Particle creation in the early Universe: achievements and problems

0.19 MB·

by  A. A. Grib

Particle creation in the early Universe:
achievements and problems
A. A. Grib1,2,†, Yu. V. Pavlov1,3,‡
1A.Friedmann Laboratory for Theoretical Physics, St.Petersburg, Russia;
6 2Theoretical Physics and Astronomy Department, The Herzen University,
1 Moika 48, St. Petersburg 191186, Russia;
0
3Institute of Problems in Mechanical Engineering, Russian Acad. Sci.,
2
Bol’shoy pr. 61, St. Petersburg 199178, Russia

n
a
J Abstract. Results on particle creation from vacuum by the gravitationalfield of the expanding Friedmann

5 Universearepresented. Finite resultsforthedensityofparticlesandtheenergydensityforcreatedparticles
2 are given for different exact solutions for different regimes of the expansion of the Universe. The results are

obtained as for conformal as for nonconformal particles. The hypothesis of the origination of visible matter
]
c from the decay of created from vacuum superheavy particles identified with the dark matter is discussed.
q
- PACS number: 04.62.+v,98.80.Cq, 95.35.+d
r
g Key words: particle creation, early Universe, dark matter
[

1 1. Introduction
nation of the fact that our Universe is isotropic.
v
8 However after these first successes the interest to
The problem of particle creationfrom vacuum by
1 the problem somehow failed. The reason for this is
6 thegravitationalfieldoftheexpandingUniversewas
theincorrectevaluationofthequantitativemeaning
6 studied intensively in the 70-ies of the past century
of the effect in books of high authority [8, 9] using
0 (see our works [1, 2] and papers of Ya.B.Zel’dovich
. thehypothesisofthecreationofonlythoseparticles
1 and A.A.Starobinsky [3, 4]). High interest to the
which are observed today. Due to the fact that the
0 problemwasexpressedbyK.P.Staniukovich[5],who
effect is proportionalto the mass of the particle the
6 always supported one of the authors of this paper
1 effect is surely negligible for small masses. However
(A.A.G.) in his activity. Really, differently from the
: if one supposes that observable particles appeared
v usualopinionthatspeakingabouttheBigBangone
aftercreationbythegravitationofmassiveparticles
i
X must consider very high densities of matter com- with the mass of the order of the Grand Unification
posed from massive particles particle creation from
r the resultis notsmallatallandevenmoreit makes
a vacuum at the certain epoch of the evolution of the
possible to explain the observable number of parti-
Universe makes possible to say that early Universe
cles in the Universe [10, 11]. The other reason of
did notcontainmassiveparticles andlookedas vac-
theneglectistheincorrectstatementinmuchpopu-
uum and possibly some fields. In works [1, 2, 6]
lar book [12] that calculations based on the method
finite values for the density of created particles and
of the diagonization of the Hamiltonian lead to the
the energy density were obtained for the first time.
infinite value of the density of the created particles
ItbecameclearthatintheUniversedescribedbythe
for nonconformalparticles. This opinion was shown
Friedmann metric some special era exists for every
to be wrong by calculations in [13]. Note that in
typeofparticlesdefinedbytheComptontimeofthe
thebook[12]aswellasinthe book[14]thereareno
particlesothatthereisnocreationoftheseparticles
anyfinite(!) valuesofphysicalevaluationsoftheef-
for the time much smaller or much larger than this
fectofparticlecreationincosmology. Thisoccurred
time passed from the beginning of the Universe. In
because of the fact that the method of the diago-
papers [3, 7] an important result was obtained that
nalization of the Hamiltonian strongly depends on
intheanisotropicUniverseparticlecreationleadsto
the form of the Hamiltonian used for calculation of
the isotropizationof the metric which can be expla-
the particle creation in Friedmann Universe. The
†andrei [email protected] authors of [1] were lucky in 1969 to use the cor-
‡[email protected] rectexpressionofthisHamiltonianthegroundstate
1
2 PARTICLE CREATION IN THE EARLY UNIVERSE
of which leads to the finite value of the observable in inflationmodels [15]. In quantumtheory of fields
quantities. This Hamiltonian is defined by the met- in curved space-time one often considers the case
ricalenergy-momentumtensorofthequantizedmas- of arbitrary ξ [12] (more general case of coupling
sivefieldandisconnectedwiththecanonicalHamil- with the curvature of Gauss-Bonnet type was con-
tonian of particles with varied mass in conformally sidered in [16, 17]). Then conformal invariance for
connected to the Friedmann metric static space. It masslessfieldsisabsent. Thestudyofnonconformal
isthisconnectionwhichisleadingtotofiniteresults case is important due to different reasons. Massive
as for conformal as for nonconformal particles. Any vector mesons [10] (longitudinal components) and
other choice of vacuum or Hamiltonian leads to the gravitons [18] satisfy equations of this type. In case
infinitevalueofthedensityofcreatedparticlesmak- of the scalar field with self interaction in general it
ing the theoryphysically meaningless. Inthis paper is impossible to conserve conformal invariance not
afterashortreviewoftheresultswhichcanbefound only ofthe effective action(the conformalanomaly)
inourbook[10]wegiveareviewofsomenewresults but of the action itself [12].
obtainedbytheauthorsinlasttime. Someunsolved Note that the Dirac equation in curved space for
problems are discussed in the Conclusion. m=0isconformalinvariantwithoutanyadditional
We use the system of units in which ¯h=c=1. modifications [10, 12].
