## 4.4 Scalar and tensor interactions

So far, we have exclusively examined scalar interactions, but it is instructive to apply our formalism to tensor interactions as well. Therefore, we extend the results of the previous section to encompass tensor perturbations. We present the cubic Hamiltonian for tensors and mixed scalar-tensor interactions. Our findings can be used for calculating correlation functions involving tensor perturbations, such as in computations related to scalar-induced gravitational waves [52] and gravitational wave anisotropies [20]. Additionally, we highlight an intriguing aspect of our approach—a nontrivial contribution of tensor perturbations to scalar interactions and vice versa, beginning at quartic order in interactions.

Including tensor perturbations, the spatial metric in the comoving gauge takes the following form:

$$h_{ij} = a^2 e^{2\zeta} (e^{\gamma})_{ij}, \qquad (4.75)$$

where the tensor perturbation $\gamma_{ij}$ is transverse and traceless, i.e. $\partial_i \gamma_{ij} = \gamma_{ii} = 0$. Let us note that (4.75) is just a shorthand and it should be understood as follows

$$h_{ij} = a^2 e^{2\zeta} (e^{\gamma})_{ij} = a^2 e^{2\zeta} \left( \delta_{ij} + \gamma_{ij} + \frac{1}{2}\gamma_{il}\gamma_{lj} + \frac{1}{6}\gamma_{il}\gamma_{lm}\gamma_{mj} + \dots \right). \qquad (4.76)$$

Using the exponential of $\gamma_{ij}$ to carry out an analysis for the tensor sector valid at all orders in perturbation theory is not as straightforward as for scalar perturbations, so that we need to use the expression (4.76) instead. Plugging it into the action (4.5) we get:

$$\mathcal{L} = \mathcal{L}_{\text{scals}} + \mathcal{L}_{\gamma} + \mathcal{L}_{\gamma\gamma} + \mathcal{L}_{\gamma\gamma\gamma} + \mathcal{L}_{\gamma\gamma\gamma\gamma} + \dots, \qquad (4.77)$$

where $\mathcal{L}_{\text{scals}}$ is given by the scalar Lagrangian in eq. (4.24), $\mathcal{L}_{\gamma}$, $\mathcal{L}_{\gamma\gamma}$, $\mathcal{L}_{\gamma\gamma\gamma}$, $\mathcal{L}_{\gamma\gamma\gamma\gamma}$ contain 1, 2, 3 and 4 powers of $\gamma_{ij}$ or its spatial or time derivatives as well as other powers of the scalar perturbation $\zeta$. Explicit expressions for each term in eq. (4.77) are provided in the appendix C. From such expressions, we can derive the on-shell conjugate momenta $\bar{p}_X \equiv \delta L/\delta \dot{X}$:

$$\bar{p}_{\zeta} = M_{\text{Pl}}^2 \left( -\frac{6a^3 e^{3\zeta} \dot{\zeta}}{N} + \frac{2ae^{\zeta} \partial_i N_i}{N} + \frac{2ae^{\zeta} N_i \partial_i \zeta}{N} - \frac{6a^3 e^{3\zeta} H}{N} - \frac{2ae^{\zeta} \partial_j N_i \gamma_{ij}}{N} \right) + \dots, \qquad (4.78)$$

$$\bar{p}_{ij} = M_{\text{Pl}}^2 \left( \frac{a^3 e^{3\zeta} \dot{\gamma}_{ij}}{4N} - \frac{ae^{\zeta} N_i \partial_j N}{2N^2} + \frac{3ae^{\zeta} N_i \partial_j \zeta}{2N} - \frac{ae^{\zeta} N_k \partial_k \gamma_{ij}}{4N} + \frac{ae^{\zeta} N_k \partial_j \gamma_{ik}}{2N} \right. \\ \left. + \frac{ae^{\zeta} \partial_k N_j \gamma_{ik}}{4N} + \frac{ae^{\zeta} \partial_j N_k \gamma_{ik}}{4N} \right)_{\text{symm}} + \dots, \qquad (4.79)$$

where the subscript “symm” denotes symmetrization with respect to the indices $i, j$, i.e. $A_{ij|\text{symm}} \equiv (A_{ij} + A_{ji})/2$, and the conjugate momenta to $N$ and $N_i$ are still zero. The dots denote cubic order terms in the expressions of the momenta $\bar{p}_{\zeta}$ and $\bar{p}_{ij}$, and only become relevant for the computation of quintic or higher order interactions, which we are not concerned with in this paper.

We immediately notice that the momentum $\bar{p}_{\zeta}$ differs from that in eq. (4.25) by the term $-2a M_{\text{Pl}}^2 e^{\zeta} \partial_j N_i \gamma_{ij} / N$. Furthermore, the tensor momentum $\bar{p}_{ij}$ receives contributions
from products of scalar perturbations. This fact will play a crucial role in deriving the quartic Hamiltonian in phase space. For example, the kinetic terms ~ $\dot{\zeta}^2$ and ~ $(\dot{\gamma}_{ij})^2$ will also generate purely tensor and scalar quartic interactions respectively, as well as mixed scalar-tensor quartic interactions. We will come back to this matter in section 4.5.

Using eqs. (4.77), (4.78) and (4.79), we can perform the Legendre transform, followed by the canonical transformation (4.30) arrive at the Hamiltonian. Up to cubic order, the Hamiltonian consists in the following contributions:

$$ \mathcal{H} = \mathcal{H}_{\text{scals}}^{(2)} + \mathcal{H}_{\text{tensors}}^{(2)} + \mathcal{H}_{\zeta\zeta\zeta}^{(3)} + \mathcal{H}_{\zeta\zeta\gamma}^{(3)} + \mathcal{H}_{\gamma\gamma\gamma}^{(3)} + \mathcal{H}_{\gamma\gamma\zeta}^{(3)} \quad (4.80) $$

where the purely scalar interactions $\mathcal{H}_{\text{scals}}^{(2)}$ and $\mathcal{H}_{\zeta\zeta\zeta}^{(3)}$ are given by eqs. (4.45) and (4.47) respectively. The evolution of the interaction picture tensor degrees of freedom is often set to be governed by their quadratic Hamiltonian, which reads:

$$ \mathcal{H}_{\text{tensors}}^{(2)} = \frac{2p_{ij}^2}{a^3 M_{\text{Pl}}^2} + \frac{M_{\text{Pl}}^2}{8} a (\partial_k \gamma_{ij})^2, \quad (4.81) $$

in which case $p_{ij}^I = a^3 M_{\text{Pl}}^2 \dot{\gamma}_{ij}^I$. The cubic interactions involving tensors take the following form

$$ \mathcal{H}_{\zeta\zeta\gamma}^{(3)} = \frac{6\zeta p_{ij} \partial_i \partial_j \theta^{(1)}}{a^2} - \frac{p_\zeta p_{ij} \partial_i \partial_j \theta^{(1)}}{a^5 H \epsilon M_{\text{Pl}}^2} - \frac{M_{\text{Pl}}^2}{2a} \theta^{(1)} \partial_i \partial_j \partial_k \theta^{(1)} \partial_k \gamma_{ij} - a M_{\text{Pl}}^2 \zeta \partial_i \partial_j \zeta \gamma_{ij} \\ - \frac{p_\zeta \partial_i \partial_j \zeta \gamma_{ij}}{a^2 H \epsilon} \quad (4.82) $$

$$ \mathcal{H}_{\zeta\gamma\gamma}^{(3)} = -\frac{6\zeta p_{ij}^2}{a^3 M_{\text{Pl}}^2} + \frac{p_\zeta p_{ij}^2}{a^6 H \epsilon M_{\text{Pl}}^4} + \frac{p_{ij} \partial_k \theta^{(1)} \partial_k \gamma_{ij}}{a^2} + \frac{M_{\text{Pl}}^2}{8} a \zeta (\partial_k \gamma_{ij})^2 + \frac{p_\zeta (\partial_k \gamma_{ij})^2}{16a^2 H \epsilon} \quad (4.83) $$

$$ \mathcal{H}_{\gamma\gamma\gamma}^{(3)} = -\frac{a M_{\text{Pl}}^2}{4} \partial_l \gamma_{ik} \partial_k \gamma_{jl} \gamma_{ij} + \frac{a M_{\text{Pl}}^2}{8} \partial_k \partial_l \gamma_{ij} \gamma_{kl}. \quad (4.84) $$

As before, we need to perform a canonical transformation to simplify the cubic Hamiltonian and make the slow-roll suppression manifest. We arrive at the following generating function:

$$ F[\tilde{p}_{\zeta}, \zeta, \tilde{p}_{ij}, \gamma_{ij}, t] = \tilde{p}_{\zeta} \zeta + \tilde{p}_{ij} \gamma_{ij} \quad (4.85) \\ - \frac{1}{4a^3 \epsilon H M_{\text{Pl}}^2} \tilde{p}_{\zeta}^2 \zeta - \frac{a \epsilon M_{\text{Pl}}^2}{H} \zeta (\partial_{\zeta})^2 - \frac{M_{\text{Pl}}^2}{6aH^3} \zeta [\zeta_{ij} \zeta_{ij} - (\partial^2 \zeta)^2] \\ + \frac{M_{\text{Pl}}^2}{2aH^2} \zeta [\zeta_{ij} \tilde{\chi}_{ij} - \partial^2 \zeta \partial^2 \tilde{\chi}] - \frac{M_{\text{Pl}}^2}{2aH} \zeta [\tilde{\chi}_{ij} \tilde{\chi}_{ij} - (\partial^2 \tilde{\chi})^2] \\ - \frac{a M_{\text{Pl}}^2}{H} \partial_i \zeta \partial_j \zeta \gamma_{ij} + \frac{1}{a^2 H^2} \partial_i \zeta \partial_j \zeta \tilde{p}_{ij} + \frac{2}{a^2 H M_{\text{Pl}}^2} \tilde{\chi} \partial_i \zeta \tilde{p}_{ij} \\ - \frac{a M_{\text{Pl}}^2}{8H} \zeta (\partial_l \gamma_{ij})^2 - \frac{2}{a^3 H M_{\text{Pl}}^2} \zeta \tilde{p}_{ij}^2. $$

