Table Of ContentSignatures of Planck-scale interactions in the cosmic microwave background?
Friedemann Queisser, Michael Uhlmann, and Ralf Schu¨tzhold
Institut fu¨r Theoretische Physik, Technische Universita¨t Dresden, D-01062 Dresden, Germany
Basedonarathergenerallow-energyeffectiveaction(interactingquantumfieldsinclassicalcurved
space-times), we calculate potential signatures of new physics (such as quantum gravity) at ultra-
highenergies(presumablythePlanckscale)intheanisotropiesofthecosmicmicrowavebackground.
These Planck-scale interactions create non-Gaussian contributions, where special emphasis is laid
on the three-point function as the most promising observable, which also allows the discrimination
between models violating and those obeying Lorentz invariance.
PACSnumbers: 98.80.Cq,04.62.+v,98.70.Vc,98.80.Qc.
Introduction In our present standard model of cos- fore, we shall consider a rather general nonlinear low-
6
0 mology (see, e.g., [1]), the anisotropies observed in energy effective action for the inflaton field (interacting
0 the cosmic microwave background (e.g., by the recent quantum fields in classical curved space-times) and cal-
2 WMAP [2] and the future PLANCK [3] mission) orig- culate the corrections due to new physics at ultra-high
n inate from quantum fluctuations of the inflaton field, energiesintheanisotropiesofthecosmicmicrowaveback-
a which were amplified and stretched by the cosmic ex- ground. For simplicity, we assume that the Planck scale
J
pansion during inflation (a very early epoch in the evo- is the characteristic scale for new physics from now on
5
lution of our universe). The expansion of our universe (but our results can easily be generalized).
2
acts as a cosmic microscope, i.e., these observed modes The Model During slow-roll inflation, the space-time
1 can be traced back to extremely short length scales canbeapproximatedbythedeSittermetric(~=c=1)
v and thus ultra-high energies (such as the Planck energy ds2 =dτ2−e2Hτdr2 =a2(t)[dt2−dr2], (1)
8 M ≈1019GeV) which are completely inaccessible with
Pl
0 present or future terrestrial experiments (e.g., particle withthe propertime τ andthe conformaltimet,respec-
1 accelerators). This observation entices the question of tively, and the Hubble constantH governingthe dynam-
1
0 whetheronemightobservesignaturesofnewinteractions icsofthescaleparametera(t). Assuminglocalityandco-
6 (suchasquantumgravity)beyondthestandardmodelof variance(non-covarianttermswillbediscussedlater),the
0 particle physics in the anisotropies. mostgenerallow-energyeffective actionfor a scalarfield
/ can be represented in a gradient expansion and reads
c
Several different approaches for such expected devia-
q tions from our known laws of physics at highest ener- L=A(ϕ)+(∂ϕ)2B(ϕ)+(∂ϕ)4C(ϕ)+O[(∂ϕ)6], (2)
-
gr gies have been investigated already. One possibility to with (∂ϕ)2 = (∂µϕ)(∂µϕ) and arbitrary coefficient
: changethephysicsatsmalldistancesisamodificationof functions A(ϕ), B(ϕ), and C(ϕ). Assuming stability
v
the dispersion relation (of the inflaton field) implying a B(ϕ) > 0 and adopting the usual field re-definition
i
X breakdown of the Lorentz invariance (which would then dφ= 2B(ϕ)dϕ, we get [10]
r be just a low-energy symmetry), see, e.g., [4]. It was
a also shown that an explicit short-distance cutoff has a L=p1 (∂φ)2−V(φ)+(∂φ)4W(φ) +O[(∂φ)6]. (3)
2
potentially measurable effect on the spectrum [5]. Possi-
The first(cid:2)two terms correspond to th(cid:3)e usual scalar field
bleback-reactioneffectsresultingfromUV-modifications
theory whereas the third contribution of this low-energy
have been considered in [6] and a possible change of the
effective action is a remnant of new physics (e.g., the in-
propagator for the inflaton while preserving the local
teraction with massive gravitons) at ultra-high energies.
