A time varying speed of light as a solution to cosmological puzzles

14 Pages·0.25 MB·English

by  Andreas Albrecht

A time varying speed of light as a solution to cosmological puzzles
Andreas Albrecht and Jo˜ao Magueijo
Theoretical Physics, The Blackett Laboratory, Imperial College, Prince Consort Road, London, SW7 2BZ, U.K.
WeconsiderthecosmologicalimplicationsoflighttravellingfasterintheearlyUniverse. Wepropose
a prescription for deriving corrections to the cosmological evolution equations while the speed of
light c is changing. We then show how the horizon, flatness, and cosmological constant problems
may be solved. We also study cosmological perturbations in this scenario and show how one may
solve the homogeneity and isotropy problems. As it stands, our scenario appears to most easily
produceextremehomogeneity,requiringstructuretobeproducedintheStandardBigBangepoch.
Producing significant perturbations during the earlier epoch would require a rather careful design
of the function c(t). The large entropy inside the horizon nowadays can also be accounted for in
this scenario.
9 PACS Numbers: 98.80.Cq, 98.80.-k, 95.30.Sf
9
9 I. THE PUZZLES OF THE BIG BANG MODEL matter content of the Universe, we change the speed of
1 light in the early Universe. We assume that the Uni-
n Cosmologists have long been dissatisfied with the verse matter content is the same as in the SBB, that is,
a “StandardBigBang”(SBB)modeloftheUniverse. This the Universe is radiation dominated at early times. We
J is not due to any conflict between the big bang theory also assume that Einstein’s gravity is left unchanged, in

5 and observations, but because of the limited scope of- a sense made precise in Section IV. The geometry and
fered by the SBB to explain certain striking features of expansion factor of the Universe are therefore the same
2 the Universe. From the SBB perspective the homogene- as in the SBB. However the local speed of light, as mea-
v
ity, isotropy,and “flatness” of the Universe, and the pri- suredbyfreefallingobserversassociatedwiththecosmic
8
1 mordial seeds of galaxies and other structure are all fea- expansion, varies in time, decelerating from a very large
0 tures which are “built in” from the beginning as initial value to its current value.
1 conditions. Cosmologistswouldlike to explainthese fea- We discuss below how Varying Speed of Light (VSL)
1 tures as being the resultofcalculable physicalprocesses. models might resolve the same cosmological puzzles as
8 A great attraction of the Inflationary Cosmologies [1] is inflation, and offer a resolution to the cosmological con-
9 thatthey addressthese issues byshowingonthe basisof stantproblemaswell. Weshallnotdwellonthe possible
/
h concrete calculations that a wide variety of initial condi- mechanisms by means of which the speed of light could
p tions evolve, during a period of cosmic inflation, to re- have changed. Rather we wish to concentrate on the
o- flectthehomogeneity,isotropy,flatnessandperturbation conditions one should impose on VSL models for their
r spectrum that we observe today. cosmological implications to be interesting. This phe-
st So far, all attempts to achieve this kind of improve- nomenological approach should be regarded as a curios-
a mentovertheSBBhavewounduptakingthebasicinfla- ity,which,wehope,willpromptfurtherworktowardsan
v: tionary form, where the observable Universe experiences actual theory in which the physical basis of VSL models
i a period of “superluminal” expansion. This is accom- is realized.
X
plished by modifying the matter content of the Universe Onemaydoubtthatsuchaself-consistenttheorycould
r in such a way that ordinary Einstein gravity becomes ever be constructed. We therefore feel forced to tran-
a
repulsive and drives inflationary expansion. This pro- scend the scope of this paper, and discuss essential as-
cessisinmanywaysremarkablystraightforwardandhas pects of such a theory. We find it befitting to start our
found numerous realizations over the years ( [2–5], etc), discussionwithanassessmentoftheexperimentalmean-
althoughit might still be arguedthat a truly compelling ing of a varying c (Section II). We also need to be more
microscopic foundation for inflation has yet to emerge. specificaboutVSLtheoriesinordertotackletheflatness,
One interesting question is whether inflation is the cosmological constant, homogeneity, and entropy prob-
right solution to the cosmological puzzles. Is inflation lems. In Section IV we state what is actually required
really what nature has chosen to do? When this mat- fromanyVSLtheorytosolvetheseproblems. Howeverin
ter is discussed there is a notable absence of any real Appendix I we layoutthe foundations forsucha theory.
competition to inflation, and this must be countedin in-
flation’s favour. However, we believe the picture would
II. THE MEANING OF A VARIABLE SPEED OF
become much clearer if some kind of debate along these
LIGHT
lines were possible. To this end, we discuss here a possi-
blealternativetoinflationarycosmologywhich,whilenot
as well developed as today’s inflationary models, might Wefirstaddressthequestionofthemeaningofavary-
lead to some illuminating discussion. ing speed of light. Could such a phenomenon be proved
In this alternative picture, rather than changing the ordisprovedbyexperiment? Physically itdoes notmake
1
sense to talk about constancy or variability of any di- [20]. Built into E 1/a and λ a is the assump-
mensional “constant”. A measurement of a dimensional tion that h¯c is consta∝nt, for otherwis∝e the wavevectorkµ
quantity must always represent its ratio to some stan- and the momentum vector pµ could not both be parallel
dard unit. For example, the length of my arm in meters transported. Hence the experimental statement that h¯c
is really the dimensionless quantity given by the ratio of is constant is circular.
thearmlengthtothelengthofameterstick. Iftheratio What would we do therefore if we were to observe
varied, one could interpret this as a variation in either changing dimensionless quantities? Any theory explain-
(orboth)ofthetwolengths. Infamiliarsituations,there ing the phenomenon would necessarily have to make use
is usually a preferred interpretation which distinguishes ofdimensionalquantities. Itwouldaprioribeamatterof
itself by giving a simpler view of the world. Choosing a choice, prejudice, or convenience to decide which dimen-
givenperson’sarmasa standardoflengthwouldrequire sional quantities are variable and which are constant (as
a whole range of simple objects to undergo peculiar dy- we mentionedin the illustrationabove). There wouldbe
namics,whereasassumingthemetersticktobeconstant a kind of equivalence, or duality between theories based
would usually give a much simpler picture. on any two choices as far as dimensionless observations
None the less, a given theory of the world requires di- are concerned. However, the equations for two theories
mensional parameters. If these parameters varied, how whichareobservationallyequivalent,butwhichhavedif-
wouldthatprocessshowupinexperiments? Suppose we ferentdimensionalparametersvarying,willingeneralnot
set out to measure the speed of light. For this one needs look the same, andagainsimplicity will end up being an
alengthmeasure(rod)andaclock. Inaworlddescribed importantfactorinmakingachoicebetweentheories. In
by a theory with time varying dimensional parameters, what follows, we will prefer to work with models which
it is quite possible that the rods and clocks, as well as have the simplicity of “minimal coupling”.
