; (37) in this case the evolution equation for the CMB temperature, written in terms of the correction term y defined in (8), is as follows We can then substitute ρ φ by the expression above, while for ρ γ we use ρ γ ∝ T 4 . The resulting differential equation for y is not in general analytically integrable, but we can write it as which may be integrated numerically.
We can obtain analytic approximations in two useful particular limits. First, for a constant equation of state (that is, w a = 0) and assuming a small k we have which corresponds to The second (and more specific) limiting case corresponds to small redshifts, z ≪ 1; here it is convenient to first change variables, integrating in redshift rather than the scale factor, that is We can now linearise the integral and then integrate, finding which corresponds to Interestingly, at first order this just depends on k, and not on w 0 (a w 0 dependency does exist at second order). Figure 3 shows the deviation of the temperature relative to the standard model as a function of redshift for the general case (left), the constant equation of state case (middle) and for the small redshift approximation (right). Given the sensitivities discussed in the previous section, for this parametrisation HIRES will be precise enough to put bounds on this type of models.
Notice that for sufficiently large values of k (in absolute value) one must have a negative k, otherwise the sign of the y(z) factor can change. One can thus infer the acceptable values that k can take (as a function of w 0 ) in order to have a y(z) non-negative for all redshifts. For the simplest case with w a = 0 and with typical values of the other relevant parameters, we find the approximate bound In the case of the small redshift approximation (assumed, quite optimistically, to hold up to redshift z ∼ 1) the temperature evolution does not depend on w 0 , and one immediately finds a limit value for k of k max < 10 −6 . Another way to set some rough constraints on the parameters is to check whether or not the temperature reaches 3000K in the modified equations. As before, it's useful to get an intuitive idea for the sensitivity of T (z) measurements on the dark energy equation of state in this class of scalar field models. Assuming that w a = 0 and the present matter density of the universe is known, we find that a set of ESPRESSO T (z) measurements could on its own constrain w 0 to a precision δw 0 ∼ 0.4, while a HIRES-like spectrograph can reach δw 0 ∼ 0.15; these numbers apply for an optimistic choice of fiducial model with k ∼ −10 −5 . (Note that there constraints are weaker than those found by [32,33], but that's due to the fact that these authors are assuming a specific model were w 0 and k are not independent, i.e. they have a single free parameter to constrain.) On the other hand, the analysis does not include T (z) measurements from clusters or from ALMA, which will be discussed in more detail elsewhere.
As a final comment we point out that we could instead assumeρ which is the particular case of Bassett and Kunz [15] for λ = 1, cf. Eq. (14). In this case the evolution of the radiation density and its temperature are trivially as before, but we have more complex evolution for the dark energy density: the dark energy equation of state effectively gets a β-correction, and therefore a constraint on β may be inferred, for example, from combining type Ia supernova measurements with other distance measure determinations probing the cosmic expansion history.

V. CONSTRAINTS FROM CURRENT DATA
Before discussing in more detail some prospects for the next generation of relevant observational facilities, we study the constraints that can be obtained from current data. These are already useful, even for the general case, given by equations (35), (39) and (30).
The evolution of the dark energy density (35) affects cosmic expansion (predominantly at smaller redshifts z 1 when dark energy starts to dominate) so all distance measures depend explicitly 3 on w 0 and w a . We can use, for example, type Ia Supernova measurements (giving d L (z)), BAO (yielding H(z) and d A (z)), galaxy ageing (providing independent measurements of H(z)) and H 0 determinations. On the other hand, equations (39) and (30) have a strong dependence on k and a different dependence on w 0 , w a , allowing degeneracies to be broken.
As was pointed out above, in the models we are considering, in which the deviation from the standard T (z) relation is due to a coupling of CMB photons with the dark energy scalar field, one would expect that the same field also couples to optical photons, thus affecting luminosity distances, as discussed in [3]. Within a given model, one can then translate T (z) deviations to violations of the distance duality relation (31). Note that on general grounds, the coupling is expected to be weaker for lower photon frequencies, so assuming a frequencyindependent coupling should yield conservative bounds on T (z) violations from SN (or other optical) data.
