Numerically evaluating the bispectrum in curved field-space - with PyTransport 2.0
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Prepared for submission to JCAP
Numerically evaluating thebispectrum in curved field-space – with PyTransport 2.0
John W. Ronayne and David J. Mulryne
School of Physics and Astronomy, Queen Mary University of London, Mile End Road,London, E1 4NS, UKE-mail: [email protected]; [email protected]
Abstract.
We extend the transport framework for numerically evaluating the power spec-trum and bispectrum in multi-field inflation to the case of a curved field-space metric. Thismethod naturally accounts for all sub- and super-horizon tree level effects, including thoseinduced by the curvature of the field-space. We present an open source implementation of ourequations in an extension of the publicly available
PyTransport code. Finally we illustratehow our technique is applied to examples of inflationary models with a non-trivial field-spacemetric. a r X i v : . [ a s t r o - ph . C O ] J a n Introduction
Recently a convenient framework was developed by Dias et al. [1] to numerically calculatethe primordial power spectrum and bispectrum of the curvature perturbation, ζ , producedby inflation with an arbitrary number of fields (see also Ref. [2–7] for earlier related works ).The essence of the approach is to set up coupled ordinary differential equations (ODEs) forthe correlations of the inflationary fields’ fluctuations. These correlations can then be relatedto the correlations of the curvature perturbation. The framework accounts for all tree leveleffects on sub- and super-horizon scales, and is referred to as the “transport approach” toinflationary perturbations.The work of Dias et al. [1] presented a rather general framework, but specific equationswere only given for inflation driven by multiple canonical scalar fields with Euclidean fieldspace metric, and only this case was implemented in two numerical codes [8, 9] which ac-companied the paper. The primary goal of the present work, therefore, is to present explicitequations for the more general case where the field-space metric of the multi-field systemis non-Euclidean. At the level of the power spectrum the transport method has alreadybeen extended to this case, and a code released in the form of a Mathematica worksheet, mTransport , by Dias, Frazer and Seery [10]. Here we extend this work to the bispectrum,presenting all the elements needed to implement the framework of Ref. [1] in this more generalsetting. An online resource for the transport method and the various codes (including thenew contributions discussed below) is available at transportmethod.com.The two numerical packages which accompanied Ref. [1] represent the first publiclyavailable tools developed to calculate the bispectrum numerically in a multi-field model.Moreover, both utilise computer algebra packages to ensure minimal work for a user . Thefirst package was developed by Seery, CppTransport , and represents a sophisticated set oftools developed in
C++ utilising the power of a number of external
C++ libraries, including
GiNaC for front end computer algebra manipulations, and
BOOST for the evolution of ODEs.It also contains a bespoke and sophisticated preprocessor, and automated data achieving andretrieval tools. On the other hand, the second package developed by Mulryne,
PyTransport ,is intended to be a more light weight product, built on a rather direct implementation of thetransport framework. A working
Python installation (with particular packages installed) anda
C++ compiler are its only dependences. The core of
PyTransport is written in
C++ to ensuregood numerical performance, but the algebraic manipulations are handled by
Python ’s SymPy package. Once an inflationary model is specified, front end functions automatically edit
C++ code that is then complied into a bespoke
Python module. This approach combines the speedof
C++ with the convenience of
Python . Data storage and analysis are left to the user. Byembedding the code in
Python , however, the power of its many packages written for thesepurposes can be readily harnessed.A second aim of the present work, therefore, is to introduce a new version of the
PyTransport package
PyTransport 2.0 , which extends the code to the case of a non-trivialfield-space metric. Our new package allows users to specify both the potential and the field- Early work on the transport approach considered only the super-horizon evolution of perturbations, how-ever it was shown in Ref. [2] that the approach could be extended to sub-horizon scales, and this work wasused as a basis for Ref. [1]. Earlier publicly available numerical packages for the power spectrum in canonical multi-field inflation are
Pyflation (pyflation.ianhuston.net [11, 12]) and
MultiModeCode [13], and a publicly available code for thebispectrum in single field inflation is
BINGO [14]. Other numerical work at the level of the bispectrum in thesingle field case includes Refs. [15–18]. – 2 –pace metric for a given model in a
Python script, and automatically takes both these functionsand generates a bespoke
Python module. This module contains a number of useful functionsincluding those needed to calculate the power spectrum and bispectrum of ζ . The packageis available at github.com/jronayne/PyTransport. Ref. [9] has also been updated such thatversion 2 details how to use this new code.Concurrently with our work, in an independent study Seery and Butchers have alsoextended the transport framework to the case of a non-Euclidean metric [19], and have in-corporated their work into a new version of the CppTransport package, which is currentlyavailable as an experimental version at github.com/ds283/CppTransport.A non-trivial field-space metric is an important feature of inflationary models that arisesin a number of contexts. First, it may be that a system of fields with a Euclidean metric maybe more easily described in an alternative coordinate system. In this case the metric remainsflat, but is nevertheless of a different form. The second possibility is that the field-space metricis curved, which arises in many circumstances. Classic examples include when non-minimallycoupled fields are rewritten as minimally coupled fields in the Einstein frame (see Refs. [20–22]), and when inflationary models are derived in supergravity. We note that a non-trivialfield-space metric can be just as important as the fields’ potential energy in determining thefields’ dynamics, and hence the observational predictions of inflationary models.Our work is structured as follows. In the first part of the paper we follow the generalframework set out in Ref. [1] closely and provide the additional calculations needed for ourmore general setting. First, in §2 we derive the second and third order action for the covariant“field” perturbations first introduced in Ref. [23] and subsequently used in Ref. [24] to ana-lytically study the bispectrum with a curved field-space metric (see also Ref. [25]). Treatingthese perturbations and their canonical momenta as operators, we calculate Hamilton’s equa-tions of motion. Then we briefly review how Hamilton’s equations can be used to calculateequations of motion for the correlations of the fluctuations in §3. These equations are thetransport equations which give our approach its name, and we provide them explicitly forthe non-trivial field-space metric case. Finally, we calculate initial conditions for this systemusing the In-In formalism in §4, and derive the relation between the covariant field perturba-tions and the curvature perturbation ζ , which allows field-space correlations to be convertedinto correlations of ζ , in §5. This completes the specific equations needed to implement theframework of Ref. [1] for the case of a non-Euclidean field-space metric. We next turn to ournumerical implementation of the equations we have derived in the PyTransport 2.0 package.After discussing briefly our implementation we showcase its utility with a number of examplesin §6. We conclude in §7.
