A (semi)-exact Hamiltonian for the curvature perturbation $\zeta$

Determining the evolution of the cosmological perturbations starting from the moment they appeared as sub-horizon quantum fluctuations to the much later epochs involving structure formation is crucial in testing the predictions of inflationary models. Although the problem is basically an expansion around a classical background solution, there are important technical issues to be tackled like the gauge invariance related to the coordinate transformations. The cosmological perturbation theory (see the classical review mfb ) properly classifies the fluctuations and determines their basic dynamical evolution like the freeze-out of superhorizon modes. In recent years, after the seminal work of mal , there has been a great interest in determining the evolution of the cosmological perturbations beyond the linear regime, which has important implications for the precision cosmology like the presence of non-gaussianities. In the approach of mal , one works in the Lagrangian formulation and takes the metric in the Arnowitt-Deser-Misner (ADM) form. One then algebraically solves for the lapse $N$ and the shift $N^{i}$ from their equations of motion and imposes gauge fixing to obtain a Lagrangian for the physical degrees of freedom, i.e the curvature perturbation $\zeta$ and the (transverse, trace-free) tensor mode $\gamma_{ij}$ describing the gravitational waves. This procedure, which can be utilized perturbatively, becomes quite cumbersome at higher orders. Yet, one can systematically work out the quantum corrections to cosmological correlation functions (including the loop effects) by using the in-in (Schwinger-Keldysh) formalism w1 ; w2 .

Although perturbation theory seems to work well in dealing with small non-linear effects, it is also desirable to have non-perturbative/exact methods to understand the implications of possible strong non-linearities. One can, for example, utilize the symmetries of the background geometry to obtain exact Ward identities for cosmological correlation functions ward1 ; ward2 , see also ward-ali . Here, the dilation symmetry gives the well-known three-point consistency condition mal ; 3pt . The stochastic approach of np1 ; np2 can be used to calculate arbitrary large correlation functions in de Sitter space, which involve the so called infrared (IR) logarithms (see also ir1 ; ir2 , and also ak2 which shows the existence of IR logarithms in the minisuperspace approximation). It is also possible to utilize the exact renormalization group flow techniques for quantum fields to infer the structure of the effective interacting potential of a real scalar field in de Sitter space akr .

More recently, using powerful physical principles like locality, unitarity and symmetry, the cosmological bootstrap strategy has been used to constrain correlation functions, for review see e.g. boot1 ; boot2 . This program offers a very broad, model independent picture of inflationary physics and allows one to infer various exact results and new insights. There are numerous important findings that follow from the bootstrap program like proving a cosmological optical theorem, which implies an infinite set of relations among the correlators ot1 , and obtaining cosmological cutting rules that fix the discontinuity of a loop diagram in terms of lower order loops and tree diagrams ot2 . Another interesting line of research involves fixing the correlators of massless spin 1 and spin 2 fields in de Sitter space by solving the Ward identities related to background symmetries nr1 .

There are also proposals for a possible holographic dual description of single scalar field inflationary models, where the dual is a three dimensional quantum field theory (a conformal field theory deformed by a nearly marginal operator) that flows from an IR fixed point to a UV fixed point corresponding in the dual picture to the (hilltop) inflaton rolling from a local maximum at past infinity to a local minimum at future infinity hi1 ; hi2 .

