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Units, Constants and Useful Formulas
Hubble's Law
------------
Hubble compared recession velocities of galaxies by measuring their
redshift and comparing this with their distance. The distances were
obtained using Cepheid variable stars found in the galaxies. Cepheid
variable stars were discovered by Henrietta Leavitt and are objects
whose period of variability is related to their average luminosity.
By comparing stars at assumed comparable distances, she found that
the ones with longer periods had higher luminosities. Subsequent to
Leavitt's discovery, Harlow Shapley used parallax methods to measure
the distance to one of these cepheid variables. The allowed him to
determine the relationship between the period of the variable and its
absolute luminosity. Now the distance to any cepheid variable could
be found by measuring its period, determining its absolute luminosity,
comparing this to its apparent luminosity and applying the inverse
square law.
Hubble found that there was a positive relationship between the
recession velocity, v, and the proper distance, D, and associated
this with the expansion of the universe. Thus:
v = H_{0}D
Where H_{0} is the HUBBLE CONSTANT. This is HUBBLE'S LAW.
Hubble's law tells us that velocity is dependent on distance.
Gravitationally interacting galaxies move relative to each other
independent of the expansion of the Universe. These relative
velocities need to be accounted for separately.
Hubble's Law can be derived in terms of the expansion of space in
the following manner:
O a a a a a
o----o----o----o----o----> x
x y
Point O is fixed. a is the scale factor and is a function of t. It
represents the relative expansion of the universe. It is easy to
see that as the system is stretched in the x direction, the further
away the points are from O, the faster they must be moving.
D = an where n is an integer that represents the number of a's.
So x->y = 2. D is the distance between the objects.
. .
D = v = an
.
= a(a/a)n
.
= (a/a)D
= H_{0}D where H_{0} is the HUBBLE CONSTANT (3 x 10^{-18} s^{-1})
In 3D
D = a√{(Δx)^{2} + (Δy)^{2} + (Δz)^{2}}
. .
D = (a/a)D as before
Since a is function of time, the coordinate mesh expands with
time.
---------------
/ / / /
/----/----/----/
/ / / /
---------------
^
------------ |
/ / / / | time
/---/---/---/ |
/ / / /
------------
a(t)
---------
/ / / / a(t)
/--/--/--/
/ / / /
---------
H_{0} corresponds to the LHS of the FRW equations at the
time of observation.
The current estimate of H_{0} is about 72 (km/s)/Mpc (1 Mpc
~ 3.26 x 10^{6} light years). So at 3.26 mly the velocity is
72 km/s, at 7.52 mly it is 144 km/s and so on.
Cosmological Redshift
---------------------
Consider a light wave and an observer moving away from
the source with velocity, v. Therefore,
λ_{S} + vt = ct
or,
t = λ_{S}/(c - v) = c/(c - v)f_{S} = 1/(1 - β)f_{S} where β = v/c
The observer will measure this time to be:
t_{O} = t/γ where γ = 1/√(1 - β^{2})
Now,
f_{O} = 1/t_{O} = γ(1 - β)f_{S}
Therefore,
f_{S}/f_{O} = √[(1 + β)/(1 - β)]
The resulting redshift, z, is:
z = (λ_{O} - λ_{S})/λ_{S} ≡ (f_{S} - f_{O})/f_{O}
= λ_{O}/λ_{S} - 1
= f_{S}/f_{O} - 1
= √[(1 + v/c)/(1 - v/c)] - 1
= (1 + v/c + v^{2}/2c^{2} + ...) - 1
For v<<c we can write:
z ~ v/c
Therefore, in terms of the Hubble constant we get:
z = H_{0}D/c
.
= (a/a)D/c
Now z = (f_{S} - f_{S})/f_{S} ~ df/f = v/c
= Hdt since D = cdt
= da/a
Equating, and solving we get:
a_{0}/a_{S} = z + 1
It is important to note that cosmological redshift is different
to Doppler shift. In Doppler Shift, the wavelength of the
emitted radiation depends on the motion of the object at the
instant the photons are emitted. In cosmological redshift,
the wavelength at which the radiation is originally emitted is
stretched as it travels through expanding space. Cosmological
redshift results from the expansion of space itself and not
from the motion of an individual body.
Distance Measurement
--------------------
We can use the above relationship to estimate the distance to
objects.
D = cz/H_{0}
The modern discovery of TYPE 1A supernovae has enabled further
refinement to these techniques by allowing measurements at much
greater distances. Type Ia supernovae occur in a binary system
where 2 stars orbit one another. One of the stars is a white
dwarf. The other can be a giant star or even a smaller white
dwarf. The white dwarf pulls material off its companion star,
adding that matter to itself. Eventually, when the white dwarf
reaches a certain mass a nuclear reaction occurs, causing the
white dwarf to explode and produce a burst of light 5 billion
times brighter than the Sun. Because the chain reaction always
happens in the same way, and at the same mass, the brightness
of these Type Ia supernovae are also always the same. Using
this knowledge, the distance to the galaxy containing the
supernova can be computed by comparing the observed and absolute
intensities and again applying the inverse square law.
Comoving and Proper Distance
----------------------------
Comoving and proper distance are two closely related distance
measures used by cosmologists to define distances between
objects. Proper distance corresponds to where a distant object
would be at a specific moment of cosmological time, which can
change over time due to the expansion of the universe. Comoving
distance factors out the expansion of the universe, giving a
distance that does not change in time due to the expansion of
space. Comoving distance and proper distance are defined to be
equal at the present time.
Age of Universe
---------------
Hubble time: 1/H_{0} ~ 14 billion years.