The in-in formalism and cosmological perturbations
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This post is written by two great guys Peter Adshead (his photo is on the left) and Eugene Lim (on the right). Peter is a PhD student of Richard Easther at Yale U., while Eugene is Richard’s former postdoc now working at Columbia U. Dmitry. |  |
The discovery of the anisotropies in the cosmic microwave background (CMB) by the COBE satellite in 1992 heralded in a new era of cosmology. Instead of simply studying the gross evolution of the universe (usually termed “background evolution”), cosmologists now study the structure and evolution of the tiny perturbations about this background. The anisotropies observed in the CMB are believed to be the seeds from which all of the structure (stars, galaxies etc.) we observe today eventually grew. These perturbations, detected as an average of over/under-density of 
, are thought to be generated during an early period of accelerated expansion – inflation.
Until recent years, studies of cosmological perturbations have been mainly focused on the linear component – which is not so surprising given its minute amplitude. However, our instruments have advanced to a stage where we can actually begin to probe structure beyond the linear order. On the other hand, the theory of calculating perturbations beyond the linear order is an extremely challenging problem.
Cosmological perturbations are characterized by correlation functions (or moments). Given a field 
that lives on some spatial foliation, we can describe it by an infinite tower of N-point correlation functions. For example the power spectrum is the 2-point correlation 
the bispectrum is the 3-pt correlation 
the trispectrum is the 4-pt correlation 
, and so on. If these fields have initial amplitudes and phases which are drawn from a Gaussian distribution, then linear evolution (modulo any non-local interactions) will preserve this property. In this case, the statistics of the field will be entirely described by the 2-pt correlation function, or power spectrum. N-pt correlation functions beyond the 2-pt will simply be a power of the 2-pt function for N even (N/2 unconnected copies of the propagator), and will vanish if N is odd. Beyond linear order, however, interactions generally mix the modes, resulting in non-zero higher moments. These are generically called “non-Gaussianities”. Hints of a non-zero bispectrum by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite in the WMAP year 5 data release has generated enormous interest in the calculation of these correlations.
The cosmological perturbations are believed to be generated at very early times during a period inflation. During inflation, quantum fluctuations at very short scales (deep inside the Hubble horizon) are stretched by the accelerated expansion and freeze out well outside the horizon. Linear perturbation theory allows us to track the evolution of these perturbations and compute the power spectrum of fluctuations (scalar fluctuations in the curvature of the spatial hypersurfaces and gravitational waves) that is frozen outside the horizon. Beyond the linear theory, there are two important corrections to this picture: quantum corrections and non-linear couplings between modes. The former will in general correct the amplitude and running (in 
space) of any spectra we compute using linear theory, while the latter will lead to non-trivial higher correlation functions.
For many years, the calculation of such corrections were based on a hodge-podge of methods which were generally messy and poorly controlled. A breakthrough occurred in 2002 when J. M. Maldacena combined an approach to quantum field theory called the Schwinger-Keldysh “in-in” formalism with the Arnowitt-Deser-Misner (ADM) formulation of Einstein’s general relativity, to exactly calculate the 3-pt correlation function of perturbations predicted by standard slow roll inflation (arXiv:astro-ph/0210603). This formalism was further refined and extended to quantum corrections by S. Weinberg in 2005 (arXiv:hep-th/0506236). As the name “Schwinger-Keldysh” indicates, this method is not new – it was formulated in 1960’s to calculate finite time matrix elements. It has been used by condensed matter physicists and finite temperature field theorists for many years. However, its application to cosmology is limited, groundbreaking papers were written by R. D. Jordan (1986) and E. Calzetta and B. L. Hu (1987), and it was not applied to cosmological perturbations until Maldacena and Weinberg.
The “in-in” formalism is just standard quantum field theory rigged to compute correlation functions instead of transition amplitudes between states (“in-out”). In other words, we want to calculate the expectation value of any number of products of fields at some time t, given initial conditions at an earlier time 
. Mathematically, one begins with the action of general relativity coupled to a slowly rolling scalar field 
which is driving the inflationary expansion
![S=\frac{1}{2}\int d^{4}x\sqrt{-g}\left[R -(\partial \varphi)^{2}+2V(\varphi)\right].](http://www.nonequilibrium.net/latexrender/pictures/633400ccd0233ba9b8cdfef400de932f.gif "S=\frac{1}{2}\int d^{4}x\sqrt{-g}\left[R -(\partial \varphi)^{2}+2V(\varphi)\right].")
