On the slow roll expansion of one-field cosmological models

We study the infrared scale expansion of single field cosmological models using the Hamilton-Jacobi formalism, showing that its specialization at unit scale parameter recovers the slow roll expansion. In particular, we show that the latter coincides with a Laurent expansion of the Hamilton-Jacobi function in powers of the Planck mass, whose terms are controlled by certain recursively-defined polynomials. This allows us to give an explicit recursion procedure for constructing all higher order terms of the slow roll expansion. We also discuss the corresponding effective potential and the action of the universal similarity group.


Introduction
An important problem in scalar field cosmology is to develop useful approximation methods for various regimes. Ideally, such methods should have a sound conceptual basis which is grounded in the dynamical features of the model. They should also be effective in the sense that they lead to systematic higher order expansion schemes which are computable in an efficient manner.
In this paper, we consider the scale expansion for single field scalar cosmology, following the proposal made in [4,5] for the much more general case of multifield models. We show that the scale expansion recovers the slow roll expansion of single field models while presenting it in a new conceptual light, which allows one to give an explicit recursion procedure for computing its higher order terms.
An important special feature of one-field models (as compared to their multifield counterparts) is the existence of a straightforward description of cosmological flow orbits through the so-called Hamilton-Jacobi formalism of Salopek and Bond [1]. This encodes the geometry of regular flow orbits through regular Hamilton-Jacobi functions, which are solutions of a certain first order non-autonomous and non-linear ODE. In this approach, the cosmological curves of the model can be recovered (up to a constant translation of cosmological time) by integrating the speed equation of a given flow orbit. The formalism was used for various purposes in references [2,6,7] (see [3,Sec. 18.6] for a brief introduction) and we give a mathematically precise treatment in the present work.
The scale transformations and universal similarities of [4] have a natural action on Hamilton-Jacobi functions, which leads to the consideration of scale expansions of the latter. These expansions can be constructed by looking for solutions of an appropriate scale deformation of the Hamilton-Jacobi equation which admit an appropriate expansion in the scale parameter. In this paper, we focus on the infrared expansion, which is relevant in the slow motion regime. We show that the latter is essentially equivalent with the small Planck mass expansion, which produces an infrared Hamilton-Jacobi function H IR , distinguished by the property that it admits a Laurent expansion in powers of the Planck mass. This follows from the structure of the cosmological and Hamilton-Jacobi equations and matches the Wilsonian expectation that infrared dynamics is controlled by the Planck mass, which sets the natural energy scale of the problem.
The coefficients of the Laurent expansion of H IR are governed by a recursion relation, which encodes the condition that H IR formally satisfies the Hamilton-Jacobi equation. Studying this recursion, we show that all terms can be expressed in terms of certain universal multivariate polynomials Q n with rational coefficients which are themselves determined by an explicit recursion. These Hamilton-Jacobi polynomials encode all relevant information required for describing the expansion of H IR , whose coefficients are recovered by evaluating certain universal rational functions v n con-structed from Q n on the scalar potential of the model and its successive derivatives. These results reduce the problem of constructing H IR to the recursive computation of Hamilton-Jacobi polynomials, which can be achieved effectively on a computer. As an application, we list the polynomials Q n and rational functions v n up to tenth order inclusively.
We also study the homogeneity properties of Q n and v n with respect to two relevant gradings on the algebra A = R[N] of polynomials in a countable number of variables, showing that these properties allow one to rewrite the expansion of H IR as an infinite sum of monomials in the potential slow roll parameters of the model as defined in [2]. This shows that the small Planck mass expansion is equivalent with the slow roll expansion of [2], which therefore also describes the infrared expansion of one field models in the sense of [4] up to an appropriate rescaling of the Planck mass by the scale parameter. Performing the corresponding substitutions to fifth order in the square of the Planck mass, we show explicitly that our expansion recovers the fifth order slow roll expansion as computed in [2]. Unlike the approach of loc. cit., which is rather involved and somewhat inefficient due to the lack of a master recursion formula, our procedure uses the recursive construction of Hamilton-Jacobi polynomials and hence allows for the efficient computation of higher order terms in the slow roll expansion. This opens the way for precision studies of one-field cosmological models, thus removing one of the technical limitations in the subject.
The paper is organized as follows. In Section 1, we give a mathematical review of one-field cosmological models, introducing the notation and terminology used in latter sections. In Section 2, we discuss the one-field realization of the universal similarities and classical RG transformations which were introduced in a much more general setting in reference [4]. Section 3 gives a careful treatment of the Hamilton-Jacobi formalism of [1] and describes the action of universal similarities and RG transformations on Hamilton-Jacobi functions. Section 4 introduces the deformed Hamilton-Jacobi equation and studies the formal infrared scale expansion of Hamilton Jacobi functions. We express the terms of this expansion through Hamilton-Jacobi polyomials and study their bihomogeneity properties, showing that the infrared expansion at unit scale parameter is equivalent with the slow roll expansion of [2], which turns out to be equivalent with a Laurent expansion in powers of the Planck mass. Section 5 presents our conclusions and some directions for further research. Appendix A recalls some basic facts about jets of univariate real-valued functions while Appendix B lists the Hamilton-Jacobi polynomials Q n and associated rational functions v n up to order 10.

Single field cosmological models
By a single field cosmological model we mean a classical cosmological model with one real scalar field (called inflaton) and target space R, which is derived from the following action on a spacetime with topology R 4 : (1.1) Here vol(g) is the volume form of the Lorentzian spacetime metric g defined on R 4 (which we take to be of "mostly plus" signature) and: where the positive constant M is the reduced Planck mass, R(g) is the Ricci scalar of g, the coordinatized scalar field ϕ : R 4 → R is a smooth map and the smooth function Φ : R → R >0 is the coordinatized scalar potential. For simplicity, we assume throughout this paper that Φ is strictly positive everywhere. The action (1.1) is well-defined when g and ϕ obey appropriate conditions at infinity. The Lagrangian density of the scalar field can be written in the standard form of a sigma model with potential: where G is a Riemannian metric on an abstract one-dimensional manifold M which is diffeomorphic with the real line and plays the role of target space of the uncoordinatized scalar field, which is a smooth functionφ : R 4 → M. The uncoordinatized scalar potentialΦ : M → R >0 is a smooth function. The coordinatized formulation given above is recovered as follows. Let x : M ∼ → R be a global coordinate (uniquely determined by G and by a choice of orientation for M) for which G takes the standard Euclidean form: Then the coordinatized scalar field ϕ and coordinatized scalar potential Φ used in (1.2) are recovered as: In particular, the formulation in terms of ϕ and Φ has a hidden dependence on the choice of the Riemannian metric G on the target space M (which, together with the orientation of M, determines the choice of the Euclidean coordinate x). This fact will play a role in the discussion of universal similarities given in Subsection 2 below, namely those similarities differ from the general treatment given in [4] when expressed in the coordinatized formulation.

