DIPARTIMENTO DI FISICA “G. GALILEI”

Scuola di dottorato di ricerca in fisica, ciclo XXII

Cosmological correlation functions in scalar and vector inflationary models

Supervisori:                                                                                        Dottoranda: Prof. Sabino Matarrese                                              Emanuela Dimastrogiovanni Dr. Nicola Bartolo

to Matteo

1 Introduction

In the standard cosmological model, at very early times the Universe undergoes a quasi de Sitter exponential expansion driven by a scalar field, the inflaton, with an almost flat potential. The quantum fluctuations of this field are thought to be at the origin of both the Large Scale Structures and the Cosmic Microwave Background (CMB) fluctuations that we are able to observe at the present epoch [1]. CMB measurements indicate that the primordial density fluctuations are of order , have an almost scale-invariant power spectrum and are fairly consistent with Gaussianity and statistical isotropy [2, 3, 4, 5, 6, 7, 8, 9]. All of these features find a convincing explanation within the inflationary paradigm. Nevertheless, deviations from the basic single-(scalar)field slow-roll model of inflation are allowed by experimental data. On one hand, it is then important to search for observational signatures that can help discriminate among all the possible scenarios; on the other hand, it is important to understand what the theoretical predictions are in this respect for the different models.

Non-Gaussianity and statistical anisotropy are two powerful signatures. A random field is defined “Gaussian” if it is entirely described by its two-point function, higher order connected correlators being equal to zero. Primordial non-Gaussianity [10, 11] is theoretically predicted by inflation: it arises from the interactions of the inflaton with gravity and from self-interactions. However, it is observably too small in the single-field slow-roll scenario [12, 13, 152]. Alternatives to the latter have been proposed that predict higher levels of non-Gaussianity such as multifield scenarios [15, 16, 17, 18, 19, 20, 21], curvaton models [22, 23, 24, 25, 26, 27] and models with non-canonical Lagrangians [28, 29, 30, 31, 32]. Many efforts have been directed to the study of higher order (three and four-point) cosmological correlators in these models [33, 34, 35, 36, 37, 31, 152, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48] and towards improving the prediction for the two-point function, through quantum loop calculations [49, 50, 51, 13, 52, 53, 54, 55, 56]. From WMAP, the bounds on the bispectrum amplitude are given by [8] and by [9] at CL, respectively in the local and in the equilateral configurations. For the trispectrum, WMAP provides [57] ( is the “local” trispectrum amplitude from cubic contributions), whereas from Large-Scale-Structures data [58], at CL. Planck [59] is expected to set further bounds on primordial non-Gaussianity.
Statistical isotropy has always been considered one of the key features of the CMB fluctuations. The appearance of some “anomalies” [60, 61, 62] in the observations though, after numerous and careful data analysis, suggests a possible a breaking of this symmetry that might have occurred at some point of the Universe history, possibly at very early times. This encouraged a series of attempts to model this event, preferably by incorporating it in theories of inflation. Let us shortly describe the above mentioned “anomalies”. First of all, the large scale CMB quadrupole appears to be “too low” and the octupole “too planar”; in addition to that, there seems to exist a preferred direction along which quadrupole and octupole are aligned [63, 64, 60, 65, 66]. Also, a “cold spot”, i.e. a region of suppressed power, has been observed in the southern Galactic sky [61, 67]. Finally, an indication of asymmetry in the large-scale power spectrum and in higher-order correlation functions between the northern and the southern ecliptic hemispheres was found [68, 62, 69]. Possible explanations for these anomalies have been suggested such as improper foreground subtraction, WMAP systematics, statistical flukes; the possibilities of topological or cosmological origins for them have been proposed as well. Moreover, considering a power spectrum anisotropy due to the existence of a preferred spatial direction and parametrized by a function as

 P(→k)=P(k)(1+g(k)(^k⋅^n)2), (1.1)

the five-year WMAP temperature data have been analyzed in order to find out what the magnitude and orientation of such an anisotropy could be. The magnitude has been found to be and the orientation aligned nearly along the ecliptic poles [70]. Similar results have been found in [71], where it is pointed out that the origin of such a signal is compatible with beam asymmetries (uncorrected in the maps) which should therefore be investigated before we can find out what the actual limits on the primordial are.