Compared to these non-perturbative methods, our approach in this paper is more direct, i.e. we only consider single scalar field inflationary models and work in the context of classical general relativity. As mentioned above, the action for the cosmological perturbations arises from an expansion around a background solution and it contains infinitely many interaction terms that can in principle be determined at any desired order by a standard but lengthy procedure. Moreover, the expansion is naturally carried out in the Lagrangian formulation and one should obtain the corresponding Hamiltonian for the in-in perturbation theory, which is an additional involved computation. Obviously, it is desirable to have a closed exact expression for the Hamiltonian of the physical fluctuation variables. In this paper we show that one can indeed obtain a non-perturbative Hamiltonian for the curvature perturbation $\zeta$ by paying attention how the expansion around the background cosmological solution yields a non-zero fluctuation Hamiltonian. As we will see both the special structure of the constraints in general relativity and the particular form of the cosmological background become important in this all-order derivation, which in the first step amounts to solving the Hamiltonian constraint exactly. The notorious problem of time that appears when dealing with the Hamiltonian constraint is eliminated by referring to the background solution. To fix the dynamics of the physical degrees of freedom, one must also solve the momentum constraint and we show that this can still be done exactly for $\zeta$ and perturbatively in $\gamma_{ij}$. After gauge fixing, the procedure yields a semi-exact Hamiltonian for $\zeta$ that only gets corrections from $\zeta$-$\gamma_{ij}$ interactions and hence becomes exact when one truncates the system by setting $\gamma_{ij}$ (and the corresponding momentum field) to zero.

As in the original derivation given in mal , here also the auxiliary variables coming from the momentum constraint contain non-local expressions involving the Green function of the spatial Laplacian. In this work, we will elaborate on the importance of boundary conditions in obtaining a long wavelength limit or a derivative expansion. As we will see, in the naive long wavelength limit one should treat the momentum constraint exactly since all the auxiliary terms it generates in the action have derivative dimension zero. Nevertheless, we show that the zeroth order derivative expansion can be related to the minisuperspace approximation when suitable boundary conditions are imposed. We determine the late time form of the exact Hamiltonian and obtain simple non-trivial classical solutions of the $\zeta$ zero-mode corresponding to appropriate initial conditions. Our exact treatment reveals how the so called infrared logarithms emerge in the evolution of $\zeta$, which has been a subtle point and a source of discussion in the literature, see e.g. zz2 ; zz1 . We observe that the exact $\zeta$ Hamiltonian contains a linear term that explicitly generates $\ln(a)$ dependence for $\zeta$, however this term is canceled out when one makes a perturbative expansion in the number of fields, hence it is only effective non-perturbatively. On the other hand, the nonlinear evolution of $\zeta$ in the minisuperspace model involves highly nontrivial dynamics depending on the initial conditions and $\ln(a)$ type of time dependence in some cases.

We consider general relativity in the presence of a minimally coupled self-interacting real scalar field $\phi$ that has the potential $V(\phi)$. When the metric is taken in the ADM form

$$ds^{2}=-N^{2}dt^{2}+h_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt),$$

(1)

the Einstein-Hilbert action can be written as (we set $8\pi G=1$)

$$S=\int dt\,d^{3}x\left[\Pi^{ij}\dot{h}_{ij}+P_{\phi}\dot{\phi}-N\Phi-N^{i}\Phi_{i}\right],$$

(2)

	$\displaystyle\Phi=\frac{2}{\sqrt{h}}\left[\Pi^{ij}\Pi_{ij}-\frac{1}{2}\Pi^{2}\right]+\frac{1}{2\sqrt{h}}P_{\phi}^{2}+\sqrt{h}\left[V(\phi)+\frac{1}{2}h^{ij}\partial_{i}\phi\partial_{j}\phi-\frac{1}{2}R^{(3)}\right],$
	$\displaystyle\Phi_{i}=-2\sqrt{h}D_{j}\left[\frac{1}{\sqrt{h}}\Pi^{j}{}_{i}\right]+P_{\phi}\partial_{i}\phi,$		(3)

where the dot denotes time derivative, $D_{i}$ is the covariant derivative and $R^{(3)}$ is the Ricci scalar of $h_{ij}$, $h=\det(h_{ij})$ and $\Pi=\Pi^{ij}h_{ij}$. The spatial indices (like $i$ and $j$) are manipulated by $h_{ij}$. In the classical theory, the canonical pairs obey the Poisson brackets

	$\displaystyle\left\{h_{ij}(\vec{x}),\Pi^{rs}(\vec{y})\right\}=\frac{1}{2}\left(\delta^{r}_{i}\delta^{s}_{j}+\delta^{r}_{j}\delta^{s}_{i}\right)\delta^{3}(\vec{x}-\vec{y}),$
	$\displaystyle\left\{\phi(\vec{x}),P_{\phi}(\vec{y})\right\}=\delta^{3}(\vec{x}-\vec{y}).$		(4)

The lapse $N$ and the shift $N^{i}$ are Lagrange multipliers enforcing the Hamiltonian and the momentum constraints; $\Phi=0$ and $\Phi_{i}=0$.