Parameterization, phase space and the rescaled Hubble function
The model is parameterized by the pair (M 0 , Φ), where 1 : 1 We prefer to use M 0 instead of M since this simplifies various formulas later on.
is the rescaled Planck mass and the scalar potential Φ ∈ C ∞ (R) is an everywhere positive function. We denote by: = R \ CritΦ the critical and noncritical sets of Φ. By definition, the phase space T of the model is the total space of the tangent bundle of the target space R: whose bundle projection we denote by π 1 : T → R. We denote by π 2 : T R → R the vertical projection of the trivial vector bundle T R. The position and speed coordinates x, v are the base and fiber coordinates, which give the Cartesian coordinate system (x, v) of R 2 . Thus π 1 and π 2 are the ordinary Cartesian projections: Moreover, we denote by: the image of the zero section of T R. The total space of the slit tangent bundleṪ R is the complement of the set Z inside T : This open set has two connected components (namely the upper and lower open half-planes in R 2 ), which we denote by: Definition 1.1. The canonical phase space lift of a smooth curve ϕ : I → R is the smooth curve c(ϕ) : I → R 2 defined through: The phase space manifold of ϕ is the smooth one-dimensional submanifold O ϕ of the phase space T = R 2 defined through: (1.9)

The cosmological equation
The classical single field cosmological model parameterized by (M 0 , Φ) is obtained by assuming that g is a Friedmann-Lemaitre-Robinson-Walker (FLRW) metric with flat spatial section: (where a ∈ is a smooth positive function) and that ϕ depends only on the cosmological time t def.
= x 0 ∈ R. Define the Hubble parameter through: , (1.11) where the dot denotes derivation with respect to t.
When H > 0 (which we assume throughout this paper), the variational equations of the action (1.1) reduce to the system formed by the cosmological equation: together with the condition: Proof. Follows by direct computation.

Cosmological curves and cosmological orbits
An interval I ⊂ R is called non-degenerate if it is nonempty and not reduced to a point. Let Int be the set of all non-degenerate intervals in R and: be the set of all smooth curves in R.
where C > 0 and t 0 ∈ I are arbitrary. We denote by Sol(M 0 , Φ) ⊂ C(R) the set of cosmological curves of the model (M 0 , Φ) and by Sol m (M 0 , Φ) the subset of pointed maximal cosmological curves. Given T ∈ R, the T -translate of a smooth curve ϕ ∈ C(R) is the curve ϕ T : I T → R defined through: This gives an action of the abelian group (R, +) on the set C(R). Since the cosmological equation (1.12) is autonomous, the T -translate of any cosmological curve is again a cosmological curve for all T ∈ R and hence the set Sol(M 0 , Φ) is invariant under this action. Notice that any cosmological curve admits translates which are pointed. A cosmological curve ϕ : I → R need not be an immersion. Accordingly, we define the singular and regular parameter sets of ϕ through: When ϕ is non-constant, it is easy to see that I sing is an at most countable set and that the restriction of ϕ to each connected component of I reg is a homeomorphism onto its image and hence an embedded curve in R.
The sets of critical and noncritical times of a cosmological curve ϕ : I → R are defined through: (1.17) The cosmological curve ϕ is called noncritical if I crit = ∅, which means its orbit is contained in the non-critical set of Φ. It is easy to see that a cosmological curve ϕ is constant iff its image coincides with a critical point of Φ, which in turn happens iff there exists some t ∈ I sing such that ϕ(t) ∈ CritΦ. Hence for any non-constant cosmological curve we have:

The cosmological dynamical system, cosmological flow curves and cosmological flow orbits
The cosmological equation (1.12) is equivalent with the following first order system of ODEs:ẋ which describes the integral curves of the vector field S ∈ X (T ) defined through: which is the single field model incarnation of the cosmological semispray of [4]. The flow of this vector field on the phase space T = R 2 is called the cosmological flow of the model (M 0 , Φ). The planar dynamical system (R 2 , S) is called the cosmological dynamical system (see [8,9] for an introduction to geometric dynamical systems theory).
Definition 1.5. An integral curve γ : I → R 2 of the vector field S is called a cosmological flow curve while its image γ(I) (considered as a submanifold of T = R 2 ) is called a cosmological flow orbit.
Notice that S vanishes precisely at the trivial liftsx def.
= (x, 0) ∈ Z of the critical points x ∈ CritΦ of the scalar potential. Accordingly, a cosmological flow curve is constant iff its orbit coincides with a point of the trivial lift: of the critical set of Φ; in this case, we say that the cosmological flow orbit is trivial. It is easy to see that any non-constant cosmological flow curve γ is an immersion. Since the cosmological energy is strictly decreasing along such a curve (see Proposition 1.14 below), it follows that γ(I) has no self-intersections and hence is an embedded submanifold of R 2 . The canonical phase space lift c(ϕ) of any cosmological curve ϕ : I → R is a cosmological flow curve c(ϕ) : I → R 2 . Conversely, any cosmological flow curve γ : I → R 2 determines a cosmological curve ϕ def. = γ(I) is the cosmological flow orbit determined by γ. Also notice that a cosmological flow orbit O determines cosmological curves ϕ : I → R such that O = c(ϕ)(I) through the first order ODE: Since this ODE is autonomous, it determines ϕ only up to translation of t by an arbitrary constant. Let ϕ : I → R be a regular smooth curve whose image (which is an interval in R) we denote by J def.
= ϕ(I). Thenφ has constant sign on I, which we denote by ζ ϕ and call the signature of ϕ. Since the map ϕ : I → J is a diffeomorphism, we can use x ∈ J as a parameter for ϕ. Set: =φ : I → R × be the speed of ϕ.
Definition 1.7. The speed function of the regular curve ϕ : I → R is the map: where the prime denotes derivation with respect to x. The regular curve ϕ can be recovered up to a time translation from its speed function v := v ϕ by solving the speed equation: which determines ϕ up to a shift of t by an arbitrary constant. The speed equation amounts to: and has general solution: where C ∈ R and x 0 ∈ J are arbitrary.
where v is this speed function.
Proof. Follows by noticing that (1.21) is the ratio of the second equation in (1.18) to the first equation.
By Proposition 1.9, a cosmological flow curve γ : I → T = R 2 and its cosmological flow orbit O = γ(I) ⊂ R 2 are regular iff the cosmological curve ϕ = π • γ : I → R is regular. In this case, we define the signature of γ and O to equal the signature ζ of ϕ. By Proposition 1.9, we have O = γ(I) ⊂ T ζ . The map which takes a regular one-dimensional submanifold O of R 2 to its speed function v : π(O) → R × restricts to a bijection between regular cosmological flow orbits and solutions of the regular flow orbit equation, which presents the former as the graphs of the latter. = H(x, v(x)) which appears in the right hand side of (1.21) is called the Hamilton-Jacobi function of the cosmological flow orbit O (see Subsection 3.2). We will see in Section 3 that the regular flow orbit equation (1.21) is equivalent with the Hamilton-Jacobi equation by the change of function v(x) → H(x), which leads to the Hamilton-Jacobi parameterization of regular cosmological flow orbits. This corresponds to using Hamilton-Jacobi coordinates (x, h) on T ± instead of the Cartesian coordinates (x, v) used above.

