Let us denote by the energy density of the component
. Perturbations are purely adiabatic when
for each component
the quantity
is the same [949, 617]. Let us consider for instance
cold dark matter and photons. When fluctuations are adiabatic it follows that
. Using the energy
conservation equation,
with
and
, one finds that the density
contrasts of these species are related by
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 and
as
, so that, for the example above one has
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 [419] or the [10
] 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 . Constraints on the amplitude of the
non-adiabatic component are given in terms of the parameter
, defined at a given scale
, by
, see e.g., [120
, 113, 525
]. As discussed in [542], adiabatic and entropy
perturbations may be correlated. To measure the amplitude of the correlation one defines a cross-correlation
coefficient,
. Here
is the cross-correlation power-spectrum between
and
and for the definition of
we have adopted the sign convention of [525
]. Observables, such as for instance
the CMB anisotropies, depend on linear combinations of
and
. Thus, constraints on
will considerably depend on the cross-correlation coefficient
(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 , are produced, as discussed for instance in [336, 582
].
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
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 [546]. 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 . 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].
Even if pure isocurvature models have been ruled out, current observations allow for mixed adiabatic and
isocurvature contributions (e.g., [267, 896, 525, 914]). As shown in [902
, 40, 914, 544, 184
, 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 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, [621] 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 , become negligible with respect to the statistical errors. The combination of CMB
and LSS gives a statistical error
, 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.
http://www.livingreviews.org/lrr-2013-6 |
Living Rev. Relativity 16, (2013), 6
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