**B. The development of ideas**

In the classic book, *The Classical Theory of Fields*,
Landau and Lifshitz
(1951, p. 338)
second Einstein's opinion of the cosmological constant
, stating there is
"no basis whatsoever" for adjustment of the
theory to include this term. The empirical side of cosmology is
not much mentioned in this book, however (though there is a
perceptive comment on the limited empirical support for the
homogeneity assumption; p. 332). In the Supplementary Notes to
the English translation of his book, *Theory of Relativity*,
Pauli (1958, p. 220)
also endorses Einstein's position.

Discussions elsewhere in the literature on how one might find empirical constraints on the values of the cosmological parameters usually take account of . The continued interest was at least in part driven by indications that might be needed to reconcile theory and observations. Here are three examples.

First, the expansion time is uncomfortably short if
= 0.
Sandage's recalibration of the distance scale in the 1960s indicates
*H*_{0} 75
km s^{-1} Mpc^{-1}. If
= 0 this says
the time of expansion from densities too high for stars to have
existed is < *H*_{0}^{-1}
13 Gyr, maybe less than
the ages of the oldest stars, then estimated to be greater than about 15
Gyr.
Sandage (1961a)
points out that the problem is removed by adding a
positive . The
present estimates reviewed below
(Sec. IV.B.3)
are not far from these numbers, but still too uncertain for a
significant case for
.

Second, counts of quasars as a function of redshift show a peak at
*z* ~ 2, as would be produced by the loitering epoch in
Lemaître's
model
(Petrosian, Salpeter, and
Szekeres, 1967;
Shklovsky, 1967;
Kardashev, 1967).
The peak is now well established, centered at *z* ~ 2.5
(Croom et al., 2001;
Fan et al., 2001).
It usually is interpreted as
the evolution in the rate of violent activity in the nuclei of
galaxies, though in the absence of a loitering epoch the
indicated sharp variation in quasar activity with time is
curious (but certainly could be a consequence of astrophysics
that is not well understood).

The third example is the redshift-magnitude relation.
Sandage's (1961a)
analysis indicates this is a promising method of
distinguishing world models. The
Gunn and Oke (1975)
measurement
of this relation for giant elliptical galaxies, with
Tinsley's (1972)
correction for evolution of the star population from
assumed formation at high redshift, indicates curvature away from
the linear relation in the direction that, as
Gunn and Tinsley (1975)
discuss, could only be produced by
(within
general relativity theory). The new application of the
redshift-magnitude test, to Type Ia supernovae
(Sec. IV.B.4), is
not inconsistent with the Gunn-Oke measurement; we
do not know whether this agreement of the measurements is
significant, because Gunn and Oke were worried about galaxy
evolution. ^{16}

**2. The coincidences argument against
**

An argument against an observationally interesting value of , from our distrust of accidental coincidences, has been in the air for decades, and became very influential in the early 1980s with the introduction of the inflation scenario for the very early universe.

If the Einstein-de Sitter model in Eq. (35) were a
good approximation at the present epoch, an observer measuring
the mean mass density and Hubble's constant when the age of the
universe was one tenth the present value, or at ten times the
present age, would reach the same conclusion, that the
Einstein-de Sitter model is a good approximation. That is, we
would flourish at a time that is not special in the course of
evolution of the universe. If on the other hand two or more
of the terms in the expansion rate equation (11) made
substantial contributions to the present value of the expansion
rate, it would mean we are present at a special epoch, because
each term in Eq. (11) varies with the expansion factor
in a different way. To put this in more detail, we imagine that the
physics of the very early universe, when the relativistic
cosmological model became a good approximation, set the values
of the cosmological parameters. The initial values of the
contributions to the expansion rate equation had to have been
very different from each other, and had to have been exceedingly
specially fixed, to make two of the
_{i0}'s have
comparable values. This would be a most remarkable and
unlikely-looking coincidence. The
multiple coincidences required for the near vanishing of
and
at a redshift not
much larger than unity makes an even stronger case against
Lemaître's coasting model, by this line of argument.

The earliest published comment we have found on this point
is by
Bondi (1961, p. 166),
in the second edition of his book *Cosmology*. Bondi notes the
"remarkable property" of the
Einstein-de Sitter model: the dimensionless parameter we now call
_{M} is
independent of the time at which it is computed
(since it is unity). The coincidences argument follows and extends
Bondi's comment. It is presented in
McCrea (1971, p. 151).
When Peebles was a postdoc, in the early 1960s, in
R. H. Dicke's gravity research group, the coincidences argument
was discussed, but published much later
(Dicke, 1970, p. 62;
Dicke and Peebles,
1979).
We do not know its
provenance in Dicke's group, whether from Bondi, McCrea, Dicke,
or someone else. We would not
be surprised to learn others had similar thoughts.

