Redshift Academy


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 (brightness).  
By comparing stars at assumed comparable distances 
in the Small Magellanic Cloud, 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.  This allowed him 
to determine its actual or absolute brightness.  
Knowing the relationship between distance, apparent 
brightness and absolute brightness, allowed for the 
distance to any cepheid variable to be calculated.

Hubble discovered the following relationship 
between the recession velocity, v, and the proper 
distance, D, and associated this with the expansion 
of the universe:

           v = H₀D

Where H₀ is the HUBBLE CONSTANT.  This is HUBBLE'S 
LAW.  

The proper distance, D, is the separation measured 
at a specific time.  It can be considered as the
physical or actual measurement made by a giant 
cosmological-sized ruler between the star and the 
Earth.  Since the Universe is expanding, the proper 
distance between two distant objects (which aren’t 
held together by gravity) will increase with time.  

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 COSMIC 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, and 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

      = H₀D where H₀ is the HUBBLE CONSTANT 

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)             
          /--/--/--/
         /  /  /  /
         ---------

The current estimate of H₀ is about 72 (km/s)/Mpc 
(1 Mpc ~ 3.26 x 10⁶ light years).  So at 3.26 
Mly the velocity ~ 72 km/s, at 7.52 mly it is 
144 km/s and so on, i.e.

   <- 3.26 Mly ->
  o              o -> v ~ 72 km/s
  O

Note:  H₀ is a function of time and therefore 
is not actually a constant in the traditional 
sense.

Comoving Distance
-----------------

There is another distance commonly referred used by
cosmologists to define distances between objects.
This is called to comoving distance.  The comoving 
distance, D_CM, is defined as the proper distance 
divided by the scale factor.  Thus,

D_CM = D_P(t)/a(t)

For objects which only get further apart as a result 
of the expansion of the Universe, the comoving distance 
between them does not change over time.  The comoving 
distance changes if the proper distances change due 
to a factor other than the expansion of the Universe
(i.e. gravitational attraction).  Comoving distance 
and proper distance are defined to be equal at the 
present time. 

Relativistic Redshift
---------------------

Consider a stationary source and an observer 
moving away from it with velocity, v.  Therefore,

λ_s + vt_s = ct_s

Where t_s is the period of the light in the
source's frame.  Rearranging gives:

t_s = λ_s/(c - v) 

   = λ_s/c(1 - β) where β = v/c

   = 1/f_s(1 - β)

The observer sees clocks in the source's frame 
running slower relative to clocks in his own 
frame.  The observer will measure this time to 
be:

t_o = t_s/γ where γ = 1/√(1 - β²)

Now f_o = 1/t₀ = γ/t_s = 1/t_s√(1 - v²/c²)

       = f_s(1 - β)/√(1 - v²/c²)

f_o/f_s = √[(1 - β)/((1 + β)]
    
The redshift, z, is defined as:

z = (f_o - f_s)/f_s

  = (f_o/f_s) - 1

  = (λ_s/λ_o) - 1  (f = c/λ)

  = (1 - β)/√(1 - v²/c²) - 1

  =  √[(1 - β)/(1 + β)] - 1

Using the binomial expansion, this can be written
as:

z = (1 - β)(1 + v²/2c²) - 1

For v << c

z = -β

  = -v/c

Cosmological Redshift
---------------------

The above analysis describes Doppler shift at
relativistic velocities.  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.

Using the v << c aproximation, z can be written 
in terms of the Hubble constant as:

z = v/c

  = H₀D/c  (v = H₀D)
 
  = H₀cΔt/c   (D = c.dt)

  = H₀Δt
     .
  = (a/a)Δt

  = (Δa/Δt)Δt/a

  = Δa/a

  = (a(t₀) - a(t_s))/a(t_s)

  = a(t₀)/a(t_s) - 1

∴ z + 1 = a(t₀)/a(t_s)

Distance Measurement
--------------------

We can use the above relationships to estimate the 
distance to objects.

D = v/H_o 

  = βc/H_o 

  = [((z + 1)² - 1)/((z + 1)² + 1)](c/H_o)

If v << c we can use:

v = H_oD ∴ D = v/H_o

z = v/c ∴ v = cz

D = cz/H₀

The modern discovery of TYPE 1A supernovae has 
enabled further refinement of 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 (Distance Modulus
Equation).

Age of Universe
---------------

The reciprocal of H₀ provides an estimate of the 
age of the universe.

Note [H₀] = [(L/t)/L] = [1/t] ∴ [1/H₀] = [t]

Using the current estimate of H₀ ~ 72 (km/s)/Mpc

1 Mpc = 3.086 x 10¹⁹ km

So,

1 km/1 MPc = 1/3.086 x 10¹⁹ = 3.24 x 10^-20

1/H₀ = 1/(72 x 3.24 x 10^-20) s

    = 1/233 x 10^-20 s

    = 0.0043 x 10²⁰ s

    = 4.3 x 10¹⁷ s

    ~ 13.6 x 10⁹ years.