In this section we review the theoretical motivations and implications for looking into primordial non-Gaussianity; the readers less theoretically oriented can go directly to Section 3.4.
The non-Gaussianities generated in the conventional scenario of inflation (single-field with standard kinetic
term, in slow-roll, initially in the Bunch–Davies vacuum) are predicted to be extremely small. Earlier
calculations showed that would be of the order of the slow-roll parameters [781, 353, 376]. More
recently, with an exact calculation [616
] confirmed this result and showed that the dominant
contribution to non-Gaussianity comes from gravitational interaction and it is thus independent
of the inflaton potential. More precisely, in the squeezed limit, i.e. when one of the modes is
much smaller than the other two, the bispectrum of the primordial perturbation
is given by
Furthermore, [264] showed that irrespective of slow-roll and of the particular inflaton Lagrangian or
dynamics, in single-field inflation, or more generally when only adiabatic fluctuations are present, there
exists a consistency relation involving the 3-point function of scalar perturbations in the squeezed
limit (see also [806, 226, 227]). In this limit, when the short wavelength modes are inside the
Hubble radius during inflation, the long mode is far out of the horizon and its only effect on the
short modes is to rescale the unperturbed history of the universe. This implies that the 3-point
function is simply proportional to the 2-point function of the long wavelength modes times the
2-point function of the short wavelength mode times its deviation from scale invariance. In
terms of local non-Gaussianity this translates into the same
found in [616]. Thus, a
convincing detection of local non-Gaussianity would rule out all classes of inflationary single-field
models.
To overcome the consistency relation and produce large local non-Gaussianity one can go
beyond the single-field case and consider scenarios where a second field plays a role in generating
perturbations. In this case, because of non-adiabatic fluctuations, scalar perturbations can
evolve outside the horizon invalidating the argument of the consistency relation and possibly
generating a large as in [582
]. The curvaton scenario is one of such mechanisms. The
curvaton is a light scalar field that acquires scale-invariant fluctuations during inflation and
decays after inflation but well before nucleosynthesis [661, 664
, 598, 342]. During the decay
it dominates the universe affecting its expansion history thus imprints its perturbations on
super-horizon scales. The way the expansion history depends on the value of the curvaton field at
the end of the decay can be highly nonlinear, leading to large non-Gaussianity. Indeed, the
nonlinear parameter
is inversely proportional to the curvaton abundance before the decay
[597
].
Models exists where both curvaton and inflaton fluctuations contribute to cosmological perturbations
[545]. Interestingly, curvaton fluctuations could be negligible in the 2-point function but detectable through
their non-Gaussian signature in the 3-point function, as studied in [155]. We shall come back on this point
when discussing isocurvature perturbations. Other models generating local non-Gaussianities are the so
called modulated reheating models, in which one light field modulates the decay of the inflaton field
[329, 515]. Indeed, non-Gaussianity could be a powerful window into the physics of reheating and
preheating, the phase of transition from inflation to the standard radiation dominated era (see
e.g., [148, 222]).
In the examples above only one field is responsible for the dynamics of inflation, while the others are
spectators. When the inflationary dynamics is dominated by several fields along the e-foldings of
expansion from Hubble crossing to the end of inflation we are truly in the multi-field case. For instance,
a well-studied model is double inflation with two massive non-interacting scalar fields [739].
In this case, the overall expansion of the universe is affected by each of the field while it is
in slow-roll; thus, the final non-Gaussianity is slow-roll suppressed, as in single field inflation
[768, 19, 926].
Because the slow-roll conditions are enforced on the fields while they dominate the inflationary
dynamics, it seems difficult to produce large non-Gaussianity in multi-field inflation; however, by tuning the
initial conditions it is possible to construct models leading to an observable signal (see [191, 876]).
Non-Gaussianity can be also generated at the end of inflation, where large-scale perturbations may have a
nonlinear dependence on the non-adiabatic modes, especially if there is an abrupt change in the equation of
state (see e.g., [126, 596]). Hybrid models [580], where inflation is ended by a tachyonic instability triggered
by a waterfall field decaying in the true vacuum, are natural realizations of this mechanism
[343, 88].