We begin by deriving the action to cubic order, and the Hamiltonian equations of motion,for covariant field-space perturbations defined on flat hypersurfaces. As we have discussed,the calculations mirror those presented in Ref. [1] but generalised to the case of a trivialfield-space metric.We begin with the action for N scalar fields minimally coupled to gravity S = 12 (cid:90) d x √− g (cid:2) M R − G IJ g µν ∂ µ φ I ∂ ν φ J − V (cid:3) , (2.0.1)– 3 –here R is the Ricci scalar associated with the spacetime metric g µν , G IJ is the N dimensionalfield-space metric, and where upper case Roman indices run from to N , which are raisedand lowered by G IJ . G IJ is a function of the fields.For a flat Friedmann-Robertson-Walker (FRW) cosmology this action leads to the back-ground equations of motion, M H = 12 G IJ ˙ φ I ˙ φ J + V ,D t ˙ φ I + 3 H ˙ φ I = − V I , (2.0.2)where the covariant time derivative of a field-space vector, U I , is defined as D t U I = ˙ U I + ˙ φ M Γ IMN U N , (2.0.3)and t indicates cosmic time, with a over-dot indicating differentiation with respect to cosmictime. The connection Γ IMN is the Levi-Civita connection compatible with the field-spacemetric G IJ .We now consider perturbations about the FRW background. It proves convenient tofollow Refs. [24, 26–28] and employ the (3+1) ADM decomposition of spacetime, such that g = − ( N − N i N i ) , g i = N i , g ij = h ij , (2.0.4)where N is the lapse function, N i is the shift vector, h ij the spatial metric, and lower caseRoman indices run over the spatial coordinates. With this choice of variables, the action(2.0.1) is written as S = 12 (cid:90) d x √ h (cid:18) M (cid:20) N R (3) + 1 N ( E ij E ij − E ) (cid:21) + 1 N π I π I − N G IJ ∂ i φ I ∂ i φ J − N V (cid:19) , (2.0.5)where R (3) is the Ricci scalar of the 3-metric h ij . The quantity E ij is proportional to theextrinsic curvature on slices of constant t , with E ij = 12 ( ˙ h ij − N i | j − N j | i ) , (2.0.6)where a bar denotes covariant derivatives with respect to the three metric. The quantity π I is defined as π I = ˙ φ I − N j φ I | j . (2.0.7) Working in the spatially flat gauge, and considering only scalar perturbations , one has R (3) =0 and h ij = a δ ij , and the only perturbations to the spacetime metric are given by N = 1 + Φ + Φ + · · · N i = θ ,i + θ ,i + · · · , (2.1.1)where Φ and Φ are the first and second order perturbations in the lapse, and θ and θ arethe first and second order perturbations in the shift. Although beyond linear order vector and tensor perturbations do couple to the scalar perturbations, theydo not affect the calculation of the scalar three point function which follows from the third-order actioninvolving only scalar perturbations. – 4 – .2 Field perturbations
Next we consider the perturbations to the matter sector and hence to the scalar fields present.The field perturbations, δφ I ( x, t ) , are defined by the expression φ I = φ I ( t ) + δφ I ( x, t ) . Thesefield-space perturbations are not, however, covariant under relabelling of field-space, and itproves convenient to work with a different set of perturbations that are covariant, which welabel Q I . These were first introduced by Gong & Tanaka [23]. The idea is to consider thegeodesic that links together the position in field-space labelled by φ I and that labelled by φ I ,and an affine parameter parametrising this trajectory denoted λ . The coordinate displacement δφ I can then be expressed by the series expansion about the point λ = 0 as δφ I = dφ I dλ (cid:12)(cid:12)(cid:12)(cid:12) λ =0 + 12! d φ I dλ (cid:12)(cid:12)(cid:12)(cid:12) λ =0 + · · · . (2.2.1)We can then form the geodesic equation D λ φ I = d φ I dλ + Γ IJK dφ J dλ dφ K dλ = 0 , (2.2.2)and define Q I = d φ I / d λ | λ =0 and D λ = Q I ∇ I (where ∇ I is the covariant derivative). Usingthis geodesic equation, the expansion (2.2.1) can be rewritten as δφ I = Q I −
12! Γ
IJK Q J Q K , (2.2.3)which relates field perturbations to the covariant perturbations. The time derivative of fieldfluctuations, δ ˙ φ I , can also be written in terms covariant quantities as δ ˙ φ I = D t Q I − ˙ φ M Γ IMN Q N −
12 Γ
IJK,M ˙ φ M Q J Q K − Γ I ( JK ) D t Q J Q K + Γ I ( JK ) Γ JMN Q K ˙ φ M Q N , (2.2.4)as can a perturbation to the field-space metric, and using (2.2.3) we find δG IJ = 2Γ ( IJ ) K Q K − Γ ( IJ ) K Γ KMN Q M Q N + Γ ( IM ) L Γ MJK Q K Q L + Γ ( JM ) L Γ MIK Q K Q L + 12 ( G IM Γ MJK,L + G JM Γ MIK,L ) Q K Q L . (2.2.5)Here we have adopted the notation of using ( IJ ) parenthesis to illustrate symmetrizationover the indices I and J . A bar | is used to excluded certain indices from the symmetrizationprocedure, for example, ( I | J | K ) symmetrizes I and K but not J . The next step is to insert our perturbed expressions for N , N i and φ I into (2.0.5) to calculatethe perturbed action. Expanding order by order, the first order action simply leads back tothe background equations, while the action at second and higher order lead to the dynamicsof the perturbations.After some integration by parts and discarding total derivatives, one finds the action atsecond and third order can be written in the form given by Elliston et al. [24] S (2) = 12 (cid:90) d xa (cid:16) Φ (cid:104) − M H Φ + G IJ ˙ φ I ˙ φ J Φ − G IJ ˙ φ I D t Q J − V ; I Q I (cid:105) − a ∂ θ (cid:104) M H Φ − G IJ ˙ φ I Q J (cid:105) + R KIJL ˙ φ K ˙ φ L Q I Q J + G IJ D t Q I D t Q J − G IJ ∂ i Q I ∂ j Q J − V ; IJ Q I Q J (cid:17) , (2.3.1)– 5 –nd S (3) = 12 (cid:90) d xa (cid:32) M H Φ + 4 M Ha Φ ∂ θ − M Φ a ( ∂ i ∂ j θ ∂ i ∂ j θ − ∂ θ ∂ θ ) − G IJ ˙ φ I ˙ φ J Φ + 2Φ ˙ φ I D t Q J + 2 a Φ G IJ ˙ φ I ∂ i θ ∂ i Q J − Φ R L ( IJ ) M ˙ φ L ˙ φ M Q I Q J − Φ (cid:18) G IJ Q I Q J + 1 a G IJ ∂ i Q I ∂ j Q J (cid:19) − a ∂ i θ G IJ D t Q I ∂ i Q J + 43 R I ( JK ) L ˙ φ L D t Q I Q J Q K + 13 R ( I | LM | J ; K ) ˙ φ L ˙ φ M Q I Q J Q K − V ;( IJK ) Q I Q J Q K − V ;( IJ ) Φ Q I Q J (cid:19) , (2.3.2)where R IJKL is the Riemann tensor compatible with the field-space metric G IJ , and R IJKL ; M it’s covariant derivative. Varying the action with respect to the lapse and shift leads to two constraint equations thatcan be used to provide expressions for the perturbations in the lapse and shift in terms ofthe covariant Q I perturbations [29]. These can be substituted back into the action to expressthe perturbed action only in terms of Q I . To do so we only need the constraint equations atlinear order (as explained in [26]), but later we will also need them at second order too, sowe provide the full expressions here.Considering first variation with respect to the shift, at linear order one finds Φ = 12 M H G IJ ˙ φ I Q J , (2.3.3)while at second order Φ = Φ ∂ − M H (cid:34) − M a ∂ i ∂ j Φ ∂ i ∂ j θ + M a ∂ Φ ∂ θ + G IJ ( ∂ i D t Q I ) ∂ i Q J + G IJ D t Q I ∂ Q J (cid:3) . (2.3.4)On large scales where spatial gradients decay, one then finds that Φ = Φ ∂ − M H (cid:2) G IJ ( ∂ i D t Q I ) ∂ i Q J + G IJ D t Q I ∂ Q J (cid:3) . (2.3.5)Next varying the action with respect to the lapse, at linear order we have ∂ θ = − a H Φ + a M H G IJ Φ ˙ φ I ˙ φ J − a M H G IJ ˙ φ I D t Q J − a M H V ; I Q I , (2.3.6)and at second order ∂ θ =2Φ ∂ θ − a H (cid:0) ∂ i ∂ j θ ∂ i ∂ j θ − ∂ θ ∂ θ (cid:1) + a M H G IJ Φ ˙ φ I D t Q J + 12 M H G IJ ˙ φ I ∂ i θ ∂ i Q J − a M G IJ D t Q I D t Q J − M H G IJ ∂ i Q I ∂ i Q J − a M V ;( IJ ) Q I Q J + a H − )( (cid:15) − − a M R L ( IJ ) M ˙ φ L ˙ φ M Q I Q J , (2.3.7)– 6 –here (cid:15) = − ˙ H/H is the slow-roll parameter. Using these latter expressions and again takingthe large scale superhorizon limit one finds the additional relation H Φ = 1 M H G IJ Φ ˙ φ I ˙ φ J − M H G IJ ˙ φ I D t Q J − M H V ; I Q I , (2.3.8)at first order, and G MN D t Q M D t Q N =2Φ G IN ˙ φ I D t Q N − V ;( MN ) Q M Q N − M H (3Φ − )( (cid:15) − − R I ( MN ) J ˙ φ I ˙ φ J Q M Q N , (2.3.9)at second order. Finally, using the equations for Φ (2.3.3) and θ (2.3.6) in terms of Q I one can write thequadratic and cubic parts of the action (2.3.1) and (2.3.2) solely in terms of Q I . It is con-venient at this stage to move from real space to Fourier space. After doing so, to keep ourexpressions to a manageable size, we follow the extended summation convention introducedin Ref. [1]. When considering Fourier space quantities we use bold font indices, I , J , . . . toindicate that the usual summation over fields is accompanied by an integration over Fourierspace. For example, A I B I = (cid:90) d k I (2 π ) A I ( k I ) B I ( k I ) , (2.3.10)where the subscript I on k I indicates that this is the wavenumber associated with objectsthat carry the I index. Using this notation the action reads S (2) = 12 (cid:90) dta (cid:0) G IJ ( k I , k J )( D t Q I ( k I ) D t Q J ( k J ) + M IJ ( k I , k J ) Q I ( k I ) Q J ( k J ) (cid:1) , (2.3.11)at second order and S (3) = 12 (cid:90) dta (cid:0) A IJK ( k I , k J , k K ) Q I ( k I ) Q J ( k J ) Q K ( k K )+ B IJK ( k I , k J , k K ) D t Q I ( k I ) Q J ( k J ) Q K ( k K )+ C IJK ( k I , k J , k K ) D t Q I ( k I ) D t Q J ( k J ) Q K ( k K ) (cid:1) , (2.3.12)at third order, where we have defined G IJ ( k I , k J ) = (2 π ) δ ( k I + k J ) G IJ (2.3.13) M IJ ( k I , k J ) = (2 π ) δ ( k I + k J ) (cid:18) k I a G IJ − m IJ (cid:19) (2.3.14) A IJK ( k I , k J , k K ) = (2 π ) δ ( k I + k J + k K ) a IJK (2.3.15) B IJK ( k I , k J , k K ) = (2 π ) δ ( k I + k J + k K ) b IJK (2.3.16) C IJK ( k I , k J , k K ) = (2 π ) δ ( k I + k J + k K ) c IJK . (2.3.17)with m IJ = V ; IJ − R IKLJ ˙ φ K ˙ φ L − (cid:15)M ˙ φ i ˙ φ J − ( ˙ φ I D t ˙ φ J + ˙ φ J D t ˙ φ I ) HM , (2.3.18)– 7 –nd a IJK = − V ; IJK − ˙ φ I V ; JK HM + ˙ φ I ˙ φ J ξ K H M + ˙ φ I ξ J ξ K H M (cid:18) − ( k J · k K ) k J k K (cid:19) + ˙ φ I ˙ φ J ˙ φ K HM (cid:32) G MN ˙ φ M ˙ φ N H M (cid:33) + ˙ φ I G JK HM k J · k K a − G NK ˙ φ L ˙ φ M ˙ φ N KR L ( IJ ) M M H + 13 ˙ φ L ˙ φ M R ( I | LM | J ; K ) , (2.3.19) b IJK = ˙ φ I ˙ φ J ˙ φ K H M − ˙ φ I ξ J ˙ φ K H M (cid:18) − ( k J · k K ) k J k K (cid:19) − ξ I G JK HM k I · k J k I + 43 ˙ φ L R I ( JK ) L , (2.3.20) c IJK = − G IJ ˙ φ K HM + ˙ φ I ˙ φ J ˙ φ K H M (cid:18) − ( k I · k J ) k I k J (cid:19) + G IJ ˙ φ K HM k I · k K k I , (2.3.21)where ξ I = 2 D t ˙ φ I + ˙ φ I H G NM ˙ φ N ˙ φ M M . (2.3.22)Here a IJK is to be symmetrised over all three indices, b IJK over J & K and c IJK over I & J .Each index permutation will have a corresponding exchange of wavenumber associated withthe indices. From the action we can derive equations of motion for the perturbations Q I ( k ) . Perturbationsbehave quantum mechanically on subhorizon scales, and to account for this we introduce theconjugate momenta to Q I , P I , and treat Q I and P I as Heisenberg picture operators whichobey Hamilton’s equations.The canonical momentum is defined as P I = δSδ ( D t Q I ) , (2.4.1)and obeys the relation, (cid:2) Q I ( k I , t ) , P J ( k J , t (cid:48) ) (cid:3) = i (2 π ) δ IJ ( k I + k J ) δ ( t − t (cid:48) ) . (2.4.2)Utilising Eqs. (2.3.11) & (2.3.12) one finds P I = a (cid:18) D t Q I + 12 B JKI Q J Q K + C IJK P J Q K (cid:19) . (2.4.3)At this stage it is helpful to rescale P I such that P I → a P I , where for convenience weemploy the same symbol for the rescaled momentum, and use it solely from here on. In termsof the rescaled momentum D t Q I = P I − B JKI Q J Q K − C IJK P J Q K + · · · . (2.4.4)– 8 –he Hamiltonian is then given by H = (cid:90) dt a G IJ P I P J − M IJ Q I Q J (cid:124) (cid:123)(cid:122) (cid:125) H − A IJK Q I Q J Q K − B IJK Q I Q J P K − C IJK P I P J Q K (cid:124) (cid:123)(cid:122) (cid:125) H int , (2.4.5)where have labelled the ‘free’ part of the Hamiltonian H , and the interaction part, H int .Finally Hamilton’s equations provide us with evolution equations for Q I and P I whichare D t Q I = − i [ Q I , H ] (2.4.6) D t P I = − i [ P I , H ] − HP I , (2.4.7)where the evolution of P I takes a slightly non-canonical form due to the rescaling of thecanonical momenta. Once equations of motion are known for the Heisenberg operators, these can immediately beconverted into equations of motion for expectation values of products of these operators usingEhrenfest’s theorem [2] . This is the idea behind the Transport approach and was explored indetail in Ref. [1], where the reader can turn for further details. For convenience, we first labelthe full phase space of Heisenberg operations with the symbol δX a , where δX a = ( Q I , P J ) and where lower case Roman indices run from to N . The expectation values we areinterested in are then the two and three-point functions of δX a (cid:104) δX a ( k a ) δX b ( k b ) (cid:105) = (2 π ) δ ( k a + k b )Σ ab ( k a ) (3.0.1) (cid:104) δX a ( k a ) δX b ( k b ) δX c ( k c ) (cid:105) = (2 π ) δ ( k a + k b + k c ) B abc ( k a , k b , k c ) . (3.0.2)As described, the equations of motion for these correlation functions follow directly fromEqs. (2.4.6)-(2.4.7) together with Ehrenfest’s theorem, and can be presented in terms ofequations of motion for Σ ab and B abc . In our covariant setting these take the form D t Σ ab ( k ) = u ac ( k )Σ cb ( k ) + u bc ( k )Σ ac ( k ) , (3.0.3)and D t B abc ( k a , k b , k c ) = u ad ( k a ) B dbc ( k a , k b , k c ) + u bd ( k b ) B adc ( k a , k b , k c ) + u cd ( k c ) B abd ( k a , k b , k c )+ u ade ( k a , − k b , − k c )Σ db ( k b )Σ ec ( k c )+ u bde ( k b , − k a , − k c )Σ ad ( k a )Σ ec ( k c )+ u cde ( k c , − k a , − k b )Σ ad ( k a )Σ be ( k c ) , (3.0.4)where the covariant time derivative acts on Σ ab in the following way D t Σ ab ( k ) = ∂ t Σ ab ( k ) + Γ ac ( k )Σ cb ( k ) + Γ bc ( k )Σ ac ( k ) , (3.0.5)– 9 –nd on B abc as D t B abc ( k a , k b , k c ) = ∂ t B abc ( k a , k b , k c ) + Γ ad ( k ) B dbc ( k a , k b , k c )+ Γ bd ( k ) B adc ( k a , k b , k c ) + Γ cd ( k ) B abd ( k a , k b , k c ) , (3.0.6)with Γ ab is defined as Γ ab = (cid:18) Γ IJK ˙ φ K
00 Γ
IJK ˙ φ K (cid:19) , (3.0.7)The u -tensors take the form u ab = (cid:18) δ IJ ˜ m IJ − Hδ IJ (cid:19) , (3.0.8)where ˜ m IJ = − k a G IJ − m IJ , (3.0.9)and u abc = (cid:18) − b JK I − c I JK a I JK b I KJ (cid:19)(cid:18) − c I KJ b I JK c KJ I (cid:19) . (3.0.10) The two-point function will in general be complex, and can be divided into its real andimaginary parts Σ ad = Σ ad Re + i Σ ad Im , (3.1.1)with the real part symmetric under interchange of its indices, and the imaginary part anti-symmetric. Both parts independently satisfy Eq. (3.0.3). On superhorizon scales the imagi-nary part decays to zero, indicating that on large scales the statistics of inflationary pertur-bations follow classical equations of motion. B abc , is in general also complex, but is real when only tree-level effects are included. Inour numerical implementation of the transport system we evolve the real and imaginary partsof Σ ab separately using Eq. (3.0.3), and evolve B abc according to the equation D t B abc ( k a , k b , k c ) = u ad ( k a ) B dbc ( k a , k b , k c ) + u bd ( k b ) B adc ( k a , k b , k c ) + u cd ( k c ) B abd ( k a , k b , k c )+ u ade ( k a , k b , k c )Σ db Re ( k b )Σ ec Re ( k c ) − u ade ( k a , k b , k c )Σ db Im ( k b )Σ ec Im ( k c )+ u bde ( k b , k a , k c )Σ ad Re ( k a )Σ ec Re ( k c ) − u bde ( k b , k a , k c )Σ ad Im ( k a )Σ ec Im ( k c )+ u cde ( k c , k a , k b )Σ ad Re ( k a )Σ be Re ( k b ) − u cde ( k c , k a , k b )Σ ad Im ( k a )Σ be Im ( k b ) , (3.1.2)which follows from Eq. 3.0.4 once Σ ab is broken into real and imaginary parts, and whichmakes it clear that B abc remains real if its initial conditions are real.– 10 – Initial conditions for the two and three-point functions
