ModelIndependent Signatures of New Physics in SlowRoll Inflation
Abstract
We compute the universal generic corrections to the power spectrum in slowroll inflation due to unknown highenergy physics. We arrive at this result via a careful integrating out of massive fields in the “inin” formalism yielding a consistent and predictive lowenergy effective description in timedependent backgrounds. The density power spectrum is universally modified at leading order in , the ratio of the scale of inflation to the scale of new physics; the tensor power spectrum receives only subleading corrections. In doing so, we show how to make sense of a physical momentumcutoff in loop integrals despite dynamical redshifts, and how the result can be captured in a combined effective action/effective density matrix, where the latter contains nonadiabatic terms which modify the boundary conditions.
pacs:
04.62.+v, 98.80.k, 98.70.VcI Introduction
The dawning of the era of precision cosmology demands that we understand the history of our universe theoretically with the same accuracy as experiment. The WMAP determination of acoustic peaks in the CMB spectrum to 1% accuracy has given strong support to the existence of an era of inflation wmap7 . Inflation, famously, predicts the primordial power spectrum of density fluctuations underlying all structure in the universe.
This ability of WMAP and future Planck data planck to constrain theoretical models of the early Universe has set off a scramble to delineate a theoretically controlled computation of the primordial inflationary power spectrum. The textbook approach makes a number of explicit and implicit assumptions, each of which can affect the power spectrum at the accuracy measured. The main assumption is that the density spectrum can be reduced to that of an adiabatic fluid whose excitations are always weakly coupled to gravity. This can be conveniently encoded in a single scalar field Lagrangian with BunchDavies initial conditions in a fixed inflationary cosmology without metric fluctuations. The most obvious issue with these assumptions is that the redshifts implied by the 60 folds of inflation necessary to solve the horizon and flatness problem, place the relevant momentum scales in an energy regime far beyond the Planckscale, where in principle gravitational backreaction and quantumgravity corrections cannot be ignored Brandenberger:1999sw . Turning the issue around, it also implies a window of opportunity that quantum gravity or any other New Physics arising at high energy scales could have a measurable effect on the power spectrum. This question was actively pursued some time ago with the conclusion that in toy models Niemeyer:2000eh ; Kempf:2000ac ; Niemeyer:2001qe ; Kempf:2001fa ; Martin:2000xs ; Brandenberger:2000wr ; Brandenberger:2002hs ; Martin:2003kp ; Easther:2001fi ; Easther:2001fz ; Easther:2002xe ; Kaloper:2002uj ; Kaloper:2002cs ; Danielsson:2002kx ; Danielsson:2002qh ; Shankaranarayanan:2002ax ; Hassan:2002qk ; Goldstein:2002fc ; Bozza:2003pr ; Alberghi:2003am ; Schalm:2004qk ; porrati ; Porrati:2004dm ; Hamann:2008yx ; Achucarro:2010da ; Starobinsky:2002rp ; Chen:2010ef ; Ashoorioon:2010xg ; Ashoorioon:2004vm one can obtain measurable corrections of the order , comparable to intrinsic cosmic variance, with the Hubble scale and the scale of New Physics Bergstrom:2002yd ; Martin:2003sg ; Martin:2004iv ; Martin:2004yi ; Easther:2004vq ; Greene:2005aj ; Easther:2005yr ; Spergel:2006hy .
To truly connect with the data one needs the universal generic modelindependent corrections to the power spectrum in terms of an effective field theory that encodes order by order the corrections due to New Physics at high scales as well as deviations from adiabaticity. How to account for New Physics is formally wellunderstood in terms of Wilsonian effective actions, and for the adiabatic mode this has been actively pursued recently Cheung:2007st ; Senatore:2010wk ; Weinberg:2005vy ; Weinberg:2006ac ; Weinberg:2008hq ; Weinberg:2010wq ; vanderMeulen:2007ah ; Achucarro:2010da . However, Wilsonian effective actions are only consistent provided adiabaticity is maintained Burgess:2002ub ; Burgess:2003zw ; Burgess:2003hw . In practice this has meant that energy is assumed to be a conserved quantity. But precisely this is impossible in a cosmological timedependent background. Redshifts continuously mix the regimes of various scales and strictly speaking a welldefined separation of energy cannot be maintained. This longstanding paradox has fundamentally hampered the construction of low energy effective theories in cosmological spacetimes, literally since energy is not a conserved quantity.
