Model-Independent Signatures of New Physics in Slow-Roll Inflation

0.81 MB·English

by  Mark G. Jackson

Model-Independent Signatures of New Physics in Slow-Roll Inflation
Mark G. Jackson and Koenraad Schalm
Instituut-Lorentz for Theoretical Physics, University of Leiden, Leiden 2333CA, The Netherlands
(Dated: January 22, 2013)
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 H/M, the ratio of the scale of inflation to the scale of new physics; the tensor power
spectrumreceivesonlysubleadingcorrections. Indoingso,weshowhowtomakesenseofaphysical
momentum-cut-offinloopintegralsdespitedynamicalredshifts,andhowtheresultcanbecaptured
inacombinedeffectiveaction/effectivedensitymatrix,wherethelattercontainsnon-adiabaticterms
3 which modify the boundary conditions.
1
0 PACSnumbers: 04.62.+v,98.80.-k,98.70.Vc
2

n
INTRODUCTION To truly connect with the data one needs the univer-
a
J sal generic model-independent corrections to the power
spectrumintermsofaneffectivefieldtheorythatencodes
9
The dawning of the era of precision cosmology de-
1 orderbyorderthecorrectionsduetoNewPhysicsathigh
mands that we understand the history of our universe
scales as well as deviations from adiabaticity. How to
] theoreticallywiththesameaccuracyasexperiment. The account for New Physics is formally well-understood in
h
WMAP determination of acoustic peaks in the CMB
t termsofWilsonianeffectiveactions,andfortheadiabatic
- spectrumto1%accuracyhasgivenstrongsupporttothe
p mode this has been actively pursued recently [28, 41–
existence of an era of inflation [1]. Inflation, famously,
e 46, 57]. However, Wilsonian effective actions are only
h predicts the primordial power spectrum of density fluc- consistent provided adiabaticity is maintained [47–49].
[ tuations underlying all structure in the universe.
In practice this has meant that energy is assumed to be

2 This ability of WMAP and future Planck data [2] to aconservedquantity. Butpreciselythisisimpossibleina
v constraintheoreticalmodelsoftheearlyUniversehasset cosmologicaltime-dependentbackground. Redshiftscon-
7
offascrambletodelineateatheoreticallycontrolledcom- tinuously mix the regimes of various scales and strictly
8
putation of the primordial inflationary power spectrum. speaking a well-defined separation of energy cannot be
8
0 The textbook approach makes a number of explicit and maintained. This long-standing paradox has fundamen-
. implicit assumptions, each of which can affect the power tally hampered the construction of low energy effective
4
0 spectrum at the accuracy measured. The main assump- theoriesincosmologicalspacetimes,literallysinceenergy
1 tion is that the density spectrum can be reduced to that is not a conserved quantity.
1 ofanadiabaticfluidwhoseexcitationsarealwaysweakly In a previous letter [50] we provided an algorithmic
:
v coupled to gravity. This can be conveniently encoded in solution to this obstacle to compute the generic new
i asinglescalarfieldLagrangianwithBunch-Daviesinitial physics corrections to the inflationary power spectrum.
X
conditionsinafixedinflationarycosmologywithoutmet- One can generate the universal low energy effective ac-
r
a ric fluctuations. The most obvious issue with these as- tion by integrating out a massive field in any particular
sumptionsisthattheredshiftsimpliedbythe60e-foldsof New Physics model. This is sensible in a cosmological
inflationnecessarytosolvethehorizonandflatnessprob- setting, as long as one computes late time expectation
lem, place the relevant momentum scales in an energy values directly in non-equilibrium real-time QFT via the
regime far beyond the Planck-scale, where in principle Schwinger-Keldysh approach. One of the principal dif-
gravitational backreaction and quantum-gravity correc- ficulties found in previous approaches was the correct
tions cannot be ignored [3]. Turning the issue around, it procedure to implement an energy and momentum scale
alsoimpliesawindowofopportunitythatquantumgrav- cutoff in a time-dependent background; in intermediate
ityoranyotherNewPhysicsarisingathighenergyscales steps the order of energy and momentum integrals now
could have a measurable effect on the power spectrum. no longer commute. We have overcome this through the
This question was actively pursued some time ago with realization that the contributions from the massive field
the conclusion that in toy models [4–32] one can obtain may be reliably captured in a stationary-phase approxi-
measurablecorrectionsoftheorderH/M,comparableto mation for vertex evaluation. This localizes interactions
intrinsic cosmic variance, with H the Hubble scale and to fixed moments in time and allows us to unambigously
M the scale of New Physics [33–40]. implement the cutoff in momenta.
2
There are several noteworthy features that this algo- SINGLE FIELD SLOW-ROLL INFLATION PLUS
rithmic solution reveals: A MASSIVE FIELD
Theminimalactiondescriptionforinflationisascalar
Most importantly, New Physics and non-adiabatic
• field coupled to gravity,
corrections no longer separate. Integrating out the
heavy field not only causes changes in the action, (cid:90) (cid:20)1 1 (cid:21)
but also in the effective density matrix of the low- Sinf[φ]= d4x√g 2Mp2lR− 2(∂φ)2−Vinf(φ) . (1)
energy field. The former can account for non-
adiabatic effects. Weassumeaspatiallyhomogeneousansatzfortheback-
ground metric
The non-adiabatic terms are non-local in position ds2 = dt2+a(t)2dx2.
• −
space, but localized on a New Physics Hypersur-
Defining H a˙/a, the background equations of motion
face where the physical momentum p(t) = k/a(t) ≡
are
equals the mass of the heavy field p(t) = M. In-
tuitively this is what should happen and was the M2
basis of many earlier ad hoc models. Here it is a φ˙ = plH(cid:48)(φ), (2)
− 4π
consequence of integrating out the heavy field.
