On the possibility of Dark Energy from corrections to the Wheeler-De Witt equation

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by  William Nelson

On the possibility of Dark Energy from corrections to the Wheeler-DeWitt equation
William Nelson and Mairi Sakellariadou
King’s College London, Department of Physics, Strand WC2R 2LS, London, U.K.
We present a method for approximating the effective consequence of generic quantum gravity
corrections to the Wheeler-DeWitt equation. We show that in many cases these corrections can
producedeparturesfromclassical physicsatlargescalesandthatthisbehaviourcanbeinterpreted
as additional matter components. This opens up the possibility that dark energy (and possible
darkmatter) couldbelargescale manifestations ofquantumgravitycorrectionstoclassical general
relativity. As a specific example we examine the first order corrections to the Wheeler-DeWitt
equation arising from loop quantum cosmology in the absence of lattice refinement and show how
theultimate breakdown in large scale physicsoccurs.
PACS numbers: 98.80.-k, 95.36.+x, 95.35.+d, 95.85.Ry
8
0
0 Quantum cosmology can be studied within the con- be well approximated by the classicalbehaviour of addi-
2
text of mini-superspace models, reducing the full quan- tional matter sources.

n tum field theory to a quantum mechanical system of fi-
a nite degrees of freedom. Applying this to the evolution By considering the mini-superspace model of an ho-
J ofathreemetricresultsintheWheeler-DeWittequation mogeneous and isotropic universe, the WDW equation

3 (WDW) equation[1],whichbreaksdownnearthe classi- reads
2
calbig-bangsingularity. LoopQuantumGravity(LQG),
1 d 1 d 9k 3
another approach to canonical quantisation, uses triads [aψ(a)] aψ(a)+ ψ(a)=0
] ada ada −16π2l2 2πl2 Hφ
c andconnectionsratherthanmetricsandextrinsiccurva- (cid:20) (cid:21) Pl Pl
q tures asthe basicvariables,withmoresuccess [2], whilst (1)
- whereaisthescalefactor,k=0, 1isthecurvature,
gr sbtyrinthgeaenxdtrbaradnyenathmeiocrailesstraetmesovtehatthebesicnogmuelaarvbaeilhaabvleiouatr is the matter Hamiltonian; in our±units ~=c=1[11].Hφ
[
small scales. Irrespective of the full underlying theory,

2 theWDWequationmustberecoveredasasemi-classical In general, Hφ will have several terms, with different
dependenceonaandφ. Beingonlyinterestedinthelarge
v approximationto the dynamicalequations of the theory,
scale (a l ) behaviour of , we expect one term
5 if classical general relativity is to be produced at large ≫ Pl Hφ
62 scales. WecanthenaskwhetherQuantumGravity(QG) tǫo(φd)oamδ.inWatiethatnhdiswaepparpopxrimoxaimtioantethitebWyD[3W/(e2qπulP2atl)i]oHnφca=n
corrections can have any effects on large scale physics.
1 be written in as
. At firstsightit would appearunlikely that this is pos-
9
sible. However,thewave-functionsolutionstotheWDW 1 d 1 d 9k
0 [aψ(a)] aψ(a)+ǫ(φ)aδψ(a)=0.
7 equationoscillateandsotheirderivativescanbecomesig- ada ada −16π2l2
(cid:20) (cid:21) Pl
0 nificant at large scales. If the QG corrections depend on (2)
: the derivatives of the wave-function, then it is possible Weonlyconsidertheflatcase,sothatsolutionstoEq.(2)
v
that they become significantat macroscopicscales. This areanalyticallytractable. Ifthisequationbreaksdownat
i
X is the case for Loop Quantum Cosmology (LQC), which small scales, due to some QG effects, extra terms should
r predicts that the evolution of the universe is dictated by becomesignificantonthesescales. ThesenewQGcorrec-
a a difference, rather than a differential equation. If the tion terms will be a function of ψ(a) and its derivatives,
lattice size of this difference equation is fixed, as was as- so the full underlying equation can be written as
sumed until recently [3], not all the oscillations of the
wave-functioncanbe supported, leading to a breakdown 1 d 1 d
[aψ(a)] +ǫ(φ)aδψ(a)
in large scale classical physics [3, 4]. ada ada
(cid:20) (cid:21)
We investigate the effects of a general class of cor- +a f ψ(a),∂ ψ,∂2ψ =0 , (3)
rections to the WDW equation, considering all possi- 0 a a ···
ble derivative terms and show that, at least initially, where a is the sc(cid:0)ale at which the n(cid:1)ew terms become
0
thesecorrectionsmimicthe behaviourofadditionalmat- important. At large scales, a a , one might naively
0
ter components. This raises the exciting possibility that assume that the QG correction≫s can simply be ignored,
QG corrections could, at late times, produce a cosmo- however this assumes that ψ(a) and all its derivatives
logical constant like effect. It is also possible that such remain small at all scales, which is easily shown not to
corrections may produce additional matter components always be consistent. This has been explored within the
suchasdarkmatter,howeverthis wouldonly be true for context of LQC [3, 4, 5], in which the exact form of the
a limited class of corrections. What is generally true is a corrections are known.