In a previous letter Jackson:2010cw we provided an algorithmic solution to this obstacle to compute the generic new physics corrections to the inflationary power spectrum. One can generate the universal low energy effective action by integrating out a massive field in any particular New Physics model. This is sensible in a cosmological setting, as long as one computes late time expectation values directly in nonequilibrium realtime QFT via the SchwingerKeldysh approach. One of the principal difficulties found in previous approaches was the correct procedure to implement an energy and momentum scale cutoff in a timedependent background; in intermediate steps the order of energy and momentum integrals now no longer commute. We have overcome this through the realization that the contributions from the massive field may be reliably captured in a stationaryphase approximation for vertex evaluation. This localizes interactions to fixed moments in time and allows us to unambigously implement the cutoff in momenta.
There are several noteworthy features that this algorithmic solution reveals:

Most importantly, New Physics and nonadiabatic corrections no longer separate. Integrating out the heavy field not only causes changes in the action, but also in the effective density matrix of the lowenergy field. The former can account for nonadiabatic effects.

The nonadiabatic terms are nonlocal in position space, but localized on a New Physics Hypersurface where the physical momentum equals the mass of the heavy field . Intuitively this is what should happen and was the basis of many earlier ad hoc models. Here it is a consequence of integrating out the heavy field.

These terms can be interpreted as modified initial conditions for the lowenergy adiabatic inflaton. This again confirms the qualitative insights gained from toy models, but now quantitatively.
In this article we apply this cosmological effective field theory approach to the cosmologically relevant scenario of slowroll inflation. We compute the corrections to the scalar power spectrum resulting from interactions with a heavy field. Parametrizing the corrections in a manner expedient to comparison with observational data, the universal generic correction to the scalar power spectrum due to unknown highenergy physics equals
where is the first slowroll coefficient. The power of the approach, however, is that each of the variables can be computed in terms of the parameters of whichever theoretical model for the unknown New Physics one has in mind. For the power spectrum the result is merely an amplitude correction, but it will be very interesting to see the effect in higherorder correlation functions.
This article is organized as follows. In §2 we present a simple model of slowroll inflation containing new physics at high energies and analyze the field fluctuations in two gauges which will prove useful. In §3 we review the inin formalism and how to compute field fluctuation correlations for our theory. In §4 we calculate the scalar and tensor power spectra in the spatially flat gauge, then convert these to the uniform density gauge and parametrize the answer in a way which facilitates easy comparison against observation. In §5 we discuss observational prospects and conclude.
Ii Single Field SlowRoll inflation plus a Massive field
The minimal action description for inflation is a scalar field coupled to gravity,
(1) 
We assume a spatially homogeneous ansatz for the background metric
Defining , the background equations of motion are
(2)  
(3) 
Primes denote derivatives with respect to the field, while overdots denote derivatives with respect to coordinate time.
To facilitate analysis of inflating solutions, one defines the (first) Hubble slowroll parameter
(4) 
Inflation occurs for and ends when , which we use to define . It will prove useful to also define a second slowroll parameter,
One could continue defining an infinite hierarchy of such parameters , but this will be sufficient for our purposes.
Of course there is a direct relationship between and . This is given by
It will be helpful to define the socalled “potential slow roll” parameters
We can then invert this to express in terms of , with exact solutions possible in the case of powerlaw inflation.
ii.1 Perturbative Interactions
We may take the initial value of the inflaton to be , so that the potential near this point is approximated as
where are all constants. Thus , and hence , can be solved for in terms of these coupling constants.
To the inflationary action (1) we then add a massive field with renormalizable interactions to the inflaton:
We will be working to cubic order in field fluctuations, but are only interested in single contractions of the heavy field fluctuations, justifying our expansion in to second order. The combined action will then produce an inflationary period but contain New Physics at scales determined by the couplings and mass scale .