12π 32π2
[H (φ)]2 H(φ)2 = V(φ). (3)
(cid:48) − M2 − M4
pl pl
These terms can be interpreted as modified ini-
• Primesdenotederivativeswithrespecttothefield,while
tial conditions for the low-energy adiabatic infla-
overdots denote derivatives with respect to coordinate
ton. This again confirms the qualitative insights
time.
gained from toy models, but now quantitatively.
To facilitate analysis of inflating solutions, one defines
the (first) Hubble slow-roll parameter (cid:15)
1
Inthisarticleweapplythiscosmologicaleffectivefield
theory approach to the cosmologically relevant scenario (cid:15) (φ) Mp2l (cid:20)H(cid:48)(φ)(cid:21)2. (4)
of slow-roll inflation. We compute the corrections to the 1 ≡ 4π H(φ)
scalar power spectrum resulting from interactions with
Inflation occurs for (cid:15) < 1 and ends when (cid:15) = 1, which
a heavy field. Parametrizing the corrections in a man- 1 1
we use to define t . It will prove useful to also define a
nerexpedienttocomparisonwithobservationaldata,the end
second slow-roll parameter,
universalgenericcorrectiontothescalarpowerspectrum
due to unknown high-energy physics equals (cid:15) (φ) dln(cid:15)1 = Mp2lH(cid:48)(cid:48)(φ).
2 ≡ dN 4πH(φ)
∆P H
ζ (C +C (cid:15) + ).
P ∼ M 0 1 1 ··· One could continue defining an infinite hierarchy of such
parameters(cid:15) ,butthiswillbesufficientforourpurposes.
n
where (cid:15)1 is the first slow-roll coefficient. The power of OfcoursethereisadirectrelationshipbetweenVinf(φ)
the approach, however, is that each of the variables Ci and (cid:15)1,(cid:15)2. This is given by
canbecomputedintermsoftheparametersofwhichever
3M2 (cid:16) (cid:15) (cid:17)
theoretical model for the unknown New Physics one has V(φ) = plH2 1 1 ,
8π2 − 3
in mind. For the power spectrum the result is merely an
(cid:114)
amplitude correction, but it will be very interesting to dV = 3H2M (cid:15)1 (cid:16)1 (cid:15)1 + (cid:15)2(cid:17),
see the effect in higher-order correlation functions. dφ − pl 4π − 3 6
This article is organized as follows. In 2 we present a d2V = 3H2(cid:16)2(cid:15) (cid:15)2(cid:17).
simplemodelofslow-rollinflationcontain§ingnewphysics dφ2 1− 2
at high energies and analyze the field fluctuations in two
It will be helpful to define the so-called “potential slow-
gauges which will prove useful. In 3 we review the in-in
§ roll” parameters
formalism and how to compute field fluctuation correla-
tionsforourtheory. In 4wecalculatethescalarandten- M2 (cid:18)V (cid:19)2 M2 V
sorpowerspectrainthe§spatiallyflatgauge,thenconvert (cid:15)V(φ)≡ 16pπl V(cid:48) , ηV(φ)≡ 8πpl V(cid:48)(cid:48).
these to the uniform density gauge and parametrize the
answerinawaywhichfacilitateseasycomparisonagainst We can then invert this to express (cid:15) ,(cid:15) in terms of
1 2
observation. In 5wediscussobservationalprospectsand (cid:15) ,η ,withexactsolutionspossibleinthecaseofpower-
V V
§
conclude. law inflation.
3
Perturbative Interactions We therefore also consider tensor fluctuations. Utilizing
the ADM formalism, we parametrize the metric as
We may take the initial value of the inflaton to be
ds2 = N2dt2+h (dxi+Nidt)(dxj +Njdt)
φ(t ) = 0, so that the potential near this point is ap- − ij
in
proximated as with N the lapse function and Ni the shift vector. Sub-
stituting this into the action produces
m2 g
Vinf(φ)≈V0+jφ+ 2 φ2+ 30!φ3+··· S = (cid:90) dtd3x√hN(cid:34)Mp2lR+ Mp2l(E Eij E2)
2 2N2 ij −
where V ,j,m2,g are all constants. Thus (cid:15) ,η , and
0 0 V V
hence (cid:15) ,(cid:15) , can be solved for in terms of these coupling (φ˙ Ni∂ φ)2 1
1 2 + − i hij∂ φ∂ φ (5)
constants. 2N2 − 2 i j
To the inflationary action (1) we then add a massive (χ˙ Ni∂ χ)2 1 (cid:21)
field χ with renormalizable interactions to the inflaton: + −2N2i − 2hij∂iχ∂jχ−V(φ,χ)
(cid:90) (cid:20)1 1 where
S [φ,χ] = d4x√g (∂χ)2+ M2χ2
new − 2 2
h deth ,
+g1φ2χ+ g2φχ2(cid:105). R ≡= Ricciijcurvature of spatial metric,
2 2
1(cid:16) (cid:17)
˙
We will be working to cubic order in field fluctuations, Eij ≡ 2 hij −∇iNj −∇jNi ,
butareonlyinterestedinsinglecontractionsoftheheavy
E = E i.
fieldfluctuations,justifyingourexpansioninχtosecond i
order. The combined action S S +S will then TheequationsofmotionforN andNiarejusttheHamil-
inf new
≡
produce an inflationary period but contain New Physics tonian and momenta constraints,
at scales determined by the couplings g and mass scale
i M2 M2 (φ˙ Ni∂ φ)2
M. 0 = plR pl(E Eij E2) − i (6)
A naive approach, based on static background quan- 2 − 2N2 ij − − 2N2
tum field theory intuition, might be the following: since 1hij∂ φ∂ φ (χ˙ −Ni∂iχ)2 1hij∂ χ∂ χ V,
we are presumed to be below the energy scale where χ- −2 i j − 2N2 − 2 i j −
(cid:34) (cid:35)
quanta can be created, we can simply integrate out this E i δ iE
field to yield the following effective ‘New Physics’ poten- 0 = Mp2l∇j j −N j (7)
tial for φ:
∂ φ(φ˙ Nj∂ φ) ∂ χ(χ˙ Nj∂ χ)
i j i j
g2 1 − − .
