3.3 Probing the early universe with non-Gaussianities

The workhorse for primordial non-Gaussianity has been so far the “local model” [781Jump To The Next Citation Point, 376Jump To The Next Citation Point, 924Jump To The Next Citation Point, 524, 93]:
( 2 2 ) Φ = ϕ + fNL ϕ − ⟨ϕ ⟩ . (3.3.1 )
Here ϕ is a Gaussian random field while Φ denotes Bardeen’s gauge-invariant potential, which, on sub-Hubble scales reduces to the usual Newtonian peculiar gravitational potential, up to a minus sign. On large scales it is related to the conserved variable ζ by
ζ = 5-+-3w-Φ, (3.3.2 ) 3 + 3w
where w is the equation of state of the dominant component in the universe. The amount of primordial non-Gaussianity is quantified by the nonlinearity parameter fNL. Note that, since −5 Φ ≃ ϕ ≃ 10, fNL ∼ 100 corresponds to relative non-Gaussian corrections of order −3 10. While ζ is constant on large scales, Φ is not. For this reason, in the literature there are two conventions for Eq. (3.3.1View Equation): the large-scale structure (LSS) and the cosmic microwave background (CMB) one. In the LSS convention, Φ is linearly extrapolated at z = 0; in the CMB convention Φ is instead primordial: thus LSS CMB CMB fNL = g(z = ∞ )∕g(0)fNL ∼ 1.3fNL, where g (z ) denotes the linear growth suppression factor relative to an Einstein–de Sitter universe. In the past few years it has become customary to always report f CMB NL values even though, for simplicity as it will be clear below, one carries out the calculations with LSS fNL.

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.

3.3.1 Local non-Gaussianity

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 f NL would be of the order of the slow-roll parameters [781, 353, 376]. More recently, with an exact calculation [616Jump To The Next Citation Point] 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 ζ is given by

local B ζ(k1 ≪ k2, k3) = 4f NL P ζ(k2)Pζ(k3), (3.3.3 )
where floNcLal is proportional to the tilt of scalar fluctuations, fNloLcal = − (5∕12)(ns − 1), a value much too small to be observable. Thus, any deviation from this prediction would rule out a large class of models based on the simplest scenario.

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, 226Jump To The Next Citation Point, 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 local fNL 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 local fNL as in [582Jump To The Next Citation Point]. 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, 664Jump To The Next Citation Point, 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 local fNL is inversely proportional to the curvaton abundance before the decay [597Jump To The Next Citation Point].

Models exists where both curvaton and inflaton fluctuations contribute to cosmological perturbations [545Jump To The Next Citation Point]. 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 ∼ 60 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., [126Jump To The Next Citation Point, 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].

3.3.2 Shapes: what do they tell us?

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 [70Jump To The Next Citation Point].

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 B (k ,k ,k ) ζ 1 2 3, defined as

⟨ζ ζ ζ ⟩ = (2π)3δ(k + k + k )B (k ,k ,k ). (3.3.4 ) k1 k2 k3 1 2 3 ζ 1 2 3
Relaxing the assumption of a local fNL, this function is a rich object which can contain a wealth of information, depending on the size and shape of the triangle formed by k1, k2 and k3. Indeed, the dependence of the bispectrum on configuration in momentum space is related to the particular inflationary model generating it. Namely, each third-order operator present in the field action gives rise to a particular shape of the bispectrum.

An example of models containing large derivative interactions has been proposed by [830, 25Jump To The Next Citation Point]. Based on the Dirac–Born–Infeld Lagrangian, ∘ ----------- ℒ = f(ϕ)−1 1 − f (ϕ)X + V(ϕ ), with X = − gμν∂μϕ ∂νϕ, it is called DBI inflation. This Lagrangian is string theory-motivated and ϕ describes the low-energy radial dynamics of a D3-brane in a warped throat: −1 f (ϕ ) is the warped brane tension and V (ϕ) 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 ℒ 3 and ℒ 2, divided by the amplitude of scalar fluctuations. This gives −1 2 fNL ∼ (ℒ3∕ℒ2 )Φ ∼ − 1∕cs, where 2 2 2 −1 cs = [1 + 2X (∂ ℒ∕ ∂X )∕(∂ℒ ∕∂X )] < 1 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 cs ≪ 1.

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

( P (k )P (k ) [P (k )P (k )P (k )]23 B ζ(k1,k2, k3) = 6feNqLuil --ζ--1--ζ--2-+ --ζ--1--ζ--2--ζ--3--- 2 3 ) 1 2 − P ζ(k1 )3Pζ(k2)3P ζ(k3) + 5 perms. , (3.3.5 )
where, up to numerical factors of order unity, feNquLil≃ − 1∕c2s. The function of momenta inside the parenthesis is the equilateral shape [260], a template used to approximate a large class of inflationary models. It is defined in such a way as to be factorisable, maximized for equilateral configurations and vanishing in the squeezed limit faster than the local shape, see Eq. (3.3.3View Equation).

To compare two shapes F1 and F2, it is useful to define a 3-dimensional scalar product between them as [70]

F ⋅ F = ∑ F (k ,k ,k )F (k ,k ,k )∕(P (k )P (k )P (k )), (3.3.6 ) 1 2 1 1 2 3 2 1 2 3 ζ 1 ζ 2 ζ 3
where the sum is over all configurations forming a triangle. Then, ∘ ----------------- cos 𝜃 = F1 ⋅ F2∕ (F1 ⋅ F1 )(F2 ⋅ F2 ) defines a quantitative measure of how much two shapes “overlap” and their signal is correlated. The cosine is small between the local and equilateral shapes. Two shapes with almost vanishing cosine are said to be orthogonal and any estimator developed to be sensitive to a particular shape will be completely blind to its orthogonal one. Note that the observable signal could actually be a combination of different shapes. For instance, multi-field models base on the DBI action [543] can generate a linear combination of local and equilateral non-Gaussianities [763].

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 ([228Jump To The Next Citation Point]; 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 ℒ3 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 c ≥ 0.01 s [228, 814Jump To The Next Citation Point]. 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 [814Jump To The Next Citation Point]. 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-k sides are equal exactly to half of the highest-k 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].

3.3.3 Beyond shapes: scale dependence and the squeezed limit

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 [809Jump To The Next Citation Point, 690, 807].

3.3.4 Beyond inflation

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 V(ϕ ) = exp (− cϕ), with c ≫ 1. 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 c1 and c2, the non-adiabatic component has negative mass and acquires a quasi invariant spectrum of fluctuations with tilt ns − 1 = 4 (c−1 2+ c−2 2), with c1,c2 ≫ 1. 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 ϕ1. In this case [527] have found that local 2 fNL = − (5∕12)c1. Current constraints on local fNL (WMAP7 year data imposes local − 10 < fNL < 74 at 95% confidence) gives an unacceptably large value for the scalar spectral index. In fact in this model, even for fNL = − 10, c2 ≃ 5 which implies a too large value of the scalar spectral index (ns − 1 > 0.17) which is excluded by observations (recall that WMAP7 year data implies ns = 0.963 ± 0.014 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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