140
Baryogenesis
where V is the potential obtained by combining the contributions from Vnr in
(4.254), VA in (4.256) and the mass term for t/>. It has the form
2 2
(a}"HI4J n
)
214J1 2n - 2
V = -cHI It/> I + nMn-3 + h.c. + IAI M2n-6
(4.258)
where HI is the (approximately constant) value of the Hubble parameter during
inflation and a, c are constants of O( 1). This, of course, is just the equation of a
damped oscillator and the important point is that, during inflation, it is close to
being critically damped. If c < 0, which corresponds to positive mass-squared for
t/>, V has a minimum at t/> = O. In this case, the average value of the field evolves
exponentially to t/> = 0 and the large value at the end of inflation needed to get
a baryon asymmetry is not achieved [72]. However, if c > 0, which, in general,
requires non-minimal Kiihler terms, V has a single minimum at t/>o, where
RH Mn-3 )1/(n-2)
It/>ol = ( ~~~l_ _
(4.259)
A
with f3 a numerical constant which depends on a, c. n. Thus, It/>o I is parametrically
between HI and M, which can easily be large. In the angular direction. the
potential varies as cos(arg a + arg A. + n arg t/» and has n degenerate minima.
Further. again because of the near critical damping. the field evolves rapidly to
one of these minima, providedc is not too small [72]. Thus, at the end of inflation,
the average value of the field has a large value with a well-defined phase, which
is constant over scales large compared with the horizon.
This sets the boundary condition for the next era. After inflation the universe
enters a matter-dominated era in which the Hubble constant is explicitly timedependent:
2
(4.260)
H= 31
(see section 1.4). The equation of motion is still given by (4.257), with V of
the form (4.258) but with H now given by (4.260). As t increases, H decreases,
so that the instantaneous minimum of V also decreases. Solving the equation of
motion reveals that t/> tracks just behind this decreasing minimum.
This evolution continues until H ..... m3/2 '" I TeV, where m3/2 is the
gravitino mass. At that point, the soft supersymmetry-breaking terms from the
hidden sector become comparable with those arising from inflation and, at later
times, dominate the evolution. The bidden-sector terms contribute a positive
mass-squared term for t/> as well as an A-term, both baving scales determined
by m3/2. Thus, the additional contribution to the potential V has the form
2 2
( Am 3/2 A t/>"
)
Vhs = m;It/>1 + nMn-3 + h.c.
(4.261)
where m", ..... m3/2 and A = 0(1). Consequently the equation of motion (4.257)
becomes underdamped as H decreases below m3/2. 1\vo important effects now
Baryogenesis
where V is the potential obtained by combining the contributions from Vnr in
(4.254), VA in (4.256) and the mass term for t/>. It has the form
2 2
(a}"HI4J n
)
214J1 2n - 2
V = -cHI It/> I + nMn-3 + h.c. + IAI M2n-6
(4.258)
where HI is the (approximately constant) value of the Hubble parameter during
inflation and a, c are constants of O( 1). This, of course, is just the equation of a
damped oscillator and the important point is that, during inflation, it is close to
being critically damped. If c < 0, which corresponds to positive mass-squared for
t/>, V has a minimum at t/> = O. In this case, the average value of the field evolves
exponentially to t/> = 0 and the large value at the end of inflation needed to get
a baryon asymmetry is not achieved [72]. However, if c > 0, which, in general,
requires non-minimal Kiihler terms, V has a single minimum at t/>o, where
RH Mn-3 )1/(n-2)
It/>ol = ( ~~~l_ _
(4.259)
A
with f3 a numerical constant which depends on a, c. n. Thus, It/>o I is parametrically
between HI and M, which can easily be large. In the angular direction. the
potential varies as cos(arg a + arg A. + n arg t/» and has n degenerate minima.
Further. again because of the near critical damping. the field evolves rapidly to
one of these minima, providedc is not too small [72]. Thus, at the end of inflation,
the average value of the field has a large value with a well-defined phase, which
is constant over scales large compared with the horizon.
This sets the boundary condition for the next era. After inflation the universe
enters a matter-dominated era in which the Hubble constant is explicitly timedependent:
2
(4.260)
H= 31
(see section 1.4). The equation of motion is still given by (4.257), with V of
the form (4.258) but with H now given by (4.260). As t increases, H decreases,
so that the instantaneous minimum of V also decreases. Solving the equation of
motion reveals that t/> tracks just behind this decreasing minimum.
This evolution continues until H ..... m3/2 '" I TeV, where m3/2 is the
gravitino mass. At that point, the soft supersymmetry-breaking terms from the
hidden sector become comparable with those arising from inflation and, at later
times, dominate the evolution. The bidden-sector terms contribute a positive
mass-squared term for t/> as well as an A-term, both baving scales determined
by m3/2. Thus, the additional contribution to the potential V has the form
2 2
( Am 3/2 A t/>"
)
Vhs = m;It/>1 + nMn-3 + h.c.
(4.261)
where m", ..... m3/2 and A = 0(1). Consequently the equation of motion (4.257)
becomes underdamped as H decreases below m3/2. 1\vo important effects now