The ADM formalism then separates out the dynamical degrees of freedom from the gauge degrees of freedom and gives an action containing only dynamical variables. Maldacena’s insight was to realize that one could express the action entirely in terms of tensor fluctuations (gravitational waves) and, depending on the choice of gauge, either the Bardeen curvature, or fluctuations in the scalar field. This is achieved by solving constraint equations for the auxiliary fields, that appear in the action as Lagrange multipliers, and substituting the solutions back into the action.
The Hamiltonian is then constructed in the usual way, defining canonical momenta, performing the Legendre transformation and eliminating time derivatives of the dynamical quantities in favor of the canonical momenta. One then separates the Hamiltonian into a piece that is quadratic in the fluctuations 
, and a higher order part 
and works in an interaction picture. In this scheme, drives the evolution of the operators while drives the evolution of the states. The correlation of an operator, 
, is then
,
where 
is the time ordering symbol.
One can then use Feynman-diagram like expansions to calculate the correlation to any order. The power spectrum can be thought of as the propagator and so lowest order results look like trees while corrections appear as loops. The key difference between these diagrams and the standard Feynman diagrams is that there is no time-flow. While each vertex is associated with both a momentum (or space) and a time integration, the propagators carry only 3-momentum.
In our recent paper (arXiv:0904.4207) we clarify and extend Weinberg’s approach (which is markedly different from the original “doubling of fields” approach, see for example Calzetta and Hu (1987)). In particular, we explore a subtlety that occurs in “in-in” calculations. To project out the correct initial conditions, one allows a small amount of evolution in imaginary time in the time integrals appearing above. This can be understood as insisting that the fields start off in the Bunch-Davies vacuum at the initial time. The subtlety is that the operators on the left and right of the expression are conjugates of each other. While the fields themselves are of course Hermitian, the deformation of the integration contour is not. This contour information has to be carefully treated throughout the calculation.
In our paper, we argue that the most efficient method for performing such calculations is to explicitly define contours and carry them around in the time integrals. There are alternative ways of tracking this information, for example one can analytically continue the integration variable to contain the appropriate imaginary piece. However, this prescription allows manipulations which can lead to erroneous results. Specifically, if one uses analytic continuation, the above correlation can be written as
![\ldots\int_{t_{0}}^{t_{2}}dt_{1}\left\langle\left[H_{\rm int}(t_{1}),\left[H_{\rm int}(t_{2}),\ldots\left[H_{\rm int}(t_{N}), \mathcal{O}(t)\right]\ldots\right]\right]\right\rangle](http://www.nonequilibrium.net/latexrender/pictures/e9912cd72e6ec94212abac6920c4470b.gif "\ldots\int_{t_{0}}^{t_{2}}dt_{1}\left\langle\left[H_{\rm int}(t_{1}),\left[H_{\rm int}(t_{2}),\ldots\left[H_{\rm int}(t_{N}), \mathcal{O}(t)\right]\ldots\right]\right]\right\rangle")
.
This formulation splits diagrams up into pieces which, considered alone, are unphysical (if one compares the expressions at second order, the 
term has been split into two pieces). If one is not careful to keep track of the imaginary pieces of the various time variables, one can run into spurious divergences or obtain erroneous results. The divergences manifest themselves when one is working at second order or above (order here refers to the number of vertices), and appear if one considers a limit where the momenta flowing through the vertices is not distinct. In reality, the small imaginary pieces introduced to select the vacuum prevent this divergence.
While this expression may appear to be a simplification of the previous one, it actually makes the computation much more algebraically intensive. To obtain results one must first combine all terms to cancel any divergences.
With the impending launch of the PLANCK satellite, we will be eventually be able to beat down the noise: PLANCK’s signal-to-noise limit in mode space is expected to be at least 5-10 times more sensitive than WMAP, allowing us to probe the CMB map to 10 times the current resolution. This will allow us to construct higher correlation functions and probe the running of the power spectrum, greatly increasing our ability to distinguish between different models of inflation. Thus the development of a controlled and consistent theory of calculating cosmological perturbations beyond linear order is very timely.