Furthermore, let us consider an N-dimensional
2. Scalar field in curved Fried- homogeneous isotropic space-time, choosing the
metric in the form
mann space-time
ds2 =g dxidxk =a2(η)(dη2 dl2), (5)
ik
Consider the complex scalar field ϕ(x) of the −
mass m in curved Friedmann space-time with the where dl2 =γ dxαdxβ is the metric ofan (N 1)-
αβ
Lagrangian dimensional space of constant curvature K =0−, 1.
±
The complete set of solutions to Eq. (2) in the
L(x)= g gik∂ ϕ∗∂ ϕ (m2+ξR)ϕ∗ϕ , (1)
| | i k − metric (5) may be found in the form
leading topEule(cid:2)r equations (cid:3)
ϕ˜(x)
ϕ(x)= =a−(N−2)/2(η)g (η)Φ (x),
( i +ξR+m2)ϕ(x)=0, (2) a(N−2)/2(η) λ J
i
∇ ∇ (6)
where are covariant derivatives in metric g of where
i ik
the N-∇dimensional space-time, g = det(gik), R — gλ′′(η)+Ω2(η)gλ(η)=0, (7)
the curvature scalar. For m=0 and ξ =ξ (N
c
2)/[4(N 1)] equation (2) is conformal in≡varian−t. Ω2(η)=(m2+(ξ ξc)R)a2+λ2, (8)
− −
This means that in mapping of the space-time with
2
N 2
metric gik on the space-time with metric g˜ik: ∆N−1ΦJ(x)= λ2 − K ΦJ(x), (9)
− − 2
(cid:18) (cid:18) (cid:19) (cid:19)
g g˜ =exp[ 2σ(x)]g , (3)
ik → ik − ik the prime denotes a derivative with respect to
the conformal time η, and J is the set of indices
where σ(x) is arbitrary smooth function of coordi-
(quantum numbers) numbering the eigenfunctions
nates there exists such scale transformation
of the Laplace-Beltrami operator ∆N−1 in (N 1)-
−
N 2 dimensional space.
ϕ(x) ϕ˜(x)=exp − σ(x) ϕ(x), (4)
→ 2 To perform quantization, let us expand the field
(cid:20) (cid:21)
ϕ(x) in the complete set of solutions (6)
that the waveequationconservesits form(see [10]).
The physical sense of the conformal invariance is
ϕ(x)= dµ(J) ϕ(+)a(+)+ϕ(−)a(−) , (10)
thatmasslessfieldhasnoitsproperlengthscale(for J J J J
the massive case such a scale is given by the Comp- Z (cid:20) (cid:21)
ton wave length λC = 1/m) and so it must behave where dµ(J) is a measure on the set of quantum
equallyinconformallyconnectedspaces(3). Incase numbers,
ξ =ξ equation(2)iscalledequationwithconformal
c
coupling(ξc =1/6forN =4). Thecaseξ =0corre- ϕ(+)(x)= gλ(η)Φ∗J(x) , ϕ(−)(x)= ϕ(+)(x)∗,
sponds to minimal coupling. Such type of coupling J √2a(N−2)/2(η) J J
ofthescalarfieldwithcurvatureusuallyissupposed (cid:0) ((cid:1)11)
GRIB, PAVLOV 3
andrequirethatthestandardcommutationrelations + F (η)a∗(+)a(+)+F∗(η)a∗(−)a(−) , (15)
hold for a(±)and a∗(±). J J J¯ J J¯ J
J J (cid:21)
A corpuscular interpretation of a quantized field
where
basedonthemethodofHamiltoniandiagonalization E = |gλ′|2+Ω2|gλ|2, F = ϑJ g′2+Ω2g2 ,
byBogoliubov’stransformationshavebeenproposed J 2 J 2 λ λ
for the gravitationalexternal field in paper [1]. (cid:2) (cid:3)(16)
We givethefollowingdefinitionofparticlesinthe and we have chosensuch eigenfunctions ΦJ(x) that,
external field [10], p. 46: “In the framework of this for arbitrary J, there is such J¯ that ΦJ∗(x) =
interpretation one calls as particles (quasiparticle) ϑJΦJ¯(x), |ϑJ|=1,(J¯¯=J, ϑJ¯=ϑJ). Suchachoice
creation-annihilation operators at moment t those is possible due to completeness and orthonormality
operators in terms of which the Hamiltonian of a of the set ΦJ(x).