A naive approach, based on static background quantum field theory intuition, might be the following: since we are presumed to be below the energy scale where quanta can be created, we can simply integrate out this field to yield the following effective ‘New Physics’ potential for :
This is incorrect, for a variety of reasons. In the remainder of this article we will delineate the correct procedure to obtain the effective description.
ii.2 Fluctuations
Let us now assume that the spatiallyhomogeneous component satisfies its own equations of motion to lowest order in and consider fluctuations around this background ^{1}^{1}1The interactions are tadpole terms, and we are therefore not expanding about a saddle point of the total action. However for the purposes of deriving a low energy Wilsonian action this is not an obstacle, as we address later in detail.,
Although our observable of interest is the power spectrum which does not usually require gaugefixing, we wish to incorporate the effects of interactions for which the inflaton field and metric fluctuations mix together, requiring the construction of gaugeinvariant quantities. We therefore also consider tensor fluctuations. Utilizing the ADM formalism, we parametrize the metric as
with the lapse function and the shift vector. Substituting this into the action produces
where
The equations of motion for and are just the Hamiltonian and momenta constraints,
In solving for and we can ignore the effect of ; as we assume a singlefield inflation model, rather the the hybridinflation model of and . This is valid because the corrections to the free equations of motion will be . Since this is the same order as the fluctuations we are computing, we may trust our unperturbed background solution.
There are two choices of gauge we will employ: spatially flat and uniform density. That is, using different gauges we may exchange perturbations in for (certain) perturbations in and viceversa. Both will be useful at different points in our calculation.
ii.2.1 Spatially Flat Gauge
We will perform the interaction calculations inside the horizon using the spatially flat gauge. This corresponds to the choice
Fluctuations are then parametrized by and . We now need the solution for in terms of these fluctuations. Since we are interested in quadratic fluctuations of and , it suffices to compute the background corrections to first order. This is because any third order terms for the background would multiply the zeroorder terms for the fields, which are automatically satisfied. We denote so that is a firstorder perturbation, as is . The solutions are given by
Substituting this back into the action (II.2) yields (to lowest relevant order)
Note the appearance of a small inflaton mass induced from the gaugefixing. In writing this we have omitted the massmixing term which is subleading in . Additionally, we have neglected all interactions which do not contain , since we are only interested in the effect of this heavy field. One of these neglected terms, the coupling, produces an scale interaction and it would be interesting to compare this to the result obtained here. We save this for a future study.
ii.2.2 Uniform Density Gauge
At the moment of scalar fluctuation horizon crossing we then convert these to the uniform density gauge since these will then stay constant and are good observables. This gauge corresponds to the choice
Note that the and fluctuations remain identical, while scalar fluctuations are now parametrized by . The conversion between gauges is
(8) 
Of course, were we to include the effects of on the background evolution, this hybridinflation model would not have constant scalar fluctuations. But as addressed previously, the leadingorder fluctuations employ a singlefield background and hence these fluctuations will still have constant superhorizon .
Iii The InIn Formalism
iii.1 InOut Amplitudes Versus InIn Expectation Values
Quantum field theory in a static background most often employs the “inout” formalism to produce scattering amplitudes. Defining the states and in the asymptotic past and future, respectively, amplitudes are defined as
Crosssections are then obtained by squaring the amplitude. For nonequilibrium systems, such as a cosmological background, the fundamentally sound approach to computing expectation values such as the power spectrum is the SchwingerKeldysh approach Calzetta:1986cq .
The procedure is the following. At some early time (in the present context, the onset of inflation) we begin with a pure state , then evolve the system for the bra and ketstate separately until some late time , when we evaluate the expectation value:
Traditionally, the instate is taken to be the BunchDavies vacuum state Bunch:1978yq , but this is not necessarily so. Expanding cosmological backgrounds allow for a more general class of vacua, which can be heuristically considered to be excited states of inflaton fluctuations. In the present context, we will find that integrating out highenergy physics generically results in boundary terms in the effective action, which represent such excited states.
If we denote the fields representing the “evolving” ket to be and those for the “devolving” bra to be , the inin expectation value (III.1) can be computed from the action
(10) 
together with the constraint that , and . It is then helpful to transform into the Keldysh basis,
In this basis the total action (10) equals
(11) 
iii.2 Density Matrices
Although we started with a pure state in (III.1), in general we could take expectation values with respect to a mixed state,
Here the density matrix is normalized so that . In the path integral language, the density matrix is equal to the logarithm of the imaginary component of the action, so that
We will find that integrating out will generically lead to a mixed state for and , since we are removing states from a unitary process.
iii.3 Perturbative Solution of Fluctuations
The equations of motion for the fluctuations are
(12) 
We will now solve these perturbatively. The Feynman rules are summarized by Figure 1, where we have used the same diagrammatic notation as in vanderMeulen:2007ah .
iii.3.1 Zeroth Order
We first consider the free fluctuation equations of motion. To do this it is helpful to switch to conformal time defined by
Neglecting the interactions in equations (12), then Fourier transforming into the comoving momentum basis, they become
Here we have introduced the conformal Hubble parameter and suppressed all indices on . The solution for is given by the Hankel function of the first kind,
(13) 
If we expand in terms of the slowroll parameters, this is the familiar expression
The linearly independent solution is simply the complex conjugate and hence uses .
