Comoving suppression mechanism and cosmological constant problem

0.12 MB·English

by  Jian Qi Shen

Comoving suppression mechanism and cosmological constant problem
∗
Jian Qi Shen
Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, P.R. China
(February 7, 2008)
theoretical cosmological constant, Λth, to the observed
In this paper, we assume that the observer is fixed in a
one, Λex, is
comovingframeofreferencewithg = λ,whereλandΛde-
00 Λ
note thecomoving parameter and thecosmological constant, 2
respectively. By using the comoving suppression mechanism Λth = 1 rUni 10120, (1.1)
andMach’sprinciple(thelatterofwhichisusedtodetermine Λex 3π (cid:18) rPl (cid:19) ≃
the comoving parameter λ), we calculate the vacuum energy
where the orders of magnitude of the Plank length and
density of quantum fluctuation field in the above-mentioned
4 comovingframeofreference. Itisshownthatinsuchacomov- cosmologicalradius (i.e., rPl ≃10−35 m and rUni ≃1026
0 m) have been inserted.
ing frame of reference, thecosmological constant will greatly
0
decrease by many orders of magnitude (if Mach’s principle During the past 40 years, in an attempt to deal with
2
is applied to this calculation, then it will be shown that Λ the cosmological constant problem, many theoretical

n is reduced by about 120 orders of magnitude). Additionally, works such as adjustment mechanism, changing grav-
a
we briefly discuss the related topics such as the varying ob- ity, quantum cosmology and the viewpoint of super-
7 J soefravnedomspaleoeudsoafcclieglhetra(tiddoctn=(∼O101−09−m9m/s/2s)2a)cqaunirdedthbeymtyhsetePriy- s[3y]m. mMeotrrey,rescuepnetrlyg,rasvinitcyeasonmdesuapsterrosptrhiynsgiscawleorbesperrvoaptoiosends
1 (cid:0) (cid:1)
oneer 10/11, Galileo and Ulysses spacecrafts. show that the large scale mean pressure of our present
1 Keywords: Comoving suppression mechanism, observed universe is negative suggesting a positive cosmological
v speed of light, cosmological constant constant,andthattheuniverseisthereforepresentlyun-
7
dergoinganacceleratingexpansion[4],alargenumberof
7
theoriesandviewpoints (suchasback reactionofcosmo-
0
1 logical perturbations [5], QCD trace anomaly [6], con-
0 tribution of Kaluza-Klein modes to vacuum energy [7],
4 I. INTRODUCTION five-dimensionalunificationofcosmologicalconstantand
0
photon mass [8], nonlocal quantum gravity[9], quantum
/
c Historically, the problem of vacuum energy density or micro structure of spacetime [10], relaxation of the cos-
q
cosmological constant [1] has been considered for many mological constant in a movable brane world [11], the
-
r years. There are now two cosmological constant prob- effect of minimal length uncertainty relation (under the
g
lems. One is that why the observedvalue of the vacuum modified commutation relation [q,p]) on the density of
:
v energydensityissosmall(theratioofexperimentalvalue states[12],andsoon)areputforwardtoresolvethecos-
Xi to the theoretical one is only 10−120). The other one is mological constant problem. In this paper, we will sug-
to understand why the observed vacuum energy density gest a so-called comoving suppression mechanism to re-
r
a is not only small, but also, as current Type Ia super- solve the cosmological constant problem. Differing from
nova observations seem to indicate, of the same order of the conventional viewpoint that the observer is fixed at
magnitude as the present mass density of the universe the comovingframe ofreferencewith the metric g00 =1,
[2]. It is readily verified that the quantum vacuum fluc- here we argue that the observer may be located in a co-
tuation energy density of one field is U = 8π2ch¯7G2 and moving coordinate system of the metric g00 = Λλ with λ
the cosmologicalconstantis thereforeΛ= 8πGU = c3 , being a certain parameter that will be discussed in the
1 1 c4 πh¯G present paper. It will be shown that in such a comov-
which can be rewritten as Λth = πrP2l, where the Planck ing frame of reference the vacuum energy density can be
lengthisrPl = Gc3h¯. However,itisalsowellknownthat greatly suppressed by the comoving compressionmecha-
the observedcoqsmologicalconstant can be expressed1 as nism. In order to determine the comoving parameter λ,
Λex = 3cH22 ≃ rU23ni, where rUni denotes the cosmological weTwhiellpmapakeer udsisecuofssMtahcrhee’stporpiniccsi:pl(ei)inthtehecodniscceupstsioofn.λ-
radius (length scale of the universe). So, the ratio of the
de Sitter metric and the comoving parameter λ; (ii) the
calculation of vacuum energy density in the λ- de Sitter
universe; (iii) the varying observed speed of light and
the potential relation to the anomalous acceleration (
1Thisexpression meansthateventhoughΛisverysmall, it 10−9m/s2) acquired by the Pioneer 10/11, Galileo an∼d
is comparable to thepresent matter density.
