Measurement of activity

where ? implies that we evaluate at horizon crossing which is where the k-dependence of the power spectrum arises from. By ignoring the non-Gaussian contributions to the bispectrum we can arrive at a k-independent expression for the local fnl [120]

(as defined in Sec (2.6.2)),

fnl = 5 6

NINJN^IJ

[NKN^K]² . (5.6)

Note this result is analytical if one can calculate analytically NI and NIJ.

5.2. Numerical Methods for Calculating the Statistics of Curvature Perturbations

We could employ the δN , in–in or analytical methods for calculating statistics of inflation but as we will now discuss this is hard. Separate universe methods are successful at estimating statistics for simple models and a main motivation for using them is to make progress in obtaining analytical results.

δN provides one approach where analytics may be possible but for more complex models (models with large numbers of field) it’s no longer straightforward to imple-ment and possibly more cumbersome than if numerics were impleimple-mented. In order to calculate the two- and three- point statistics of ζ a model containing N -fields one will need to computed 2N coefficients of the NI tensors, 2N (N + 1/2) coefficients of the NIJ tensor from Eqn. (5.3). In addition this method is only valid for light fields (less massive than the Hubble scale). These is a limiting factor in analytical approaches of δN .

In the in–in formalism calculating the correlation function may be hard as they may contain rapidly oscillating components [106, 116, 125] in the integrand of Eqn. (4.14). If this is the harmonic oscillator in Eqn. (4.14) we may ignore it up to horizon crossing but if there are oscillations in the potential then we cannot. In an adiabatic evolution under certain analytical prescriptions this will decay and leave insignificant contributions to the two- and three- point functions a few e-folds after horizon crossing. If the super-horizon evolution is characterized by non-adiabatic perturbations then components of the integrand will no longer be insignificant on super-horizon scales. If this is the case the integrals will no longer be rapidly oscil-lating but as long wavelength modes are large compared to the Hubble scale there may be a sensitivity to the mass spectrum and decay channel of the model [78].

However, it has been shown that numerical methods [116] are often the only way to calculate the correlation functions when these oscillations are significant; one such situation is when non-Gaussiantity is generated well before horizon crossing due to a rapid oscillation of the slow-roll parameters [126]. In addition analytical calculations in the in–in formalism become even more complex when we allow for a hierarchy of the external wavenumbers ki in the three point functions hδX^aδXbδXci (i.e. if we

5.2: Numerical Methods for Calculating the Statistics of Curvature Perturbations78

wish to examine configurations other than equilateral) and the methods which do accommodate this involve lengthy factorizations which further increase the number of terms in our calculation

In addition this problem, the in–in formalism calculation of the correlation func-tions relies on the massless approximation whereby all fields are less massive than the Hubble scale. When fields are light in comparison to the Hubble scale the esti-mates of the correlation functions we obtain are universally applicable to any model of inflation. If the mass spectrum extends above the Hubble scale then this approx-imation breaks down and more specialized approaches must be made by keeping a subset of terms that capture the possible effects. However, it can be difficult to identify which terms are important as it is analytical not tractable. In both δN and the in–in formalism it is difficult to incorporate heavy fields.

5.2.1. Calculating Statistic for Models with Heavy Fields

While inflation is driven by light scalar fields the effects of additional heavy fields on the dynamics of inflation has recently been of interest [127]. There is a strong case for considering such fields if models have a UV completion in fundamental particle physics, such as Supergravity and String theory [83]. In Ch. (8) we examine a new class of model which features the non-trivial effects of heavy scalar fields. We can classify these models into three categories depending on both the influence of the heavy and light modes as well as how these modes affect one another.

In a Minkowski background, massive fields with mi  H are suppressed by the inverse of the mass [59,128]. By virtue of the decoupling theorem, for scales below the mass of the heavy particle the full theory may be approximated arbitrarily closely by an effective theory of the light fields alone [129]. The heavy physics becomes negligible and one can integrate out the massive field leaving an effective single-field model whereby the light field tracks the minimum of the heavy field’s potential. In a time dependent background, however more care needs to be taken, as dynamical effects arising from choices of initial conditions in the parameter space may compensate for how small 1/mi is. If there is a bending in the field-space, as in Fig. (5.2), then there is an associated angular velocity ˙θ [131]. Bending terms like the angular velocity appear as couplings between adiabatic and isocurvature modes in the system, meaning care is required when integrating out the heavy mode as O( ˙θ/mⁱ) could be unity or larger in the case where turning is rapid. In the case where the turning effects are small enough that the heavy modes track the minimum of the potential then only adiabatic excitations are relevant but due to the kinetic mixing of the fields the potential will be modified. The ‘Gelaton’ models [132] features a heavy field and non-canonical kinetic coupling, “the heavy field ‘gels’ to the light one”, effectively getting dragged along by it and altering the light field dynamics.