We introduce the following decomposition of the spatial metric variable

$$h_{ij}=e^{2\hat{\zeta}}\,\hat{\gamma}_{ij},$$

(5)

where $\det\hat{\gamma}_{ij}=1$ (and thus $h=e^{6\hat{\zeta}}$). Similarly, the conjugate momenta can be decomposed into the trace and the trace-free parts as

$$\Pi^{ij}=\frac{1}{6}h^{ij}\,\hat{\pi}+e^{-2\hat{\zeta}}\,\hat{\sigma}^{ij},$$

(6)

where $\hat{\sigma}^{ij}h_{ij}=0$. In these new variables the action becomes

$$S=\int dt\,d^{3}x\left[\hat{\pi}\dot{\hat{\zeta}}+\hat{\sigma}^{ij}\dot{\hat{\gamma}}_{ij}+P_{\phi}\dot{\phi}-N\Phi-N^{i}\Phi_{i}\right],$$

(7)

Note that $(\hat{\zeta},\hat{\pi})$ and $(\hat{\gamma}_{ij},\hat{\sigma}^{kl})$ are conjugate variables, and one can as usual denote $N^{\mu}=(N,N^{i})$ and $\Phi_{\mu}=(\Phi,\Phi_{i})$.

Assume now that there is a cosmological Friedmann-Robertson-Walker (FRW) type solution given by the scale factor $a(t)$ and the scalar field $\phi(t)$, and furthermore $N=1$ and $N^{i}=0$. The background equations of motion

	$\displaystyle 3\frac{\dot{a}^{2}}{a^{2}}=\frac{1}{2}\dot{\phi}^{2}+V,$
	$\displaystyle\frac{\ddot{a}}{a}=-\frac{1}{3}\dot{\phi}^{2}+\frac{1}{3}V,$		(8)
	$\displaystyle\ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}+\frac{\partial V}{\partial\phi}=0,$

are thus assumed to hold. The cosmological perturbations can be introduced as fluctuations over the classical background

	$\displaystyle\hat{\zeta}=\ln a(t)+\zeta,$
	$\displaystyle\phi=\phi(t)+\varphi,$
	$\displaystyle\hat{\gamma}_{ij}=\delta_{ij}+\gamma_{ij},$
	$\displaystyle\hat{\pi}=-6a^{2}\dot{a}+\pi,$
	$\displaystyle P_{\phi}=a^{3}\dot{\phi}+p_{\varphi},$		(9)
	$\displaystyle\hat{\sigma}^{ij}=\sigma^{ij},$
	$\displaystyle N=1+n,$
	$\displaystyle N^{i}=n^{i}.$

The main variables are given by $(\zeta,\varphi,\gamma_{ij},\pi,p_{\varphi},\sigma^{ij})$ and $(n,n^{i})$ are Lagrange multipliers (one must recall that $\det(\hat{\gamma})=\det(\delta_{ij}+\gamma_{ij})=1$ and $\hat{\gamma}_{ij}\sigma^{ij}=0$). From (4) one can determine the Poisson brackets of these variables as