The second and third lines of the generating functions are the same as in eq. (4.57), and only affect $\mathcal{H}_{\zeta\zeta\zeta}^{(3)}$, recasting it in the form (4.68). Terms in the fourth and fifth lines are new. Like in the scalar case, when evaluated on interaction picture fields, the fourth line turns out to be equal to the argument of total time derivative terms in the Lagrangian, as can be easily checked by comparing with ref. [30], where total time derivative tensor interactions were highlighted for the first time.
From the generating function, we can derive the relations between the new and the old variables:

$$ \tilde{\zeta} = \text{Eq. (4.58)} - \frac{1}{a^3 H} \partial^{-2} (\partial_i \partial_j \zeta \tilde{p}_{ij}) \quad (4.86) $$

$$ \begin{aligned} p_{\zeta} = \text{Eq. (4.59)} &+ \frac{2aM_{\text{Pl}}^2}{H} \partial_i \partial_j \zeta \gamma_{ij} - \frac{2}{a^2 H^2} \partial_j (\partial_i \zeta \tilde{p}_{ij}) + \frac{2}{a^2 H M_{\text{Pl}}^2} \partial_i \partial_j (\tilde{\chi} \tilde{p}_{ij}) \\ &- \frac{aM_{\text{Pl}}^2}{8H} (\partial_k \gamma_{ij})^2 - \frac{2}{a^3 H M_{\text{Pl}}^2} \tilde{p}_{ij}^2, \end{aligned} \quad (4.87) $$

$$ \tilde{\gamma}_{ij} = \gamma_{ij} + \frac{M_{\text{Pl}}^2}{a^2 H^2} \partial_i \zeta \partial_j \zeta + \frac{2}{a^2 H} \tilde{\chi} \partial_i \partial_j \zeta - \frac{4}{a^3 H} \zeta \tilde{p}_{ij}, \quad (4.88) $$

$$ p_{ij} = \tilde{p}_{ij} - \frac{a M_{\text{Pl}}^2}{H} \partial_i \zeta \partial_j \zeta + \frac{a M_{\text{Pl}}^2}{4H} \partial_k (\zeta \partial_k \gamma_{ij}). \quad (4.89) $$

We can invert the relations above perturbatively to find the old variables in terms of the new one, and compute the new Hamiltonian $\tilde{\mathcal{H}}$. As previously announced, we see that the canonical transformations mix scalar and tensor perturbations, which implies that correlation functions of the old scalar or tensor variables will be found from both new scalar and tensor variables. One may derive diagrammatic rules following the procedure outlined in appendix B to find the precise relations, although we do not do it here.

Using the transformations above, we simplify the tensor interactions:

$$ \begin{aligned} \tilde{\mathcal{H}}_{\zeta\zeta\gamma} = &-M_{\text{Pl}}^2 \left[ -\frac{2\epsilon}{a^2 M_{\text{Pl}}^2} \tilde{\chi} \partial_i \tilde{\zeta} \partial_j \tilde{\zeta} \tilde{p}_{ij} + \frac{1}{4aM_{\text{Pl}}^2} \partial_i \tilde{\chi} \partial_j \tilde{\chi} \partial^2 \tilde{\gamma}_{ij} + a\epsilon \partial_i \tilde{\zeta} \partial_j \tilde{\zeta} \tilde{\gamma}_{ij} \right], \quad (4.90) \\ &-\frac{1}{a^2} \left[ \frac{\epsilon}{H} \partial_j (\tilde{\zeta}^2) - \frac{2}{M_{\text{Pl}}^2} \partial_j \left( 3\tilde{\zeta} - \frac{\tilde{p}_{\zeta}}{2a^3 H \epsilon M_{\text{Pl}}^2} \right) \tilde{\chi} \right] \partial_i \tilde{p}_{ij}, \end{aligned} $$

$$ \begin{aligned} \tilde{\mathcal{H}}_{\zeta\gamma\gamma} = &-M_{\text{Pl}}^2 \left[ \frac{2\epsilon}{a^3 M_{\text{Pl}}^4} \tilde{\zeta} \tilde{p}_{ij}^2 - \frac{1}{a^2 M_{\text{Pl}}^4} \partial_l \tilde{\chi} \tilde{p}_{ij} \partial_l \tilde{\gamma}_{ij} + \frac{1}{8} a \epsilon \tilde{\zeta} (\partial_l \tilde{\gamma}_{ij})^2 \right] \\ &+ \frac{2}{a^2} \partial_k \tilde{\theta}^{(1)} \gamma_{jk} \partial_i \tilde{p}_{ij}, \end{aligned} \quad (4.91) $$

$$ \tilde{\mathcal{H}}_{\gamma\gamma\gamma} = -M_{\text{Pl}}^2 \left[ \frac{1}{4} a \partial_m \tilde{\gamma}_{il} \partial_l \tilde{\gamma}_{jm} \tilde{\gamma}_{ij} + \frac{1}{8} a \partial_i \tilde{\gamma}_{lm} \partial_j \tilde{\gamma}_{lm} \tilde{\gamma}_{ij} \right]. \quad (4.92) $$

Upon inserting the interaction picture fields $\tilde{\gamma}_{ij}^I$ and $\tilde{p}_{ij}^I = M_{\text{Pl}}^2 a^3 \tilde{\gamma}_{ij}^I$, we have that $\partial_i \tilde{p}_{ij}^I = 0$ and the equation above simply reduces to the bulk interactions found in ref. [30], as expected.

One may want to seek for an additional canonical transformation to remove the interactions proportional to $\partial_i \tilde{p}_{ij}$ so that the Hamiltonian do not contain at all such terms, even without evaluating it on interaction picture fields. This can be done by modifying the generating function as follows:

$$ F[\tilde{p}_{\zeta}, \zeta, \tilde{p}_{ij}, \gamma_{ij}, t] = \text{Eq. (4.85)} \quad (4.93) \\ + \frac{a M_{\text{Pl}}^2}{4} \gamma_{ij} \partial_i \left\{ 2 \partial_k \left[ \left( \frac{\tilde{\chi}}{M_{\text{Pl}}^2} - \frac{\zeta}{H} \right) \gamma_{jk} \right] - \left[ \partial_j \left( \frac{\epsilon \zeta^2}{H} \right) - 2 \partial_j \left( 3 \zeta - \frac{\tilde{p}_{\zeta}}{2a^3 H \epsilon M_{\text{Pl}}^2} \right) \right] \frac{\tilde{\chi}}{M_{\text{Pl}}^2} \right\}. $$
Adding this term only modifies the canonical transformation law for the tensor conjugate momentum as

$$p_{ij} = \text{Eq. (4.89)} \qquad (4.94) \\ - \frac{a M_{\text{Pl}}^2}{4} \left\{ -2 \partial_i \partial_k \left[ \left( \frac{\tilde{\chi}}{M_{\text{Pl}}^2} - \frac{\zeta}{H} \right) \gamma_{jk} \right] + \partial_i \left[ \partial_j \left( \frac{\epsilon \zeta^2}{H} \right) - 2 \partial_j \left( 3\zeta - \frac{\tilde{p}_\zeta}{2a^3 H \epsilon M_{\text{Pl}}^2} \right) \right] \frac{\tilde{\chi}}{M_{\text{Pl}}^2} \right\}$$

and does not induce any additional transformations for $\zeta$, $p_\zeta$ and $\gamma_{ij}$. Furthermore, eq. (4.93) is easily seen to be a total spatial derivative, and so is its partial time derivative. Therefore, the sole effect of adding this term is the removal of interactions proportional to $\partial_i \tilde{p}_{ij}$ in eqs. (4.90) and (4.92), and does not introduce additional terms.

## 4.5 Quartic Hamiltonian

Let us finally comment on the quartic Hamiltonian. Quartic vertices are relevant for the calculations of tree-level trispectra and loop-corrections to primordial correlators. The procedure to arrive at a manifestly slow-roll suppressed Hamiltonian is conceptually the same as outlined in the previous sections. However, the calculation is significantly more involved, due to the large number of SR unsuppressed terms that need to be removed via canonical transformations. For this reason, here we will simply outline the steps of the calculation, leaving a full simplification for future works. To show how the procedure works in practice, we apply it to a simplified situation in which the size of only a subset of interactions is much larger than the other ones, so that a simplification becomes feasible.