Basic cosmological observables and the dissipation equation
Definition 1.12. A basic local cosmological observable is a real-valued function F : T = R 2 → R. The dynamical reduction of F along a cosmological curve ϕ : I → R is the function F ϕ : I → R defined through: Notice that the rescaled Hubble function H is a basic local cosmological observable. Its dynamical reduction H ϕ along a cosmological curve ϕ is also called the rescaled Hubble parameter of that curve.
Definition 1.13. The cosmological energy function is the basic local cosmological observable E : R 2 → R >0 defined through: The cosmological kinetic energy function E kin : R 2 → R ≥0 and the cosmological potential energy function E pot : R 2 → R >0 are the basic local cosmological observables defined through: With these definitions, we have: (1.23) Proposition 1.14. The dynamical reduction of E along any cosmological curve ϕ : I → R satisfies the cosmological dissipation equation: (1.24) In particular, E ϕ and H ϕ are strictly decreasing functions of t if ϕ is a non-constant cosmological curve.
Proof. Follows immediately by using the cosmological equation (1.12).
Corollary 1.15. The Hubble inequality: holds for any cosmological curve ϕ : I → R and any t 1 , t 2 ∈ I such that t 1 ≤ t 2 .
= ϕ(I). Since ϕ is a regular curve, the map ϕ : I → J is a diffeomorphism and hence the cosmological equation (1.12) for ϕ is equivalent with the regular flow orbit equation (1.21) by Proposition 1.10. In turn, the regular flow orbit equation can be written as: where we used the fact that H > 0. This is equivalent with the dissipation equation (1.24) when combined with the speed equation (1.19) of ϕ.