Lorentz invariance has been studied in [7]. Several au-
Identifying the scalar field φ with the inflaton field, we
thors have investigated different effects that result from
assume the potential for chaotic inflation V(φ) = m2φ2,
amodificationofthecommutationrelationsathighener-
giessuchas[x ,p ]=i~δ (1+ηp2),see,e.g.,[8]. Finally, butourmainresultscaneasilybegeneralizedtootherpo-
i j ij
tentials(see discussionbelow). Nowwesplitthe inflaton
theimpactofdifferentchoicesoftheinitialvacuumstate
field φ into a classicalbackgroundfield φ plus its quan-
have been considered in [9], for example. 0
tum fluctuations φ(t,r) = φ (t)+Φ(t,r). In the usual
0
These references [9] already illustrate the importance slow-rollapproximationm≪H,theslowlydecayingclas-
ofconsideringthecorrectinitialstate. (Onecouldobtain sicalbackgroundsolutionreads φ (τ)∝exp(−m2τ/3H),
0
basically arbitrary results by changing the initial state and the Lagrangianfor the fluctuations is given by
correspondingly.) Furthermore, the aforementioned in-
a2
vestigationsweremainlydevotedtolinearfields,whereas L = d3r Φ˙2−(∇Φ)2 −ag Φ˙3−Φ˙(∇Φ)2
2
one would expect new physics such as quantum gravity Z (cid:26) h i h i(cid:27)
toimposenonlinearcorrections,i.e.,interactions. There- +O[(∂φ)6]+O[Φ4]+O[g2], (4)
2
with g = 2W(φ )∂φ /∂τ ≈ −2φ W(φ )m2/(3H) denot- cosmic expansion is slow compared to the internal dy-
0 0 0 0
ing a roughly constant effective coupling strength. Note namics. If we now choose an intermediate time t such
0
thattheinteractionterm(∂φ)4intheLagrangian(3)also that H≪k/a(t )≪M , the effects of both, the cosmic
0 Pl
implies smallcorrectionssuchas Φ˙2 and thus a smallre- expansion HH and the interaction Hg are small and can
definitionofthe propagationspeedc , the massm, and be treated as perturbations [12]. As a result, the quan-
eff
theamplitudeoftheinflatonfieldΦetc. Asusualduring tumstate(adiabaticvacuum)ofthesemodesatthattime
inflation, the mass term m2Φ2 can be neglected for the t can be derived via the adiabatic expansion [13]
0
fluctuations Φ due to m ≪ H (terms like λΦ3 will be
discussed below). Since we are mainly interested in the hΨ |H˙ |Ψ i
n S 0
|in(t )i = N |Ψ i−i |Ψ i +
three-point function, we omitted the higher-order terms 0 0 0 n ∆E2
O[Φ4], but the associatedfour-pointcontributioncan be nX>0 n
calculatedviathesameformalism(seediscussionbelow). + i d hΨn|H˙S|Ψ0i +O[H3], (7)
For the sake of brevity, we omit stating the higher-order ∆E dt ∆E2
n n !
corrections O[(∂φ)6]+O[Φ4]+O[g2] from now on.
Hamiltonian AfteraLegendretransformation,weob- where |Ψ i are the instantaneous eigenvectors of the
n
tain the low-energy effective Hamiltonian density Hamiltonian (6) with appropriate phases and N =
0
1+O[H2]isanormalization. Forg =0,theHamiltonian
1 Π2 Π3 Π
H= +a2(∇Φ)2 +g − (∇Φ)2 . (5) H0+HH canbediagonalizedexactly(i.e.,toallordersin
2(cid:20)a2 (cid:21) (cid:20)a5 a (cid:21) H) via a Bogoliubov transformation ck =αkbk+βkb†−k.
