This insert explicates cosmological constant and dark energy.
In modern cosmological models, they are both effects used to reproduce the observed acceleration of the universe.
However, cosmological constant has been used since the beginning of modern cosmology in 1917 for other reasons.
There is a horrible problem in nomenclature using cosmological constant and dark energy.
The two quantities are NOT the same thing, but they are often conflated because dark energy (i.e., the simplest version thereof) has the same effect in cosmological models as the cosmological constant though it is different in other respects.
So people often use the terms interchangeably as if they were synonyms as a simplification. They rely on context to specify the meaning. But this way of specifying the meaning is lousy for intro astronomy courses.
Yours truly has tried different ways of being clear. One way is:
If you think this is complexity over nothing, you are right.
The bottom line is yours truly will often just use the terms cosmological constant and dark energy interchangeably as if they were synonyms as simplification. Context must specify the meaning.
However, for various reasons of historical interest,
Einstein wanted
a STATIC cosmological model---one
that was full of stars distributed
uniformly on average throughout
space
Note stars NOT
galaxies since no one
knew for sure about galaxies
in 1917 and
Einstein may have known nothing.
For why he wanted these things,
see IAL 30: Cosmology:
The Static Universe Assumption.
In fact, Einstein
could NOT find a
static cosmological model
with general relativity
as he had originally proposed it (No-520).
Also, in fact, he could NOT find any
cosmological model
with general relativity
as he had originally proposed it
(Cormac O'Raifeartaigh et al.,
Einstein's 1917 Static Model of the Universe:
A Centennial Review, 2017;
Cormac O'Raifeartaigh,
Historical and Philosophical Aspects of the Einstein World, 2019).
As Einstein
used it, cosmological constant
is NOT a form of energy
at all
in a strict general relativity sense.
It is a quantity that affects
the geometry of
spacetime.
However, the geometry of
spacetime does encode
energy in a non-local way: i.e., in a way
that does NOT allow you to say there is so much
energy here or there.
For a further discussion of this intriguing point, see
Non-Local Encoding of Energy in General Relativity
in Cosmology file:
cosmological_redshift.html.
The repulsion could be considered at kind
of anti-gravity,
but its NOT what one ordinarily means by that term and no one calls it that.
In fact, Einstein had NO
accurate value for the fine tuning.
So Einstein's STATIC MODEL
had, in fact, an undetermined free parameter.
What if the cosmological constant
was NOT fine tuned for a STATIC MODEL.
The model would evolve in time, but Einstein
did NOT investigate that evolution
in 1917.
We call Einstein's STATIC MODEL
the Einstein universe (1917).
If he had derived it, then the
Einstein universe (1917)
and all simple
general relativistic
cosmological models
including the
Einstein universe (1917)
would have been easy to find.
In fact, the
Einstein universe (1917)
is the only STATIC MODEL among the simple
general relativistic
cosmological models.
All others are in general expansion or contraction.
Alexander Friedmann (1888--1925)
and, independantly,
Georges Lemaitre (1894--1966)
derived the Friedmann equation
in the 1920s.
You must understand that
Einstein's
research in 1917
was a pioneering effort before all the mathematical tricks for dealing with
general relativity had
been developed by Einstein
and many others in the decades after
1917.
Einstein described his
search for the
Einstein universe (1917)
as a "rough and winding road"
(Cormac O'Raifeartaigh et al.,
Einstein's 1917 Static Model of the Universe:
A Centennial Review, 2017, p. 18).
The cosmological constant,
whether true or NOT,
has a definite singular meaning as discussed in the figure above
(local link /
general link: general_relativity_field_equations.html).
Dark energy, however,
is a word for our ignorance since there are many possible
versions of dark energy
and which if any are true we do NOT know.
However, when people say
dark energy,
they mean the simplest version of
dark energy unless context
says they mean something else: e.g.,
any kind of dark energy.
Note we call the
dark energy
by that name
because we do NOT see it so far in any other way than through its effect on
the universal expansion.
It's very "dark".
For another thing very dark, see the figure below
(local link /
general link: vantablack.html).
