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Baryon asymmetry in the very early universe
Volume 81B, number 3,4
PHYSICS LETTERS
26 February 1979
BARYON ASYMMETRY 1N THE VERY EARLY UNIVERSE S. DIMOPOULOS Physics Department, Columbia University, New York, NY, USA and L. SUSSKIND Department of Physics, Stanford University, Stanford, CA 94305, USA Received 27 November 1978
We suggest a three stage mechanism for the development of a net baryon excess in the universe. The first phase is an initial heating due to strong gravitational fluctuations during the first 10 .43 s. This is followed by a stage in which baryon and CP violating processes produce a large excess via a mechanism suggested by Weinberg. Finally thermalization dissipates most of the initially produced baryon number leaving a small excess.
Several authors have recently proposed that the excess of matter to antimatter in the universe may arise because o f baryon and CP violation during the very early stages of expansion [ 1 - 3 ] . In this note we will suggest a possible framework for such a phenomenon based on baryon violating processes in grand unified theories o f the kind suggested by Georgi and Glashow, Pati and Salam and others [3]. In these theories, baryon violation is mediated b y superheavy gauge bosons called X which typically have mass ~ 1015-1019 GeV. To begin with we remind the reader of the necessary conditions for baryon asymmetry to occur [2]. In addition to baryon violating processes it is obviously essential to have C and CP violation. Otherwise the baryon violating processes will have random direction in different, causally unconnected regions and no net excess will occur. TCP invariance imposes important constraints on the rate of expansion during the phase where the excess is produced [2]. In fact TCP insures that in thermal equilibrium the average baryon density must vanish. To indicate the importance of this remark, let us suppose that the universe was born with some nonvanishing baryon density. Furthermore, suppose the expansion ceased at a temperature, high enough to allow baryon violations to occur ( T ~ M x ) . The system 416
would shortly come to thermal equilibrium due to scattering and would lose any excess baryon number through thermalization. Therefore the universe must cool sufficiently rapidly to shut down the baryon violation before it is lost through dissipation. In this paper we will suggest a three stage evolution during which the baryon number of the universe is created. We will see that the observed baryon excess places a limit on the mass o f the x-boson which is consistent with the indications from grand unified schemes. Stage I: The hot initial soup. At times of order the Planck time (10 -44 s) the universe is a hot soup consisting o f all elementary particle species including gravitons. The strong gravitational interactions bring the system to equilibrium at a temperature o f order Mp, the Planck mass (1019 GeV). No significant baryon excess yet exists. Stage II: The universe cools to a temperature ~ M x. As T passes through this region thermal equilibrium is not maintained if the baryon violating forces are weak. Specifically, if the time A t for the temperature to pass through a region of width ~ M x is small by comparison with the time needed to relax to equilibrium, a net excess of baryons may exist at the end o f this phase. A particular mechanism for this excess has been proposed by Weinberg [4]. Weinberg supposes that the only
Volume 81B, number 3, 4
PHYSICS LETTERS
important effect of cooling through the M x region is to lose the x particles by decays. When T < M x there is not enough kinetic energy among the decay products of x's to allow them to produce new x's. This mechanism alone, gives a baryon density per particle [4] A=(Nx/NT)6B
,
(1)
where A is the baryon excess per particle, N x is the number o f x ' s in the hot thermal soup at T > M x , 6 B is the net baryon gain per x decay and N T is the total number of particles. In thermal equilibrium N x / N T is of order the inverse of the total number o f elementary particle types with mass < T times the number of species of x-bosons. Including leptons, photons, intermediate vector bosons, quarks, gluons and x's this number is ~ 10 -1 . The quantity 5 B would vanish in a C or CP symmetric theory. Indeed, in lowest order perturbation theory it vanishes since the amplitudes for positive and negative change in B are complex conjugates. In higher order, interference between the first order and processes with final state CP violations can yield ~B =/=0. If we assume CP violations are of order a at high energy then by the end of stage 2 Weinberg's mechanism gives + 1 A = ( N x / N T ) f B ~ c~/10 ~ 10 - 3 .
