The Cosmological Constant Problem

TLDR: In this article, five different approaches to the cosmological constant problem are described, and a brief review of the history of this problem is given. But none of the approaches are considered in this paper.

Abstract: Astronomical observations indicate that the cosmological constant is many orders of magnitude smaller than estimated in modern theories of elementary particles. After a brief review of the history of this problem, five different approaches to its solution are described.

Full text

The
cosmological
constant
problem
Steven
Weinberg
Theory
Group,
Department
of
Physics,
University of
Texas,
Austin,
Texas
7871Z
Astronomical observations indicate that the cosmological
constant
is
many
orders
of
magnitude
smaller
than estimated
in modern theories
of elementary particles.
After
a
brief
review of the
history
of this
prob-
lem,
five different
approaches
to its solution are described.
CONTENTS
I. Introduction
II.
Early
History
III.
The
Problem
IV.
Supersymmetry,
Supergravity,
Superstrings
V.
Anthropic
Considerations
A. Mass
density
8.
Ages
C. Number counts
VI.
Adjustment
Mechanisms
VII.
Changing
Gravity
VIII.
Quantum
Cosmology
IX. Outlook
Acknowledgments
References
As Iwas
going
up
the
stair,
I
neet
a
man who
wasn't
theve.
He
wasn't
there
again
today,
I
wish,
Iwish
he'd
stay
away.
1
1
2
3
6
8
8
8
9
11
14
20
21
21
Hughes
Mearns
R„—
—,
'g
R
—
A,
g„=
—
8e
GT„
(2.
1)
Now,
for
A,
&0,
there was a static solution for a universe
filled
with dust of zero
pressure
and mass
density
8+6
(2.
2)
Its
geometry
was
that of a
sphere
S3,
with
proper
cir-
cumference
2m.
v,
where
II.
EARLY HISTORY
After
completing
his
formulation of
general
relativity
in
1915
—
1916,
Einstein
(1917)attempted
to
apply
his
new
theory
to
the
whole universe.
His
guiding
principle
was
that the universe is static:
"The
most
important
fact that
we draw
from
experience
is that the relative velocities
of
the stars are
very
small
as
compared
with
the
velocity
of
light.
"
No such static solution of his
original
equations
could
be
found
(any
more than
for
Newtonian gravita-
tion),
so
he
modified
them
by
adding
a
new
term
involv-
ing
a free
parameter
A.
,
the
cosmological
constant:
I. INTRODUCTION
r
=
1/VSmpG
so
the mass of
the
universe was
(2.
3)
Physics
thrives on
crisis. We all
recall
the
great
pro-
gress
made
while
finding
a
way
out
of various crises of
the
past:
the failure to detect
a motion of the
Earth
through
the
ether,
the
discovery
of the
continuous
spec-
trum of beta
decay,
the
~-0
problem,
the
ultraviolet
divergences
in electromagnetic
and then
weak
interac-
tions,
and
so
on.
Unfortunately,
we
have run short of
crises
lately.
The
"standard
model"
of
electroweak and
strong
interactions
currently
faces
neither internal
incon-
sistencies nor
conflicts with experiment.
It has
plenty
of
loose
ends;
we
know no reason
why
the
quarks
and lep-
tons should
have
the
masses
they
have,
but then we
know
no
reason
why
they
should
not.
Perhaps
it is
for want of
other crises to
worry
about
that
interest is
increasingly
centered
on one veritable
crisis: theoretical
expectations
for
the
cosmological
con-
stant
exceed
observational
limits
bP
some
120
orders of
magnitude.
'
In these
lectures
I will first review the
histo-
ry
of
this
problem
and then
survey
the various
attempts
that
have
been made
at
a
solution.
*Morris
Loeb Lectures
in
Physics,
Harvard
University,
May
2,
3, 5,
and
10,
1988.
For a
good
nonmathematical
description of
the
cosmological
constant
problem,
see
Abbott
(1988).
M=2mr
p=
—
k
'
6
(2.
4)
4
In
some
popular
history
accounts,
it
was
Hubble'
s
discovery
of the
expansion
of the universe that led
Ein-
stein to
retract his
proposal
of
a
cosmological
constant.
The real
story
is more
complicated,
and
more
interesting.
One
disappointment
came almost
immediately.
Ein-
stein had been
pleased
at the
connection
in his model
be-
tween the mass
density
of the
universe and its
geometry,
because, following
Mach's
lead,
he
expected
that the
mass distribution of the universe
should set inertial
frames. It was
therefore
unpleasant
when
his friend
de
Sitter,
with
whom Einstein remained
in touch
during
the
war,
in
1917
proposed
another
apparently
static
cosmo-
logical
model with no matter
at all.
(See
de Sitter,
1917.
)
Its line element
(using
the same coordinate
system
as de
Sitter,
but in
a
difterent
notation) was
dv
=
[dt
dr—
1
cosh
Hv
H tanh
Hr(dO
—
+
sin
Odg
)],
(2.
5)
2The
notation used
here for
metrics,
curvatures,
etc.
,
is the
same
as
in
W'einberg
(1972).
Reviews of
Modern
Physics,
Vol.
61,
No.
1,
January
1989
Copyright
1988
The American
Physical
Society