	$\displaystyle\{\zeta(\vec{x}),\pi(\vec{y})\}=\delta^{3}(\vec{x}-\vec{y}),$
	$\displaystyle\{\gamma_{ij}(\vec{x}),\sigma^{kl}(\vec{y})\}=\frac{1}{2}\left(\delta^{k}_{i}\delta^{l}_{j}+\delta^{l}_{i}\delta^{k}_{j}\right)\delta^{3}(\vec{x}-\vec{y})-\frac{1}{3}\hat{\gamma}_{ij}(\vec{x})\hat{\gamma}^{kl}(\vec{y})\delta^{3}(\vec{x}-\vec{y}),$		(10)
	$\displaystyle\{\sigma^{ij}(\vec{x}),\sigma^{kl}(\vec{y})\}=\frac{1}{3}\hat{\gamma}^{kl}(\vec{x})\sigma^{ij}(\vec{y})\delta^{3}(\vec{x}-\vec{y})-\frac{1}{3}\hat{\gamma}^{ij}(\vec{x})\sigma^{kl}(\vec{y})\delta^{3}(\vec{x}-\vec{y}),$

where $\hat{\gamma}^{ij}$ is the inverse of $\hat{\gamma}_{ij}$, and all other brackets vanish; in particular one has

$$\{\gamma_{ij}(\vec{x}),\pi(\vec{y})\}=0,\hskip 8.53581pt\{\zeta(\vec{x}),\sigma^{ij}(\vec{y})\}=0,\hskip 8.53581pt\{\pi(\vec{x}),\sigma^{ij}(\vec{y})\}=0.$$

(11)

Using the expansion (9) in the action, one obtains

$$S=\int dt\,d^{3}x\left[\pi\dot{\zeta}+\sigma^{ij}\dot{\gamma}_{ij}+p_{\varphi}\dot{\varphi}-\Phi^{(\geq 2)}-n\left(\Phi^{(1)}+\Phi^{(\geq 2)}\right)-n^{i}\Phi_{i}^{(\geq 1)}\right],$$

(12)

where we define

	$\displaystyle\Phi=\Phi^{(1)}+\Phi^{(\geq 2)},$
	$\displaystyle\Phi_{i}=\Phi_{i}^{(\geq 1)}.$		(13)

Here, we group the terms in the constraints according to their order; $\Phi^{(1)}$ is linear in the perturbations and $\Phi^{(\geq 2)}$ has quadratic and higher order terms. Similarly, $\Phi_{i}^{(\geq 1)}$ contains linear and higher order terms. Note that the zeroth order values of the constraints, which can be denoted by $\Phi^{(0)}$ and $\Phi_{i}^{(0)}$, vanish by the background equations of motion. By the same reason, the linear fluctuation terms in (12) also cancel each other.

The action (12) governing the dynamics of the perturbations $(\zeta,\varphi,\gamma_{ij},\pi,p_{\varphi},\sigma^{ij})$ has the Hamiltonian (density)¹¹1With some abuse of language, here and below we call $H$ to be the Hamiltonian, although it is actually the Hamiltonian density while the Hamiltonian is given by the integral $\int d^{3}x\,H$. $H=\Phi^{(\geq 2)}$ and four constraints $\Phi^{(1)}+\Phi^{(\geq 2)}=0$ and $\Phi_{i}^{(\geq 1)}=0$, where $(n,n^{i})$ are Lagrange multipliers. We see that on the constraint surface $\Phi^{(1)}+\Phi^{(\geq 2)}=0$, the Hamiltonian simplifies, i.e it only contains the ”linear terms”

$$H=-\Phi^{(1)}.$$

(14)

Of course, this is an apparently false oversimplification since the constraint surface must also be sliced by gauge fixing conditions. Still, the slicing can be done in such a way that the Hamiltonian can be determined exactly. Indeed, in the $\varphi=0$ gauge one can explicitly determine $p_{\varphi}$ from the Hamiltonian constraint $\Phi^{(1)}+\Phi^{(\geq 2)}=0$ as this is simply a quadratic polynomial in $p_{\varphi}$. In this way the system can be deparametrized; setting $\varphi=0$ and determining $p_{\varphi}$ in terms of other variables correspond to choosing a time gauge. As we will discuss in the next section, the momentum constraint $\Phi_{i}^{(\geq 1)}=0$ can be dealt with perturbatively in $\gamma_{ij}$ and $\sigma^{ij}$ while keeping $\zeta$ and $\pi$ dependence exact.

