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Cosmological particle production
Volume 74A, number 3,4
PHYSICS LETTERS
12 November 1979
COSMOLOGICAL PARTICLE PRODUCTION ~ Angelika MULLER, Berndt MULLER and Walter GREINER Institut fir Theoretische Physik der Johann Wolfgang Goethe-Universität, Frankfurt am Main, Germany Received 13 July 1979
We apply the WKB approximation of path integrals to pair creation in expanding universes. Friedman cosmologies for arbitrary equations of state p = np are discussed. Observational data are in favor of a stiff equation of state (p p).
Many attempts have been made to calculate particle creation in the early universe [1—10]. Recently, Chitre and Hartle [11] have applied the method of path integral quantization to a universe with expansion law R(t) = t. To obtain explicit solutions, this. method requires analytic continuation techniques in
backward in time (as an antiparticle), then forward (as particle). If the functional S [x(a)] has a stationary point, i.e. if there exists a path 1(a), such that öS [1(a)] = 0, then this path alone describes the dominant contribulion to the full amplitude K(x0, x~). 1(a) is the path
the complex space—time manifold, which restricts the exactly solvable models to a few analytically tractable ones. In this note, we apply the WKB approximation to the evaluation of the path integral. We show that by choosing an appropriate path in the complex time plane, particle creation may be calculated with realistic expansion laws. In particular, we shall treat the Friedman cosmology with an equation of state p = ap, 0 ~ a < 1, covering the incoherent matter (a = 0) and ultrarelativistic Fermi gas (a = 1/3) as special cases. We now discuss the mathematical method. In the path-integral formalism the propagation of a particle from x~to x0 is described by an amplitude
taken in the classical limit of the quantum theory, and S [1(a)] is just the classical action Sd, i.e. the approximation is equivalent to the WKB-method. In the case of classically forbidden processes such as pair creation, of course, no such stationary path exists. However, it can be shown [12,13] that the path integral possesses a certain contour-independence in the complex coordinate manifold. In particular, the path may be chosen anywhere in the lower half complex a-plane (t-plane in the nonrelativistic case) [12] The greater freedom now allows finding stationary paths in many classically forbidden cases. This method was applied to tunneling [13] ,pair creation in electric fields [14] and to instanton solutions in field non-abelian gaugetime theories [15] .When the external varies with only, it suffices to treat the time-dependent part of the classical action and write
K(x0,x~)
j.—
E
paths
/ 1-S [x(a)] \~ exp~-
-
I
(1) .
where the summation runs over all possible paths x (a) connecting x~and x 0 ‘S stands for the classical action along each individual a denotes the to proper time. The amplitude for pairpath, creation (relative persistence of the vacuum) may be obtained by taking both x~, x 0 at t °° and lettmg the particle first propagate .
—~
*
This work has been supported by the Gesellschaft für Schwerionenforschung (GSI), and by the Deutsche
Forschungsgemeinschaft (DFG).
.
5cl
aS~1dt f—~---
=
+
real constant,
-
where the integration runs over a properly chosen -
path in the lower half of the t-plane. In the following we shall treat the expansion of an isotropic, homogeneous universe described by the Robertson—Walker metric 281
Volume 74A, number 3,4
PHYSICS LETTERS
(2) da2 = dt2 R(t)2{d~2+ ~2(K)df22}, with ~(— 1,0,1) = sinh(~,x~sin x), respectively. The time-differential of the action 5cl’ —
aS~
1/at
=
2/R2 (t) + ~2)
(3)
1/2,
±(k
obtained from the Hamilton—Jacobi equation, is independent of ic. Here u is the mass of the produced particle and the constant of motion, k, is defined so that k/R (t) is the physical momentum p (t) of the partide. The plus sign describes motion forward in time (particles), the minus sign motion backward in time (antiparticles). Then the calculation of particle creation reduces to the evaluation of the expression
w wo exp =
(_2 Im
f± (k2/R2
+ p2)1/2
dt).