The explicite time-dependence of this Hamiltonian can Up to second order in H (the accuracy we are inter-
partly be removed via the (time-dependent) canonical ested in), the Bogoliubov coefficients αk and βk read
transformation Π → aΠ and Φ→ Φ/a generated by the αk = 1+iaH/2k and βk = iaH/2k −a2H2/4k2. The
squeezing operator S, giving rise to an additional term remaining corrections can be calculated with stationary
to the Hamiltonian: In the squeezed representationwith perturbation theory
the Schr¨odinger equation id(S|Ψi)/dt=H (S|Ψi), the
S
Ψ0 H Ψ0
Hamilton density reads |Ψ i= Ψ0 + Ψ0 m g n +O[g2], (8)
n n m E0 −E0
H = 1 Π2+(∇Φ)2 − aH[ΦΠ+ΠΦ] (cid:12) (cid:11) mX6=n(cid:12) (cid:11)(cid:10) n(cid:12)(cid:12) (cid:12)(cid:12)m (cid:11)
S (cid:12) (cid:12)
2 2
+(cid:2)ag2 Π3−Π(∇(cid:3)Φ)2 =H0+HH+Hg. (6) weihgeenrest(aHte0s+obHtaHin)e(cid:12)Ψd0nv(cid:11)ia=thEen0a(cid:12)fΨor0ne(cid:11)maenretiothneeduBnpoegrotluiurbbeodv
transformation. (cid:12) (cid:12)
(cid:2) (cid:3)
This form allows for an intuitive physical interpretation:
Due to the cosmic expansion a = eHτ, the modes are Final state The impact of the cosmic expansion HH
on the modes increases with time and their evolution fi-
being continuouslystretchedandhence the effects ofthe nallybecomesnon-adiabaticwhenk/a=O[H],i.e.,when
interactions H ∝ 1/a2 decrease with time whereas the
g they cross the horizon and freeze. Therefore, in order to
influence of the expansion HH ∝a increases. calculate the final state, we switch from the Schr¨odinger
Initial state Sincetheinteractionsarestrongatearly
representation (thereby undoing the canonical transfor-
times, the derivation of the correct initial state requires
mation S etc.) to the interaction picture: The Heisen-
the treatment of strongly interacting quantum fields in
berg operators carry full dynamics for the unperturbed
curved space-times, which is a nontrivial task, cf. [11]. problem H0+HH to all orders in H (cf. [1])
Letusconsidertheevolutionofthemodesinmoredetail:
Due to spatial homogeneity, the co-moving wavenumber d3k 1 H
k correspondingto aspatialdependence ofeik·r remains Φ= +i eik·r−ik/(Ha)ak+H.c. (9)
2k(2π)3 a k
constant – but the physical wavenumber k/a decreases Z (cid:18) (cid:19)
(stretching). Hence modes with different k leave the Note thapt ak annihilates the adiabatic vacuum (7) for
Planck scale at different times, i.e., when k/a=O[MPl]. vanishing g, which can be used as a consistency check.
Assuming that the rate of the cosmic expansion given TheremainingsmallinteractionHamiltonianH actson
g
by the Hubble parameter H is much slower than the thequantumstate,whichenablesustocalculatethefinal
internal evolution rate of these Planckian modes (scale state via time-dependent perturbation theory
separation H ≪ M ), the most natural initial state is
Pl
the adiabatic vacuum state, cf. [11]. During the further t
expansion, the wavelength increases until k/a ≪ MPl |out(t)i= 1−i dt′Hg(t) |in(t0)i+O[g2]. (10)
holds and the low-energy effective action (3) becomes Z
t0
valid [12]. In view of the adiabatic theorem, the modes
evolve adiabatically and hence stay in the adiabatic vac- Asanothertestofthe consistencyofthe usedformalism,
uum as long as the external time-dependence due to the one may check that the resulting final state |out(t)i is
3
indeed independent of t as it should be. Since we are Three-point function With the final state obtained
0
mainly interested on modes which freeze well before the after performing the integration, one can calculate the
end of inflation, we may approximately extend the inte- three-point function of the frozen fluctuations and ob-
grationto infinite proper time τ =∞ whichcorresponds tains (after some algebra) the following spectrum
to vanishing conformal time t=0.
d3k d3k d3k δ3(k +k +k )
hΦ(r )Φ(r )Φ(r )i = −gH5 1 2 3 exp{i(k ·r +k ·r +k ·r )} 1 2 3 ×
1 2 2 (2π)6 1 1 2 2 3 3 k k k
Z 1 2 3
3 e ·e 1 e ·e e ·e 1 1
× − 2 3 + 2 3 + 2 3 + +perm. (1+O[H]+O[g]) ,(11)
k3 k3 k 2k k 2k2 k k
(cid:18) + (cid:26) + + 2 3 + (cid:18) 2 3(cid:19) (cid:27)(cid:19)
withtheabbreviationsk =k +k +k ande =k /k . independent interaction terms become important at late
+ 1 2 3 j j j
The remaining permutations (denoted by “perm.”, i.e., times. Hence the three-point spectrum generated by the
k →k ,k →k ,andk →k etc.) havetobeincluded λΦ3-termcanbe derivedfromtime-dependent perturba-
1 2 2 3 3 1
as well. The spectrum is scale invariant as one would tiontheoryandbehavesask /(k k k )3. Thisspectrum
+ 1 2 3
expect since the sequence: leaving of Planck scale and is dominated by large length scales and thus not scale
stretching → horizon crossing and freezing occurs on all invariant – which should make it easy to distinguish be-
scales (just at different times). The first contribution tween the effects of the momentum-independent interac-
3/k3 onthesecondlineresultsfromtheΠ3-termwhereas tionλΦ3 andthosegeneratedbyhigh-energy(Planckian)
+
the remaining ones appear because of the Π(∇Φ)2-term interactions such as the spectrum (11).