### 4.5.1 All contributions to the quartic Hamiltonian

Let us remind the reader that the quartic Hamiltonian is not obtained by simply expanding the original Hamiltonian up to fourth order in perturbations. Indeed, following the notation of section 4.3, the quartic Hamiltonian receives several contributions that look schematically as follows:

$$\tilde{\mathcal{H}}^{(4)} = \mathcal{H}^{(4)}(\tilde{\zeta}, \tilde{p}_{\tilde{\zeta}}, \tilde{\gamma}_{ij}, \tilde{p}_{ij}) + \Delta\tilde{\mathcal{H}}^{(4)\text{from (2)}} + \Delta\tilde{\mathcal{H}}^{(4)\text{from (3)}} + \Delta\tilde{\mathcal{H}}^{(4)\text{from F}}. \qquad (4.95)$$

We now explain the origin of each contribution in this equation, and clarify some subtleties associated to their calculation.

* $\mathcal{H}^{(4)}(\tilde{\zeta}, \tilde{p}_{\tilde{\zeta}}, \tilde{\gamma}_{ij}, \tilde{p}_{ij})$. This is the standard contribution from the Legendre transform, expanded at fourth order in perturbations, and written in terms of the new variables. Had we not performed the canonical transformation (4.93), this would be the final result for $\mathcal{H}^{(4)}$, and the starting point for subsequent simplifications. As well known, it is in general not equal to $-\mathcal{L}^{(4)}$ [36]. Computing this term is already quite demanding as, in general, the relation between the time derivative of the coordinate and its conjugate momentum is not always analytically invertible. One could need to invert it perturbatively, as explained in e.g. [37, 38]. However, in our case we get a simple analytical expression of $\dot{\zeta}$ and $\dot{\gamma}_{ij}$ in terms of their momenta (4.78) and (4.79), thereby significantly simplifying its calculation.
Let us note that varying the quartic Hamiltonian with respect to the second order lapse and shift we can get the second order constraint equations needed for this calculation. Actually, terms involving $\alpha^{(2)}$, and shift $N_i^{(2)}$ only appear in the first term in eq. (4.95), as all other ones are generated by the canonical transformation (4.93). Before moving on to the next contribution, let us finally stress that in order to obtain the correct $\mathcal{H}^{(4)}$ we have to take into account both scalar and tensor perturbations, as the conjugate scalar and tensor momenta in (4.78) and (4.79) contain second order tensor and scalar contributions, respectively.

* $\Delta \tilde{\mathcal{H}}^{(4) \text{ from } (2)}$. Inserting the expression for the old variables in terms of the new ones in the quadratic Hamiltonian we get a correction to $\tilde{\mathcal{H}}^{(4)}$, which can be calculated as:

$$ \Delta \tilde{\mathcal{H}}^{(4) \text{ from } (2)} = \sum_A \left. \frac{\delta \mathcal{H}^{(2)}}{\delta z^A} \right|_{\tilde{z}} (z^A)^{(3)} + \frac{1}{2} \sum_{A,B} \left. \frac{\delta^2 \mathcal{H}^{(2)}}{\delta z^A \delta z^B} \right|_{\tilde{z}} (z^A)^{(2)} (z^B)^{(2)}. \quad (4.96) $$

To derive the cubic contributions $(z^A)^{(3)}$, eqs. (4.86), (4.87), (4.88) and (4.94) need to be expanded to third order in the new $\tilde{z}$ variables. Let us also emphasize once more that it is crucial not to plug the solutions for the linear constraints into the quadratic Hamiltonian before performing the canonical transformation. The first term in eq. (4.96) yields the same contribution to $\Delta \tilde{\mathcal{H}}^{(4) \text{ from } (2)}$ irrespective of whether one starts from the off-shell quadratic Hamiltonian (4.38) or the on-shell one (4.45). However, this consistency does not extend to the second term in (4.96). Indeed, it can be easily checked that quartic interactions made up of products of two quadratic terms in the perturbative expansion (4.60) differ depending on whether one uses the off-shell or the on-shell Hamiltonians as starting points for the canonical transformation.

* $\Delta \tilde{\mathcal{H}}^{(4) \text{ from } (3)}$. Similarly, inserting the expression for the old variables in terms of the new ones in the cubic Hamiltonian we get the following correction to $\tilde{\mathcal{H}}^{(4)}$:

$$ \Delta \tilde{\mathcal{H}}^{(4) \text{ from } (3)} = \sum_A \left. \frac{\delta \mathcal{H}^{(3)}}{\delta z^A} \right|_{\tilde{z}} (z^A)^{(2)}. \quad (4.97) $$

Also in this case, the starting point for the simplification should be a priori the off-shell cubic Hamiltonian (4.46), before any on-shell solution for the linear constraints is enforced.

* $\Delta \tilde{\mathcal{H}}^{(4) \text{ from } F}$. Finally, there is a fourth contribution from plugging the canonical transformation into $\partial F / \partial t$:

$$ \Delta \tilde{\mathcal{H}}^{(4) \text{ from } F} = \left( \frac{\delta}{\delta \zeta} \frac{\partial F(\zeta, \gamma_{ij}, \tilde{p}_{\zeta}, \tilde{p}_{ij}, t)}{\partial t} \zeta^{(2)} + \frac{\delta}{\delta \gamma_{ij}} \frac{\partial F(\zeta, \gamma_{ij}, \tilde{p}_{\zeta}, \tilde{p}_{ij}, t)}{\partial t} \gamma_{ij}^{(2)} \right) \Big|_{\tilde{z}^A} . \quad (4.98) $$

We remind the reader that our type II generating function is already a function of the new momenta, so only the new coordinates need to be expanded perturbatively to calculate this contribution. It is important to first take the partial time derivative, and then only expand in terms of the new phase-space variables.
The process of extracting manifestly slow-roll suppressed quartic interactions from $\tilde{\mathcal{H}}^{(4)}$ is the same as in section 4.3. The difference lies in the much larger number of initial interactions to be simplified, which makes the simplification computationally demanding. Rather than performing all such kinds of simplifications, which goes beyond the scopes of this paper, we will focus on a subset of interactions to show how the algorithm described at the end of section 4.3 works in practice for quartic interactions.

### 4.5.2 Dominant interactions with a large $\eta$

In this section, we focus on a subset of the quartic interactions, that dominate whenever $\eta$ and its derivatives are large. This may be relevant for inflationary scenarios with a transient stage during which the usual slow-roll approximation breaks down, with higher order slow-roll parameters become of order one ($\epsilon_{i>1} \sim \mathcal{O}(1)$) while $\epsilon$ is still remaining small. We already encountered some interactions proportional to $\epsilon_2 = \eta$ in the simplified cubic Hamiltonian (4.46). Those are indeed the dominant cubic interactions in this regime, and we can estimate their sizes as, for example,<fnref n="8" />

$$ \text{Large } \eta \text{ cubic vertex: } \quad \left. \frac{\partial}{\partial \tilde{\zeta}} \right|_{\tilde{\zeta} \sim \tilde{\eta}} \sim \left[ \frac{\epsilon \eta a M_{\text{Pl}}^2 \tilde{\zeta} (\partial \tilde{\zeta})^2}{\tilde{\zeta}} \right] / \tilde{\zeta} \sim \eta. \qquad (4.99) $$

The other cubic interaction with $\eta$, schematically of the form $(\eta/\epsilon)\tilde{p}_{\zeta}^2\tilde{\zeta}$, contributes roughly the same order, although the precise relative size of these two contributions depends also on the details of the hierarchies between higher order slow-roll parameters $\epsilon_{i>1}$ and one, as encoded in the ratio $\dot{\zeta}/(H\zeta)$. We also stress already that in addition to these two leading interactions, the canonical transformation used to simplify the cubic order Hamiltonian may lead to non-negligible contributions in scenarios with large $\eta$ where the momentum $\tilde{p}_{\zeta}$ does not necessarily decay on super-horizon scales, in contrast to the usual slow-roll attractor scenario.

The lowest order correlation functions to which quartic interactions contribute are the tree-level trispetrum, and the 1-vertex correction to the 1-loop power spectrum. The same correlation functions will receive a contribution from two insertions of (4.99). We can use this to estimate the typical size of a quartic interactions as important as, or more important than, the cubic ones, as follows:

$$ \text{Large } \eta \text{ quartic vertex: } \quad \left. \frac{\partial}{\partial \tilde{\zeta}^2} \right|_{\tilde{\zeta} \sim \tilde{\eta}} \sim \left( \left. \frac{\partial}{\partial \tilde{\zeta}} \right|_{\tilde{\zeta}} \right)^2 \sim \eta^2. \qquad (4.100) $$

For example, a quartic Hamiltonian interaction of the form $a\epsilon\eta^2 M_{\text{Pl}}^2 \tilde{\zeta}^2 (\partial\tilde{\zeta})^2$ would parametrically contribute as much as the leading cubic interactions. Quartic interactions with an estimated size smaller than (4.100) would provide a subdominant contribution during a stage of $\eta \sim \mathcal{O}(1)$.

Let us now isolate such dominant interactions in $\tilde{\mathcal{H}}^{(4)}$. The Lagrangian for single-field inflation (4.5) does not contain any interactions with $\epsilon_{i>1}$ in its original form and the only way such interactions can appear in the Hamiltonian is by performing canonical transformations. There are therefore only two possibilities. Either such interactions are generated by the

<footnote n="8">⁸We estimate the size of interactions around the time of Hubble crossing using $dt \sim 1/H$, $a \sim k/H$, $\partial\zeta \sim k\zeta$, $p_\zeta \sim \epsilon a^3 M_{\text{Pl}}^2 \dot{\zeta}$ and $\zeta \sim H/(k^{3/2}\epsilon^{1/2}M_{\text{Pl}})$. How to relate $\dot{\zeta}$ to $\zeta$ depends on the details of this non-slow-roll phase.</footnote>
canonical transformation in (4.85) to simplify the cubic interactions, or they are generated by a new canonical transformation to render manifest the true size of quartic interactions.