The coincidences argument is sensible but not a proof, of course.
The discovery of the 3 K thermal cosmic microwave background radiation
gave us a term in the expansion rate equation that is down from the
dominant one by four orders of magnitude, not such a large factor by
astronomical standards. This might be
counted as a first step away from the argument. The evidence from
the dynamics of galaxies that
_{M0} is
less than unity is another step
(Peebles, 1984,
p. 442; 1986).
And yet another is the development of the evidence that the
and dark
matter terms differ by only a factor of three
(Eq. [2]).
This last is the most curious, but the community has
come to accept it, for the most part. The precedent makes
Lemaître's coasting model more socially acceptable.

A socially acceptable value of
cannot be such as to
make life impossible, of course.
^{17}
But perhaps the most productive interpretation of the coincidences
argument is that it demands a search for a more fundamental
underlying model. This is discussed further in
Sec. III.E and the Appendix.

Another tradition to consider is the relation between
and the vacuum or dark energy density. In one approach to the
motivation for the Einstein field equation, taken by
McVittie (1956)
and others,
appears as a
constant of integration (of the
expression for local conservation of energy and momentum).
McVittie (1956, p. 35)
emphasizes that, as a constant of integration,
"cannot
be assigned any particular value on *a priori* grounds."
Interesting variants of this line of thought are still under discussion
(Weinberg, 1989;
Unruh, 1989,
and references therein).

The notion of as a constant of integration may be related to the issue of the zero point of energy. In laboratory physics one measures and computes energy differences. But the net energy matters for gravity physics, and one can imagine represents the difference between the true energy density and the sum one arrives at by laboratory physics. Eddington (1939) and Lemaître (1934, 1949) make this point.

Bronstein (1933)
^{18}
carries the idea further, allowing
for transfer of energy between ordinary matter and that represented
by . In our
notation, Bronstein expresses this picture
by generalizing Eq. (9) to

(36) |

where and
*p* are the energy density and pressure of
ordinary matter and radiation. Bronstein goes on to propose a
violation of local energy conservation, a thought that no longer
seems interesting.
North (1965, p. 81)
finds
Eddington's (1939)
interpretation of the zero of energy also somewhat hard to defend.
But for our purpose the important point is that the idea of
as a form of energy has been in the air, in at least some circles, for
many years.

The zero-point energy of fields contributes to the dark energy
density. To make
physical sense the sum over the zero-point mode energies must be
cut off at a short distance or a high
frequency up to which the model under consideration is valid.
The integral of the zero-point energy (*k* / 2) of normal modes (of
wavenumber *k*) of a massless real bosonic scalar field
(),
up to the wavenumber cutoff *k*_{c}, gives the vacuum
energy density quantum-mechanical expectation value
^{19}

(37) |

Nernst (1916)
seems to have been the first to write down this
equation, in connection with the idea that the zero-point energy
of the electromagnetic field fills the vacuum, as a light aether,
that could have physically significant properties.
^{20}
This was before Heisenberg and Schrödinger: Nernst's
hypothesis is that each degree of freedom, which classical
statistical mechanics assigns energy *kT*/2, has
"Nullpunktsenergie"
*h* / 2. This would mean
the ground state energy of a one-dimensional harmonic oscillator is
*h*, twice
the correct value. Nernst's expression for the energy density
in the electromagnetic field thus differs from
Eq. (37) by a factor of two (after taking account
of the two polarizations), which is
wonderfully close. For a numerical example, Nernst noted that if
the cutoff frequency were
= 10^{20} Hz, or
~ 0.4 MeV, the energy density of the "Lichtäther" (light aether)
would be
10^{23} erg cm^{-3}, or about 100 g cm^{-3}.