This results in a sound speed of less than 1 in the effective single-field description,

5.2: Numerical Methods for Calculating the Statistics of Curvature Perturbations79

Figure 5.2.: A diagram of a bending of the trajectory in field space [130]. High-lighted is the adiabatic projection ~T and the non-adiabatic (isocurvature) projection N along the evolution of the fields.~

resulting in large equilateral non-Gaussianity. There is a range of validity for this effective description to work, however, and in the simplest of scenarios predictions can be calculated analytically. In general however it is necessary to track the full evolution of both the heavy and light modes to account for not just the adiabatic fluctuations.

In the second case where the time dependent background effects are varying rapidly, such as a sharp turn where ˙θ/mi  1, the heavy field can undergo a non-adiabatic evolution. With non-adiabatic evolution comes particle production and the excitation of both the heavy and light modes. The effects of particle pro-duction during inflation can largely impact the homogeneous background as well as the inflationary trajectory and the Hubble parameter, thus significantly altering the statistics of inflation. Such scenarios have been examined under special circum-stances in the past [133, 134]. Three main contributions to the correlations of ζ are the particle production in the light modes (examined in Ref. [133] as the dom-inant contribution), the conversion of heavy modes into light modes (examined in Refs. [133, 134] but by neglecting particle production) and the response of heavy modes to light mode fluctuations. At the level of two-point statistics [127] it was found that features of heavy physics results in dampened superimposed oscillations onto the power spectrum. At the level of three-point statistics [135] the bispec-trum was calculated under the effect of the periodic production of heavy degrees of freedom. Only in cases which limit to ‘Gelaton’ like behaviour can be studied analytically by use of effective field theories. In all non-adiabatic cases numeri-cal methods are required when any of these effects are important, with effects like particle production making any analytical progress hard.

The third and final situation, the ‘Quasi-single field inflation’ model [136, 137], represents a mixture of the previous two. The heavy field has a mass of roughly

5.2: Numerical Methods for Calculating the Statistics of Curvature Perturbations80

the same size as H and the trajectory turning rate is small enough that the heavy modes do not displace away from the minimum meaning there is negligible particle production from coupling to the background field configuration. However as the mass is small enough, particle production may still occur due to the metric couplings.

The massive field will also have a cubic self-coupling which will parameterize the angular velocity ˙θ and the bispectrum can be generated through the conversion from the isocurvature modes to the curvature mode, once the inflaton trajectory turns.

If V⁰⁰⁰ > H then non-Gaussanities can be sizable [136]. We therefore wish to use an approach to calculating the effects of heavy fields during inflation, such an approach is the transport method.

5.2.2. The Transport Method

In this section we will review the framework for a method to numerically evolve the inflationary statistics. The method, developed by D. J. Mulryne et. al. [78, 121–

123, 138], is called the Moment Transport method for inflationary statistics. This method evolves the statistical quantities (the moments or correlation functions) themselves rather than evolving a perturbed quantity ζ. The moments are evolved through a system of coupled ordinary differential equations called the transport equations. These equations were originally constructed for purely super-horizon evolutions for N -canonical fields [121,123,139] (and for non-canonical fields [104]) and then were later adapted for sub-horizon evolution [78, 122].

There are some strong advantages to this numerical approach. The first is that the treatment of Feynman integrals that rapidly oscillate one avoids in many other setups [140] as we evolve the statistics rather than evaluate the integrals (in Eqn. (4.14)) themselves. When compared with analytical or approximate methods a second advantage is that all effects up to tree-level are included so there is no need to make assumptions by discarding terms that may be relevant. By tracking each correlation function directly it is easier to pinpoint what are the terms that contribute any non-Gaussianity. A third advantage, and perhaps the strongest selling point of this method, is that we can track the evolution of these statistics from deep within the horizon in the quantum regime through horizon crossing to super-horizon scales where the correlation functions are classical statistical quantities and where we can evaluate observable quantities of the gauge invariant curvature perturbations. This makes the transport method a useful tool when calculating observables for different inflationary models.

