Could thermal fluctuations seed cosmic structure?

9 Pages·0.17 MB·English

by  Joao Magueijo

Could thermal fluctuations seed cosmic structure?
Jo˜ao Magueijo and Levon Pogosian
Theoretical Physics, The Blackett Laboratory, Imperial College,
Prince Consort Road, London SW7 2BZ, United Kingdom
We examine the possibility that thermal, rather than quantum, fluctuations are responsible for
seeding the structure of our universe. We find that while the thermalization condition leads to
nearly Gaussian statistics, a Harrisson-Zeldovich spectrum for the primordial fluctuations can only
be achieved in very special circumstances. These depend on whether the universe gets hotter or
colder in time, while themodes are leaving thehorizon. In the latter case we find a no-go theorem
which can only be avoided if the fundamental degrees of freedom are not particle-like, such as in
stringgases neartheHagedorn phasetransition. Theformer caseisless forbidding,andwesuggest
two potentially successful “warming universe” scenarios. One makes use of the Phoenix universe,
3 the otherof “phantom” matter.
0
0
2
I. INTRODUCTION Big Bang cosmology all observed fluctuations spanned
n
causally disconnected regions when the universe was at
a
J A major success of the Big Bang theory is its abil- the required high temperatures to induce the appropri-
6 ity to account for the detailed structure of the Universe, ate level of fluctuations. In addition, simply postulating
1 such as galaxy clustering, or the temperature fluctua- exact thermalization over all scales, say at Planck time,
tions in the cosmic microwave background. Paramount leads to grossly inappropriate results (thermal fluctua-

2 to this picture is the phenomenon of gravitational in- tionsarewhite-noiseratherthanscaleinvariant). Hence,
v stability, by means of which primordialsmall departures we shouldassume that exactthermalizationonly applies
7 fromhomogeneitycangrowintotheobservedstructures. to sub-horizonmodes, andthatthere is a mechanismfor
3
WithintheBigBangtheorytherequiredprimordialfluc- pushing sub-horizon thermal modes outside the horizon,
3
1 tuationsaretreatedmerelyasinitialconditions. Vacuum where they freeze andbecome non-thermal. Since differ-
1 quantum fluctuations in inflationary scenarios have been entmodesfreezeatdifferenttemperatures,itmaybethat
2 shown to lead to the required initial conditions, fitting the final super-horizon spectrum is indeed of HZ type.
0 current large scale structure data. It is, however,impor- We shall considerthree types of mechanisms for push-
h/ tanttoknowjusthowunavoidableisthe conclusionthat ing modes outside the horizon: accelerated expansion,
p cosmic structures have a quantum origin. Could these varying speed of light (VSL) and a contracting universe.
- primordial fluctuations have a thermal origin instead? In the first case we consider inflation models driven by
o The possibility that we are descendent from thermal a dominant component of thermal radiation, as in [3],
r
t fluctuations was advanced by Peebles, in his book Prin- where deformed dispersion relations affect common ra-
s
a ciples of Physical Cosmology, pp. 371-373 [1]. Peebles diation at high temperatures. This is not to be con-
: pointed out that if the Universe was in thermal equilib- fused with inflationary models in which there is a finite
v
rium on the comoving scale of 10 Mpc when its temper- radiation component during inflation [4,5]. There the
i
X ature was T =1011 Gev, then the observed value of σ dominant component is always the inflaton field (even
10
r could be explained. This fascinating remark leaves sev- though the thermal bath drives inflaton fluctuations).
a eral questions unanswered. Such a scenario may explain The second possibility is a varying speed of light (VSL),
the observed value of σ , but what about fluctuations either in the form of a space-time field (c(xµ)) [6–9],
10
on other scales? Furthermore, thermal fluctuations are or as an energy-dependent effect (c(E)) [11]. There
not strictly Gaussian - does this scenario conflict with are VSL models [8] in which quantum vacuum fluctu-
observations? ations can produce the HZ spectrum [10]. However,in a
Motivated by these unsolved issues, in this paper we largeclassofVSLscenariostheuniverseisnevervacuum
go further and examine under which conditions primor- dominated. Hence subhorizonscalethermalizationisthe
dial thermal fluctuations lead to a Harrison-Zeldovich reason for the apparent homogeneity of the Universe -
(HZ) [2] spectrum of approximately Gaussian fluctua- and likewise thermal fluctuations are responsible for the
tions. Such a scenario would fit all existing data in the primordial fluctuations. In the third case, we consider
same way that the usual inflationary quantum fluctua- the possibility of thermal fluctuations seeding structures
tion scenario does. Indeed, the only potentially distin- within Lemaitre’s Phoenix universe [12].