thephotonspeeds,couldallvary. Becausemeasurements Let us illustrate this point with a topical example.
arefundamentallydimensionless,theexperimentalresult There has been a recent claim [22] of experimental ev-
willonlymeasuresomedimensionlesscombinationofthe idence for a time changing fine structure constant α =
fundamental constants. Let us sketch a simple illustra- e2/(4π¯hc). Although the ongoing chase for systematics
tion: Suppose we measure time with an atomic clock. precludesanydefinitiveconclusions,letusassumeforthe
Taking the Rydberg energy (E = m e4/2(4πǫ )2¯h2) to purpose of the argument that the effect is real.
R e o
represent the dependence of all atomic energy levels on In building a theory which explains a variable α we
the fundamental constants, the oscillation period of the must make a decision. We could postulate that electric
atomic clock will be ¯h/E . Likewise, taking the Bohr charge changes in time, or, say, that h¯c must change in
R
radius(a =4πǫ ¯h2/∝m e2)toreflecttherelationshipbe- time. Bekenstein [23] constructs a theory based on the
0 0 e
tween the lengths of ordinary objects (made of atoms) first alternative. He postulates a Lorentz invariant ac-
and the fundamental constants, the length of our rod is tion, which does not conserve electric charge. Our the-
a . Thus a measurement of c with our equipment is oryisbasedonthesecondchoice. Wepostulatebreaking
0
∝
really a measurement of the dimensionless quantity Lorentzinvariance,achangingh¯c,andconsequentlynon-
conservation of energy. Any arguments against the ex-
c 8πǫ
0
= (1) perimental meaning of a changing c can also be directed
a /(h¯/E ) α
0 R at Bekensteins’ changing e theory, and such arguments
essentiallythefinestructureconstant. Wecouldofcourse areinbothcasesmeaningless. Inbothcasesthechoiceof
use other equipment which depends in different ways on a changing dimensional “constant” reverts to the postu-
the fundamental dimensionless constants. For example, latesofthetheoryandisnot,apriori,anexperimentalis-
pendulum clocks will necessarily involve Newton’s con- sue. Theobservablesarealwaysdimensionless. However,
stantG. Differentexperimentswillresult,whichmeasure theminimally coupledtheoriesbasedoneitherchoiceare
different dimensionless combinations of the fundamental not dual (as we shall point out in Appendix I). For this
dimensional constants. Our conclusion that physical ex- reason one might prefer one formulation over the other.
periments are only sensitive to dimensionless combina- Finally, and on a different tone, suppose that future
tions of dimensional constants is hardly a new one. This experiments were to confirm that not only α changes in
idea has been often stressed by Dicke (eg. [21]), and we time, but also that there are time variations in dimen-
believe this is not controversial. sionless coupling constants based on other interactions,
Thus,speakingintheoreticaltermsoftimevaryingdi- α = g2/(h¯c)1. Suppose further that the ratios between
i i
mensional constants can lead to problems. To give an the various constants, r = α /α , were observed to be
ij i j
historical example, papers [18,19] were written claiming
stringent experimental upper bounds on the time vari-
abilityofthedimensionalquantityh¯c. Inthesetheprod-
uctEλwasfoundtobethesameforlightemittedatvery 1In writing these constants we have assumed that the cou-
different redshifts. From the deBroglie relation h¯c=Eλ plings of these interactions are defined in terms of “charges”
oneinferstheconstancyofh¯c. Bekensteingivesanillumi- (with dimensions of [E]1/2[L]1/2).
nating discussion of the fallacy built into this argument
2
TIME TIME
US NOW US NOW
t
TWWHOIC RHE AGRIOEN CSA IUNS OAULRLY PAST PAST LIGHT CONE HORIZON AT c PAST LIGHT CONE
DISCONNECTED
t
c
SOME TIME IN OUR PAST
BIG BANG
BIG BANG SPACE
FIG.1. Conformaldiagram(lightat45◦)shSoPwACiEngthehori- moFdIeGl.in2.wDhiicahgraatmtimsheowtinthgethspeehedoroizfolnighsttrcuhcatunrgeedinfraomSBcB−
c
zonstructureintheSBBmodel. Ourpastlightconecontains to c+ ≪ c−. Light travels at 45◦ after t but it travels at
c
regions outside each others’ horizon.
a much smaller angle with the space axis before t . Hence
c
it is possible for the horizon at t to be much larger than
c
the portion of the Universe at t intersecting our past light
constant. Choosing what dimensionalconstants were in- c
cone. Allregions inourpasthavethenalwaysbeenincausal
deedconstantswouldstillbeamatteroftaste. Onecould
contact.
stilldefineatheoryinwhichthevariouschargesg change
i
intime,withfixedratios,andh¯cremainsconstant. How-
ever it would perhaps start to make more sense, merely Suppose there was a “phase transition” at time t when
c
for reasons of simplicity, to postulate instead a changing the speed oflight changedfromc− to c+. Our pastlight
¯hc. cone intersects t = t at a sphere with comoving radius
c
Therefore, even though a variable c cannot be made a r =c+(η η ),whereη andη aretheconformaltimes
0 c 0 c
−
dimensionlessstatement,evidenceinfavouroftheoretical now and at t . This is as much of the Universe after the
c
models with varying c could be accrued if the other α phase transition as we can see today [6]. On the other
i
changed, with fixed ratios. hand the horizon size at t has comoving radius r =
c h
c−η . If c−/c+ η /η , then r r , meaning that the
c 0 c h
≫ ≪
whole observable Universe today has in fact alwaysbeen
III. COSMOLOGICAL HORIZONS incausalcontact(seeFig.2). Somesimplemanipulations
show that this requires
Perhaps the most puzzling feature of the SBB is the
c− 1 1 T+
presence of cosmological horizons. At any given time log 32 log z + log c (2)
any observercan only see a finite regionof the Universe, 10 c+ ≫ − 2 10 eq 2 10 T+
P
with comoving radius r = cη, where η denotes confor-
h
wherez istheredshiftatmatterradiationequality,and
maltime, andc the speedoflight. Sincethe horizonsize eq
T+ and T+ are the Universe and the Planck tempera-
increases with time we can now observe many regions in c P
ourpastlightconewhicharecausallydisconnected,that tures after the phase transition. If Tc+ ≈TP+ this implies
is, outside each others’ horizon (see Fig. 1). The fact light travelling more than 30 orders of magnitude faster
that these regions have the same properties (eg. Cosmic beforethephasetransition. Itistempting,forsymmetry
−
Microwavebackgroundtemperaturesequaltoafewparts reasons,simply to postulate that c = but this is not
∞
in105)ispuzzling astheyhavenotbeeninphysicalcon- strictly necessary.
tact. This is a mystery one may simply relegate to the
setting up of initial conditions in our Universe.
