# Effective field theory and non-Gaussianity from general inflationary states

###### Abstract

We study the effects of non-trivial initial quantum states for inflationary fluctuations within the context of the effective field theory for inflation constructed by Cheung et al. which allows us to discriminate between different initial states in a model-independent way. We develop a Green’s function/path integral based formulation that incorporates initial state effects and use it to address questions such as how state-dependent is the consistency relation for the bispectrum, how many e-folds beyond the minimum required to solve the cosmological fine tunings of the big bang are we allowed so that some information from the initial state survives until late times, among others. We find that the so-called consistency condition relating the local limit of the bispectrum and the slow-roll parameter is a state-dependent statement that can be avoided for physically consistent initial states either with or without initial non-Gaussianities.

## 1 Introduction

Both theoretical Maldacena:2002vr ; Komatsu:2001rj ; Dalal:2007cu as well as observational studies Komatsu:2010fb ; Slosar:2008hx of non-Gaussianity in the statistics of metric perturbations have the potential to open up a veritable treasure trove of insights concerning the nature and evolution of quantum fluctuations generated during inflation.

One of the basic questions about quantum fluctuations of the inflaton concerns their initial state. The usual logic for determining this state makes use of the fact that at short enough distances or equivalently, early enough (conformal) times, the space-time appears approximately flat. Then we can choose the linear combination of solutions to the full mode equation that matches to positive frequency flat space modes. This choice leads to the usual Bunch-Davies vacuum Bunch:1978yq . While this procedure seems reasonable on the face of it, its basic premise is that the inflaton field will exist as a fundamental degree of freedom down to arbitrarily short distances/early times. This is a radical assumption; it is much more likely that the description of inflation as being driven by a scalar field is an effective one, valid only up to an energy scale . If this is indeed the case, the choice of initial state becomes a far more complex issue Martin:2000xs ; Danielsson:2002kx ; Kaloper:2002uj ; Collins:2005nu ; Collins:2006bg .

There have been a number of works Babich:2004gb ; Creminelli:2005hu ; Chen:2006nt ; Holman:2007na ; Tolley:2009fg ; Senatore:2009gt ; Baumann:2011su attempting to use the so-called shape of the bispectrum, i.e. whether the three-point function of the perturbation is peaked in momentum space when the triangle the three momenta form is squeezed, equilateral, flattened, or orthogonal (peaked on both equilateral and flattened triangles), to constrain models of inflation. More recently, the effects of more general initial states of inflaton perturbations on cosmological observables has been studied Chen:2006nt ; Holman:2007na ; Collins:2009pf ; Meerburg:2009ys ; Agullo:2010ws ; Ashoorioon:2010xg ; Ganc:2011dy ; Dey:2011mj ; Chialva:2011hc ; Kundu:2011sg ; Dey:2012qp , with a particular emphasis on the halo bias Agullo:2012cs ; Ganc:2012ae ; Agarwal:2012 .

Our aim in the present work is to make use of the recently developed effective field theory (EFT) of inflation Cheung:2007st , which incorporates the interpretation of metric perturbations as the near Goldstone mode of spontaneously broken time translations together with an effective description of the initial state for this mode. Doing this naturally incorporates the fact that there may be a limit to the domain of validity of this description via a cutoff indicating when the Goldstone theory becomes strongly coupled. We develop a method for integrating the initial density matrix into the effective action within the context of the in-in formalism Schwinger:1960qe ; Keldysh:1964ud ; Bakshi:1962dv ; Bakshi:1963bn ; this allows us to use methods of in-in perturbation theory in evaluating the effects of non-trivial initial states on correlation functions of the metric perturbations.

This combination of formalisms allows us to answer a variety of questions, such as whether the so-called consistency relation Creminelli:2004yq ; Cheung:2007sv for the bispectrum in the squeezed limit is true for an arbitrary state. We find that within the context of the EFT of inflation, non Bunch-Davies initial states can in fact violate this condition allowing for a loophole in the claim that the observation of local non-Gaussianity would necessarily rule out single field inflation.

In section 2 we review the in-in formalism for general initial states. We take particular care in showing how to construct Green’s functions that incorporate the choice of a non-standard initial state. Section 3 then reviews the EFT of inflation described in Cheung:2007st . Section 4 uses these techniques to impose constraints that the initial state must satisfy in order to be a state consistent with an inflationary phase as well as with the painstaking WMAP observations of the power spectrum.