In order to solve for Σ ab Re , Σ ab Im and B abc numerically, the last element we need are initial con-ditions. Following the approach of Ref. [1] (which is closely related to that of Ref. [16]), theseare fixed at some early time at which all the wavenumbers of a given correlation are far insidethe horizon during inflation, and where m IJ is subdominant to ( k/a ) G IJ in Eq. (2.3.14). Inthis limit it is reasonable to assume that the solution for the two-point correlation function of Q I is well approximated by the de-Sitter space solution and we can use this solution to pro-vide initial conditions for our numerical evolution. We note that it is only required that thissolution be valid at some point long before all scales of interest cross the horizon, and more-over, that the numerical evolution is then free to evolve away from this solution, accountingfor the complex dynamics that can subsequently occur in general inflationary models.The two-point function in de Sitter space is typically written in conformal time τ andtakes the form, (cid:104) Q I ( k , τ ) Q J ( k , τ ) (cid:105) = (2 π ) δ ( k + k )Π IJ H k (1 − ikτ )(1 + ikτ ) e ik ( τ − τ ) , (4.0.1)where Π IJ is given by [24] Π IJ ( τ , τ ) = T exp (cid:18) − (cid:90) τ τ dτ Γ IKL (cid:2) φ M ( τ ) (cid:3) dφ K dτ (cid:19) G LJ ( τ ) , (4.0.2)which transforms as a bitensor with the first index I transforming in the tangent spaceat point φ M ( τ ) and the second index J in the tangent space at point φ M ( τ ) . The two-point functions (cid:104) Q I ( τ ) P J ( τ ) (cid:105) , and (cid:104) P I ( τ ) P J ( τ ) (cid:105) can then be calculated by differentiatingEq. (4.0.1), using the definition of P I and accounting for the use of conformal time. For ourpurposes we only need to consider the limit τ → τ with − τ (cid:29) , which corresponds to equaltime correlations on sub-horizon scales. In this limit Π IJ → G IJ , and one finds Σ ab ∗ Re = 12 a k (cid:18) aG IJ − aHG IJ − aHG IJ ( k /a ) G IJ (cid:19) (4.0.3) Σ ab ∗ Im = 12 a k (cid:18) kG IJ − kG IJ (cid:19) , (4.0.4)where we denote values at the initial time long before horizon crossing with an asterisk. Theinitial conditions for Σ ab Re where also given by Dias, Frazer and Seery [10]. Some further detailsare given in appendix A.1.In order to calculate the initial conditions for B abc we need to calculate the three-pointcorrelation functions for Q I and P I for Fourier modes on sub-horizon scales. As argued inRef. [1], these can be calculated using the In-In formalism. By writing the interaction partof the Hamiltonian given in Eq. (2.4.5) in the form H int = H abc δX a δX b δX c , the generalexpression for the three-point function can compactly be written as (cid:104) δX a δX b δX c (cid:105) ∗ = − i (cid:90) τ init −∞ d τ (cid:68)(cid:104) δX a ∗ δX b ∗ δX c ∗ , H efg δX e δX f δX g (cid:105)(cid:69) , (4.0.5)which leads to B abc ∗ = − i (cid:90) τ init −∞ dτ H efg Σ ae ( τ ∗ , τ )Σ bf ( τ ∗ , τ )Σ cg ( τ ∗ , τ ) + c.c. , (4.0.6)– 11 –here we have defined H abc as H abc = 13! (cid:18) − a IJK − b IKJ − b KJI − c IJK (cid:19)(cid:18) − b IJK c KJI − c IKJ (cid:19) , (4.0.7)and Σ ab ( τ , τ ) with dependence on two times as (cid:104) δX a ( k , τ ) δX b ( k , τ ) (cid:105) = (2 π ) δ ( k + k )Σ ab ( τ , τ ) . (4.0.8)The explicit integrals which result for the different elements of B abc are similar in structureto those of the canonical field-space metric case presented in Ref. [1], where one can turnfor a full discussion. When performing the integrations explicitly we must understand thetime dependence of the terms which enter. The time dependence of the b IJK and c IJK tensors which appear in the interaction Hamiltonian is slow-roll suppressed and their timedependence can be neglected. On the other hand, the a IJK tensor contains ‘fast’ changingterms proportional to ( k/a ) ∼ ( kτ ) which grow exponentially into the past and whosetime dependence must be included. It is also assumed that H and Π IJ which appear in theexpression for Σ( τ , τ ) are also sufficiently slowly varying that their time dependence can beneglected. The integral is dominated by its upper limit, and these assumptions mean thatwhen evaluating it one takes Π IJ → G IJ ( τ ∗ ) and H → H ( τ ∗ ) . The assumptions need onlybe true for a short period around the time the initial conditions are fixed. In the resultingexpressions for the initial conditions for B abc , we keep both the terms which grow fastest as τ → −∞ as well as the sub-leading terms. The results are rather long to present, and so aregiven in appendix A.1 together with some further details of the calculation.We note that all the initial conditions are the simply covariant versions of those for thecanonical case presented in Ref. [1] with no new terms appearing (except through the extraRiemann terms in the a and b tensors). Thus far we have discussed the framework in which the power spectrum and bispectrumof covariant field perturbations can be calculated. These are however not directly related toobservations. A quantity often used to make the connection between primordial perturbationsand observational constraints is the curvature perturbation on uniform density slices, ζ .To calculate the statistics of ζ we need to know how it is related to the set of pertur-bations { Q I , P J } . We require only the form of this relation on super-horizon scales, and wewrite it in the form ζ ( k ) = N a δX a + 12 N ab δX a δX b , (5.0.1)where N a ( k ) =(2 π ) δ ( k − k a ) N a N ab ( k , k a , k b ) =(2 π ) δ ( k − k a − k b ) N ab ( k a , k b ) . (5.0.2)In this notation the two and the three-point function of ζ are given by (cid:104) ζ ( k ) ζ ( k ) (cid:105) =(2 π ) δ ( k + k ) P ( k ) (cid:104) ζ ( k ) ζ ( k ) ζ ( k ) (cid:105) =(2 π ) δ ( k + k + k ) B ( k , k , k ) , (5.0.3)– 12 –ith P ( k ) = N a N b Σ ab Re ( k ) B ( k , k , k ) = N a N b N c B abc ( k , k , k ) + ( N a N b N cb ( k , k )Σ ac Re ( k )Σ bd Re ( k ) + 2 cyc. ) . (5.0.4)For the case of multi-field inflation with canonical kinetic terms, N a and N ab werecalculated in Ref. [30] (also see Refs. [31, 32]). Here we extend the calculation to the case ofa non-trivial field-space metric.A first step in the calculation of ζ in terms of field-space fluctuations on a flat hypersur-face is to relate ζ to the total density perturbation on the flat hypersurface. This calculationwas performed in Ref. [30], and is unchanged in our new setting. One finds ζ = − H δρ ˙ ρ + H ˙ δρδρ ˙ ρ − H ρδρ ˙ ρ + ˙ H δρ ˙ ρ . (5.0.5) The new element for the non-trivial field-space case is therefore to calculate δρ in this setting.In general, one finds that ρ = − T /g [33], where T µν is the energy momentum