0
that QG correction terms that dominate at small scale
canalsoproducesignificantlargescaleeffects, whichcan Assuming a ǫ(φ), whilst f(ψ(a),∂ ψ, ) is
0 a
≪ ···
2
ρ δ terms grow, in agreement with Ref. [7]. In addition, for
φ
H
Matter a−3 a0 0 δ > 1, the terms containing the highest powers of ǫ
−
grow fastest and hence are always dominant, implying
Radiation a−4 a−1 1
−
Vacuum energy a0 a3 3 ψ∗(a)
∂ ψ(a) √ǫ lim a(1+δ)/2ψ(a) ,
a
≈ − a→∞ ψ(a)
TABLEI:ThescalingoftheenergydensityρandtheHamil- (cid:20) (cid:21)
tonian H = a3ρ for matter, radiation and vacuum energy ∂2ψ(a) ǫa1+δψ(a) ,
φ a ≈ −
with respect to both a and µ. Theparameter δ is also given. ψ∗(a)
NotethattheenergydensityandhenceH ofvacuumenergy ∂3ψ(a) ǫ3/2 lim a3(1+δ)/2ψ(a) ,
φ a ≈ a→∞ ψ(a)
are negative. (cid:20) (cid:21)
∂4ψ(a) ǫ2a2(1+δ)ψ(a) , (9)
a ≈
etc.
small, we can find approximate solutions to Eq. (3) as
This allows us to approximate the initial effect of large
ψ(a) C J 2√ǫ a(3+δ)/2 scale break down of classical physics (breakdown in pre-
≈ 1 3√+2δ (cid:20)3+δ (cid:21) classicality [6]) as a second matter component. In gen-
+ C2 Y 3√+2δ (cid:20)32+√ǫδa(3+δ)/2(cid:21) , (4) ceaordardrliettcihoteinoaβnlthtteerdrmmerssivliainkteivEaeqα.i∂s(aβ2pψ)r,o(ptaho)arcttaiosnncaablleetaolipkapeβr(oa1xα+i+mδ)β/a(21t,+edtδh)/bu2ys.
where J and Y are Bessel functions of the first kind and It is then possible to consider these correction terms as
secondkind,respectively,C andC areintegrationcon- being produced from an effective Hamiltonian. A useful
1 2
stants;forclaritytheφdependencehasbeensuppressed. pedagogicalexampleistoconsiderclassicalmatterterms
We can thus evaluate approximations to the derivatives: (although the method is, of course, general), where the
effectiveHamiltonianthatmimicsthecorrectiontermsis
∂ ψ(a) 1 √ǫZ (a)a(3+δ)/2 a−1ψ(a) ,
a ≈ − 1 cor aα+β(1+δ)/2 . (10)
(cid:16) (cid:17) H ∝
∂2ψ(a) Z (a)a(3+δ)/2 √ǫa3+δ √ǫa−2ψ(a) ,
a ≈ 1 − Anexampleofhowtheseapproximatesolutionscompare
∂3ψ(a) (cid:16)ǫZ a(9+3δ)/2 3Z a(3+δ)(cid:17)/2 √ǫa−3ψ(a) to the exact case is given in Fig. (1).
a ≈ 1 − 1 We have shown how hypothetical correction terms to
(cid:16)√ǫ(1+δ)a3+δ√ǫa−3ψ(a)(cid:17), (5) the WDW equation can mimic the behaviour of addi-
−
etc. tionalmattersources. Inparticular,suchcorrectionscan
be used to produce an effective cosmological constant.