Finally, in section 5 we turn to the calculation of the bispectrum in the presence of both, the cubic operators appearing in the EFT action for the Goldstone mode describing the fluctuations, as well as those from initial state non-Gaussianities. We then conclude in section 6 with some further directions this work could be used for.

## 2 The in-in formalism for general initial states

In order to evaluate time-dependent expectation values of operators as opposed to S-matrix elements connecting in and out states, we use the in-in or closed time path formalism Schwinger:1960qe ; Keldysh:1964ud ; Bakshi:1962dv ; Bakshi:1963bn . We review this formalism here, with an emphasis on the dependence of these correlators on the initial state. We perform most of the analysis here for a scalar field in flat space for simplicity; the conformal flatness of Friedmann-Robertson-Walker (FRW) spacetimes makes the translation from the flat spacetime case to the FRW one rather simple.

Suppose we want to evaluate the expectation value of an operator at a time in the Schrödinger picture. This is given by

(1) |

where denotes the density matrix of the system involved, satisfying the Liouville equation

(2) |

with the full Hamiltonian of the system. If we define the time evolution operator satisfying the Schrödinger equation

(3) |

we can solve eq.(2) as

(4) |

Inserting this into eq.(1) we then arrive at

(5) |

which can be read as follows: start with the initial state and then evolve it to a time at which point the operator is inserted. Finally, evolve back to . This is the origin of the closed time path required to describe in-in expectation values.

We can arrive at a more useful formulation of this result by explicitly evaluating the trace in the field representation where we use the eigenstates of the field operator ,

(6) |

We can write the expectation value now as

Using the standard representation of matrix elements of the time evolution operator in terms of the path integral we can write the above expression as

(8) |

where the fields in are evaluated at the initial time and is the action of the system. The subscript indicates that the path integrals are taken only over the configurations satisfying , ensuring continuity of the field viewed as being defined over the entire closed contour.

We can abstract a generating functional from eq.(8),

(9) |

where we have added sources to the appropriate actions. We can also extend the contour to by turning the sources off after the latest time in the string of field operators we may be computing the correlators of.

This generating functional can be used to construct a perturbative expansion that takes the existence of non-trivial initial conditions into account. In the Appendix, we show how to derive the in-in Green’s functions for an arbitrary Gaussian initial state; initial state non-Gaussianity can then be treated as an interaction term with support only at the initial time.

For use in the next few sections, we give here the parameterization of the initial density matrix,

(10) |

with

(11) |

being the normalization chosen so that and the subscript indicating higher order interactions in the initial state. The action is in general complex, with hermiticity of the density matrix imposing the condition . We consider the quadratic and cubic part of what can be thought of as a boundary action on the initial time hypersurface,

(12) | |||||

(13) | |||||

where we used spatial homogeneity explicitly in the quadratic part; it is implied in the cubic action. The kernel is fully symmetric in its spatial arguments, while is only necessarily symmetric in its first two. Note that all kernels are defined at the initial time . Further the kernels can include non-local interactions, which is not in contradiction with the fact that the field theory is local. For future reference, we rewrite the above terms in momentum space,

(14) | |||||

(15) | |||||

with a similar term corresponding to the kernel in the cubic action.

## 3 The effective field theory of inflation

The EFT developed by Cheung et al. Cheung:2007st mimics the construction in spontaneously broken gauge theories. At sufficiently high energies, the dynamics of the longitudinal gauge degree of freedom can be described by that of the would-be Goldstone boson. In the cosmological case, instead of an internal gauge symmetry, time reparameterization symmetry is spontaneously broken by the choice of inflaton zero mode configurations ; inflaton fluctuations are then defined via

(16) |

Time reparameterization invariance is realized non-linearly on the field; if is an infinitesimal time diffeomorphism, then the linear realization on is now enforced non-linearly on through

(17) |

The logic is to first choose a gauge for which and construct the most general form of the action, which is invariant under *time-dependent spatial* diffeomorphisms organized as fluctuations about a background solution,