Several fairly recent works have taken the direction of analysing the consequences, in terms of dynamics of the Universe and of cosmological fluctuations, of an anisotropic pre-inflationary or inflationary era. A cosmic no-hair conjecture exists according to which the presence of a cosmological constant at early times is expected to dilute any form of initial anisotropy [72]. This conjecture has been proven to be true for many (all Bianchi type cosmologies except for the the Bianchi type-IX, for which some restrictions are needed to ensure the applicability of the theorem), but not all kinds of metrics and counterexamples exist in the literature [73, 74, 75]. Moreover, even in the event isotropization should occur, there is a chance that signatures from anisotropic inflation or from an anisotropic pre-inflationary era might still be visible today [76, 77, 78, 79]. In the same context of searching for models of the early Universe that might produce some anisotropy signatures at late time, new theories have been proposed such as spinor models [80, 81, 82, 83], higher p-forms [84, 85, 86, 87, 88, 89] and primordial vector field models.
Within vector field models, higher order correlators had been computed in [90, 91, 92, 93, 94] and, more recently, in [95, 96] for vector fields. We considered vector field models in [97, 98]. Non-Abelian theories offer a richer amount of predictions compared to the Abelian case. Indeed, self interactions provide extra contributions to the bispectrum and trispectrum of curvature fluctuations that are naturally absent in the Abelian case. We verified that these extra contributions can be equally important in a large subset of the parameter space of the theory and, in some case, can even become the dominant ones.

The promising perspective of achieving more and more precise measurements for the cosmological observables thanks to Planck and future experiments and the search for signatures that may help identify the correct inflationary model, have also motivated studies of higher order corrections to cosmological correlation functions and to the power spectrum in particular. Indeed, loop corrections to the correlators arise from the interactions involving the fields during inflation and therefore carry some important information about the physics of the very early Universe.
Loop corrections may lead to interesting effects which scale like the power of the number of e-folds between horizon exit of a given mode and the end of inflation [99, 100, 101, 102, 103]. The interest in loop corrections to the correlators of cosmological perturbations generated during an early epoch of inflation has been recently stimulated by two papers of Weinberg [104, 105]. The reason is that one-loop corrections to the power spectrum of the curvature perturbation seem to show some infra-red divergences which scale like , where is some infra-red comoving momentum cut-off [106, 107, 108, 109]. However, it has been discussed in [110, 111] (see also [112]) that such potentially large corrections do not appear in quantities that are directly observable.
As to the power spectrum of curvature perturbations, one-loop corrections have been computed in single-field slow-roll inflation by D. Seery [54, 55] and by N. Bartolo and myself [56], in single-field slow-roll inflation. In [56] we completed the analysis carried out in [54, 55], where the metric tensor fluctuations had been neglected for simplicity, by including them in the calculations and proving that their contribution is as important as the one from the scalar perturbations. In the context of loop-calculations, we have also been working on corrections to the power spectrum in theories with non-canonical Lagrangians, which allow for higher and possibly observable corrections [113].
It can be safely stated that in standard single-field slow-roll inflation, the perturbative expansion is well-behaved, in the sense that the agreement with observations found at tree-level for the power-spectrum is not spoiled by the radiative corrections and, on a more general basis, higher order loop corrections introduce smaller and smaller corrections as the perturbation series expansion progresses. This is not generically true in more general theories, such as for instance models with non-canonical Lagrangians, for which bounds need to be requested on the parameters of the theory in order to preserve the validity of the perturbative approach [114, 115].

This thesis collects the main results of our work on loop corrections to the power spectrum in theories of scalar inflation (Secs. 2 to 6) [56, 113], on anisotropic pre-inflationary cosmologies (Sec. 7) [78] and on primordial non-Gaussianity and anisotropy predictions from theories of inflation where vector fields can play a role in the production of the late time cosmological fluctuations (Secs. 8 to 12) [97, 98]. The N and the Schwinger-Keldysh formalisms are some of the main tools of our computation and will be briefly reviewed.

2 Schwinger-Keldysh formalism

The temperature fluctuations in the CMB are rather small, of order . Theoretical predictions for the power-spectrum of curvature perturbations during inflation provide a very good match at tree level: this suggests that it is correct to use perturbation theory to evaluate cosmological correlators. A formalism conveniently employed to implement the perturbative approach is the Schwinger-Keldysh, also dubbed as “in-in”, formalism. It was first formulated in [49, 50, 51], later applied by J. Maldacena in [13] to the calculation of the bispectrum of curvature fluctuations and revived by S. Weinberg in [52, 53]. In this formalism the expectation value of a field operator is given by

 Missing or unrecognized delimiter for \left (2.1)

where represents the vacuum of the interacting theory, and are time-ordering and anti-ordering operators, the subscript indicates the fields in interaction picture and is the interaction Hamiltonian. The interaction picture has the advantage of allowing to deal with free fields only; the fields can be thus Fourier expanded in terms of quantum creation and annihilation operators

 δϕ(→x,t)=∫d3kei→k→x[a→kδϕk(t)+a+−→kδϕ∗k(t)],

with commutation rules

 [a→k,a+→k′]=(2π)2δ(3)(→k−→k′).