In addition to this, in our implementation of the transport method we have ex-cellent control of the accuracy, even for a large numbers of fields. The Transport method has been utilized in publicly available codes, for the non-canonical two-point correlators in mTransport [118], for the canonical three-two-point correlators [78]

in PyTransport [141] and CppTransport [142], for non-canonical three-point

cor-5.2: Numerical Methods for Calculating the Statistics of Curvature Perturbations81

relators in PyTransport 2.0 [143] (as we will review in Ch. (6)) and more recently CppTransport [144].

Here, an important result of in this thesis is to derive the full transport equations that are valid during the quantum phase [122] and adapt them to the covariant field perturbations. Here we write the evolution equation of the Fourier modes of the covariant field and momentum fluctuations and for convenience we first label the full phase space of Heisenberg operations with the symbol δXa(ka) = (Q^I(kI), P^I(kI) where a runs from 1 to 2N for N fields.

The expectation values we are interested in are then the two- and three-point functions of δX^a defined in Eqns. (4.19) and (4.29). As described, the equations of motion for these correlation functions follow directly from Eqns. (3.24) and (3.25) together with Ehrenfest’s theorem, and can be presented in terms of equations of motion for Σ^ab and B^abc. By using Ehrenfest’s theorem and from Eqn. (4.3) we may define the evolution of the expectation value of quantum operators as,

DthδQ^Ii = h−iδQ^I,Hi ,

DthδP^Ii = h−iδP^I,Hi − 3HhP^Ii . (5.7) Equivalently these two expressions are valid without the expectation brackets. In general, we can reformulate the above equation into a product of a matrix and the expectation value,

DthδX^ai = u^abhδX^bi + u^abchδX^bδX^ci · · · , (5.8) where u^a_b is a matrix to be computed from background quantities and the dots indicate higher order terms. The u^a_band higher order u^a_bctensors satisfy the relations,

u^a_b= (2π)³δ(ka− k^b)u^a_b(ka, kb) ,

u^a_bc= (2π)³δ(ka− k^b− k^c)u^a_bc(ka, kb, kc) . (5.9) The evolution equation of the two-point function is also derived by the same theorem except now we apply the chain rule which takes the form,

DthδX^aδX^bi = h(DtδX^a)δX^bi + hδX^a(DtδX^b)i . (5.10) In our covariant setting these take the form

DtΣ^ab(k) = u^ac(k)Σ^cb(k) + u^bc(k)Σ^ac(k), (5.11)

5.2: Numerical Methods for Calculating the Statistics of Curvature Perturbations82 where the covariant time derivative acts on Σ^ab in the following way

DtΣ^ab(k) = ∂tΣ^ab(k) + Γ^a_c(k)Σ^cb(k) + Γ^b_c(k)Σ^ac(k) , (5.13)

The u-tensors take the form

u^ab =

5.2: Numerical Methods for Calculating the Statistics of Curvature Perturbations83

The two-point function will in general be complex, and can be divided into its real and imaginary parts

Σ^ad = Σ^ad_Re+ iΣ^ad_Im, (5.19) with the real part symmetric under interchange of its indices, and the imaginary part anti-symmetric. Both parts independently satisfy Eqn. (5.11). On super-horizon scales the imaginary part decays to zero, indicating that on large scales the statistics of inflationary perturbations follow classical equations of motion.

B^abc, is in general also complex, but is real when only tree-level effects are included.

In our numerical implementation of the transport system we evolve the real and imaginary parts of Σ^ab separately using Eqn. (5.11), and evolve B^abc according to the equation which follows from Eqn. (5.12) once Σ^ab is broken into real and imaginary parts, and which makes it clear that B^abc remains real if its initial conditions are real. In Sec. (4.2) we calculated the initial conditions for the two-point correlation function deep within the horizon. It is now possible to restructure these in tensorial form Σ^ab,

Likewise the initial conditions for B^abc can be obtained from Sec. (4.3).

We now calculate the statistics of ζ (as we defined in Eqn. (3.49)) in the notation of Eqn. (3.50). In this notation the two and the three-point function of ζ are given