As explained above, local non-Gaussianity is expected for models where nonlinearities develop outside the
Hubble radius. However, this is not the only type of non-Gaussianity. Single-field models with derivative
interactions yield a negligible 3-point function in the squeezed limit, yet leading to possibly observable
non-Gaussianities. Indeed, as the interactions contain time derivatives and gradients, they vanish outside
the horizon and are unable to produce a signal in the squeezed limit. Correlations will be larger for other
configurations, for instance between modes of comparable wavelength. In order to study the observational
signatures of these models we need to go beyond the local case and study the shape of non-Gaussianity
[70].
Because of translational and rotational invariance, the 3-point function is characterized by a function of
the modulus of the three wave-vectors, also called the bispectrum , defined as
An example of models containing large derivative interactions has been proposed by [830, 25]. Based on
the Dirac–Born–Infeld Lagrangian,
, with
, it is called
DBI inflation. This Lagrangian is string theory-motivated and
describes the low-energy radial dynamics
of a D3-brane in a warped throat:
is the warped brane tension and
the interaction
field potential. In this model the non-Gaussianity is dominated by derivative interactions of
the field perturbations so that we do not need to take into account mixing with gravity. An
estimate of the non-Gaussianity is given by the ratio between the third-order and the second order
Lagrangians, respectively
and
, divided by the amplitude of scalar fluctuations. This gives
, where
is the speed of sound of
linear fluctuations and we have assumed that this is small, as it is the case for DBI inflation. Thus, the
non-Gaussianity can be quite large if
.
However, this signal vanishes in the squeezed limit due to the derivative interactions. More precisely, the particular momentum configuration of the bispectrum is very well described by
where, up to numerical factors of order unity, To compare two shapes and
, it is useful to define a 3-dimensional scalar product between
them as [70]
The interplay between theory and observations, reflected in the relation between derivative interactions
and the shape of non-Gaussianity, has motivated the study of inflation according to a new approach, the
effective field theory of inflation ([228]; see also [951]). Inflationary models can be viewed as
effective field theories in presence of symmetries. Once symmetries are defined, the Lagrangian will
contain each possible operator respecting such symmetries. As each operator leads to a particular
non-Gaussian signal, constraining non-Gaussianity directly constrains the coefficients in front of
these operators, similarly to what is done in high-energy physics with particle accelerators. For
instance, the operator
discussed in the context of DBI inflation leads to non-Gaussianity
controlled by the speed of sound of linear perturbations. This operator can be quite generic in
single field models. Current constraints on non-Gaussianity allow to constrain the speed of
sound of the inflaton field during inflation to be
[228, 814
]. Another well-studied
example is ghost inflation [57], based on the ghost condensation, a model proposed by [56]
to modify gravity in the infrared. This model is motivated by shift symmetry and exploits
the fact that in the limit where this symmetry is exact, higher-derivative operators play an
important role in the dynamics, generating large non-Gaussianity with approximately equilateral
shape.
Following this approach has allowed to construct operators or combination of operators leading to
new shapes, orthogonal to the equilateral one. An example of such a shape is the orthogonal
shape proposed in [814]. This shape is generated by a particular combination of two operators
already present in DBI inflation. It is peaked both on equilateral-triangle configurations and on
flattened-triangle configurations (where the two lowest-
sides are equal exactly to half of the highest-
side) – the sign in this two limits being opposite. The orthogonal and equilateral are not an
exhaustive list. For instance, [258] have shown that the presence in the inflationary theory of an
approximate Galilean symmetry (proposed by [686] in the context of modified gravity) generates
third-order operators with two derivatives on each field. A particular combination of these
operators produces a shape that is approximately orthogonal to the three shapes discussed
above.
Non-Gaussianity is also sensitive to deviations from the initial adiabatic Bunch–Davies vacuum of inflaton fluctuations. Indeed, considering excited states over it, as done in [226, 444, 653], leads to a shape which is maximized in the collinear limit, corresponding to enfolded or squashed triangles in momentum space, although one can show that this shape can be written as a combination of the equilateral and orthogonal ones [814].