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Dear Peter and Eugene,
here is the first question of mine: suppose I select a different vacuum state – say, an Allen-Mottola vacuum that differs from B.-D. or arbitrary adiabatic vacuum for quasi-dS, to be as general as possible. To which extent will it affect the final result for NG? For example, will I generate spurious divergences choosing a different initial vacuum state?
Cheers,
Dmitry.
Hi Dmitry,
The effect of vacuum choice is essentially the same as the choice initial conditions. Choosing the adiabatic, or Bunch-Davies vacuum amounts to selecting the initial state that has no particle content. The epsilon prescription is basically a way of preparing the initial state. Inflation didn’t really start in the infinite past, but here we merely use this as a trick to ensure that any initial transients are well damped away. Choosing a different initial state can be loosely thought of as choosing a state that initially has some particle content. Although we haven’t given this a tremendous amount of thought, it is our opinion that with different initial conditions there should be no spurious divergences in the loops – provided the calculation is done properly. Each choice of vacuum will have an analogous prescription which prevents these spurious divergences from arising.
The effect on nan-Gaussianities of choosing an excited initial state built over a Bunch-Davies vacuum was considered by Tolley and Holman (arXiv:0710.1302). Since one has particle content initially, one sees effects which are proportional to the number of e-foldings of inflation – or the time elapsed since inflation began. This is not surprising since the interactions can no longer be thought of as being adiabatically turned on, the amount of interaction will be proportional to the amount of time elapsed. As well as these initial state signatures, excited states allow different momentum configurations. One can pick up so-called “folded triangles.” 3pt correlations functions with folded triangle momenta are those with momentum amplitude k1=2k2=2k3, hence one can only construct a triangle where all three modes essentially point in either the same or directly opposite directions.
I hope this answers your question!
Best,
Peter
Dear Peter,
let me slightly reformulate the question. B.-D., Allen-Mottola and adiabatic vacua describe similar physics at subhorizon scales and differ at superhorizon scales. Since, as you say, inflation starts at some particular moment of time
, you don’t want to specify the state of modes which are superhorizon at that moment of time, since you’ll never have access to them (am I correct?) If so, probably the best choice of vacuum in the initial moment of time would be adiabatic vacuum – since choosing this vacuum you only specify corr. properties of modes which are subhorizon at 
. On the other hand, if you want to glue a contracting patch of dS to the expanding patch, then the appropriate initial condition would be in vacuum state (one of the A.M. vacua, see Bousso, Maloney, Strominger for example).
I understood the part about
prescription, but the question I wanted to ask is really the following: why do you say that this particular prescription corresponds exactly to Bunch-Davies vacuum (and not, say, some other 
vacuum)?
Cheers,
Dmitry.
Hi Dmitry,
As I remember, the BD vacuum used to just mean the alpha=0 (in the Allen parameterization) choice of the dS invariant alpha vacuum before 2004/5 when the whole transplanckian business was all the rage. But these days (for better or for worse), people generally use the term BD vacuum to mean the usual positive frequency Minkowski vacuum in the limit of (k/a)>>H when these modes are created to sto speak, i.e. the one that is adiabatic and does not lead to particle production when the universe expands, even if we are not really in dS space but instead just in near-dS inflationary spacetime. The prescription we used picks up this vacuum — one can show this by using our prescription to calculate the 2-pt correlation. We use the term BD vacuum to refer to this.
I hope this answers your question! Apologies for a bit tardy!
Best,
Eugene
Dear Peter and Eugene,
Sure, if you really say that your prescription picks up adiabatic vacuum, then I agree (that’s what I was aiming to
).
Cheers and thanks for explanations, I think your paper is quite a piece of work!
Cheers,
Dmitry.
What is the hypothesis of the CMBR experiment?
And how long did the experiment take?
Please Reply A.S.AP.
What is the hypothesis of the CMBR experiment?
How long did the experiment last?
Hi Jessica,
which one in particular? There was a dozen of them at this point already – COBE, WMAP (still orbiting), ACBAR, Planck was just launched etc. etc.
Cheers,
Dmitry.