V (φ) = 1φ2 φ2 − N − N
new − 4 (cid:3) M2
− In solving for N and Ni we can ignore the effect of χ; as
g2
1 φ4+ (naive). we assume a single-field inflation model, rather the the
≈ 4M2 ··· hybrid-inflation model of φ and χ. This is valid because
This is incorrect, for a variety of reasons. In the remain- the corrections to the free equations of motion will be
der of this article we will delineate the correct procedure (g2). Since this is the same order as the fluctuations
O
to obtain the effective description. we are computing, we may trust our unperturbed back-
ground solution.
There are two choices of gauge we will employ: spa-
Fluctuations tially flat and uniform density. That is, using different
gauges we may exchange perturbations in φ for (certain)
Let us now assume that the spatially-homogeneous perturbations in gµν and vice-versa. Both will be useful
component φ (t) satisfies its own equations of motion to at different points in our calculation.
0
lowest order in g and consider fluctuations around this
i
background [70],
Spatially Flat Gauge
ϕ(t,x) φ(t,x) φ (t).
0
≡ −
Wewillperformtheinteraction calculations insidethe
Although our observable of interest is the ϕ-power spec- horizon using the spatially flat gauge. This corresponds
trum which does not usually require gauge-fixing, we to the choice
wish to incorporate the effects of interactions for which
the inflaton field and metric fluctuations mix together, hij = a(t)2hˆij, hˆij =δij +γij,
requiring the construction of gauge-invariant quantities. ∂ γ = 0, γ =0.
i ij ii
4
Fluctuations are then parametrized by ϕ,γ and χ. We THE IN-IN FORMALISM
ij
nowneedthesolutionforN,Ni intermsofthesefluctua-
tions. Sinceweareinterestedinquadraticfluctuationsof
In-Out Amplitudes Versus In-In Expectation Values
ϕ and γ , it suffices to compute the background correc-
ij
tionstofirstorder. Thisisbecauseanythirdorderterms
for the background would multiply the zero-order terms Quantum field theory in a static background most of-
for the fields, which are automatically satisfied. We de- ten employs the “in-out” formalism to produce scatter-
noteN =1+N1 sothatN1 isafirst-orderperturbation, ing amplitudes. Defining the states in and out in the
as is Ni. The solutions are given by asymptotic past and future, respecti|vel(cid:105)y, amp|litu(cid:105)des are
(cid:114) defined as
(cid:15) ϕ
N = 1+ 1 ,
2 M
pl
(cid:18) (cid:19)
Ni = ∂in, ∂i∂in= (cid:15)1 d 1 ϕ . AO ∼(cid:104)out|O|in(cid:105).
−√2dt √(cid:15) M
1 pl
Substitutingthisbackintotheaction(5)yields(tolowest
Cross-sections are then obtained by squaring the ampli-
relevant order)
tude. For non-equilibrium systems, such as a cosmolog-
(cid:90) (cid:34) M2 1 ical background, the fundamentally sound approach to
S = − dtd3x a3 − 2pl (∂γij)2+ 2(∂ϕ)2+jϕ computing expectation values such as the power spec-
trum is the Schwinger-Keldysh approach [55].
(cid:18) (cid:19)
+ m2 2 d (cid:0)a3H(cid:15) (cid:1) ϕ2+ 1(∂χ)2+ 1M2χ2 The procedure is the following. At some early time t
− a3dt 1 2 2 in
(in the present context, the onset of inflation) we begin
(cid:21)
+ g1ϕ2χ+ g2ϕχ2+ 1γ ∂ χ∂ χ . withapurestate in(tin) ,thenevolvethesystemforthe
2 2 2 ij i j bra-andket-state|separa(cid:105)telyuntilsomelatetimet,when
Note the appearance of a small inflaton mass induced we evaluate the expectation value:
from the gauge-fixing. In writing this we have omitted
the ϕχ mass-mixing term which is subleading in H/M.
(t) in(t) (t)in(t) (9)
Additionally, wehaveneglectedallinteractionswhichdo (cid:104)O (cid:105) ≡ (cid:104) |O | (cid:105)
not contain χ, since we are only interested in the ef- = in(tin)ei(cid:82)ttindt(cid:48)H(t(cid:48)) (t)e−i(cid:82)ttindt(cid:48)(cid:48)H(t(cid:48)(cid:48)) in(tin) .
(cid:104) | O | (cid:105)
fect of this heavy field. One of these neglected terms,
the hij∂ ϕ∂ ϕ coupling, produces an M -scale interac-
i j pl
tion and it would be interesting to compare this to the Traditionally, the in-state in is taken to be the Bunch-
| (cid:105)
result obtained here. We save this for a future study. Davies vacuum state [56], but this is not necessarily so.