guishing feature would be a different signature (or ab- The paper is organizedasfollows. In SectionII we ex-
sence thereof) of gravitational waves (tensor modes) in amine the statistics of thermal fluctuations. In Section
the thermal scenario. III we derive the necessary conditions for thermal fluc-
A matter not addressed by Peebles is how to es- tuations to have a scale-invariant spectrum. In Section
tablish thermalization on the relevant scales. In pure IV we show that these conditions cannot be fulfilled in
1
universes dominated by radiation comprised of conven- where prime denotes differentiation with respect to tem-
tional particles and in which the temperature decreases perature. We see that whereas the amplitude of these
with time. At the end of Section IV and in Section V fluctuations ismodeldependent, theirspectrum(i.e. the
we propose a set of models that may bypass this no-go fact that σ2 1/V) is not. The white noise nature of
U ∝
theorem. We summarize our results in Section VI. thermal fluctuations follows from the fact that energy is
an extensive quantity (i.e. proportional to the volume).
Thefactthatσ2 1/V leadstoaninterestingheuris-
U ∝
II. THE STATISTICS OF THERMAL ticinterpretationofthermalfluctuations. Itseemstoim-
FLUCTUATIONS ply that thermal fluctuations may be seen as a Poisson
processinvolvingasetofregionswithcoherencelengthλ
Most of the currently available data is consistent with dependent only on the temperature (and not the sample
the hypothesis thatprimordialfluctuationsareGaussian volume). InavolumeV therearen=V/λ3 suchregions,
distributed [13]. A possible exception is the report of so that a Poissonprocess results in variance
non-vanishing “inter-scale”components of the CMB bis-
σ2(n) 1 λ3
pectrumcalculatedfromthefouryearCOBE-DMRdata σ2 = = = (6)
[14]. NewCMBmeasurements,inparticularbytheMAP U n2 n V
satellite, are expected to provide tighter constraints on
implying that any white noise spectrum of fluctuations
the amount of cosmologicalnon-Gaussianity.
may be seen as a Poissonprocess. The dependence λ(T)
Based on current observational evidence, one has to
canbeinferredfromeq.(5)andingeneraltakestheform
requirethatatleastonverylargescales,primordialfluc-
tuationsmustbesufficientlywelldescribedbyaGaussian T2ρ′
distribution. Here we encounter the first obstacle, since λ3 = (7)
ρ2
strictly speaking thermal fluctuations are not Gaussian.
In whatfollowswe shallshow that these fluctuations are
translating, for (3), into
Gaussian to a very good approximation under the same
set of conditions which assure thermalization. γ 1
λ3 (8)
Fluctuations inathermal(canonical)ensemble canbe ≈ T3(l T)γ−4
T
determined from the partition function
We see that λ T−1 only for γ = 4. For γ = 1 (re-
Z = e−βEr , (1) alized in non-co∝mmutative geometry [11]) λ is temper-
Xr ature independent and equals the length scale of non-
where β =T−1. This is true even if deformed dispersion commutativity. In general the thermal coherence length
decreaseswithincreasingtemperature,exceptifγ <1. In
relations are introduced [11]. Hence the total energy U
theexceptionalcaseγ <1thecoherencelengthincreases
inside a volume V is given by:
withthetemperatureandtherelativeenergyfluctuations
U =hEi= PrEer−e−βEβrEr =−dldoβgZ (2) awneosmhaallolurusllyeionuctretahsieswexitchepthtieotneaml pcaesrea.ture;inSectionIV
r The non-Gaussianity of thermal fluctuations may now
P
In general this integral needs not be proportional to T4, be studied in terms of the cumulants of the distribution.