quantized field is diagonal at the moment t. There- According to the Hamiltonian diagonalization
foreaquasiparticleisinterpretedasanenergyquan- method [10] (the nonconformal case see in [13, 19]),
tum and the measurement of the number of quasi- the functions gλ(η) should obey the followinginitial
particles is connected with the measurement of en- conditions:
ergy. Indeed, the measurement theory in quantum g′(η )=iΩ(η )g (η ), g (η ) =Ω−1/2(η ).
mechanics requires that in the result of measuring λ 0 0 λ 0 | λ 0 | 0
(17)
of some physical quantity a system would found it-
Ifthequantizedscalarfieldisinthevacuumstate
self in a proper state of the corresponding opera-
at the instant η , then the number density of the
tor. Thereforethemeasurementofenergyinevitably 0
pairs of particles created up to the instant η can be
transfers the system in a proper state of the Hamil-
evaluated (for K =0) as [10]
tonian. To find this state the Hamiltonian must be
diagonalized.” ∞
B
Let us build the Hamiltonian as the canonical n(η)= N S (η)λN−2dλ, (18)
one for the variables ϕ˜(x) and ϕ˜∗(x), for which the 2aN−1 Z λ
0
equationofmotiondoesnotcontaintheirfirst-order
derivatives with respect to the time η. Recall that where B = 2N−3π(N−1)/2Γ((N 1)/2) −1, Γ(z)
N
−
the equations of motion do not change after adding is the gamma function, and
(cid:2) (cid:3)
a full divergence ∂Ji/∂xi to the Lagrangian den-
g′(η) iΩg (η)2
sity L(x). Let us choose, in the coordinate system S (η)= | λ − λ | . (19)
λ
(η,x), the vector 4Ω
(Ji)=(√γcϕ˜∗ϕ˜(N 2)/2,0,...,0), (12) As shownin[13], Sλ ∼λ−6, andthe integralin(18)
− converges for N < 7. So the density of particles
where γ = det(γαβ), c = a′/a. Then, using the createdinfourdimensionalFriedmannspace-timeis
Lagrangian density L∆(x) = L(x) + ∂Ji/∂xi, we finite. Notethatthefirstuseofthemethodofdiago-
obtain for the momenta canonically conjugate to ϕ˜ nalizationoftheHamiltonianinpaper[20]ledtothe
∗
and ϕ˜ : infinite density of created quasiparticles. However
∂L∆ ∂L∆ there not the canonical Hamiltonian for fields ϕ˜(x),
∗′ ′
π ≡ ∂ϕ˜′ =√γϕ˜ , π∗ ≡ ∂ϕ˜∗′ =√γϕ˜, (13) buttheHamiltonianwhichisobtainedfromthemet-
rical energy momentum tensor for the scalar field
respectively. Integrating the Hamiltonian density
with minimal coupling was used. In the work [21]
h(x) = ϕ˜′π+ϕ˜∗′π∗ −L∆(x) over the hypersurface created particle were defined for the so called adi-
Σ: η =const, we obtainthe followingexpressionfor
abatic vacuum state in such a way that the corre-
the canonical Hamiltonian:
sponding Hamiltonian was not diagonal at any mo-
H(η)= dN−1x√γ ϕ˜∗′ϕ˜′+γαβ∂ ϕ˜∗∂ ϕ˜+ mentoftime. Thispreventedtoobtainfiniteresults
α β
for particle creation in Friedmann space-time.
ZΣ (cid:26)
N 2 Note that the equation (7) by use of the trans-
(m2+ξR)a2 − 2c′+(N 2)c2 ϕ˜∗ϕ˜ .(14)
formation g(η) = expz(η) is going to the following
− 4 −
(cid:20) (cid:21) (cid:27) equation for the function v(η) z′(η)
(cid:0) (cid:1)
The Hamiltonian (14) may be written in terms of ≡
the operators a(±) and a∗(±) in the following way: v′(η)+v2(η)+Ω2(η)=0, (20)
J J
H(η) = dµ(J) E (η) a∗(+)a(−)+a∗(−)a(+) + beingtheRiccatiequationofthegeneraltypetheso-
J J J J¯ J¯ lutions of which cannot be expressed in finite form
Z (cid:20) (cid:16) (cid:17)
4 PARTICLE CREATION IN THE EARLY UNIVERSE
in elementary functions. So the number of scale bH0L‰103
factors a(η) leading to exact solutions is relatively 4
small. In cases when exact solution can be found
3.5
it usually is expressed through special functions —
3
hypergeometric functions, Bessel functions etc.
2.5
2
3. Particle creation in cosmological
1.5
models with p = wε
1
0.5
The Einstein equations
Α
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
R g R= 8πGT , (21)
ik ik ik
− 2 −
Figure 1: b(0) for conformal scalar particles and
with the energy-momentum tensor of the back- a(t) tα.