in the interaction Hamiltonian Hg. The derivation of the four-point function can be ac-
An estimate of the relative size of this non-Gaussian complished by means of the same formalism – but ne-
contribution can be obtained by comparison to the cessitates higher orders in the adiabatic expansion etc.
(Gaussian) two-point function The relativesize ofthe non-Gaussiancontribution(com-
pared to the Gaussian part of the four-point function,
hΦΦΦi
=O[gH2]=O[φ W(φ )m2H]≪1. (12) for example) can be estimated to O[W(φ )H4]. Hence
0 0 0
hΦΦi3
the comparison of the relative sizes of the non-Gaussian
Notepthat m can be eliminated with the classical equa- contributions of the three-point and the four-point func-
tions of motion and the Friedman equation, i.e., gH2 = tions, respectively, yields the unknown ratio Hφ0/MP2l.
O[φ W(φ )m2H] = O[W(φ )H3M2/φ ]. Usually the Lorentz invariance So far, we assumed that the
0 0 0 Pl 0
typical scale of new physics such as quantum gravity is Planckian interactions respect the local Lorentz invari-
expected to be the Planck scale. Therefore, one would anceandhencestartedfromacovariantaction(3). How-
naturally assumethat the coupling behaves accordingto ever, it may well be that the usual local Lorentz invari-
W ∼ 1/M4 if W did not depend on φ . In this case, ance is just a low-energy symmetry and broken at the
Pl 0
the relative size of the effect would be suppressed with Planck scale. In this case, many more terms can occur
H3/M3 and thus probably hard to measure. in the low-energy effective action – even if we still de-
Pl
However, one should also bear in mind that the usual mand spatialhomogeneity and isotropyaccordingto the
picture of inflation involves a large value of the inflaton cosmological principle: First of all, bilinear terms such
field φ ≫ M and hence W(φ ) could be much bigger as Φ¨2 and (∇2Φ)2 exactly correspondto a change in the
0 Pl 0
than 1/M4. Similarly, the scale for new physics might dispersion relation mentioned in the introduction. As-
Pl
be below the Planck scale or the inflaton potential V(φ) suming adiabaticity (which may not by valid for all dis-
could differ from m2φ2. In all these cases, the effect persion relations), we may treat those corrections with
could be far stronger than H3/M3. On the other hand, the presented formalism and it turns out that they ba-
Pl
theemployedformalismonlyworksifgH2 ≪1holds,i.e., sically generate a global shift of the two-point spectrum
if the relative size of the effect (12) is small compared to to lowest order, which is probably hard to measure.
one (which is consistent with the observations [2]). The next terms occurring in an expansion into powers
Generalizations Let us discuss the impact of ofΦandthenumbersofderivativesareexactlytheterms
momentum-independent interaction terms such as λΦ3 Π3 andΠ(∇Φ)2 discussedbefore [14]. However,without
[in contrast to contributions like (∂Φ)4] which occur if Lorentzinvariance,thepre-factorsofthesetwotermsare
the potential V is not just given by the mass term no longer necessarily equal. Therefore, measuring and
V(φ) 6= m2φ2. In contrast to H , these momentum- comparing the relative strength of the two contributions
g
4
in the above three-point spectrum (11) provides a pos- 3663 (2002); Mod. Phys. Lett. A 16, 999 (2001); Jens
sibility to detect violations of the Lorentz invariance at C.Niemeyer,RenaudParentani,ibid.64,101301(2001).
the Planck scale. [5] Jens C. Niemeyer, Phys.Rev.D 63, 123502 (2001).
[6] J´erˆome Martin, RobertH.Brandenberger,Phys.Rev.D
Summary Assuming locality and covariance, we de-
71, 023504 (2005).
rived the most general low-energy effective action of the
[7] S. Shankaranarayanan, L. Sriramkumar, Phys. Rev. D
scalar inflaton field (including possible remnants from 70, 123520 (2004).