Ulysses spacecrafts [13].
1
II. COMOVING FRAME OF REFERENCE AND it will be verified that λ =10−30. This, therefore,im-
Λ
λ- DE SITTER METRIC 30
plies that the fundameqntal constant c is 10 times more
than the observed speed of light (c′ = 3.00 108 m/s).
In this section, first we propose the concept of λ- de ×
Thus it is believed that the above viewpoint cannot in-
Sitter world, the line element of which takes the form
terpret the cosmologicalconstant problem.
The previous discussion shows that the observed cos-
λ λ
ds2 = c2dt2 exp 2 ct [dr2+r2 dθ2+sin2θdϕ2 ], mological constant is not the physical meanings of λ,
Λ − r3 ! namely,λcannotbeconsideredaso-calledobserved cos-
(cid:0) (cid:1) mologicalconstant. The observed cosmologicalconstant
(2.1)
is still Λ. In what follows we will consider the compres-
sion mechanism of vacuum energy density in the λ- de
which can be solved via Einstein field equation with the
Sitter world (i.e., the comoving frame of reference with
cosmological constant term. In this comoving frame of
reference with g00 = Λλ (where the coordinate time is no g00 = Λλ).
longer the proper time), the cosmologicalexpansionrate
equations are of the form
III. SUPPRESSION OF Λ IN λ- DE SITTER
8πGλ λ 3k 3R˙2 WORLD
ρ+λ= + ,
c2 Λ ΛR2 c2R2
8πGλ λ k R˙2 2R¨ In this section, the vacuum energy density in the λ-
p=λ , (2.2) de Sitter worldwillbe calculatedand,by takingaccount
c4 Λ − ΛR2 − c2R2 − c2R
of Mach’s principle, the reduction of Λ by 120 orders of
where R denotes the cosmological scale factor. If we magnitude will be demonstrated.
