Adiabatic and Isocurvature Perturbation Projections in Multi-Field Inflation

Current data are in good agreement with the predictions of single field inflation. However, the hemispherical asymmetry seen in the cosmic microwave background data, may hint at a potential problem. Generalizing to multi-field models may provide one possible explanation. A useful way of modeling perturbations in multi-field inflation is to investigate the projection of the perturbation along and perpendicular to the background fields' trajectory. These correspond to the adiabatic and isocurvature perturbations. However, it is important to note that in general there are no corresponding adiabatic and isocurvature fields. The purpose of this article is to highlight the distinction between a field redefinition and a perturbation projection. We provide a detailed derivation of the evolution of the isocurvature perturbation to show that no assumption of an adiabatic or isocurvature field is needed. We also show how this evolution equation is consistent with the field covariant evolution equations for the isocurvature perturbation in the flat field space limit.


Introduction
One of the main aims of modern cosmology is to determine if there was a period of inflation in the early Universe. A crucial property of inflation is the number of fields involved. The recent Planck cosmic microwave background (CMB) results, while on the whole being consistent with general single field inflation predictions [1,2], may show the need for multiple fields as there are hints of a possible hemispherical asymmetry in the data [3]. Similar observations were made in the WMAP data [4], but their statistical significance was questioned [5]. Confirmation by Planck makes a systematic or foreground explanation less likely. Also, the Planck data shows indications of the modulation extending to smaller scales than was seen in the lower resolution WMAP data. If the primordial fluctuations really are asymmetrical, this isotropy breaking could conceivably result from a large scale mode modulating the smaller scale modes [6]. Such a scenario could be realized in a multi-field inflation model [7]. To provide a good fit, the amplitude of the dipole modulation would have to be scale dependent [8], especially to accommodate the quasar data [9].
In this article we revisit and clarify a useful method of analyzing inhomogeneity in multi-field inflation models. In Ref. [10] it was proposed to decompose multi-field perturbations into an adiabatic direction along the field trajectory (δσ) and an isocurvature direction perpendicular to the field trajectory (δs). This was found very useful in solving problems related to the preheating of the Universe at the end of inflation and also in understanding how correlated adiabatic and isocurvature perturbations may be generated. Subsequently, it has also been used extensively in other multi-field inflation investigations. However, in Ref. [11] it was argued that this procedure leads to an inconsistency.
Here we show that the apparent inconsistency is due to a conflict of notation within [10], and that in order to derive the evolution equations of the adiabatic and isocurvature perturbations it is not necessary to assume the existence of an adiabatic and isocurvature field, meaning that the inconsistency does not apply to the results of [10].

Perturbation Projections vs Field Transformations
For presentation reasons we will consider the two canonical scalar field (ϕ 1 = φ, ϕ 2 = χ) case as was done in [10]. An extension of the formalism developed in [10] for a larger number of fields and non-canonical couplings can be found in [12]. As in [10], we consider a background plus a first order perturbation: ϕ(t)+δϕ(x, t). The perturbation variables satisfy a system of second order linear differential equations. Therefore, new variables defined as linear combinations of the φ and χ will also be described by a system of second order linear equations. In [10] it was proposed to look at linear combinations that would project the perturbation parallel (δσ) and perpendicular (δs) to the background fields' direction of motion in field space, as shown in Figure 1: where θ is defined byσ and a dot denotes a derivative with respect to time. It was shown that δσ is associated with the adiabatic part of the fields' perturbation and δs is the isocurvature (also known as "entropy") part of the perturbation.
If we combine the above two equations we get the apparently nonsensical result that tan 2 θ = −1. This contradiction can be avoided if we make θ a constant in Eqs. (5) and (6).
Although this apparent contradiction may be interesting, it does not have any direct impact on the evolution equations derived for δσ and δs in [10]. The reason being that at no point during that analysis was the assumption made that there was an adiabatic and isocurvature field. All that was done was the projection defining Eqs. (1) and (2) were differentiated with respect to time and then the evolution equations of δϕ were substituted into the result to simplify the equations.
Some of the terminology and notation in [10] gave the impression that the existence of adiabatic and isocurvature fields was assumed. For example it was stated that subscript notation was used for derivatives, V x = ∂V /∂x. However, the quantity V ss ≡ cos(θ) 2 V χχ − 2 cos(θ)V φχ sin(θ) + V φφ sin(θ) 2 was also defined, but it was not specified that this was just a definition and not the second partial derivative of the potential with respect to s. This form of notation was used as it is suggestive in that if θ were constant then it would be the correct form for the partial derivative, as then one could actually define an isocurvature field. In fact, any subscripts of the potential which have σ or s in them were just meant to be variable definitions, rather than denoting partial derivatives.