Let us start by the former possibility, i.e. by inspecting the canonical transformations derived from the generating function (4.85). They only depend on the background quantities $(a, H, \epsilon)$, so that interactions involving $\eta$ can only be generated through the partial time derivative of the generating function, i.e. $\Delta\tilde{H}^{(4)}_{\text{from } F}$. Among such interactions, the ones least suppressed by $\epsilon \ll 1$ are given by:

$$ \tilde{\mathcal{H}}_{\eta, \text{dominant}}^{(4)} = \frac{\eta}{8a^6 H M_{\text{Pl}}^4 \epsilon^2} \tilde{p}_{\zeta}^3 \tilde{\zeta} + \frac{\eta}{2a^2 H} \tilde{p}_{\zeta} \tilde{\zeta} (\partial \tilde{\zeta})^2 + \frac{\eta}{a^2 H} \tilde{p}_{\zeta} \tilde{\zeta}^2 \partial^2 \tilde{\zeta} + \mathcal{H}_{\eta, \text{inv.deriv.}}^{(4)}, \quad (4.101) $$

with $\mathcal{H}_{\eta, \text{inv.deriv.}}^{(4)}$ involving interactions of the form $\epsilon\eta D(\tilde{\zeta}^4)$ and $(\eta/\epsilon)D(\tilde{\zeta}^2\tilde{p}_{\zeta}^2)$ with $D$ a complicated operator involving inverse spatial derivatives. Interactions above only contain one power of $\eta$, thus providing an apparently smaller size than our estimate in (4.100) when $\eta$ is large. However, as in section 4.3, we can perform a canonical transformation to incorporate the interactions not involving inverse spatial derivatives in (4.101) into the kinetic term for a new momentum variable. This will result in interactions of the form $\zeta^4$ or $p_{\zeta}^2\zeta^2$ but with additional factors of $\eta$ and its derivatives, which would decrease the size of the interaction in slow roll, while here it increases it. In practice, we define the following generating function:

$$ F_{\eta}[\tilde{\zeta}, \tilde{p}_{\zeta}, t] = \tilde{p}_{\zeta} \tilde{\zeta} - \frac{\eta}{8a^3 H M_{\text{Pl}}^2 \epsilon} \tilde{p}_{\zeta}^2 \tilde{\zeta}^2 - M_{\text{Pl}}^2 \frac{a\epsilon\eta}{6H} \tilde{\zeta}^3 \partial^2 \tilde{\zeta}, \quad (4.102) $$

from which we can derive the expressions relating old and new variables as follows

$$ \tilde{p}_{\zeta} = \tilde{p}_{\zeta} - \frac{\eta}{4a^3 H M_{\text{Pl}}^2 \epsilon} \tilde{p}_{\zeta}^2 \tilde{\zeta} - M_{\text{Pl}}^2 \frac{a\epsilon\eta}{H} \tilde{\zeta} (\partial \tilde{\zeta})^2 - M_{\text{Pl}}^2 \frac{a\epsilon\eta}{H} \tilde{\zeta}^2 \partial^2 \tilde{\zeta} \quad (4.103) $$

$$ \tilde{\zeta} = \tilde{\zeta} + \frac{\eta}{8a^3 H M_{\text{Pl}}^2 \epsilon} \tilde{p}_{\zeta} \tilde{\zeta}^2. \quad (4.104) $$

Let us note that, being the generating function quartic (besides the trivial term), not only does this canonical transformation leave the quadratic action invariant, but also the cubic one. Computing the new quartic Hamiltonian, and keeping only terms quadratic in $\eta$ and its logarithmic derivative we get:

$$ \tilde{\mathcal{H}}_{\eta^2, \text{dominant}}^{(4)} = \frac{\tilde{p}_{\zeta}^2 \tilde{\zeta}^2 \eta^2}{8a^3 M_{\text{Pl}}^2 \epsilon} - \frac{\tilde{p}_{\zeta}^2 \tilde{\zeta}^2 \eta \eta_2}{8a^3 M_{\text{Pl}}^2 \epsilon} + \frac{M_{\text{Pl}}^2}{2} a \epsilon \tilde{\zeta}^2 \eta^2 (\partial \tilde{\zeta})^2 + \frac{M_{\text{Pl}}^2}{2} a \epsilon \tilde{\zeta}^2 \eta \eta_2 (\partial \tilde{\zeta})^2, \quad (4.105) $$

where we have defined $\eta_2 \equiv \dot{\eta}/H\eta$. We have also consistently neglected $\mathcal{H}_{\eta, \text{inv.deriv.}}^{(4)}$ as it is less boosted by $\epsilon_{i>1}$ parameters. Interestingly, this expression exactly matches the dominant $\eta$ interactions derived using the EFT of inflation in ref. [53]. However, to be consistent when computing a given observable generated by such large quartic interactions, one should also consider contributions given by the canonical transformation relating correlation functions of $\zeta$ to those of $\tilde{\zeta}$. One can check that the only term containing $\eta$ is the one shown in eq. (4.104), leading to $\zeta \simeq \tilde{\zeta} - \eta/(8a^3\epsilon H M_{\text{Pl}}^2)\tilde{p}_{\zeta}\tilde{\zeta}^2$. Whether it can give a contribution to correlation functions of $\zeta$ as sizable as the dominant cubic and quartic interactions should be decided depending on the details of the modelling of the large $\eta$ phase, and in particular depending
on the behaviour of the momenta which are super-horizon and the time of evaluation of the correlation functions (see discussion around eq. (4.71)).

Next, let us explore the second potential source of large $\eta$ contributions, namely, additional canonical transformations aimed at elucidating the magnitude of the entire quartic Hamiltonian. As mentioned earlier, we do not undertake this entire calculation, as it is beyond the scope of this work. Instead, our goal is to present a general argument explaining why such transformations are unlikely to yield dominant contributions to the large $\eta$ interactions. Let $O_{p_{\zeta}^n \zeta^m}^{(i)}$ denote an interaction consisting solely of operators with $n$ powers of $p_{\zeta}$, $m$ powers of $\zeta$, and $i$ spatial derivatives, suitably contracted. Inverse Laplacian operators contribute with a factor of $-2$ to the sum for computing the index $i$. We find the leading contribution to our on-shell quartic Hamiltonian (4.95) to be superficially of slow-roll order $\epsilon^0$, and of the form:

$$ \tilde{\mathcal{H}}_{\epsilon^0}^{(4)} = \frac{1}{a^7 H^4 \epsilon^2 M_{\text{Pl}}^2} \mathcal{O}_{\tilde{p}_{\zeta}^2 \tilde{\zeta}^2}^{(4)} + \frac{1}{a^6 H^5 \epsilon} \mathcal{O}_{\tilde{p}_{\zeta}^3 \tilde{\zeta}^3}^{(6)} + \frac{M_{\text{Pl}}^2}{a^4 H^3 \epsilon} \mathcal{O}_{\tilde{p}_{\zeta} \tilde{\zeta}^3}^{(4)} + \frac{M_{\text{Pl}}^2}{a^3 H^4} \mathcal{O}_{\tilde{\zeta}^4}^{(6)} + \frac{M_{\text{Pl}}^2}{a H^2} \mathcal{O}_{\tilde{\zeta}^4}^{(4)}. \quad (4.106) $$

One can follow the usual procedure and determine the canonical transformation to simplify the interactions above by incorporating the first two terms in (4.106) into the quadratic kinetic term. Only the generating function corresponding to the first term above includes factors of $\epsilon$ and eventually leads to an $\eta$ dependent term upon taking the partial time derivative. The resulting $\eta$-dependent contribution is of the form $\eta \mathcal{F}_{\tilde{p}_{\zeta} \tilde{\zeta}}^{(4)}/a^4 H^3 \epsilon$, and is superficially of slow-roll order $\sim \epsilon$.