and indeed under deformeddispersion relationsU Tγ, The third order centered cumulant is given by [15]
∝
with 1 < γ < 4. However, because energy is an exten-
d3logZ d2U
sive quantity, we always have that U = ρV, that is U is κ = E3 3 E2 E +2 E 3 = = (9)
proportional to the volume. If γ =4 there is a preferred 3 h i− h ih i h i − dβ3 dβ2
6
length scale l , and we can choose units so that
T so that the relative skewness is
ρ≈T4(lTT)γ−4. (3) κ3 γ(γ+1)(lTT)2−γ2
s = (10)
(specificallyweuseunitssuchthatG=h¯ =c =k =1, 3 σ3 ≈ γ3/2 (VT3)1/2
0 B
wherec iscurrentvalueofc,andneglectfactorsoforder
0
Hence for large volumes s 1. Likewise for higher
1). 3
≪
cumulants
The energy variance is given by
σ2(E)= E2 E 2 = d2logZ = dU (4) κn =(−1)ndndlβongZ =(−1)n−1ddnβ−n1−U1 (11)
h i−h i dβ2 −dβ
and so the relative variance is and so
σU2 = σ2U(E2 ) = Tρ22ρ′V1 (5) sn = σκnn ≈cn(γ)(lT(TV)T(n23−)1n2)−(41−γ) (12)
2
withtheproportionalityconstantc (γ)=γ(γ+1)...(γ+ outside the horizon and are allowed to re-enter it at a
n
n 2)/γn/2. Setting V = L3 we find the unsurprising later time1. Then, at first horizon crossing we have
−
result that this can be written as
a˙
k | | (17)
3(n−1) h ∼ c
cn(γ) λ 2
s (13)
n ≈ γn2−1(cid:18)L(cid:19) where k arecomovingwavenumbers,a is the scale factor
andcisthespeedoflight2. Themodulussignineq. (17)
and so if the volume under study is much larger than accounts for the possibility of such a crossing happening
the thermal coherence length as defined above we have during a contracting phase in a bouncing universe - a
indeed that s 1. model discussedin subsectionVB. We parameterizethe
n
≪
WeconcludethatthermalfluctuationsareGaussianto temperaturedependenceofthefirsthorizoncrossingwith
a very good approximation if, when they leave the hori- k =Tµλµ−1 , (18)
zon, the thermal coherence length is much smaller than h 1
thehorizonsize. ThedeparturesfromGaussianityalways where λ1 is some characteristic length scale. We will
leadtopositivecumulantsandthesedecaywiththeorder consider two types of scenarios: those in which the tem-
nasineq.(13). Sincetheconditionforthermalizationis peratureoftheradiationisdecreasingwiththeevolution,
thatthecoherencelengthismuchsmallerthanthescales and those in which the temperature is increasing.
under study, we may conclude that Gaussianity is part Iftheuniversecoolsasitexpandsthensolvingthehori-
andparcelofthe self-consistencyconditionsfor studying zon problem and the existence of a first crossing require
thermal fluctuations in thermal equilibrium. that µ < 0. More generally, one needs dkh/dT < 0 or,
from eq. (18),
a¨ c˙
III. THE POWER SPECTRUM >0 (19)
a˙ − c
Hence, while modes are being forced outside the horizon
Gaussianfluctuations arefully describedby their two-
one musthaveeither acceleratedexpansionora decreas-
point function. This can be encoded in σ2, but is more
U ing speed of light, or a combination of both.
often expressed in terms of the power spectrum P(k) =
Iftheuniverseheatsupasitevolvesintime,thenµ>0
δ 2 , where δ are the Fourier modes of the density
hc|onkt|raist δρ/ρ. Tkhe two can be related via the integral (or, dkh/dT > 0). We consider this case in some detail
in Section V.