Universal similarities and the dynamical renormalization group
Cosmological models with an arbitrary number of scalar fields admit a two-parameter group of similarities which relate the cosmological curves of models with different parameters (see reference [4]). This group acts naturally on the rescaled Planck mass, scalar field metric and scalar potential of multifield models, as well as on their phase space and basic observables. As we shall see below, the realization of these actions in single field cosmological models is somewhat special since our formulation uses the coordinatized scalar field and scalar potential -unlike the abstract formulation (1.3), whose multifield version was used in loc. cit. Let Pot(R) be the set of smooth and everywhere positive functions defined on R. Then the space of parameters of single-field models with smooth and positive scalar potential is: Par where the first factor parameterizes the rescaled Planck mass M 0 . Since most physically interesting basic observables depend on the parameters M 0 and Φ, in this section we will view them as functions F : Par × R 2 → R rather than as functions from T = R 2 to R.
is the smooth map f : I → R defined through: Notice that f and f have the same image. Scale transforms define an action of the multiplicative group R >0 on the set C(R) of smooth curves in R. The parameter is called the scale parameter.
Remark 2.2. Let ψ be the scale transform of ϕ at scale parameter = M 0 : Then: This form of the cosmological equation has the property that M 0 "counts the number of derivatives of ψ". When M 0 is small, one can look for solutions ψ of (2.1) which are expanded as a power series in M 0 : where ϕ n (t) def.
= ψ n (M 0 t) for all n ≥ 0. Substituting this expansion into (2.1) gives a differential recursion relation for ψ n , together with the zeroth order condition: is the "leading classical effective potential" of reference [4]. This shows that ϕ 0 is a gradient flow curve of V 0 . In latter sections, we will construct the rescaled Planck mass expansion (2.2) indirectly, namely by expanding the Hamilton-Jacobi function of the cosmological flow orbit of ϕ as a Laurent series in M 0 . Notice that (2.2) can be viewed as a particular instance of the infrared scale expansion proposed in [4], namely the scale expansion considered at scale parameter = M 0 . This is natural from the Wilsonian perspective adopted in loc. cit., since M 0 sets the natural energy scale of cosmological dynamics.
Following [4], we define: Remark 2.4. Notice that T is isomorphic with the additive group (R 2 , +) through the componentwise exponential map: However, the multiplicative formulation is more convenient for our purpose.
Definition 2.5. The parameter action ρ par is the action of T on the set of parameters Par = R >0 × Pot(R) given by: The curve action ρ 0 is the action of T on the set C(R) given by: The similarity action is the action ρ def.
The transformations ρ(λ, 1) are called parameter homotheties, while the transformations ρ(1, ) are called scale similarities. As in [4], scale similarities fix the abstract scalar fieldφ : R 4 → M of (1.3) while acting as follows on the system (M 0 , G,Φ), where G is the target space metric (1.4) of the abstract single field model andΦ : M → R >0 is the abstract scalar potential: Then the formulation given above follows by noticing that under such a transformation the Euclidean coordinate x of (1.4) changes as: x → λ 1/2 x and hence the coordinatized scalar field ϕ = x •φ and potential Φ =Φ • x −1 of (1.5) change as: In particular, a parameter homothety induces a rescaling of the coordinatized scalar field ϕ, because the latter is defined using the Euclidean coordinate x on M, which itself depends on the target space metric G. Notice that the action ρ par of T on the parameter set Par is free. Let Sol(M 0 , Φ) ⊂ C(R) be the set of cosmological curves of the model parameterized by (M 0 , Φ) ∈ Par. Proposition 2.6. We have: In particular, the set: Proof. Let ϕ be a solution of the cosmological equation of the model (M 0 , Φ). Then it is easy to see that λ 1/2 ϕ is a solution of the cosmological equation of the model In other words, the cosmological equation is invariant under the universal similarity action.
Remark 2.7. The parameter homothety ρ(λ, 1) with parameter λ = M −2 0 can be used to eliminate the rescaled Planck mass M 0 . For this, let: and define the reduced scalar potential Φ 0 : R → R >0 and the reduced Hubble function ) and the cosmological equation (1.12) of the model (M 0 , Φ) is equivalent with: which is the cosmological equation of the model (1, Φ 0 ). In particular, the positive homothety classes of the cosmological curves of the model (M 0 , Φ) depend only the reduced scalar potential Φ 0 . Accordingly, one can set M 0 = 1 without loss of generality. Every orbit of the action ρ par intersects the subset: The latter is stabilized by the scale subgroup: of T, whose action on Par 1 identifies with the action of R >0 given by: . Hence the quotient Par/T is in bijection with the quotient of Pot(R) through this action, which coincides with the set: of positive homothety classes of smooth positive functions defined on R. In particular, the overall scale of the scalar potential can also be eliminated by performing a scale transform of ϕ.
The ρ 0 (λ, )-transform of a curve ϕ ∈ C(R) induces the transformationφ(t) → λ 1/2 φ(t/ ) of the tangent vector field to that curve. Since the pointed maximal In particular, the set Sol m of pointed maximal cosmological curves of the model (M 0 , Φ) identifies with T = R 2 and the ρ 0 (λ, )-transform of such curves identifies with the transformation: of R 2 . Accordingly, we define: Definition 2.8. The phase space scaling action is the action ρ s of T on the phase space T = R 2 defined through: The remarks above are encoded by the following: = γ ϕ (I) its cosmological flow orbit. Then the following relations hold for all (λ, ) ∈ T: The restriction of ρ(λ, ) to the subset Par × Sol m Par × R 2 identifies with the map: 6) which we call the phase space similarity action. Following [4], we define: Definition 2.10. The cosmological renormalization group T ren is defined through: Definition 2.11. The similarity action of T ren is the restriction of ρ to T ren , which is given by: The parameter action of T ren is the restriction of ρ par to T ren , which is given by: The curve action of T ren is the restriction of ρ 0 to T ren , which is given by: Proposition 2.6 implies: The restriction of ρ ren to Par × Sol m identifies with the action of T ren on Par × R 2 given by the corresponding restriction ofρ: Definition 2.12. The phase space renormalization group action is the action of T ren on T = R 2 obtained by restricting (2.5): Remark 2.13. Notice that the rescaled Planck mass can also be eliminated by performing a renormalization group transformation with parameter = 1 M 0 , since:

The Hamilton-Jacobi formalism
In this section, we discuss the Hamilton-Jacobi equation of regular cosmological flow orbits, which is equivalent with the regular flow orbit equation (1.21) of Subsection 1.5. We also discuss the action of the universal similarity group T on Hamilton-Jacobi functions.

The Hamilton-Jacobi transformation
By definition, the Hamilton-Jacobi half-plane is the open upper half-plane R × R >0 endowed with Cartesian coordinates (x, h), where x ∈ R and h ∈ R >0 . The restriction of the rescaled Hubble function H to the zero section Z of T = T R gives the positive smooth function: Definition 3.1. The Hamilton-Jacobi region of the model (M 0 , Φ) is the closed overgraph of the function H 0 inside the Hamilton-Jacobi half-plane: The Hamilton-Jacobi locus of the model is the frontier of the Hamilton-Jacobi region: where H is the rescaled Hubble function of the model.
The Hamilton-Jacobi transformation is a double cover of the Hamilton-Jacobi region U which is ramified above the Hamilton-Jacobi locus and whose ramification set is the image Z of the zero section of T R: The positive sheet of this cover is the open upper half-plane T + while its negative sheet is the open lower half-plane T − . The closures of the two sheets meet along Z. Since and hence consists of two points of the phase space T = R 2 which are symmetric with respect to the x axis Z. The preimage reduces to a point which lies on the x axis when (x, h) ∈ B. The Galois group of the cover is Z 2 , whose action by (x, v) → (x, −v) on T permutes the two sheets. The fixed point set of this action coincides with the ramification set Z. The restriction of h to any of the sheets T ± is a diffeomorphism between that sheet and IntU and hence gives coordinates (x, h) ∈ IntU defined on that sheet. Given a non-trivial cosmological flow orbit O ⊂ R 2 , its image through h is a one-dimensional submanifold of the Hamilton-Jacobi half-plane which is contained in the Hamilton-Jacobi region U. We denote this image by:    Let ϕ : I → R be a regular curve of signature ζ ϕ and image ϕ(I) = J. Since the map ϕ : I → J is a diffeomorphism, we can use x ∈ J as a parameter for ϕ. As in Subsection 1.5, set: Then the H-function of ϕ is given by: where H ϕ : I → R is the rescaled Hubble parameter of ϕ: We have sign(v ϕ ) = sign(t ϕ ) = ζ ϕ .
of the phase space. Writing NoncritΦ = R \ CritΦ as an at most countable disjoint union of intervals: = Φ(K j ) ⊂ R located along the φ-axis. This map has inverse given by: In particular, one can express the restriction of any basic cosmological observable to a connected component of N as a function of Φ and H. The derivative of Φ can be expressed as a function of Φ, which we denote by W j : which has the advantage that the parameter M 0 appears only in the term involving the first derivative ofĤ. One can seek solutions of (3.4) which expand as a power series in which amounts to seeking solutions of (3.3) which expand as a Laurent series of the form: We will see below (see Remark 3.13) that this is equivalent with the rescaled Planck mass expansion of cosmological curves mentioned in Remark 2.2.
The following proposition gives a bijection between regular Hamilton-Jacobi functions and regular cosmological flow orbits. The bijection is obtained by writing the regular flow orbit equation (1.21) in terms of Hamilton-Jacobi coordinates on that sheet of the Hamilton-Jacobi map which contains a given regular flow orbit. This presents the h-image of that flow orbit as the graph of the corresponding regular Hamilton-Jacobi function. = π(O) → R is a regular Hamilton-Jacobi function of signature ξ = −ζ. In this case, we have: In particular, the regular Hamilton-Jacobi manifolds of the model (M 0 , Φ) coincide with the graphs of its regular Hamilton-Jacobi functions.
We stress that the flow orbit O and its associated Hamilton-Jacobi function H have opposite signatures. The proposition shows that the speed function v of a regular flow orbit can be recovered from the Hamilton-Jacobi function H of that orbit through the formula: while H can be recovered from v as: Proof. Since O is regular, we have v O (x) = 0 for all x ∈ J. The regular flow orbit equation (1.21) is equivalent with: where we used the relation v(x) 2 + 2Φ(x) = M 2 0 H(x, v(x)) 2 and the fact that H > 0.
, the second equation in (3.7) amounts to: = H(x, v(x)). The remaining statements of the proposition are obvious.
Remark 3.11. The signed Hamilton-Jacobi equation for a regular orbit of signature ζ which is contained in the open subset U j,ζ = K j × R ζ of the phase space can be written as follows: where W j was defined in Remark 3.6.