tensor. Theperturbation in the density up to second order is therefore δρ = δT + ρδg + (cid:0) δT + ρδg (cid:1) δg . (5.1.1)For an arbitrary number of scalar fields with non-trivial field-space metric the energy mo-mentum tensor is given by T µν = G IJ ∂ µ φ I ∂ ν φ J − G IJ g µν ∂ λ φ I ∂ λ φ J − g µν V. (5.1.2)This leads to the background energy density ρ = G IJ ˙ φ I ˙ φ J + V as expected. PerturbingEq. (5.1.2) and using Eq. (5.1.1) and recalling that g + δg = − + 2Φ − g i + δg i = ∂ i θ + ∂ i θ − ∂ i θ g ij + δg ij = h ij − ∂ i θ ∂ j θ , (5.1.3)one finds that δρ = 12 G IJ ( ˙ φ I ˙ δφ J + ˙ φ J ˙ δφ I ) − Φ G IJ ( ˙ φ I ˙ δφ J + ˙ φ J ˙ δφ I ) + 12 δG IJ ( ˙ φ I ˙ δφ J + ˙ φ J ˙ δφ I )+ 12 G IJ ˙ δφ I ˙ δφ J − Φ G IJ ˙ φ I ˙ φ J + 12 (3Φ − ) G IJ ˙ φ I ˙ φ J + 12 δG IJ ˙ φ I ˙ φ J − Φ δG IJ ˙ φ I ˙ φ J + V ; I δφ I + 12 V ;( IJ ) δφ I δφ J . (5.1.4)Finally, we need to rewrite this expression in terms of the covariant perturbations, Q I instead of the raw field perturbations δφ I . Collecting some terms together and applying therelations (2.2.3), (2.2.4) and (2.2.5) we obtain a neat expression which at linear order gives δρ = − Φ G IJ ˙ φ I ˙ φ J + G ( IJ ) ˙ φ I D t Q J + V ; I Q I , (5.1.5)– 13 –nd at second order δρ = 12 R L ( IJ ) M ˙ φ L ˙ φ M Q I Q J + 12 V ;( IJ ) Q I Q J − G ( IJ ) ˙ φ I D t Q J + 12 G IJ ˙ φ I ˙ φ J (3Φ − ) + 12 G IJ D t Q I D t Q J . (5.1.6)Moreover, one can use Eqs. (2.3.3) and (2.3.4) to substitute for Φ and Φ and write δρ entirely in terms of the covariant perturbations Q I . There are in fact a number of equivalentways to write δρ as a function of the field-space perturbations using Eq. (2.3.8) and (2.3.9),which on substitution into Eq. (5.0.5) lead to equivalent ways to write ζ in terms of Q I .Different possibilities were discussed at length in Ref. [30] for the canonical case. For thenumerical implementations of Ref. [1] the simplest of these was used, which follows from theuse of Eq. (2.3.3) and (2.3.4), and in the non-trivial field-space case leads to δρ = − HG IJ ˙ φ I Q J , (5.1.7)and δρ = 3 M H (3Φ − )= 32 M ˙ φ I ˙ φ J Q I Q J − H∂ − (cid:0) G IJ ( ∂ i D t Q I ) ∂ i Q J + G IJ D t Q I ∂ Q J (cid:1) . (5.1.8) N tensors Substituting Eqs. (5.1.7) and (5.1.8) into Eq. (5.0.5) one finds ζ (1) = − M H(cid:15) G IJ ˙ φ I Q J , (5.2.1)and ζ (2) = 16 M H (cid:15) (cid:20)(cid:18) M ˙ φ I ˙ φ J (cid:20) −
32 + 92 (cid:15) + 34 (cid:15) M H V ; K ˙ φ K (cid:21)(cid:19) Q I Q J + (cid:18) M H(cid:15) ˙ φ I ˙ φ J (cid:19) Q I D t Q J − H∂ − (cid:0) G IJ ( ∂ i D t Q I ) ∂ i Q J + G IJ ( D t Q I ) ∂ Q J (cid:1)(cid:21) . (5.2.2)On moving to Fourier space we can identify expressions for the N tensors defined above, andwe find that N a = − M H(cid:15) ˙ φ I (cid:32) (cid:33) (5.2.3) N ab = − M H (cid:15) M ˙ φ I ˙ φ J (cid:104) − + (cid:15) + (cid:15) M H V ; K ˙ φ K (cid:105) H(cid:15) ˙ φ I ˙ φ J M − G IJ Hk (cid:0) k a · k b + k a (cid:1) H(cid:15) ˙ φ I ˙ φ J M − G IJ Hk (cid:0) k a · k b + k b (cid:1) . (5.2.4)We note that these equations are simply the covariant from of the canonical case presented inRef. [30] with no new terms appearing. It should be noted, however, that additional Riemannterms do appear in intermediate expressions, for example for δρ (5.1.6).– 14 – Numerical implementation
So far in this paper we have developed the theoretical framework necessary to perform anumerical evolution of the power spectrum and bispectrum for models of inflation with anon-Euclidean field-space metric. Now we turn to their practical application.The equations presented have been implemented in an new version of the open source
PyTransport [9] package,
PyTransport 2.0 . To use this package, an end user is requiredto specify the model they wish to analyse (in terms of the potential and the field-space),then the code compiles a bespoke python module which contains functions that enable theuser to calculate the evolution of the background fields, the evolution of the covariant field-space correlations, and the power-spectrum and bispectrum of ζ . The code is released atgithub.com/ds283/CppTransport with accompanying user manual explaining in detail thesteps needed to set up the package and apply it to models of interest. To demonstrate the utility of our framework and numerical implementation, here we presentresults we have generated for a number of models.In order to illustrate these numerical results we define some quantities that are usefulwhen studying a model of inflation. The dimensionless power spectrum, P , of the curvatureperturbations, ζ , is defined by P ( k ) = k π P ( k ) , (6.2.1)where P ( k ) was defined in Eq. (5.0.4), and the reduced bispectrum of ζ by f nl ( k , k , k ) = B ( k , k , k ) P ( k ) P ( k ) + P ( k ) P ( k ) + P ( k ) P ( k ) . (6.2.2)For one triangle of wavevectors in the bispectrum, it is often convenient to use a parameterto describe the overall scale, k s = k + k + k , and two further parameters for the shape, α and β , defined as k = k s α + β ) k = k s − α + β ) k = k s − β ) , (6.2.3)with the allowed values of ( α, β ) falling inside a triangle in the α , β plane with vertices ( − , , (1 , and (0 , . Ref. [1] attempted to construct a model in which the field-space trajectory was curved in suchas way as to exhibit Gelaton [34] or QSFI [35] behaviour. For reasons presented there, thisbehaviour was difficult to achieve, but the model presented there is still a useful example,and in the present context provides a useful check of our code.– 15 – a) (b)
Figure 1 : The time evolution of the polar coordinate fields θ and R with metric (6.3.2) onthe left, and the cartesian coordinates, X and Y on the right. (a) (b) Figure 2 : The time evolution of correlation functions. On the left the time evolution of thetwo-point function of the curvature perturbation, ζ , and on the right the evolution of thethree-point function for an equilateral configuration. Both were taken for modes exiting thehorizon 21 e-folds before the end of inflation.The model is defined by the action for two fields R and θ as S = − (cid:90) d x √− g (cid:2) ( ∂R ) + R ( ∂θ ) + 2 V ( R, θ ) (cid:3) , (6.3.1)where the potential (defined below in Eq. (6.3.3)) represents a circular valley at a fixed valueof R – and hence is naturally written in terms of these ‘polar coordinate’ fields. However, asthe codes developed for Ref. [1] only dealt with canonical kinetic terms, in that work it wasnecessary to perform a field redefinition to cartesian coordinates X and Y . Here we evolvethe statistics directly for the fields R and θ and compare results, using this as a test case tobenchmark our code against its canonical precursor.