Cosmology for string theorists, a crash course

93 Pages·2002·0.937 MB·English

by  Garcia-Bellido J.

ARNOWITT-DESER-MISNER FORMALISM
In the Arnowitt{Deser{Misner formalism the four dimensional metric
g is parametrized by the three-metric h and the lapse and shift func-
(cid:22)(cid:23) ij
tions N and Ni, which describe the evolution of time-like hypersurfaces,
2 ij
g = N + h N N ; g = g = N ; g = h : (1)
00 i j 0i i0 i ij ij
(cid:0)
The action for the in(cid:13)aton scalar (cid:12)eld with potential V ((cid:30)) in the ADM
formalism has the form
1 1
= d4xp g R (@(cid:30))2 V ((cid:30))
2 3
S Z (cid:0) 2(cid:20)2 (cid:0) 2 (cid:0)
4 5
1 1
4 p (3) ij 2 (cid:30) 2 i
= d xN h R + K K K + ((cid:5) ) (cid:30) (cid:30) V ((cid:30)) ;
2 ij i j 3
Z 2(cid:20)2 (cid:18) (cid:0) (cid:19) 2 (cid:20) (cid:0) (cid:21)(cid:0)
j
4 5
(2)
where (cid:20)2 = 8(cid:25) G = 8(cid:25)=M2 is the gravitational coupling de(cid:12)ning the
P
Planck mass and (cid:5)(cid:30) is the scalar-(cid:12)eld’s conjugate momentum
1
(cid:30) _ i
(cid:5) = ((cid:30) N (cid:30) ): (3)
i
N (cid:0)
j
Vertical bars denote three-space-covariant derivatives with connections
derived from h ; (3)R is the three-space curvature associated with the
ij
metric h , and K is the extrinsic curvature three-tensor
ij ij
1
_
K = (N + N h ); (4)
ij i j j i ij
2N (cid:0)
j j
where a dot denotes di(cid:11)erentiation with respect to the time coordinate.
The traceless part of a tensor is denoted by an overbar. In particular,
1 1
(cid:22) i p
K = K Kh ; K = K = N @ ln h : (5)
ij ij (cid:0) 3 ij i N (cid:20) i i (cid:0) t (cid:21)
j
The trace K is a generalization of the Hubble parameter, as will be shown
below. Variation of the action with respect to N and Ni yields the energy
and momentum constraint equations respectively
2
(3) (cid:22) (cid:22)ij 2 2
R + K K K + 2(cid:20) (cid:26) = 0;
ij
(cid:0) (cid:0) 3
(6)
2
(cid:22)j 2 (cid:30)
K K + (cid:20) (cid:5) (cid:30) = 0:
i j i i
(cid:0) 3
j j j
Variation with respect to h gives the dynamical gravitational (cid:12)eld
ij
equations, which can be separated into the trace and traceless parts
1 3 1 (cid:20)2
_ i i (3) (cid:22) (cid:22)ij 2
K N K i = Nji + N 0 R + KijK + K + T1 ; (7)
(cid:0) (cid:0) 4 4 2 2
j j B C
@ A
1
K(cid:22)_ i NkK(cid:22) i + Ni K(cid:22) k NkK(cid:22)i = N i + N k(cid:14)i
j j
j j k k j j k j k j
(cid:0) j j (cid:0) j (cid:0) j 3 j (8)
(3)(cid:22)i (cid:22) i 2 (cid:22)i
+ N R + KK (cid:20) T :
(cid:18) j j (cid:0) j(cid:19)
Variation with respect to (cid:30) gives the scalar-(cid:12)eld’s equation of motion
1 1 @V
_ (cid:30) i (cid:30) (cid:30) i i
((cid:5) N (cid:5) ) K(cid:5) N (cid:30) (cid:30) + = 0: (9)
i i j ij
N (cid:0) (cid:0) (cid:0) N (cid:0) @(cid:30)
j j j
The energy density on a constant-time hypersurface is
1
(cid:30) 2 i
(cid:26) = ((cid:5) ) + (cid:30) (cid:30) + V ((cid:30)); (10)
i j
2 (cid:20) (cid:21)
j
and the stress three-tensor
1
(cid:30) 2 k
T = (cid:30) (cid:30) + h ((cid:5) ) (cid:30) (cid:30) V ((cid:30)) : (11)
ij i j ij 2 k j 3
2 (cid:20) (cid:0) (cid:21) (cid:0)
j j j
4 5
It is extremely di(cid:14)cult to solve these highly nonlinear coupled equa-
tions in a cosmological scenario without making some approximations.