Reconstruction of regular cosmological curves
Let ϕ : I → R be a regular smooth curve of signature ζ ϕ and image ϕ(I) = J. Since the map ϕ : I → J is a diffeomorphism, we can use x ∈ J as a parameter for ϕ. As above, set: =φ • t ϕ : J → R × be the speed function of ϕ.
In this case, H ϕ coincides with the Hamilton-Jacobi function H O of the cosmological flow orbit O determined by ϕ.
Remark 3.13. Notice that the speed equation of H ϕ can be written as: with general solution: , (3.10) where C ∈ R and x 0 ∈ J are arbitrary. This relation can be inverted to find the equation x = ϕ(t) of ϕ. In particular t ϕ and hence ϕ depend non-locally on H ϕ . Also notice that the scale transform ψ def.
= ϕ M 0 of Remark 2.2 has speed function: and hence is obtained by inverting the function: where C = M 0 C . Relation (3.11) shows that the expansion of ψ in powers of M 0 mentioned in Remark 2.2 corresponds to the power series expansion ofĤ ϕ in M 0 and hence to the Laurent expansion of H ϕ in M 0 .
Proof. Since ϕ is a regular curve, the cosmological equation for ϕ is equivalent with the dissipation equation (1.24) by Proposition 1.16. We have: Hence the dissipation equation (1.24) is equivalent with:  = π(O) → U defined through: where H O is the Hamilton-Jacobi function of O.
where C ∈ R and x 0 ∈ J are arbitrary.
Remark 3.15. The first slow roll approximation for a regular cosmological flow orbit O corresponds to taking: M 0 is independent of O. This approximation is accurate for a cosmological flow orbit O which is very close to the zero set Z (and hence the corresponding portion of the Hamilton-Jacobi manifold V O is very close to the Hamilton-Jacobi locus B). In this case, V O reduces to the leading classical effective potential: of [4] while H O reduces to H 0 . Notice that V 0 and H 0 depend only on the parameters (M 0 , Φ) of the model, being independent of the orbit O. Also notice that H 0 is a Hamilton-Jacobi function only when restricted to a connected component of the interior of the critical locus CritΦ (assuming that Int(CritΦ) is non-empty), in which case it corresponds to a trivial cosmological flow orbit. In the next section, we explain how one can construct a better approximant for cosmological orbits which are close to the zero section Z of T R.

Similarities of Hamilton-Jacobi functions
Under a phase space scaling action ( Accordingly, the Hamilton-Jacobi function H( of O maps to the Hamilton-Jacobi function ofÕ, which is given by: Setting λ = 2 gives the action of the renormalization group T ren on Hamilton-Jacobi functions: Accordingly, we define: Definition 3.16. The Hamilton-Jacobi scaling action is the action r 0 of T on C(R) given by: The Hamilton-Jacobi similarity action is the action r = ρ par × r 0 of T on Par × C(R): Let H(M 0 , Φ) and H reg (M 0 , Φ) be the sets of all (respectively regular) Hamilton-Jacobi functions of the model (M 0 , Φ) and H ± reg (M 0 , Φ) the subsets of regular Hamilton-Jacobi functions of positive and negative signature. We have: Proposition 3.17. For any (λ, ) ∈ T and (M 0 , Φ) ∈ Par, we have: Proof. It is clear that H ∈ C(R) satisfies the Hamilton-Jacobi equation (3.3) of the model (M 0 , Φ) iff r(λ, )(H) satisfies the Hamilton-Jacobi equation of the model ρ par (λ, )(M 0 , Φ) = (λ 1/2 M 0 , λ 2 Φ λ 1/2 ). Moreover, the derivative of H is non-vanishing iff that of r 0 (λ, )(H) is, in which case the two derivatives have the same sign.
Definition 3.18. The Hamilton-Jacobi RG scaling action is the restriction r 0 ren of r 0 to T ren , which is given by: The Hamilton-Jacobi RG similarity action is the restriction r ren of r to T ren , which is given by:

The IR Hamilton-Jacobi function and potential
In this section, we discuss the construction of a universal formal IR Hamilton-Jacobi function H IR as a Laurent power series in the rescaled Planck mass M 0 and of the associated formal Hamilton-Jacobi potential V IR , which admits a power series expansion in M 0 . For technical reasons, we start by constructing slightly more general objects which also depend on the scale parameter and reduce to H IR and V IR when this parameter is formally set to one. We show that the coefficients of the corresponding expansions are controlled by certain quasi-homogeneous polynomials in the scalar potential and its derivatives, which are determined by a recursion relation. Using the homogeneity properties of these polynomials, we show that the resulting expansion can be written in terms of the slow-roll parameters of [2] and coincides with the slow roll expansion of loc. cit. This provides a conceptually clear explanation of the slow roll expansion as an expansion in powers of the rescaled Planck mass. In particular, we give an explicit recursion procedure for constructing the slow roll expansion, thus providing an efficient method for computing its higher order terms. In Appendix B, we list the the coefficients of this expansion up to 10th order, thus improving markedly on the results of loc. cit.  M 0 for → 0. Notice that the parameter "counts the number of derivatives" and that (4.1) is equivalent with the equation: satisfies the Hamilton-Jacobi equation of (M 0 , Φ) for all M 0 ∈ (0, µ).
We also have the following result, which relates the solutions of (4.1) to scale transforms of the Hamilton-Jacobi functions of a different model.
Proof. The first statement is obvious. The second statement follows by noticing that we have: where r is the Hamilton-Jacobi similarity action of Subsection 3.5.
When is small, the behavior of H away from the origin captures the behavior of H(x) for x very large in absolute value. Indeed, we have H (±1) = H(±1/ ) and 1/ tends to infinity when → 0. Below, we construct a formal solution u of (4.1) for all M 0 > 0 as a power series in 2 . Performing the transformation (4.5) (or (4.4)) will then produce a "universal" formal solution H IR of the Hamilton-Jacobi equation of the model (M 0 , Φ) such that H IR /H 0 is given by a power series in M 2 0 whose coefficients are rational functions of Φ and its derivatives (more precisely, they are ratios between a polynomial in Φ and its derivatives and a power of Φ). This also produces a series for the infrared Hamilton-Jacobi potential V IR def.
= M 2 0 H IR . Conceptually, this constructs successive approximations for a "formal infrared flow orbit" O IR of the model (M 0 , Φ) -which corresponds to an actual flow orbit only when the series are uniformly convergent on some interval of the real axis. The resulting formal power series expansion of u will have the form: where U n ∈ C ∞ (R) are expressed as functions of Φ and its derivatives (as we confirm in Proposition 4.13 below). Accordingly, H IR has the expansion:

Formal IR Hamilton-Jacobi families
One can extract a formal equation from (4.1) by taking J = R and expanding the right hand side as a formal power series in . Recall that the element 1 + X ∈ R[[X]] admits a square root given by the binomial expansion: where a 0 = 1 and: (4.8) are the generalized binomial coefficients, the first of which are: Using (4.7) in (4.1) produces the formal deformed Hamilton-Jacobi equation: where the unknown: is a formal power series in whose coefficients are smooth univariate functions u n ∈ C ∞ (R) and we defined: The formal IR Hamilton-Jacobi function of the model (M 0 , Φ) is the following series with coefficients in the ring C ∞ (R): 14) The formal IR effective potential of the model (M 0 , Φ) is the following series with coefficients in the ring C ∞ (R): Here, the right hand sides of (4.14) and (4.15) are understood as the sequences of their partial sums.
We will see in a moment that u is uniquely determined by M 0 and Φ -and hence so are V eff , H IR and V IR . where s j ∈ N for all j ≥ 1.
The first two relations in (4.17) read: Proof. Using the expansions (4.11) and (4.12) in equation (4.10) and rearranging terms gives (4.16) and the recursion relation: The latter is equivalent to (4.17) upon using (4.16).

Expressing u through u 0 and its derivatives
Preparations. Let C ∞ x denote the space of germs of smooth real-valued univariate functions at x ∈ R and J ν x denote the space of jets of order ν of such functions x has a natural ring structure which we recall in Appendix A. For any non-empty open interval K ⊂ R, any point x ∈ J and any smooth real-valued function f ∈ C ∞ (K), let f x ∈ C ∞ x and j ν x (f ) ∈ J ν x denote respectively the germ and jet of order ν of f at x (see Appendix A for the precise definition of j ν x (f )). We identify J ∞ x with the infinite-dimensional vector space R ∞ def. = R N by sending j ∞ x (f ) to the reduced infinite jet j ∞ x (f ) ∈ R N , which we define as the infinite vector with components: Accordingly, for any n ∈ N, we identify J n x with the vector space R n+1 by sending j n x (f ) to the reduced n-th order jet j n x (f ) ∈ R n+1 , which we define as the (n+1)-vector with components: [j n x (f )] k = f (k) (x) ∀k = 0, . . . , n . This gives linear isomorphisms: where surjectivity follows from Borel's lemma. For any function f ∈ C ∞ (K) as above and all ν ∈ N ∪ {∞}, we denote by j ν (f ) : K → R ν the function defined through: For later reference, we define the following cones in R n and R N : = (w 0 , . . . w n ) (4.20) denote the n-th truncation of the infinite sequence w = (w j ) j≥0 ∈ R N and let τ n : R N → R n+1 be the corresponding truncation map: The Hamilton-Jacobi polynomials. Since the sum in the right hand side forces s 1 + . . . + s 2k = n − k < n and hence s j < n, relation (4.17) expresses u n in terms of u 0 and u 0 , . . . , u n−1 . Together with (4.16), this determines u n recursively for all n ≥ 1 as a function of u 0 = √ 2Φ M 0 and its derivatives of order ≤ n: for some rational function f n : R n+1 0 → R. The expansion (4.11) becomes: To describe the structure of f n , we will use the algebra A of polynomials with real coefficients in a countable set of variables X k (k ∈ N). Formally, A def.