One may however try to explain these very peculiar IV. A PRESCRIPTION FOR MODIFYING
PHYSICAL LAWS WHILE THE SPEED OF
initial conditions. The horizon problem is solved by in-
LIGHT IS VARYING
flationaryscenariosbypostulatingaperiodofaccelerated
or superluminal expansion, that is, if a is the expansion
factor of the Universe, a period with a¨ > 0. The Fried- Hidden in the above argument is the assumption that
man equations require that the strong energy condition the geometry of the Universe is not affected by a chang-
ρ+3p/c2 0 must then be violated, where ρc2 and p ing c. We have allowed a changing c to do the job nor-
aretheene≥rgydensityandpressureofthecosmicmatter. mally done by “superluminal expansion”. To enhance
This violation is achieved by the inflaton field. If a¨ > 0 this effect we have forced the geometry to still be the
for a sufficiently long period one can show that cosmo- SBB geometry. We now elaborate on this assumption.
logicalhorizonsareapost-inflationillusion,andthatthe We will propose a prescription for how, in general, to
whole observed Universe has in fact been in causal con- modify gravitational laws while c is changing. This pre-
tact since an early time. scription is merely the one we found the most fertile. In
A more minimalistic way of solving this problem is to Appendix I we describe in detail a theory which realizes
postulatethatlighttravelledfasterintheEarlyUniverse. this prescription.
3
The basic assumption is that a variable c does not in- proportional to c2. If, however, K = 0 not even mass is
6
duce corrections to curvature in the cosmological frame, conserved.
andthatEinstein’sequations,relatingcurvaturetostress In Eqn. 5 we have included the effects of G˙ under the
energy,arestillvalid. Therationalebehindthispostulate samepostulate merelyfor completness. Insuchaformu-
is that c changes in the local Lorentzian frames associ- lation VSL does not reduce to Brans Dicke theory when
ated with cosmologicalexpansion. The effect is a special c˙ = 0, and G˙ = 0. This is because we postulate that
6
relativistic effect, not a gravitational effect. Therefore Friedmann equations remain unchanged, which implies
curvature should not feel a changing c. that the conservation equations acquire terms in c˙ and
The previous statement is not covariant. However in- G˙. In Brans Dicke theory one postulates exactly the op-
troducing a function c(t) is not even Lorentz invariant. posite: the conservationequations must still be valid, so
So it is not surprising that a favoured gauge, or coor- thattheweakequivalenceprincipleissatisfied. Whilewe
dinate choice, must be made, where the function c(t) is couldhavetakenthisstanceforcaswellwefeelthatvio-
specified, and in which the above postulate holds true. lationofenergyconservationisthe hallmarkofchanging
The cosmological frame (with the cosmological time t) c. Variable c must break Poincare invariance, for which
provides such a preferred frame. energy is the Noether current. Barrow[25]has proposed
In a cosmological setting the postulate proposed im- a formulation of VSL which has the correct Brans Dicke
plies that Friedman equations remain valid even when limit.
c˙=0:
6
a˙ 2 8πG Kc2 V. THE FLATNESS PUZZLE
= ρ (3)
a 3 − a2
(cid:18) (cid:19)
a¨ 4πG p Wenowturntotheflatnesspuzzle. Theflatnesspuzzle
= ρ+3 (4)
a − 3 c2 canbeillustratedasfollows. Letρcbethecriticaldensity
(cid:16) (cid:17) of the Universe:
where, we recall, ρc2 and p are the energy and pressure
densities, K =0, 1 and G the curvature and the gravi- 3 a˙ 2
±
tationalconstants,andthe dotdenotesa derivativewith ρc = (6)
8πG a
respectto propertime. If the Universe is radiationdom- (cid:18) (cid:19)
inated, p = ρc2/3, and we have as usual a t1/2. We that is, the mass density corresponding to K = 0 for a
∝
have assumed that a frame exists where c = c(t), and givenvalueofa˙/a. Letusdefineǫ=Ω 1withΩ=ρ/ρ .
c
identified this frame with the cosmologicalframe. Then −
TheassumptionthatEinstein’sequationsremainunaf-
fected by decelerating light carries with it an important ρ˙ ρ˙c
ǫ˙=(1+ǫ) (7)
consequence. Bianchi identities apply to curvature, as ρ − ρ
(cid:18) c(cid:19)
a geometrical identity. These then imply stress energy
conservation as an integrability condition for Einstein’s If p = wρc2 (with w˙ = 0), using Eqns.(3), (4), and (5)
equations. If c˙ =0, however, this integrability condition we have:
6
is not stress energy conservation. Source terms, propor-
ρ˙ a˙ G˙ c˙ ǫ
tional to c˙/c, come about in the conservation equations. = 3 (1+w) +2 (8)
Seen in another way, the conservation equations im- ρ − a − G c1+ǫ
ply an equation of motion for free falling point particles. ρ˙ a˙ G˙
c
= (2+(1+ǫ)(1+3w)) (9)
This is normally the geodesic equation, but now source
ρ −a − G
c
terms will appear in the geodesic equation. Clearly a vi-
olationoftheweakequivalenceprincipleisimpliedwhile and so
c is changing [7]. This, of course, does not conflict with
a˙ c˙
experiment, as we take c˙=0 only in the EarlyUniverse, ǫ˙=(1+ǫ)ǫ (1+3w)+2 ǫ (10)
6 a c
possiblyforonlyaveryshorttime (suchasaphasetran-
sition). IntheSBBǫgrowslikea2 intheradiationera,andlikea
Althoughthisisageneralremarkweshallbeconcerned
in the matter era, leading to a total growthby 32 orders
mostlywithviolationsofenergyconservationinacosmo-
of magnitude since the Planck epoch. The observational
logicalsetting. Friedmanequationscanbecombinedinto
fact that ǫ can at most be of order 1 nowadays requires
a “conservation equation” with source terms in c˙/c and
thateitherǫ=0strictly,oranamazingfine tuningmust
G˙/G: have existed in the initial conditions (ǫ < 10−32 at t =
a˙ p G˙ 3Kc2 c˙ tP). This is the flatness puzzle.
