3.5 Isocurvature modes

At some time well after inflation but deep into the radiation era the universe is filled with several components. For instance, in the standard picture right before recombination there are four components: baryons, cold dark matter, photons and neutrinos. One can study the distribution of super-Hubble fluctuations between different species, which represent the initial conditions for the subsequent evolution. So far we have investigated mostly the adiabatic initial conditions; in this section we explore more generally the possibility of isocurvature initial conditions. Although CMB data are the most sensitive to constrain isocurvature perturbations, we discuss here the impact on Euclid results.

3.5.1 The origin of isocurvature perturbations

Let us denote by ρα the energy density of the component α. Perturbations are purely adiabatic when for each component α the quantity ζα ≡ − 3H δρα ∕˙ρα is the same [949, 617]. Let us consider for instance cold dark matter and photons. When fluctuations are adiabatic it follows that ζcdm = ζγ. Using the energy conservation equation, ˙ρα = − 3H (ρα + pα) with pcdm = 0 and pγ = ργ∕3, one finds that the density contrasts of these species are related by

δρcdm 3δργ ------= -----. (3.5.1 ) ρcdm 4 ργ
Using that ncdm ∝ ρcdm and n γ ∝ ρ3γ∕4, this also implies that particle number ratios between these species is fixed, i.e., δ(ncdm ∕nγ) = 0.

When isocurvature perturbations are present, the condition described above is not satisfied.18 In this case one can define a non-adiabatic or entropic perturbation between two components α and β as 𝒮 ≡ ζ − ζ α,β α β, so that, for the example above one has

δρcdm- 3δρ-γ δ(ncdm∕n-γ) 𝒮cdm,r = ρ − 4 ρ = n − n . (3.5.2 ) cdm γ cdm γ

A sufficient condition for having purely adiabatic perturbations is that all the components in the universe were created by a single degree of freedom, such as during reheating after single field inflation.19 Even if inflation has been driven by several fields, thermal equilibrium may erase isocurvature perturbations if it is established before any non-zero conserving quantum number was created (see [950]). Thus, a detection of non-adiabatic fluctuations would imply that severalvscalar fields where present during inflation and that either some ofvthe species were not in thermal equilibrium afterwards or that some non-zero conserving quantum number was created before thermal equilibrium.

The presence of many fields is not unexpected. Indeed, in all the extension of the Standard Model scalar fields are rather ubiquitous. In particular, in String Theory dimensionless couplings are functions of moduli, i.e., scalar fields describing the compactification. Another reason to consider the relevant role of a second field other than the inflaton is that this can allow to circumvent the necessity of slow-roll (see, e.g., [331]) enlarging the possibility of inflationary models.

Departure from thermal equilibrium is one of the necessary conditions for the generation of baryon asymmetry and thus of the matter in the universe. Interestingly, the oscillations and decay of a scalar field requires departure from thermal equilibrium. Thus, baryon asymmetry can be generated by this process; examples are the decay of a right-handed sneutrino [419Jump To The Next Citation Point] or the [10Jump To The Next Citation Point] scenario. If the source of the baryon-number asymmetry in the universe is the condensation of a scalar field after inflation, one expects generation of baryon isocurvature perturbations [664]. This scalar field can also totally or partially generate adiabatic density perturbations through the curvaton mechanism.

In summary, given our ignorance about inflation, reheating, and the generation of matter in the universe, a discovery of the presence of isocurvature initial conditions would have radical implications on both the inflationary process and on the mechanisms of generation of matter in the universe.

Let us concentrate on the non-adiabatic perturbation between cold dark matter (or baryons, which are also non-relativistic) and radiation 𝒮 = 𝒮cdm,γ. Constraints on the amplitude of the non-adiabatic component are given in terms of the parameter α, defined at a given scale k0, by P 𝒮 − Pζ ≡ α − (1 − α ), see e.g., [120Jump To The Next Citation Point, 113, 525Jump To The Next Citation Point]. As discussed in [542], adiabatic and entropy perturbations may be correlated. To measure the amplitude of the correlation one defines a cross-correlation coefficient, ∘ ----- β ≡ − P 𝒮,ζ∕ P𝒮P ζ. Here P𝒮,ζ is the cross-correlation power-spectrum between 𝒮 and ζ and for the definition of β we have adopted the sign convention of [525Jump To The Next Citation Point]. Observables, such as for instance the CMB anisotropies, depend on linear combinations of ζ and 𝒮. Thus, constraints on α will considerably depend on the cross-correlation coefficient β (see for instance discussion in [403]).

If part of the cold dark matter is created out of equilibrium from a field other than the inflaton, totally uncorrelated isocurvature perturbations, with β = 0, are produced, as discussed for instance in [336, 582Jump To The Next Citation Point]. The axion is a well-known example of such a field. The axion is the Nambu–Goldstone boson associated with the [715] mechanism to solve the strong-CP problem in QCD. As it acquires a mass through QCD non-perturbative effects, when the Hubble rate drops below its mass the axion starts oscillating coherently, behaving as cold dark matter [742, 2, 316]. During inflation, the axion is practically massless and acquires fluctuations which are totally uncorrelated from photons, produced by the inflaton decay [805, 578, 579, 910]. As constraints on α β=0 are currently very strong (see e.g., [118, 526]), axions can only represent a small fraction of the total dark matter.