The top left panel of Fig. 4 shows 2-parameter joint constraints (68% and 95%, having marginalised over k and Ω m ) for the SN (blue filled contours), H(z) (dashed lines) and combined SN+H(z) data (solid line contours).
Having allowed for violation of photon number conservation through the parameter k, the SN constraints on the dark energy equation of state are weak, but the constraints improve dramatically with the inclusion of H(z) data that are not affected by k. The region near w 0 = 0, favoured (at the 1σ level) by the SN data, corresponds to negative values of k (i.e. photon dimming due to decay to scalar particles) as is evident from the top right panel of Fig. 4. The bottom level panel then shows that corresponds to large values of Ω m , so low values for the dark energy density parameter. This is the wellknown degeneracy between dark energy and photon number non-conservation, which gets broken by including the H(z) data favouring Ω m near 0.25. Finally, in the bottom right panel we show the corresponding constraints in the w 0 − Ω m plane, having marginalised over k and w a . The parameter w a is currently weakly constrained (ranging from w a ≃ −4 to w a ∼ 2) and it is not shown here.
For comparison, we also show constraints on the dark energy equation of state w for a flat wCDM model, again allowing for k ∈ [−5, 5]×10 −5 (Fig. 5, upper panels), and also constraints in the w 0 − w a and w − Ω m planes for a CPL-CDM and wCDM models respectively, but this time assuming photon conservation, k = 0 (Fig. 5, lower panels). The last two plots in figure 5 highlight the dramatic effect of k for the SN data. On the contrary, the effect of k for the H(z) data is insignificant; that is, H(z) alone does not significantly constrain k as mentioned aboverefer to equation (35) and the ranges of w 0 and k.
Let us now consider a more realistic prior on Ω m , using results from CMB WMAP9 [44]. We first consider the WMAP9 constraint 0.15 ≤ Ω m ≤ 0.53 at 95% C.L. based on the wCDM model. Thus, we allow Ω m in the interval [0, 1] but assume a Gaussian prior (centred at 0.34 and with standard deviation 0.095) when marginalising over Ω m . For k, we take a flat prior over the range k ∈ [−5, 5] × 10 −5 as above. Fig. 6 (upper panels) shows our current SN+H(z) constraints for flat CPL-CDM (left) and wCDM (right) models for this choice of priors. The stronger prior on Ω m now disfavours the large Ω m region, leading to an extension of the 68% SN contour towards the region with {w 0 ∼ −1, Ω m ∼ 0.3, k ∼ 0} (also refer to Fig. 4). However, the large variation in k allowed by our flat prior makes the area of the allowed SN contours much larger than what one would obtain if the coupling of the scalar field to photons was not taken into account (cf. Fig. 5). In the same figure, we also show the corresponding SN contours that one would obtain if k was constrained at the ∼ 10 −5 level. The SN contours shrink a little, but the total constraint is still dominated by the H(z) data. For the Union2.1 SN dataset to become competitive with the current H(z) determinations, k must be constrained at the < 10 −6 (cf. Fig 4). We plan to study in detail constraints on k from SZ T(z) measurements of clusters (cf. [10]) in a follow-up publication.
As an alternative approach, we also use the WMAP9+ACT+SPT+BAO+H0 result Ω m = 0.263 ± 0.015 for the CPL-CDM model (waCDM in the WMAP data tables) to set a much stronger prior on Ω m . In this case, we also remove the corresponding BAO datapoints from our H(z) sample (the 3 WiggleZ datapoints at z = 0.44, 0.60 and 0.73). The resulting constraints are shown in Fig. 6 (lower panels). Again, the overall constraint is dominated by the H(z) data, but the bound on w a (left) is now much stronger.
Note that we have chosen to show constraints for the above Ω m priors as extreme examples (conservative vs strong priors). The corresponding Planck low ℓ+BAO prior, namely 0.225 ≤ Ω m ≤ 0.362 at 95% C.L. for the CPL-CDM model (called wwa model in the Planck parameter tables), gives constraints at a level between those of Fig. 4 and Fig. 6 (but closer to Fig. 6).