(2)
The observed specific entropy per baryon in the present universe is of order 10 -9 . Assuming a quasistatic expansion and baryon conservation, the entropy per baryon is conserved, indicating that the ratio A may have been ~ 10 - 9 from the time at which baryon violation became insignificant. Accordingly, between the end of stage II and the time baryon violation ceased completely, the net baryon number may have diminished by a factor 10 -6. (Alternatively heating due to dissipative processes such as viscosity may have caused the specific entropy to increase by 106.) Thus we come to stage II1. Stage IIl: The dissipation of baryon number. In thermal equilibrium the net baryon number vanishes as a consequence of C P T [2]. As long as baryon violating processes have non-negligible strength, they
26 February 1979
will tend to dissipate the existing baryon number [2] through the inevitable second law which requires the system to return to equilibrium. For temperatures somewhat below Mx, real x's are not produced but virtual x's mediate baryon violating 4-Fermi interactions. These processes are effectively described by 4-Fermi couplings with coupling constants G ~ a/M2x .
(3)
These interactions are the ones which can potentially allow the system to thermalize and lose its baryon excess [2]. To estimate this effect we note that the baryon excess (deviation from equilibrium) satisfies dA/dt = -7(T) A.
(4)
The process is analogous to the decay of a current in a conductor. The decay constant 3' is like electrical resistance. In general it is temperature dependent and can be calculated from absorptive parts of thermal green's functions. It is proportional to baryon violating cross sections and therefore to G 2. From dimensional analysis 3" ~ G Z T 5 = ( a 2 / M 4 ) T 5 .
(5)
To integrate eq. (4) we must know how temperature depends on time. For this we assume a standard expand. ing radiation dominated universe satisfying R -1 d R / d t ~ [ ( 8 n / 3 ) G p ] l / 2 ,
(6)
where p = energy density and R is the cosmological scale parameter. In such a universe R ~ X/~-/14p ,
(7)
T~R
-1 ,
(8)
T ~ (Mp/t) 1/2 .
(9)
and
Thus eq. (4) takes the form dA/d t ~ - (a 2/M 4) ~ A ,
(1 O)
which integrates to A(t) = A(t0)
,1 This is probably an upper bound. In general it is possible for the lowest nonvanishing order to be of higher order in c~. In addition the strength of CP violating effects is proportional to CP violating phases which may be small.
× exp [ ( a 2 / M 4 ) M 5 [ 2 ( t - 3 / 2 - t0-3/2)]
(11)
(we have ignored factors of order 1 in the exponent). If we assume that t o corresponds to the end of stage 417
Volume 81B, number 3, 4
PHYSICS LETTERS
II and therefore to a temperature ~ M x then eq. (9) gives
to ~ M p / M 2 .
(12)
The excess baryon number left over as t -~ oo is then dx(oo) = A(t0) exp [-aMp/Mx] .
(13)
Eq. (2) gives an estimate for dx(t0) ~ 10 - 3 . Observationally A ( o o ) ~ 10 - 9 . If we assume most of the discrepancy arises from the effects of baryon violating thermalization then
26 February 1979
nitude too large. Secondly, thermalization will dissipate the produced baryon excess. An important result is that a nonvanishing excess remains at t ~ ~o. This excess is very sensitive to M x. For M x ~ ot2Mp the baryon excess becomes completely negligible; for M x >> cz2Mp it seems that too large an excess may result. We are indebted to S. Weinberg for suggesting the mechanism o f x decay as the primary mechanism for baryon excess.
exp [-a2Mp/Mx] ~ 10 - 6 ,
References
or
a2mp/Mx~ 10,
M x~mp/lOa 2 ~ lO-5Mp.
(14)
Conclusions. In view of the computational uncertainties the estimate o f eq. (14) should not be taken as a serious quantitative result. The lesson to learn is qualitative. First of all, Weinberg's mechanism gives a baryon excess during the stage when the temperature decreases below M x. This excess is independent o f M x as noted by Weinberg. It may be several orders of mag-
418
[1] L. Parker, private communication; M. Yoshimura, Tohoko Univ. preprint, TU/78/179 (March 1978). [21 S. Dimopoulos and L. Susskind, SLAC Pub. 2126 (June 1978). [3] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438; J.C. Pati and A. Salam, Phys. Rev. D8 (1973) 1240. [4] S. Weinberg, private communication.