Steven
Weinberg:
The
cosmological
constant
problem
with
H
related to
the
cosmological
constant
by
H
=&A,
/3
(2.
6)
and
p=p
=0.
Clearly
matter
was
not needed to
produce
inertia.
At about
this
time,
the redshift of distant
objects
was
being,
discovered
by
Slipher.
Over the
period
from 1910
to
the
mid-1920s,
Slipher (1924)
observed that
galaxies
(or,
as then
known, spiral
nebulae)
have
redshifts
z
=
b,A, /A,
ranging
up
to
6%,
and
only
a
few have
blue-
shifts.
Weyl
pointed
out
in 1923 that de
Sitter's
model
would
exhibit such a redshift, increasing
with
distance,
because
although
the metric in
de
Sitter's
coordinate sys-
tem is time independent,
test bodies
are not at
rest;
there
is a nonvanishing
component
of the
afBne connection
III. THE PROBLEM
Unfortunately,
it was not so
easy
simply
to
drop
the
cosmological
constant,
because
anything
that
contributes
to
the
energy
density
of
the
vacuum
acts
just
like a
cosmological
constant.
Lorentz invariance tells
us
that
in the
vacuum
the energy-mornenturn
tensor
must
take
the
form
&
T„.
&=
—
(p&g„.
. (3.
1)
X„=X+8~G(p)
. (3.
2)
(A
minus
sign appears
here
because we
use
a metric
which for flat
space-time
has
goo=
—
1.
)
Inspection
of
Eq.
(2.
1)
shows
that this has the same
efFect
as
adding
a
term
8m G
(p
)
to the
effective
cosmological
constant
I
«
=
—
H
sinhHr
tanhHr
giving
a
redshift
proportional
to distance
z=Hr for
Hr
(&1
.
(2.7)
(2
8)
Equivalently
we
can
say
that
the Einstein
cosmological
constant
contributes a term A,
/8mG
to the total
efFective
vacuum
energy
In
his
influential
textbook, Eddington
(1924) interpreted
Slipher's
redshifts
in
terms of de
Sitter's
"static"
universe.
But of
course, although
the
cosmological
constant was
needed
for
a
static universe,
it was not
needed
for an
ex-
panding
one.
Already
in
1922,
Friedmann (1924)
had
de-
scribed
a
class of cosmological models,
with line
element
(in
modern notation)
2
2
dr
dr
=dt
R(t)
—
+r
(d6
+
sin
Hdy
)
1
—
kr
(2.9)
These
are
comoving
coordinates;
the
universe
expands
or
contracts as
R
(t)
increases or decreases,
but the
galaxies
keep
fixed
coordinates
r, o,
y.
The
motion of
the cosmic
scale
factor
is
governed
by
an
energy-conservation
equa-
tion
2
p
=&p&+X/8
G=A,
,
/8
G .
(3.
3)
A
crude experimental
upper
bound on
A,
,&
or
pz
is pro-
vided
by
measurements of
cosmological
redshifts
as
a
function of
distance,
the
program
begun
by
Hubble in the
late 1920s.
The
present
expansion
rate is
today
estimated
as
1
dR
=Ho
=50
—
100
km/sec
Mpc
R dt
now
=(
—,
'
—
1)X10
'
/yr
.
Furthermore,
we do not
gross
effects
of
the curvature of
the
universe,
so
very roughly
ik
i/R'„.
„SH',
.
Finally,
the
ordinary
nonvacuum
mass
density
of
the
universe is not
much
greater
than its critical value
Ip
—
(p
&
I
-3H,
'/8~G
.
dR
=
—
k+
—
'R
(8m
Gp+
A,
)
.
dt
(2
10)
Hence
(2.10)
shows
that
The
de
Sitter model is
just
the
special
case with
k
=0
and
p=O;
in order
to
put
the
line element
(2.5)
in the more
general
form
(2.
9),
it
is
necessary
to
introduce
new
coor-
dinates,
t'=
t
—
H
'
ln
coshHr,
r'=H
'
exp(
Ht)
sinhHr,
—
(2.
11)
and
then
drop
the
primes.
However,
we
can
also
easily
find
expanding
solutions
with A,
=O
and
p
)
0. Pais
(1982)
quotes
a
1923 letter of
Einstein to
Weyl,
giving
his
reac-
tion
to
the
discovery
of
the expansion
of the
universe:
"If
there is
no
quasi-static
world,
then
away
with
the
cosmological
term&"
or,
in physicists'
units,
ipvi
510
g/cm
=10
GeV
(3.
4)
( )
JA4vrk
dk
1
~k2+
2
A
(2m)'
2
16~'
(3.5)
A
more
precise
observational
bound
will
be
discussed in
Sec.
V,
but this one
will
be
good
enough
for our
present
purposes.
As
everyone knows,
the
trouble
with
this
is
that the
en-
ergy
density
(p)
of
empty
space
is
likely
to
be
enormous-
ly
larger
than
10 GeV
. For one
thing,
summing
the
zero-point
energies
of all normal
modes
of some
field of
mass m
up
to
a
wave
number
cutoff
A))
m
yields
a
vacu-
um
energy
density
(with fi
=
c
=
1
)
Rev.
Mod.
Phys.
,
Vol.
61,
No.
1,
January
1989