where Forexample,consider a0, i.e., amatter dominated
φ
H ∝
universe(seeTableI);δ =0. Ifwewantthequantumcor-
Z1(a)≡C1 J3√+2δ+1(cid:20)32+√ǫδa(3+δ)/2(cid:21)hψ(a)i−1 are3c,tiiomnpslytoinmgiαm+ic βa/v2ac=uu3m. eTnheurgsy, ctohrernecwtieonneteedrmHscolrik∝e
+C Y 2√ǫ a(3+δ)/2 ψ(a) −1 . (6) a0a3, a0a2d2ψ/da2, a0ad4ψ(a)/da4, a0d6ψ(a)/da6,etc.,
2 3√+2δ+1(cid:20)3+δ (cid:21)h i wenitche oaf0a<m0a,tatlelrrdesoemmibnlaeteadvuancuivuemrseen(seeregyFiing.t(h1e))p.reIsn-
Taking the large argumentexpansion of the Bessel func- the presence of other types of matter δ changes and so
tions, which is the large a limit for δ > 3, one obtains theformofthecorrectiontermsthatgivevacuumenergy
−
like behaviour are different. In particular, for a universe
C1sin(A(a)) C2cos(A(a)) dominated by radiation, we have δ = 1 and the only
lim Z1(a) = − , (7) −
a→∞ C1cos(A(a))+C2sin(A(a)) correction term that can mimic the behaviour of dark
ψ∗(a) energy is a0a3. Considering a universe dominated by a
= lim , (8) field that scales like a2, i.e., δ = 2, then a cor-
a→∞(cid:20) ψ(a) (cid:21) rection term like a a−H3φd4∝ψ(a)/da4 mimics a (negative)
0
whereψ∗(a)isthesolutioncorrespondingtochoosingthe cosmological constant like term. This is precisely the
dominant correction from LQC.
integrationfactorsC C andC C comparedto
1 2 2 1
→− →
thesolutiongiveninEq.(4). Thisapproximationisvalid Up to now we have only discussed how the correction
only when the correction terms are still small compared termsscalewitha,howeverthereis clearlyalsoadepen-
to the matter component, however this method will give denceontheconstantpartofthematterHamiltonian,ρ
0
the correct effective initial behaviour of QG corrections. (throughǫ(φ)). In principle this differentiates these cor-
In particular, the corrections become non-negligible for rection terms from other models of dark energy (cosmo-
scales at which the corrections are of the order of a−1 logicalconstant,quintessence,etc.),asdoesthefactthat
0
andtheapproximationbreaksdownwhentheyareofthe as these correction terms begin to dominate the WDW
orderofǫ(φ)aδ. Itisclearthatforδ > 1thederivative equationthisapproximationwillbreakdownandthebe-
−
3
1
lutions and so we cannot say that these dark matter like
≀≀ ≀≀
correctionterms would produce the necessary behaviour
0.8
to explain structure formation, galaxy and cluster dy-
0.6 namics, etc. Another difficulty with this type of correc-
tion comes from the fact that the energy density of dark
0.4 Bare Approx.(dots) matterisapproximatelyfivetimesthatofstandardmat-
ց ց ter. If correction terms were to produce dark matter in
0.2
the presence of a matter dominated universe, a would
0
0 have to be fine tuned to ensure that a ǫ(φ) (the ap-
≀≀ ≀≀ 0 ≪
proximation used here) and a ǫ(φ)β/2−1 5. Similar
-0.2 0 ≈
dark matter like correction terms can be produced for a
-0.4 Eտxact(dashes) radiation,orvacuumenergy,dominateduniverse;ifthese
correctionsaroseduetothethepresenceofavacuumen-
-0.6
ergy,one may explainthe coincidence problem,i.e., that
thedarkmatterdegreesoffreedomwouldbedictatedby
-0.8
30 35 40 45 50 ≀≀ 105 110 115 120≀≀ 155 160 165 170 thoseofthedarkenergy,howeveragainsignificanttuning
a would probably be required.