The simplest version of dark energy
(hereafter in this subsection usually just
dark energy)
is a sort of stuff that is
CONSTANT in
space
(NO matter how that expands/contracts
via general relativity)
and time
(most especially
cosmic time).
And it has formally NEGATIVE pressure which
we will explicate shortly.
The simplest theory of what the
dark energy is is that it is
vacuum energy
which is a special case of
zero-point energy
of quantum mechanics.
However, the most natural estimate of
vacuum energy is 50 to 120
orders of magnitude
larger than what the
dark energy has to be to
match observations: i.e., be replacement for
the cosmological constant.
This profound discrepancy between theory
and observation is the
cosmological constant problem.
OK, what is
NEGATIVE pressure?
Well, it's a
pressure that pulls in rather than pushes out.
This means that as a universe expands,
you need to put mass-energy
into it in order to expand it against the pulling in
NEGATIVE pressure.
In order to keep the
density
of the dark energy
constant in space
and time
as the universe expands,
the dark energy
pressure has to be constant
in space
and time also.
This means there is NEVER any imbalance and the
dark energy
pressure
NEVER actually pulls on itself or anything.
Nothing feels it in theory.
In fact, this mysterious
dark energy
pressure p_dark must
obey the
formula p_dark = - ρ_dark/c**2,
where the density of the
dark energy is ρ_dark.
It seems to yours truly that
So we have some requirements for simplest version
the dark energy, but
we still do NOT know what it is.
The expression
dynamical dark energy
is a catchall expression for
all versions of dark energy
that are NOT simplest version where
dark energy
is constant in space
and time.
There are many theories
of dynamical dark energy,
but which if any are true, we do NOT know.
Theorists keep coming
up with new
theories
of dynamical dark energy
in the hope of solving all problems in
cosmology.
And that is all we need to say
about dynamical dark energy.
UNDER RECONSTRUCTION BELOW to section 4
Now constant
dark energy
has a constant energy density in time and space.
It is about the simplest kind of
dark energy
imaginable.
That the cosmological-constant dark energy density
is constant in time is UNUSUAL. For example, matter density
must decrease because of the
expansion of the universe.
In fact, as
cosmic scale factor a(t)
increases, volumes increase by a(t)**3, and matter density
falls as a(t)**(-3).
The cosmological-constant dark energy density
can be treated as a contribution to
Omega: we denote this
contribution by
Omega_Lambda.
But the dark energy
need NOT have constant density in either time or space and
it may interact with other forms of mass-energy
in ways we do NOT know.
Until circa
2018,
nothing in the observations told us to go beyond the simple theory of
cosmological-constant dark energy.
But currently, there is
Hubble tension:
see the discussion in the figure below
(local link /
general link: hubble_constant_problem.html).
Yes, quantum field theory
(i.e., relativistic quantum mechanics) suggests there could be
cosmological-constant dark energy,
but alas
predicts its density to be 10**120 times bigger than needed to fit the
observed acceleration
(e.g.,
Carroll, S. 2003, p. 3, Why is the Universe Accelerating?).
This remarkable OVERESTIMATE suggests that the
dark energy
is more complex than the simple
cosmological-constant dark energy
with a density which is constant in time and space.
Another remarkable thing about
dark energy is
that
the cosmological-constant dark energy density
from the analysis of the
CMB data
is
Omega_Lambda
= 0.6853(74),
which is COMPARABLE to the
matter density
Omega_matter
= 0.3147(74).
The reference for these values is
Planck 2018 results. I. Overview and the cosmological legacy
of Planck 2018.
There may be some deep reason why the
dark energy
and matter should be COMPARABLE in which case the
dark energy
CANNOT be simply a
cosmological-constant dark energy.
On the other hand, possibly the COMPARABILITY of
Omega_Lambda
= 0.6853(74)
and
Omega_matter
= 0.3147(74)
(Planck 2018 results. I. Overview and the cosmological legacy
of Planck 2018)
is explained
by the anthropic principle.
(See IAL 0:
A Philosophical and Historical Introduction to Astronomy: The Anthropic Principle.)
Say there are an infinity or a quasi-infinity of
pocket universes
in a multiverse
with different parameters set by some probability distribution of parameters.