[16]:
Inbothtypesofscenarioswewillbeinterestediniden-
∞ tifying conditions for a HZ spectrum of density fluctua-
1
σ2 = P(k)W2(kL)k2dk , (14) tions to be left outside the horizon. If there is no sig-
U 2π2 Z F
0 nificantevolutionofthegravitationalpotentialφoutside
thehorizonwhilethemodesarebeingpushedout(thisis
where W (kL) is a filter function and L V1/3 is the
F ∼ usually enforced by requiring that the equation of state
smoothing scale (so that U = ρV). Assuming a power
remainsmoreorlessconstant),thendensityfluctuations
law dependence for the power spectrum, P(k) = A2kn,
haveaHZspectrumwhentheequal-timepowerspectrum
and, for instance, a Gaussian filter function, W (kL) =
F of φ has a form
e−k2L2/2, integration of eq. (14) gives
k3P =B2 , (20)
φ
σ2 = A˜2 = A˜2 , (15) with B 10−5. If in addition at horizon crossing we
U L3+n V1+n3 have φ ≈δ , then we need3:
h h
≈
where A˜2 = A2Γ n+3 /(4π2), that is, A˜ A. By com-
2 ≈
paring eqns. (5) a(cid:0)nd (1(cid:1)5) one can see that thermal den-
sity fluctuations have a white noise spectrum (n = 0)
1This condition is more restrictive than requiring that the
with amplitude:
horizon problem be solved – for instance the Milne universe
T2ρ′ (with a ∝ t) does not have horizons, and yet sub-horizon
P(k)= δ 2 k0 (16) modesarenotpushedoutsidethehorizon. Thereaderisalso
h| k| i≈ ρ2
referred to Ref. [17] for a simple condition for solving the
horizon and flatness problems.
where we have ignored factors of order 1. This result
2c is the speed of light at the time of the horizon crossing,
only applies to modes which are in causal contact, i.e.
which could be different from c0
sub-horizon modes. More precisely it only applies when 3This may be seen as an independent assumption, or justi-
self-gravity is negligible, and so to modes smaller than
fied using Einstein’s equations. The perturbed Friedmann
the Jeans length.
equations in the comoving longitudinal gauge imply that
Suppose we have a model in which a certain range of k2φ ≈ a2ρδ and, using the Friedmann equation, a˙2 ≈ ρa2,
Fourier modes of thermal density fluctuations are forced and eq. (17), we obtain φ≈(kh/k)2δ. Henceφh≈δh.
3
k3δ2 B2 10−10 , (21) (wherewistheequationofstate),andifweparameterize
h h ≈ ≈
the dependence of the speed of light on the temperature
whichissometimesusedasthedefinitionoftheHZspec- as:
trum. Sinceφ δ ,thisexpressionistrueinanygauge
h h
because the δ d≈efined in the various gauges are all pro- c Tα . (29)
h
∝
portional to each other.
If we identify these fluctuations with thermal fluctu- Then from eqns. (17), (18), (28) and (29) the condition
ations about to leave the horizon, then, using eq. (16), for scale invariance reduces to:
this is equivalent to
γ 1+w
=2 (30)
T2 dρ 1 3α 1 w
k3(T) B2 . (22) − −
ρ2 dT h ≈ or, alternatively, to:
(whereagainwehaveneglectedfactorsoforder1). eq.(3) (2 γ)+(2+γ)w
then leads to α= − . (31)
6(1+w)
γ
k3T1−γ B2 . (23)
lγ−4 h ≈
T
IV. A NO-GO THEOREM FOR COOLING
Taking the time derivative of the above expression leads
UNIVERSES
to
k˙h T˙ If at all times the universe has been cooling (T˙ < 0),
3 +(1 γ) =0 . (24)
k − T thenwearriveataforbiddingsetofconditionsforascale-
h
invariant spectrum. In this case k must decrease with
h
Since k˙ >0 is the conditionformodes to be leavingthe the temperature (µ<0). As already noted, from (25) it
h
horizon, it follows from eq. (24) that then follows that this requires
T˙ γ <1 (32)
(1 γ) <0 . (25)
− T
Thisconditionisastrictinequality: γ =1leadstoeither
Hence scale invariance requires that γ < 1 or γ > 1 n = 4 (if the modes are indeed being pushed out of the
depending on whether the Universe is getting colder or horizon), or to n=0 (if the Hubble length stagnates).