For the massive fluctuations the free field solution is also a Hankel function but can written more transparently using the WKB approximation as
(14) 
The timedependent frequency here is , and the WKB approximation is always valid for .
For fluctuations the free solution is again given by Hankel function but of a different order,
iii.3.2 Higher Order
We may then iteratively solve for higherorder solutions to the fluctuations,
(15) 
Fourier transforming into comoving momentum, these vertices can be evaluated using the retarded Green’s function can be written in terms of the fluctuation solutions,
These are shown in Figure 2. The advanced Green’s function is then simply the timereversal of this:
Note that comoving momentum is conserved at vertices. A similar procedure applies for and using their corresponding retarded Green’s function and ,
Of course the solution may then be iterated again to obtain yet even higherorder solutions,
Note that for fluctuations the slowroll parameters are implicitly included, and that the order of the solution refers to the order of the couplings .
iii.3.3 Statistical Correlations
After obtaining to our desired order, we may then take statistical averages of the zeroth order solutions via
to get correlations. This can be heuristically thought of as gluing the crosses together in all possible ways. An example is shown in Figure 3.
While this may produce Feynman diagrams with loops, representing integrals over comoving momentum, they are nonetheless completely classical. An important difference of the inin formalism is that loops may represent statistical, but not quantum, fluctuations.
iii.4 SelfConsistency of Background Solution
The introduction of fluctuations means that the background solution will be slightly modified due to backreaction. This is an correction to the linear coupling , as shown in Figure 4, having the value
The apparent timedependence is illusory, as follows. Let us convert the integral over comoving momentum into one over physical momentum. In staticbackground QFT loop regularization one imposes a UV cutoff on the Wickrotated 4momentum, thus respecting Lorentz symmetry. Since the quaside Sitter background of inflation breaks Lorentz symmetry (‘energy’ is illdefined) we cannot impose such a cutoff, but comoving momentum is conserved, and therefore we can perform an equivalent procedure as follows. Via (14) the field’s conjugate variable to time is given by
Thus we may place a cutoff on at some scale, effectively imposing a cutoff on . Let us momentarily assume that is constant, allowing us to estimate the correction as
This is independent of , assuring us that it is the same answer one would get from a static spacetime answer had we simply truncated . This timeindependent correction can then be cancelled by appropriate redefinition of . A similar procedure can of course be done for the tadpole of , as shown in Figure 3.
Iv Power Spectrum Evaluation
Here we will give the explicit formula for computing the scalar and tensor power spectrum in our highenergy model.
iv.1 Inflaton Fluctuations
We will need to know the correlation of inflaton fluctuations at the moment of horizon crossing,
To evaluate this perturbatively, substitute the classical solution into this expression. In the decoupling limit or , the inflaton fluctuation power spectrum is simply
where is the horizoncrossing time of mode , . The first order corrections are then:
There are no interactions in the action (11) which will produce this, and so we turn to the second order contributions,
(19) 
There are a total of four diagrams which could possibly contribute at , shown in Figure 5. We will give explicit expressions for the first two,
iv.2 Inflaton Vertex Evaluation
We first dissect the interaction at the vertex before evaluating the full diagram. Since the time coordinate is integrated over, vertex evaluation is very different than in standard quantum field theory. Writing out the Green’s and Wightman’s functions in terms of ’s and ’s, we see there are three types of cubic vertices. We will evaluate each.
The first is
(20) 
The superscript indicates there is some dependence upon the moment that the mode leaves the horizon. Since we assume all interactions happen well inside the horizon, a good approximation is
where is given by (13). To leading order in the slowroll parameters, the scale factor is given by
Substituting this into (20) results in the expression
(21) 
where the phase is given by