∼
ground matter Ti = diag(ε, p,..., p) in met-
k − −
ric (5) have the form
much smaller and later than the Compton one. On
c2+K 16πGε Fig. 1 the results of numerical calculations of the
= ,
a2 (N 1)(N 2) coefficient b(0) for N = 4 for different scale factors
− − a(t) = a tα are given. From (24) one can see the
1 N 3 8πGp 0
c′+ − c2+K = . (22) effect of connection of the number of created parti-
−a2 2 N 2
h (cid:0) (cid:1)i − cles with the number of causally disconnected parts
From (22) it follows that for p = wε where w = on the Friedmann Universe at the Compton time of
const, the energy density of the background matter its evolution.
is changing according to the law ε a−(1+w)(N−1), The results for nonconformalscalar field can rad-
∼
i.e. is decreasing with the increasing of a and if ically depend on the initial moment of time (see
w > 1 it is constant, for w = 1 it increases with Eqs.(7),(8)). Howeverforthecondition(ξ ξ)R>
c
− − −
increasing a for w < 1. 0, for the case of the minimally coupled scalar field
−
For K = 0 and w > (N 3)/(N 1) from (22) in the dust Universe p = 0 (a(t) t2/3), the initial
one obtains − − − momentcanbedefinedbythe con∼dition: Ω2 0for
≥
any λ. In the other case (negative squareof energy)
2
a=a tq =a ηβ, q = , onecanobtainchangeofthevacuumtakingintoac-
0 1
(N 1)(w+1)
− count self interaction as it occurs for spontaneous
q
β = , q (0,1). (23) breaking of symmetry.
1 q ∈
− Here we take for nonconformal scalar field initial
valueoft sothatm2+(ξ ξ )R(t )=0. Theresults
ExpressionforthenumberofpairsNa(t)=n(t)a3(t) of numer0ical calculations−[2c2] of0the coefficient b(0)
created in the volume a3(t) up to the moment of
for the scalar field with minimal coupling and the
time t can be written as
scale factor a(t)=a tα are given on Fig. 2.
0
a(t ) N−1 In the interval q (0,1) the following two cases
Na(t)=b(q0)(t)· tC , (24) has the exact soluti∈ons of equations for the scalar
C
(cid:16) (cid:17) field.
where t = 1/m is the Compton time. Then
C
b(0)(t)/(1 q)N−1 is the coefficient of proportion-
alqity for t−he number of created particles and the 3.1. The exact solution for a0√t = a1η
number of causally disconnected regions Nc(t) = The Friedmann model with such scale factor is very
((1 q)a(t)/t)N−1 at the Compton time after the important for applications because for K = 0 in
−
Big Bang. four-dimensionalspace-timeitcorrespondstothera-
The results of calculations [22] of the number of diationally dominated Universe. The initial condi-
createdparticlesforthescalarparticleswithconfor- tions for the conformally coupled scalar field can be
mal coupling don’t depend on the moment of time put at η 0 i.e. close to the singularity a=0
→
whenoneputstheinitialconditionaswellasontime
of observation of created particles if these times are g (0) =1/√λ, g′(0)=iλg (0). (25)
| λ | λ λ
GRIB, PAVLOV 5
bH0L‰103 3.3. The exact solution for a(t) t
∼
4 For w = (N 3)/(N 1) one obtains from the
− − −
3.5 Einstein equations (22) that a = a0t = a1ea0η. If
3 a0 = 1 and K = 1, then ε = 0 and the metric (5)
−
2.5 with such scale factor is flat while coordinates xk
describe the part of the Minkowski space. The four
2
dimensional hyperbolic space-time with a(t) = t is
1.5
known as the Milne Universe.
1
In metric(5) with a = a t the solution of equa-
0.5 0
tion (7) with ξ = ξ and initial conditions (17) put
Α c
0.6 0.7 0.8 0.9 1 at the moment mt 0 has the form
→
√λ iλ
Figure 2: b(0) for scalar particles with minimal cou- g (t)= Γ J (mt)eiα0, (29)
pling and a(t) tα. λ a0 a0 iλ/a0
∼ (cid:16) (cid:17)
where Γ(z) is the gamma function, α is the ar-
0
Thesolutionofequation(7)withconditions(25)can bitrary real constant. Independently from a0 and
be written in the form K the asymptotic at mt 1 value of the den-
≫
sity of created quasiparticles for N = 4 is equal
ei(mt+α0) 1 i 1
g (t)= Φ δ2, ; i2mt + to n(t) m/(512πt2). So different from zero re-
λ √λ (cid:20) (cid:16)4 − 2 2 − (cid:17) sult take≈s place even for the Milne Universe where
3 i 3 gravitationalfieldisabsentandonecannothavecre-
i2δ√mt Φ δ2, ; i2mt , (26)
4 − 2 2 − ation of real particles! Really the space-time anal-
(cid:16) (cid:17)(cid:21) ysis of particle creation using correlation function
where Φ(a, b; z) is the degenerate hypergeometric
of the created pair of particles introduced in [23]
Kummer function, δ λ/(ma(t )) has the sense of
≡ C shows that the corresponding correlation function
thephysicalmomentattheComptontimet =1/m
C of the created pair is exponentially small at dis-
inunitsm. α isarbitraryrealconstant. Therepre-
0 tances larger than the Compton length of the par-
sentationfor(26)throughfunctionsoftheparabolic
ticle. This means that created quasiparticles in this
(0)
cylinderisgivenat[6]. Theasymptoticvalueb
1/2 ≈ case are virtual pairs with the characteristic corre-
5,3 10−4 [6]. lation length 1/m.