There is a way out to generate large non-Gaussianity in single-field inflation. Indeed, one can temporarily break scale-invariance, for instance by introducing features in the potential as in [225]. This can lead to large non-Gaussianity typically associated with scale-dependence. These signatures could even teach us something about string theory. Indeed, in axion monodromy, a model recently proposed by [831] based on a particular string compactification mechanism, the inflaton potential is approximately linear, but periodically modulated. These modulations lead to tiny oscillations in the power spectrum of cosmological fluctuations and to large non-Gaussianity (see for instance [366]).
This is not the only example of scale dependence. While in general the amplitude of the non-Gaussianity signal is considered constant, there are several models, beside the above example, which predict a scale-dependence. For example models like the Dirac–Born–Infeld (DBI) inflation, e.g., [25, 223, 224, 111] can be characterized by a primordial bispectrum whose amplitude varies significantly over the range of scales accessible by cosmological probes.
In view of measurements from observations it is also worth considering the so-called squeezed limit of non-Gaussianity that is the limit in which one of the momenta is much smaller than the other two. Observationally this is because some probes (like, for example, the halo bias Section 3.4.2, accessible by large-scale structure surveys like Euclid) are sensitive to this limit. Most importantly, from the theoretical point of view, there are consistency relations valid in this limit that identify different classes of inflation, e.g., [262, 257] and references therein.
The scale dependence of non-gaussianity, the shapes of non-gaussianity and the behavior of the squeezed
limit are all promising avenues, where the combination of CMB data and large-scale structure surveys such
as Euclid can provide powerful constraints as illustrated, e.g., in [809, 690, 807].
As explained above, the search of non-Gaussianity could represent a unique way to rule out the simplest of the inflationary models and distinguish between different scenarios of inflation. Interestingly, it could also open up a window on new scenarios, alternative to inflation. There have been numerous attempts to construct models alternative to inflation able to explain the initial conditions of our universe. In order to solve the cosmological problems and generate large-scale primordial fluctuations, most of them require a phase during which observable scales today have exited the Hubble size. This can happen in bouncing cosmologies, in which the present era of expansion is preceded by a contracting phase. Examples are the pre-big bang [385] and the ekpyrotic scenario [498].
In the latter, the 4-d effective dynamics corresponds to a cosmology driven by a scalar field with a steep
exponential potential , with
. Leaving aside the problem of the realization of the
bounce, it has been shown that the adiabatic mode in this model generically leads to a steep blue
spectrum for the curvature perturbations [595, 261]. Thus, at least a second field is required
to generate an almost scale-invariant spectrum of perturbations [365, 263, 182, 528]. If two
fields are present, both with exponential potentials and steepness coefficients
and
,
the non-adiabatic component has negative mass and acquires a quasi invariant spectrum of
fluctuations with tilt
, with
. Then one needs to convert the
non-adiabatic fluctuation into curvature perturbation, similarly to what the curvaton mechanism
does.
As the Hubble rate increases during the collapse, one expects nonlinearities in the fields to become more
and more important, leading to non-Gaussianity in the produced perturbations. As nonlinearities grow
larger on super-Hubble scales, one expects the signal to be of local type. The particular amplitude of the
non-Gaussianity in the observable curvature perturbations depends on the conversion mechanism from the
non-adiabatic mode to the observable perturbations. The tachyonic instability itself can lead to a phase
transition to an ekpyrotic phase dominated by just one field . In this case [527] have found that
. Current constraints on
(WMAP7 year data imposes
at
95% confidence) gives an unacceptably large value for the scalar spectral index. In fact in this model, even
for
,
which implies a too large value of the scalar spectral index (
)
which is excluded by observations (recall that WMAP7 year data implies
at
68% confidence). Thus, one needs to modify the potential to accommodate a red spectrum or
consider alternative conversion mechanisms to change the value of the generated non-Gaussianity
[183, 554].
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Living Rev. Relativity 16, (2013), 6
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