Expanding cosmological backgrounds allow for a more
general class of vacua, which can be heuristically consid-
Uniform Density Gauge ered to be excited states of inflaton fluctuations. In the
present context, we will find that integrating out high-
At the moment of scalar fluctuation horizon crossing energy physics generically results in boundary terms in
wethenconvertthesetotheuniformdensitygaugesince the effective action, which represent such excited states.
these will then stay constant and are good observables.
If we denote the fields representing the “evolving” ket
This gauge corresponds to the choice
to be ϕ ,χ ,γ and those for the “devolving” bra
+ + ij,+
{ }
ϕ = 0, hij =a(t)2[(1+2ζ)δij +γij], tboebceom{ϕp−ut,eχd−f,rγoimj,−t}h,ethacetiino-ninexpectationvalue(9)can
∂ γ = 0, γ =0.
i ij ii
Notethattheχandγfluctuationsremainidentical,while
scalar fluctuations are now parametrized by ζ. The con- S[ϕ+,χ+,γij,+] S[ϕ ,χ ,γij, ]. (10)
S ≡ − − − −
version between gauges is
(cid:114)
4π ϕ togetherwiththeconstraintthatϕ (t)=ϕ (t),χ (t)=
ζ = . (8) + +
(cid:15)1 Mpl χ (t) and γij,+(t)=γij, (t). It is then hel−pful to trans-
Ofcourse,werewetoincludetheeffectsofχontheback- fo−rm into the Keldysh ba−sis,
ground evolution, this hybrid-inflation model would not
have constant scalar fluctuations. But as addressed pre-
ϕ¯ (ϕ +ϕ )/2, Φ ϕ ϕ ,
viously,theleading-orderχ-fluctuationsemployasingle- ≡ + − ≡ +− −
field background and hence these fluctuations will still χ¯ (χ++χ )/2, X χ+ χ ,
≡ − ≡ − −
have constant superhorizon ζ. γ¯ (γ +γ )/2, Γ γ γ .
ij ij,+ ij, ij,+ ij,
≡ − ≡ − −
5
In this basis the total action (10) equals GR F
[ϕ¯,Φ,χ¯,X,γ¯,Γ]=
S (cid:90)
d3xdt a(t)3(cid:2) M2∂γ¯ ∂Γ +∂ϕ¯∂Φ+jΦ R
− − pl ij ij G F
(cid:18) (cid:19)
+ m2 2 d (cid:0)a3H(cid:15) (cid:1) ϕ¯Φ+∂χ¯∂X+M2χ¯X
− a3dt 1
g (cid:18) Φ2(cid:19) g (cid:18) X2(cid:19) GR F
+g ϕ¯Φχ¯+ 1 ϕ¯2+ X+g χ¯Xϕ¯+ 2 χ¯2+ Φ
1 2 4 2 2 4
(cid:18) (cid:19)(cid:21)
1 1
+γ¯ij∂iχ¯∂jX+ 2Γij ∂iχ¯∂jχ¯+ 4∂iX∂jX . (11) g1 g1 g1
Density Matrices
Althoughwestartedwithapurestatein(9),ingeneral g2 g2 g2
wecouldtakeexpectationvalueswithrespecttoamixed
state,
(cid:88)
(t) = in (t) (t)in (t) ρ .
i j ij
(cid:104)O (cid:105) (cid:104) |O | (cid:105)
i,j Mp−l2 Mp−l2 Mp−l2
Here the density matrix ρ is normalized so that Trρ =
(cid:80)
ρ = 1. In the path integral language, the density
i i,i
matrix is equal to the logarithm of the imaginary com-
ponent of the action, so that
FIG. 1: Feynman rules for our theory in the in-in formalism.
Singlesolidlinesindicatecontractionsofϕ¯,dashedsinglelines
=Re ilnρ.
S S− indicate those of Φ, with analogous notation for double lines
indicating the heavy field components {χ¯,X}, wiggly lines
We will find that integrating out χ will generically lead
indicating the graviton component γ¯ and circled dots for Γ.
toamixedstateforϕandγ,sinceweareremovingstates
from a unitary process.
τ defined by
Perturbative Solution of Fluctuations
ds2 =a(τ)2( dτ2+dx2).
The equations of motion for the fluctuations are −
(cid:18) (cid:19) Neglecting the interactions in equations (12), then
∂2ϕ¯+3Hϕ¯˙ −j− m2− a23ddt(cid:0)a3H(cid:15)1(cid:1) ϕ¯=g1ϕ¯χ¯+ g41ΦX, Fourier transforming into the comoving momentum ba-
sis, they become
g (cid:18) Φ2(cid:19) g
∂2χ¯+3Hχ¯˙ M2χ¯= 1 ϕ¯2+ + 2ΦX
− 2 (cid:18) 4 1 4 (cid:19) ϕ¯(k0)(cid:48)(cid:48)+2Hϕ¯(k0)(cid:48)−(cid:18)k2+m2a2− a23ddt(cid:0)a3H(cid:15)1(cid:1)(cid:19)ϕ¯(k0) = 0,
∂ γ¯ ∂ χ+ Γ ∂ χ ,
∂2γ¯ij +3Hγ¯˙ij = M2p−l2−(cid:18)j∂iχ¯∂ijjχ¯i+ 14∂4iXi∂jjXi(cid:19). (12) χ¯(k0)(cid:48)(cid:48)+2Hγχ¯¯i((kj00,)k)(cid:48)(cid:48)(cid:48)−+(cid:0)2kH2γ¯+i(j0,M)k(cid:48)−2ak22(cid:1)γ¯χ¯i(j(k0,0)k) == 00,.