hotter. We recall that γ < 1 is equivalent to saying If γ < 1 the fractional amplitude of thermal fluctu-
that the thermal coherence length λ increases with the ations anomalously increases with the temperature. As
temperature (see eq. (8)), or that the fractional energy equation(8)shows,thisalsoimpliesthatthethermalcor-
fluctuations increasewith the temperature. Thisis quite relationlengthincreaseswiththetemperature. Belowwe
anomalous and we shall rule it out explicitly in the next present a no-go theorem which shows that it is unlikely
section. Hence we are left with warming universes as a that this condition is satisfied assuming that the weak
possibility for HZ fluctuations of thermal origin. energy condition is satisfied and that the fundamental
Theseresultsmaybeexpressedmorequantitativelyby degrees of freedom are particle-like.
noting that Eqs. (18) and (23) lead to the condition for Consider radiation in thermal equilibrium with a cer-
scale invariance: tain dispersion relation p(E) which becomes the usual
p2 =E2 at sufficiently low energies. The energy density
γ =1+3µ , (26) is proportional to the integral:
or more generally the expression for the tilt E p2(E) dp
ρ(T) I(T)= dE , (33)
∝ Z eE/T 1 (cid:12)dE(cid:12)
1 γ − (cid:12) (cid:12)
n=4+ − . (27) (cid:12) (cid:12)
µ wherethe integrationisoverallallowed(cid:12)valu(cid:12)esofenergy.
For convenience, let us define F(E) Ep2(E) dp/dE
In our argument so far we have abstained from using ≡ | |
and re-write I(T) as:
Einstein’s equations. Instead, we have treated eq (21)
asanindependentassumptionandalsoassumedthatthe
dE F(E)
gravitational potential would freeze outside the horizon. I(T)=T Tf(T) . (34)
Z T eE/T 1 ≡
HoweverwecangofurtherifwearepreparedtouseFried- −
mann equations: In order to have γ <1, f(T) must be a decreasing func-
tion of temperature at sufficiently high values of T, i. e.
a˙ 1
′ ′
√ρ and ρ , (28) f (T)<0 . Let us evaluate f (T) :
a ∝ ∝ a3(1+w)
4
′ dE F(E) E eE/T String-driven inflationary models, making use of the
f (T)= 1 . (35)
Z T2 eE/T 1(cid:20)T eE/T 1 − (cid:21) existenceofalimitingtemperature,havebeenconsidered
− −
in the late 1980’s [24,25]. More recently, in Ref. [26], it
Ifweonlyconsidernon-negativeenergies,thenthefactor was proposed that winding modes of open strings on D-
undertheintegralappearinginfrontofthesquarebrack- branes above the Hagedorn phase transition can provide
etsisnon-negative,whiletheexpressioninsidethebrack- thenegativepressurenecessarytodriveinflation. Inpar-
ets is a non-negative, monotonically increasing function ticular, it was suggested that one could achieve a period
′
ofE/T. Hence,f (T)>0forallT. Thus,the inequality of exponential inflation (with w=-1) if not all transverse
(32) cannot be satisfied. dimensions supported winding modes [26].
The above proof is quite general and is valid for all It would be very interesting to investigate if thermal
models that would aim to achieve γ < 1 by modify- fluctuations could indeed be viable candidates for struc-
ing the dispersion relations without altering the statis- ture formation in these models and the degree of fine-
tical properties of the gas. In particular, this proof im- tuning it would involve.
plies that the modified dispersionrelationsconsideredin The other possibility, that of a VSL theory with α =
Refs. [18,11,3]) could not result in a HZ spectrum. 1/3, is currently lacking a specific model realization.
The no-go argument assumed that the thermalized
radiation was made of particles obeying Bose-Einstein
statistics. A radically different statistics is likely to be V. WARMING UNIVERSES
needed in order to obtain the desired exponent in the
Stefan-Boltzmann relation. Thus the only way we see of
Butitcouldbethatattheearlystagewhenthemodes
bypassing this no-go theorem is to allow for non-particle
are being pushed out of the horizon the universe is get-
likedegreesoffreedom. Weconcludethissectionbysug-
ting hotter. Such is the case of thermal radiation with
gestingascenariowhichmaymakeuseofthispossibility.
γ(1+w) < 0. If γ > 0, denser radiation means hotter
radiation; however if w < 1 the universe gets denser
−
(and so hotter) as it expands. Alternatively we could
A. Saturating temperature
have w > 1, so that the universe gets less dense as it
−
expands; but then if γ < 0 this translates into a higher
An example of a system in whichit is possible to have temperature.