The in-in formula has many similarities with the S-matrix in quantum field theory in terms of mathematical structure and perturbative approach, but they also have fundamental differences: the S-matrix corresponds to a transition amplitude between an initial and a final state; a cosmological correlation function is instead the expectation value of a given observable at a given time; moreover, asymptotic states in cosmology are only defined at very early times, when the same initial conditions as in Minkowsky spacetime apply for the free fields.
Using the positive and negative path technique of the in-in formalism [52, 53], the expectation value above can be recast in the form

 ⟨Ω|Θ(t)|Ω⟩=⟨0∣∣∣T(ΘI(t)e−i∫t0dt′(H+I(t′)−H−I(t′)))∣∣∣0⟩, (2.2)

where the plus and minus signs indicate modified Feynman propagators, i.e. modified rules of contraction between interacting fields; schematically we have

 ⟨T(ϕ1ϕ2...ϕn)⟩=∑ij,lm,...[ˆϕiϕj,ˆϕlϕm,...], (2.3)

where the sum is taken over all of the possible sets of field contractions and

 ˆϕ+(η′)ϕ+(η′′)=G>(η′,η′′)Θ(η′−η′′)+G<(η′,η′′)Θ(η′′−η′), ˆϕ+(η′)ϕ−(η′′)=G<(η′,η′′), ˆϕ−(η′)ϕ+(η′′)=G>(η′,η′′), ˆϕ−(η′)ϕ−(η′′)=G<(η′,η′′)Θ(η′−η′′)+G>(η′,η′′)Θ(η′′−η′).

In momentum space we have

 G>k(η′,η′′)≡δϕk(η′)δϕ∗k(η′′), G

It is important to remember that, when we apply this formalism, the external fields are always supposed to be treated like fields.

3 Scalar loop corrections to Pζ

The power spectrum for the comoving curvature perturbation is defined by

 ⟨ζ→k1(t)ζ→k2(t)⟩=(2π)3Pζ(k)δ(3)(→k1+→k2), (3.1)

This and all other correlation functions presented in this thesis are computed using the formula. at a given time can be interpreted as a geometrical quantity indicating the fluctuations in the local expansion of the universe; in fact, if is the number of e-foldings of expansion evaluated between times and , where the initial hypersurface is chosen to be flat and the final one is uniform density, we have

 ζ(→x,t)=N(→x,t∗,t)−N(t∗,t)≡δN(→x,t). (3.2)

The number of e-foldings depends on all the fields and their perturbations on the initial slice. In principle, since the fields are governed by second order differential equations, it should also depend on their first time derivatives, but if we assume that slow-roll conditions apply, then the time derivatives will not count as independent quantities.

Let us apply Eq. (3.2) to the computation of in single-field slow-roll inflation (the Lagrangian for the scalar field is given by )

 ⟨ζ→k1(t)ζ→k2(t)⟩=∫d3x1(2π)3d3x2(2π)3e−i(→k1→x1+→k2→x2) ⟨(∑nN(n)(t∗,t)n!(δϕ(→x1,t∗))n),(∑mN(m)(t∗,t)m!(δϕ(→x2,t∗))m)⟩. (3.3)

The sums can be expanded to the desired order. Up to one loop we have

 ⟨ζ→k1(t)ζ→k2(t)⟩ = N(1)2⟨δϕ→k1δϕ→k2⟩∗ (3.4) + 12!N(1)N(2)∫d3q⟨δϕ→k1δϕ→qδϕ→k2−→q⟩∗+(→k1↔→k2) + 13!N(1)N(3)∫d3qd3p⟨δϕ→k1δϕ→qδϕ→pδϕ→q+→p−→k2⟩∗+(→k1↔→k2) + 1(2!)2(N(2))2∫d3qd3p⟨δϕ→qδϕ→k1−→qδϕ→pδϕ→k2−→p⟩∗.