Steven
Weinberg:
The
cosmological
constant
problem
If we
believe general
relativity
up
to
the
Planck
scale,
then we
might
take A=(SmG)
',
which would
give
&p)
=2-"~-'G-'=2X
10"
GeV'.
(3.6)
Casimir
(1948}
showed
that
quantum
fluctuations
in
the
space
between two
Aat
conducting plates
with
separation
d
would pro-
duce a
force
per
unit area
equal
to Ac+
/240d,
or 1.
30X 10
dyn
cm
/d
.
This was
measured
by
Sparnaay (1957),
who
found
a
force
per
area of
(1
—
4)X10
'
dyncm
/d,
when d
was
varied
between
2 and
10
pm.
But we saw
that
~
&
p)
+1,
/ SAG~
is
less
than
about
10
GeV,
so
the
two
terms here
must cancel to better
than
118 decimal
places.
Even if we
only
worry
about
zero-point
energies
in
quantum
chromo
dynamics,
we
would
expect
&p)
to
be
of
order
AocD/16m,
or 10
GeV, requiring
I,
/SmG
to cancel
this
term to
about
41
decimal
places.
Perhaps
surprisingly,
it
was a
long
time
before
particle
physicists began
seriously
to
worry
about
this
problem,
despite
the
demonstration in
the
Casimir effect of the
reality
of zero-point
energies.
Since the
cosmological
upper
bound
on
~
&
p
)
+A,
/Sm
G
~
was
vastly
less than
any
value
expected
from
particle
theory,
most
particle
theor-
ists
simply
assumed that for some
unknown reason this
quantity
was
zero. But
cosmologists
generally
continued
to
keep
an
open
mind,
analyzing
cosmological
data
in
terms of models with
a
possibly
nonvanishing
cosmologi-
cal constant.
In
fact,
as
far
as
I
know,
the
first
published
discussion
of the
contribution of
quantum
Auctuations to
the
effective
cosmological
constant
was
triggered
by
astro-
nomical observations.
In the late 1960s it seemed that an
excessively
large
number of
quasars
were
being
observed
with redshifts
clustered
about
z
=1.
95. Since
1+z
is the
ratio
of
the cosmic
scale factor
R(t)
at.
present
to its
value
at
the time the
light
now observed
was emitted,
this
could
be explained
if the universe loitered
for
a
while
at a
value
of R
(t) equal
to
1/2. 95
times the
present
value.
A
number
of
authors
[Petrosian,
Salpeter,
and
Szekeres
(1967);
Shklovsky
(1967);
Rowan-Robinson
(1968)j
pro-
posed
that such a
loitering
could be
accounted for in
a
model
proposed
by
Lemaitre
(1927,
1931). In
this model
there
is
a
positive
cosmological
constant
X,
z
and
positive
curvature
k
=+1,
just
as
in the static
Einstein
model,
while the
mass of the universe is
taken close
to
the
Ein-
stein value
(2.4).
The scale
factor
R
(t)
starts
at
R
=0
and
then
increases;
however,
when
the mass
density
drops
to near
the
Einstein
value
(2.
2),
the
universe
behaves for
a
while like
a
static Einstein
universe,
until
the
instability
of this model
takes
over
and
the universe
starts
expanding
again.
In order
for this
idea to
explain
a
preponderance
of redshifts at
z
=-1.
95,
the vacuum
ener-
gy
density
pv
would
have
to
be
(2.
95)
times the
present
nonvacuum
mass
density
po.
