In cosmology, the cosmological constant (usually
denoted by the Greek capital letter lambda:
Λ) is the energy density of space, or vacuum
energy, that arises in Albert Einstein's field
equations of general relativity. It is closely
associated to the concepts of dark energy
and quintessence.Einstein originally introduced
the concept in 1917 to counterbalance the
effects of gravity and achieve a static universe,
a notion which was the accepted view at the
time. Einstein abandoned the concept in 1931
after Hubble's discovery of the expanding
universe. From the 1930s until the late 1990s,
most physicists assumed the cosmological constant
to be equal to zero. That changed with the
surprising discovery in 1998 that the expansion
of the universe is accelerating, implying
the possibility of a positive nonzero value
for the cosmological constant.Since the 1990s,
studies have shown that around 68% of the
mass–energy density of the universe can
be attributed to so-called dark energy. The
cosmological constant Λ is the simplest possible
explanation for dark energy, and is used in
the current standard model of cosmology known
as the ΛCDM model. While dark energy is poorly
understood at a fundamental level, the main
required properties of dark energy are that
it functions as a type of anti-gravity, it
dilutes much more slowly than matter as the
universe expands, and it clusters much more
weakly than matter, or perhaps not at all.According
to quantum field theory (QFT) which underlies
modern particle physics, empty space is defined
by the vacuum state which is a collection
of quantum fields. All these quantum fields
exhibit fluctuations in their ground state
(lowest energy density) arising from the zero-point
energy present everywhere in space. These
zero-point fluctuations should act as a contribution
to the cosmological constant Λ, but when
calculations are performed these fluctuations
give rise to an enormous vacuum energy. The
discrepancy between theorized vacuum energy
from QFT and observed vacuum energy from cosmology
is a source of major contention, with the
values predicted exceeding observation by
some 120 orders of magnitude, a discrepancy
that has been called "the worst theoretical
prediction in the history of physics!". This
issue is called the cosmological constant
problem and it is one of the greatest unsolved
mysteries in science with many physicists
believing that "the vacuum holds the key to
a full understanding of nature".
== History ==
Einstein included the cosmological constant
as a term in his field equations for general
relativity because he was dissatisfied that
otherwise his equations did not allow, apparently,
for a static universe: gravity would cause
a universe that was initially at dynamic equilibrium
to contract. To counteract this possibility,
Einstein added the cosmological constant.
However, soon after Einstein developed his
static theory, observations by Edwin Hubble
indicated that the universe appears to be
expanding; this was consistent with a cosmological
solution to the original general relativity
equations that had been found by the mathematician
Friedmann, working on the Einstein equations
of general relativity. Einstein reportedly
referred to his failure to accept the validation
of his equations—when they had predicted
the expansion of the universe in theory, before
it was demonstrated in observation of the
cosmological red shift—as his "biggest blunder".In
fact, adding the cosmological constant to
Einstein's equations does not lead to a static
universe at equilibrium because the equilibrium
is unstable: if the universe expands slightly,
then the expansion releases vacuum energy,
which causes yet more expansion. Likewise,
a universe that contracts slightly will continue
contracting.However, the cosmological constant
remained a subject of theoretical and empirical
interest. Empirically, the onslaught of cosmological
data in the past decades strongly suggests
that our universe has a positive cosmological
constant. The explanation of this small but
positive value is an outstanding theoretical
challenge, the so-called cosmological constant
problem.
Some early generalizations of Einstein's gravitational
theory, known as classical unified field theories,
either introduced a cosmological constant
on theoretical grounds or found that it arose
naturally from the mathematics. For example,
Sir Arthur Stanley Eddington claimed that
the cosmological constant version of the vacuum
field equation expressed the "epistemological"
property that the universe is "self-gauging",
and Erwin Schrödinger's pure-affine theory
using a simple variational principle produced
the field equation with a cosmological term.
== Equation ==
The cosmological constant
Λ
{\displaystyle \Lambda }
appears in Einstein's field equation in the
form
R
μ
ν
−
1
2
R
g
μ
ν
+
Λ
g
μ
ν
=
8
π
G
c
4
T
μ
ν
,
{\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}Rg_{\mu
\nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu
\nu },}
where the Ricci tensor/scalar R and the metric
tensor g describe the structure of spacetime,
the stress-energy tensor T describes the energy
and momentum density and flux of the matter
in that point in spacetime, and the universal
constants G and c are conversion factors that
arise from using traditional units of measurement.
When Λ is zero, this reduces to the field
equation of general relativity usually used
in the mid-20th century. When T is zero, the
field equation describes empty space (the
vacuum).
The cosmological constant has the same effect
as an intrinsic energy density of the vacuum,
ρvac (and an associated pressure). In this
context, it is commonly moved onto the right-hand
side of the equation, and defined with a proportionality
factor of 8π: Λ = 8πρvac, where unit conventions
of general relativity are used (otherwise
factors of G and c would also appear, i.e.
Λ = 8π(G/c2)ρvac = κρvac, where κ is
Einstein's constant). It is common to quote
values of energy density directly, though
still using the name "cosmological constant",
with convention 8πG = 1. The true dimension
of Λ is a length−2.
Given the Planck (2018) values of ΩΛ = 0.6889±0.0056
and H0 = 67.66±0.42 (km/s)/Mpc = (2.1927664±0.0136)×10−18
s−1, Λ has the value of
Λ
=
1.1056
×
10
−
52
m
−
2
,
{\displaystyle \Lambda =1.1056\times 10^{-52}\,{\text{m}}^{-2},}
or 2.888×10−122 in reduced Planck units
or 4.33×10−66 eV2 in natural units.