After having properly chosen the generating function so as to cancel out the leading order Hamiltonian (4.106), the remaining interactions are superficially of slow-roll order $\sim \epsilon$, and we find their form to be:

$$ \tilde{\mathcal{H}}_{\epsilon^1}^{(4)} = \frac{1}{a^8 H^3 \epsilon^2 M_{\text{Pl}}^4} \mathcal{O}_{\tilde{p}_{\zeta}^3 \tilde{\zeta}^3}^{(2)} + \frac{1}{a^6 H \epsilon^2 M_{\text{Pl}}^4} \mathcal{O}_{\tilde{p}_{\zeta}^3 \tilde{\zeta}^3}^{(0)} + \frac{1}{a^5 H^2 \epsilon M_{\text{Pl}}^2} \mathcal{O}_{\tilde{p}_{\zeta}^2 \tilde{\zeta}^2}^{(2)} + \frac{1}{a^3 \epsilon M_{\text{Pl}}^2} \mathcal{O}_{\tilde{p}_{\zeta}^2 \tilde{\zeta}^2}^{(0)} \\ + \frac{1}{a^7 H^4 \epsilon M_{\text{Pl}}^2} \mathcal{O}_{\tilde{p}_{\zeta}^2 \tilde{\zeta}^2}^{(4)} + \frac{1}{a^4 H^3} \mathcal{O}_{\tilde{p}_{\zeta}^3 \tilde{\zeta}^3}^{(4)} + \frac{1}{a^2 H} \mathcal{O}_{\tilde{p}_{\zeta} \tilde{\zeta}^3}^{(2)} + \frac{\epsilon M_{\text{Pl}}^2}{a H^2} \mathcal{O}_{\tilde{\zeta}^4}^{(4)} - \frac{\epsilon M_{\text{Pl}}^2}{a^3 H^4} \mathcal{O}_{\tilde{\zeta}^4}^{(6)} \\ + \frac{1}{a^4 H^3} \frac{\eta}{\epsilon} \mathcal{F}_{\tilde{p}_{\zeta} \tilde{\zeta}^3}^{(4)}, \quad (4.107) $$

where the last term is precisely the one mentioned above, produced by the first canonical transformation. As in the previous step, one could then find another canonical transformation to reduce the size of the interactions. It is easy to see check that of all the terms in eq. (4.107), only the first two terms and the last one can possibly produce interactions proportional to $\eta$ through this procedure. The former generate a term of the form $\eta \mathcal{F}_{\tilde{p}_{\zeta}^2 \tilde{\zeta}^2}^{(2)}/a^5 H^2 M_{\text{Pl}}^2$, which is, however, subdominant with respect to those in the leading interactions in the large $\eta$ regime derived in eq. (4.105). On the other hand, simplifying the last term in eq. (4.107), may lead to an interaction of the form $\eta(\eta_2 - H)M_{\text{Pl}}^2 \mathcal{F}_{\tilde{\zeta}^4}^{(4)}/aH^2 M_{\text{Pl}}^2$. The interaction $\propto \eta\eta_2$ would be of the same order as those in eq. (4.105). However, the one $\propto \eta$ would be of slow-roll order $\epsilon^1$, and cannot be cancelled by any other contribution. This would be problematic, as such interaction would lead to a large trispectrum of order unity even during standard slow-roll inflation, while it is known to start at the next order in this regime [54, 55]. Therefore, we conclude that the complicated operator $\mathcal{F}_{\tilde{\zeta}^4}^{(4)}$ must vanish upon careful computation of its
various contributions and uses of spatial integration by parts. Based on this argument, we conclude that the dominant interactions in the large $\eta$ limit take the form that we have derived in eq. (4.105). Moreover, the canonical transformations needed to reach the stage with the operator $\mathcal{F}_{\tilde{\zeta}^4}^{(4)}$ do not introduce $\eta$-dependent terms so we can consider $\tilde{\zeta} = \zeta$ in the large $\eta$ regime, and the only relevant canonical transformation is still given by eq. (4.104) alone.

This example serves primarily for illustration, providing a simplified setting to demonstrate the method for manipulating $\mathcal{H}^{(4)}$ using canonical transformations. However, let us mention that our exercise has already interesting applications on its own. In particular, a heated discussion has recently emerged regarding whether models with a transient stage of large $\eta$ can produce large loop-corrections to the spectrum of $\zeta$ (see refs. [48, 56–67] for an incomplete list of papers). One of the controversies in this context is related to how to deal with total time derivative interactions terms, recently discussed in refs. [68–70]. We hope that our identification of the dominant interactions in the large $\eta$ regime, together with our thorough explanation of how to deal with total time derivatives in the approach with integrations by parts, our equivalently with canonical transformations at the level of the full Hamiltonian in phase space, can contribute to solving this matter.

### 4.5.3 Order $n$ Hamiltonian

Before concluding, let us thus generalize the learnt lesson, and provide a prescription to simplify interactions at order $n$ in perturbations. To be fully general we reintroduce a collection of multiple fields $\psi^A$ and associated momenta $p_\psi^A$ with $A = 1, \dots, N$. Starting from a Hamiltonian with interactions of superficially slow-roll size $\epsilon_i^m$ ($m$ powers of either of the slow-roll parameters) after introducing interaction picture fields, we propose the following prescription to make explicit the true size of interactions:

* Take the interactions with the highest powers of momenta, and superficial size $\epsilon_i^m$. At order $n$ this would be $(p_\psi^A)^n$.

* Define new momenta $\tilde{p}_\psi^A$ to incorporate this interaction into the kinetic term for this new momentum.

* Find the corresponding generating function and new position variable $\tilde{\psi}^A$, possibly perturbatively, and derive the new Hamiltonian $\tilde{\mathcal{H}}$ for the new phase-space variables. The generating function is of order $n$, and lower order interactions of order $< n$ are not modified.

* Repeat this procedure for the interaction of order $(p_\psi^A)^{n-1} \psi^A$, etc. The last step is the order $p_\psi^A (\psi^A)^{n-1}$, after which one is left with only terms of the form $(\psi^A)^n$. At this point there are three possibilities. (i) the remaining interactions of the form $(\psi^A)^n$ do not cancel out among each others: the *true* size of the Hamiltonian is $\epsilon_i^m$. (ii) the remaining interactions of the form $(\psi^A)^n$ cancel out among each others: the *true* size of the Hamiltonian is $< \epsilon_i^m$, and we will have to repeat the procedure above until we reach the true size of the interaction Hamiltonian at order $n$.
The perturbative canonical transformations obtained to simplify the Hamiltonian of order $n$ way will also generate higher order interactions, which will add to other interactions of the same order. One can then go to the order $n+1$ and apply again the algorithm we have just proposed.

## 5 Conclusions

Despite the established use of in-in perturbation theory for computing cosmological correlation functions, unresolved ambiguities were persisting in handling total time derivative interactions in the Lagrangian. In the context of inflation, these interactions primarily emerge from integrations by parts designed to reveal the correct slow-roll suppression of non-linear interactions in the comoving gauge. Although the importance of not overlooking total time derivatives was already emphasized, a systematic approach to such terms beyond the single vertex order was still lacking in the literature. Our work precisely clarifies the impact of total time derivatives and terms proportional to the linear equations of motion in the in-in perturbation theory and formalizes their analysis.

In this work, we systematize the treatment of total time derivative interactions in the interaction Hamiltonian of the in-in perturbation theory and we elucidate their possible contributions to correlation functions. We show that they lead to i) boundary terms in the form of equal-time nested commutators with the external operator $O$ whose correlation functions are sought for; ii) modifications of the usual bulk interactions that need to be integrated over time, in the form of effective interactions. Only the former contribution was investigated before our work, while we show that the latter is also crucial to cancel superficially large contributions from higher-order bulk interactions in the Hamiltonian generated from the Legendre transformation of the Lagrangian that includes total time derivative interactions. We provide a general formula valid at any order in perturbation theory that enables to recast all contributions from total time derivatives in the interaction Hamiltonian as either a redefinition of the external operator $O$ or of the bulk interactions. Explicitly expanding the formula up to second order, as required for practical calculations of the most common primordial correlation functions, bispectrum, contact and exchange trispectrum and 1-loop corrections to the power spectrum, we demonstrate its practical use beyond the single-vertex order. For completeness and sake of comparison, we also explored how the Lagrangian path integral approach of the in-in formalism performs in dealing with total time derivatives and terms proportional to linear equations of motion, and showcased concrete calculations proving equivalence with the Hamiltonian operator approach. Although perfectly suitable for performing calculations, these two versions of the in-in perturbation theory with total time derivative interactions that we have developed could be judged cumbersome, with either fine cancellations to be sought for between different vertex-orders or interactions with complicated operators to take into account. We thus propose another route to the calculation of primordial correlation functions, that avoids altogether the generation of total time derivative interactions.

Instead of performing integrations by parts in the Lagrangian to reveal the correct size of interactions, our proposed method operates directly in phase space, where the Hamiltonian is defined. By utilizing canonical transformations, we are able to simplify interactions without generating total time derivatives, so that the canonically transformed Hamiltonian exclusively
comprises bulk interactions. The canonical transformations establish a relationship between correlation functions of old and new variables, and we explain how to retrieve the former from the knowledge of the latter that can be computed using the transformed Hamiltonian. With this approach the interaction Hamiltonian is significantly simpler, which greatly trivializes the computation of correlation functions beyond the single-vertex order. We demonstrate the equivalence of this new method with the in-in perturbation theory that includes total time derivatives for the two toy models that we had already introduced. In order to build intuition on the possible contributions from the inverse canonical transformations relating correlation functions of the old phase-space variables to those of the new ones, we also introduce diagrammatic rules of thumb.

As an application, we utilized our formalism to compute the cubic Hamiltonian for scalar and tensor perturbations in canonical single-field inflation. Using canonical transformations, we are able to extract the correct slow-roll order suppression of the phase-space interactions without introducing total time derivatives nor terms proportional to the linear equations of motion. Importantly, this cubic Hamiltonian is relevant beyond interaction picture calculations, and can also be used for Schrödinger picture non-linear evolutions. When evaluated on interaction picture fields and momenta though, it exactly reduces to the cubic interaction Hamiltonian found from the cubic scalar and tensor Lagrangians derived in earlier works, with the exception of total time derivative interactions which are now absent. In comparison to the more traditional approach with integration by parts, our method proves simpler to compute the primordial bispectrum and illuminates the connection between the use of either boundary terms and field redefinitions a la Maldacena, the latter representing simply a subset of the canonical transformations proposed in our approach. We also discuss the generalization of our formalism to quartic and higher-order interactions. To illustrate practicality, we applied our formalism to recover the subset of phase-space quartic interactions that are dominant in the large $\eta$ limit. When evaluated on interaction picture fields and momenta, this quartic interaction picture Hamiltonian exactly coincides with findings using the Lagrangian of the EFT of inflation beyond the slow-varying background approximation. However, we additionally provide with the crucial information about the inverse canonical transformations to be performed in order to retrieve correlation functions of the exact primordial curvature perturbation seeding the large-scale structures of our universe.