By the end of the 1920s Nernst's hypothesis was replaced with the
demonstration that in quantum mechanics the zero-point energy of
the vacuum is as real as any other. W. Pauli, in unpublished
work in the 1920s, ^{21}
repeated Nernst's calculation, with the correct factor of 2, taking
*k*_{c} to correspond to the
classical electron radius. Pauli knew the value of
_{}
is quite unacceptable: the radius of the static Einstein universe
with this value of
_{}
"would not even reach to the moon"
(Rugh and Zinkernagel,
2000,
p. 5). ^{22}
The modern version of this "physicists' cosmological constant problem" is
even more acute, because a natural value for *k*_{c} is
thought to be much larger than what Nernst or Pauli used.
^{23}

While there was occasional discussion of this issue in the middle of the
20^{th} century (as in the quote from N. Bohr in
Rugh and Zinkernagel,
2000, p. 5),
the modern era begins with the paper by
Zel'dovich (1967)
that convinced the community to consider the possible
connection between the vacuum energy density of quantum physics
and Einstein's cosmological constant.
^{24}

If the physics of the vacuum looks the same to any inertial
observer its contribution to the stress-energy tensor is the
same as Einstein's cosmological constant (Eq. [19]).
Lemaître (1934)
notes this: "in order that absolute motion, i.e., motion relative to the
vacuum, may not be detected, we must associate a pressure
*p* = -
*c*^{2} to the energy density
*c*^{2} of vacuum".
Gliner (1965)
goes further, presenting the relation
between the metric tensor and the stress-energy tensor of a
vacuum that appears the same to any inertial observer. But it was
Zel'dovich (1968)
who presented the argument clearly enough and
at the right time to catch the attention of the community.

With the development of the concept of broken symmetry in the now
standard model for particle physics came the idea that the
expansion and cooling of the universe is accompanied by a
sequence of first-order phase transitions accompanying the
symmetry breaking. Each first-order transition has a latent heat
that appears as a contribution to an effective time-dependent
(*t*) or dark
energy density. ^{25}
The decrease in value of the dark energy density at
each phase transition is much larger than an acceptable present
value (within relativistic cosmology); the natural presumption is
that the dark energy is negligible now. This final
condition seems bizarre, but the picture led to the very
influential concept of inflation. We discussed the basic elements
in connection with Eq. (27); we turn now to some implications.

^{16} Early measurements of the
redshift-magnitude relation were meant
in part to test the Steady State cosmology of
Bondi and Gold (1948)
and Hoyle (1948).
Since the Steady State cosmology assumes
spacetime is independent of time its line
element has to have the form of the de Sitter solution with
_{K0} = 0
and the expansion parameter in Eq. (27). The
measured curvature of the redshift-magnitude relation is in the
direction predicted by the Steady State cosmology. But this
cosmology fails other tests discussed in
Sec. IV.B.
Back.

^{17} If
were negative
and the magnitude too large there would not be enough time
for the emergence of life like us. If
were positive and
too large the universe would expand too rapidly to allow galaxy
formation. Our existence, which requires something resembling the
Milky Way galaxy to contain and recycle heavy elements, thus
provides an upper bound on the value of
.
Such anthropic considerations are discussed by
Weinberg (1987,
2001),
Vilenkin (2001),
and references therein.
Back.

^{18}
Kragh (1996, p. 36)
describes Bronstein's motivation and history. We
discuss this model in more detail in
Sec. III.E, and comment on why
decay of dark energy into ordinary matter or radiation would be
hard to reconcile with the thermal spectrum of the 3 K cosmic
microwave background radiation. Decay into the dark sector may be
interesting.
Back.