Those NOT somewhat like our own
in mass-energy
contents
may NOT be able to support life.
Too small a
cosmological-constant dark energy density
and the universe may NOT have formed the right
kind of galaxies
and stars.
Too large a
cosmological-constant dark energy density
and the universe would have expanded too quickly
ever to form galaxies and
stars.
So the ratio of
dark energy
and matter may just be a roll of the
pocket universe
parameter
dice.
Now it is very hard to prove an argument based on the
anthropic principle.
But such an argument could be falsified if the
dark energy density
and matter density
(the sum of dark matter density
and
baryonic matter density)
were fine-tuned beyond
the needs (so far as we can tell) of making our
pocket universes
suitable for us to be here.
For example, the ratio is now of
dark energy density
to matter density is approximately 2 to 1.
If the ratio were exactly 2 to 1, that is more exactness
than is needed for a
pocket universe
to that can support life like us,
and strongly suggests
dark energy
and matter---NOT
just that random throw of the
pocket universe
parameter
dice.
But there is NOT such exactness, and
so the multiverse
passes this significant
falsification test.
As noted above in section A Horrible Problem in Nomenclature,
the cosmological constant
symbol
is the capital
capital Greek Lambda Λ
which display in
the figure below
(local link /
general link: greek_letter_lambda.html).
Often one just says Lambda instead
of the cosmological constant
or the simplest version of
dark energy which is
constant in space
and time.
We sometime say
the cosmological constant Λ
for clarity.
Is the cosmological constant
an ad hoc hypothesis?
First, what is an ad hoc hypothesis?
An ad hoc hypothesis is a
hypothesis
invented JUST to make a theory give a very specific
theory often without any other
consequences.
Because theories
theories should be general
ad hoc hypotheses
are likely to be wrong.
An especially egregious
kind of ad hoc hypothesis is
a fudge factor.
A fudge factor is a
quantity
one just inserts into a calculation to get the answer one
expects---fudge factor
because one cooks it up.
Students are quite adept at creating
fudge factors on
tests.
For example, leprechauns
are the cause of ...
For leprechauns
see the figure below
(local link /
general link: leprechaun.html).
The argument is
that cosmological constant
satisfies Occam's razor in that
is the simplest and most natural way of getting a
STATIC MODEL of the universe
from the Einstein field equations.
It is NOT a case of a hodge-podge of hypotheses that have little chance of being right.
So it could be argued that cosmological constant
STATIC cosmological constant
had some chance of being right if
Einstein's assumption of a
STATIC
universe had been right---which it wasn't, of course.
One could also say
that cosmological constant
is by far the simplest modification of the
Einstein field equations
that one could imagine
and the specific specific effect it was usef for applies to
whole universe.
Thus, the modified
Einstein field equations
are a very interesting alternative to the
original Einstein field equations,
and so cosmological constant
NO matter what one thinks of
STATIC cosmological models.
The upshot is the cosmological constant
is NOT an
ad hoc hypothesis in the
opinion of yours truly.
In fact,
Alexander Friedmann (1888--1925)???
and Georges Lemaitre (1894--1966)
in their work on
expanding universe
cosmological models did NOT
dispense with the
cosmological constant.
This shows that they too thought it was NOT
an ad hoc hypothesis.
For an interesting historical tidbit on dark energy,
see the discussion below
on Erwin Schroedinger (1887--1961)
in the figure below
(local link /
general link: erwin_schroedinger.html).
Question: Density
parameter Omega is:
From the analysis of the
CMB data, galaxy cluster data, and
Type Ia supernovae ASSUMING the dark energy
is a cosmological-constant dark energy,
one finds
Omega_Lambda=0.6853(74)
(see Planck 2018 results. I. Overview and the cosmological legacy
of Planck 2018).
Answer 2 and 3 are right. But answer 3 is best in this context.
Is there any reason for believing there could be
just a
cosmological-constant dark energy
from physical theory?
Yours truly believes that
the
cosmological constant
is NOT an
ad hoc hypothesis
even though it was introduced originally by
Albert Einstein (1879--1955)
to get a specific effect:
a STATIC cosmological model
which we call the
Einstein universe (1917).