Totally uncorrelated isocurvature perturbations can also be produced in the curvaton mechanism, if the dark matter or baryons are created from inflation, before the curvaton decay, and remain decoupled from the product of curvaton reheating [546Jump To The Next Citation Point]. This scenario is ruled out if the curvaton is entirely responsible for the curvature perturbations. However, in models when the final curvature perturbation is a mix of the inflaton and curvaton perturbations [545], such an entropy contribution is still allowed.

When dark matter or baryons are produced solely from the curvaton decay, such as discussed by [597], the isocurvature perturbations are totally anti-correlated, with β = − 1. For instance, some fraction of the curvaton decays to produce CDM particles or the out-of-equilibrium curvaton decay generates the primordial baryon asymmetry [419, 10].

If present, isocurvature fields are not constrained by the slow-roll conditions imposed on the inflaton field to drive inflation. Thus, they can be highly non-Gaussian [582, 126]. Even though negligible in the two-point function, their presence could be detected in the three-point function of the primordial curvature and isocurvature perturbations and their cross-correlations, as studied in [495, 546].

3.5.2 Constraining isocurvature perturbations

Even if pure isocurvature models have been ruled out, current observations allow for mixed adiabatic and isocurvature contributions (e.g., [267, 896, 525, 914Jump To The Next Citation Point]). As shown in [902Jump To The Next Citation Point, 40, 914, 544, 184Jump To The Next Citation Point, 847], the initial conditions issue is a very delicate problem: in fact, for current cosmological data, relaxing the assumption of adiabaticity reduces our ability to do precision cosmology since it compromises the accuracy of parameter constraints. Generally, allowing for isocurvature modes introduces new degeneracies in the parameter space which weaken constraints considerably.

The cosmic microwave background radiation (CMB), being our window on the early universe, is the preferred data set to learn about initial conditions. Up to now, however, the CMB temperature power spectrum alone, which is the CMB observable better constrained so far, has not been able to break the degeneracy between the nature of initial perturbations (i.e., the amount and properties of an isocurvature component) and cosmological parameters, e.g., [538, 902]. Even if the precision measurement of the CMB first acoustic peak at ℓ ≃ 220 ruled out the possibility of a dominant isocurvature mode, allowing for isocurvature perturbations together with the adiabatic ones introduce additional degeneracies in the interpretation of the CMB data that current experiments could not break. Adding external data sets somewhat alleviates the issue for some degeneracy directions, e.g., [903, 120, 323]. As shown in [184], the precision polarization measurement of the next CMB experiments like Planck will be crucial to lift such degeneracies, i.e., to distinguish the effect of the isocurvature modes from those due to the variations of the cosmological parameters.

It is important to keep in mind that analyzing the CMB data with the prior assumption of purely adiabatic initial conditions when the real universe contains even a small isocurvature contribution, could lead to an incorrect determination of the cosmological parameters and on the inferred value of the sound horizon at radiation drag. The sound horizon at radiation drag is the standard ruler that is used to extract information about the expansion history of the universe from measurements of the baryon acoustic oscillations. Even for a CMB experiment like Planck, a small but non-zero isocurvature contribution, still allowed by Planck data, if ignored, can introduce a systematic error in the interpretation of the BAO signal that is comparable if not larger than the statistical errors. In fact, [621Jump To The Next Citation Point] shows that even a tiny amount of isocurvature perturbation, if not accounted for, could affect standard rulers calibration from CMB observations such as those provided by the Planck mission, affect BAO interpretation, and introduce biases in the recovered dark energy properties that are larger than forecast statistical errors from future surveys. In addition it will introduce a mismatch of the expansion history as inferred from CMB and as measured by BAO surveys. The mismatch between CMB predicted and the measured expansion histories has been proposed as a signature for deviations from a DM cosmology in the form of deviations from Einstein’s gravity (e.g., [8, 476]), couplings in the dark sector (e.g., [589]) or time-evolving dark energy.

For the above reasons, extending on the work of [621], [208] adopted a general fiducial cosmology which includes a varying dark energy equation of state parameter and curvature. In addition to BAO measurements, in this case the information from the shape of the galaxy power spectrum are included and a joint analysis of a Planck-like CMB probe and a Euclid-type survey is considered. This allows one to break the degeneracies that affect the CMB and BAO combination. As a result, most of the cosmological parameter systematic biases arising from an incorrect assumption on the isocurvature fraction parameter fiso, become negligible with respect to the statistical errors. The combination of CMB and LSS gives a statistical error σ(fiso) ∼ 0.008, even when curvature and a varying dark energy equation of state are included, which is smaller than the error obtained from CMB alone when flatness and cosmological constant are assumed. These results confirm the synergy and complementarity between CMB and LSS, and the great potential of future and planned galaxy surveys.

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