PHYSICS LETTERS
26 February 1979
BARYON ASYMMETRY 1N THE VERY EARLY UNIVERSE S. DIMOPOULOS Physics Department, Columbia University, New York, NY, USA and L. SUSSKIND Department of Physics, Stanford University, Stanford, CA 94305, USA Received 27 November 1978
We suggest a three stage mechanism for the development of a net baryon excess in the universe. The first phase is an initial heating due to strong gravitational fluctuations during the first 10 .43 s. This is followed by a stage in which baryon and CP violating processes produce a large excess via a mechanism suggested by Weinberg. Finally thermalization dissipates most of the initially produced baryon number leaving a small excess.
Several authors have recently proposed that the excess of matter to antimatter in the universe may arise because o f baryon and CP violation during the very early stages of expansion [ 1 - 3 ] . In this note we will suggest a possible framework for such a phenomenon based on baryon violating processes in grand unified theories o f the kind suggested by Georgi and Glashow, Pati and Salam and others [3]. In these theories, baryon violation is mediated b y superheavy gauge bosons called X which typically have mass ~ 1015-1019 GeV. To begin with we remind the reader of the necessary conditions for baryon asymmetry to occur [2]. In addition to baryon violating processes it is obviously essential to have C and CP violation. Otherwise the baryon violating processes will have random direction in different, causally unconnected regions and no net excess will occur. TCP invariance imposes important constraints on the rate of expansion during the phase where the excess is produced [2]. In fact TCP insures that in thermal equilibrium the average baryon density must vanish. To indicate the importance of this remark, let us suppose that the universe was born with some nonvanishing baryon density. Furthermore, suppose the expansion ceased at a temperature, high enough to allow baryon violations to occur ( T ~ M x ) . The system 416
would shortly come to thermal equilibrium due to scattering and would lose any excess baryon number through thermalization. Therefore the universe must cool sufficiently rapidly to shut down the baryon violation before it is lost through dissipation. In this paper we will suggest a three stage evolution during which the baryon number of the universe is created. We will see that the observed baryon excess places a limit on the mass o f the x-boson which is consistent with the indications from grand unified schemes. Stage I: The hot initial soup. At times of order the Planck time (10 -44 s) the universe is a hot soup consisting o f all elementary particle species including gravitons. The strong gravitational interactions bring the system to equilibrium at a temperature o f order Mp, the Planck mass (1019 GeV). No significant baryon excess yet exists. Stage II: The universe cools to a temperature ~ M x. As T passes through this region thermal equilibrium is not maintained if the baryon violating forces are weak. Specifically, if the time A t for the temperature to pass through a region of width ~ M x is small by comparison with the time needed to relax to equilibrium, a net excess of baryons may exist at the end o f this phase. A particular mechanism for this excess has been proposed by Weinberg [4]. Weinberg supposes that the only
Volume 81B, number 3, 4
PHYSICS LETTERS
important effect of cooling through the M x region is to lose the x particles by decays. When T < M x there is not enough kinetic energy among the decay products of x's to allow them to produce new x's. This mechanism alone, gives a baryon density per particle [4] A=(Nx/NT)6B
,
(1)
where A is the baryon excess per particle, N x is the number o f x ' s in the hot thermal soup at T > M x , 6 B is the net baryon gain per x decay and N T is the total number of particles. In thermal equilibrium N x / N T is of order the inverse of the total number o f elementary particle types with mass < T times the number of species of x-bosons. Including leptons, photons, intermediate vector bosons, quarks, gluons and x's this number is ~ 10 -1 . The quantity 5 B would vanish in a C or CP symmetric theory. Indeed, in lowest order perturbation theory it vanishes since the amplitudes for positive and negative change in B are complex conjugates. In higher order, interference between the first order and processes with final state CP violations can yield ~B =/=0. If we assume CP violations are of order a at high energy then by the end of stage 2 Weinberg's mechanism gives + 1 A = ( N x / N T ) f B ~ c~/10 ~ 10 - 3 .