where a star indicates evaluation around the time of horizon crossing. Eq. (3.4) can finally be rewritten as [116, 109]

 ⟨ζ→k1(t)ζ→k2(t)⟩ = (2π)3δ(3)(→k1+→k2)[(N(1))2(Ptree(k1)+Pone−loop(k1)) (3.5) + N(1)N(2)∫d3q(2π)3Bϕ(k1,q,|→k1−→q|) + 12(N(2))2∫d3q(2π)3Ptree(q)Ptree(|→k1−→q|) + N(1)N(3)Ptree(k)∫d3q(2π)3Ptree(q)],

is the tree level power spectrum (3.6)

 ⟨δϕ→k1δϕ→k2⟩∗ = (2π)3P(k)δ(3)(→k1+→k2)=(2π)3H2∗2k3δ(3)(→k1+→k2), (3.6)

where is the Hubble parameter evaluated at horizon exit (when ). The variance per logarithmic interval in is given by . The one loop contribution to the power spectrum is given by

 Pone−loop(k)=Pscalar(k)+Ptensor(k), (3.7)

where the first term on the right-hand side, , accounts for the contributions coming from the inflaton self-interactions and were computed by D. Seery in  [108, 109]

 Pscalar=H4∗k3[g1ln(k)+g2], (3.8)

where and are numerical factors. Their diagrammatic representation is given in Fig. for the leading order and in Fig. for the next-to-leading order corrections. The loop corrections , arising from interactions between the tensor (graviton) modes and the scalar field, were ignored for simplicity in [108, 106], however they should be included since they are not slow-roll suppressed compared to loops of scalar modes. Their computation was presented for the first time in our paper [56] and will be reviewed in Secs. 4 to 6 of this thesis.

Both and are evaluated at around the time of horizon crossing and as such they are due to genuine quantum effects.

The contributions in the third and fourth lines of Eq. (3.5), also dubbed as “classical one-loop”, can be considered as classical loop contributions arising after the perturbation modes leave the horizon. The distinction between classical and quantum loops is intended as for example in [109]: quantum loops find their origin in the Lagrangian interaction terms between the inflaton perturbations and the gravitational modes or from self-interaction of ; classical loops are corrections merely coming from the expansion of using the formula and originate from zeroth order terms in the Schwinger-Keldysh formula.

Finally, the second line of (3.5) includes the integral of , the bispectrum of the scalar field defined by

 ⟨δϕ→k1δϕ→k2δϕ→k3⟩≡(2π)3δ(3)(→k1+→k2+→k3)Bϕ(k1,k2,k3) (3.9)

and from [13] we have

 Bϕ≃√ϵH4∗F(ki)mP, (3.10)

where is the Planck mass, is the slow-roll parameter () and is a function of the momenta moduli of dimension .

4 Perturbative expansion of the Lagrangian in P(X,ϕ) theories

In this and in the next two sections, we will review the computation of the tensor loop corrections to . For our purposes, the exponentials in Eq. (2.2) need to be expanded up to second order in the interaction Hamiltonian

 ⟨Ω|Θ(η)|Ω⟩1L = i⟨0∣∣T[Θ∫η−∞dη′(H+I(η′)−H−I(η′))]0⟩ (4.1) + (−i)22⟨0|T[Θ∫η−∞dη′(H+I(η′)−H−I(η′))∫η−∞dη′′(H+I(η′′)−H−I(η′′))]∣∣0⟩,

where . One-loop power-spectrum diagrams require an expansion of the interaction Hamiltonian to third and fourth order in the field fluctuations, i.e. . We provide in Figs. (3) and (4) the diagrammatic representation of the leading order corrections that we will find for the diagrams with tensor loops in single-field slow-roll inflation. The continuos lines represent scalar propagators, whereas the dotted lines indicate tensor propagators. In order to derive this result and the analytic expressions for these diagrams, we need to first calculate and expand up to fourth order in the field perturbations and . The starting point is the Lagrangian of the system.

We will begin with a more general Lagrangian for the scalar field than the usual , by introducing a non-conventional kinetic term, i.e.