assume that the Hubble parameter H = R˙ is constant In the comoving coordinate system with g00 = Λλ, the
R
and we can ignore the matter density and pressure2 in action of a particle is S = mc ds with
−
Eq.(2.2), we may obtain λ = 3H2. In this step, some-
c2 R
one may think that in Eq.(2.2) the comoving parameter λ λ λ
ds= c2dt2 exp 2 ct dr2 = c2 v2dt,
λ acts just as the cosmological constant. This, however, vuΛ − r3 ! rΛ −
may be not the true case, the reason for which will be u
t
consideredasfollows: theHubbleparameterH wasmea- (3.1)
sured by the Doppler redshift observation3, where the
redshift z = R˙ L with L being the physical distance wherethetestparticlevelocitysquaredisdefinedasv2 =
between the oRbcs√erΛvλed star and the observer. Thus the exp 2 λ3ct ddrt 2. Thus the Lagrangian of the test
observed Hubble parameter Hobs = RR˙ = zc ΛλL1. By part(cid:18)iclequnde(cid:19)r(cid:0)con(cid:1)sideration is L = mc λc2 v2. It
using the above-obtained relation λ = 3H2, oqne can ar- − Λ −
c2 follows that the canonical momentum andqthe canonical
rive at
Hamiltonian read
Λ= 3z2c2. (2.3) ∂L mv mΛλc2
L2 p= = , H =pv L= . (3.2)
∂v λ v2 − λ v2
ItshouldbenotedthateventhoughEq.(2.3)isafamiliar Λ − c2 Λ − c2
q q
expression, it is of no help for overcoming the cosmolog- Let us define the observed mass and the observed speed
ical constant problem, since here the speed of light c is of light as follows: m′ = m , c′ = λc. Clearly, the
no longer the observed one. Instead, the observed speed √λ Λ
Λ
of light is c′ =c λ (rather than c). In the next section expressions (3.2) can be rewritten asq
Λ
q m′v m′c′2
p= , H = , (3.3)
1 v2 1 v2
− c′2 − c′2
2Recent observation shows that the components of regular q q
whichtakethe formfamiliarto us all. Forthe extremely
matter, dark matter and dark energy in the universe is as
′ ′
relativistic case v c, we have pc H. I assume that
follows: 4% (regular matter), 23% (dark matter), 73% (dark
→ →
energy). This, therefore, means that the component that is theobservers(say,we)arefixedactuallyinthecomoving
described by the cosmological constant term is more than 23 frame of reference′with g00 = Λλ, and that the observed
of the total matter in thewhole universe. speed of light is c rather than c. The calculated result
3Theredshiftisz= △λ withλbeingthewavelengthoflight. for the vacuum energy density in the comoving frame of
λ
reference with g00 = Λλ is
2
2 pPl 21 where the expression for c, i.e., c = c′ 1038m/s has
U = (2π¯h)3 4πp 2¯hωpdp √Λλ ∼
Z been inserted. In Eq.(3.8) T denotes the cosmological
2 pPl 1
4πp2 (pc′)dp, (3.4) age of the present universe. In accordance with the re-
≃ (2π¯h)3 2 sult in Eq.(3.8), one can conclude without fear that the
Z
comoving parameter λ taking the form (3.7) just agrees
where the cutoff momentum is pPl = h¯c′3. Note that with Mach’s principle.
G
here the fundamental constant c has bqeen replaced with Additionally, it should be pointed out that the ob-
the observed speed of light c′ (c′ = 3.00 108 m/s). served (and measured) mass m′ of a particle is not
Further calculationyieldsU = c′7 , andc×onsequently the real mass m, the relation between which is given
8π2h¯G2
the calculated cosmologicalconstant reads m = λm′ 10−30m′. It may be believed that this
Λ ≃
8πG c′7 relatioqnalsohas close relationto Mach’s principle which
Λ= . (3.5) holds that the inertial properties of a body are deter-
c4 8π2¯hG2
mined by the mass distribution in the universe, and the
It should be noted that in Eq.(3.5) the coefficient 8πG inertial force acting upon a body arises from an inter-
need not to be replaced with 8πG, since here c is a fuc4n- action between it and the matter-energy content of the
c′4 whole universe. It is of physical interest that the real
damentalconstantratherthananobservedspeedoflight.
mass m is actually merely one part of 1030 of the ob-
By using the relation (i.e., c= Λc′) between c and c′,
λ served one, which, therefore, means that the real total
Eq.(3.5)canbe rewrittenas Λ3q= c′3 λ2. With the help cosmologicalmass is only about 1023 Kg that is surpris-
πh¯G
ingly less than that of the Earth (the observed mass of
of rPl = Gc′h¯3, one can arrive at the Earth is 6.0 1024 Kg)!