Our formalism can be extended and applied in several directions. An evident example is a comprehensive computation of quartic interactions in single-field inflation in the comoving gauge, a task that remains unexplored in the existing literature, even with the more traditional approach at the level of the Lagrangian and using integrations by parts. Although this computation will be demanding, our work establishes all the essential tools for undertaking such an endeavor. Another intriguing avenue consists in the simplification of the phase-space Hamiltonian interactions in multifield models of inflation that encompass both the adiabatic curvature perturbation that we have considered in this work, and a set of entropic — or isocurvature — fluctuations. We also anticipate that our insights into large $\eta$ interactions will contribute to clarifying the recent debate surrounding loop corrections to the primordial power spectrum arising from a temporary non-attractor stage of inflation. Lastly, we stress again that the phase-space interactions elucidated in this work can be applied beyond interaction
picture calculations of primordial correlation functions. For instance, they may prove useful in studying the quantum properties of cosmological perturbations beyond the linear order. Concrete applications could range from the calculation of self-decoherence in single-field inflation from non-linear interactions between different scales, to Schrödinger picture non-linear evolutions. We plan to explore ourselves some of these directions in future works.

## Acknowledgments

We are grateful to Xingang Chen, David Langlois, Jérôme Martin, Sébastien Renaux-Petel, David Seery and Denis Werth for useful discussions at various stages of this project. We also thank Xingang Chen, Guillem Domenech, Jacopo Fumagalli, Sebastian Garcia-Saenz, Sébastien Renaux-Petel and Denis Werth for comments on a draft of this paper. Some calculations in this paper were performed with the Mathematica software MathGR [71]. We would like to acknowledge CERN and the EuCAPT Consortium for the 2023 EuCAPT International Travel Award that helped foster this work. L.P. acknowledges funding support from the Initiative Physique des Infinis (IPI), a research training program of the Idex SUPER at Sorbonne Université. L.P. would also like to thank New York University for hospitality during the latest stages of this project.

## A Canonical quantization and quantum anomalies

In most cosmological applications, a semi-classical and perturbative treatment is used. Semi-classical because only vacuum fluctuations around a homogeneous background are quantified. Perturbative because the theory is first assumed to be free, and therefore the Hamiltonian is truncated at quadratic order before canonical quantization. Then only, non-linear interactions expressed in terms of the quantized free fields are added to the theory in a perturbative scheme. We will therefore focus here on a free quadratic Hamiltonian, for simplicity still with a single degree of freedom and no mass term as in eq. (2.51). Roughly speaking, canonical quantization amounts to promoting the phase-space variables $(\psi, p_{\psi})$ in $\mathcal{H}_{\text{free}}$ to quantum operators verifying the canonical commutation relations inherited from the Poisson brackets in the corresponding classical theory:

$$ \begin{gathered} \psi(t, \vec{x}) \rightarrow \hat{\psi}(t, \vec{x}), \qquad (A.1) \\ p_{\psi}(t, \vec{x}) \rightarrow \hat{p}_{\psi}(t, \vec{x}), \\ \forall t, \quad \{\psi(t, \vec{x}), p_{\psi}(t, \vec{y})\} = \delta^{(3)}(\vec{x} - \vec{y}) \rightarrow [\hat{\psi}(t, \vec{x}), \hat{p}_{\psi}(t, \vec{y})] = i\hbar \delta^{(3)}(\vec{x} - \vec{y}), \end{gathered} $$

where we kept the $\hbar$-factor exceptionally. Then, one must define the vacuum state of the theory, $|0\rangle$, often found by minimization of the energy $\langle 0|\mathcal{H}|0\rangle$. The creation and annihilation quantum operators are used to construct the corresponding Fock space made of n-particles states, e.g. $|1_k\rangle = \hat{a}_k^{\dagger}|0\rangle$, etc. The creation and annihilation operators must verify the commutation relations

$$ [\hat{a}_{\vec{k}}, \hat{a}_{\vec{k}'}^{\dagger}] = (2\pi)^3 \delta^{(3)}(\vec{k} - \vec{k}'). \qquad (A.2) $$
Since they form a complete set of quantum states, the position and momentum operators in Fourier space can be decomposed in this basis, and the coefficients of the decomposition are the mode functions ($\psi_k(t), p_{\psi,k}(t)$):

$$ \hat{\psi}(t, \vec{x}) = \int \frac{d^3\vec{k}}{(2\pi)^3} e^{i\vec{k}\cdot\vec{x}} (\psi_k(t)\hat{a}_{\vec{k}} + \psi_k^*(t)\hat{a}_{-\vec{k}}^\dagger), \quad (A.3) $$

$$ \hat{p}_{\psi}(t, \vec{x}) = \int \frac{d^3\vec{k}}{(2\pi)^3} e^{i\vec{k}\cdot\vec{x}} (p_{\psi,k}(t)\hat{a}_{\vec{k}} + p_{\psi,k}^*(t)\hat{a}_{-\vec{k}}^\dagger). \quad (A.4) $$

From the canonical commutation relations in real space, those for the creation and annihilation operators, and the above expressions, the mode functions must verify the Wronskian condition: $\forall t, \psi_k p_{\psi,k}^* - \psi_k^* p_{\psi,k} = i\hbar$.

A quantum canonical transformation must leave invariant all these properties. In practice, the difference with classical mechanics is the presence of non-commutating operators. This implies for example that the order in which quantum canonical transformations are performed now matters, and that the inverse transformations must be defined with care. Also, a canonical transformation should a priori be assigned a given operator ordering, e.g. $\hat{p}_{\psi} = \hat{p}_{\psi} + \hat{\psi}\hat{p}_{\psi}$ is different from $\hat{p}_{\psi} = \hat{p}_{\psi} + \hat{p}_{\psi}\hat{\psi}$, etc., see ref. [72] in the context of quantum mechanics. One way to go around the operator ordering ambiguities is to consider a path-integral approach where all instances of the fields and their conjugate momenta are simply dummy variables of integration, instead of operators: $\mathcal{Z} = \int \mathcal{D}\psi\mathcal{D}p_{\psi}e^{i\int[\psi, p_{\psi}]}$. However, then, equivalent subtleties arise at the level of the discrete time path. The main point is that, for quantum path integrals, the “jump” between two values $\psi_j$ and $\psi_{j-1}$ (resp. $p_{\psi}^j$ and $p_{\psi}^{j-1}$) may not be proportional to the time elapsed $t_j - t_{j-1} \equiv \Delta t$ during the jump, but to its square root $\sqrt{\Delta t}$. This property, ubiquitous in classical but stochastic processes, result in so-called quantum anomalies in the context of canonical transformations.

* Given a generating function $F$ for a canonical transformation at the time $t_j$, the relation between old and new variables is affected by corrections proportional to higher-order derivatives of $F$ and to the jumps $\Delta\psi_j \equiv \psi_j - \psi_{j-1}$, $\Delta p_{\psi}^j \equiv p_{\psi}^{j+1} - p_{\psi}^j$. For example, for a type III generating function, eq. (3.7) would now read [73]:

$$ \begin{aligned} \psi_j &= -\frac{F_3(p_{\psi}^{j+1}, \tilde{\psi}_j, t_j) - F_3(p_{\psi}^j, \tilde{\psi}_j, t_j)}{\Delta p_{\psi}^j} \quad &(A.5) \\ &= -\frac{\partial F_3}{\partial p_{\psi}^j}(p_{\psi}^j, \tilde{\psi}_j, t_j) - \frac{1}{2}\frac{\partial^2 F_3}{\partial p_{\psi}^{j-2}}(p_{\psi}^j, \tilde{\psi}_j, t_j)\Delta p_{\psi}^j + \dots \end{aligned} $$

$$ \begin{aligned} \tilde{p}_{\psi}^j &= -\frac{F_3(p_{\psi}^j, \tilde{\psi}_j, t_j) - F_3(p_{\psi}^j, \tilde{\psi}_{j-1}, t_j)}{\Delta \tilde{\psi}_j} \quad &(A.6) \\ &= -\frac{\partial F_3}{\partial \tilde{\psi}_j}(p_{\psi}^j, \tilde{\psi}_j, t_j) + \frac{1}{2} \frac{\partial^2 F_3}{\partial \tilde{\psi}_j^2}(p_{\psi}^j, \tilde{\psi}_j, t_j) \Delta \tilde{\psi}_j + \dots \end{aligned} $$

where usually one discards the contributions explicitly proportional to the jumps in the limit $\Delta t \to 0$. However in general in a quantum theory those contribute finite corrections, though suppressed by an additional factor of $\hbar$, not explicit here. In practice,
this has the consequence that the new Hamiltonian that describes the system in terms of the new phase-space variables is not only a function of the fields and momenta, but also of their time derivatives — the equivalent of the “jumps” in the continuous time description — $\tilde{H}(\tilde{\psi}, \tilde{p}_{\psi}, \tilde{\dot{\psi}}, \tilde{\dot{p}}_{\psi})$.