γ <1 is a gas of strings at temperatures close to the so- Another possibility is a stage of radiation injection,
calledHagedorntemperature,TH [19]. Inagasofstrings, either from particle/antiparticle annihilation, from false
the number of degenerate states increases exponentially vacuumdecay,orfromacosmologicalconstantdischarge.
with energy [20] and the canonical partition function di- Yet another possibility is the Phoenix universe of
verges for all T > TH. This does not necessarily mean Lemaitre[12],wheremodeswouldbe pushedoutsidethe
thattemperaturehigherthanTH areunphysical. Infact, horizonwithtemperatureincreasingduringthecontract-
all physical quantities, such as energy density and spe- ing phase.
cific heat are actually finite at T ≥ TH [21]. In [22] it If the universe gets hotter in time, we need kh to in-
was suggested that TH corresponds to a phase transi- crease with time, and with temperature, so that µ > 0.
tion, somewhat analogous to the deconfining transition A necessary condition for scale invariance is then
in QCD. At temperatures close to T , the canonical en-
H
semble description of string gases becomes invalid due γ >1 , (36)
to increasingly large energy fluctuations [23]. One must
use the microcanonical ensemble instead, which is well- bypassig the no-go theorem in the previous section.
defined only if all spatial dimensions were compactified Again, from eqns. (17), (18), (28) and (29) we obtain
[21]4.
At least within the canonical ensemble formalism, T 1 1
H µ=γ α>0 (37)
canbeinterpretedasthelimiting temperatureofthe gas (cid:20)2 − 3(1+w)(cid:21)−
– as energy is increased, the temperature remains con-
stant. In the language of eq. (3) this corresponds to or equivalently
γ 0,inagreementwiththeconstraint(32). Astraight-
→ α 1+3w
forward examination of eq. (30) with γ = 0 shows that < . (38)
γ 6(1+w)
scale-invariance can be satisfied if either w = 1, as in
−
inflation, or if α=1/3, as in VSL models.
Conditions (30) and (31) still apply.
We now consider particular solutions to these condi-
tions.
4In Ref. [21] it is further suggested that in this picture one
needsa mechanism which would later makethreeof thespa-
tial dimensions sufficiently large for usto live in.
5
A. A phantom phase goes back to Lemaitre [12], it has often been forgotten
and revived a few times, e.g. more recently in [28].
“Phantom” matter [27] exhibits an equation of state It is questionable that this model dispenses with fine-
with w < 1, and it may constitute the dark energy of tuning,speciallywhendealingwiththehomogeneityand
theunivers−e. Ithasalsobeenconjecturedthatnormalra- entropyproblems. Itmaybethatjusttoomuchjunk(en-
diation at high temperatures could behave like phantom tropy)is left overfromprevious cycles. This can usually
matter [3]. For these models there is a critical density, be avoided combining the bouncing universe with infla-
ρ , such that w > 1 for ρ < ρ , while for ρ > ρ one tion, such as in its ekpyrotic incarnation [29].
c c c
has w < 1. If the−Universe starts off with ρ > ρ and Another possibility is that the speed oflight decreases
c
− 2 at the bounce, so that all the relevant scales we see to-
expandingthen a ( t)3(1+w). As the universeexpands
∝ − day were sub-horizon modes at some point deep in the
it gets denser and hotter. Eventually a phase transition
radiation epoch in the contracting phase. Hence homo-
brings it to the sub-Planckian regime.
geneity(andabsenceofpervadingcoalescingblackholes)
Insuchascenariothereishyper-inflation,sothemodes
could have been established on all the relevant scales
are pushed out of the horizon without a VSL. However,
for the current cycle. But more importantly in this sce-
in orderfor density fluctuations to havea scale invariant
nario thermal fluctuations could become the Harrisson-
spectrum, the condition in eq. (30) must be satisfied.
Zeldovich spectrum of initial conditions which we ob-
Since γ >1 and w< 1, we find that
− served in our current cycle. We now illustrate how this
1 1 1 could happen.