– 16 – a) (b) Figure 3 : The reduced bispectrum f nl ( k , k , k ) for equilateral configurations. On the leftthe evolution of f nl versus time for an equilateral configuration with modes leaving the horizon21 e-folds prior to the end of inflation. On the right the bispectrum over a range of equilateralconfigurations as a function of exit time of the scale k s / .The field-space metric of the model can be read off from Eq. (6.3.1), and is G IJ = (cid:18) R (cid:19) . (6.3.2)The potential is V = V (cid:18) π θ + 12 η R M ( R − R ) + 13! g R M ( R − R ) + 14! λ R M ( R − R ) (cid:19) , (6.3.3)and we choose parameters V = 10 − M , η R = 1 / √ , g R = M V − / , ω = π/ , λ R =0 . M ω − / V − / and R = √ − / π √ − . With these choices, the radial direction representsa heavy mode confining the inflationary trajectory to the valley, with angular direction light.We further choose initial conditions R ini = (cid:113) R + (10 − R ) and θ ini = arctan (cid:18) − R R (cid:19) . (6.3.4)Generating results using our new code for the field evolution and correlations in the { R, θ } basis, and then subsequently using a coordinate transformation to translate the resultsto the { X, Y } basis, we can compare our results to the output of the canonical code. We findexcellent agreement. The evolution of correlation functions of the curvature perturbation, ζ ,are coordinate invariant, and also match that generated using the canonical code. In Fig. 1athe background field evolution in the non-canonical case is plotted. Under the coordinatetransformation to the canonical fields X and Y we get the evolution in Fig. 1b. In Fig. 2a& 2b one can clearly see that after horizon crossing the curvature perturbation freezes in,becoming constant on large scales as expected. The evolution of the reduced Bispectrum f nl for one equilateral triangle is shown in Fig. 3a. The reduced bispectrum in the equilateralconfiguration as a function of horizon crossing time is given in Fig. 3b, and can be comparedwith Fig. 11 of Ref. [1]. – 17 – a) (b) Figure 4 : The time evolution of the fields φ , φ and φ on the left, and the time evolutionof the two-point function of ζ for a k-mode exiting the horizon 60 e-folds before the end ofinflation on the right. The turn in field-space occurs 13 e-folds into inflation when the field φ experiences excitations from its coupling to the lighter field φ via the field-space metric.After roughly 30-e-folds the φ field reaches the minimum and the amplitude of the powerspectrum increases at this time. Figure 5 : The power-spectrum of the curvature perturbation for a range of modes whichexit the horizon over a window of 7 e-folds. The scale k pivot is taken to be when the modeleaves the horizon at 58 e-folds prior to the end of inflation. Both the scales and amplitudesare normalised to the spectrum at the pivot scale. Next we consider the quasi-two field model introduced in Ref. [10] where the power spectrumwas calculated. In this model there are two light scalar fields which drive inflation and oneheavy field which interacts with the light ones through a coupling in the kinetic terms. Thisleads to a fast turn in the plane of the lighter two fields resulting in the well known feature ofoscillations in the power spectrum and bispectrum (see for example [36–43]). In this paper wereproduce the power spectrum presented in Ref. [10] as a test of our code and then calculate– 18 – a) (b)
Figure 6 : The evolution of the three-point function for one equilateral configuration, andthe reduced bispectrum, f nl , for equilateral configurations over a range k s . The reducedbispectrum is plotted for modes leaving the horizon between 59 and 51 e-folds before the endof inflation. The highly oscillatory behaviour is a result of the excitations to the heavy fieldaround horizon crossing. Figure 7 : Amplitude over shape configurations of the reduced bispectrum f nl ( α, β ) at a fixed k t
53 e-folds before the end of inflation, corresponding to log( k/k pivot ) = 4 . .the bispectrum for the first time. The three fields are labelled φ , φ and φ , and model hasa metric which takes the form G IJ = φ ) 0Γ( φ ) 1 00 0 1 . (6.4.1)The function Γ( φ ) has the following φ dependence [44], Γ( φ ) = Γ cosh (cid:16) (cid:16) φ − φ ∆ φ (cid:17)(cid:17) , (6.4.2)– 19 – a) (b) Figure 8 : The evolution of the fields θ and ψ on the left and the evolution two-point functionof the curvature perturbation on the right for a mode leaving the horizon 50 e-folds prior tothe end of inflation. From 30 e-folds into inflation until the end there is no further evolutionof the two-point function.with Γ = 0 . the maximum value attained by Γ( φ ) . φ = 7 M p is the value of φ at theapex of the turn in field-space and ∆ φ = 0 . is the range of φ over which the turn occursThe potential is defined as V = 12 g m φ + 12 g m φ + 12 g m φ , (6.4.3)with parameters g = 30 , g = 300 , g = 30 / and m = 10 − . The initial conditions of thefields are φ = 10 . M p φ = 0 . M p φ = 13 . M p . (6.4.4)In Fig. 4a the background field evolution is plotted. At 13 e-folds into the evolution the turnin the inflationary trajectory occurs, as can be seen by the increase in the amplitude of theheaviest field. In Figs. 4b & 6a the evolution of both the two and three-point correlationfunctions of curvature perturbations are plotted. The power spectrum obtained in Fig. 5matches that seen in Ref. [10] illustrating that the code is in good agreement with thisearlier implementation. We produce the reduced bispectrum over equilateral configurationsin Fig. 6b, the structure of which is defined by a pulse of large and rapidly oscillating valuesof the three-point function. Finally, for a fixed scale k t we plot the reduced bispectrum inFig. 7 as a function of the α and β parameters discussed in §6.2 for a fixed k t . In the models considered above the field-space metrics were non-trival, but flat. As a furthertest of our code, therefore, we now introduce a model with a constant non-zero Ricci curvature.We construct a toy model containing two fields θ and ψ , where the action is defined as S = − (cid:90) d x √− g (cid:2) r ( ∂θ ) + r sin θ ( ∂ψ ) + 2 V ( θ, ψ ) (cid:3) , (6.5.1)where r is the radius of the surface of the sphere which the field trajectory is confined to. Thecurvature of the field-space, defined by the Ricci Scalar, is related to the radius, R = r . The– 20 – a) (b) Figure 9 : Evolution of the reduced bispectrum in an equilateral configuration on the leftand the reduced bispectrum for an equilateral configuration versus the radius of the metricsphere on the right. From 30 e-folds into inflation until the end there is no further evolutionof f nl . The evolution of f nl was taken for a mode leaving the horizon at 26 e-folds fromthe beginning of inflation. The bispectrum on the right is taken for a range of modes in thewindow between 25 and 30 e-folds and for a radius between 9 and 11.5. It illustrates a largeamplitude correlation over scales for a small radius (or rather large field-space curvature).field-space metric which describes the line element along the surface of a sphere is therefore G IJ = (cid:18) r r sin θ (cid:19) . (6.5.2)For the potential we use the same potential given for the axion-quartic model studied in Ref.