• A second, related anomaly concerns the Jacobian of the transformation in the path integral, taking into account all canonical transformations at the discrete times $t_j$. Indeed, the equivalent of eq. (3.13) now becomes [73]:

$$J = \prod_{j=0}^{\infty} \left[ 1 + A_j \Delta\tilde{\psi}_j + B_j \Delta\tilde{p}_{\psi}^j \right] \neq 1, \quad (A.7)$$

with contributions proportional to the jumps at each time step, $A_j$ and $B_j$ being given by third derivatives of the generating function. In the continuous limit, one can exponentiate the Jacobian and finds $J \to \exp\{\int dt A(t) \tilde{\psi}(t) + B(t) \tilde{p}_{\psi}(t)\}$, contributing as external source terms for the time derivatives.

Ref. [73] shows examples of physical systems and canonical transformations including all relevant corrections in a quantum system, and proves that the quantum anomalies cancel each other. In those cases, one can therefore consider a Hamiltonian path integral in terms of the new phase-space variables and independent of the time derivatives of the fields and momenta. Although definitely a fascinating direction for future work, we defer the study of such quantum anomalies in the cosmological context. In practice, we will therefore overlook the possibility of having quantum anomalies and use canonical transformations at the level of the classical Hamiltonian, before canonical quantization. It remains to be known whether quantum anomalies may affect the calculation of loop-level diagrams or purely quantum effects like entanglement, etc., in the context of quantum canonical transformations.

## B Diagrammatic rules

We present here some diagrammatic rules of thumb that we found convenient to develop, in order to classify the possible contributions to an observable $\langle O(\psi) \rangle$, where $\psi$ is related to a variable $\tilde{\psi}$ via a canonical transformation. These are only approximate diagrammatic rules in the sense that we pay no attention to numerical factors, permutations, etc. Moreover, for simplicity, we overlook the presence of momenta $\tilde{p}_{\psi}$ in the expression for $\psi$, as well as the possibility to compute expectation values of an operator including $p_{\psi}$. More generally, as in the rest of this section, we only consider explicitly a single degree of freedom $\psi$ instead of a collection $\{\psi^a\}_{a=1,...,N}$. However, we already propose a straight generalization of these diagrammatic rules to a multifield setup in phase space: by the inclusion of indices $A$ such that $\{\psi^A\}_{A=1,...,2N} = \{\psi^a, p_{\psi}^a\}_{a=1,...,N}$, with both propagators and vertices mixing them.

For definiteness, we now focus on a transformation $\psi = g_2\tilde{\psi} + g_3\tilde{\psi}^2 + g_4\tilde{\psi}^3 + ...$ where we wrote explicitly only the relevant contributions for the calculations of the tree-level bispectrum and trispectrum, as well as the one-loop power spectrum. In the diagrams below, a white dot simply corresponds to the transition $\psi \to g_2\tilde{\psi}$ (quadratic vertex), while a black dot represents a non-linear transition $\psi \to g_3\tilde{\psi}^2$ (cubic vertex), $\psi \to g_4\tilde{\psi}^3$ (quartic vertex), etc.
$$ + g_2^4 \times \int d^3 \vec{q} [P_{\tilde{\psi}}(q) P_{\tilde{\psi}}(|\vec{q} + \vec{k}_1|) P_{\tilde{\psi}}(|\vec{q} + \vec{k}_1 + \vec{k}_2|) P_{\tilde{\psi}}(|\vec{q} - \vec{k}_4|) + \text{perm.}] + \dots $$

The one-loop power spectrum of $\psi$ is given by:

$$ \begin{aligned} P_{\tilde{\psi}}^{1-\text{loop}}(k) &= [P_{\tilde{\psi}}(k) - P_{\tilde{\psi}}^{\text{tree}}(k)]^{1-\text{loop}} && (\text{B.4}) \\ &= [\text{[red line} \rightarrow \text{red line} - P_{\tilde{\psi}}^{\text{tree}}(k)]^{1-\text{loop}} + \text{[blue line} \rightarrow \text{blue line} + \text{red line} \rightarrow \text{blue line} + \text{blue line} \rightarrow \text{red line}]^{1-\text{loop}} + \text{[green line} \rightarrow \text{green line} + \text{red line} \rightarrow \text{green line} + \text{red line} \rightarrow \text{green line}]^{1-\text{loop}}] \\ &= P_{\tilde{\psi}}^{1-\text{loop}}(k) + 2g_3^2 \times \int \frac{d^3\vec{q}}{(2\pi)^3} P_{\tilde{\psi}}^{\text{tree}}(q) P_{\tilde{\psi}}^{\text{tree}}(|\vec{q} + \vec{k}|) \\ &\quad + 6g_4 \times \int \frac{d^3\vec{q}}{(2\pi)^3} P_{\tilde{\psi}}^{\text{tree}}(q) P_{\tilde{\psi}}^{\text{tree}}(k) + 2g_3 \times \int \frac{d^3\vec{q}}{(2\pi)^3} B_{\tilde{\psi}}^{\text{tree}}(q, |\vec{k} + \vec{q}|, k), \end{aligned} $$

where $[\dots]^{1-\text{loop}}$ means “truncated at one loop”. This expression for the one-loop power spectrum is exact, though we used that $g_2 = 1$ explicitly. We see that the one-loop power spectrum of $\psi$ is given both by the one-loop power spectrum of $\tilde{\psi}$ as well as the one-loop corrections from the non-linear canonical transformation $\psi(\tilde{\psi})$.

### C Complete expressions for the scalar and tensor Lagrangian

$$ \begin{aligned} \frac{\mathcal{L}_{\gamma}}{M_{\text{Pl}}^2} = & - \frac{ae^{\zeta} \partial_j N \dot{\gamma}_{ij} N_i}{2N^2} + \frac{3ae^{\zeta} \partial_j \zeta \dot{\gamma}_{ij} N_i}{2N} - \frac{e^{-\zeta} \partial_k N_j \partial_k \gamma_{ij} N_i}{2aN} - \frac{e^{-\zeta} \partial_k N_j \partial_j \gamma_{ik} N_i}{2aN} \\ & + \frac{e^{-\zeta} \partial_k N_j \partial_i \gamma_{jk} N_i}{2aN} + \frac{e^{-\zeta} \partial_k \zeta \partial_k \gamma_{ij} N_i N_j}{aN} + ae^{\zeta} N \partial_i \zeta \partial_j \zeta \gamma_{ij} + 2ae^{\zeta} N \partial_i \zeta \partial_j \gamma_{ij} \\ & - \frac{2ae^{\zeta} H \partial_j N_i \gamma_{ij}}{N} - \frac{2ae^{\zeta} \zeta \partial_j N_i \gamma_{ij}}{N} - \frac{e^{-\zeta} \partial_k N_i \partial_k N_j \gamma_{ij}}{4aN} - \frac{e^{-\zeta} \partial_k N_i \partial_j N_k \gamma_{ij}}{2aN} \\ & - \frac{e^{-\zeta} \partial_i N_k \partial_j N_k \gamma_{ij}}{4aN} + \frac{e^{-\zeta} \partial_j N_i \partial_k N_k \gamma_{ij}}{aN} - \frac{2ae^{\zeta} H \partial_j \zeta N_i \gamma_{ij}}{N} - \frac{2ae^{\zeta} \zeta \partial_j \zeta N_i \gamma_{ij}}{N} \\ & + \frac{e^{-\zeta} \partial_k \zeta \partial_k N_j N_i \gamma_{ij}}{aN} + \frac{e^{-\zeta} \partial_k \zeta \partial_j N_k N_i \gamma_{ij}}{aN} + \frac{e^{-\zeta} \partial_j \zeta \partial_k N_i N_j \gamma_{jk}}{aN} \\ & + \frac{e^{-\zeta} \partial_j \zeta \partial_i N_k N_i \gamma_{jk}}{aN}, \end{aligned} \quad (\text{C.1}) $$