Although our main interest in this paper is cosmology, one can generalize the above construction to any solution in general relativity (or to other theories with first class constraints and Lagrange multipliers). One can denote the background values and the fluctuations generically as $Q_{i}=\bar{Q}_{i}+\delta Q_{i}$, $P_{i}=\bar{P}_{i}+\delta P_{i}$ and $N^{\mu}=\bar{N}^{\mu}+\delta N^{\mu}$, where $N^{\mu}$ collectively denotes the lapse and the shift (or Lagrange multipliers in the theory which do not have conjugate momenta). The constraints can be expanded in the fluctuation fields as above $\Phi_{\mu}=\Phi_{\mu}^{(1)}+\Phi_{\mu}^{(\geq 2)}$. Using these expansions in the action should yield

$$S=\int d^{4}x\left[\delta P_{i}\delta\dot{Q}_{i}\ -\bar{N}^{\mu}\Phi_{\mu}^{(\geq 2)}-\delta N^{\mu}\left(\Phi_{\mu}^{(1)}+\Phi_{\mu}^{(\geq 2)}\right)\right],$$

(15)

where the linear terms cancel and $\Phi_{\mu}^{(0)}=0$ since the background is assumed to be a solution. Then, the Hamiltonian for the fluctuations becomes $H=\bar{N}^{\mu}\Phi_{\mu}^{(\geq 2)}$ and the new constraints imposed by $\delta N^{\mu}$ field equations read $\Phi_{\mu}^{(1)}+\Phi_{\mu}^{(\geq 2)}=0$. Again, the Hamiltonian on the constraint surface simplifies to

$$H=-\bar{N}^{\mu}\,\Phi_{\mu}^{(1)}.$$

(16)

Note that for a generic background, the new Hamiltonian (16) gets contributions from both the original Hamiltonian and the momentum constraints ($\Phi_{0}$ and $\Phi_{i}$), while in the cosmological case (14), only $\Phi_{0}$ contributes to $H$ since $\bar{N}^{i}=0$.

As we will see below, the rest of the computation depends on how one solves the constraints and imposes the gauge fixing to identify a set of basic physical degrees of freedom (in our case, these would be $\zeta$, transverse-traceless $\gamma_{ij}$ and their conjugate momenta) and to determine the rest of the variables in terms of this basic set. Usually this is done perturbatively as first performed in the original work mal . Here, we would like to go beyond the perturbation theory, therefore we will try to solve the equation $\Phi^{(1)}+\Phi^{(\geq 2)}=0$ exactly without assuming any order between the terms. Of course, among different ways of solving this equation we would like to fix $\Phi^{(1)}$ in terms of the main physical variables we identified since this determines the Hamiltonian (14). One may see (19) below to observe how this works out, where $p_{\varphi}$, which is in $\Phi^{(1)}$, is determined in terms of other variables; corresponding to the equation $\Phi^{(1)}=-\Phi^{(\geq 2)}$.

We now carry out the above computation explicitly. In terms of the variables introduced in (9), the Hamiltonian constraint becomes

	$\displaystyle\Phi=$		$\displaystyle\frac{2e^{-3\zeta}}{a^{3}}\left[-\frac{1}{24}(\pi-6a^{2}\dot{a})^{2}+\sigma^{ij}\sigma^{kl}\hat{\gamma}_{ik}\hat{\gamma}_{jl}\right]+\frac{e^{-3\zeta}}{2a^{3}}\left[p_{\varphi}+a^{3}\dot{\phi}\right]^{2}$		(17)
			$\displaystyle+a^{3}e^{3\zeta}\left[V(\phi(t)+\varphi)+\frac{e^{-2\zeta}}{2a^{2}}\hat{\gamma}^{ij}\partial_{i}\varphi\partial_{j}\varphi-\frac{1}{2}R^{(3)}\right].$