These considerations led
Zeldovich
(1967)
to
attempt
to
account for a
nonzero vacuum
energy
density
in terms
of
quantum
Auctuations. As we
have
seen,
the zero-point
energies
themselves
gave
far too
large
a value for
&
p
),
so
Zeldovich
assumed that these
were canceled
by
A,
/Sn.
G,
leaving only
higher-order effects:
in
particular,
the gravi-
tational
force
between the
particles
in
the
vacuum
Quc-
tuations. (In Feynman diagram
terms,
this
corresponds
to throwing
away
the
one-loop
vacuum
graphs,
but keep-
ing
those with two
loops.
)
Taking
A
particles
of
energy
A
per
unit volume
gives
the gravitational self-energy
den-
sity
of
order
&p)=(GA
/A
')A
=GA
(3.
7)
4Veltman
(1975)
attributes
this view
to
Linde
(1974),
himself
(quoted
as
"to
be
published"
),
and
Dreitlein
(1974). However,
Linde's
paper
does not seem
to
me
to
take this
position.
Dreitlein's
paper
proposed
that
Eq.
(3.
9)
could
give
an
accept-
ably
small value
of
&p),
with
p/i/g
fixed
by
the Fermi
cou-
pling
constant of
weak
interactions, if
p
is
very
small,
of order
10
MeV.
Veltman's
paper
gives experimental arguments
against
this
possibility.
For no
clear
reason,
Zeldovich
took the cutoff
A as 1
GeV,
which
yields
a density
&
p
)
=
10
GeV,
much
smaller than
from zero-point
energies
themselves, but
still
larger
than
the
observational bound
(3.
4)
on
~&p)+A,
/Sm.
G~
by
some 9
orders of
magnitude.
Neither
Zeldovich
nor
anyone
else felt
encouraged
to
pursue
these ideas.
The
real
beginning
of
serious
worry
about
the vacuum
energy
seems to date
from the success
of the idea of spon-
taneous
symmetry
breaking
in the
electroweak
theory.
In this
theory,
the
scalar field
potential
takes
the form
(with
p
&0,
g
&0)
V=
Vo
pY0+g—
(A)'
.
(3.
8)
At its
minimum this
takes the value
4
&p)
=V,
„=V,
—"
(3.
9)
Apparently
some
theorists felt that
V
should
vanish
at
$
=0,
which would
give
Vo
=
0,
so that
&
p
)
would
be
negative
definite.
In the electroweak
theory
this would
give
&
p)
=
—
g(300
GeV),
which even for
g
as
small
as
a would
yield
~
&
p
)
~
=10
GeV,
larger
than the
bound
on
p~
by
a
factor 10 . Of course we know
of no reason
why
Vo
or
A, must
vanish,
and
it is
entirely possible
that
Vo
or
A, cancels the
term
—
p
/4g
(and
higher-order
corrections),
but
this
example
shows
vividly
how
un-
natural it
is
to
get
a
reasonably
small effective cosmologi-
cal
constant.
Moreover,
at
early
times the
effective
temperature-dependent
potential
has a
positive
coefficient
for
P
P,
so the
minimum
then
is
at
/=0,
where
V(P)=
Vo.
Thus,
in order to
get
a
zero
cosmological
constant
today,
we have to
put
up
with an enormous
cosmological
constant at times before the
electroweak
phase
transition.
[This
is
not in conflict with
experiment;
in
fact,
the
phase
transition occurs
at a temperature
T of
order
p/&g,
so the
black-body radiation
present
at
that
Rev. Mod.
Phys.
,
Vol.
61,
No.
1,
January
1989