We will now solve these perturbatively. The Feynman Here we have introduced the conformal Hubble parame-
rules are summarized by Figure 1, where we have used ter (τ) a(cid:48)/a2 and suppressed all indices on γ¯. The
H ≡
the same diagrammatic notation as in [57]. solution for ϕ¯(0) is given by the Hankel function of the
k
first kind,
Zeroth Order √ πτ
U (τ) = − H(1)( kτ),
k − 2a(τ) ν −
We first consider the free fluctuation equations of mo- 3 (cid:15)
tion. To do this it is helpful to switch to conformal time ν ≈ 2 +(cid:15)1+ 22. (13)
6
If we expand in terms of the slow-roll parameters, this is
the familiar expression
H
U (τ) (1 ikτ)e ikτ +
k −
≈ √2k3 − ···
The linearly independent solution U (τ) is simply the FIG. 2: Inflaton fluctuation corrections ϕ¯(1) induced by in-
k∗
complex conjugate and hence uses H(2). teractions. Crosses indicate the zeroth-order solution ϕ¯(0)
ν or χ¯(0).
Forthemassiveχ-fluctuationsthefreefieldsolutionis
also a Hankel function but can written more transpar-
ently using the WKB approximation as
exp(cid:104) i(cid:82)τ dτ (cid:113)k2+ M2 (cid:105) ×
V (τ) 1 − τin (cid:48) H(τ(cid:48))2τ(cid:48)2 . (14)
k ≈ a(τ) (cid:16) (cid:17)1/4
√2 k2+ M2
H(τ)2τ2
The time-dependent frequency here is =⇒ +
(cid:113)
ω(τ) k2+ M2 , and the WKB approxi-
≡ H(τ)2τ2
mation ω˙ /ω2 1 is always valid for H/M 1.
For γ¯| -|fluctu(cid:28)ations the free solution is again(cid:28)given by FIG.3: Statisticalaveragingofatree-levelsolutionmaypro-
ij
duceloops,despiteremainingclassical. Notethatthesecond
Hankel function but of a different order,
diagram can always be cancelled with an appropriate coun-
√ πτ 3 terterm.
W (τ) = − H(1)( kτ), µ +(cid:15) .
k − 2a(τ) µ − ≈ 2 1
Of course the solution may then be iterated again to
Higher Order
obtain yet even higher-order solutions,
We may then iteratively solve for higher-order solu- (cid:90) (cid:16) (cid:17)
tions to the fluctuations, ϕ¯(2)(x)= d4y√g GR(x,y) g1ϕ¯(0)χ¯(1)(y)+ .
···
(cid:90) (cid:16) g (cid:17)
ϕ¯(1)(x)= d4y√gGR(x,y) g1ϕ¯(0)χ¯(0)(y)+ 22χ¯(0)(y)2 . Note thatfor fluctuations theslow-rollparameters (cid:15)i are
implicitly included, and that the order of the solution
(15)
refers to the order of the couplings g .
Fourier transforming into comoving momentum, these i
verticescanbeevaluatedusingtheretardedGreen’sfunc-
tion GR can be written in terms of the fluctuation solu-
Statistical Correlations
tions,
GRk(τ1,τ2) ≡ i(cid:104)ϕ¯(k0)(τ1)Φ(0k)(τ2)(cid:105) (16) After obtaining ϕ¯ to our desired order, we may then
−
= 2θ(τ τ )Im[U (τ )U (τ )]. take statistical averages of the zeroth order solutions via
− 1− 2 k 1 k∗ 2
TheseareshowninFigure2. TheadvancedGreen’sfunc- F (τ ,τ ) = ϕ¯(0)(τ )ϕ¯(0)(τ )
tion GA is then simply the time-reversal of this: k 1 2 (cid:104) k 1 −k 2 (cid:105)
= Re[U (τ )U (τ )],
k 1 k∗ 2
GAk(τ1,τ2)≡GRk(τ2,τ1). 0 = (cid:104)Φ(k0)(τ1)Φ(0k)(τ2)(cid:105),
−
Note that comoving momentum is conserved at ver- (τ ,τ ) = χ¯(0)(τ )χ¯(0)(τ )
tices. A similar procedure applies for χ¯(1) and γ¯(1) using Fk 1 2 (cid:104) k 1 −k 2 (cid:105)
= Re[V (τ )V (τ )],
their corresponding retarded Green’s function R and k 1 k∗ 2
GR, G 0 = (cid:104)X(k0)(τ1)X(0k)(τ2)(cid:105),
−
GkR(τ1,τ2) =≡ −i(cid:104)χ2¯θ(k0()τ(1τ1−)Xτ2(−)0kI)m(τ2[V)(cid:105)k(τ1)Vk∗(τ2)], (17) Fk(τ1,τ2) == R(cid:104)γ¯ek(0[W)(τk1()τγ¯1−()0Wk)(kτ∗2()τ(cid:105)2)],
A(τ ,τ ) R(τ ,τ ), 0 = Γ(0)(τ )Γ(0)(τ )
Gk 1 2 ≡ Gk 2 1 (cid:104) k 1 k 2 (cid:105)
−
GRk(τ1,τ2) ≡ i(cid:104)γ¯k(0)(τ1)Γ(0k)(τ2)(cid:105) (18) to get correlations. This can be heuristically thought
−
= −2θ(τ1−τ2)Im[Wk(τ1)Wk∗(τ2)], of as gluing the crosses together in all possible ways.
GAk(τ1,τ2) ≡ GRk(τ2,τ1). An example is shown in Figure 3. While this may pro-
7
POWER SPECTRUM EVALUATION
Here we will give the explicit formula for computing
the scalar and tensor power spectrum in our high-energy
model.