= R[Σ] R[N]
is the free associative and commutative algebra on the countable set: This algebra admits an exhaustive ascending filtration by the subspaces A n = R[X 0 , . . . , X n ]: and admits gradings defined by the following degrees: • The ordinary polynomial degree deg, with respect to which: • The order-weighted degree wdeg, with respect to which: We combine these into the N×N-valued bigrading which associates to each monomial T the bidegree (deg(T ), wdeg(T )) (in this order).
Let ∂ X j be the canonical derivations of A. Given a polynomial T ∈ R[X 0 , . . . , X n ] and any α ∈ N, define T ∈ R[X 0 , . . . , X n+1 ] and DT ∈ R[X 0 , . . . , X n+1 ] by: Then (−) and D(α) are differential operators on A of bidegrees (0, 1) and (1, 1) respectively; in particular, they map bi-homogeneous polynomials to bi-homogeneous polynomials. Also notice the following relation in the algebra of rational functions which implies: Definition 4.7. The Hamilton Jacobi polynomials Q n ∈ Q[X 0 , . . . , X n ] are defined through the recursion relation: with the initial condition Q 0 = 1, where a k are the generalized binomial coefficients (4.8).
Remark 4.8. For all n ≥ 0, we have: (4.25) The first nontrivial polynomials Q n are (see Appendix B for a list of these polynomials up n = 10): Proposition 4.9. For all n ≥ 0, the Hamilton-Jacobi polynomial Q n is homogeneous of bidegree (2n, 2n). If n ≥ 1, then the largest power of X 0 in the constituent monomials of Q n is strictly smaller than n.
Proof. The bi-homogeneity statement is true for n = 0. Let n ≥ 1 and assume that the statement holds for all s < n. Then relation (4.24) implies that the statement also holds for n and hence it holds for all n by induction. The statement about the largest power of X 0 in Q n also follows by induction over n. The statement is obviously true for n = 1 since Q 1 = 1 2 X 2 1 . If s ≥ 1 and the largest power of X 0 in Q s is strictly smaller than s, then relation (4.25) shows that the largest power of X 0 in D(2s − 1)Q s is at most s. The latter is also true if s = 0 since D(−1)Q 0 = X 1 . Now if n ≥ 2 and the statement holds for all 0 ≤ s < n, then the previous remark and the recursion relation (4.24) shows that it also holds for n. = X k 0 0 . . . X kn n ∈ Q[X 0 , . . . , X n ] where n, k 0 , . . . , k n ∈ N. This monomial has bidegree (2n, 2n) iff k 0 , . . . , k n satisfy: (4.27) If n ≥ 1 and q appears as a monomial in Q n , then Proposition 4.9 shows that we also have k 0 < n, i.e. n j=1 k j > n.
The functions f n .
Proposition 4.11. For any n ≥ 0, we have: . (4.28) Proof. We proceed by induction over n. Relations (4.28) hold for n = 0 since Q 0 = 1. They also hold for n = 1 by the first equation in (4.18). Assume that n ≥ 2 and that (4.28) holds for all s = 0, . . . , n − 1. Then the second relation in (4.28) gives: Now the recursion relation (4.17) and the definition of Q n implies that the first relation in (4.28) holds for u n . In turn, this implies that the second relation in (4.28) holds for u n . By induction we conclude that relations (4.28) hold for all n ≥ 0. In particular, we have f 0 (w 0 ) = w 0 .
In the proposition below and in what follows, we use the same notation for a polynomial and the polynomial function which it induces by evaluation over a field.
Proposition 4.13. We have: and: (4.30) Proof. Relations (4.29) and (4.30) follow immediately from Corollary 4.12 and the expansion (4.11) upon using (4.16) and the fact that Q n and D(2n − 1)Q n are homogeneous of degrees 2n and 2n+1 respectively with respect to the ordinary polynomial grading.
Notice that S 0 = id R .
Moreover, they behave as follows under a scale transformation of Φ: and hence they satisfy: M 0 is homogeneous of degree 1/2 respectively −1 under a positive constant rescaling of Φ or M 0 : and that it transforms as follows under a scale transformation of Φ: The last relation implies: Proposition 4.13 now implies the conclusion upon using (4.31) and (4.32) and the fact that Q n is homogeneous of bidegree (2n, 2n).
Proposition 4.16. The formal IR Hamilton-Jacobi function and formal IR effective potential can be obtained respectively by substituting = 1 in u and V eff : and: where the series in the right hand side are understood as sequences of their partial sums.
Let us momentarily denote H IR by H to indicate its dependence on the parameters of the model. The following result shows that the set: is invariant under the Hamilton-Jacobi similarity action r of the group T = R 2 >0 (see Subsection 3.5) and in particular under the action r ren of the renormalization group T ren .
where (λ, ) ∈ T and r 0 (λ, ) (which was given in Definition 3.16) is applied to the partial sums of series. In particular, H IR and V IR satisfy: for all > 0.

Expansion in powers of Φ and its derivatives
Lemma 4.18. For any n ≥ 1, we have: and hence: where: is a homogeneous polynomial of bidegree (n, n). = 1 ∈ Q[X 0 ]. If n ≥ 1, then notice that the largest power of X 0 which appears in R n is strictly smaller than n.
Proof. The Taylor expansion of Φ around x and the binomial expansion of the square root give: The conclusion now follows by comparing this with the Taylor expansion of √ Φ around x: The first nontrivial polynomials R are: Corollary 4.20. For any n ≥ 0, we have: where g n : R n+1 0 → R n+1 is a function whose components g k n are rational and homogeneous of bidegree (0, k): Proof. Follows immediately from Lemma 4.18.
Proof. Using (4.37) in (4.29) gives (4.39) with v n as in (4.40) upon recalling that Q n is homogeneous of degree 2n with respect to the ordinary polynomial grading. Relations (4.40) and (4.38) give: This is equivalent with (4.41) with S n as in (4.42) since Q n is homogeneous of degree 2n with respect to the order-weighted grading. To prove the last statement, let us assume that n ≥ 1. Then Remark 4.19 shows that the largest power of X 0 in each polynomial R j with j ∈ {1, . . . , n} is strictly smaller than n and hence for all such j the rational function is a sum of Laurent monomials in which X 0 appears with power ≤ −1. Since Q n has ordinary polynomial degree 2n and X 0 appears in Q n with exponent strictly smaller than n (see Proposition 4.9), relation (4.43) shows that v n is a sum of Laurent monomials of the form: Together with the observation above, this implies that X 0 enters the Laurent monomials of v n with powers which are strictly smaller than −n and hence enters the monomials of S n with powers which are strictly smaller than n.
We have: and hence: Corollary 4.22. The formal IR Hamilton-Jacobi function and formal IR effective potential of the model (M 0 , Φ) are given by: and: = M 0 √ 2Φ and the series in the right hand side are understood as the sequences of their partial sums.
Remark 4.23. The series in Corollary 4.22 can also be viewed as formal power series in the rescaled Planck mass M 0 (with coefficients in the ring C ∞ (R)).