·
3.2. The exact solution for a0t1/3 = 3.4. De Sitter space-time
a √η
1
Thisisaspaceofconstantcurvature. Itisasolution
The model with such scale factor for K = 0 and of Einstein equations in empty space with nonzero
N = 4 is the Universe with limiting rigid p = ε cosmological constant. Using the special choice of
equation of state. The solution of the equation (7) coordinatesinDeSitterspaceonecanwriteitsmet-
withξ =ξc andinitialconditions(25)for suchscale ric in the form (5) with scale factor 1–3) for the De
factor [6]: Sitterspaceofthe firsttypeand4)forthe DeSitter
space of the second type:
πδ2eiα0
g (t)= (mt)2/3+δ2
λ
√3λ × 1 coshHt 1
q 1)a eHt = − , 2) = ,
1
C (δ)J (mt)2/3+δ2 3/2 + Hη H Hsinη
1 1/3
sinhHt 1 cosHt 1
+hC2(δ)J−1/3(cid:16)(cid:0) (mt)2/3+δ2(cid:1) 3/2(cid:17) , (27) 3) H = Hs−inhη, 4) H = Hcoshη. (30)
where J (x) — Bes(cid:16)se(cid:0)l functions (cid:1) (cid:17)i Inallthesecasestheequationforthescalarfield(7)
ν
hastheexactsolution. Incase1),t ( ,+ )
C1(δ)=J2/3 δ3 +iJ−1/3 δ3 , η ( , 0), the solution of equati∈on−(7∞) wit∞h co⇔n-
∈ −∞
C2(δ)=J−2/3 δ3 −iJ1/3(cid:0)δ3(cid:1), δ ≡ ma(cid:0)λ(t (cid:1)). (28) ditions (17) for η0 →−∞ has the form
C
The asymptotic(cid:0)va(cid:1)lue for b(cid:0)(10/)3(cid:1)≈8,1·10−4. gλ(η)=r−π2η eπ2ImνHν(2)(−λη)eiα0,
6 PARTICLE CREATION IN THE EARLY UNIVERSE
1 m2+(ξ ξ )R metrics
c
ν = − , (31)
4 − H2
r
a(η)=A+Btanhγη, a(η)= A+Btanhγη,
where H(2)(z) is the Hankel function, α is arbi-
ν 0 (33)
trary real constant. As it was shown in [23] (and p
A, B, γ = const are expressed through the hyper-
for nonconformal case for m2 (ξ ξ)R in [24])
≥ c − geometric function (see [10], 9.5, [12], 3.4).
by the method of the space-time correlation func- § §
Metric with the scale factor a(η)= aη2+bη in
tionthe createdpairsmustbe interpretedasvirtual
fourdimensionalspace-timeforη b/acorresponds
pairs. AbsenceofcreationofrealparticlesinDeSit- ≪ p
to the limiting rigid equation of state p = ε. For
ter space is confirmed by the local form of the vac-
η b/a(K =0)thisscalefactorcorrespondstothe
uum energy-momentum tensor and zero imaginary ≫
radiationdominatedUniversep=ε/3. Thesolution
part of the effective Lagrangian[10].
of equation (7) with ξ = ξ for this scale factor can
In cases 2 – 4 also for the same scale factors but c
beexpressedthroughthedegeneratehypergeometric
taking η γη, γ = const the solution also can
→ Kummer function.
be expressedthrough hypergeometric functions (see
Homogeneous isotropic space-time with the scale
9.5 in [10] and [17]). For γ = 1 these scale fac-
§ 6 factor
tors don’t describe the De Sitter space. For ex-
ample the space-time with the scale factor a(η) =
2γt
1/(Hcoshγη) = sin(γHt)/H has the evolution be- a(η)=a tanγη=a exp 1,
1 1
tween two singularities. s (cid:18) a1 (cid:19)−
π
η 0, , t (0, + ) (34)
3.5. Particle creation in cosmology ∈ 2γ ∈ ∞
(cid:18) (cid:19)
with phantom matter
is describing the the radiation dominated Universe
Forw < 1the exactsolutionofequation(7)exists a(t) √2γa tforsmalltimes,forlargertimes—ex-
− ≈ 1
for the value w = (N +1)/(N 1) (w = 5/3, ponentiallyexpandingUniversea(t) a expγt/a .
