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Hubble Units:
The Hubble constant H_0
is the relative rate of
expansion of the universe
(or the rate of
expansion of the universe
per unit length).
It is a fundamental parameter
of modern
cosmology.
The
Hubble units
are characteristic quantities
for the observable universe
dervied the
Hubble constant.
Characteristic quantities
are quantities that characterize
a system.
A characteristic quantity
may be just a vague scale size (except for its precise definition)
or it can have a precise meaning (beyond what its definition suggests).
The
Hubble units
are of the former class, except that
for the critical density
and except for
certain
Friedmann-equation (FE) models
where a precise meaning many occur.
The
Hubble units
(except that
for the critical density)
are of less interest since circa
1998
when a
standard model of cosmology (SMC, Λ-CDM model)
appeared for which
characterizing parameters with
precise meanings exist.
Even if the
Λ-CDM model needs revision
or replacement
(which is may due to the
Hubble tension),
those characterizing parameters are likely
to endure.
What is the Hubble tension?
Beginning circa
and continuing to the present (circa 2021),
there has been an disagreement
in the determination of the
Hubble constant
between its
indirect measurement (H_0 ≅ 68 (km/s)/Mpc) and its
direct measurement (H_0 ≅ 74 (km/s)/Mpc)
(see Wikipedia:
Hubble's law: Measured values of the Hubble constant).
Circa 2021 the disagreement
is ∼ 4
standard deviations
(i.e., 4)
σ).
The disagreement is called the
Hubble tension.
Resolving the
Hubble tension
may as aforesaid require revising or replacing the
Λ-CDM model.
For more details, see
hubble_tension.html
and
Wikipedia:
Hubble's law: Measured values of the Hubble constant.
Given the
Hubble tension,
it seems best in calculating
Hubble units
to adopt a
fiducial value
for Hubble constant
that CANNOT be wrong by more than a few percent and is a nice
round number.
So we adopt H_0_fiducial=70 (km/s)/Mpc.
We then define the reduced
Hubble constant
h_70 = H_0/[70 (km/s)/Mpc] and write the
Hubble units
with explicit h_70 factors.
Hubble Unit List:
- Hubble constant H_0 = 70*h_70
(km/s)/Mpc: relative rate of
expansion of the universe
(or the rate of
expansion of the universe
per unit length).
- Hubble time = 1/H_0
= (4.4081*10**17 s)/h_70 = (13.968 Gyr)/h_70:
For cosmological models
with Big Bang singularity,
the Hubble time
should be of order the
age of the observable universe.
In the
Λ-CDM model,
current best value for
age of the observable universe = 13.797(23) Gyr (Planck 2018).
It is actually just a coincidence that
Hubble time
and the current value are so close.
- Hubble length = L_H = c/H_0 = 4.2827 Gpc/h_70 = 13.968 Gly/h_70:
For cosmological models
with Big Bang singularity,
the Hubble length
should be of order the
proper radius of the observable universe.
In the
Λ-CDM model,
current best value for
proper radius of the observable universe = 14.25 Gpc = 46.48 Gly
which, in fact, is 3.327 times
the Hubble length.
So there no close agreement between
Hubble length
and
proper radius of the observable universe.
- critical density ρ_c = 3H_0**2/(8πG) = (9.20387*10**(-27))*h_70**2 kg/m**3 = (1.35983*10**11)*h_70**2 M_☉/Mpc**3:
In the Friedmann-equation (FE) models
(which include the Λ-CDM model),
the ratio of average density ρ of
all the mass-energy
of the universe
(i.e., baryonic matter,
electromagnetic radiation (EMR)
(most importantly the
cosmic background radiation),
cosmic neutrino background,
dark matter)
to the critical density
determines the
spatial geometry.
This ratio is the
density parameter
Ω = ρ/ρ_c:
- Ω < 1:
hyperbolic space or
negative curvature space.
- Ω = 1:
Euclidean space or
flat space.
- Ω > 1:
hyperspherical space or
positive curvature space.
If the
Friedmann-equation (FE) models
are taken to apply to the
whole universe
(which formally they do)
rather than to a part (in which the
observable universe is embedded),
then
negative curvature space
and
flat space
are imply an infinite
universe or
open universe
and the
positive curvature space
implies a finite
universe or
closed universe,
but with
NO boundary: the 2-dimensional analogue is the surface of
a sphere.
Currently, the best
observed Ω = 0.000(5) (Planck 2015)
which is means the
observable universe
has flat space
to within observational error, but
the possibility of slight
negative or positive
curvature is open.
Local file: local link: hubble_units.html.
File: Cosmology file:
hubble_units.html.