FIG.4: Correctiontothebackgroundlinearcouplingcoming
from χ-fluctuation backreaction. This can be absorbed into
the definition of j.
Inflaton Fluctuations
We will need to know the correlation of inflaton fluc-
duce Feynman diagrams with loops, representing inte- tuations at the moment of horizon crossing,
grals over comoving momentum, they are nonetheless
k3
completely classical. Animportantdifferenceofthein-in P (k) ϕ¯ (τ)ϕ¯ (τ) .
formalismisthatloopsmayrepresentstatistical,butnot ϕ ≡ 2π2 (cid:104) k −k (cid:105)|k=aH
quantum, fluctuations. To evaluate this perturbatively, substitute the classical
solution ϕ¯ = ϕ¯(0) +ϕ¯(1) + into this expression. In
···
the decoupling limit g 0 or M , the inflaton
i
Self-Consistency of Background Solution → → ∞
fluctuation power spectrum is simply
The introduction of fluctuations means that the back- P(0)(k) = k3 ϕ¯(0)(τ )ϕ¯(0)(τ )
ground solution will be slightly modified due to backre- ϕ 2π2(cid:104) k k k k (cid:105)
−
action. Thisisan (g )correctiontothelinearcoupling k3
O 2 = F (τ ,τ )
j, as shown in Figure 4, having the value 2π2 k k k
∆j = (cid:90) d3q g2χ¯(0)χ¯(0)(τ) = g2 (cid:90) d3q (τ,τ). where τk is the horizon-crossing time of mode k,
(cid:104) (2π)3 2 q −q (cid:105) 2 (2π)3Fq τk ∼ − 1/k. The first order corrections are then:
The apparent time-dependence is illusory, as follows. P(1)(k)= k3 2 ϕ¯(1)ϕ¯(0) .
Let us convert the integral over comoving momentum ϕ 2π2 × (cid:104) k k(cid:105)
−
into one over physical momentum. In static-background
There are no interactions in the action (11) which will
QFT loop regularization one imposes a UV cutoff Λ on
produce this, and so we turn to the second order contri-
the Wick-rotated 4-momentum, thus respecting Lorentz
butions,
symmetry. Since the quasi-de Sitter background of infla-
tion breaks Lorentz symmetry (‘energy’ is ill-defined) we k3 (cid:16) (cid:17)
cannot impose such a cutoff, but comoving momentum Pϕ(2)(k)= 2π2 (cid:104)ϕ¯(k1)ϕ¯(1k)(cid:105)+2(cid:104)ϕ¯(k2)ϕ¯(0k)(cid:105) . (19)
− −
is conserved,andthereforewecanperformanequivalent
There are a total of four diagrams which could possibly
procedure as follows. Via (14) the χ-field’s conjugate
variable to time is given by contribute at O(g12), shown in Figure 5. We will give
explicit expressions for the first two,
(cid:112)
E q2τ2H2+M2.
∼ k3 (cid:90) τk (cid:90) τk
P(A)(k) = ( ig )2 dτ a(τ )4 dτ a(τ )4
ThuswemayplaceacutoffonEatsomescale,effectively ϕ 2π2 − 1 1 1 2 2 ×
τin τin
iHmpisoscionngstaancut,toafflloowninqg. Lusettousesmtiommaetnettahrielycoarsrseucmtioenthaast (cid:90) (d2π3q)3 (cid:2)−iGRk(τk,τ1)(cid:3)Fq+k(τ1,τ2)Fq(τ1,τ2)(cid:2)−iGAk(τ2,τk)(cid:3),
∆j g2 (cid:90) √Λ2−M2/H|τ| d3q (cid:113)H2τ2 Pϕ(B)(k) = 2kπ32 (−ig1)2(cid:90) τkdτ1 a(τ1)4(cid:90) τkdτ2 a(τ2)4×
=≈ g22 (cid:20)0Λ(cid:112)Λ2 M2+(2Mπ)23l2n(cid:18)q2+ HMM2τ22 (cid:19)(cid:21). (cid:90) (d2π3q)3 (cid:2)−iGRk(τk,τ1)(cid:3)[−iGτRkin(τ1,τ2)](cid:2)−iG0Rτin(τ1,τ2)(cid:3)[−iGAq(τ2,τk)].
2π − Λ+√Λ2 M2
−
This is independent of H, assuring us that it is the same
answeronewouldgetfromastaticspacetimeanswerhad Inflaton Vertex Evaluation
we simply truncated E Λ. This time-independent cor-
≤
rection can then be cancelled by appropriate redefinition We first dissect the interaction at the vertex before
of j. A similar procedure can of course be done for the evaluating the full diagram. Since the time coordinate is
tadpole of χ, as shown in Figure 3. integratedover,vertexevaluationisverydifferentthanin
8
whereτ issomereferencetimeandH issometheHub-
ble scale∗ at this time. Then ∗
H
A B du= (1+(cid:15)1)dτ.
M
In this variable it is clear that admits a stationary
1
A
phaseapproximationattheenergy-conservationmoment
dφ M/H dφ
C D 0= =
du 1+(cid:15) dτ
1
FIG. 5: Power spectrum corrections mediated by the heavy M/H (cid:20) dν (cid:113) (cid:21)
sfiienlgdl.e lSininesgliendsoiclaidtelitnheosseinodficΦa,tewcitohntarnaacltoiognosusofnoϕ¯t,atdioanshfeodr = 1+(cid:15)1 πdτ +k1+k2− |k1+k2|2+u−c2 .
double lines indicating the heavy field components {χ¯,X}.
