- De Sitter, who
was a colleague and sometimes friendly disputant
with Albert Einstein (1879--1955),
presented
the de Sitter universe---as we call it---in
1917 just after the
Einstein universe
and in the same year.
The de Sitter universe
is a general relativity
cosmological model
that assumes the
cosmological principle
and cosmological constant.
- Remarkably both Einstein
and De Sitter failed to
discover the
Friedmann equation
which was discovered by
Alexander Friedmann (1888--1925) in
1922
and independently by
Georges Lemaitre (1894--1966)
by 1927
(see Wikipedia: Georges Lemaitre: Career;
Wikipedia: Georges Lemaitre: Work).
The Friedmann equation
is dervied from
general relativity
assuming
cosmological principle
and the perfect fluid
for the mass-energy contents
of the universe
and allows many
cosmological models
(including
the Einstein universe
and the de Sitter universe)
do be easily derived.
Einstein
and De Sitter
had to use klutzier means to derive their
cosmological models
more directly from
the Einstein field equations.
Their cosmological models were
pioneering works.
- The de Sitter universe
was the first
expanding universe model.
But it has with NO
mass-energy,
and thus zero
density for
the perfect fluid.
It just has
a cosmological constant
tuned to give
exponential expansion
for space.
Recall space has a structure
according to general relativity:
e.g., Euclidean space and
many kinds of
non-Euclidean space
(i.e., curved space).
- Exponential expansion
is expansion obeying an
exponential function
as illustrated in Image 2 below.
- Image 2 Caption: "The
exponential function exp(x)
(i.e.,
(e = 2.71828 ... raised to the power x),
on the interval -5
to 5.
Grid
unit
is 1 along both
axes."
(Somewhat edited.)
Note despite appearances in Image 2, the
exponential function only goes
asymptotically
to zero
as the x coordinates
goes to negative
infinity: i.e., as x → -∞, one has
exp(x) → 0.
- Image 3 Caption: See the blue
curve
for the cosmic scale factor a(t)
for the de Sitter universe (1917).
The formula for this
cosmic scale factor a(t) is
a(t) = a_0 exp(t/t_H) ,
where
exp(x) is exponential function,
t is cosmic time
measured from t = -∞ (i.e., the de Sitter universe has NO
Big Bang and is
eternal in both
directions of
cosmic time),
a_0 is some fiducial scale distance (usually set to 1 for the present epoch),
and
t_H is the Hubble time
for the de Sitter universe.
- The original de Sitter universe
had the same geometry
as the
Einstein universe.
Thus, the
original de Sitter universe
is a 3-dimensional surface of a 4-dimensional
hypersphere: i.e.,
it is finite, but unbounded, hyperspherical space
(Bo-98;
No-520;
CL-28--29,159;
O'Raifeartaigh 2019, p. 16).
In modern times, we generally think of the
de Sitter universe as being
infinite with
Euclidean geometry (i.e. flat geometry).
- The de Sitter universe
had a vogue for 15
years
or so after 1917
since it had
universal expansion
which was observationally discovered by
Edwin Hubble (1889--1953)
in 1929 and had been partially anticipated
earlier in the 1920s by others
(most importantly by
Georges Lemaitre (1894--1966)
in 1927:
see
Hubble 1929,
No-523--524,
Livio 2011,
Steer 2012,
Way 2013,
Trimble 2012,
Trimble 2013,
Elizalde 2018).
But since the
de Sitter universe
had
NO mass-energy,
it was assumed RULED OUT by reality as being exactly true.
The need for more realistic
expanding universe models
was filled by models with
mass-energy,
some with and some without the
cosmological constant.
- However, de Sitter universe
predicted the
cosmological redshift
(O'Raifeartaigh et al. 2017, p. 39)
which made it of interest circa the
1920s when
most galaxies
(known to be
galaxies
after 1924) exhibited
redshifts.
The prediction of the
cosmological redshift
by the
de Sitter universe
may have encouraged Hubble
in the interpretation of his data.
- The de Sitter universe
has NOT gone away.
The Λ-CDM model
is asymptotically evolving toward being
a de Sitter universe
with either a
cosmological constant
or some more complex form
dark energy---where we count
the cosmological constant
as the simplest form of dark energy
even if it is NOT really
an energy in order to save on
words though we arn't saving on them at this
moment.
Also the inflationary era
(cosmic time t = 10**(-36)--10**(-32) estimated)
of inflation cosmology
is a de Sitter universe phase
with some form of
mass-energy
filling the role of a
cosmological constant.
- Note
that the steady state universe,
which is of historical interest,
is a de Sitter universe
in which the continuous creation of
mass-energy
fills the role of the
cosmological constant.
- Note also that the
Einstein universe (1917)
and
de Sitter universe (1917)
are NOT either the same as the
Einstein-de Sitter universe
which
Einstein and
De Sitter presented
in 1932.
The
Einstein-de Sitter universe
was probably the favored
cosmological model
circa 1965--circa
1998.
The
Einstein-de Sitter universe
is Friedmann-equation Λ=0 model:
i.e., it is a Friedmann-equation (FE) model
with the
cosmological constant (AKA Lambda, Λ)
set to zero.
It can have any
universe geometry: i.e.,
hyperbolic space (Ω < 1),
Euclidean space (Ω = 1), or
or hyperspherical space (Ω > 1).
People tended to theoretically favor
Euclidean space (Ω = 1)
because that was simplest
universe geometry, but
in fact, in those days the observations tended to favor
hyperbolic space (Ω < 1).
See also
universe_geometry.html.