$$ \begin{aligned} \frac{\mathcal{L}_{\gamma\gamma}}{M_{\text{Pl}}^2} = & \frac{a^3 e^{3\zeta} \dot{\gamma}_{ij}^2}{8N} - \frac{1}{8} ae^{\zeta} N (\partial_k \gamma_{ij})^2 - \frac{1}{4} ae^{\zeta} N \partial_k \gamma_{ij} \partial_j \gamma_{ik} + \frac{1}{4} ae^{\zeta} N \partial_k \gamma_{ij} \partial_i \gamma_{jk} \\ & + \frac{ae^{\zeta} \partial_k \gamma_{ij} \dot{\gamma}_{jk} N_i}{2N} - \frac{ae^{\zeta} \dot{\gamma}_{jk} \partial_k \gamma_{ij} N_i}{4N} + \frac{e^{-\zeta} \partial_l \gamma_{ik} \partial_l \gamma_{jk} N_i N_j}{4aN} \\ & + \frac{e^{-\zeta} \partial_l \gamma_{ik} \partial_k \gamma_{jl} N_i N_j}{4aN} - \frac{e^{-\zeta} \partial_l \gamma_{ik} \partial_j \gamma_{kl} N_i N_j}{2aN} + \frac{e^{-\zeta} \partial_i \gamma_{kl} \partial_j \gamma_{kl} N_i N_j}{8aN} \\ & - ae^{\zeta} N \partial_k \zeta \partial_j \gamma_{ik} \gamma_{ij} + \frac{ae^{\zeta} \partial_k N_i \dot{\gamma}_{jk} \gamma_{ij}}{4N} + \frac{ae^{\zeta} \partial_i N_k \dot{\gamma}_{jk} \gamma_{ij}}{4N} \\ & - \frac{ae^{\zeta} \partial_k \zeta \dot{\gamma}_{jk} N_i \gamma_{ij}}{2N} + \frac{e^{-\zeta} \partial_l N_k \partial_l \gamma_{jk} N_i \gamma_{ij}}{4aN} + \frac{e^{-\zeta} \partial_l N_k \partial_k \gamma_{jl} N_i \gamma_{ij}}{4aN} \\ & - \frac{e^{-\zeta} \partial_l N_k \partial_j \gamma_{kl} N_i \gamma_{ij}}{2aN} + \frac{ae^{\zeta} H \partial_k \gamma_{ij} N_i \gamma_{jk}}{N} + \frac{ae^{\zeta} \dot{\zeta} \partial_k \gamma_{ij} N_i \gamma_{jk}}{N} \end{aligned} $$
$$
\begin{align}
& - \frac{e^{-\zeta} \partial_l N_l \partial_k \gamma_{ij} N_i \gamma_{jk}}{2aN} - \frac{ae^{\zeta} \partial_j \zeta \dot{\gamma}_{ik} N_i \gamma_{jk}}{2N} + \frac{e^{-\zeta} \partial_l N_j \partial_l \gamma_{ik} N_i \gamma_{jk}}{4aN} \nonumber \\
& + \frac{e^{-\zeta} \partial_j N_l \partial_l \gamma_{ik} N_i \gamma_{jk}}{4aN} + \frac{e^{-\zeta} \partial_l N_j \partial_k \gamma_{il} N_i \gamma_{jk}}{2aN} + \frac{e^{-\zeta} \partial_j N_l \partial_k \gamma_{il} N_i \gamma_{jk}}{2aN} \nonumber \\
& - \frac{e^{-\zeta} \partial_l N_j \partial_i \gamma_{kl} N_i \gamma_{jk}}{4aN} - \frac{e^{-\zeta} \partial_j N_l \partial_i \gamma_{kl} N_i \gamma_{jk}}{4aN} - \frac{1}{2} ae^{\zeta} N \partial_i \zeta \partial_k \zeta \gamma_{ij} \gamma_{jk} \nonumber \\
& - ae^{\zeta} N \partial_i \zeta \gamma_{ij} \gamma_{jk} + \frac{ae^{\zeta} H \partial_k N_i \gamma_{ij} \gamma_{jk}}{N} + \frac{ae^{\zeta} \zeta \partial_k N_i \gamma_{ij} \gamma_{jk}}{N} \nonumber \\
& + \frac{e^{-\zeta} \partial_l N_i \partial_l N_k \gamma_{ij} \gamma_{jk}}{8aN} + \frac{e^{-\zeta} \partial_l N_i \partial_k N_l \gamma_{ij} \gamma_{jk}}{4aN} + \frac{e^{-\zeta} \partial_i N_l \partial_k N_l \gamma_{ij} \gamma_{jk}}{8aN} \nonumber \\
& - \frac{e^{-\zeta} \partial_k N_i \partial_l N_l \gamma_{ij} \gamma_{jk}}{2aN} + \frac{ae^{\zeta} H \partial_k \zeta N_i \gamma_{ij} \gamma_{jk}}{N} + \frac{e^{-\zeta} \partial_k N_i \partial_l N_j \gamma_{ij} \gamma_{kl}}{4aN} \nonumber \\
& - \frac{e^{-\zeta} \partial_j N_i \partial_l N_k \gamma_{ij} \gamma_{kl}}{2aN} + \frac{e^{-\zeta} \partial_k N_i \partial_j N_l \gamma_{ij} \gamma_{kl}}{4aN}, \tag{C.2}
\end{align}
$$

$$
\begin{align}
\frac{\mathcal{L}_{\gamma\gamma\gamma}}{M_{\text{Pl}}^2} ={}& \frac{1}{12} ae^{\zeta} N \partial_l \gamma_{ak} \partial_k \gamma_{jl} \gamma_{ij} - \frac{1}{12} ae^{\zeta} N \partial_l \gamma_{ik} \partial_j \gamma_{kl} \gamma_{ij} + \frac{1}{8} ae^{\zeta} N \partial_i \gamma_{kl} \partial_j \gamma_{kl} \gamma_{ij} \nonumber \\
& - \frac{ae^{\zeta} \partial_l \gamma_{jk} \dot{\gamma}_{kl} N_i \gamma_{ij}}{4N} + \frac{ae^{\zeta} \dot{\gamma}_{kl} \partial_j \gamma_{kl} N_i \gamma_{ij}}{4N} - \frac{ae^{\zeta} \partial_j \gamma_{il} \dot{\gamma}_{kl} N_i \gamma_{jk}}{4N} \nonumber \\
& + \frac{1}{3} ae^{\zeta} N \partial_l \zeta \partial_k \gamma_{il} \gamma_{ij} \gamma_{jk} - \frac{ae^{\zeta} \partial_l N_i \dot{\gamma}_{kl} \gamma_{ij} \gamma_{jk}}{12N} - \frac{ae^{\zeta} \partial_i N_l \dot{\gamma}_{kl} \gamma_{ij} \gamma_{jk}}{12N} \nonumber \\
& + \frac{1}{12} ae^{\zeta} N \partial_j \partial_l \gamma_{ik} \gamma_{ij} \gamma_{kl} + \frac{1}{6} ae^{\zeta} N \partial_k \zeta \partial_j \gamma_{il} \gamma_{ij} \gamma_{kl} + \frac{1}{6} ae^{\zeta} N \partial_i \zeta \partial_l \gamma_{jk} \gamma_{ij} \gamma_{kl} \nonumber \\
& - \frac{ae^{\zeta} \partial_k N_i \dot{\gamma}_{jl} \gamma_{ij} \gamma_{kl}}{12N} - \frac{ae^{\zeta} H \partial_l \gamma_{jk} N_i \gamma_{ij} \gamma_{kl}}{3N} - \frac{ae^{\zeta} H \partial_l N_i \gamma_{ij} \gamma_{jk} N_i \gamma_{jk} \gamma_{kl}}{3N} \nonumber \\
& - \frac{ae^{\zeta} H \partial_l N_i \gamma_{ij} \gamma_{jk} \gamma_{kl}}{6N} - \frac{ae^{\zeta} H \partial_i N_l \gamma_{ij} \gamma_{jk} \gamma_{kl}}{6N} + \frac{1}{3} ae^{\zeta} N \partial_i \partial_k \zeta \gamma_{ij} \gamma_{jl} \gamma_{kl}, \tag{C.3}
\end{align}
$$

$$
\begin{align}
\frac{\mathcal{L}_{\gamma\gamma\gamma}}{M_{\text{Pl}}^2} ={}& \frac{a^3 e^{3\zeta} \dot{\gamma}_{il} \dot{\gamma}_{kl} \gamma_{ij} \gamma_{jk}}{48N} - \frac{1}{48} ae^{\zeta} N \partial_m \gamma_{il} \partial_m \gamma_{kl} \gamma_{ij} \gamma_{jk} + \frac{1}{24} ae^{\zeta} N \partial_m \gamma_{il} \partial_k \gamma_{lm} \gamma_{ij} \gamma_{jk} \nonumber \\
& - \frac{1}{16} ae^{\zeta} N \partial_i \gamma_{lm} \partial_k \gamma_{lm} \gamma_{ij} \gamma_{jk} - \frac{a^3 e^{3\zeta} \dot{\gamma}_{ik} \dot{\gamma}_{jl} \gamma_{ij} \gamma_{kl}}{48N} + \frac{1}{48} ae^{\zeta} N \partial_m \gamma_{ik} \partial_m \gamma_{jl} \gamma_{ij} \gamma_{kl} \nonumber \\
& + \frac{1}{8} ae^{\zeta} N \partial_m \gamma_{ik} \partial_l \gamma_{jm} \gamma_{ij} \gamma_{kl} - \frac{1}{48} ae^{\zeta} N \partial_j \gamma_{im} \partial_l \gamma_{km} \gamma_{ij} \gamma_{kl} \nonumber \\
& - \frac{5}{24} ae^{\zeta} N \partial_m \gamma_{ik} \partial_j \gamma_{lm} \gamma_{ij} \gamma_{kl} + \frac{1}{24} ae^{\zeta} N \partial_k \gamma_{im} \partial_j \gamma_{lm} \gamma_{ij} \gamma_{kl} \nonumber \\
& + \frac{1}{48} ae^{\zeta} N \partial_l \partial_m \gamma_{ik} \gamma_{ij} \gamma_{jm} \gamma_{kl} + \frac{1}{3} ae^{\zeta} N \partial_k \partial_m \zeta \gamma_{ij} \gamma_{il} \gamma_{jm} \gamma_{kl} \nonumber \\
& - \frac{1}{16} ae^{\zeta} N \partial_j \partial_m \gamma_{ik} \gamma_{ij} \gamma_{kl} \gamma_{lm} - \frac{5}{12} ae^{\zeta} N \partial_i \partial_k \zeta \gamma_{ij} \gamma_{jm} \gamma_{kl} \gamma_{lm}. \tag{C.4}
\end{align}
$$

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