[1]. The potential is of the form, V = 14 g θ θ + Λ (cid:18) − cos (cid:18) πψf (cid:19)(cid:19) , (6.5.3)where the field ψ is our “2-sphere-axion” and our parameters are g θ = 10 − , Λ = (25 / π ) gM , ω = 30 /π and f = M p . The initial conditions of the fields are set to θ ini = 2 . M p and φ ini = f / − − M p , (6.5.4)which is sufficient for inflation for 64 e-folds. The background evolution of the fields are plottedin Fig. 8a, with the corresponding evolution of correlations of the curvature perturbations fortwo-point (Fig. 8b) and three-point (Fig. 9a) functions. We study the effects of curvature onquantities like the bispectrum by varying the radius r . Figure 9b is a contour graph of thebispectrum as a function of r . We see that for a radius r > . the bispectrum is small,but for r < . the bispectrum begins to increase. This indicates a correlation betweenlarge curvature and a value of large f nl in this model. Finally we consider a more realistic case inspired by models of D-brane inflation. Such modelshave recently been the subject of considerable interest, with a number of groups statistically– 21 – igure 10 : The evolution of the 6 moduli fields during inflation. Rich dynamics exist owingto the couplings in the conifold metric. Inflation ends when the branes collide at a value of r = 0 . (a) (b) Figure 11 : On the left, the power spectrum of curvature perturbation and on the right thebispectrum of curvature perturbations over an equilateral configuration for modes exiting thehorizon after a large range of times between 12 and 64 e-folds.probing their realisations [45–47]. In one such scenario two D3-branes are attracted by aCoulomb force. Compactification induces a warping of the 6-D manifold where the D3-branesits, resulting in a non-trivial field-space metric in the Lagrangian of the system. Both thegeometry of the metric and structure of the potential affect the inflationary dynamics. Initialwork [45] looked at the background dynamics of this system, while more recent studies lookedinto the distribution of 2-point statistics [46, 47]. Here we illustrate how our new code couldbe used to obtain information about the bispectrum, though we defer realistic studies tofuture work.We consider the Lagrangian of D3-brane inflation as S = − (cid:90) d x √− g (cid:0) G IJ dφ I dφ J + 2 V ( φ , . . . φ ) (cid:1) , (6.6.1)– 22 –here a is the scale factor. The scalar fields represent the 6 brane coordinates, one radial r and five angular dimensions θ , θ , φ , φ and ψ . The field-space metric G IJ corresponds tothe Klebanov-Witten conifold geometry [48]. The metric is of the form, G IJ dφ I dφ J = dr + r d Ω , (6.6.2)with the metric of the cone d Ω [49] is given by d Ω = 16 (cid:88) i =1 (cid:0) dθ i + sin θ i dφ i (cid:1) + 19 (cid:32) dψ + (cid:88) i =1 cos θ i dφ i (cid:33) , (6.6.3)which is a non-compact geometry built over the five-dimensional ( SU (2) × SU (2)) /U (1) coset space T , . As a toy example we do not generate a realistic potential (motivated byany attractive forces between branes or contribution from either the homogeneous or theinhomogeneous bulk), instead, for simplicity, we take a quadratic potential for the 6 fields V ( φ ) = (cid:88) i =1 m i φ i , (6.6.4)where m i are the randomised masses of the fields. A randomised set of masses and initial con-ditions are selected with the criteria that 64 e-folds of inflation occur. With these parametersthe evolution of the dynamics and statistics can be run and the background trajectory foreach of the six fields is plotted in Fig. 10. The power spectrum is plotted in Fig. 11a and thebispectrum in the equilateral configuration is plotted in Fig. 11b. It would be interesting torun a more realistic analysis including the full potential of the system but this is beyond thescope of our work. We have, however, demonstrated that this is possible using the transportmethod and its implementation in code via PyTransport . In PyTransport 2.0 , one can opt to specify explicitly a field space metric. If this option is notselected the code defaults to assuming that the metric is Euclidean and the code reverts backto the previous canonical code. The simplicity of a Euclidean metric means that a number ofinternal loops do not need to be performed, and hence the canonical code is expected to befaster than when a metric is specified explicitly (even if the metric is the Euclidean one). Todemonstrate this effect and also to benchmark the speed of the new code in Fig. 12 we showhow the speed of the new code compares with that of the canonical one. We also show how thespeed of the code is sensitive to the number of e-folds before horizon crossing (of the shortestscale in the triangle being evaluated) at which initial conditions are fixed, and to differenttolerances which fix the accuracy of the code. For this purpose we use the double quadraticpotential used to calculate performance data in Ref. [1]. As can be seen, the new code isroughly a factor of 2 slower for this two field model. We find that introducing a simple fieldspace metric, such as the 2-sphere metric used in § 6.5, leads to very similar timing data to theEuclidean metric (though more complicated metrics will inevitably slow down the code as theterms in the metric need to be evaluated at each time step). A more significant effect comesfrom increasing the number of fields. The size of the arrays which store information about theRiemann tensor and its derivative scale as N and N respectively (for the canonical codethe largest arrays scale as N ), and therefore memory issues and overheads resulting formaccessing and looping over these arrays grow rapidly as field number increases.– 23 – igure 12 : Top panel: scaling of integration time with increasing number of massless (orsubhorizon) e-folds (using relative and absolute tolerances of − ) for an equilateral triangleof the bispectrum and a squeezed triangle ( α = 0 , β = 0 . ). Timings were performed usingthe canonical code and the new non-canonical code setting a Euclidean metric explicitly.Bottom panel: scaling of integration time with integration tolerance with 5 e-folds of masslessevolution. The double quadratic model used to analysis performance in Ref. [1] is timed usingthe canonical PyTransport package and compared to the same model using PyTransport 2.0.The computer used for timings contained an 3.1 GHz Intel i7-4810MQ processor. We have extended the method of calculating the power spectrum and bispectrum developedin Ref. [1] for canonical multi-field inflation to include models which contain a non-trivialfield-space metric. First in §2 the equations of motion and the conservation equations forperturbations were derived for our non-canonical multi-field system. We reviewed how thesystem of equations can be written as an autonomous system for a set of covariant “field” per-turbations. Next we reviewed how the transport method is applied in combination with theseequations to give equations for the evolution of the correlations of the covariant perturbationsduring inflation. To use this system in practice we needed to calculate both initial conditionsfor our new system of equations, and the relation between covariant field-space perturbationsand the curvature perturbation ζ . A neat result we found is that our expressions for thesequantities take the form of the covariant versions of the expressions presented in Ref. [1], withno additional Riemann terms appearing (except through the new terms that appear in the a ,– 24 – and c tensors which define the equations of motion).We have demonstrated explicitly that our method is successful in evaluating the ob-servable statistics of inflationary models with many fields and a curved field-space metric.The code we have developed to do this is the second iteration of the PyTransport package,
PyTransport 2.0 , and agrees with its predecessor in the case of models which can be writtenin Euclidean and non-Euclidean coordinates (as discussed in § 6.3). Moreover, we have shownthat for simple 2-field models that the speed of the new code compares with well with that ofthe canonical model. It should be noted, however, that the the code has not been tested formodels exceeding more than six fields, and that we expect time taken to scale poorly withnumber of fields. Our hope is that this new code will be useful to the inflationary cosmologycommunity.