− − − ≈ 1 1
if N = 4). The scale factor of metric in this case is The solutionofequation(7)forthe scalarfieldwith
a = a0/( t) = a1/√ η. For t 0 there is a sin- conformalcouplingandinitialvaluesatη 0isex-
− − → − →
gularityoftheBigRip. Thesolutionofequation(7) pressedthroughthegeometricfunctionF(a, b, c; z):
satisfying the conditions (17) for η has the
0
→ −∞
form g(η)= eiα0(cosγη) 1+√1−4m2a21/γ2 /2
πm2a2 √λ
gλ(η)= 2iη√λ exp 1 +i(λη+α0) (cid:0) (cid:1)
− − 4λ 1 λ
×Ψ(cid:16)1+ im22λa21 ,2;−2iλη(cid:17), (32) (cid:20)F(cid:16)α,β,2; s1in2γη(cid:17)1+3iγ sinγη×
(cid:16) (cid:17) F α+ ,β+ , ;sin2γη ,
where Ψ(a, b; z) is the degenerate hyperbolic Tri- × 2 2 2
comifunction, α isthe arbitraryrealconstant. For (cid:16) (cid:17)(cid:21)
0
tohneedoebntsaitinysof(lpoaorkti[c2le5s])crnea=tedmin3/N24π=2.4fIonrtsp→ite−o0f, α,β = 41"1+s1− 4mγ22a21 ± γ2 λ2−m2a21#. (35)
q
the fact that the general number of created parti-
cles Na(t) = n(t)a3(t) in Lagrange volume a3(t) is Note that if one finds the general solution (7)
unboundedly increasing for t 0, the back reac- for the scalar conformally coupled to the metric
→ −
tion of particles creation and vacuum polarization with the scale factor a(η) one can obtain by redef-
ofa massive,conformallycoupledscalarfieldonthe inition λ the general solution for the scale factor
space-time metric can be neglected in the whole re- a˜ = a2(η)+b2, where b = const. For example
gion where one can apply the approach of quantum take the model with the scale factor
p
field theory in curved space-time [25].
a(η)= a2η2+b2, <η < , (36)
4. Exact solutions in cosmological 1 −∞ ∞
q
models with p/ε = const which in asymptotic regions η corresponds
6 (for N = 4) to radiation dom→in±at∞ed cosmology.
Letusmakeashortreviewofscalefactorsleading The space is contracting to the minimal scale fac-
to exact solutions of the equation (7). Solutions for tor a(0) = b, is reflecting and again is expanding.
GRIB, PAVLOV 7
The solution of the equation (7) with initial condi- for creation from vacuum pairs of superheavy par-
tions (17) defined by the condition of the diagonal- ticles with the mass of the Grand Unification scale
ization of the Hamiltonian at the moment η = 0 in the early radiation dominated Universe give the
like (26) has the form surprisingresult— its number is of the order ofthe
observed Eddington-Dirac number. So putting the
expi α0+ma1η2/2 hypothesis of their decay on quarks and leptons in
g (η)=
λ (λ(cid:0)2+m2b2)1/4 (cid:1) × the earlyUniverseone obtains the observednumber
of protons and electrons. All this shows that one
1 i 1
Φ δ2, ; ima η2 + cannot neglect the effect of particle creation in the
1
×" 4 − 2 2 − ! early Universe!
The radiation dominance in the end of inflation
+iη λ2+m2b2 Φ 3 i δ2, 3; ima η2 ,(37) era is important for our calculations. However this
1
4 − 2 2 − !# radiationis formednotbyour visibleparticles. Itis
p quintessence or some dark light particles not inter-
δ2 = λ2+m2b2. The influence of the parameter b on acting with ordinary particles.
the inte2nmsait1y ofparticle creationfor the case m=1, Forthetimet m−1thereisaneraofgoingfrom
≫
a=1/2is shownonFig. 3. It is seenthat for b 0 theradiationdominatedmodeltothe dustmodelof
→
superheavy particles,
N(cid:144)N0
1 3 2 M 4 1
Pl
t , (39)
0.8 X ≈ 64πb(s) m m
(cid:18) (cid:19) (cid:18) (cid:19)
0.6
where M 1,2 1019GeV is Planck mass. If m
0.4 1014GeVP,lt≈ 1·0−15sforscalarand t 10−17∼s
X X
0.2 ∼ ∼
for spinor particles.
b Let us define d, the permitted part of long-living
0.2 0.4 0.6 0.8 1
X-particles, from the condition: on the moment of
Figure3: Therelationofnumberofthecreatedpar- recombination trec in the observable Universe one
ticlesinmodel(36)withparameterbtotheradiation has dεX(trec)=εcrit(trec). It leads to
dominated case (b=0).