ρ˙+3 ρ+ = ρ + (5) The ǫ = 0 solution is in fact unstable for any matter
a c2 − G 4πGa2c
field satisfying the strong energy condition 1+3w > 0.
(cid:16) (cid:17)
In a flat Universe (K = 0) a changing c does not vio- Inflationsolvestheflatnessproblemwithaninflatonfield
late mass conservation. Energy, on the other hand, is which satisfies 1+3w < 0. For such a field ǫ is driven
4
towards zero instead of away from it. Thus inflation can S = dx4√ g c4(R+2Λ1) + + (11)
solve the flatness puzzle. − 16πG LM LΛ2
Z (cid:18) (cid:19)
As Eqn.10showsadecreasingspeedoflight(c˙/c<0)
where isthematterfieldsLagragian. TheterminΛ
wouldalso driveǫ to 0. If the speed oflight changesin a M 1
L
isageometricalcosmologicalconstant,asfirstintroduced
sharp phase transition, with c˙/c a˙/a, we can neglect
| |≫ by Einstein. The term in Λ represents the vacuum en-
the expansiontermsinEqn.10. Thenǫ˙/ǫ=2c˙/csothat 2
ǫ c2. A short calculation shows that the condition (2) ergy density of the quantum fields [11]. Both tend to
∝ dominate the energy density of the Universe, leading to
also ensures that ǫ 1 nowadays, if ǫ 1 before the
≪ ≈ the so-called cosmological constant problem. However
transition.
they represent two rather different problems. We shall
The instability of the K = 0 Universes while c˙/c < 0
6 attempt to solve the problem associated with the first,
can be expected simply from inspection of the non con-
not the second, term. Ususally one hopes that the sec-
servation equation Eq. (5). Indeed if ρ is above its criti-
ond term will be cancelled by an additional couter-term
cal value, then K = 1, and Eq. (5) tells us that mass is
in the Lagrangian. In the rest of this paper it is the
taken out of the Universe. If ρ < ρ , then K = 1, and
c
− geometricalcosmologicalconstantthatisunderscrutiny.
then mass is produced. Either way the mass density is
If the cosmologicalconstant Λ=0 then the argument
pushedtowardsits criticalvalueρ . Incontrastwiththe
c 6
in the previous section still applies, with ρ = ρ +ρ ,
Big Bang model, during a period with c˙/c < 0 only the m Λ
where ρ is the mass density in normal matter, and
K =0 Universe is stable. m
Note that with the set of assumptions we have used a
Λc2
changingGcannotsolvetheflatnessproblem(cf.[8–10]). ρ = (12)
Λ
8πG
We have assumed in the previous discussion that we
are close, but not fine-tuned, to flatness before the tran- is the mass density in the cosmological constant. One
sition. It is curious to note that this need not be the still predicts Ω +Ω =1, with Ω =ρ /ρ and Ω =
m Λ m m c Λ
case. Suppose instead that the Universe acquires “natu- ρ /ρ . However now we also have
Λ c
ral initial conditions” (eg. ǫ 1) well before the phase
transition occurs. If such Uni≈verses are closed they rec- a˙ pm G˙ 3Kc2 c˙
ρ˙ +3 ρ + = ρ˙ ρ + (13)
ollapse before the transition. If they are open, then they m a m c2 − Λ− G 4πGa2c
approach ǫ = 1. This is the Milne Universe, which is (cid:16) (cid:17)
− If Λ is indeed a constant then from Eq. (12)
our case (constant G) may be seen as Minkowski space-
time. SuchacurvaturedominatedUniverseisessentially
ρ˙ c˙ G˙
empty,andacoordinatetransformationcantransformit Λ =2 (14)
ρ c − G
into Minkowski space-time. Inflation cannot save these Λ
empty Universes, as can be seen from Eqn. 10. Indeed
Ifwedefineǫ =ρ /ρ wethenfind,aftersomestraight-
Λ Λ m
even if 1 + 3w < 0 the first term will be negligible if
forward algebra, that
ǫ 1. This is not true for VSL: the second term will
≈ −
still push an ǫ= 1 Universe towards ǫ=0. a˙ c˙1+ǫ
Λ
− ǫ˙ =ǫ 3 (1+w)+2 (15)
Heuristically this results from the fact that the viola- Λ Λ a c 1+ǫ
tions of energy conservation responsible for pushing the (cid:18) (cid:19)
Universe towards flatness do not depend on there being Thus, in the SBB model, ǫ increases like a4 in the ra-
Λ
any matter in the Universe. This can be seen from in- diation era, like a3 in the matter era, leading to a total
spection of Eqn. (5). growthby64ordersofmagnitudesincethePlanckepoch.
In this type of scenario it does not matter how far be- Again it is puzzling that ǫ is observationally known to
Λ
fore the transition the “initial conditions” are imposed. beatmostoforder1nowadays. Wehavetofaceanother
We end up with a chaotic scenario in which Darwinian fine tuning problem in the SBB model: the cosmological
selection gets rid of all the closed Universes. The open constant problem.
Universesbecomeemptyandcold. Inthewinterofthese If c˙ =0 the solution ǫ =0 is in fact unstable for any
Λ
Universes a phase transitionin c occurs, producing mat- w > 1. Hence violating the strong energy condition
−
ter, and leaving the Universe very fine tuned, indeed as 1+3w > 0 would not solve this problem. Even in the
an Einstein deSitter Universe (EDSU). limitingcasew = 1thesolutionǫ =0isnotanattrac-
Λ
−
tor: ǫ would merely remain constant during inflation,
Λ
then starting to grow like a4 after inflation. Therefore
VI. THE COSMOLOGICAL CONSTANT inflationcannot“explain”thesmallvalueofǫ ,asitcan
Λ
PROBLEM withǫ,unlessoneviolatesthedominantenergycondition
w 1.