(4)
12 November 1979
The classical action 5c1’ eq.with (3), acorresponds a quantum equation of motion conformallytoinvariant kinetic energy term. Rumpf [9] has pointed out that conformally coupled wave equations have WKB-type solutions near the singularity, whereas other equations have not. This has important quences: (1) Conformal theories allow for the definition of a WKB-vacuum state close to a singularity. (2) Particle production is finite in a universe with singularity. This is the finding of all authors who have investigated conformal theories [2,6,9—11] Minimally coupled scalar fields give infmite production rates [181. Since all fundamental particles of nature (leptons, photons, quarks, etc.; note that gravitons cannot be discussed in this context) obey wave equations of the conformal type, we have chosen to use expression (3). For a nonsingular universe approaching an adiabatic -
limit for
t -÷ —
00,
the meaning of eq. (4) is clear: it de-
The vacuum persistence probability W 0 is determined by the overall normalization condition [11] We may put W0 = 1, because the deviation from unity is of the~ same order as the relative error in the WKB-approximation. If i~does not have poles within the region {Im t < 0} of the complex plane, the branch point(s) of as/at is (are) always given by R = ik/~u.The contour is chosen as in fig. 1. The path runs backward from t =00 on the (antiparticle) Riemann sheet with the negative sign of aS/at, passes around the branch point onto the (particle) sheet with the positive sign and back to t = oo• Whenever an analytical expansion law is given, the t-integration may be replaced by an integration over R, and it suffices to know the expansion velocity A = f(R) as a function of the curvature radius R. .
—
rn R (I)
__________________
~ReR(t)
-
SingulQrity I - —
‘
lJThe integration path starts at R
Fig. 1. = + on the antipartide-sheet (dashed line). Below the branch point —ik/~the to R passes = + oo. Observe the singularity R = and 0 isthen avoided. path onto the that particie-sheet (solidatline) back
282
scribes pair production when no particles were present in the infinite past. A universe with singularity will be described as a limit of nonsingular cosmologies: going backward in time, we assume that the present expansion halts at t = Oat a minimal radiusR0, and is preceded by a contraction phase governed by the analytically continued expansion law R (t). If R (t) is singu. lar, i.e.R (0) = 0, this may bewith achieved formally by 2 + t~)hI2 t replacing t by ±(t 0 defined by R (t0) = R0. One finds that the particular choice of R0 affects the creation rate only for particles with k ~ I.LR0. For R0 —~0this becomes a vanishing part of the total production rate, so that the transition to the limiting singular cosmology is well delmed. Obviously, our calculation does not correspond to starting from a vacuum state in the WKB sense at the singularity. Selecting an initial vacuum state at t = 0 would result in a spectrum [9] falling off with some power of k, superimposed expression. additionalofspectrum would on notour originate from This the dynamics the expansion process, but would be due to the uncertainty introduced by the short time interval available for the definition of the initial state. We now apply eq. (4) to a number of special cases: (a) Friedman universe filled with ultra-relativistic Fermi gas. From the Friedman equations and the equation of state,2 p— =K]~p, the expansion law 1/2 (5) A = [(D0/R)
Volume 74A, number 3,4
is obtained. Do is an integration constant related to the mass of the universe. Insertion in eq. (4) leads to In It’/& = - nDopz2 2F1 (5, ; ; 2;-KZ2),
(6)
with z = k/Do& The asymptotic number of particles per unit coordinate volume is obtained by integration over k (with IV0 = 1): _+!. = 2s+l Sk2 dk W(k) d3x 2n2 = (2st 1)(2n)-3
(D,&3/2
_#k(k/@+,)(3”+1)/2 W In To = ry(3-3~)/4)ry(7t3cl)/4)
I+
15K
32(~rD&~
L
_@.!I-!I-.-.-_-.-.-.-. paiii inur!Mse
80
60
u!trarelativistic Fermigos
I,
I
0
+
*”I ’
(f-0
’
over k
(@u)(34+1)l(oL+l) F(2Lul+l)) (9)
r((7t 3cg4))aa+l).