(18) |

where we use the FRW metric, . For simplicity, we neglect functions of the perturbed extrinsic curvature since, in the simplest scenarios, these are higher order (see Cheung:2007st for a discussion of exceptions to this). This form is clearly sufficient to capture the models, being the canonical kinetic term, for instance. By re-introducing the time reparameterization gauge transformation a la Stückelberg, the action can be made fully diffeomorphism invariant, albeit in a non-manifest way, as follows,

(19) | |||||

with overdots denoting cosmic time derivatives. The functions and are determined by the requirement that the one-point function for vanishes. At the classical tree level this implies

(20) | |||||

(21) |

being the Hubble parameter. However, at the quantum level, for a given quantum state, there will be tadpole corrections. They encode how a given quantum state backreacts onto its background. For now we will keep and arbitrary, anticipating that both will receive quantum corrections.

The clear advantage of the EFT approach comes from recognizing that physics at higher energies is dominated by the dynamics of . In the high energy limit we can neglect fluctuations in the geometry, and neglect all non-derivative suppressed interactions of (except the linear ones which are necessary to determine the tadpole cancellation condition), giving in this limit the following effective action,

(22) | |||||

where we have integrated by parts and defined

(23) |

This action gives a general EFT for inflation, from which we can recover specific models by a judicious choice of parameters. At cubic and even quadratic order it is also possible to have additional terms with higher spatial derivatives, as considered, for example, in Senatore:2009gt . However, these terms are always expected to be subdominant to the above ones, unless the coefficients of the terms in eq.(22) happen to be unnaturally small.

The restriction to high energies means that the decoupling effective theory will break down shortly after horizon crossing. However, the simple relation between the Goldstone mode and the curvature perturbation , valid in single field inflation, tells us that we can use the EFT action to follow a given mode up to and just beyond horizon crossing, and then use the constancy of outside the horizon Cheung:2007st .

In order to compute correlation functions such as the power spectrum and the bispectrum, we can truncate the action, keeping only terms out to cubic order in and its derivatives. We also remove the tadpole term so as to ensure that quantum corrections to the slow-roll parameters are small and the action is at least quadratic in , as discussed in the next section. Moreover we include leading contributions from the mixing of with gravity, by expanding the lapse and shift to first order in . We can then rewrite the Lagrangian density to cubic order following Cheung:2007sv as

(24) |

where is the tadpole term above while

(25) | |||||

at next to leading order in slow-roll. The final term is a non-local interaction, which is just an artifact of the choice of gauge; all gauge-invariant observables will be local. Here , being the usual slow-roll parameter, and we have defined the effective sound speed for perturbations as

(27) |

The coefficients of the cubic terms are given by

(28) | |||||

(29) | |||||

(30) | |||||

(31) | |||||

(32) |

where and are other slow-roll parameters.

For use in later sections we convert the action to conformal time (related to cosmic time via ), with primes denoting conformal time derivatives,

(33) | |||||

(34) | |||||

The overall factor of comes from the change from cosmic to conformal time in the measure.

We can rewrite the Lagrangians above in terms of a field with the correct mass dimension and kinetic term. Thus define

(35) |

Then we can write, with ,

(36) | |||||

(37) | |||||

at next to leading order in slow-roll. We have defined

(38) |

where the coefficients with tildes are dimensionless.

This is an effective field theory and as such, has a limited domain of validity. There are two important scales for the EFT Baumann:2011su ; the symmetry breaking scale at which time translations are spontaneously broken by the background evolution and a description in terms of a Goldstone boson first becomes applicable and the strong coupling scale where perturbative unitarity is lost. The strong coupling scale can be computed Cheung:2007st ; Baumann:2011su and is given by

(39) |

## 4 Constraints on general initial states

Our formalism allows for the use of any state as an initial state. However, not all states are physically acceptable. There are two types of constraints that must be imposed on the , or equivalently the theory. The first type is a consistency condition due to the fact that the theory is an effective one and all of the physics we use this theory for must remain consistent with the precepts of effective field theory. In particular, we should not be able to excite modes near the cutoff . One way to do this is to absolutely forbid that such modes appear in the state, and this is the approach we will take here. It’s worth noting, though, that this may be too strict. We could imagine a scenario where we have some particles with energies near the cutoff in the initial state, but their contribution to the energy density is small and the time evolution of the system is such that we do not produce any more such particles. The exponential expansion associated with inflation only helps to enforce this weaker criterion. This may be worth pursuing in future work.

