### In-In

My colleague, Steve Weinberg, has a new paper out, calculating loop corrections to the correlations of fluctuations in inflationary cosmology.

Departures from Gaussianity of the perturbations has been a subject of much interest. A few years ago, Maldacena computed the contribution to the 3-point function of the fluctuations, which follows from the nonlinearities of the GR action. As you might suspect, Steve shows that Juan’s result is tantamount to computing some tree-graphs, not in standard perturbation theory, but in Schwinger’s In-In formalism. he then proceeds to calculate the one-loop corrections.

Maldacena’s result was already unobservably small, so these loop corrections are not of great observational interest. Still, it’s nice to see that there is a consistent formalism for calculating them, and that the late-time result, say for the loop correction to the 2-point function, depends only on the behaviour of the inflaton field at horizon-exit, and not on the whole intervening history^{1}.

The “In-In” formalism, as the name suggests, consists of computing the expectation values of operators in the $|in\rangle$ vacuum, in which the fluctuations behave as free fields in the far past. Take the Hamiltonian, expand it about the classical solution, and drop the terms linear in the fluctuations. This yields the fluctuation Hamiltonian, $\tilde{H}$. Working in the interaction picture for $\tilde{H}$, $\langle in| Q(t) |in\rangle \equiv \left\langle \overline{T}\exp\left(i\int^t_{-\infty} H_I dt\right)\, Q^I(t)\, T \exp\left(-i\int^t_{-\infty} H_I dt\right) \right\rangle$ where $T$ denotes time-ordering, $\overline{T}$ denotes anti-time-ordering and $Q^I(t)$ is a string of interaction picture fields.

Because there’s both time-ordering and anti-time-ordering, the Feynman rules that follow from this formalism are a bit baroque. Steve applies them to calculate the 1-loop correction to the 2-point function of scalar fluctuations. The well-known tree-level answer is $\int d^3x e^{i\vec{q}\cdot(\vec{x}-\vec{x}')} \langle in|\zeta(\vec{x},t)\zeta(\vec{x}',t')|in\rangle_{\text{tree}} = \frac{8\pi G H^2(t_q)}{4(2\pi)^3 |\epsilon(t_q)|q^3}$ The 1-loop correction is $\int d^3x e^{i\vec{q}\cdot(\vec{x}-\vec{x}')} \langle in|\zeta(\vec{x},t)\zeta(\vec{x}',t')|in\rangle_{\text{1-loop}} = -\frac{\pi(8\pi G H^2(t_q))^2 N}{15(2\pi)^3 q^3}[\log q +C]$ where $\epsilon =-\frac{\dot{H}}{H^2},\qquad H= \frac{\dot{a}}{a}$ $t_q$ is the time when the mode with wave number $q$ exits the horizon, and $N$ is a numerical constant.

Note that this is suppressed, relative to the tree-level answer, by a factor of the slow-roll parameter, $\epsilon$, and an additional factor of $H^2/M_p^4$. That makes it *really tiny*. But, interestingly, even in the very slow-roll limit, where $H(t_q),\epsilon(t_q)$ are nearly constant, it departs from the scale-invariant tree-level answer by a factor of $\log(q)$.

Anyway, lots of interesting discussion of the subtleties of field theory in (quasi-)de Sitter space. You *know* you meant to spend some time understanding that stuff.

^{1} This question doesn’t really arise for tree-graphs, where one only has to consider some fixed external wave number $q$. For loop graphs, we need to *integrate* over the wave numbers, $p$ on internal lines. Only if the integral is dominated by $p\sim q$, can we talk about a definite “time of horizon exit,” when $p/a\sim q/a\sim H$.

## Re: In-In

Hi Jacques,

I have a question for you. How do you compute such in-in amplitudes in string theory? Isn’t this important if one is to extract cosmological predictions from string theory? Looking at books like Polchinski, all they explain is how to compute the ordinary scattering amplitudes.

Marc