 S=12∫d3xdt√−g[M2PR+2P(X,ϕ)], (4.2)

where is a generic function of the scalar field and of and is the Ricci scalar in four dimensions. Notice that the action (4.2) reduces to the standard case if , where is the potential for the scalar field.
Theories of inflation where the Lagrangian kinetic term is a generic function of the scalar field and its first derivatives, like in Eq. (4.2), are string theory-inspired. They represent interesting alternatives to the basic inflationary scenario because of their non-Gaussianity predictions. The crucial quantity in this sense is represented by the speed of sound , which is allowed to vary between and . The perturbative expansion of the interaction Hamiltonian in this kind of models has coefficients proportional to inverse powers of the sound speed and therefore, for small values of , allows both for non-negligible loop corrections to the power spectrum of the curvature fluctuations [113] and for large values for the amplitudes of three [31] and four [41, 42, 43, 44, 45, 46] point functions. In this thesis, we will carry out the calculations of the interaction Hamiltonian for these general theories up to a certain point and then, for simplicity in the presentation, focus on the canonical case (the remaining computations for more general Lagrangians will be found in [113]).

Let us list the background equations for the system

 2˙H+3H2=−P, (4.3) 3H2=2XPX−P, (4.4) ˙X(PX+2XPXX) + 2√3(2XPX−P)1/2XPX (4.5) = √2X(Pϕ−2XPXϕ),

where a dot indicates a derivative w.r.t. cosmic time and, to zeroth order, we have .
The so called flow-parameters are defined as

 ε≡−˙HH2, (4.6) η≡˙εεH. (4.7)

These quantities reduce to the slow-roll parameters in the standard case, so it is natural to assume and . It is not correct to talk about slow-roll if is left as a generic function of and , since the smallness of and does not necessarily indicate that and . It can be convenient to decompose as the sum , where

 εϕ≡−˙ϕH2∂H∂ϕ, (4.8) εX≡−˙XH2∂H∂X. (4.9)

The parameters that are expected to appear in the perturbative expansion of the Lagrangian are

 c2s=PXPX+2XPXX, (4.10) s≡˙cscsH, (4.11) u≡1−1c2s, (4.12) Σ≡XPX+2X2PXX, (4.13) λ≡X2PXX+23X3PXXX, (4.14) Π≡X3PXXX+25X4PXXXX, (4.15)

where is the sound speed. is allowed to vary between and , so the quantity can freely range between and . The only assumption we make is , from being constant in the standard case.

4.1 Arnowitt-Deser-Misner (ADM) decomposition for P(X,ϕ) theories

The Lagrangian in Eq. (4.2) will now undergo a perturbative expansion in terms of the field fluctuations ( is the homogeneous background value for the field) and of the metric fluctuations.
It is convenient to adopt the Arnowitt-Deser-Misner (ADM) splitting for the metric. In the spatially flat gauge the perturbed metric is

 ds2=−N2dt2+hij(dxi+Nidt)(dxj+Njdt), (4.16)
 hij=a2(t)(eγ)ij, (4.17)

where is the scale factor, is a tensor perturbation with (traceless and divergenceless) and det. Notice that repeated lower indices are summed up with a Kronecker delta, so stands for and .
In the ADM formalism, the action (4.2) becomes [13]

 S=12∫dtd3x√h[NR(3)+2NP+N−1(EijEij−E2)], (4.18)

where is the curvature scalar associated with the three dimensional metric and

 Eij=12(˙hij−▽iNj−▽jNi), E=hijEij.

A dot indicates derivatives w.r.t. time , all the spatial indices are raised and lowered with and units of will be from now on employed. To order we have

 R(3)=−14∂iγal∂iγal. (4.19)

The lapse and shift functions, and , can be written as

 N=1+α, Nj=∂jθ+βj,

where , and are functions of time and space ( is divergenceless). We have exploited the gauge freedom to set two scalar and two vector modes to zero, thus leaving one scalar mode from , one scalar and two vector modes from and two tensor modes (the two independent polarizations of the graviton) from , together with the inflaton field perturbation . and are non-dynamical degrees of freedom and can be expressed in terms of the other modes ( and ), once the Hamiltonian and the momentum constraints (we derive them in the next section) are solved.

