Further mathematical detail
Relativistic redshift formula
If the velocity (v) is comparable with the speed of light (c), then the formula relating the velocity and redshift (z) is
Where θ is the angle the light source is moving with respect to the observer. θ = 0 degrees means the light source is moving directly away from the observer, θ = 180 degrees means the light source is moving directly towards the observer and θ = 90 degrees means the light source is moving at right angles to its direction from the observer.
The deceleration parameter
As discussed previously, the distance d(t) at a time t of an object moving away from us due to the expansion of the Universe, is given by:
d(t) = do.a(t)
where do is the distance of the object at the current age of the Universe (to) and a(t) is the cosmic scale factor.
The deceleration parameter is a dimensionless number and is defined as:
Where a’(t) is the first derivative of a(t), a’’(t) is the second derivative. The minus sign means that if the second derivative is negative, which means the rate of increase of a(t) is slowing down, then q(t) will be positive.
Over the last sixty years or so, most models of the Universe have had a deceleration parameter between -1 (which is a rapidly increasing exponential acceleration) and +3 which is a rapid deceleration. If the deceleration parameter is lower than -1 it cause some interesting challenges because it would mean that there is a faster than exponential expansion.
The Hubble parameter H(t) is equal to the recessional velocity divided by the distance. If we have an object a distance d(t) away then
d(t) = doa(t)
The recessional velocity v(t) at a distance d(t) is given simply by differentiating which gives
d’(t) = doa’(t)
Therefore, the Hubble parameter is given by.
If we differentiate the above expression, using the product rule, with u = a’(t) and v= 1 / a(t), then we get
Multiplying both sides of equation 3 by the following factor
Using the definitions of q(t) and H(t) given in equations 1 and 2 gives
A simple example
A simple example
If we have a Universe, with an accelerating expansion in which the scale factor increases as the time squared:
a(t) = (t/to)2 (In reality the scale factor will be a far more complex function of time than this).
Then a’(t) = 2t/to2 and a’’(t) = 2/to2
Then from equation 1 the deceleration parameter q(t) is:
Which simplifies to q(t) = – ½ . The value is negative indicating an accelerating expansion
The Hubble parameter H(t) from equation 2 is:
Which simplifies to H(t) = 2/t. As the value of the Hubble parameter is inversely dependent on t, its value decreases with time.