The momentum $p_{\varphi}$ can be solved exactly from $\Phi=0$ as

$$p_{\varphi}=-a^{3}\dot{\phi}\pm\left[\frac{1}{6}(\pi-6a^{2}\dot{a})^{2}-4\sigma^{ij}\sigma^{kl}\hat{\gamma}_{ik}\hat{\gamma}_{jl}-2a^{6}e^{6\zeta}\left(V(\phi(t)+\varphi)+\frac{e^{-2\zeta}}{2a^{2}}\hat{\gamma}^{ij}\partial_{i}\varphi\partial_{j}\varphi-\frac{1}{2}R^{(3)}\right)\right]^{1/2},$$

(18)

where $\pm$ corresponds to the two different roots of the quadratic equation. This ambiguity is precisely related the so called notorious problem of time that arises in the Hamiltonian formulation of general relativity. To solve it, we first assume $\dot{\phi}<0$ in the classical solution, which is the case for the slow-roll inflation, and demand $p_{\varphi}$ vanishes when other perturbations are turned off, which implies

$$p_{\varphi}=-a^{3}\dot{\phi}-\left[\frac{1}{6}(\pi-6a^{2}\dot{a})^{2}-4\sigma^{ij}\sigma^{kl}\hat{\gamma}_{ik}\hat{\gamma}_{jl}-2a^{6}e^{6\zeta}\left(V(\phi(t)+\varphi)+\frac{e^{-2\zeta}}{2a^{2}}\hat{\gamma}^{ij}\partial_{i}\varphi\partial_{j}\varphi-\frac{1}{2}R^{(3)}\right)\right]^{1/2}.$$

(19)

Therefore, the problem of time is solved by referring to the specific (inflationary) background. Note that (19) is valid as long as the square root is meaningful, which is the only restriction (that is automatically satisfied in perturbation theory). We will always assume that (19) is well defined although obviously one can imagine field configurations where the term in the square root becomes negative. In that case one should revise the solution (19) and reconstruct the Hamiltonian below.

As discussed in the previous section, one also needs the linearized constraint $\Phi^{(1)}$ that gives the Hamiltonian (14). For that one may note

$$R^{(3)}=\frac{e^{-2\zeta}}{a^{2}}\left[R^{(3)}(\hat{\gamma})-4\hat{\gamma}^{ij}\hat{\nabla}_{i}\hat{\nabla}_{j}\zeta-2\hat{\gamma}^{ij}\partial_{i}\zeta\partial_{j}\zeta\right],$$

(20)

where $R^{(3)}(\hat{\gamma})$ and $\hat{\nabla}$ are the Ricci scalar and the covariant derivative of $\hat{\gamma}_{ij}$. A straightforward calculation then gives

$$\Phi^{(1)}=-\frac{1}{2}\,a\,\partial_{i}\partial_{j}\gamma_{ij}+2\,a\,\partial^{2}\zeta+6Va^{3}\zeta+a^{3}\,\frac{\partial V}{\partial\phi}\,\varphi+\dot{\phi}p_{\varphi}+\frac{\dot{a}}{a}\pi,\\$$

(21)

where $V$ and $\partial V/\partial\phi$ are functions evaluated on the background field $\phi(t)$, e.g. $V=V(\phi(t))$.

At this point one can impose the gauge

$$\varphi=0,$$

(22)

and use (14), (19) and (21) to obtain

	$\displaystyle H=$		$\displaystyle a^{3}\dot{\phi}^{2}+\frac{1}{2}\,a\,\partial_{i}\partial_{j}\gamma_{ij}-2\,a\,\partial^{2}\zeta-6Va^{3}\zeta-\frac{\dot{a}}{a}\pi$		(23)
			$\displaystyle+\dot{\phi}\left[a^{6}\dot{\phi}^{2}+\frac{1}{6}\pi^{2}-2a^{2}\dot{a}\pi-4\sigma^{ij}\sigma^{kl}\hat{\gamma}_{ik}\hat{\gamma}_{jl}-2a^{6}V(e^{6\zeta}-1)+a^{6}e^{6\zeta}R^{(3)}\right]^{1/2}.$

One may expand this Hamiltonian in the fluctuation fields to see that the zeroth and the first order terms cancel each other in $H$, as it should be.