×
q In order to close this section, we briefly conclude the
2
λ3 comoving suppression mechanism for the cosmological
Λ= , (3.6)
π13rP23l constantwithsomeremarks. Inviewoftheabovediscus-
sion,the comovingsuppressionmechanismis established
which is expressed in terms of the Plank length and the based on the following four points: (i) the assumption
comoving parameter λ. Thus it follows from Eq.(3.6) that the observer is fixed in a comoving frame of refer-
that the vacuum energy density (or cosmological con- ence with g00 = Λλ; (ii) Mach’s principle that can de-
stant Λ) is determined only by the comoving parameter termine the comoving parameter λ; (iii) the observed
λ. If λ is taken to be very small (or nearly vanishing), speed of light c′ = c λ; (iv) the vacuum energy den-
Λ
then the corresponding Λ is also very small. This is just
the comoving suppression mechanism for the cosmologi- sity U = 8π2c′h¯7G2 in thqe comoving frame of reference with
calconstant. Eventhoughaccordingto the conventional g00 = Λλ. Thus the comoving suppression mechanism
choice where it is assumed that the observer is fixed in will inevitably lead to the relaxation of the cosmological
the comoving frame of reference with g00 = 1, the vac- constant by 120 orders of magnitude.
uumenergydensityisverylarge(evennearlydivergent),
in the comoving frame with g00 = Λλ, Λ is greatly com-
IV. THE VARYING OBSERVED SPEED OF
pressed by many orders of magnitude.
LIGHT
Nowtheproblemlefttousistodeterminethevalueof
the comoving parameter λ. Brief analysis indicates that
it is in connection with Mach’s principle: specifically, According to the above comoving suppression mecha-
substitution of the observed cosmological constant Λ nism,itisreadilyverifiedthattheobservedspeedoflight
3 into Eq.(3.6) yields ≃ c′ is not constant. In fact, its variation (rate of change)
r2
Uni at present is
λ= rrU3Pnli, Λλ ≃ rrUPnli ∼10−61. (3.7) ddct =O 10−9m/s2 . (4.1)
(cid:16) (cid:17)
How can we understand the physical meanings of the In 1998, Anderson et al. reported that, by ruling out
obtainedλinEq.(3.7)? Theexpression2 λctintheex- anumber ofpotentialcauses,radiometric datafromthe
3
Pioneer10/11,GalileoandUlyssesspacecraftindicatean
ponentialfactorofthelineelement(2.1)oqfthecomoving
frame with g00 = Λλ reads asppapcaercernatftanwoitmhaalomusa,gcnointsutdaent,a8c.c5ele1ra0t−io8cnma/ctsi2ndgiorencttehde
∼ ×
′ towards the Sun [13]. Is it the effects of dark matter or
λ 2 cT
2 cT = = (1), (3.8) a modification of gravity? Unfortunately, neither easily
r3 √3rUni O works. It is interesting that by taking the cosmic mass,
3
M =1053 kg,andcosmicscale,R=1026 m,ourcalcula- In the conventional comoving frame of reference, the
tionshowsthatthis acceleration is just equal to the value metric of de-Sitter world is
of field strength on the cosmic boundary due to the total
Λ
cosmic mass. This fact leads us to consider a theoreti- ds2 =c2dt2 exp 2 ct dr2 (5.1)
calmechanismto interpretthis anomalousphenomenon. − r3 !
The author favors that the gravitational Meissner effect
rather than Eq.(2.1), i.e.,
may serve as a possible interpretation. Here we give a
rough analysis, which contains only the most important
λ λ
features rather than the precise details of this theoreti- ds2 = c2dt2 exp 2 ct dr2. (5.2)
cal explanation. Parallelto London,s electrodynamics of Λ − r3 !