(2)
The observed specific entropy per baryon in the present universe is of order 10 -9 . Assuming a quasistatic expansion and baryon conservation, the entropy per baryon is conserved, indicating that the ratio A may have been ~ 10 - 9 from the time at which baryon violation became insignificant. Accordingly, between the end of stage II and the time baryon violation ceased completely, the net baryon number may have diminished by a factor 10 -6. (Alternatively heating due to dissipative processes such as viscosity may have caused the specific entropy to increase by 106.) Thus we come to stage II1. Stage IIl: The dissipation of baryon number. In thermal equilibrium the net baryon number vanishes as a consequence of C P T [2]. As long as baryon violating processes have non-negligible strength, they
26 February 1979
will tend to dissipate the existing baryon number [2] through the inevitable second law which requires the system to return to equilibrium. For temperatures somewhat below Mx, real x's are not produced but virtual x's mediate baryon violating 4-Fermi interactions. These processes are effectively described by 4-Fermi couplings with coupling constants G ~ a/M2x .
(3)
These interactions are the ones which can potentially allow the system to thermalize and lose its baryon excess [2]. To estimate this effect we note that the baryon excess (deviation from equilibrium) satisfies dA/dt = -7(T) A.
(4)
The process is analogous to the decay of a current in a conductor. The decay constant 3' is like electrical resistance. In general it is temperature dependent and can be calculated from absorptive parts of thermal green's functions. It is proportional to baryon violating cross sections and therefore to G 2. From dimensional analysis 3" ~ G Z T 5 = ( a 2 / M 4 ) T 5 .
(5)
To integrate eq. (4) we must know how temperature depends on time. For this we assume a standard expand. ing radiation dominated universe satisfying R -1 d R / d t ~ [ ( 8 n / 3 ) G p ] l / 2 ,
(6)
where p = energy density and R is the cosmological scale parameter. In such a universe R ~ X/~-/14p ,
(7)
T~R
-1 ,
(8)
T ~ (Mp/t) 1/2 .
(9)
and
Thus eq. (4) takes the form dA/d t ~ - (a 2/M 4) ~ A ,
(1 O)
which integrates to A(t) = A(t0)
,1 This is probably an upper bound. In general it is possible for the lowest nonvanishing order to be of higher order in c~. In addition the strength of CP violating effects is proportional to CP violating phases which may be small.
× exp [ ( a 2 / M 4 ) M 5 [ 2 ( t - 3 / 2 - t0-3/2)]
(11)
(we have ignored factors of order 1 in the exponent). If we assume that t o corresponds to the end of stage 417
Volume 81B, number 3, 4
PHYSICS LETTERS
II and therefore to a temperature ~ M x then eq. (9) gives
to ~ M p / M 2 .
(12)
The excess baryon number left over as t -~ oo is then dx(oo) = A(t0) exp [-aMp/Mx] .
(13)
Eq. (2) gives an estimate for dx(t0) ~ 10 - 3 . Observationally A ( o o ) ~ 10 - 9 . If we assume most of the discrepancy arises from the effects of baryon violating thermalization then
26 February 1979
nitude too large. Secondly, thermalization will dissipate the produced baryon excess. An important result is that a nonvanishing excess remains at t ~ ~o. This excess is very sensitive to M x. For M x ~ ot2Mp the baryon excess becomes completely negligible; for M x >> cz2Mp it seems that too large an excess may result. We are indebted to S. Weinberg for suggesting the mechanism o f x decay as the primary mechanism for baryon excess.
exp [-a2Mp/Mx] ~ 10 - 6 ,
References
or
a2mp/Mx~ 10,
M x~mp/lOa 2 ~ lO-5Mp.
(14)
Conclusions. In view of the computational uncertainties the estimate o f eq. (14) should not be taken as a serious quantitative result. The lesson to learn is qualitative. First of all, Weinberg's mechanism gives a baryon excess during the stage when the temperature decreases below M x. This excess is independent o f M x as noted by Weinberg. It may be several orders of mag-
418
[1] L. Parker, private communication; M. Yoshimura, Tohoko Univ. preprint, TU/78/179 (March 1978). [21 S. Dimopoulos and L. Susskind, SLAC Pub. 2126 (June 1978). [3] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438; J.C. Pati and A. Salam, Phys. Rev. D8 (1973) 1240. [4] S. Weinberg, private communication.