Rewriting the expansion in term of slow roll functions
Suppose that n ≥ 1. Since the polynomial S n of Proposition 4.21 has bidegree (2n, 2n), the rational function v n ∈ Q(X 0 , . . . , X n ) of (4.41) is a finite linear combination of Laurent monomials of the form: where k 0 , . . . , k n ∈ N satisfy conditions (4.27) and we have k 0 = 2n − n j=1 k j < n. These conditions allow us to write W as: where k 1 , . . . , k n ∈ N satisfy n j=1 jk j = 2n and n j=1 k j > n. Definition 4.24. The slow roll rational functions (β j ) j≥0 at rescaled Planck mass M 0 are defined through: We have β 0 ∈ R(X 0 , X 1 ), β 1 ∈ R(X 0 , X 2 ) and β j ∈ R(X 0 , X 1 , X j+1 ) for all j ≥ 2. Define: where: Then the following identities are satisfied: and we have: (4.52) Proof.
Definition 4.27. The polynomials (σ n ) n≥1 are called the slow roll polynomials.
Definition 4.28. The potential slow roll functions of the model (M 0 , Φ) are the smooth functions β n ∈ C ∞ (R) defined through: Notice that reference [2] prefers to work with the reduced potential slow roll functions: It is customary to use the notations: def.
The following result shows in particular that the expansions of the functions H IR and V IR coincide with their slow roll expansions in the sense of reference [2].
Proposition 4.29. We have: and: is related to M 0 through m = 2 √ 3πM 0 . We stress that the computational procedure proposed in [2] is rather impractical and far less efficient that the approach of the present paper, which relies on the recursive construction of Hamilton-Jacobi polynomials. We illustrate this in Appendix B, where we list the rational functions v n up to n = 10. These determine the expansions of H IR and V IR up to that order through relations (4.46) and (4.47).

4.6
The N -th order approximate IR flow orbit where C, x 0 ∈ R.

The formal IR Hamilton-Jacobi jet expansion family
For all n, k ∈ N, let ∆ n,k : A → A be the differential operator defined through We have: Proposition 4.32. For any n ≥ 1 and k ≥ 0 we have: and: Proof. Relation (4.61) follows immediately from Proposition 4.11 upon using (4.23). Now (4.61) implies (4.62) upon using Corollary 4.20 and the fact that the polynomial ∆ n,k Q n is homogeneous of degree 2n + k with respect to the polynomial grading.

Corollary 4.33 implies:
Corollary 4.36. For any x ∈ R, we have: j ∞ x (u) = e(j ∞ x (Φ)) . where P n ∈ R[ ][X 0 , . . . , X n+1 ] = R[ , X 0 , . . . , X n+1 ] is the polynomial given by: Moreover, we have: For example, we have: and hence: Proof. Recall that u satisfies (4.2). Repeatedly differentiating this equation with respect to x allows us to express Φ (n) M 2 0 as a function of and the derivatives u (j) with j = 0, . . . , n + 1: where P n ∈ R[ , X 0 , . . . , X n+1 ] is a polynomial. The higher order differentiation formula for functions f, g ∈ C ∞ (R): where ι : is the injective map given by: Proof. Follows immediately from Proposition 4.37 and Corollary 4.36 using the fact that Φ : R → R >0 is an arbitrary positive smooth function.

Conclusions and further directions
Following the ideas of [4,5], we constructed the infrared scale expansion of single field cosmological models using the Hamilton-Jacobi formalism of [1], finding that it corresponds to seeking a solution of the Hamilton-Jacobi equation of such models which admits a Laurent expansion in powers of the Planck mass. We showed that this recovers the celebrated slow roll expansion of Liddle, Parsons and Barrow [2], which it identifies with the small Planck mass expansion of such models. We also described the single field realization of the universal similarity group action of [4,5] both at the level of cosmological curves and in the Hamilton-Jacobi formalism.
Unlike the approach of [2], the method of the present paper leads to an explicit recursive construction of the coefficients of the slow roll expansion. We showed that the latter are obtained by evaluating certain model-independent rational functions v n on the scalar potential of the model and its successive derivatives. In turn, these rational functions are obtained by explicit universal formulas from the Hamilton-Jacobi polynomials Q n . The latter are model-independent multivariate polynomials with rational coefficients defined by a recursion relation which allows for their efficient computation. As an application, we listed the coefficients of the Hamilton-Jacobi slow roll/small Planck mass expansion up to order 10 inclusively (see Appendix B). The identification of the small Planck mass and slow roll expansions follows from the fact that the polynomials Q n are bihomogeneous with respect to a certain bigrading of the algebra of polynomials in a countable number of variables.
The present paper opens up a few avenues for further research. First, one could perform a systematic study of Hamilton-Jacobi polynomials and their deeper properties, on which we only touched upon in the present paper. This would make it possible to extract asymptotic bounds for the small Planck mass/slow roll expansion which would enable one to analyze its summability and resummation. In this regard, we notice that the deformed Hamilton-Jacobi equation considered in the present paper can be approached with the well-established methods of singular perturbation theory for ODEs with a small parameter. Second, one can apply similar methods to the UV expansion of [4], which in the single field case should be related to the psi series expansion discussed in [10,11]. Finally, one can study the IR and UV expansions proposed in [4] for multifield cosmological models, which should provide a generalization of the slow roll and psi series expansions to the multifield case. We hope to report on these and related questions in future work.

A Jets of univariate real-valued functions
Let C ∞ x be the commutative R-algebra of germs of smooth univariate real-valued functions at a point x ∈ R. Let id R , 1 R ∈ C ∞ (R) be the identity and unit functions of R and e x ∈ C ∞ (R) be the germ at x of the function id R − x1 R ∈ C ∞ (R). Then C ∞ x is a (non-Noetherian) commutative local ring with maximal ideal: and residue field R and for all n ∈ N we have: x be the commutative R-algebra of jets of order ν of univariate real-valued functions at x and x ∈ J ∞ x be the infinite order jet of the identity function id R . Then J ∞ x is a Noetherian local ring with maximal ideal: which identifies with the R-algebra R[[ x ]] of formal power series in the variable x . Accordingly, J n x identifies with the space R[[ x ]]//µ n+1 x of polynomials of degree n in the variable x . The infinite jet j ∞ x (f ) of a germ f ∈ C ∞ x identifies with the formal Taylor series of f at x, while j n x (f ) identifies with its n-th Taylor polynomial: The higher order differentiation formula for functions f, g ∈ C ∞ (R): (f g) (n) = k+l=n (k + l)! k!l! f (k) g (l) ∀n ≥ 0 implies that infinite jet prolongation gives a morphism of local rings: which satisfies j ∞ x (e x ) = x . This morphism is surjective by Borel's lemma and induces isomorphisms of rings: B.2 The rational functions v n for n ≤ 10