≥−
Therearetwotypesofcosmologicalconstantproblems, However, as Eqn. (15) shows, a period with c˙/c 0
≪
and we wish to start our discussion by differentiating would drive ǫΛ to zero. If the speed of light changes
them. Let us write the action as: suddenly (c˙/c a˙/a) then we can neglect terms in
| | ≫
a˙/a, and so
5
ǫ˙ c˙ 1
Λ many regions which have apparently been causally dis-
=2 (16)
ǫΛ(1+ǫΛ) c1+ǫ connected. Although solving the horizon problem is a
necessaryconditionforsolvingthehomogeneityproblem,
which when combined with ǫ˙/ǫ=2c˙/c leads to
inagenericinflationarymodelsolvingthefirstcausesse-
ǫ ǫ riousproblemsinsolvingthelatter. Earlycausalcontact
Λ
(17)
betweentheentireobservedUniverseallowsequilibration
1+ǫ ∝ 1+ǫ
Λ
processestohomogenizethe wholeobservedUniverse. It
Theexactconstraintontherequiredchangeincdepends iscrucialtotheinflationpicturethatbeforeinflationthe
on the initial conditions in ǫ and ǫΛ. In any case once observable universe in well inside the Jeans length, and
both ǫ 1 and ǫΛ 1 we have ǫΛ c2. Then we thus equilibrates toward a homogeneous state. However
≈ ≈ ∝
cansolve the cosmologicalconstant problemin a sudden nosuchprocessisperfect, andsmalldensityfluctuations
phase transition in which tend to be left outside the Hubble radius, once the Uni-
verse resumes its standard Big Bang course. These fluc-
log c− 64 1log z +2log Tc+ (18) tuations then grow like a2 during the radiation era, like
10 c+ ≫ − 2 10 eq 10 T+ aduringthematterera,usuallyentailingaveryinhomo-
P
geneous Universe nowadays. This is a common flaw in
This condition is considerably more restrictive than (2),
early inflationary models [12] which requires additional
andmeans achangein c by morethan 60ordersofmag-
fine-tuning to resolve.
nitude, if T+ T+. Note that once again a period with
c ≈ P In order to approach this problem we study in Ap-
G˙/Gwouldnotsolvethe cosmologicalconstantproblem. pendix II the effects of a changing c on the theory of
Equations (10) and (15) are the equations one should scalar cosmological perturbations [13]. The basic re-
integrate to find conditions for solving the flatness and sult is that the comoving density contrast ∆ and gauge-
cosmologicalconstant problems for arbitrary initial con- invariant velocity v are subject to the equations:
ditions and with arbitrary curves c(t). They generalize
′ ′ ′
the conditions (2) and (18) which are valid only for a ′ a c a
∆ 3w + ∆= (1+w)kv 2 wΠ (19)
starting point with ǫ 1 and ǫΛ 1 and for a step − a c − − a T
≈ ≈ (cid:18) (cid:19)
function c(t). a′ c′ c2k 3 a′ a′ c′
As in the case of the flatness problem we do not need v′+ 2 v = s + ∆
a − c 1+w − 2k a a c
to impose “natural initial conditions” (ǫΛ 1) just be- (cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
fore the transition. These could have exis≈ted any time kc2w 2/3 3 a′ 2
+ Γ kc + wΠ (20)
before the transition, and the argument would still go 1+w − 1+w k2c2 a T
through, albeit with a rather different overallpicture for (cid:18) (cid:19) !
the history of the Universe.
where k is the wave vector of the fluctuations, and Γ is
If ǫ 1 well before the transition, then the Uni-
Λ the entropy production rate, Π the anisotropic stress,
≈ T
verse soon becomes dominated by the cosmological con-
andc thespeedofsound,accordingtodefinitionsspelled
s
stant. We have inflation! The curvature and matter will
out in Appendix II.
be inflated away. We end up in a de-Sitter Universe.
In the case of a sudden phase transition Eqn. (19)
When the transitionis aboutto occur it finds a flatUni-
shows us that ∆ c, regardless of the chosen equations
verse (ǫ = 0), with no matter (ρ = 0), and with a ∝
m of state for Γ and Π . Hence
T
cosmological constant. If we rewrite Eqn.(15) in terms
of ǫm = ρm/ρΛ, for ǫ = 0 and c˙/c a˙/a, we have ∆+ c+
ǫ˙m = 2(c˙/c)(1+ǫm). Integrating|lea|ds≫to1+ǫm c−2. ∆− = c− (21)
− ∝
We conclude that we do not need the presence of any
matter in the Universe for a VSL transition to converta meaning a suppression of any fluctuations before the
cosmological constant dominated Universe into a EDSU phase transition by more than a factor of 10−60 if con-
Universe full of ordinary matter. This can be seen from dition (18) is satisfied. The suppression of fluctuations
Eqns. (13)-(14). A sharp decline in c will always dis- induced by a sudden phase transition in c can be intu-
charge any vacuum energy density into ordinary matter. itively understood in the same fashion as the solution to
Westressthecuriouspointthatinthistypeofscenario theflatnessproblem. Massconservationviolationensures
the flatness problemis not solvedby VSL, but rather by that only a Universe at critical mass density is stable, if
the period of inflation preceding VSL. c˙/c 0. But this process occurs locally, so after the
≪
phase transition the Universe should be left at critical
density locally. Hence the suppression of density fluctu-
VII. THE HOMOGENEITY OF THE UNIVERSE ations.
We next need to know what are the initial conditions
for∆andv. Supposethatatsomeveryearlytimet one
Solving the horizon problem by no means guarantees i
hasc˙/c=0andthewholeobservableUniversenowadays
solving the homogeneity problem, that is, the uncanny
homogeneity of the currently observed Universe across is inside the Jeans length: η0 ciηi/√3. The latter
≪
6
condition is enforced as a byproduct of solving the hori- but since ∆ c we have
∝
zon problem. The whole observable Universe nowadays
is then initially in a thermal state. What is more each ∆+(k) 4kbT+c+ (27)
portionoftheUniversecanbedescribedbythecanonical ≈sρ+c2+ c−
ensembleandsotheUniverseishomogeneousapartfrom
thermalfluctuations[14]. Thesearecharacterizedbythe Even if T+ = T+ = 1019Gev these fluctuations would
P
mass fluctuation stillbenegligiblenowadays. ThereforealthoughtheUni-
verse ends up in a thermal state after the phase transi-
δM2 4k T
σ2 = h i = b i (22) tion,its thermalfluctuations, associatedwiththe canon-
M M 2 Mc2i ical ensemble, are strongly suppressed.