Detailed Derivation of Isocurvature Projection Evolution Equation
In this section we provide a much more detailed derivation of the evolution equation for δs that was given in [10]. This is done so as to show explicitly that there is no need to propose the existence of an adiabatic or isocurvature field.
Using ϕ 1 = φ and ϕ 2 = χ in Eq. (11) gives δs + 3Hδs + k 2 a 2 δs +θ 3Hδσ + 2δ σ − δsθ + δσθ + C s = 0, where We rewrite Eq. (12) as where From Eqs. (19), (20) and (21) we find where C sσV ≡ (−V φφ + V χχ ) cos(θ) sin(θ) + V φχ cos(θ) 2 − sin(θ) 2 , (23) and the expressions for C sσA and C ssA are obtained by replacing V ϕ 1 ϕ 2 with A ϕ 1 ϕ 2 in the corresponding expressions for C sσV and C ssV . In [10], instead of notation such as C ssV , V ss was used. We avoid this here so as not to lead to the incorrect impression that we are taking the derivative of the potential with respect to an adiabatic or isocurvature field. This interpretation would only be correct in the constant θ case. Using the background Klein-Gordon equations in Eq. (21) gives Using the "A" version of Eq. (23) and Eq. (24) leads to which may be used in Eq. (22) so that the evolution equation Eq. (17) for δs becomes δs + 3Hδs + k 2 δs a 2 +θ 3Hδσ + 2δ σ − δsθ + δσθ Upon using Eq. (4) this may be expressed as δs + 3Hδs As shown in [14] the three-curvature perturbation, ζ, is related to the curvature perturbation in the longitudinal gauge, Ψ, and adiabatic, isocurvature perturbations byζ giving which allows us to substitute for δσ in Eq. (29). This, along witḣ and the background Klein-Gordon equation, means we can simplify the right-hand-side of Eq. (29) to Finally, we can use Eq. (24) to simplify Eq. (29) tö δs + 3Hδs + δs k 2 a 2 + 3θ 2 + C ssV = Eq. (35) matches the evolution equation derived in [10]. Therefore, we have explicitly shown how to derive the evolution equation for the isocurvature perturbation projection without having to define an isocurvature or adiabatic field. A similar calculation can be used to derive the evolution equation for the adiabatic perturbation projection.

Comparison with the Covariant Approach
Having made explicit the calculation appearing in [10] we should check that the approach of [11] gives consistent answers. In that paper the author wrote down the multi-field formalism in a manner that maintained explicit covariance under field re-definitions, whilst also allowing for the field-space to be curved -see also [18]. In order to apply that formalism to the current case we shall take the field-space to be flat, and we shall use Cartesian coordinates φ and χ to represent the ϕ α . The proper-speed in field space is denotedσ and is given byσ where the ( , ) denotes the field-space inner-product. The adiabatic and isocurvature perturbations were then defined as where the antisymmetrization is denoted by [x, y] ≡ 1 2 (xy − yx). We note that the appropriate field-perturbation variable identified in [11], i.e. the one that transforms as a tensor under field re-definitions, reduces to δϕ α in the Cartesian co-ordinates we are using. For the case of two fields there is a single isocurvature mode, and so we identify Having done that, we work on the {α, β} = {φ, χ} component of (3.16) in [11], using our Eqs. (24) and (10) to yield δs + 3Hδs + k 2 a 2 δs + (C ssV −θ 2 )δs = 2 Here we have used (3.7) of [11] for a spatially-flat background geometry, in spatially-flat gauge, to give 4πG in order to replace δσ with the scalar metric perturbation A 1 . We then relate this to the co-moving density perturbation m [19][10] Using (B.7) and (B.8) of [11] we find that the gauge invariant intrinsic curvature perturbation Ψ is given by and putting all this together in Eq. (40) gives precisely (35).

Conclusions
When studying multi-field inflation models it is often necessary to make use of both field transformations and perturbation projections [15,16,17]. For example Byrnes and Wands (2006) [17] found that the adiabatic and isocurvature perturbations can be correlated at horizon exit and so it is useful to first transform to uncorrelated fields at horizon exit and then use a projection to follow the adiabatic and isocurvature components evolution after horizon exit. But, as we have highlighted in this article, it is not necessary to define a field transformation when one wants to derive the evolution equations of a perturbation projection.
We have also shown that the field-covariant evolution equations and definitions of the adiabatic and isocurvature perturbations in [11] tend to the definitions and equations of motion in [10] in the flat field space limit. The main conclusion of this article is that whilst some of the terminology and notation in [10] led to the impression that the projection Eqs. (1) and (2) were the result of a transformation to an "isocurvature" and "adiabatic" field, the actual derivation and use of the evolution equations of δσ and δs never made this incorrect assumption.