The usual approach is to assume homogeneity of the (cid:12)elds to give a
background solution and then linearize the equations to study deviations
from spatial uniformity. The smallness of cosmic microwave background
anisotropies gives some justi(cid:12)cation for this perturbative approach at
least in our local part of the Universe. However, there is no reason to
believe it will be valid on much larger scales. In fact, the stochastic ap-
proach to in(cid:13)ation suggests that the Universe is extremely inhomogeneous
on very large scales. Fortunately, in this framework one can coarse-grain
over a horizon distance and separate the short- from the long-distance
behavior of the (cid:12)elds, where the former communicates with the latter
through stochastic forces. The equations for the long-wavelength back-
ground (cid:12)elds are obtained by neglecting large-scale gradients, leading to
a self-consistent set of equations, as we will discuss in the next section.
SPATIAL GRADIENT EXPANSION
It is reasonable to expand in spatial gradients whenever the forces aris-
ing from time variations of the (cid:12)elds are much larger than forces from
spatial gradients. In linear perturbation theory one solves the perturba-
tion equations for evolution outside of the horizon: a typical time scale
is the Hubble time H 1, which is assumed to exceed the gradient scale
(cid:0)
a=k, where k is the comoving wave number of the perturbation. Since
we are interested in structures on scales larger than the horizon, it is
reasonable to expand in k=(aH). In particular, for in(cid:13)ation this is an
appropriate parameter of expansion since spatial gradients become ex-
ponentially negligible after a few e-folds of expansion beyond horizon
crossing, k = aH.
It is therefore useful to split the (cid:12)eld (cid:30) into coarse-grained long-
wavelength background (cid:12)elds (cid:30)(t;xj) and residual short-wavelength (cid:13)uc-
tuating (cid:12)elds (cid:14)(cid:30)(t;xj). There is a preferred timelike hypersurface within
the stochastic in(cid:13)ation approach in which the splitting can be made con-
sistently, but the de(cid:12)nition of the background (cid:12)eld will depend on the
choice of hypersurface, i.e. the smoothing is not gauge invariant. For
stochastic in(cid:13)ation the natural smoothing scale is the comoving Hubble
length (aH) 1 and the natural hypersurfaces are those on which aH is
(cid:0)
constant. In that case a fundamental di(cid:11)erence between between (cid:30) and
(cid:14)(cid:30) is that the short-wavelength components are essentially uncorrelated
at di(cid:11)erent times, while long-wavelength components are deterministi-
cally correlated through the equations of motion.
In order to solve the equations for the background (cid:12)elds, we will have
to make suitable approximations. The idea is to expand in the spatial
gradients of (cid:30) and to treat the terms that depend on the (cid:13)uctuating (cid:12)elds
as stochastic forces describing the connection between short- and long-
wavelength components. In this Section we will neglect the stochastic
forces due to quantum (cid:13)uctuations of the scalar (cid:12)elds and will derive the
approximate equation of motion for the background (cid:12)elds. We retain only
those terms that are at most (cid:12)rst order in spatial gradients, neglecting
such terms as (cid:30) i, (cid:30) (cid:30) i, (3)R, (3)Rj, and T(cid:22)j.
ij i j i i
j j
We will also choose the simplifying gauge Ni = 0 [Note that for the
evolution during in(cid:13)ation this is a consequence of the rapid expansion,
more than a gauge choice]. The evolution equation (8) for the traceless
part of the extrinsic curvature is then K(cid:22)_ i = NKK(cid:22)i. Using NK =
j j
@ lnph from (5), we (cid:12)nd the solution K(cid:22) i h 1=2, where h is the
t j (cid:0)
(cid:0) /
determinant of h . During in(cid:13)ation h 1=2 a 3, with a the overall
ij (cid:0) (cid:0)
(cid:17)
expansion factor, therefore K(cid:22)i decays extremely rapidly and can be set
j
to zero in the approximate equations. The most general form of the
three-metric with vanishing K(cid:22)i is
j
2 k k k k
h = a (t;x ) (cid:13) (x ); a(t;x ) exp[(cid:11)(t;x )]; (12)
ij ij
(cid:17)
where the time-dependent conformal factor is interpreted as a space-
dependent expansion factor. The time-independent three-metric (cid:13) , of
ij
unit determinant, describes the three-geometry of the conformally trans-
formed space. Since a(t;xk) is interpreted as a scale factor, we can sub-
stitute the trace K of the extrinsic curvature for the Hubble parameter
1 1
i i i
H(t;x ) (cid:11)_(t;x ) = K(t;x ): (13)
(cid:17) N(t;xi) (cid:0)3
The energy and momentum constraint equations (6) can now be writ-
ten as
(cid:20)2 1
2 (cid:30) 2
H = ((cid:5) ) + V ((cid:30)) ; (14)
2 3
3 2
4 5
(cid:20)2
(cid:30)
H = (cid:5) (cid:30) ; (15)
i i
(cid:0) 2
j j
together with the evolution equation (7)
1 3 (cid:20)2 (cid:20)2
_ 2 (cid:30) 2
H = H + T = ((cid:5) ) ; (16)
(cid:0)N 2 6 2
where T = 3 1((cid:5)(cid:30))2 V ((cid:30)) .