α> > . (39)
As long as w > 1 (and γ > 0) the universe heats
2 − 3(ω+1) 2
−
up in time during the contracting phase. Hence a nec-
Hence, in order to obtain scale-invariance, one needs a essary condition for scale invariance is that γ >1, evad-
VSL. ing the no-go theorem presented in Section IV. Modes
Regardingthespectrum’samplitude,fromeq.(23)one must leave the horizon as the universe contracts, and as
can obtain the requirement eq. (37) shows this can be achieved without a VSL if
w > 1/3, i.e. the universe undergoes the necessary ac-
γ λ(1−µ)µ(1−γ) B2 10−10 , (40) celera−ted contraction when the strong energy condition
lγ−4 1 ≈ ≈ is satisfied. A general condition for scale invariance is
T
then eq. (30). We find the notable result that standard
or, using eq. (26) to express µ in terms of λ, radiation (w = 1/3, γ = 4, α = 0) satisfies the condi-
tion for scale invariance. Hence, as long as the scales we
γ−4
γ λ1 B2 . (41) can observe today were sub-horizon deep in the radia-
(cid:18)l (cid:19) ≈ tion epoch during the contracting phase of the previous
T
cycle, thermal fluctuations in standard radiation lead to
Thus, given a value of γ, eq. (41) constrains the ratio of a Harrisson-Zeldovichspectrum of initial fluctuations in
the two characteristic length scales lT and λ1. Within the new phase5.
anyspecific modelprovidingarelationbetweenthescale A problem arises with the amplitude, which may be
factor a and the temperature T it should be possible to remedied if there is a drop in the speed of light at the
expressλ1 intermsoflT. Consequently,thesetwolength bounce. For γ =4 the scale lT drops out of the problem
scales should not be considered as independent. (cf. eq. (3)). Also for w = 1/3 and α = 0 we have
We may expect one of the models based upon the for- k T, so µ = 1 and the scale λ disappears from
h 1
malismpresentedin[11]tosatisfytheseconditions,plac- eq. (∝18). Hence the ratio of these parameters disappears
ing further restrictions upon deformations of dispersion from the expression for the amplitude, as can be seen
relations. The spectrum’s amplitude in this case will be directly from Eqn. (41); the amplitude becomes then a
fixed by the ratio between the non-commutative length combinationofnumericalfactorsoforder1clearlyincon-
scale (related to lT) and the Planck length (related to tradictionwithobservations. Thiscanalsobeillustrated
λ1). using the 10 Mpc comoving scale as the normalization
point, as in Peebles’ example [1]. Rephrasing Peebles
argument in terms of this model, we know that in the
B. A bouncing universe case of a perfectly symmetric bounce the right normal-
izationσ canbeobtainedifthe10Mpccomovingscale
10
In the bouncing model a closed universe goes through
aseriesofcyclesstartingwithaBigBangandexpansion,
followedbyre-contractionandaBigCrunch. Everytime
theuniverseapproachesthecrunchitbouncesintoanew 5This assumes certain mode matching at the bounce. In
bang, and if entropy increases at the bounce the new general, the picture will be much more complicated. This
cycle lasts longer (i.e. expands to a larger and colder issue is discussed in subsection VB1
turnaround point) in each new cycle. Although the idea
6
betweentimeandtemperatureissymmetricalaroundthe
bounce, but for c−/c+ 1 the 10 Mpc comoving scale
≪
would leave the horizon in the contracting phase much
laterandatahighertemperature,asillustratedinFig.1.
Using t/(1sec) (T/(1Mev))−2, we find that t =
⋆
time c H−1 10−28secbeforeth≈ebouncethetemperatureis1011Gev.
+
t The comoving horizon at this time is 10 Mpc across if
10
a
⋆
c−t⋆ =10Mpc (42)
a
0
Since a /a = T /1011Gev (where T 2.7 K)
⋆ 0 CMB CMB
≈
we get
10 1 1 GeV (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)o(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)u(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)tside c− 1021 (43)
the horizon 10 Mpc a(t )/a c+ ≈
(cid:0)(cid:1) 10 today
(cid:0)(cid:1) Note that unlike other VSL arguments, which lead to
(cid:0)(cid:1) physical distance
(cid:0)(cid:0)(cid:1)(cid:1) lower bounds on c−/c+, this argument leads to an iden-
10 1 1 GeV (cid:0)(cid:0)(cid:1)(cid:1) tity: thespectrumamplituderesultsdirectlyfromagiven
valueofc−/c+. Itmaybe possibletoobtainasimilaref-
c H−1 fectbyanotherwayofmakingthe bounceasymmetrical.