We note that the expansion law R(t) = t is formally obtained by the unphysical value (Y= -l/3. Our result in this case is identical with that of Chitre and Hartle [ 121 obtained from an integration over all paths. The dependence on Do and p may be qualita-
I
I,
0.5
1
I,
1’ I
Pi. 2. The number of created pairs in a closed universe is plotted versus the value of the parameter cyin the equation of state p = cyp.
tively understood by reasoning that pair production occurs when the tidal forces are able to supply the necessary energy 21.(over a distance of the compton wavelength h = p-l. Taking the largest radius of the universe where this condition is satisfied and counting the number of cells with volume X3 one finds precisely the combination of Do and p in eq. (9). In fig. 2 the function N = N(o) is plotted for a closed universe (K = t 1). In order to account for the observed total particle number, the equation of state must be p > 4p/5, which can only hold in strongly interacting systems [ 171 . From our discussion of the validity of the approximation it is clear that pair creation, observed in real time, mainly occurs for R
containing eq. (6) as a special case. Integration gives the total numberof particles:
x (n-3/2 r ((3 -3~q4)
log N
(7)
where s is the spin of the produced particles. For fermions in a closed universe (K = t 1) we have /d3x = 2n2 and N- 1063. Since Dop - 1041 for all known baryons, the correction term is negligible, rendering diV/d3x independent of K. We conclude that the global topological nature of the universe (closed, open, etc.) is irrelevant for the particle production. It is therefore sufficient to treat the case K = 0. It is also clear that the equation of state p = p/3 cannot account by far for the actual particle content of the universe (~10~~). (b) Friedmann universe (K = 0) with p = arp. The independence of the global topology (i.e. K) allows discussing more general equations of state. We investigate the class p = cupwith 0 < (Y< 1, thus covering the whole range of physically allowed sound velocities 0 < (dp/dp)1/2 = us < 1. Under these assumptions the probability W(k) for pair production is found to be:
-5-&
12 November 1979
PHYSICS LETTERS
(10)
This value may be converted into an average radius i? for particle creation and hence to the density at that time. Using recent numbers [ 191 we find that production occurred at nuclear density p. for 0.8 < (Y< 1, and at much higher density (1015po) for cr = l/3. Also, H may be converted into the average energy (today) per particle E~/&~JD~)~/(~~+~). In general, the original spectrum is not thermal, but the average energy corresponds to about 3 K for (Y= 1 or slightly below, and a far too high temperature for (Y= l/3. As we discussed above, the influence of a minimal radius R. of the universe may be estimated by restricting the spectrum of eq. (8) to k 2 p R,. When R. Q E/p, the effect is negligible, but for larger R. 283
Volume 74A, number 3,4
PHYSICS LETTERS
the total particle number becomes effectively damped 3(°~)I2). This factor by a factor exp(—pD may be interpreted as0a(R0/D0) chemical potential related to the mass of the produced particles [10].
12 November 1979
J. Hofmann and thank U. Heinz, D. Margetan and M. Soffel for careful reading of the manuscript,
[5] Ya. Zel’dovich and A. Starobinskii, Zh. Eksp. Teor. Fiz. 61(1972) 2161 lSov. Phys. JETP 34 (1972) 1159]. [6] S.G. Mamaev, V.M. Mostepanenko and A.A. Starobinskii, Zh. Eksp. Teor. Fiz. 70 (1976) 1577 [Soy.Phys. JETP 43(1976)]. [7] G. Schafer and H. Dehnen, Astron. Astrophys. 54 (1977) 823; 61(1977) 671. 18] H. Rumpf, Nuovo Cimento 35B (1976) 321. [9] H. Rumpf, Ph.D. Thesis, Vienna (1977). 1101 J. Audretsch and G. Schlfer, J. Phys. All (1978) 1583; Phys. Lett. 66A (1978) 459. [11] D.M. Chitre and J.B. Hartle, Phys. Rev. D16 (1977) 251. [12] D.W. McLaughlin, J. Math. Phys. 13 (1972) 1099. [13] K.F. Freed, J. Chem. Phys. 56 (1972) 692. [14] M.S. Marmnov and V.S. Popov, Fortschr. Phys. 25 (1977) 373. [15] A.M. Polyakov, Phys. Lett. 59B (1975) 82; G.’t Hooft, Phys. Rev. Lett. 37 (1976) 8. [16] For this discussion cf. any textbook on quantum mechanics, e.g. A. Messiah, Quantum mechanics, Vol. 1 (North Holland, Amsterdam, 1961) 231 ff. [17] Y.B. Zel’dovich, Zh. Eksp. Teor. Fiz.pp. 41(1961)1609 [Soy.Phys. JETP 14 (1962) 11431; G. Kalman, Phys. Rev. 158 (1967) 144; J. Walecka, Ann. Phys. 83 (1974) 491. 1181 L. Parker, The production of elementary particles by strong gravitational in: Asymptotic structure of space—time, eds. F.P.fields, Esposito and L. Witten (Plenum
References
Press, New York, 1977). [19] D.N. Schramm and R.V. Wagoner, Ann. Rev. Nuci. Sci.