(cid:18)2 (cid:0) (cid:19)
In general, H is a function of the scalar (cid:12)eld and time, H(t;xi)
(cid:17)
H((cid:30)(t;xi);t). From the momentum constraint (15) we (cid:12)nd that the
scalar-(cid:12)eld’s momentum must obey
2 @H
(cid:30)
(cid:5) = : (17)
0 1
(cid:0)(cid:20)2 @(cid:30)
@ At
Comparing Eq. (16) with the time derivative of H((cid:30);t),
1 @H @H 1 @H
(cid:30)
= (cid:5) +
0 1 0 1 0 1
N @t @(cid:30) N @t
@ Ax @ At @ A(cid:30) (18)
(cid:20)2 1 @H
(cid:30) 2
= ((cid:5) ) + ;
0 1
(cid:0) 2 N @t
(cid:30)
@ A
@H
we (cid:12)nd = 0.
0 1
@t
(cid:30)
@ A
In fact, we should not be surprised since this is actually a consequence
of the general covariance of the theory.
On the other hand, the scalar (cid:12)eld’s equation (9) can be written to
(cid:12)rst order in spatial gradients as
1 @V
_ (cid:30) (cid:30)
(cid:5) + 3H(cid:5) + = 0: (19)
N @(cid:30)
We can also show that the conjugate momentum (cid:5)(cid:30) does not depend
explicitly on time, its only dependence comes through (cid:30). For this, di(cid:11)er-
entiate Eq. (14) w.r.t. (cid:30) to obtain
@(cid:5)(cid:30) @V
(cid:30) (cid:30)
(cid:5) 0 1 + 3H (cid:5) + = 0
@(cid:30) @(cid:30)
B Ct
@ A
and compare with (19), where
1 @(cid:5)(cid:30) @(cid:5)(cid:30)
_ (cid:30) (cid:30)
(cid:5) = (cid:5) 0 1 + 0 1 ; (20)
N @(cid:30) @t
B Ct B C(cid:30)
@ A @ A
@(cid:5)(cid:30)
which implies 0 1 = 0.
@t
B C(cid:30)
@ A
HAMILTON-JACOBI FORMALISM
We can now summarise what we have learned. The evolution of a
general foliation of space-time in the presence of a scalar (cid:12)eld (cid:13)uid can
be described solely in terms of the rate of expansion, which is a function
of the scalar (cid:12)eld only, H H((cid:30)(t;xi)), satisfying the Hamiltonian
(cid:17)
constraint equation:
2
2 @H
2 2
3H ((cid:30)) = + (cid:20) V ((cid:30)); (21)
0 1
(cid:20)2 @(cid:30)
@ A
together with the momentum constraint and the evolution of the scale
factor,
1 (cid:20)2 @H
_ (cid:30)
(cid:30) = = (cid:5) (22)
0 1
N (cid:0) 2 @(cid:30)
@ A
1
(cid:11)_ = H((cid:30)); (23)
N
as well as the dynamical gravitational and scalar (cid:12)eld evolution equations
1 2 @H 2 (cid:20)2
_ (cid:30) 2
H = = ((cid:5) ) ; (24)
0 1
N (cid:0) (cid:20)2 @(cid:30) (cid:0) 2
@ A
1
_ (cid:30) (cid:30)
(cid:5) = 3H (cid:5) V ((cid:30)): (25)
0
N (cid:0) (cid:0)
Therefore, H((cid:30)) is all you need to specify (to second order in (cid:12)eld
gradients) the evolution of the scale factor and the scalar (cid:12)eld during
in(cid:13)ation.