VI. CONSTRAINTS FROM FUTURE DATASETS
Let us now proceed to present forecasts for future datasets. We will be particularly interested in studying the impact of the Euclid mission, and will start by considering one of its probes, BAO from the Euclid wide survey, which is expected to cover ∼15000 deg 2 of extragalactic sky down to a redshift of order 2. Assuming the same conservative priors as in Fig. 4, we repeat the above analysis with simulated data for Euclid BAO and a SNAP-like SN mission. BAO radial distance errors have been estimated using the code developed by Seo & Eisenstein [45] adapted for Euclid estimated parameters [12]. For SN we follow [46] for a Dark Energy Task Force Stage IV SN mission.
The relevant constraints are shown in Fig. 7. As before, we show 68% and 95% likelihood contours for the SN data in blue (filled contours), while the transparent dashed line contours are for H(z) data. The combined constraints are shown as solid line, transparent contours. Note in Fig. 7 that the constraint on w a starts to become interesting with Euclid BAO+SNAP-like SN, even allowing for photon number non-conservation. However, in the absence of an independent constraint on k, much Having allowed for photon number non-conservation, the SN data alone do not strongly constrain dark energy (and favour w0 ∼ 0 at the 1σ level), but the inclusion of the H(z) data strongly improves dark energy constraints. Top Right-Bottom Left: The SN data favoured region w0 ∼ 0 corresponds to negative k (photon dimming) and large Ωm, exemplifying the well-known dark energy-photon dimming degeneracy. This gets broken by using the H(z) data which favour Ωm ≃ 0.25. Bottom Right: In the CPL parametrisation, and having allowed for non-zero k, the SN constraints are weak even for the most strongly constrained dark energy parameter, w0. Cf. of the observed SN dimming can be explained by photon conversion to scalar particles, so the SN contours grow (compared to the case k = 0) and the total constraint is dominated by H(z).
To highlight this point, we also show in the middle and bottom left panels (dotted, transparent contours) the SN constraints achievable if the photon number violation parameter, k, could be constrained at the level 10 −6 . This can be seen to have an important effect on the joint contours, as the SN-only constraints become competitive to the H(z)-only ones. Note, in particular, that the horizontal band around w 0 (bottom left plot) for the SN data disappears, as it corresponds to a region with k ≃ −10 −5 (cf. top right plot). We will discuss current and future constraints on k, and their quantitative effect on dark energy parameter bounds, in a follow-up publication.
It has also been proposed that Euclid carries out a dedicated SN survey, which could yield up to a few thousand SNeIa up to redshift 1.5 [47]. This makes Euclid an ideal instrument to constrain the models we are studying, capable of delivering both radial/angular diameter distance measurements and luminosity distances, and thus minimising systematics. Based on the recently studied 6-month 'AAA' Euclid survey [48], one can expect more than 1700 SNe Ia in the redshift range 0.75 < z < 1.55. We adopt the assumptions in this study and repeat our forecast analysis, now using only Euclid for both BAO and Supernovae. (We neglect the correlation between errors in different redshift bins; the effects or doing this are expected to be relatively small, given the other approximations we are also making.) Our results are shown in Fig. 8.
From Fig. 8 we see that Euclid can, even on its own, provide useful constraints on Dark Energy allowing for photon number non-conservation, especially for wCDM models. Note however, that the SN-only constraints (blue filled contours) are weak and the joint constraint is dominated by the BAO H(z). In particular, k is not constrained in this prior range. Naturally, these constraints become much stronger by combining the Euclid SNe with a low-redshift sample, e.g. from a SNAP-like mission (Fig. 9).
Supernova measurements with Euclid (or SNAP), can only reach a maximum redshift of around z ∼ 1.7. However, the next generation of ground and space-based optical-IR telescopes will significantly extend this redshift range, which will bring about major improvements in terms of dark energy constraints. Specifically, JWST (through NIRcam imaging), should find about 50 supernovae and measure their light curves [49], and with E-ELT spectroscopy provided by HARMONI [50] the redshift and supernova type can be confirmed. The redshift range of this high-z sample is expected to be 1 < z < 5. The redshift distribution of these supernovas is not easy to extrapolate, since even the most detailed current studies such as those of the SNLS team [51] only reach out to z ∼ 1. In the absence of a specific redshift distribution, we will simply assume it to be uniform in the above range. With these assumptions, our forecasts are shown in Fig. 10.