superconductivity, it shows that gravitational field may
give rise to an effective rest mass m = h¯ √8πGρ due Itisbelievedthatthecomovingparameterλin(5.2)pos-
g c2 m sessesphysicalmeanings and therefore deservesessential
to the self-induced charge current [14], where ρm is the consideration. Note that since in Eq.(5.2), λc2 = c′2
mh¯ass de1n0si2t6ymofthtahte aupnpivreorxsiem.aTtehleyneoqnuealcsaRn,obwthaeinreththaet with c′ = 3.00 × 108 m/s, our experimentalΛmeasure-
mgc ≃ ments (astrophysical observations) cannot demonstrate
mass density of the universe is taken to be ρ = 0.3ρ
m c whether we are just in a comoving frame of reference
[15] with ρc ≃ 2×10−26kg/m3 being the critical mass with g00 = Λλ rather than with g00 =1, which, therefore,
density. An added constant acceleration, a, may result
means that the comoving parameter λ should be deter-
4
from the Yukawa potential and can be written as
mined on by Mach’s principle. But, what is the physical
GM mgc 2 GM −9 2 origin of Mach’s principle? This problem seems to be of
a= 2 ¯h ≃ R2 ≃10 m/s . (4.2) no satisfactorysolutions upto now. So, the key problem
(cid:16) (cid:17) of the comoving suppression mechanism for the cosmo-
Hence we demonstrated that the value of 10−9m/s2 is logicalconstantprobleminthepresentpapermaybethe
averytypicalandinteresting“acceleration”,whicharises resolutionof the physical meanings and originof Mach’s
in (i) the anomalous acceleration acting on the Pioneer principle, which is now under consideration and will be
10/11, Galileo and Ulysses spacecrafts [13], (ii) gravi- published elsewhere.
tational Meissner effect [16], (iii) the gravitational field ∗Shen’selectronicaddress: [email protected]
strengthatthe cosmicboundary,and(iv)eventhe vary-
ingobservedspeedoflight. Itisreasonablybelievedthat
thesefacts mayclue physicistsonthe mysteryofanoma-
lous acceleration acquired by the Pioneer 10/11, Galileo
and Ulysses spacecrafts [13]. Since this subject is be-
yondthescopeofthe presentpaper,wewillnotconsider
[1] onthehistoryofthecosmological constantproblem,see,
it further.
for example, Straumann,N., (2002). gr-qc/0208027.
[2] Weinberg, S., (2000). astro-ph/0005265.
[3] Weinberg, S., (1988). Rev. Mod. Phys. 61, 1 and refer-
V. CONCLUDING REMARKS
ences therein.
[4] Perlmutter,S.etal.,(1998).Nature391,51;Perlmutter,
The comoving suppression mechanism for the cosmo- S. et al., (1999). Astrophys. J. 517, 565; Riess, A.G. et
logicalconstantproblemassumesthatwe(observers)are al., (1998). Astrophys.J. 116, 1009.
fixed in a comoving coordinate system with g00 = Λλ [5] Brandenberger, R.H., (2002). hep-th/0210165.
rather than with g00 = 1, and that the comoving pa- [6] Schu¨tzhold, R., (2002). gr-qc/0204018.
[7] Gupta, A., (2002). hep-th/0210069.
rameter λ can be determined by Mach’s principle. In
[8] Kohler, C., (2002). gr-qc/0202076.
such a theoretical framework, the calculation of the vac-
[9] Moffat, J.W., (2002). hep-th/0207198; hep-th/0210140.
uum energy density in this comoving frame of reference
[10] Padmanabhan, T., (2002). gr-qc/0204020.
indicates that the cosmological constant is reduced by
[11] Khlebnikov,S., (2002). hep-th/0207258.
120 orders of magnitude.
[12] Chang, L.N.,Minic, D.,Okamura,N.,andTakeuchi,T.,
(2002). hep-th/0201017.
[13] Anderson, J.D., Laing, P.A., Lau, E.L., et al., (1998).
Phys. Rev.Lett. 81, 2858.
4It may also be calculated as follows: a = GM mgc 2 = [14] Hou,B.Y.andHou,B.Y.,(1979).Phys.Ener.Fort.Phys.
GcM2 (4πρmG) = Gc2MRG(4πRR23ρm) ≃ GRM2 , where us2e i(cid:0)s mh¯ad(cid:1)e of [15] PNeuacclo.c3k,,2J5.5A.., Cole, S., Norberg, P., et al., (2001). Na-
Gc2MR ≃ O(1),4πR3ρm ≃ M, which holds when the approxi- ture 410, 169.
mate evaluation is performed.
[16] Shen, J.Q., (2003). gr-qc/0305094.
4