These equations are still too complicated to solve for arbitrary po-
tentials V ((cid:30)). In the next section we will (cid:12)nd solutions to them in the
slow-roll approximation.
SLOW-ROLL APPROXIMATION AND ATTRACTOR
Given the complete set of constraints and evolution equations (21) -
(25), we can construct the following parameters,
_ 2
H 2 H ((cid:30)) @ lnH
0
(cid:15) = 0 1 = ; (26)
(cid:17) (cid:0) H2 (cid:20)2 H((cid:30)) (cid:0) @ lna
B C
@ A
(cid:127)
(cid:30) 2 H ((cid:30)) @ lnH
00 0
(cid:14) = 0 1 = ; (27)
(cid:17) (cid:0) H(cid:30)_ (cid:20)2 H((cid:30)) (cid:0) @ lna
B C
@ A
in terms of which we can de(cid:12)ne the number of e-folds N as
e
a (cid:20)2 H((cid:30))d(cid:30)
end tend (cid:30)end
N ln = Hdt = : (28)
e
(cid:17) a(t) Zt (cid:0) 2 Z(cid:30) H ((cid:30))
0
In order for in(cid:13)ation to be predictive, you need to ensure that in(cid:13)ation
is independent of initial conditions. That is, one should ensure that there
is an attractor solution to the dynamics, such that di(cid:11)erences between
solutions corresponding to di(cid:11)erent initial conditions rapidly vanish.
Let H ((cid:30)) be an exact, particular, solution of the constraint equation
0
(21), either in(cid:13)ationary or not. Add to it a homogeneous linear pertur-
bation (cid:14)H((cid:30)), and substitute into (21). The linear perturbation equation
reads H ((cid:30))(cid:14)H ((cid:30)) = (3(cid:20)2=2)H (cid:14)H, whose general solution is
00 0 0
3(cid:20)2 H ((cid:30))d(cid:30)
(cid:30) 0
(cid:14)H((cid:30)) = (cid:14)H((cid:30) ) exp0 1 = (cid:14)H((cid:30) ) exp( 3(cid:1)N);
i i
2 Z(cid:30)i H ((cid:30)) (cid:0)
B 00 C
@ A
(29)
where (cid:1)N = N N > 0, and we have used (28) with the particular
i
(cid:0)
solution H ((cid:30)). This means that very quickly any deviation from the
0
attractor dies away. This ensures that we can e(cid:11)ectively reduce our two-
dimensional space ((cid:30);(cid:5)(cid:30)) to just a single trajectory in phase space.
As a consequence, regardless of the initial condition, the attractor
behaviour implies that late-time solutions are the same up to a constant
time shift, which cannot be measured.
AN EXAMPLE: POWER-LAW INFLATION
An exponential potential is a particular case where the attractor can
be found explicitly and one can study the approach to it, for an arbitrary
initial condition.
Consider the in(cid:13)ationary potential
(cid:12)(cid:20)(cid:30)
V ((cid:30)) = V e ; (30)
0 (cid:0)
with (cid:12) 1 for in(cid:13)ation to proceed. A particular solution to the Hamil-
(cid:28)
tonian constraint equation (21) is
1
(cid:12)(cid:20)(cid:30)
H ((cid:30)) = H e 2 ; (31)
att 0 (cid:0)
1
(cid:20)2 (cid:12)2
(cid:0)
2
H = V 01 1 : (32)
0 0
3 (cid:0) 6
B C
@ A
This model corresponds to an in(cid:13)ationary universe with a scale factor
that grows like
2
p
a(t) t ; p = 1: (33)
(cid:24) (cid:12)2 (cid:29)
The slow-roll parameters are both constant,
2 H ((cid:30)) 2 (cid:12)2 1
0
(cid:15) = 0 1 = = 1; (34)
(cid:20)2 H((cid:30)) 2 p (cid:28)
B C
@ A
2 H ((cid:30)) (cid:12)2 1
00
(cid:14) = 0 1 = = 1: (35)
(cid:20)2 H((cid:30)) 2 p (cid:28)
B C
@ A
All trajectories tend to the attractor (31), while we can also write down
the solution corresponding to the slow-roll approximation, (cid:15) = (cid:14) = 0,
(cid:20)2
2 (cid:12)(cid:20)(cid:30)
H ((cid:30)) = V e ; (36)
SR 0 (cid:0)
3
which di(cid:11)ers from the actual attractor by a tiny constant factor, 3p=(3p
(cid:0)
1) 1, responsible for a constant time-shift which cannot be measured.