−
For instance if G and h¯ vary differently at the bounce a
different constraint is obtained.
Certainly,the validityofthe abovediscussiondepends
upon the matching of growing and decaying modes dur-
c H−1
+ ing the bounce dynamics - something which is unknown
in the absence of a specific model for the bounce. We
conclude with a discussionof the two main uncertainties
faced by this model.
FIG.1. The normalization problem in bouncing universes
1. Mode matching at the bounce
and a way to solve it. In the case of a perfectly symmetrical
bounce, density perturbations on 10 Mpc scale exit thehori-
zonduringthecontractingphaseattemperaturesmuchlower Bounce dynamics will in general mix the modes in ei-
than 1011 GeV. The problem could be fixed if the speed of ther phase [34,35], a matter which may upset our previ-
lightinthecontractingphase(c−)hadamuchbiggerconstant ous arguments. In the contracting phase we have
valuethan it has today (i. e. c− ≫c+). B−
φ=A−φ0+ (44)
t
| |
leavesthehorizoninthecontractingphasewhentheuni- and in the expanding phase
verse was at 1011 Gev. However, if the relation between
B
temperature and time on either side of the bounce is φ=A φ + + (45)
+ 0
symmetric, this comoving scale would leave the horizon t
muchearlierinthe contractingphase, whenthe universe where φ is the gravitational potential and t = 0 at the
wasmuchcolderandthereforethefractionalfluctuations bounce. Matching these modes depends on the bounce
were much larger. dynamics, and everything we have said previously as-
Theproblemwiththenormalizationmentionedabove, sumed that B− mode was not excited and that the A−
couldbe resolvedif the bounce wasasymmetrical,allow- mode was matched onto the A+ mode.
ing for the 10 Mpc comoving scale to leave the horizon If the B− mode is excited, but then matches onto the
when the universe was indeed at 1011 Gev. One possi- B+ mode, then the earlier discussion still applies. How-
ble scenariois whenthe speedof lightis constantduring everiftheB− modematchesontotheA+ modethenthe
the previous (c = c−) and current (c = c+) cycles, but conclusions in this paper have to be revised: in addition
c− c+, i.e. the speed of light decreases at the bounce. totheHZspectrumpredictedhere,therewouldbeavery
≫
Onecanmakeadditionalsimplifyingassumptionsjustto red component (n = 3). Another pathological case is
−
illustrate the point. Letus assumethat the fundamental when A− and B+ are excited, while B− =A+ =0.
constants G and h¯ also change at the bounce in such a We defer to a future publication a more careful study
way so as to preserve the relation between the temper- ofthissituation. Itcertainlydependsonhowthebounce
ature and time (this can be achieved if h¯3c5/G is the is actually produced. Mode matching in pre-big-bang
same before and after the bounce). Hence the relation [36]and ekpyrotic[29]cosmologieshas been discussedin
considerable detail in Refs. [34,35]
7
2. Entropy production at the bounce Harrisson-Zeldovichspectrumplusgravitationalinstabil-
ityratherthana“proof”ofthe inflationaryoriginofthe
Another matter which may change the spectrum and initial fluctuations.
normalization of the fluctuations is entropy production
at the bounce (which almost certainly occurs). Indeed
for normal radiation, i. e. with γ = 4, the amplitude of ACKNOWLEDGMENTS
thermal density fluctuations on a given scale L can be
expressed in terms of the entropy contained in a sphere We are grateful to Ruth Durrer for pointing out sev-
of radius L (eq. (15.26) of [1]): eral errors in an earlier draft of the manuscript, and to
Pedro Ferreira for sharing his ideas on the subject and
δ2 = 16 . (46) for bringing Ref. [30] to our attention. We also thank
L 3S Andrew Liddle for clarifying comments on the definition
of HZ spectra and choice of gauge.
Thus,ifentropyisproducedatdifferentratesondifferent
scalesthe spectrumoffluctuationscouldbe modified. In
additionthisissueisboundtointerferewithanynormal-
ization condition for the amplitude.
VI. CONCLUSIONS
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