h i
For a more general c(t) function the procedure is as
Converted into a power spectrum for ∆ this is a white
follows. Integrate Eqn. (24) backwards up to a time t
noise spectrum with amplitude i
whenc˙=0,tofindT(t ). Give∆(t )athermalspectrum
i i
4k T of fluctuations, according to (23), with T(t ). With this
P (k)= ∆(k)2 b i (23) i
∆ h| |i∝ ρ c2 initial condition integrate Eqns.(19) and (20) (or even
i i
better the second orderequation(64) givenin Appendix
What happens to a thermal distribution, its tempera- II), to find ∆ nowadays.
ture, and its fluctuations, while c is changing? In ther- It is conceivable that a careful design of c(t) would
mal equilibrium the distribution function of particle en- leave fluctuations, once c˙ = 0 again, with the right am-
ergies is the Planck distribution P(E)=1/(eE/kbT 1), plitude and spectrum to explain structure formation. In
−
where T is the temperature. When one integrates over particular c(t) may be designed so as to convert a white
the whole phase space, one obtains the bulk energy den- noisespectrumintoa scale-invariantspectrum. However
sity ρc2 (kbT)4/(h¯c)3. Let us now consider the time we feel that until a mechanism for inducing c(t) is found
∝
when the Universe has already flattened out sufficiently such efforts are bound to look ludicrously contrived.
for mass to be approximately conserved. To define the We feel that the power of VSL scenarios is precisely
situation more completely, we make two additional mi- in leaving the Universe very homogenous, after c has
crophysical assumptions. Firstly, let mass be conserved stopped changing. This would then set the stage for
also for individual quantum particles, so that their en- causalmechanismsofstructureformationtodotheirjob
ergies scale like E c2. Secondly, we assume particles’ [15,16].
∝
wavelengthsdonotchangewithc. Ifhomogeneityispre-
served, indeed the wavelength is an adiabatic invariant,
fixed by a set of quantum numbers, eg: λ = L/n for a VIII. THE ISOTROPY OF THE UNIVERSE
particle in a box of size L.
UnderthefirstoftheseassumptionsaPlanckiandistri- There is a sense in which there is an isotropy problem
bution with temperature T remains Planckian, but T in the SBB model, similar to the homogeneity problem.
c2. Under the second assumption, we have λ=2π¯hc/E∝, We follow closely the remark made in [13], pp.26.
and so h¯/c should remain constant. Therefore the phase In Appendix III we write down the vector Einstein’s
space structure is changed so that, without particle pro- equations in the vector gauge, and from them we derive
duction, one still has ρc2 (kbT)4/(h¯c)3, with T c2. the vorticity “conservation”equation when c˙/c=0. If v
A black body therefore r∝emains a black body, w∝ith a is the vorticity (defined in Appendix) and ΠT th6 e vector
temperature T c2. If we combine this effect with ex- stress, we have:
∝
pansion, with the aid of Eqn. (5) we have
′ ′
a c kc w
a˙ c˙ v′+(1 3w) v 2 v = ΠT (28)
T˙ +T 2 =0 (24) − a − c − 2 1+w
a − c
(cid:18) (cid:19) In the absence of driving stress, v remains constant dur-
We can then integrate this equation through the epoch ing the radiation dominated epoch, and decays like 1/a
when c is changing to find the temperature T of the in the matter epoch. In [13]it is further arguedthat the
i
initial state. This fully fixes the initial conditions for relevant dimensionless quantity is
scalar fluctuations, by means of (23).
Inthe caseofasuddenphasetransitionwehaveT+ = (k/a)v 1
ω = (29)
T−c2+/c2−, and so (a′/ca) ∝ a(1−9w)/2
4k T− 4k T+ Hence for w > 1/9 vorticity grows, leading to a further
σM2− = Mbc2− = Mbc2+ (25) fine tuning problem.
ThisismostnotablyaproblemifweacceptthePlanck
or
equipartition proposal, introduced in [17]. At Planck
4k T+ epoch there would then be a significant vorticity. De-
∆−(k)2 b (26)
≈ ρ+c2+ pendingonhowonelooksatit,this vorticitywouldthen
7
get frozen in or grow, leading to a very anisotropic Uni- horizon grows faster than t if p < ρc2/3: it grows ex-
verse nowadays. ponentially if p = ρc2, and like−tn (with n > 1) for
−
Whether or not this is a problem is clearly debatable. ρc2 < p < ρc2/3. This provides the inflationary so-
− −
In any case either inflation or VSL models could solve lution to the horizon problem. However in the latter
this prospective problem. For w < 1/3 we have that v case Eqn. (31) implies that σ decreases exponentially,
h
decays faster than 1/a2. Whatever−dimensionless quan- leading to σ (t ) 1. The way inflation bypasses this
h 0
≪
tity one chooses to look at, vorticity is therefore safely problemisbydroppingtheadiabaticassumption. Indeed
inflated away. If c˙/c = 0 we have that v c2. Again during inflation the Universe supercools,and a period of
6 ∝
anyprimordialvorticityissafelysuppressedafteraphase reheating follows the end of inflation2.
transitionincsatisfyinganyoftheconditions(2)or(18). InaVSLscenariosthedetailedsolutiontotheentropy
problemdependsonwhenandwhattypeof“naturalcon-
ditions”aregiventothepretransitionUniverse. Wefirst
IX. THE ENTROPY PROBLEM AND SETTING derive equations for the entropy under varying c. From
THE INITIAL CONDITIONS s = (4/3)ρc2/T, ρ T4/(h¯c)3, and from Eqns 5 and 13
∝
we obtain that the entropy of radiation satisfies
LetusfirstconsidertheSBBmodel. LetS betheen-
h
s˙ 3ρ˙ a˙ 3c˙ǫ(1+ǫ ) 3c˙
tropy inside the horizon, and σh = Sh/kB be its dimen- = = 3 + Λ ǫΛ (32)
sionless counterpart. σ is of order1096 nowadays. If we s 4ρ − a 2c 1+ǫ − 2c
h
assumethattheonlyscalesinthecosmologicalmodelare
If the Universe is EDSU, there are no violations of mass
the onesprovidedby the fundamentalconstants,then at
conservation, and entropy is conserved. However if the
t the temperature is T . At Planck time, σ (being
P P h Universeis open orhas a positive cosmologicalconstant,
dimensionless)is naturallyof order1. In the SBB model
thenwehaveseenthatthereiscreationofmass. Accord-
the horizon distance is d = 2t in the radiation domi-
h ingly there must be creation of particles, and entropy is
nated epoch, and ignoring mass thresholds t 1/T2. If
∝ produced. If the Universe is closed, particles are taken
evolution is adiabatic one then has (in a flat Universe)
away,and the entropy decreases.