The variation in slow-roll phase is given by
dν dN dν 3
= (cid:15) (cid:15)
1 2
dτ dτ dN ≈−τ
standard quantum field theory. Writing out the Green’s
andWightman’sfunctionsintermsofU’sandV’s,wesee and so is higher-order in slow-roll and can be neglected.
there are three types of cubic vertices. We will evaluate The solution then defines the NPH,
each.
The first is u−c1 =−(cid:112)2k1k2(1−cosθ), cosθ = kk1·kk2.
1 2
(cid:90) τk
(τ ;k ,k ) dτ a(τ)4U (τ)U (τ)V (τ). Expandingthephasenearthisstationarypointgivesthe
A1 k 1 2 ≡ τin k1 k2 −∗(k1+k2) inflection
(20)
The superscript k indicates there is some dependence d2φ(cid:12)(cid:12)(cid:12) = M/H(cid:12)(cid:12)(cid:12) u−c3 .
uponthemomentthatthemodeleavesthehorizon. Since du2(cid:12) 1+(cid:15) (cid:12) k +k
uc 1 uc 1 2
weassumeallinteractionshappenwellinsidethehorizon,
Then to leading order in H/M and slow-roll parameters
a good approximation is
the amplitude is
1 (cid:104) (cid:16) νπ π(cid:17)(cid:105)
Uk(τ)≈ a(τ)√2k exp −i kτ + 2 + 4 , k|τ|(cid:29)1 A1(τk;k1,k2) ≈ (cid:12) (22)
√πiθ(τ τ )θ(τ τ )(1+ 3(cid:15) )e iφ(cid:12)
wpahrearmeeνteisrsg,ivthenebscya(le13f)a.ctToorliesagdiivnegnobrdyerintheslow-roll − 2√k1kk2−[2kc1k2(1c−−coinsθ)]1/42√1HM− (cid:12)(cid:12)(cid:12)τc.
1+(cid:15) The physics of this is clear. This diagram accounts for
a(τ) 1. thethresholdproduction/decayofheavyparticlesathigh
≈− τH
redshift in the early universe. Although this induced
Substituting this into (20) results in the expression termislocalizedintime, itisk-dependentandthusdoes
not represent a physical boundary. Rather, it represents
1 (cid:90) τk dτ (1+(cid:15) )e iφ
(τ ;k ,k ) 1 − a ‘hypersurface’ representing the energy scale of interac-
A1 k 1 2 ≈−2√2k1k2 τin Hτ (cid:0)|k1+k2|2+ HM2τ22(cid:1)1/4 tion, precisely as expected in the New Physics Hyper-
(21) surace approach.
where the phase is given by The second possible vertex is identical to but with
1
A
one U conjugated,
(cid:90) τ (cid:114) M2
φ(τ)≡νπ+(k1+k2)τ − dτ(cid:48) |k1+k2|2+ H2τ 2. (τ ;k ,k ) (cid:90) τkdτ a(τ)4U (τ)U (τ)V (τ).
τin (cid:48) A2 k 1 2 ≡ τin k1 k∗2 −∗(k1+k2)
Letusnowintroducetherescaledtimecoordinateusuch
This has only imaginary-time saddlepoint solutions.
that
Since our τ-integral is confined to the real axis we will
H never pass over this point in our integration, and so
u τ.
≡ M this amplitude will be suppressed as A2 ∼ erf(MH) ∼
To obtain the differential, recall that the Hubble scale MHe−(M/H)2 except for cosθ ≈−1, when the saddlepoint
movestoinfinityandtheinflectionvanishes. Inallother
changes with time as
cases we may neglect such interactions. Finally we con-
(cid:18)τ (cid:19)(cid:15)1 sider which has both U’s conjugated and so admits
H(τ) H A3
≈ ∗ τ nosaddlepointsolutions, andthuscanalsobeneglected.
∗
9
1(cid:45)cosΘ The measure transforms as
E M2(1+(cid:15) )
min d(1 cosθ)dq 1 dEdτ.
2 − → H2Eτ3k
Τ
E fin The integral in (23) is then transformed as
max
Τ (cid:90) d3q
(τ ;k,q)2
E (2π)3 |A1 k |
1 (cid:90) √qdqd(1 cosθ)
= − (1+3(cid:15) )
Τ 1
in 16√2πk3/2M H√1 cosθ
1 (cid:90) dτdE −
q (1+4(cid:15) ).
→ −16πk2 τ3H4 1
FIG. 6: The phase space of diagram A in both {q,1−cosθ}
The phase φ cancels out in this amplitude and so we
and {τ,E} coordinate systems. Constant-τ contours are
need not evaluate it. We can now easily place limits on
showninblue,andaconstant-Econtourisshowninred. The
low-energyphasespaceallowedinourintegrationisshownin the region of integration in terms of the energy scale Λ:
gray. Maximal Q As done previously, we place a limit on
the physical energy to be Λ. Using the fact that the
vertex with a virtual ϕ and an external ϕ of comoving
momentumkhavepreciselytheenergyoftheχ-field,the
Inflaton Power Spectrum
bound is found to be
After the vertex evaluation it remains to perform the E = Hk τ +Λ.
max
− | |
integration over loop momentum.
Minimal E Since 1 cosθ 2 this places a minimal
− ≤
limit on E,
First scalar correction
M2
E = .
min 4H(τ)3k τ
Let us first consider diagram A,
| |
g2 k3 (cid:90) d3q Although this is time-dependent, it remains scale-
Pϕ(A)(k)= 21 2π2 |Uk(τk)|2 (2π)3 |A1(τk;k,q)|2. invariant because it contains only the combination kτ.