FIG. 1: Solutions to the WDW equation, for δ = 0 (i.e., LQG is a background independent, non-perturbative
matter dominated universe), ǫ(φ)=5×10−3, a0 =9×10−6 method of quantising gravity. Reducing the symmetries
(withlPl=1). TheQGcorrectionsarea0a2d2ψ(a)/d2a. The of the theory to a specific cosmological model makes
solutionsarecalculated numerically: thebare solution (solid)
the theory tractableandensuresa largescalecontinuum
excludes QG corrections entirely whilst the approx. solution
limit. Thetheoryisbasedonholonomiesofthestandard
(dots)approximatesthemasdiscussedinthetext. Forsmalla
Ashtekarvariables[2],namelythetriadandconnections.
theQGcorrections donot play a role, however as abecomes
larger the effects of the QG corrections become significant. Inan isotropicmodelthese canbe parameterisedby sin-
Theformofthisdeviationiswellcharacterisedbytheaddition glevariables,p˜andc˜respectively,whichintermsofstan-
ofasecondmattercomponent,whichisdemonstratedbythe dard cosmological variables are p˜ = a2,c˜ = k + γa˙,
| |
accuracy of the approximate solution (see dotted line). For (γ is an ambiguity parameter known as the Barbero-
a ≈ 165, the approximate correction term is ≈ 0.25ǫψ(a) Immirzi parameter). Define p = p˜V2/3 and c = c˜V1/3,
andhencecannolongerbeconsideredsmallcomparedtothe 0 0
where V is the volume of a fiducial cell [8], related
matter component, ǫψ(a); theapproximation breaks down. 0
via the classical identity, c,p = 8πl2 γ/3. By ana-
{ } Pl
logue with the full LQG theory, c is quantised via its
holonomies, h = exp(iµ c/2), where µ is an arbitrary
0 0
real number. Then [8], pˆµ = p µ 4πl2Plγ|µ| µ , and
haviour of the wave-function will be drastically altered. | i | || i ≡ 3 | i
In additionto correctionsthat mimic the behaviourof e\µ0c/2 µ =eµ0ddµ µ = µ+µ0 .
| i | i | i
dark energy, it is also possible to find correction terms
TheactionofthegravitationalpartoftheHamiltonian
that have a dark matter like form, for which Eq. (10) is
constraint on the basis states µ reads [9],
| i
aα+β(1+δ)/2 a0 . (11)
cor 3
H ∝ ∝ ˆ µ = S(µ)+S(µ+4µ ) µ+4µ
Hg| i 256π2l4 γ3µ3 0 | 0i
Forexample,inamatterdominateduniverse(i.e.,δ =0), Pl 0
n(cid:2) (cid:3)
4S(µ)µ + S(µ)+S(µ 4µ ) µ 4µ (,14)
dψ(a) d2ψ(a) d3ψ(a) − | i − 0 | − 0i
a0a−1/2 da , a0a−1 da2 , a0a−3/2 da3 , etc. where (cid:2) (cid:3) o
(12)
allproduceadditionalmatterliketerms. Noticethat,un- S(µ)= 4πl2 γ/3 3/2 µ+µ 3/2 µ µ 3/2 . (15)
like the dark energy case, these corrections do not scale Pl | 0| −| − 0|
faster than original matter component and so will never (cid:0) (cid:1) (cid:12)(cid:12) (cid:12)(cid:12)
dominate Eq. (3). In addition, these correction mat- TheHamiltonianconstr(cid:12)aintis ˆ + ˆ ψ =0(cid:12),where
g φ
H H | i
ter terms will be closely related to the physical matter
Hamiltonian degrees of freedom. For example, |ψi= µψµ|µi. Takingtheco(cid:16)ntinuuml(cid:17)imitψµ →ψ(µ)
and using the expansion [4]
P
d4ψ(a)
a0a−2 da4 ≈a0ǫ(φ)2ψ(a) , (13) µ+αµ 3/2 µ+βµ 3/2 =µ3/2 3µ0 (α β)
0 0
| | −| | " 2µ −
whichamounttoreplacingǫ(φ) ǫ˜(φ)=ǫ(φ)+a ǫ(φ)2
→ 0 3µ2 µ3
in Eq. (3). Within the mini-superspace model used here + 0 α2 β2 0 α3 β3 + , (16)
it is only possible to explore homogeneous, isotropic so- 8µ2 − − 16µ3 − ···#
(cid:0) (cid:1) (cid:0) (cid:1)
4
we can expand Eq. (14). ter sources, at least whilst they are smaller than the ex-
Changing variables, using a2 =4πl2 γ µ/3, we obtain isting mattercomponents. Thissimple procedurecanbe
Pl | |
used to examine, to firstorder,the effect of any QG cor-
2πl2 1 d 1 d d 1dψ(a) rectionunder consideration. Here, as a specific example,
ψ(a)= Pl aψ(a) +
g we used the first order corrections from loop quantum
H 3 ada"ada # da"a da #!