In conclusion, we have set forth a general approximation method allowing computing the number of created particles in the universe in very general cosmol-
ogies. A comparison with experimental data favors a stiff equation of state for matter at high density (p—~p).Such an equation of state is also preferred from entropy considerations, as was recently pointed out by Barrows [20] . We have not touched the question of the backreaction. This and the effect of anisotropic expansion will be the subject of a forthcoming publication. Also, the question of particle—antiparticle asymmetry is beyond the reach of our discussion, but existing attempts to include interactions violating particle—antiparticle symmetry [21,22] probably could be incorporated in the scenario. We acknowledge fruitful discussions with
27 [1] 0. Nachtmann, Z. Phys. 208 (1968) 113. [2] L. Parker, Phys. Rev. Lett. 21(1968) 562; 28 (1972) 705; Phys. Rev. 183 (1969) 1057; D3 (1971) 346. [3] Urbantke and Sexl, Phys. Rev. 179 (1969) 1247. [4] Ya. Zel’dovich, Zh. Eksp. Teor. Fiz. Pis’ma (1970) 443 [Soy. Phys. JETP Lett. 12 (1970) 307].
284
(1977) 37.
[20] J.D. Barrows, Nature 272 (1978) 211. [21] M. Yoshimura, Phys. Rev. Lett. 41(1978) 281. [22] N.J. Papastamatiou and L. Parker, preprint UWM-486779-1.
PHYSICS LETTERS
12 November 1979
COSMOLOGICAL PARTICLE PRODUCTION ~ Angelika MULLER, Berndt MULLER and Walter GREINER Institut fir Theoretische Physik der Johann Wolfgang Goethe-Universität, Frankfurt am Main, Germany Received 13 July 1979
We apply the WKB approximation of path integrals to pair creation in expanding universes. Friedman cosmologies for arbitrary equations of state p = np are discussed. Observational data are in favor of a stiff equation of state (p p).
Many attempts have been made to calculate particle creation in the early universe [1—10]. Recently, Chitre and Hartle [11] have applied the method of path integral quantization to a universe with expansion law R(t) = t. To obtain explicit solutions, this. method requires analytic continuation techniques in
backward in time (as an antiparticle), then forward (as particle). If the functional S [x(a)] has a stationary point, i.e. if there exists a path 1(a), such that öS [1(a)] = 0, then this path alone describes the dominant contribulion to the full amplitude K(x0, x~). 1(a) is the path
the complex space—time manifold, which restricts the exactly solvable models to a few analytically tractable ones. In this note, we apply the WKB approximation to the evaluation of the path integral. We show that by choosing an appropriate path in the complex time plane, particle creation may be calculated with realistic expansion laws. In particular, we shall treat the Friedman cosmology with an equation of state p = ap, 0 ~ a < 1, covering the incoherent matter (a = 0) and ultrarelativistic Fermi gas (a = 1/3) as special cases. We now discuss the mathematical method. In the path-integral formalism the propagation of a particle from x~to x0 is described by an amplitude
taken in the classical limit of the quantum theory, and S [1(a)] is just the classical action Sd, i.e. the approximation is equivalent to the WKB-method. In the case of classically forbidden processes such as pair creation, of course, no such stationary path exists. However, it can be shown [12,13] that the path integral possesses a certain contour-independence in the complex coordinate manifold. In particular, the path may be chosen anywhere in the lower half complex a-plane (t-plane in the nonrelativistic case) [12] The greater freedom now allows finding stationary paths in many classically forbidden cases. This method was applied to tunneling [13] ,pair creation in electric fields [14] and to instanton solutions in field non-abelian gaugetime theories [15] .When the external varies with only, it suffices to treat the time-dependent part of the classical action and write
K(x0,x~)
j.—
E
paths
/ 1-S [x(a)] \~ exp~-
-
I
(1) .