A positive vacuum energy density resulting
from a cosmological constant implies a negative
pressure, and vice versa. If the energy density
is positive, the associated negative pressure
will drive an accelerated expansion of the
universe, as observed. (See dark energy and
cosmic inflation for details.)
=== ΩΛ (Omega Lambda) ===
Instead of the cosmological constant itself,
cosmologists often refer to the ratio between
the energy density due to the cosmological
constant and the critical density of the universe,
the tipping point for a sufficient density
to stop the universe from expanding forever.
This ratio is usually denoted ΩΛ, and is
estimated to be 0.6889±0.0056, according
to results published by the Planck Collaboration
in 2018.In a flat universe, ΩΛ is the fraction
of the energy of the universe due to the cosmological
constant, i.e., what we would intuitively
call the fraction of the universe that is
made up of dark energy. Note that this value
changes over time: the critical density changes
with cosmological time, but the energy density
due to the cosmological constant remains unchanged
throughout the history of the universe: the
amount of dark energy increases as the universe
grows, while the amount of matter does not.
=== Equation of state ===
Another ratio that is used by scientists is
the equation of state, usually denoted w,
which is the ratio of pressure that dark energy
puts on the universe to the energy per unit
volume. This ratio is w = −1 for a true
cosmological constant, and is generally different
for alternative time-varying forms of vacuum
energy such as quintessence. The Planck Collaboration
(2018) has measured w = −1.028±0.032, consistent
with −1, assuming no evolution in w over
cosmic time.
== Positive value ==
Observations announced in 1998 of distance–redshift
relation for Type Ia supernovae indicated
that the expansion of the universe is accelerating.
When combined with measurements of the cosmic
microwave background radiation these implied
a value of ΩΛ ≈ 0.7, a result which has
been supported and refined by more recent
measurements. There are other possible causes
of an accelerating universe, such as quintessence,
but the cosmological constant is in most respects
the simplest solution. Thus, the current standard
model of cosmology, the Lambda-CDM model,
includes the cosmological constant, which
is measured to be on the order of 10−52
m−2, in metric units. It is often expressed
as 10−35 s−2 or 10−122 in other unit
systems. The value is based on recent measurements
of vacuum energy density,
ρ
vacuum
=
5.96
×
10
−
27
kg/m
3
{\displaystyle \rho _{\text{vacuum}}=5.96\times
10^{-27}{\text{ kg/m}}^{3}}
, or 10−47 GeV4, 10−29 g/cm3 in other
unit systems.
As was only recently seen, by works of 't
Hooft, Susskind and others, a positive cosmological
constant has surprising consequences, such
as a finite maximum entropy of the observable
universe (see the holographic principle).
== Predictions ==
=== 
Quantum field theory ===
A major outstanding problem is that most quantum
field theories predict a huge value for the
quantum vacuum. A common assumption is that
the quantum vacuum is equivalent to the cosmological
constant. Although no theory exists that supports
this assumption, arguments can be made in
its favor.Such arguments are usually based
on dimensional analysis and effective field
theory. If the universe is described by an
effective local quantum field theory down
to the Planck scale, then we would expect
a cosmological constant of the order of
M
p
l
2
{\displaystyle M_{\rm {pl}}^{2}}
(
6
×
10
54
eV
2
{\displaystyle 6\times 10^{54}\,{\text{eV}}^{2}}
in natural unit or
1
{\displaystyle 1}
in reduced Planck unit). As noted above, the
measured cosmological constant is smaller
than this by a factor of ~10−120. This discrepancy
has been called "the worst theoretical prediction
in the history of physics!".Some supersymmetric
theories require a cosmological constant that
is exactly zero, which further complicates
things. This is the cosmological constant
problem, the worst problem of fine-tuning
in physics: there is no known natural way
to derive the tiny cosmological constant used
in cosmology from particle physics.
=== Anthropic principle ===
One possible explanation for the small but
non-zero value was noted by Steven Weinberg
in 1987 following the anthropic principle.
Weinberg explains that if the vacuum energy
took different values in different domains
of the universe, then observers would necessarily
measure values similar to that which is observed:
the formation of life-supporting structures
would be suppressed in domains where the vacuum
energy is much larger. Specifically, if the
vacuum energy is negative and its absolute
value is substantially larger than it appears
to be in the observed universe (say, a factor
of 10 larger), holding all other variables
(e.g. matter density) constant, that would
mean that the universe is closed; furthermore,
its lifetime would be shorter than the age
of our universe, possibly too short for intelligent
life to form. On the other hand, a universe
with a large positive cosmological constant
would expand too fast, preventing galaxy formation.
According to Weinberg, domains where the vacuum
energy is compatible with life would be comparatively
rare. Using this argument, Weinberg predicted
that the cosmological constant would have
a value of less than a hundred times the currently
accepted value. In 1992, Weinberg refined
this prediction of the cosmological constant
to 5 to 10 times the matter density.This argument
depends on a lack of a variation of the distribution
(spatial or otherwise) in the vacuum energy
density, as would be expected if dark energy
were the cosmological constant. There is no
evidence that the vacuum energy does vary,
but it may be the case if, for example, the
vacuum energy is (even in part) the potential
of a scalar field such as the residual inflaton
(also see quintessence). Another theoretical
approach that deals with the issue is that
of multiverse theories, which predict a large
number of "parallel" universes with different
laws of physics and/or values of fundamental
constants. Again, the anthropic principle
states that we can only live in one of the
universes that is compatible with some form
of intelligent life. Critics claim that these
theories, when used as an explanation for
fine-tuning, commit the inverse gambler's
fallacy.
In 1995, Weinberg's argument was refined by
Alexander Vilenkin to predict a value for
the cosmological constant that was only ten
times the matter density, i.e. about three
times the current value since determined.
== See also