cosmologyto examinewhatthe breakdownoflargescale
(cid:2) (cid:3)
2πl2 a 1 d4ψ(a) 4 d3ψ(a) 47 d2ψ(a) classical physics looks like in the absence of lattice re-
+ Pl 0 +
9 a3 da4 − a4 da3 8a5 da2 finement. The fact that quantum corrections introduce
extra matter components opens up the possibility that
1 dψ(a) 135 they may be able to explain the current cosmological
+ ψ(a) , (17)
2a6 da − 16a7 ! acceleration and possibly dark matter. That these cor-
rection components become significant only at specific
scalesisencouraging,asthismayprovideanexplanation
where a = 16π2l4 γ2µ2/9. The full Hamiltonian con-
0 Pl 0 why classical general relativity is valid on sub-galactic
straint is easily written in the same form as Eq. (3). As
scales,butrequirestheinputofadditionmatteronsuper-
we have shown, for δ > 1 the higher derivative terms
− galactic scales. However it remains to be seen if it is
scale faster. Since in this case the lower derivative terms
possible for such corrections to meet observational con-
arefurther suppressedby powersofa, to a goodapprox-
straints.
imation the effective large scale equation to solve is
This work can be used in two complementary ways:
top-down and bottom-up. The former is to apply this
1 1 d 1 d d 1dψ(a)
aψ(a) + method to any fundamental theory that can produce a
2 ada ada da a da
" # " #!
WDW like equation on large scales and so characterise
a (cid:2) (cid:3)
+ 0ǫ(φ)2a2δ−1ψ(a)+ǫ(φ)aδψ(a) 0 , (18) the new, phenomenological effects of the theory. The
3 ≈ latter would estimate the types of QG corrections that
would be necessary to produce certain desirable effects
where 3 /(2πl2 ) = ǫ(φ)aδ and we used Eq. (9d) to
Hφ Pl (e.g., dark energy) at particular scales. The case of
approximatethe fourthorderderivativeterminEq.(17)
loopquantumcosmologyillustrateshowbothapproaches
bytermthatresemblesanadditionalmattercomponent.
can be used to mutual benefit. The underlying theory
LQC is a concrete example of the generalmethods for
was used to calculate the form of the QG corrections
approximating the QG corrections. For a universe dom-
to the WDW equation. The phenomenological conse-
inated by a (classical) matter content with δ = 0, the
quence(eventualdominationofthe correctiontermsand
leadingcorrectiontermactsasaneffectiveradiationfield,
hencea breakdowninlargescaleclassicalphysics)high-
= a ǫ(φ)2a−1/3, i.e., the correction term mim-
Hcor 0 lighted the importance of modelling lattice refinement
ics radiation. For a radiation dominated universe with
within cosmologicalmodels.
δ = 1, the correction term acts as =a ǫ(φ)2a−3.
cor 0
− H Considering the phenomenological consequences of a
For δ = 3, i.e., a universe dominated by a vacuum en-
theory is the first step to testing it against experimental
ergy, such as during inflation, the correction term acts
like = a ǫ(φ)2a5. Notice that in this case the andobservationaldata. WehaveshownherethatforQG
cor 0
H this principle can be invaluable in guiding our searchfor
corrections scale faster than the existing matter com-
the full theory.
ponent, which motivated the modelling of lattice refine-
ment (µ µ˜(µ) is no longer a constant) in loop quan-
0
→
tum cosmology [3, 6, 7]. For a universe dominated by a
source with δ =2, the dominant correction term acts as
=a ǫ(φ)2a3, i.e. itmimics anegativecosmological Acknowledgments
cor 0
H
constant(i.e. anacceleratedcontraction),whilstδ =1/2
leads to the correction term mimicking dark matter. This work is partially supported by the European
We have shown how generic QG corrections to the Union through the Marie Curie Research and Training
WDW equation mimic the behaviour of additional mat- Network UniverseNet (MRTN-CT-2006-035863).
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5
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