Planckian interactions) via a gradient expansion. Based [8] Amjad Ashoorioon, Achim Kempf, Robert B. Mann,
on this ansatz, a formalism for the treatment of this Phys. Rev. D 71, 023503 (2005); Stephon Alexander,
(strongly) interacting quantum field theory in a curved Robert Brandenberger, Jo˜ao Magueijo, ibid. 67, 081301
space-time and the derivation of the quantum state via (2003); Richard Easther, Brian R. Greene, William
H. Kinney, Gary Shiu, ibid. 64, 103502 (2001); Achim
theadiabaticityassumptionwaspresented. Thisenabled
Kempf, Jens C. Niemeyer, ibid. 64, 103501 (2001);
us to predict the induced spectrum of the three-point
Achim Kempf, ibid.63, 083514 (2001).
function – which is quite robust against uncertainties in [9] Seoktae Koh, Mod. Phys. Lett. A 19, 1223 (2004);
cosmic evolutions and can clearly be distinguished from
J´erˆome Martin, RobertH.Brandenberger,Phys.Rev.D
other effects such as a λΦ3-term – making it a promis- 68, 063513 (2003); Kevin Goldstein, David Lowe, ibid.
ing observable. This concrete prediction of the explicite 67,063502(2003);JensC.Niemeyer,RenaudParentani,
spectrumfacilitatesarefinedsearchinthepresent/future David Campo, ibid. 66, 083510 (2002); Ulf H. Daniels-
WMAP and PLANCK data with good statistics and son, ibid. 66, 023511 (2002).
[10] To first order (in W or g), which is the accuracy we
hence high sensitivity. Abandoning covariance, it might
are interested in,onecan eliminate terms containing2φ
even be possible to detect violations of the Lorentz in-
(whichwouldcomplicatetheHamiltoniananalysis)after
variance at the Planck scale.
an integration by parts as well as using the equation of
Acknowledgments This work was supported by the motion for φ. The same applies to contributions of the
Emmy-Noether Programme of the German Research typeφ2(∂µ∂νφ)(∂µ∂νφ) etc.
Foundation (DFG) under grant No. SCHU 1557/1-1,2. [11] N. D. Birrell and P. C. W. Davies, Quantum Fields in
Furthersupportbythe COSLABprogrammeofthe ESF Curved Space, (Cambridge UniversityPress, Cambridge,
England 1982).
and the Pacific Institute of Theoretical Physics is also
[12] Here we assume g = (1/√MPl), but it would be suffi-
gratefully acknowledged. cient to demand gH2O 1 and to replace MPl by 1/√g,
≪
i.e., H k/a(t0) 1/√g.
≪ ≪
[13] This expansion can be derived via the decomposition
′ ′
|Ψ(t)i = nan(t)exp{−i dtEn(t)}|Ψn(t)i of the
quantum state Ψ(t) into the instantaneous eigenstates
[1] A. D. Linde, Particle Physics and Inflationary Cos- Ψn(t) andPinse|rtioniof thisRansatz intotheSchr¨odinger
mology (Harwood Academic, Chur, Switzerland, 1990); |equatioinid Ψ(t) /dt=H(t) Ψ(t) .Afteranintegration
A. R. Liddle and D. H. Lyth, Cosmological Inflation byparts,th|ecoeffii cientsan(t)| canibeexpressedinterms
and Large-Scale Structure (Cambridge University, Cam- of the matrix elements Ψn H˙ Ψm divided by the en-
bridge, England, 2000); P. J. E. Peebles, Principles of ergy gap squared (En hEm|)2.|Foriadiabatic evolution,
Physical Cosmology (Princeton, 1993). this ratio must be sma−ll which facilitates an expansion
[2] http://map.gsfc.nasa.gov into powers of H.
[3] http://planck.esa.int [14] In case of multiple inflaton fields, many more combi-
[4] S. Shankaranarayanan, Musongela Lubo, Phys. Rev. D nations are possible – in both cases: with and with-
72, 123513 (2005); J´erˆome Martin, Sergio E. Jor´as, outLorentzinvariance.Furthermore,weassumethatthe
and Robert H. Brandenberger, ibid. 66, 083514 (2002); scale factors a(t) occur with the same power as in the
J´erˆome Martin, Robert H. Brandenberger, ibid. 63, Lorentz invariant situation – which is not strictly neces-
123501 (2001); ibid. 65, 103514 (2002); R. H. Bran- sary without demanding covariance.
denberger and J. Martin, Int. J. Mod. Phys. A 17,