Steven
Weinberg:
The
cosmological
constant
problem
8
=0,
(3.10)
time
has
an
energy
density
of
order
p
/g,
larger
than
the
vacuum
energy
by
a
factor
1/g
(Bludman and
Ruder-
man,
1977).
]
At
even earlier
times there were
other
tran-
sitions,
implying
an even
larger
early
value
for
the
effective
cosmological
constant.
This is
currently
regard-
ed
as
a
good thing;
the
large
early
cosmological
constant
would drive cosmic
inAation,
solving
several
of the long-
standing
problems
of
cosmological
theory
(Guth, 1981;
Albrecht
and Steinhardt,
1982;
Linde, 1982).
We
want
to
explain
why
the effective
cosmological constant
is small
now,
not
why
it
was
always
small.
Before
closing
this
section,
I want to take
up
a
peculiar
aspect
of
the
problem
of
the
cosmological
constant. The
appearance
of
an
effective
cosmological
constant makes it
impossible
to
find
any
solutions of the Einstein
field equa-
tions in which
g„
is the constant
Minkowski term
g„.
That
is,
the
original
symmetry
of
general
covariance,
which is
always
broken
by
the
appearance
of
any
given
metric
g„,
cannot,
without
fine-tuning, be
broken in
such
a
way
as
to
preserve
the
subgroup
of
space-time
translations.
This situation is unusual.
Usually
if
a.
theory
is
invari-
ant under some
group G,
we would not
expect
to have to
fine-tune
the
parameters
of the
theory
in
order to find
vacuum
solutions that
preserve
any
given
subgroup
H
C
G.
For
instance,
in the electroweak
theory,
there
is a
finite
range
of
parameters
in which
any
number of
dou-
blet scalars will
get
vacuum
expectation
values
that
preserve
a
U(1)
subgroup
of
SU(2)XU(1).
So
why
will
this
not
work for the translational
subgroup
of
the
group
of
general
coordinate
transformations?
Suppose
we look
for
a
solution of the field
equations
that
preserves
transla-
tional
invariance. With
all
fields
constant,
the field equa-
tions for matter and
gravity
are
with
c
independent
of
g„.
With
this
X,
there are
no
solutions of
Eq.
(3.
11),
unless
for some
reason
the
coefficient c vanishes
when
(3.10)
is
satisfied.
Now that the
problem
has been
posed,
we
turn
to its
possible
solution. The
next
five sections will
describe five
directions
that
have been taken
in
trying
to
solve
the
problem
of the
cosmological
constant.
IV. SUPERSYMMETRY, SUPERGRAVITY,
SUPERSTR
INGS
Shortly
after
the
development
of four-dimensional
glo-
bally supersymmetric
field
theories,
Zumino