3 The most suspicious case is therefore if the Universe
T
σ (t) σ (t ) P . (30) was Einstein de-Sitter before the phase transition. Let
h h P
≈ T −
(cid:18) (cid:19) usassumethereforethatatt=t (thePlancktime with
P
the constants before the transition) the entropy inside
Since σh(tP)∼1, one has σh(t0)∼1096. Thus the large the horizon(which has proper size c−t−) was of order 1.
entropyinsidethe horizonnowadaysis areflectionofthe P
Then the entropy inside the Hubble volume at t = t+,
lack of scales beyond the ones provided by the funda- P
before and after the transition, is
mental constants, the fact that the horizon size is much
largernowadaysthan at Plancktime, and the flatness of
c+t+ 3 a(t−) 3
theUniverse. Onemayrephrasethehorizonandflatness σ (t+)=σ (t−) P P 1 (33)
problemsintermsofentropy[2]. Howeverifoneiswilling h P h P c−t− a(t+) ≈
(cid:18) P(cid:19) (cid:18) P (cid:19)
to accept the horizon structure and flatness of the Uni-
verse simply as features of the initial conditions (rather wherewehaveusedt+/t− =(c−/c+)2. Onetakesafrac-
P P
than problems), there is no additional entropy problem. tion(c+/c−)3 ofthehorizonvolumebeforethetransition
There is a problem that arises if one tries to solve the tomaketheHubblevolumeafterthetransition. However
horizon problem, keeping the adiabatic assumption, by the entropy inside the horizon has increased since t+ by
P
means of superluminal expansion. This blows what at the same factor. Therefore entropy conservation in this
Planck time is a region much smaller than the Planck case does not conflict with σh(t+P) ≈ 1 after the transi-
size into a comovingregioncontaining the whole observ- tion. One way of understanding this is that by imposing
able Universe nowadays. This solves the horizon prob- flatness from the outset (before the transition) one has
lem. Howeverifevolutionisadiabaticsuchaprocessim- already “solved” the entropy problem. Notice that the
plies that σh(t0) 1. Stated in another way, since the above argument works for any value of tc/t+P.
number of particl≪es inside the horizon n is of the same Now consider the case where “natural” initial condi-
h −
order as σ , this implies an empty Universe nowadays. tions were also imposed at t , with Λ = 0. One should
h P
−
More mathematically, if d is the horizon proper dis- have ǫ(t ) of order 1. We have already discussed how
h P
tance, one has the flatness problem is solved in this case, when large
empty curvature dominated universes are filled with a
σ˙ 3
h
= (31)
σ d
h h
t ′
where we have used dh =a dt/a. With any standard 2 This issue has been carefully analyzed in the context of
matter (p > ρc2/3) the horizon grows like t. Accord-
inflationary models and models with time varying G in [10]
− R
ingly σ grows like a power of t. On the other hand the
h
8
(nearlyperfectly)criticalenergydensityduringthetran- tionwouldnotconflictwiththedynamicsofflatnessand
sition. Open Universes become veryempty, but they are Λ, as shown before, but now t− =t+.
P P
still pushed to EDSU at the transition. One may in-
tegrate (32) to find that s+/s− = (1 + ǫ−)−3/4. One
may also use Eqn. 10 to find that ǫ has evolved since X. CONCLUSIONS
t− to ǫ−(t+)+ 1 (a(t−)/a(t+))2 (t−/t+)2, where
P P ≈ P P ≈ P P
we have used a t for the Milne Universe. Hence we Wehaveshownhowatimevaryingspeedoflightcould
∝
have that during the transition entropy is produced like providea resolutionto the wellknowncosmologicalpuz-
s+/s− = (t+/t−)3/2 = (c−/c+)3. Given that a t for zles. These “VSL” models could provide an alternative
P P ∝
such Universes, the entropy before the transition in the tothestandardInflationarypicture,andfurthermorere-
proper volume of size c+t+ is solve the classical cosmological constant puzzle. At a
P
technical level, the proposed VSL picture is not nearly
S−(c+t+)= c+t+P 3 a(t−P) 3 c+ 3 (34) as well developed as the inflationary one, and one pur-
P c−t− a(t+) ≈ c− pose of this article is to stimulate further work on the
(cid:18) P(cid:19) (cid:18) P (cid:19) (cid:18) (cid:19)
unresolved technical issues. We are not trying to take
that is there is practically no entropy in relevantvolume an “anti-inflation” stand, but we do strongly feel that
before the transition. However we have that after the broadeningtherangeofpossiblemodelsoftheveryearly
transition Universewouldbeveryhealthyforthefieldofcosmology,
and would ultimately allow us to state in more concrete
− 3
σ (t+)=S+(c+t+)=S−(c+t+) c 1 (35) terms the extent to which one model is preferred.
h P P P c+ ≈ On a more fundamental level we hope to expand the
(cid:18) (cid:19)
phenomenologicalapproachpresentedinthispaperintoa
In such scenarios the Universe is rather cold and empty theorywherethe conceptof(Poincare)symmetrybreak-
before the transition. However the transition itself re- ingprovidesthephysicalbasisforVSL.Symmetrybreak-
heats the Universe. Notice that, like in the first case ing is also the central ingredient in causal theories of
discussed, the above argument works for any value of structure formation. We therefore hope to arrive at a
tc/t+P. scenario where symmetry breaking provides a complete
−
Ifatt=tP onealsohasǫΛ ≈1thenwehaveascenario and consistentcomplement to the SBB model which can
inwhichthe cosmologicalconstantdominates,solvesthe resolvethe standardpuzzles as wellas explainthe origin
flatness problem, and is discharged into normal matter. of cosmic structure.