Acknowledgments
DJM is supported by a Royal Society University Research Fellowship. JWR acknowledgesthe support of a studentship jointly funded by Queen Mary University of London and bythe Frederick Perren Fund of the University of London. We thank Karim Malik and PedroCarrilho for helpful discussions and for comments on a previous version of this manuscript. Wethank David Seery, Sean Butchers, Mafalda Dias and Jonathan Frazer for useful discussionsrelated to code development, and David Seery for cross checks of the output of our code withthat of
CppTransport . – 25 – ppendices
A Initial Conditions
Here we provide a few more detail of how the initial conditions for the transport system arecalculated. We recall that (cid:63) denotes a time long before horizon crossing at which − τ (cid:29) ,where τ denotes conformal time, and that for a de-Sitter expansion τ = − / ( aH ) . A.1 Two point function
First we consider initial conditions for the two point function for the various combinationsof covariant field perturbation and momenta correlations. The calculation is similar to thatpresented in Ref. [10], though in that work the time variable used for the transport systemwas e-folds N , while in this paper we use cosmic time, t . • Field-Field correlationBeginning with the expression for the two point function of Q I (4.0.1) we consider the − τ (cid:29) limit for the field-field correlations. We find (cid:104) Q I ( k , τ ) Q J ( k , τ ) (cid:105) = (2 π ) δ ( k + k ) G IJ k H ( τ )(1 − ikτ )(1 + ikτ ) ≈ (2 π ) δ ( k + k ) G IJ k H ( τ ) | kτ | ≈ (2 π ) δ ( k + k ) G IJ a k . (A.1.1)The initial condition for Σ IJ ∗ is then Σ IJ ∗ Re = G IJ a k (cid:12)(cid:12)(cid:12)(cid:12) ∗ , Σ IJ ∗ Im = 0 . (A.1.2) • Field-Momentum correlationNext recalling that at linear order P I = D t Q I and that the covariant derivative of the parallelpropagator is zero, we consider the leading term in the expression for the field-momentumcorrelation of unequal time correlations, and subsequently take equal time limit for the case − τ (cid:29) . Recalling that dτ = dt/a ( t ) we find (cid:104) Q I ( k , τ ) P J ( k , τ ) (cid:105) = (2 π ) δ ( k + k ) Π IJ k H ( τ ) H ( τ )(1 + ikτ ) (cid:18) k τ a (cid:19) e ik ( τ − τ ) = (2 π ) δ ( k + k ) G IJ k H ( τ ) (cid:18) k τa (cid:19) (1 − ikτ )= (2 π ) δ ( k + k ) (cid:18) − G IJ H ka + i G IJ a (cid:19) . (A.1.3)The real and imaginary parts of the initial conditions for this case are then Σ IJ ∗ Re = − G IJ H ka (cid:12)(cid:12)(cid:12)(cid:12) ∗ , Σ IJ ∗ Im = G IJ a (cid:12)(cid:12)(cid:12)(cid:12) ∗ . (A.1.4)– 26 – Momentum-Momentum correlationWe follow a similar procedure to consider the momentum-momentum correlation (cid:104) P I ( k , τ ) P J ( k , τ ) (cid:105) = (2 π ) δ ( k + k ) Π IJ k H ( τ ) H ( τ ) (cid:18) k τ a (cid:19) (cid:18) k τ a (cid:19) e ik ( τ − τ ) = (2 π ) δ ( k + k ) G IJ k H ( τ ) (cid:18) k τ a (cid:19) = (2 π ) δ ( k + k ) G IJ k a . (A.1.5)The initial condition for Σ IJ in this case is Σ IJ ∗ Re = G IJ k a (cid:12)(cid:12)(cid:12)(cid:12) ∗ , Σ IJ ∗ Im = 0 . (A.1.6) A.2 Three point function
For the three-point function as discussed in the main text, an integral must be evaluatedto calculate the initial condition. By substituting Eq. (4.0.7) into Eq. (4.0.6) we obtain theinitial condition B abc ∗ . To illustrate how this is evaluated in practice, let us consider thisexplicitly for the case of a field-field-field correlation. • a,b,c → Field-Field-FieldSubstituting in the expression for the two-point function we obtain B abc ∗ = − iH i k i (1 + ik τ )(1 + ik τ )(1 + ik τ ) e − ik s τ × (cid:90) τ init −∞ dηH η (cid:34) ˙ φ I G JK H ( k · k )(1 − ik η )(1 − ik η )(1 − ik η ) e ik s η + a IJKs H η (1 − ik η )(1 − ik η )(1 − ik η ) e ik s η + b IJK H η (1 − ik η )(1 − ik η ) k ηe ik s η + c IJK k k η (1 − ik η ) e ik s η + perms (cid:21) + c.c. , (A.2.1)where we assume that H and Π IJ are sufficiently slowly varying to be taken as constants andthat we can take Π IJ → G IJ .In order to perform the integration we need to know the time dependence of the tensors.As discussed in §4 the a IJK tensor contains fast and slow varying parts. The part containingterms quadratic in η vary quickly and so are included in the integral separately (the first termin Eq. (A.2.1)), the remaining parts we label a IJKs and we assume can be considered constantin time. The next step is to perform the integration recalling that the result is dominated bythe upper limit (because the integral is highly oscillatory into the past). Keeping the leading– 27 –nd sub-leading terms in τ , and writing in terms of a and H , the final result is B abc ∗ = 14 a k · k · k · k s (cid:0) − c IJK ( k , k , k ) · ( k · k ) − c IKJ ( k , k , k ) · ( k · k ) − c JKI ( k , k , k ) · ( k · k ) + a a IJKs ( k , k , k )+ a a IKJs ( k , k , k ) + a a JKIs ( k , k , k )+ a Hb IJK ( k , k , k ) (cid:18) ( k + k ) · k k · k − K k · k (cid:19) + a Hb IKJ ( k , k , k ) (cid:18) ( k + k ) · k k · k − K k · k (cid:19) + a Hb JKI ( k , k , k ) (cid:18) ( k + k ) · k k · k − K k · k (cid:19) + ˙ φ I H G JK ( − k − k + k ) + ˙ φ J H G IK ( − k − k + k ) + ˙ φ K H G IJ ( − k − k + k ) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∗ , (A.2.2)where K ≡ k k + k k + k k and k s = k + k + k . Repeating for the other correlationswe find • a,b,c → Momentum-Field-Field B abc ∗ = − H a K (cid:18) − k ( k + k ) k s · k · k · k (cid:19) (cid:0) − c IJK ( k , k , k ) · ( k · k ) − c IKJ ( k , k , k ) · ( k · k ) − c JKI ( k , k , k ) · ( k · k ) + a a IJKs ( k , k , k ) + a a IKJs ( k , k , k ) + a a JKIs ( k , k , k )+ G JK ˙ φ I H ( − k − k + k ) + ˙ φ J H G IK ( − k − k + k ) + ˙ φ K H G IJ ( − k − k + k ) (cid:33) − H a K (cid:18) − k · ( k · k ) k s (cid:19) (cid:18) c IJK ( k , k , k ) k k (cid:18) k k s (cid:19) + c IKJ ( k , k , k ) k k (cid:18) k k s (cid:19) + c JKI ( k , k , k ) k k (cid:18) k k s (cid:19) − a a IJKs ( k , k , k ) (cid:18) K − k · k · k k s (cid:19) − a a IKJs ( k , k , k ) (cid:18) K − k · k · k k s (cid:19) − a a JKIs ( k , k , k ) (cid:18) K − k · k · k k s (cid:19) + b IJK ( k , k , k ) k · k · k H + b IKJ ( k , k , k ) k · k · k H + b JKI ( k , k , k ) k · k · k H − G JK ˙ φ I H ( − k − k + k ) (cid:18) K k · k · k k s (cid:19) − G IK ˙ φ J H ( − k − k + k ) (cid:18) K k · k · k k s (cid:19) − G IJ ˙ φ K H ( − k − k + k ) (cid:18) K k · k · k k s (cid:19)(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∗ , (A.2.3)where K k + k + k . – 28 – a,b,c → Momentum-Momentum-Field B abc ∗ = − a K k · k · k ) · k · k k s (cid:0) − c IJK ( k , k , k ) · ( k · k ) − c IKJ ( k , k , k ) · ( k · k ) − c IJK ( k , k , k ) · ( k · k ) + a a IJKs ( k , k , k ) + a a IKJs ( k , k , k ) + a a JKIs ( k , k , k )+ a Hb IJK ( k , k , k ) (cid:18) ( k + k ) · k k · k + ( k · k ) · k · k · k (cid:19) + a Hb IKJ ( k , k , k ) (cid:18) ( k + k ) · k k · k + ( k · k ) · k · k · k (cid:19) + a Hb JKI ( k , k , k ) (cid:18) ( k + k ) · k k · k + ( k · k ) · k · k · k (cid:19) − G JK ˙ φ I H ( − k − k + k ) − G IK ˙ φ J H ( − k − k + k ) − G IJ ˙ φ K H ( − k − k + k ) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∗ . (A.2.4) • a,b,c → Momentum-Momentum-Momentum B abc ∗ = − H a K k k k k s (cid:18) c IJK ( k , k , k ) · ( k · k ) (cid:18) k k s (cid:19) + c IKJ ( k , k , k ) · ( k · k ) (cid:18) k k s (cid:19) + c JKI ( k , k , k ) · ( k · k ) (cid:18) k k s (cid:19) − a a IJKs ( k , k , k ) (cid:18) K − k · k · k k s (cid:19) − a a IKJs ( k , k , k ) (cid:18) K − k · k · k k s (cid:19) − a a JKIs ( k , k , k ) (cid:18) K − k · k · k k s (cid:19) + b IJK ( k , k , k ) H k k · k + b IKJ ( k , k , k ) H k · k · k + b IJK ( k , k , k ) H k · k · k − G JK ˙ φ I H ( − k − k + k ) (cid:18) K k · k · k k s (cid:19) − G IK ˙ φ J H ( − k − k + k ) (cid:18) K k · k · k k s (cid:19) − G IJ ˙ φ K H ( − k − k + k ) (cid:18) K k · k · k k s (cid:19)(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∗ . (A.2.5) References [1] M. Dias, J. Frazer, D. J. Mulryne and D. Seery,
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