where the summation runs over all possible paths x (a) connecting x~and x 0 ‘S stands for the classical action along each individual a denotes the to proper time. The amplitude for pairpath, creation (relative persistence of the vacuum) may be obtained by taking both x~, x 0 at t °° and lettmg the particle first propagate .
—~
*
This work has been supported by the Gesellschaft für Schwerionenforschung (GSI), and by the Deutsche
Forschungsgemeinschaft (DFG).
.
5cl
aS~1dt f—~---
=
+
real constant,
-
where the integration runs over a properly chosen -
path in the lower half of the t-plane. In the following we shall treat the expansion of an isotropic, homogeneous universe described by the Robertson—Walker metric 281
Volume 74A, number 3,4
PHYSICS LETTERS
(2) da2 = dt2 R(t)2{d~2+ ~2(K)df22}, with ~(— 1,0,1) = sinh(~,x~sin x), respectively. The time-differential of the action 5cl’ —
aS~
1/at
=
2/R2 (t) + ~2)
(3)
1/2,
±(k
obtained from the Hamilton—Jacobi equation, is independent of ic. Here u is the mass of the produced particle and the constant of motion, k, is defined so that k/R (t) is the physical momentum p (t) of the partide. The plus sign describes motion forward in time (particles), the minus sign motion backward in time (antiparticles). Then the calculation of particle creation reduces to the evaluation of the expression
w wo exp =
(_2 Im
f± (k2/R2
+ p2)1/2
dt).
(4)
12 November 1979
The classical action 5c1’ eq.with (3), acorresponds a quantum equation of motion conformallytoinvariant kinetic energy term. Rumpf [9] has pointed out that conformally coupled wave equations have WKB-type solutions near the singularity, whereas other equations have not. This has important quences: (1) Conformal theories allow for the definition of a WKB-vacuum state close to a singularity. (2) Particle production is finite in a universe with singularity. This is the finding of all authors who have investigated conformal theories [2,6,9—11] Minimally coupled scalar fields give infmite production rates [181. Since all fundamental particles of nature (leptons, photons, quarks, etc.; note that gravitons cannot be discussed in this context) obey wave equations of the conformal type, we have chosen to use expression (3). For a nonsingular universe approaching an adiabatic -
limit for
t -÷ —
00,
the meaning of eq. (4) is clear: it de-
The vacuum persistence probability W 0 is determined by the overall normalization condition [11] We may put W0 = 1, because the deviation from unity is of the~ same order as the relative error in the WKB-approximation. If i~does not have poles within the region {Im t < 0} of the complex plane, the branch point(s) of as/at is (are) always given by R = ik/~u.The contour is chosen as in fig. 1. The path runs backward from t =00 on the (antiparticle) Riemann sheet with the negative sign of aS/at, passes around the branch point onto the (particle) sheet with the positive sign and back to t = oo• Whenever an analytical expansion law is given, the t-integration may be replaced by an integration over R, and it suffices to know the expansion velocity A = f(R) as a function of the curvature radius R. .