Solving Hamiltonian and momentum constraint equations

Momentum and Hamiltonian constraints are derived from varying the action w.r.t. the shift and lapse functions respectively. It turns out that, in order to expand the action to a given order , it is only necessary to perturb and up to order [13, 31]. Therefore we will solve the constraints to second order in the metric and scalar field fluctuations.
Let us begin with the expansions

 α=α1+α2, βi=β1i+β2i, θ=θ1+θ2.

where and are respectively first and second order in the fields fluctuations (similarly for and , and for and ). Let us then expand to second order. is a generic function of and . We first need the expansion of

 X = Missing or unrecognized delimiter for \left (4.20) = −12[−N−2(˙ϕ+˙δϕ)2+2N−2Ni∂iδϕ(˙ϕ+˙δϕ)+(hij−NiNjN2)∂iδϕ∂jδϕ] = X0+ΔX

where , is the zeroth order part, i.e. and is the perturbation to the desired order (). Notice that , but for simplicity we will suppress the subscript ’’ in the background value of the field.
The expressions for the perturbations become

 ΔX1 = 2X0[˙δϕ˙ϕ−α1] (4.21) ΔX2 = X0[(δ˙ϕ˙ϕ)2−4α1δ˙ϕ˙ϕ−2α2+3α12−2Ni1∂jδ˙ϕ˙ϕ]−1a2˙ϕ2∂iδϕ∂iδϕ (4.22)

and so on for and higher order terms. The expansion of up to second order becomes

 P(X,ϕ)=P0+PX|0ΔX+Pϕ|0Δϕ+12!PXX|0(ΔX)2+12!Pϕϕ|0Δϕ2+PXϕ|0ΔXΔϕ

where as usual the subscript ’’ indicated the zeroth order, , and similarly for the second order derivatives, and needs to be expanded up to the needed order.

We are now ready to write the momentum and Hamiltonian contraints

 ▽i[N−1(Eij−δijE)]=N−1PX[˙ϕ−Nl∂lϕ]∂jϕ (4.24)
 R(3)+2P−4PXX−N−2(EijEij−E2)−2PXhij∂iϕ∂jϕ=0 (4.25)

The momentum constraint to first order reads

 2H∂jα1−12a2∂2β1j=PX˙ϕ∂jδϕ, (4.26)

where is the Hubble parameter. Eq. (4.26) can be solved to derive . Taking the derivative of both sides of (4.26) and using the divergenceless condition for , we have

 α1=PX˙ϕδϕ2H. (4.27)

Using the solution found for , we find , from which we can set . Here , which we will indicate in the rest of the thesis also as , and from now on we define for simplicity.

The momentum constraint to second order is

 2H∂jα2−4Hα1∂jα1−1a2∂jα1∂2θ1+1a2∂iα1∂i∂jθ1−12∂iα1˙γij −12a2∂2βj+14˙γik∂iγkj−14γik∂i˙γkj−14˙γik∂jγik+12a2∂iθ1∂2γij =PX∂jδϕδ˙ϕ+2XPXX∂jδϕδ˙ϕ−2XPXX˙ϕα1∂jδϕ−PX˙ϕα1∂jδϕ +PXϕ˙ϕδϕ∂jδϕ. (4.28)

The solutions are

 α2 = α212+12Ha2∂−2[∂2α1∂2θ1−∂i∂jα1∂i∂jθ1]+PX2H∂−2Σ (4.29) + 14H∂−2[˙γij∂i∂jα1]−14a2H∂−2[∂i∂jθ1∂2γij] + 18H∂−2[∂j˙γik∂jγik]+PXϕ˙ϕ2H∂−2[(∂jδϕ)2+δϕ∂2δϕ] + XPXXH∂−2[∂2δϕδ˙ϕ+∂jδϕ∂jδ˙ϕ−˙ϕ(∂jα1∂jδϕ+α1∂2δϕ)],

where , and

 12a2∂2βj = 2H∂jα2−4Hα1∂jα1−1a2∂jα1∂2θ1+1a2∂iα1∂i∂jθ1 (4.30) − 12∂iα1˙γij+14˙γik∂iγkj−14γik∂i˙γkj−14˙γik∂jγik + 12a2∂iθ1∂2γij−PX∂jδϕδ˙ϕ−2XPXX∂jδϕδ˙ϕ + 2XPXX˙ϕα1∂jδϕ+PX˙ϕα1∂jδϕ−PXϕ˙ϕδϕ∂jδϕ.

Let us now move to the Hamiltonian constraint which provides

 4Ha2∂2θ1=−4XPX(δ˙ϕ˙ϕ−α1)+2Pϕδϕ−8PXXX2(δ˙ϕ˙ϕ−α1)−4XPXϕδϕ−12H2α1,

to first order and

 −4Ha2∂2θ2=(−2α1)[4XPXδ˙ϕ˙ϕ+20PXXX2δ˙ϕ˙ϕ+2XPXϕδϕ+8PXXXX3δ˙ϕ˙ϕ +4PXXϕX2δϕ+4Ha2∂2θ1]−4X(PX+2X