(1975)
point-
ed
out that
supersymmetry
in these theories
would,
if
un-
broken,
imply
a vanishing
vacuum
energy.
The
argu-
ment
is
very
simple:
the
supersymmetry
generators
Q
satisfy
an anticommutation
relation
(4.
1)
where
a and
P
are
two-component
spin
indices;
o
„cr2,
and
0.
3
are
the
Pauli
matrices;
o0=1;
and
I'"
is
the
energy-momentum
4-vector
operator.
If
supersymmetry
is
unbroken,
then the vacuum state
l0&
satisfies
(4.
2)
and
from
(4.
1)
and
(4.2)
we
infer
that
the vacuum has
vanishing
energy
and momentum
y(y
yy
)
y
&8
(P)
(4.
3)
This result can also be obtained
by
considering
the
poten-
tial
V(P,
P'
)
for the
chiral scalar
fields
P'
of
a
globally
su-
persymmetric
theory:
(3.
1
1)
g„~
A
1'„A
1t;
~
D;)
(
A
)
f~;
the
Lagrangian
transforms as a
density,
X~DetAX
.
(3.12)
(3.
13)
(3.14)
When
Eq.
(3.
10)
is
satisfied,
this
implies
that
X
trans-
forms
as
in
(3.14)
under
(3.12)
alone
This
has
the
.
unique
solution
X
=c(Detg
)
'~
(3.15)
With
N
g's,
these are
N
+6
equations
for
N +
6
un-
knowns,
so one
might
expect
a solution
without
fine-
tuning.
The
problem
is that
when
(3.10)
is
satisfied,
the
dependence
of
X
on
gz
is too
simple
to
allow
a
solution
of
(3.
11).
There is
a
GL(4) symmetry
that survives
as
a
vestige
of
general
covariance even when
we
constrain
the
fields to be
constants: under
the
GL(4)
transformation
where
W(P)
is the
so-called
superpotential.
(Gauge
de-
grees
of freedom are
ignored here,
but
they
would
not
change
the
argument.
)
The condition for unbroken
su-
persymmetry
is
that
8'be
stationary
in
P,
which would
imply
that Vtake its minimum
value,
(4.
4)
Quantum
effects do not
change
this
conclusion,
because
with
boson-fermion
symmetry,
the fermion
loops
cancel
the boson ones.
The trouble with this result
is that
supersymmetry
is
broken
in
the
real
world,
and
in this case
either
(4.
1)
or
(4.
3)
shows
that
the vacuum
energy
is positive-definite.
If
this vacuum
energy
were the sole contribution
to the
effective
cosmological constant, then the effect
of
super-
symmetry
would
be
to convert
the
problem
of
the
cosmo-
logical
constant
from
a crisis into
a disaster.
Fortunately
this
is
not the whole
story.
It
is not
possi-
ble to decide the value of
the effective
cosmological
con-
stant unless
we
explicitly
introduce
gravitation
into the
theory.
Any
globally
supersymmetric
theory
that
in-
Rev.
Mod.
Phys.
,
VoI.
61,
No.
1,
January
1989

