Caption:
A cosmological lookup table for the
observable universe
based on the
Λ-CDM model
for
Λ-CDM model parameters
valid circa 2013---which are NOT
significantly different from the ones valid
circa 2021 for the purposes
of the cosmological lookup table.
For the 2021 update
of the cosmological lookup table
(which is NOT so good on-screen), see
Sergey V. Pilipenko, 2021,
Paper-and-pencil cosmological calculator, arXiv:1303.5961.
The cosmological lookup table
correlates the
cosmological distance measures:
- cosmological redshift
z = (λ_obs-λ_emis)/λ_emis = a_0/a(t)-1, where
λ_obs is the observed wavelength.
λ_emis is the emission wavelength,
a_0 = 1 is the
cosmic scale factor at
cosmic present t_0 = to the age of the observable universe = 13.797(23) Gyr (Planck 2018),
and
a(t) is the
cosmic scale factor
at the emission cosmic time t.
- Hubble parameter H(z) = (da/dt)/a whose
cosmic present value is the
Hubble constant (fiducial value 70 (km/s)/Mpc).
- comoving distance
(measured in
megaparsecs (Mpc))
which is the
physical distance (AKA proper distance)
at cosmic present which is unobservable,
except asymptotically as z → 0.
Physical distance is by definition
the spacetime interval
that can be measured at one instant in time
by a ruler.
For cosmological physical distances,
the appropriate time is
cosmic time t (with time zero being
at the mythical Big Bang singularity).
- distance modulus μ = 5*log(d_pc)-5,
where d_pc is
physical distance measured in
parsecs (pc):
1 pc = 3.08567758 ... *10**16 m = 206264.806 ... AU = 3.26156377 ... ly ≅ 3.26 ly.
Distance moduli are usually
used only for astronomical objects
in the relatively nearby
observable universe
(see Wikipedia: Distance modulus: Usuage).
For example of distance modulus,
NGC 4548 (M91) located
at 19(5) Mpc
has distance modulus μ = 31.0
(see Wikipedia: Distance modulus: Usuage).
- cosmic time t (with time zero being
at the mythical Big Bang singularity) measured in
gigayears (Gyr).
Note
cosmic present t_0 = the age of the observable universe = 13.797(23) Gyr (Planck 2018).
- lookback time = t_0 - t
measured in
gigayears (Gyr).
- size '': yours truly has no idea.
- angle 1 kpc:
This is the angle
in arcseconds ('')
subtended on
the celestial sphere
by a ruler
1 kpc in length perpendicular
to the
line of sight.
Note that the ruler decreases with
cosmological redshift z
and physical distance
until z ≅ 1.5 and then increases thereafter: the
ruler starts looking bigger as it moves
farther away, but it still looks fainter.
This behavior is a consequence of observing the
ruler in
an expanding universe:
the light from the
ruler started out toward us when the
physical distance was much smaller.
The cosmological distance measure
obained by treating the observation of the ruler
as in static universe is called the
angular diameter distance
D_A=L_ruler/θ_ruler which increases with
cosmological redshift z
until z ≅ 1.5 and then increases thereafter.
For
angular diameter distance
as a function of
cosmological redshift z,
see Image 2 in
Cosmology file:
cosmos_distance_z_10000.html.
Credit/Permission: ©
Sergey V. Pilipenko,
2013 /
No permission.
Image link: Placeholder image
alien_click_to_see_image.html.
Local file: local link: cosmological_redshift_lookup_table.html.
File: Cosmology file:
cosmological_redshift_lookup_table.html.