− −
However if ρ ρ at t t , then whatever the tran- Note added in proof:
Λ ≈ P ≈ P
sition time, after the transition the Universe will have a After this paper, as well as its sequel [26], were com-
−
density in normal matter equal to ρm = ρP. Hence the pleted J. Moffat brought to our attention two papers in
−
Hubble timeafterthe transitionwillbet ,whateverthe which he proposes a similar idea [27]. While Moffat’s
P
actualageoftheUniverse. Onemayintegrate(32)tofind work does not go as far as ours in addressing the flat-
that in this case (setting ǫ = 0) the entropy production ness, cosmological constant, and entropy problems, he
during the transition is s+/s− =(1+ǫ−)3/4. In the pe- does go considerably further than we have in terms of
Λ
riod between t=t− and the transition, ǫ increases like specific model building. Moffat’s model does not satisfy
P Λ
a4, and the entropy density is diluted like 1/a3. Hence our prescription for solving the cosmological problems,
after the transition the entropy density is what it was at but it may do so in a modified form. We are currently
t = t−, that is s+ 1/L−3. If we now follow the Uni- investigating this possibility. We regret that because we
P ≈ P
verse until its Hubble time is t+ (when its density is ρ+) were unaware of this work we did not cite it in the first
P P
wemustwaituntilthe expansionfactorhasincreasedby publicly distributed version of this paper.
a factor of (ρ+/ρ−)1/4. Given that s 1/a3 the entropy
density is diluPtedPby a factor of (ρ+/∝ρ−)3/4. Therefore
P P ACKNOWLEDGEMENTS
the entropy density when the Hubble time is t = t+ is
P
s 1/L+3. Again the dimensionless entropy inside the
Hu≈bble vPolume, when this has size L+, is of order 1. We would like to thank John Barrow, Ruth Dur-
P rer, Robert Brandenberger, Gary Gibbons, and Erick
Finally it is worth noting that treating the pre-
Weinberg for discussion. We acknowledge support from
transition universe simply as a Roberston-Walker model
PPARC (A.A.) and the Royal Society (J.M.).
is no doubt overly simplistic, and we use it simply as
a device to introducing our ideas. We expect that fur-
therdevelopmentoftheseideascouldresultinaradically
different view of the pre-transition phase (much as has
happened with the inflationary scenario). One interest-
ing observation is that one could avoid having multiple
Planck times by considering that G c4. Such assump-
∝
9
APPENDIX I: A SPECIFIC REALIZATION OF δ(3)(x x(τ))euµ
jµ = − (42)
VSL γ c
This current3 is the current which couples to the gauge
In this Appendix we set up a specific VSL theory. We
field, and in the rest frame it equals e. Therefore it can-
firstdiscussthesimplecaseoftheelectrodynamicsofthe
not be conserved, and indeed we have that
point particle in Minkowski space time. We start from
Bekenstein’s theory of variable α, and show how a VSL
jµ
alternative could be set up. We highlight the subtleties ∂ jµ = ∂ e (43)
µ e2 µ
encountered in the VSL formulation. We then perform
the same exercise with the Einstein-Hilbert action. We Let us now postulate instead that a changing α is to
brieflyconsiderthe dynamics ofthe fieldψ =c4. Finally be interpreted as c ¯h α−1/2, and that e and m are
we castthe key elements of ourconstructioninto a body to be seen as consta∝nts.∝Minimal coupling, in the above
of axioms. sense, wouldthen prompt us to consider the action(36),
but with c = c(xµ) everywhere, and e and m constants.
This action leads to equations:
A. Electrodynamics in flat space time
1 e
mx¨ = (mc2) + uνF (44)
µ ,µ µν
A changing α theory was proposed by Bekenstein [23] 2 c
based on the postulate of Lorentz invariance. The elec-
with the electromagnetic tensor defined as
trodynamics of a point particle was first analyzed. If
Lorentz invariance is to be preserved then the particle
F =c(∂ (A /c) ∂ (A /c)) (45)
µν µ ν ν µ
mass m and its charge e must be variable. In order to −
preserve“minimalcoupling”(reductiontostandardelec- However the above construction is not complete. In
tromagnetismwhenα=const)onechoosestheworldline spite of the appearance of Eqns. (36), (44), and (45),
action Lorentz invariance is broken. This boils down to the
L= mc uµu + euµA (36) fact that, say ∂µ is no long a 4-vector. Even if c were
µ µ to be regarded as a scalar, ∂ would contain c in its
− − c µ
zero component, but not in its spatial components. The
p
with uµ = x˙µ, gµν = ηµν, e = e(xµ), and m = m(xµ). usual contractions leading to S could still be taken but
Minimal coupling means simply to take the standardac- S would no longer be a scalar. This manifests itself in
tion and replace e and m by variables without breaking the equations (44) in the fact that in x¨µ there are terms
Lorentz invariance. e and m must then be scalar func- in ∂c which break Lorentz invariance.
tions. This action leads to equation: Since the action is not Lorentz invariant, a minimal
coupling prescription cannot possibly be true in every
e
(mx˙ )˙ = m c2+ uνF (37) coordinate system. Minimal coupling is now the state-
µ ,µ µν
− c
ment that there is a preferred reference frame in which
with the electromagnetic field tensor defined as the action is to be obtained from the standard action
simply by replacing c with a field. Let us call this frame
1 the “lightframe”. Inregionsin whichc changesverylit-
F = (∂ (eA ) ∂ (eA )) (38)
µν e µ ν − ν µ tle changes in the action upon Lorentz transformations
arenegligible. Henceallboostsperformeduponthelight
The electromagnetic action can therefore be defined as:
frame become nearly equivalent and Lorentz invariance
is recovered.
1
SEM = − d4xFµνFµν (39) The Maxwell equations in a VSL theory become
16π
Z
1
Also, the particle action (36) may be written as a La- ∂ (cFµν)=4πjµ (46)
µ
c
grangian density:
in the light frame. Given that Lorentz invariance is bro-
S = d4xδ(3)(x−x(τ))( mc2+(e/c)uµA ) (40) ken,onecannolongerexpectthegeneralexpressionfora
M γ − µ conservedcurrenttotakethe form∂ jµ =0. Indeedone
Z µ
inwhichγ istheLorentzfactor. Maxwell’sequationsare
then:
e∂ (Fµν/e)=4πjµ (41) 3 Thereisanalternativeviewinwhichratherthanachang-
µ
ing e one considers that the vacuum is a dielectric medium
with the current with variable ǫ. One may then identify a conserved charge,
but thisis not thecharge which couples tothe gauge field.
10