’
HOMOGENEOUS SCALAR FIELD DYNAMICS
Singlet minimally coupled scalar (cid:12)eld (cid:30), with e(cid:11)ective potential V ((cid:30))
1
= d4xp g ; = g(cid:22)(cid:23)@ (cid:30)@ (cid:30) V ((cid:30)): (1)
inf inf inf (cid:22) (cid:23)
S (cid:0) L L (cid:0)2 (cid:0)
Z
Its evolution equation in a Friedmann-Robertson-Walker metric:
1
(cid:127) 2 _
(cid:30) (cid:30) + 3H(cid:30) + V ((cid:30)) = 0; (2)
0
(cid:0) a2r
together with the Einstein equations,
(cid:20)2 1 1
2 _2 2
H = (cid:30) + ( (cid:30)) + V ((cid:30)) ; (3)
3 22 2a2 r 3
(cid:20)42 5
_ _2
H = (cid:30) ; (4)
(cid:0) 2
where (cid:20)2 8(cid:25)G. The in(cid:13)ation dynamics described as a perfect (cid:13)uid
(cid:17)
with a time-dependent pressure and energy density given by
1 1
_2 2
(cid:26) = (cid:30) + ( (cid:30)) + V ((cid:30)); (5)
2 2a2 r
1 1
_2 2
p = (cid:30) ( (cid:30)) V ((cid:30)): (6)
2 (cid:0) 6a2 r (cid:0)
The (cid:12)eld evolution equation (2) implies the energy conservation equation,
(cid:26)_ + 3H((cid:26) + p) = 0: (7)
If the potential energy density of the scalar (cid:12)eld dominates the kinetic
and gradient energy, V ((cid:30)) (cid:30)_2; 1 ( (cid:30))2, then
(cid:29) a2 r
p (cid:26) (cid:26) const: H((cid:30)) const:; (8)
’ (cid:0) ) ’ ) ’
which leads to the solution
a(cid:127)
a(t) exp(Ht) > 0 accelerated expansion: (9)
(cid:24) ) a
De(cid:12)nition: number of e-folds,
N ln(a=a ) a(N) = a exp(N)
i i
(cid:17) )
THE SLOW-ROLL APPROXIMATION
During in(cid:13)ation, the scalar (cid:12)eld evolves very slowly down its e(cid:11)ective
potential. We can then de(cid:12)ne the slow-roll parameters,
H_ (cid:20)2 (cid:30)_2
(cid:15) = 1; (10)
(cid:17) (cid:0) H2 2 H2 (cid:28)
(cid:127)
(cid:30)
(cid:14) 1: (11)
_
(cid:17) (cid:0) H(cid:30) (cid:28)
The condition which characterizes in(cid:13)ation is
a(cid:127)
(cid:15) < 1 > 0; (12)
() a
i.e. horizon distance d H 1 grows more slowly than scale factor a.
H (cid:0)
(cid:24)
The number of e-folds during in(cid:13)ation:
a (cid:20)d(cid:30)
end te (cid:30)e
N = ln = Hdt = : (13)
ai Zti Z(cid:30)i 2(cid:15)((cid:30))
r
The evolution equations (2) and (3) become
(cid:15) (cid:20)2
2 2
H 1 H = V ((cid:30)); (14)
(cid:0) 3! ’ 3
(cid:14)
_ _
3H(cid:30) 1 3H(cid:30) = V ((cid:30)): (15)
0
0 (cid:0) 31 ’ (cid:0)
@ A
_
Phase space reduction for single-(cid:12)eld in(cid:13)ation, H((cid:30); (cid:30)) H((cid:30)).
!
2
1 V ((cid:30))
0
(cid:15) = 1;
0 1
2(cid:20)2 V ((cid:30)) (cid:28)
B C
1 V@ ((cid:30)) A
00
(cid:17) = 1;
(cid:20)2 V ((cid:30)) (cid:28)
V ((cid:30))d(cid:30)
2 (cid:30)e
N = (cid:20) :
(cid:30)i V ((cid:30))
Z 0