2
3 M 1
Pl
d= . (40)
thenumberofcreatedparticlesisgoingtotheknown 64πb(s) (cid:18) m (cid:19) √mtrec
resultforb=0. Thisshowsthepossibilityofputting
Form=1013 1014GeVonehasd 10−12 10−14
theinitialconditions(17)atb=0, η →0,i.e. atthe forscalarand−d 10−13 10−15for≈spinorpa−rticles.
singularity for the radiation dominated Universe. ≈ −
So the lifetime of the main part or all X-particles
Note that for some scale factors the solutions for
must be smaller or equal than t .
the scalar field with nonconformal coupling can be X
We giveinRef.[26,27]the modelwhichcangive:
obtained by the redefinition of the mass and mo-
(a) short-living X-particles decay in time τ < t
mentum. The problem of description of such scale q X
(more wishful is τ t 10−38 10−35s, i.e.,
factors was solved in [17]. q ∼ C ≈ −
the Compton time for X-particles); (b) long-living
particles decay with τ >1/m.
l
5. Superheavy particles in the If τ is larger than the time of breaking of the
l
Grand Unification symmetry it can be that some
early Universe
quantum number can be conserved leading to some
effectivetimeτeff >t 4.3 1017s (t istheageof
The total number of massive particles created the Universe).lThe smUa≈ll d · 10−15 U10−12 part of
in Friedmann radiation dominated Universe (scale ∼ −
long-living X-particles with τ > t forms the dark
factor a(t)=a t1/2) inside the horizon is (see (24)) l U
0
matter.
Na =n(s)(t)a3(t)=b(s)m3/2a30, (38) For telff ≤1027s one must get a strong anisotropy
of ultra high energy cosmic rays in the direction to
where b(0) 5.3 10−4 for scalar and b(1/2) thecenteroftheGalaxy. However,experimentswith
3.9 10−3 fo≈r spino·r particles [10]. It occurs tha≈t cosmic rays don’t show such an anisotropy and one
·
N 1080 for m 1014GeV [10, 11]. Calculations must suppose teff > 1027s. The special conditions
a ∼ ∼ l
8 PARTICLE CREATION IN THE EARLY UNIVERSE
existing in vicinity of black holes can lead to the nonlinearterms(seepaper[30]forsomespecialcase)
collisions of superheavy particles with high energy leading to redefinition of vacuum as it is done in
and to their decay there on ordinary particles as it spontaneous symmetry breaking theories.
was in the early Universe. Our estimates [28] show Acknowledgments. This work was supported
that such processes near the black holes of active bytheRussianFoundationforBasicResearch,grant
galactic nuclei can explain the stream of ultrahigh No. 15-02-06818-a.
energy cosmic rays with energy 1019eV observed
≥
in experiments.
References
The observed entropy in this scenario originates
due to transformation of X-particles into light par-
[1] A.A. Grib and S.G. Mamayev, Yadernaya
ticles: quarks, antiquarks and some particle similar
Fizika10,1276–1281(1969)[Engl.transl.: Sov.
to Λ0 in K0-meson theory, having the same quan-
J. Nucl. Phys. (USA) 10, 722–725 (1970)]. On
tum number as X. Baryon charge is created close
field theory in Friedmann space.
to the time t , which can be equal to the Compton
q
time of X-particles tC 10−38 10−35s [22]. [2] A.A.Grib,S.G.Mamayev,andV.M.Mostepa-
∼ −
nenko, Quantum effects in intensive external
fields [in Russian] (Atomizdat, Moscow, 1980).
6. Conclusion
[3] Ya.B. Zel’dovich and A.A. Starobinskii,
Particlecreationfromvacuumby thegravitation Sov. Phys. JETP 34, 1159–1166 (1972). Parti-
of the expanding Friedmann Universe can explain cle production and vacuum polarization in an
the observable number of visible particles as well anisotropic gravitational field.
as dark matter. In our scenario it is dark matter
[4] Selected Works of Ya.B. Zeldovich, Vol. II:
consisting of superheavy particles with mass of the
Particlies, Nuclei, and the Universe (Nauka,
Grand Unification which is created first, ordinary
Moscow, 1985) [Engl. transl: Princeton Univ.
particles are created not by gravitation but by the
Press, Princeton, 1993].
conversion of particles of dark matter into them at
veryhighenergywhenhighsymmetryofGrandUni- [5] K.P.StanyukovichandV.N.Melnikov,Hydro-
fication is present. Breaking of this symmetry leads dynamics, Fields and Constants in the Theory
to conservation of some part of originally created of Gravitation [in Russian] (Energoatomizdat,
dark matter particles which are stable today. Moscow, 1983).
This hypothesis can be checked in processes with
elementary particles in ergosphere of rotating black [6] S.G. Mamayev, V.M. Mostepanenko,
holes where collisions of particles can lead to very and A.A. Starobinskii, Sov. Phys. JETP
high energy in the centre of mass frame so the situ- 43, 823–830 (1976). Particle creation from
ation of the early Universe can be reproduced. the vacuum near a homogeneous isotropic
In our papers [28, 29] it was shown 1) conversion singularity.
of dark matter particles with large mass into visi-
[7] V.N.LukashandA.A.Starobinskii,Sov. Phys.
ble particles with relatively small mass can lead to
JETP 39 742–747(1974).The isotropization of
observation of the products on the Earth as ultra
the cosmological expansion owing to particle
high energy cosmic ray particles inspite of the large
production.
red shift. 2) Estimates of the process in taking into
account observable dark matter density lead to ob- [8] Ya.B. Zel’dovichandI.D. Novikov,The Struc-
servable flaws of ultra high energy cosmic rays. ture and Evolution of the Universe (Nauka,
There are still some unsolved problems. Moskow, 1975) [Engl. transl.: University of
a) In anisotropic space-time only in some special Chicago Press, Chicago, 1983].
case [19] one can obtain the finite results by the
[9] S.Weinberg, The First Three Minutes. A Mod-
Hamiltoniandiagonalization. Sothistechnicalprob-
ern View of the Origin of the Universe (Basic
lem is still unsolved.