Steven
Weinberg:
The
cosmological
constant
problem
volves
gravity
is
inevitably a
locally
supersymmetric
su-
pergravity
theory.
In
such
a
theory
the
eff'ective
cosmo-
logical
constant is
given
by
the
expectation
value
of
the
potential,
but
the
potential
is
now
given
by
(Cremmer
et al.
,
1978, 1979;
Barbieri et QI.
,
1982;
%'itten
and
Bagger,
1982)
V(P,
P*)
=
exp(8m.
GK)[D,
.
W(g
')'j(D
W)'
—
24~G
I
Wl'~, (4.
5)
where
K
(P,
P'
)
is a real function
of
both
P
and
P'
known
as
the Kahler
potential,
D,
-S'
is
a
sort of
covariant
derivative
BS'
6
BK
aO'
aa'
'
and
(
g
')'j
is the
inverse
of
a metric
() E
j
ayieayj
(4.
6)
(4.
7)
The condition
for unbroken
supersymmetry
is
now
D,
8'=0.
This
again
yields
a
stationary
point
of the po-
tential,
but
now it
is
one at which Vis
generally
negative.
In
fact,
even
if
we
fine-tuned
8'
so
that there were
a
su-
persymmetric
stationary
point
at which W
=0
and hence
V
=0,
such a
solution would
not,
in
general,
be the state
of
lowest
energy,
though
it would
be
stable
[Coleman
and
de
Luccia
(1980),
Weinberg
(1982)].
It
should, however,
be
mentioned that if there is
a set
of field values
at
which
8'=0
and
D,
W=O
for all i
in
lowest order of perturba-
tion
theory,
then
the
theory
has
a
supersymmetric
equi-
librium
configuration
with
V=0
to all orders of pertur-
bation
theory,
though
not
necessarily
beyond
perturba-
tion
theory
(Cxrisaru,
Siegel,
and
Rocek,
1979).
The
same
is
believed to be true
in
superstring
perturbation
theory
(Dine
and
Seiberg,
1986;
Friedan, Martinec,
and
Shenker,
1986;
Martinec,
1986;
Attick,
Moore,
and
Sen,
1987;
Morozov and
Perelomov, 1987).
Without
fine-tuning,
we can
generally
find
a
nonsuper-
symmetric
set of scalar
field values at
which
V=O
and
D;
W&0,
but
this
would
not
normally
be a
stationary
+g(Sn
Snn')
while the
superpotential
is
W=
W,
(C')+
W2(S"),
(4.
8)
(4.
9)
and
T,
C',
S"
are all
chiral scalar fields.
No constraints
are
placed
on
the
functions
h
(C',
C"),
IC(S",
S"*),
Wi(c'),
or
Wz(S"),
except
that
h
and
E
are
real,
and
functions all
depend
only
on
the fields
indicated;
in par-
ticular,
the
superpotential
must be
independent
of
the
single
chiral scalar T.
With
these conditions
the
potential
(4.
5)
takes
the
form
V=
exp(8m')
88'
(~
—
1
)a
3(T+T*+h)
X
b
+(D„W)(g
')"
(D
W)*
where
(JV
')'b
is
the
reciprocal
of the matrix
ah
ac'*ac'
(4.
10)
(4.11)
The matrices
JPb
and
g"
are
necessarily
positive-
definite, because of their
role
in
the kinetic
part
of the
scalar
Lagrangian
point
of V. Thus in
supergravity
the
problem
of
the
cosmological
constant
is no
more
a
disaster, but
just
as
much
a crisis,
as in
nonsupersymmetric
theories.
On
the other
hand,
supergravity
theories
o6'er
oppor-
tunities for
changing
the context
of the
cosmological
con-
stant
problem,
if
not
yet
for
solving
it.
Cremmer
et
aI.
(1983)
have
noted that
there
is
a
class of Kahler
poten-
tials
and
superpotentials that,
for a broad
range
of most
parameters,
automatically
yield
an
equilibrium
scalar
field
configuration
in which
V=O,
even
though
super-
symmetry
is broken.
Here
is
a
somewhat
generalized
version: the
Kahler
potential
is
sc
=
3»l
T—
+
T'
h(c',
—
c'*)
I
j8~G
k1Il
J
g
p
X
p
3
aT ah
ac'
(T+T*+h)
ax"
aC'
ax"
aT ah
ac
ax& a(
b
ax&
3
IT+ T*
—
h
I
gCae
gCb
Bx„
os"'
as
gn
ax~
ax„
(4.12)
aw
=D„S"=0
.
i3C'
But this
is
not
necessarily
a
configuration, because here
(4.
13)
supersymmetric
Hence
Eq.
(4.
10)
is
positive
and
therefore,
without
fur-
ther
fine-tuning,
may
be
expected
to have
a
stationary
point
with
V
=0,
specified
by
the
conditions
D.
~=-'
+8-6'
~
aC"
aC'
3
Bh
IT+T
+hl
ac
(4.
14)
and
this does
not
necessarily
vanish.
(However,
to
have
supersymmetry
broken,
it
is
essential
that the
superpo-
tential
actually
depend
on all of
the chiral scalars
S",
be-
Rev.
Mod.
Phys.
,
Vol.
61,
No.
1,
January
1989