—
rn R (I)
__________________
~ReR(t)
-
SingulQrity I - —
‘
lJThe integration path starts at R
Fig. 1. = + on the antipartide-sheet (dashed line). Below the branch point —ik/~the to R passes = + oo. Observe the singularity R = and 0 isthen avoided. path onto the that particie-sheet (solidatline) back
282
scribes pair production when no particles were present in the infinite past. A universe with singularity will be described as a limit of nonsingular cosmologies: going backward in time, we assume that the present expansion halts at t = Oat a minimal radiusR0, and is preceded by a contraction phase governed by the analytically continued expansion law R (t). If R (t) is singu. lar, i.e.R (0) = 0, this may bewith achieved formally by 2 + t~)hI2 t replacing t by ±(t 0 defined by R (t0) = R0. One finds that the particular choice of R0 affects the creation rate only for particles with k ~ I.LR0. For R0 —~0this becomes a vanishing part of the total production rate, so that the transition to the limiting singular cosmology is well delmed. Obviously, our calculation does not correspond to starting from a vacuum state in the WKB sense at the singularity. Selecting an initial vacuum state at t = 0 would result in a spectrum [9] falling off with some power of k, superimposed expression. additionalofspectrum would on notour originate from This the dynamics the expansion process, but would be due to the uncertainty introduced by the short time interval available for the definition of the initial state. We now apply eq. (4) to a number of special cases: (a) Friedman universe filled with ultra-relativistic Fermi gas. From the Friedman equations and the equation of state,2 p— =K]~p, the expansion law 1/2 (5) A = [(D0/R)
Volume 74A, number 3,4
is obtained. Do is an integration constant related to the mass of the universe. Insertion in eq. (4) leads to In It’/& = - nDopz2 2F1 (5, ; ; 2;-KZ2),
(6)
with z = k/Do& The asymptotic number of particles per unit coordinate volume is obtained by integration over k (with IV0 = 1): _+!. = 2s+l Sk2 dk W(k) d3x 2n2 = (2st 1)(2n)-3
(D,&3/2
_#k(k/@+,)(3”+1)/2 W In To = ry(3-3~)/4)ry(7t3cl)/4)
I+
15K
32(~rD&~
L
_@.!I-!I-.-.-_-.-.-.-. paiii inur!Mse
80
60
u!trarelativistic Fermigos
I,
I
0
+
*”I ’
(f-0
’
over k
(@u)(34+1)l(oL+l) F(2Lul+l)) (9)
r((7t 3cg4))aa+l).
We note that the expansion law R(t) = t is formally obtained by the unphysical value (Y= -l/3. Our result in this case is identical with that of Chitre and Hartle [ 121 obtained from an integration over all paths. The dependence on Do and p may be qualita-
I
I,
0.5
1
I,
1’ I
Pi. 2. The number of created pairs in a closed universe is plotted versus the value of the parameter cyin the equation of state p = cyp.
tively understood by reasoning that pair production occurs when the tidal forces are able to supply the necessary energy 21.(over a distance of the compton wavelength h = p-l. Taking the largest radius of the universe where this condition is satisfied and counting the number of cells with volume X3 one finds precisely the combination of Do and p in eq. (9). In fig. 2 the function N = N(o) is plotted for a closed universe (K = t 1). In order to account for the observed total particle number, the equation of state must be p > 4p/5, which can only hold in strongly interacting systems [ 171 . From our discussion of the validity of the approximation it is clear that pair creation, observed in real time, mainly occurs for R
containing eq. (6) as a special case. Integration gives the total numberof particles:
x (n-3/2 r ((3 -3~q4)
log N
(7)
where s is the spin of the produced particles. For fermions in a closed universe (K = t 1) we have /d3x = 2n2 and N- 1063. Since Dop - 1041 for all known baryons, the correction term is negligible, rendering diV/d3x independent of K. We conclude that the global topological nature of the universe (closed, open, etc.) is irrelevant for the particle production. It is therefore sufficient to treat the case K = 0. It is also clear that the equation of state p = p/3 cannot account by far for the actual particle content of the universe (~10~~). (b) Friedmann universe (K = 0) with p = arp. The independence of the global topology (i.e. K) allows discussing more general equations of state. We investigate the class p = cupwith 0 < (Y< 1, thus covering the whole range of physically allowed sound velocities 0 < (dp/dp)1/2 = us < 1. Under these assumptions the probability W(k) for pair production is found to be:
-5-&
12 November 1979
PHYSICS LETTERS
(10)
This value may be converted into an average radius i? for particle creation and hence to the density at that time. Using recent numbers [ 191 we find that production occurred at nuclear density p. for 0.8 < (Y< 1, and at much higher density (1015po) for cr = l/3. Also, H may be converted into the average energy (today) per particle E~/&~JD~)~/(~~+~). In general, the original spectrum is not thermal, but the average energy corresponds to about 3 K for (Y= 1 or slightly below, and a far too high temperature for (Y= l/3. As we discussed above, the influence of a minimal radius R. of the universe may be estimated by restricting the spectrum of eq. (8) to k 2 p R,. When R. Q E/p, the effect is negligible, but for larger R. 283
Volume 74A, number 3,4
PHYSICS LETTERS
the total particle number becomes effectively damped 3(°~)I2). This factor by a factor exp(—pD may be interpreted as0a(R0/D0) chemical potential related to the mass of the produced particles [10].
