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July 2, 2005

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 history1.

The “In-In” formalism, as the name suggests, consists of computing the expectation values of operators in the |in|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, H˜\tilde{H}. Working in the interaction picture for H˜\tilde{H}, in|Q(t)|inT¯exp(i tH Idt)Q I(t)Texp(i tH Idt) \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 TT denotes time-ordering, T¯\overline{T} denotes anti-time-ordering and Q I(t)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 d 3xe iq(xx)in|ζ(x,t)ζ(x,t)|in tree=8πGH 2(t q)4(2π) 3|ϵ(t q)|q 3 \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 d 3xe iq(xx)in|ζ(x,t)ζ(x,t)|in 1-loop=π(8πGH 2(t q)) 2N15(2π) 3q 3[logq+C] \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 ϵ=H˙H 2,H=a˙a \epsilon =-\frac{\dot{H}}{H^2},\qquad H= \frac{\dot{a}}{a} t qt_q is the time when the mode with wave number qq exits the horizon, and NN is a numerical constant.

Note that this is suppressed, relative to the tree-level answer, by a factor of the slow-roll parameter, ϵ\epsilon, and an additional factor of H 2/M p 4H^2/M_p^4. That makes it really tiny. But, interestingly, even in the very slow-roll limit, where H(t q),ϵ(t q)H(t_q),\epsilon(t_q) are nearly constant, it departs from the scale-invariant tree-level answer by a factor of log(q)\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 qq. For loop graphs, we need to integrate over the wave numbers, pp on internal lines. Only if the integral is dominated by pqp\sim q, can we talk about a definite “time of horizon exit,” when p/aq/aHp/a\sim q/a\sim H.

Posted by distler at July 2, 2005 3:55 AM

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6 Comments & 0 Trackbacks

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

Posted by: Marc on July 2, 2005 8:11 PM | Permalink | Reply to this

Re: In-In

A very interesting question, to which I don’t know the answer.

That, among many other issues, is what makes string cosmology a murky subject at present.

Posted by: Jacques Distler on July 2, 2005 8:53 PM | Permalink | PGP Sig | Reply to this

Re: In-In

I think the formalism for that does not exist, and I am not sure whether or not it should (as the usual lore is that string perturbation theory only calculates on-shell quantities, and I am not sure these are on-shell). A related example is calculation of correlation functions (not just partition function) at finite temperature. In print, I am only aware of one attempt to face this issue, by Samir Mahtur, e.g 9306090.

Moshe

Posted by: Moshe Rozali on July 2, 2005 9:31 PM | Permalink | Reply to this

Observables

In string theory, we can calculate gauge-invariant observables. (I think that’s what you meant when you said “on-shell.”)

While, perhaps, somewhat awkwardly formulated (i.e., they are formulated in a particular gauge), these are things which are sensible observables, at least in perturbation theory.

Our inability to deal with cosmological backgrounds runs deeper than that.

Posted by: Jacques Distler on July 2, 2005 9:42 PM | Permalink | PGP Sig | Reply to this

Re: In-In

What I meant is the usual statement that only observables that are defined asymptotically (such as S-matrix elements) seem to have good definition, maybe that is just the statement of diff. invariance as part of the gauge symmetry of string PT. This excludes all local probes of the theory (such as Green’s function), but I am not sure about various quantities one calculates in closed time path formalisms. They do seem to rely only on asymptotia, so maybe they should be part of string PT.

I agree that we’ll have to understand such issues better before we apply string theory (rather than its low energy limit) to cosmology. Could be kind of fun.

Posted by: Moshe Rozali on July 2, 2005 10:01 PM | Permalink | Reply to this

Re: In-In

This is a very interesting paper; I’m not sure how I missed it the first time around. Finn and I showed that as long as the (effective) action for the fluctuations is diffeo invariant then in a quasi-de Sitter (inflating) spacetime you get something analagous to a Callan-Symanzik equation for n-point functions of the inflaton (or the appropriate gauge invariant variable). Emil Motolla pointed out to me that essentially what we were doing was the tree level calculation in Schwinger’s in-in formalism; I’m interested to see whether or not our results really hold up at the one-loop level.

Posted by: Bob McNees on July 6, 2005 9:14 PM | Permalink | Reply to this

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