Model-Independent Signatures of New Physics in Slow-Roll Inflation
We compute the universal generic corrections to the power spectrum in slow-roll inflation due to unknown high-energy physics. We arrive at this result via a careful integrating out of massive fields in the “in-in” formalism yielding a consistent and predictive low-energy effective description in time-dependent 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 momentum-cut-off 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 non-adiabatic terms which modify the boundary conditions.
pacs:04.62.+v, 98.80.-k, 98.70.Vc
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 Bunch-Davies 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 Planck-scale, where in principle gravitational backreaction and quantum-gravity 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 model-independent 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 well-understood 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 time-dependent background. Redshifts continuously mix the regimes of various scales and strictly speaking a well-defined separation of energy cannot be maintained. This long-standing 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 non-equilibrium real-time QFT via the Schwinger-Keldysh approach. One of the principal difficulties found in previous approaches was the correct procedure to implement an energy and momentum scale cutoff in a time-dependent 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 stationary-phase 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 non-adiabatic 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 low-energy field. The former can account for non-adiabatic effects.
The non-adiabatic terms are non-local 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 low-energy 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 slow-roll 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 high-energy physics equals
where is the first slow-roll 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 higher-order correlation functions.
This article is organized as follows. In §2 we present a simple model of slow-roll inflation containing new physics at high energies and analyze the field fluctuations in two gauges which will prove useful. In §3 we review the in-in 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 Slow-Roll inflation plus a Massive field
The minimal action description for inflation is a scalar field coupled to gravity,
We assume a spatially homogeneous ansatz for the background metric
Defining , the background equations of motion are
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 slow-roll parameter
Inflation occurs for and ends when , which we use to define . It will prove useful to also define a second slow-roll 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 so-called “potential slow- roll” parameters
We can then invert this to express in terms of , with exact solutions possible in the case of power-law 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.
Let us now assume that the spatially-homogeneous component satisfies its own equations of motion to lowest order in and consider fluctuations around this background 111The 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 gauge-fixing, we wish to incorporate the effects of interactions for which the inflaton field and metric fluctuations mix together, requiring the construction of gauge-invariant 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
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 single-field inflation model, rather the the hybrid-inflation 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 vice-versa. 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 zero-order terms for the fields, which are automatically satisfied. We denote so that is a first-order 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 gauge-fixing. In writing this we have omitted the mass-mixing 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
Of course, were we to include the effects of on the background evolution, this hybrid-inflation model would not have constant scalar fluctuations. But as addressed previously, the leading-order -fluctuations employ a single-field background and hence these fluctuations will still have constant superhorizon .
Iii The In-In Formalism
iii.1 In-Out Amplitudes Versus In-In Expectation Values
Quantum field theory in a static background most often employs the “in-out” formalism to produce scattering amplitudes. Defining the states and in the asymptotic past and future, respectively, amplitudes are defined as
Cross-sections are then obtained by squaring the amplitude. For non-equilibrium systems, such as a cosmological background, the fundamentally sound approach to computing expectation values such as the power spectrum is the Schwinger-Keldysh 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 ket-state separately until some late time , when we evaluate the expectation value:
Traditionally, the in-state is taken to be the Bunch-Davies 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 high-energy 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 in-in expectation value (III.1) can be computed from the action
together with the constraint that , and . It is then helpful to transform into the Keldysh basis,
In this basis the total action (10) equals
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
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,
If we expand in terms of the slow-roll 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
The time-dependent 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 higher-order solutions to the fluctuations,
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 time-reversal 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 higher-order solutions,
Note that for fluctuations the slow-roll 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 in-in formalism is that loops may represent statistical, but not quantum, fluctuations.
iii.4 Self-Consistency 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 time-dependence is illusory, as follows. Let us convert the integral over comoving momentum into one over physical momentum. In static-background QFT loop regularization one imposes a UV cutoff on the Wick-rotated 4-momentum, thus respecting Lorentz symmetry. Since the quasi-de Sitter background of inflation breaks Lorentz symmetry (‘energy’ is ill-defined) 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 time-independent 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 high-energy 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 horizon-crossing 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,
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
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 slow-roll parameters, the scale factor is given by
Substituting this into (20) results in the expression
where the phase is given by