12 November 1979
J. Hofmann and thank U. Heinz, D. Margetan and M. Soffel for careful reading of the manuscript,
[5] Ya. Zel’dovich and A. Starobinskii, Zh. Eksp. Teor. Fiz. 61(1972) 2161 lSov. Phys. JETP 34 (1972) 1159]. [6] S.G. Mamaev, V.M. Mostepanenko and A.A. Starobinskii, Zh. Eksp. Teor. Fiz. 70 (1976) 1577 [Soy.Phys. JETP 43(1976)]. [7] G. Schafer and H. Dehnen, Astron. Astrophys. 54 (1977) 823; 61(1977) 671. 18] H. Rumpf, Nuovo Cimento 35B (1976) 321. [9] H. Rumpf, Ph.D. Thesis, Vienna (1977). 1101 J. Audretsch and G. Schlfer, J. Phys. All (1978) 1583; Phys. Lett. 66A (1978) 459. [11] D.M. Chitre and J.B. Hartle, Phys. Rev. D16 (1977) 251. [12] D.W. McLaughlin, J. Math. Phys. 13 (1972) 1099. [13] K.F. Freed, J. Chem. Phys. 56 (1972) 692. [14] M.S. Marmnov and V.S. Popov, Fortschr. Phys. 25 (1977) 373. [15] A.M. Polyakov, Phys. Lett. 59B (1975) 82; G.’t Hooft, Phys. Rev. Lett. 37 (1976) 8. [16] For this discussion cf. any textbook on quantum mechanics, e.g. A. Messiah, Quantum mechanics, Vol. 1 (North Holland, Amsterdam, 1961) 231 ff. [17] Y.B. Zel’dovich, Zh. Eksp. Teor. Fiz.pp. 41(1961)1609 [Soy.Phys. JETP 14 (1962) 11431; G. Kalman, Phys. Rev. 158 (1967) 144; J. Walecka, Ann. Phys. 83 (1974) 491. 1181 L. Parker, The production of elementary particles by strong gravitational in: Asymptotic structure of space—time, eds. F.P.fields, Esposito and L. Witten (Plenum
References
Press, New York, 1977). [19] D.N. Schramm and R.V. Wagoner, Ann. Rev. Nuci. Sci.
In conclusion, we have set forth a general approximation method allowing computing the number of created particles in the universe in very general cosmol-
ogies. A comparison with experimental data favors a stiff equation of state for matter at high density (p—~p).Such an equation of state is also preferred from entropy considerations, as was recently pointed out by Barrows [20] . We have not touched the question of the backreaction. This and the effect of anisotropic expansion will be the subject of a forthcoming publication. Also, the question of particle—antiparticle asymmetry is beyond the reach of our discussion, but existing attempts to include interactions violating particle—antiparticle symmetry [21,22] probably could be incorporated in the scenario. We acknowledge fruitful discussions with
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