Books, New York, 1977).
b) Quantization of minimal coupled field in gen-
eral case leads to two types of modes: usual oscil- [10] A.A. Grib, S.G. Mamayev, and V.M. Moste-
lations and unstable growing modes. So that one panenko, Vacuum Quantum Effects in Strong
must introduce indefinite metric for their quantiza- Fields (Friedmann Lab. Publ., St.Petersburg,
tion. Correctquantizationneedstakingintoaccount 1994).
GRIB, PAVLOV 9
[11] A.A. Grib and V.Yu. Dorofeev, Int. J. Mod. [23] S.G. Mamayev and N.N. Trunov, Yadernaya
Phys. D 3, 731–738 (1994). Creation of parti- Fizika 37, 1603–1612 (1983); Sov. J. Nucl.
cles and entropy in the early Friedmann Uni- Phys. 37, 952-957 (1983). Space-time descrip-
verse. tion of particle creation in gravitational and
electromagnetic fields.
[12] N.D. Birrell and P.C.W. Davies, Quantum
fields in curved space (Cambridge University [24] Yu.V. Pavlov, Grav. Cosmol. 14, 314–320
Press, Cambridge, 1982). (2008). Space-time description of scalar par-
ticle creation by a homogeneous isotropic
[13] Yu.V.Pavlov,Teor. Matem. Fiz. 126, 115–124 gravitational field.
(2001) [Engl. transl.: Theor. Math. Phys. 126,
92–100 (2001)]. Nonconformal scalar field in a [25] Yu.V. Pavlov, Grav. Cosmol. 15, 341–344
homogeneous isotropic space and the Hamilto- (2009). On particles creation and renormal-
nian diagonalization method. ization in a cosmological model with a Big
Rip.
[14] R.M. Wald, Quantum field theory in curved
[26] A.A. Grib and Yu.V. Pavlov, Int. J. Mod.
spacetime and black hole thermodynamics (Uni-
Phys. D 11, 433–436 (2002). Superheavy parti-
versity of Chicago Press, Chicago, 1994).
cles in Friedmann cosmology and the dark mat-
[15] A.D. Linde, Particle Physics and Inflation- ter problem.
aryCosmology(HarwoodAcademic,NewYork,
[27] A.A. Grib and Yu.V. Pavlov, Int. J. Mod.
1990).
Phys. A 17, 4435–4439 (2002). Cold dark mat-
[16] Yu.V.Pavlov,Teor. Matem. Fiz. 140, 241–255 ter and primordial superheavy particles.
(2004) [Engl. transl.: Theor. Math. Phys.
[28] A.A.GribandYu.V.Pavlov,Mod. Phys. Lett.
140, 1095–1108 (2004)]. Renormalization and
A 23, 1151–1159(2008).Do active galactic nu-
dimensional regularization for a scalar field
clei convert dark matter into visible particles?
with Gauss-Bonnet-type coupling to curvature.
[29] A.A. Grib and Yu.V. Pavlov, Grav. Cosmol.
[17] Yu.V.Pavlov,Teor. Matem. Fiz. 174, 504–512
15, 44–48 (2009). Active galactic nuclei and
(2013) [Engl. transl.: Theor. Math. Phys.
transformation of dark matter into visible mat-
174, 438–445(2013)]. Exact solutions and par-
ter.
ticle creation for nonconformal scalar fields in
homogeneous isotropic cosmological models. [30] A.A. GribandV.Yu. Dorofeev,Grav. Cosmol.
16, 85–91(2010).Spontaneoussymmetrybreak-
[18] L.P. Grishchuk and V.M. Yudin, ing for long-wave gravitons in the early Uni-
J. Math. Phys. 21, 1168–1175 (1980). Con- verse.
formal coupling of gravitational wave field to
curvature.
[19] Yu.V. Pavlov Int. J. Mod. Phys. A 17,
1041–1044 (2002). Creation of the noncon-
formal scalar particles in nonstationary metric.
[20] T. Imamura, Phys. Rev. 118, 1430–1434
(1960). Quantized meson field in a classical
gravitational field.
[21] L. Parker. Phys. Rev. 183, 1057–1068 (1969).
Quantized Fields and Particle Creation in Ex-
panding Universe. I.
[22] A.A. Grib and Yu.V. Pavlov, Grav. Cosmol.
14, 1–7 (2008). Is dark matter the relic of the
primordialmatterthatcreatedthevisiblematter
of the Universe?