^{19} Eq. (37), which usually figures in
discussions
of the vacuum energy puzzle, gives a helpful indication of the
situation: the zero-point energy of each mode is real and the
sum is large. The physics is seriously incomplete,
however. The elimination of spatial momenta with magnitudes *k*
> *k*_{c}
only makes sense if there is a preferred reference frame in which
*k*_{c} is defined.
Magueijo and Smolin
(2002)
mention a related issue:
In which reference frame is the Planck momentum of a virtual
particle at the threshold for new phenomena? In both cases one
may implicitly choose the rest frame
for the large-scale distribution of matter and radiation. It
seems strange to think the microphysics cares about large-scale
structure, but maybe it happens in a sea of interacting
fields. The cutoff in Eq. (37) might be
applied at fixed comoving wavenumber
*k*_{c}
*a*(*t*)^{-1}, or
at a fixed physical value of *k*_{c}. The first
prescription can be described by an action written as a sum of terms
_{}^{2}
/ 2 + *k*^{2}
_{}^{2} /
(2*a*(*t*)^{2}) for the
allowed modes. The zero-point energy of each mode scales with the
expansion of the universe as *a*(*t*)^{-1}, and the
sum over modes scales as
_{}
*a*(*t*)^{-4}, consistent with
*k*_{c}
*a*(*t*)^{-1}. In the limit of exact spatial
homogeneity, an equivalent approach uses the spatial average of
the standard expression for the field stress-energy tensor.
Indeed,
DeWitt (1975) and
Akhmedov (2002)
show that the
vacuum expectation value of the stress-energy tensor, expressed
as an integral cut off at *k* = *k*_{c}, and computed
in the preferred coordinate frame, is diagonal with space part
*p*_{} =
_{} / 3, for
the massless field we are
considering. That is, in this prescription the vacuum zero point
energy acts like a homogeneous sea of radiation. This defines a
preferred frame of motion, where the stress-energy tensor is
diagonal, which is not unexpected because we need a preferred
frame to define *k*_{c}. It is unacceptable as a model for
the properties of dark energy, of course. For example, if the
dark energy density were normalized to the value now under
discussion, and varied as
_{}
*a*(*t*)^{-4}, it
would quite mess up the standard model for the origin of the
light elements. We get a more acceptable model for the behavior
of _{}
from the second prescription, with the cutoff
at a fixed physical momentum. If we also want to satisfy local
energy conservation we must take the pressure to be
*p*_{} = - _{}. This does not contradict the derivation of
*p*_{} in the first prescription, because the second
situation cannot be described by an action: the pressure must be
stipulated, not derived. What is worse, the known fields at
laboratory momenta certainly do not allow this stipulation; they
are well described by analogs of the action in the first
prescription. This quite unsatisfactory situation illustrates how
far we are from a theory of the vacuum energy.
Back.

^{20} A helpful discussion of Nernst's
ideas on cosmology is in
Kragh (1996, pp. 151-7).
Back.

^{21} This is discussed in
Enz and Thellung (1960),
Enz (1974),
Rugh and Zinkernagel
(2000, pp. 4-5),
and Straumann (2002).
Back.

^{22} In an unpublished letter in 1930,
G. Gamow considered the gravitational consequences of the Dirac sea
(Dolgov, 1990,
p. 230). We thank A. Dolgov for helpful correspondence on this point.
Back.

^{23} In terms of an energy scale
_{}
defined by
_{} =
_{}^{4},
the Planck energy *G*^{-1/2} is
about 30 orders of magnitude larger than the "observed" value of
_{}.
This is, of course, an extreme case, since a lot of the theories of
interest break down well below the Planck scale. Furthermore, in
addition to other contributions, one is allowed to add a counterterm
to Eq. (37) to predict any value of
_{}.
With reference to this point, it is interesting to note that while Pauli
did not publish his computation of
_{},
he remarks in his
famous 1933 *Handbuch der Physik* review on quantum mechanics that
it is more consistent to "exclude a zero-point energy for each degree
of freedom as this energy, evidently from experience, does not interact
with the gravitational field"
(Rugh and Zinkernagel, 2000, p. 5).
Pauli was fully aware that one must take account of zero-point
energies in the binding energies of molecular structure, for
example (and we expect he was aware that what contributes to the
energy contributes to the gravitational mass). He chose to drop
the section with the above comment from the second
(1958) edition of
the review
(Pauli, 1980, pp.
iv-v). In a globally supersymmetric field theory there are equal
numbers of bosonic and fermionic degrees of freedom, and the net
zero-point vacuum energy density
_{}
vanishes
(Iliopoulos and Zumino,
1974;
Zumino, 1975).
However, supersymmetry is not a
symmetry of low energy physics, or even at the electroweak unification
scale. It must be broken at low energies, and the proper setting for
a discussion of the zero-point
_{} in
this case is locally supersymmetric supergravity.
Weinberg (1989, p. 6)
notes "it is very
hard to see how any property of supergravity or superstring theory
could make the effective cosmological constant sufficiently small".
Witten (2001)
and Ellwanger (2002)
review more recent developments
on this issue in the superstring/M theory/branes scenario.
Back.

^{24} For subsequent more detailed
discussions of this issue, see
Zel'dovich (1981),
Weinberg (1989),
Carroll, Press, and Turner
(1992),
Sahni and Starobinsky
(2000),
Carroll (2001),
and Rugh and Zinkernagel
(2000).
Back.

^{25} Early references to this point are
Linde (1974),
Dreitlein (1974),
Kirzhnitz and Linde
(1974),
Veltman (1975),
Bludman and Ruderman
(1977),
Canuto and Lee (1977),
and Kolb and Wolfram
(1980).
Back.