Frequently asked questions

Q1. What are the contributions in "The cosmological constant problem" ?

In this paper, the cosmological constant was added to the original equations of general relativity to make them more stable. 

Q2. What is the cosmological constant in the static Einstein model?

In this model there is a positive cosmological constant X,z and positive curvature k =+1, just as in the static Einstein model, while the mass of the universe is taken close to the Einstein value (2.4). 

Q3. What is the real beginning of serious worry about the vacuum energy?

The real beginning of serious worry about the vacuum energy seems to date from the success of the idea of spontaneous symmetry breaking in the electroweak theory. 

Q4. Why does the standard model include parameters AO, Go, go?

This is because, in addition to charges, masses, etc., the standard model implicitly also involves parameters AO, Go, go, . . . appearing in the effective action for gravitation. 

Q5. What is the way to measure the effective cosmological constant?

and of greatest interest to astronomy, if it is only anthropic constraints that keep the effective cosmological constant within empirical limits, then this constant should be rather large, large enough to show up before long in astronomical observations. 

Q6. How many times the present nonvacuum mass density would be?

In order for this idea to explain a preponderance of redshifts at z =-1.95, the vacuum energy density pv would have to be (2.95) times the present nonvacuum mass density po. 

Q7. What is the standard model of electroweak and strong interactions?

The "standard model" of electroweak and strong interactions currently faces neither internal inconsistencies nor conflicts with experiment. 

Q8. What is the way to explain the efFective cosmological constant?

It is conceivable that in supergravity the property of having zero efFective cosmological constant does survive to low energies without any symmetry to guard it, but this would run counter to all their experience in physics. 

Q9. What is the variational derivative of Eq. (7.1) with respect to the metric?

The variational derivative of Eq. (7.1) with respect to the metric is5I 1 (R" ——,'g" R )+T"'5g„8~G (7.2)where, as usual, T is the variational derivative of I~ with respect to g„. 

Q10. What was the reason that Einstein's model would exhibit a redshift?

Weyl pointed out in 1923 that de Sitter's model would exhibit such a redshift, increasing with distance, because although the metric in de Sitter's coordinate system is time independent, test bodies are not at rest; there is a nonvanishing component of the afBne connectionUnfortunately, it was not so easy simply to drop the cosmological constant, because anything that contributes to the energy density of the vacuum acts just like a cosmological constant. 

Q11. What is the effect of creating and destroying a baby universe?

The effect of creating and destroying arbitrary numbers of baby universes of all types can thus be expressed by adding a suitable term in the actionS=S+g (a;+a; )f d x 0;(x), (8.25) where a; and a; are the annihilation and creation operators for a baby universe of type i, and O, (x) is the corresponding local operator. 

Q12. How can the authors prevent the cosmological constants from generating an unacceptable constant?

In order to prevent these quantum effects from generating an unacceptable cosmological constant, the cutoff A must be taken very small.