Initial and Final τ Of course we require
(23)
E < E . Their equality will determine the
This is non-trivial for two reasons. First, we wish to min max
earliestandlatesttimesforwhichthisconditionisvalid,
implement a physical cutoff, and so a change of coordi-
an equation which is quadratic in τ. The two solutions
natesisrequired. Second,loopmomentuminfluencesthe
are given by
time of vertex interaction via τ (q,k), and so affects the
c
Hubble scale. Thus we need to convert the integral over (cid:32) (cid:33)
τ(0)
comovingmomentumtooneovertimeofinteraction. To τ =τ(0) 1 (cid:15)1ln ± ,
accomplish this, first simplify the 3d integral to 2d, ± ± − τ
∗
(cid:90) d3q 1 (cid:90) where we have defined
q2dqd(1 cosθ).
(2π)3 → (2π)2 −
Λ √Λ2 M2
τ(0) ± − . (25)
Nowreplacethecoordinates q,1 cosθ with τ,E via ± ≡− 2kH
{ − } { } ∗
the definitions
The later root is
τ−1 H(cid:112)2kq(1 cosθ), (cid:32) τ(0)(cid:33)
≡ −M(cid:114) q − τfin =τ−(0) 1−(cid:15)1ln τ− ,
E Hq τ = M . ∗
≡ | | 2k(1 cosθ)
− whereas the earlier root is
Of course the definition of τ coincides with τc, since this (cid:32) τ(0)(cid:33)
is the stationary-phase localized time of interaction. In- τ =τ(0) 1 (cid:15) ln + .
verting these coordinate systems gives in + − 1 τ
∗
E M2 Thiswillalwaysbeafterthepointwheretheexternalen-
q = , 1 cosθ = . (24)
τ H − −2HEτk ergy parameter Ek Hk τ reaches Λ, so we are assured
| | ≡ | |
10
that all energy scales under consideration are below Λ. P(cid:106)(cid:72)k(cid:76)
Figure 6 shows the phase space in the two coordinate
systems.
This gives us a window τ <τ <τ in which to inte-
in fin
grateoutthishigh-energyinteractioninacontrolledway.
Note that choosing Λ M makes this window vanish, a
≤
reassuring fact since (by definition) there should not be
any interactions below the scale of New Physics. There
is an upper bound on Λ as well: the interactions must
ceasebeforethegauge-transformationatthehorizon-exit
near τ = 1/k (whereupon our plane-wave approxima-
k − lnk
tion would break down). This means we are allowed to
set the cutoff scale in the window FIG. 7: Power spectra for the example theory in slow-roll
inflation. Theredlineshowstheeffectofinteractionsrelative
M <Λ< 1(cid:0)H +M2/H(cid:1). tothatoftheflatspectrumofthefreetheoryinexactdeSitter
2 space shown in black. Note the corrections slightly decreases
at higher k as a result of the Hubble scale changing.
ForatypicalvalueofM/H 100,thismeanswecanset
∼
M Λ 50M +H, which should be adequate for the
≤ ≤
low-energy behavior. We now expand the Hubble rate
Total result
around the reference point lnτ = lnk , and keeping
only the leading term in Λ/M,∗the −integr∗al is evaluated
Summing diagrams A, B, C and D results in the fol-
as
lowing total shift in the scalar power spectrum,
(cid:90) τfin dττ3 (cid:18)1+4(cid:15)1−3(cid:15)1lnττ (cid:19)(cid:90) EmaxdE ∆Pϕ(k) = 9g612πk33 |Uk(τk)|2 MΛ4H3
≈ (cid:18)τ−in8Λ33MH∗24k2(cid:19)(cid:20)1− 23(cid:15)1+4∗(cid:15)1lnE(cid:12)(cid:12)(cid:12)(cid:12)m4iΛnHM2kτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:21). × (cid:20)1− 23(cid:15)1−4(cid:15)1ln(cid:18)4MΛ∗H2k∗k(cid:19)(cid:21). (27)
∗
∗ ∗
Since Λ M and g H, the correction then scales as
1
The power spectrum correction is then ∆P H∼/M. Figure∼7 shows these corrections to the
∼
power spectrum in the example theory.
g2 k3 Λ3
P(A)(k) = 1 U (τ )2
ϕ 2 24π3 | k k | M4H
(cid:20) (cid:18) ∗ (cid:19)(cid:21) The Tensor Power Spectrum
2 4ΛH k
× 1− 3(cid:15)1−4(cid:15)1ln M2k∗ . (26)
∗ We did not include a γ2χ coupling in our high-energy
model, nor are there any induced by gauge-fixing; hence
thetensorspectrumisthesameasthestandardslow-roll
Second scalar correction case. The gravitational power spectrum is
32k3
P(0)(k) γ¯(0)(τ )γ¯(0) (τ )
Diagram B can be evaluated in an identical fashion, h,ij ≡ πM2 (cid:104) ij,k k ij, k k (cid:105)
pl −
except for two subtleties: the Heaviside function in the
32k3
Gtiorneesnt’oscfuonntcrtiibountea,llaonwdotnhleythwaolfi’osfptrhoeduGcaeuasssiiagnnflchuacntugae-. = πMp2l(cid:15)ijF(τk,τk). (28)
Thus
1 Comments and Interpretation
P(B)(k)= P(A)(k).
ϕ −2 ϕ
InthedeSitterlimitof(cid:15) 0,theresultisaperfectly
i
→
scale-invariant correction,
Third and fourth scalar correction g2Λ3H
∆Pϕ = 1912π3M∗4.
Diagrams C and D can also be evaluated in a similar Let us verify that we should have anticipated this by
fashion, and to leading order in H/M they cancel. simplycomputingthephysicalmomentumpattheNPH