For comparison, we also consider the alternative case of the TMT (also with JWST support) [52], which expects to find about 250 Supernovas in the range 1 < z < 3; this is shown in Fig. 11. We empahsise that the numbers we use for the E-ELT and the TMT come from assumptions made in Phase A studies of their relevant instruments; the amounts of telescope time required for gathering each dataset are not necessarily comparable.
We can see that these high redshift supernovae lead to significantly improved constraints, compared to the previous Euclid+SNAP case. On the other hand, the constraints from the E-ELT and the TMT are comparable, indicating that the larger redshift lever arm partially compensates the smaller number of supernovae. Table I provides a comparison of the uncertainties in the various model parameters obtainable in each case.

VII. CONCLUSIONS
We have studied two typical classes of phenomenological scenarios involving scalar fields coupled to radiation, specifically considering their effects on the redshift evolution of the cosmic microwave background temperature. In the first of these a BSBM-type field provides a time variation of the fine-structure constant α, while in the  other the dynamical scalar field is responsible for the recent acceleration of the universe.
Our analysis shows that the effects of the coupling of scalar fields to photons, which include effects on luminosity distances, dramatically weaken current and future constraints on cosmological parameters. In particular, our results strongly suggest that in order to fully exploit forthcoming SN data one must also independently constrain photon-number non-conservation arising from the possible coupling of SN photons to the dark energy scalar field. In this context, direct measurements of the background temperature at different redshifts (such as those provided by ALMA and HIRES) can be used in combination with distance measures to break parameter degeneracies and significantly improve constraints on physical processes in the early universe.
Nevertheless, our analysis demonstrates that Euclid can, even on its own, provide useful dark energy constraints while allowing for the possibility of photon number non-conservation. Naturally, stronger constraints can be obtained in combination with other probes. In this context its worth emphasising that the only Euclid probes we considered are BAO and the proposed SN survey. The Euclid mission includes further probes which can be used to tighten the constraints. In this sense our results are conservative (but an analysis of this more gen-eral case is left for future work).
We have also considered the role of the increased redshift lever arm provided by type Ia supernovas at high redshift (z > 2), such as can be found by JWST and ground-based extremely large telescopes. Specifically, we have considered two different samples which are meant  Fig. 7. While Euclid alone cannot strongly constrain k (violations of photon number conservation) it will still provide useful constraints on dark energy parameters, especially within the wCDM model. However, the joint constraints (solid transparent contours) are determined predominantly by the BAO data (dashed transparent contours), while the SN-only bounds (blue filled contours) are weak. Therefore, these constraints become much stronger when a low-redshift SN sample is a also included, cf. Fig. 9.
to be representative of the E-ELT and TMT, with the former going deeper into the matter era while the latter has five times more supernovas. The constrains from both datasets (in combination with lower-redshift measurements) are quite comparable: the E-ELT provides a better constraint on the matter density (for which the increased redshift lever arm is the dominant factor) while the TMT provides better constraints on the dark energy parameters w 0 and w a (since, at least in the models we considered, dark energy is negligible at the higher redshifts, the larger number of supernovas provides the dominant effect).
Finally, let us point out that HIRES exquisite precision and stability will give it two other abilities of note: it will be able to make the first measurements of the cosmological redshift drift and also to map out the behaviour of fundamental couplings from about z ∼ 0.5 to z ∼ 4 and possibly well beyond. As discussed in [11,53,54], both of these will provide further constraints on dynamical dark energy as well as key consistency tests of many of these scenarios. Thus the combination of Euclid and the E-ELT offers us the prospect of a complete mapping of the dynamics of the dark sector of the universe all the way up to redshift z ∼ 4, and our work highlights the point that mapping the bright sector of the universe-through T (z) measurements-also plays a role in this endeavour. of Nottingham. We acknowledge useful comments and suggestions from Isobel Hook and other members of the Euclid Cosmology Theory and Transients & Supernovae Science Working Groups. We also acknowledge use of the Planck Mission Cosmological Parameters Products, which are publicly available from the Planck Legacy Archive (http://www.sciops.esa.int/Planck).