It is important to emphasize that the gauge fixing (22) breaks down when the scalar background obeys $\dot{\phi}=0$ (yet it is possible to eliminate $\varphi$ and $p_{\varphi}$ via other smooth gauges which are valid even when $\dot{\phi}=0$, see ak1 ). In many models $\dot{\phi}=0$ happens repeatedly while the scalar oscillates about the minimum of its potential during rehating. In that case the expansion of the square-root in (23) around $a^{6}\dot{\phi}^{2}$ fails, which is directly related to the breakdown of the gauge fixing condition (22).

To proceed one should work out the momentum constraint. In terms of the fluctuation fields introduced in (9), one can see that the momentum constraint (3) becomes

$$\hat{\gamma}_{ik}\hat{\nabla}_{j}\sigma^{kj}=-\frac{1}{6}\partial_{i}\pi+\frac{1}{2}(-6a^{2}\dot{a}+\pi)\partial_{i}\zeta,$$

(24)

where we have also utilized the gauge (22) (recall that $\hat{\nabla}$ is the derivative operator of $\hat{\gamma}_{ij}$). We would like to underline that no approximation is done in obtaining (24). The momentum tensor $\sigma^{ij}$ can be decomposed as

$$\sigma^{ij}=\sigma^{ij}_{TT}+\hat{\nabla}^{i}\sigma^{j}_{T}+\hat{\nabla}^{j}\sigma^{i}_{T}+\hat{\nabla}^{i}\hat{\nabla}^{j}\sigma-\frac{1}{3}\hat{\gamma}^{ij}\hat{\nabla}^{k}\hat{\nabla}_{k}\sigma;$$

(25)

where $\hat{\nabla}_{i}\sigma^{ij}_{TT}=0$ and $\hat{\nabla}_{i}\sigma^{i}_{T}=0$. One can see that $\sigma^{ij}_{TT}$ drops out from the constraint equation (24) and thus it becomes the physical momentum conjugate to $\gamma_{ij}$. The auxiliary fields $\sigma_{T}^{i}$ and $\sigma$ can (in principle) be determined from (24) uniquely once appropriate boundary conditions are imposed. The evolution will be completely fixed once a corresponding gauge condition is imposed. We set

$$\partial_{i}\gamma_{ij}=0,$$

(26)

which is the standard choice for the tensor field.

The conditions (22) and (26) fix the coordinate gauge freedom completely. Since we are trying to make an exact computation, extra care is needed when switching to a different gauge choice. In the linear theory, the gauge transformations related to coordinate changes are given by the Lie derivative of the background. In an exact treatment, this should be modified to include all higher order terms, which can be expressed as the exponentiation of the Lie derivative, see e.g. revmalik .

Obviously, it is difficult if not impossible to use (35) in an exact way in the quantum theory. Yet (35) is still useful since one can expand it to obtain all interactions at any given order. It is important to note that the quantum field theory of cosmological perturbations has issues, maybe the less severe one being the ordering ambiguity. The theory is also non-renormalizable (since it involves gravity) and there is no natural way of dealing with loop infinities; one simply subtracts, somehow arbitrarily, the large/infinite contributions to estimate the regularized loop corrections. On the other hand, (35) can be useful in determining the non-perturbative classical dynamics; especially the evolution of the long-wavelength superhorizon modes which are known to become classical to all orders cls . However, there are subtleties in using (35) because of the non-local expressions involving $\partial^{-2}$ given in (30). In a naive derivative expansion of the action, the spatial Ricci scalar $R^{(3)}$ can be neglected but $\sigma^{ij}$ has derivative order zero, see (41) and (42), hence one should deal with the momentum constraint exactly if no other approximation is utilized.