For now, we will assume that the state satisfies the Hadamard condition Fulling:1989nb , which demands that the Bogoliubov coefficients fall off faster than at large . As in Holman:2007na , we enforce this by demanding that for (the factor of comes from choosing as the physical cutoff), that is to say, as the mode momentum is redshifted across the cutoff of the theory at , it starts off in its vacuum state.

Beyond this, the next set of constraints enforces the fact that inflation should have occurred. This takes two forms; the backreaction of the energy density in the initial state should be subdominant relative to and corrections to the slow-roll equations of motion should be much smaller than the scale of the original terms themselves. We will take these up in turn.

### 4.1 Backreaction

To impose the backreaction constraint we demand that , where the energy density is given by . This will generally involve divergent mode integrals and a more sophisticated analysis would involve using something like an adiabatic expansion birrelldavies of the modes to isolate the divergent terms. The higher dimension terms in the action would then induce divergences beyond those of the free theory, but these would be absorbed in higher dimensional counter terms in the usual way when dealing with effective field theories. To arrive at the estimates we are looking for, we will default to a cruder method where we compute the expectation value of various operators, subtract off their values in the Bunch-Davies vacuum and then cut the integrals off at .

The quadratic part of the Lagrangian will give rise to contributions to the energy density such as

(40) |

When the are expanded in terms of the Bunch-Davies modes , there will be terms proportional to , as well as cross-terms proportional to with their complex conjugates. We want to argue that we can neglect these latter terms. They are proportional to and we would like to be able to use the Riemann-Lebesgue lemma to show that the oscillatory nature of these terms will wash out the integrals. This requires a large parameter in the exponential; at early times we can replace with and ferret out this large parameter by writing

(41) |

Once the cutoff becomes smaller than , the use of the EFT to describe inflation is no longer valid, so we expect that . At late times, on the other hand, the factor of in the energy density will exponentially redshift the cross-terms.

As an example consider the contribution from the spatial gradient part of the kinetic term,

(42) | |||||

where we use the notation in the Appendix for the modes. We should note that the above result is for a pure state; for a mixed initial state, we need only multiply the integrand by (see eq.(144)). We can replace , likewise for a mixed state we can replace , since the remainder is the Bunch-Davies contribution we are subtracting, and write the modes in the de Sitter limit,

(43) |

We also model as , where and falls to zero faster than at high . Putting all of this together we find a constraint of the form

(44) |

with and an infrared cutoff. The left hand side can be bounded above by

(45) |

For the second integral we can use the fact that , while we can model the result of the first integral by saying that to arrive at the result that the integral can be approximated as up to factors of order unity. We can then satisfy the backreaction constraint by demanding that

(46) |

But the left hand side can be rewritten as so that the constraint becomes . Of course, this will change if different models for the Bogoliubov coefficient are chosen, but we can see that the constraint is relatively easy to satisfy even with of order unity.

For a mixed initial state, we can model as , with decaying faster than at high . Then taking (with positive) in , the above constraint becomes . We see that this constraint can again be satisfied with both and of order unity.

The cubic interaction terms in also contribute to the energy density. While we would have to go to higher order in cubic interactions to find an effect within the Gaussian part of the initial density matrix, we can find a first order contribution from these terms when we include initial non-Gaussianity, as encoded in the cubic action of eq.(15). In fact we can use cubic interactions to place bounds on the high behavior of the kernel . Let’s show how this works by considering the contribution to the energy density of the operator appearing in the cubic Lagrangian of eq.(37). We have

(47) | |||||

(48) |

We then use the cubic part of the initial state action to compute the momentum space expectation value as

(49) |

The expectation value can be computed in the Gaussian theory by taking the time to lie on the part of the time contour and then using the Green’s functions found in the Appendix. Using the symmetry of the kernel in its arguments we find for example that

(50) |

Inserting this back into eq.(48) and doing the integrals we see that the cubic contribution to the energy density is

The strongest backreaction constraint on the kernel will come when we only keep the terms in the mode functions containing the Bogoliubov coefficients, since these are order unity. Doing this we can bound the cubic contribution as

(52) | |||||