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The Hubble parameter is defined by:

H(t) = a'(t) / a(t)

where a is the scale factor which is a function of cosmological time t.

This definition is equivalent to the Hubble relation:

v(t) = H(t) r(t)

where v(t) and r(t) are the proper velocity and distance of an object at cosmological time t.

I assume that this relationship is exact provided that we acknowledge that the Hubble parameter H(t) is constant only in space and *not* in time. It can be interpreted as simply asserting that space is expanding uniformly at a rate H(t).

Now let us assert that a universal speed limit exists, namely the velocity of light c. We know this is true observationally as the redshift diverges as v -> c.

This assumption allows us to define the Hubble radius R(t) by

c = H(t) R(t).

Then the Hubble radius, R_0, at the present time t_0 is given by:

c = H_0 R_0

where H_0 is the present Hubble parameter value.

As we have:

R(t) = R_0 a(t)

R(t) = (c / H_0) * a(t)

Therefore

a'(t) / a(t) = c / R(t)

a'(t)/ a(t) = H_0 / a(t)

So that finally

a(t) = H_0 * t

or equivalently

R(t) = c * t

Thus we seem to have derived a scale factor with a linear time dependence using only two facts:

1. Space is expanding uniformly.

2. A cosmological speed limit, namely c, exists.

Is this correct?

I expect I'm also assuming that space is flat.

This very simple scale factor time dependence, the so-called coasting cosmology, is actually close to what is observed.

H(t) = a'(t) / a(t)

where a is the scale factor which is a function of cosmological time t.

This definition is equivalent to the Hubble relation:

v(t) = H(t) r(t)

where v(t) and r(t) are the proper velocity and distance of an object at cosmological time t.

I assume that this relationship is exact provided that we acknowledge that the Hubble parameter H(t) is constant only in space and *not* in time. It can be interpreted as simply asserting that space is expanding uniformly at a rate H(t).

Now let us assert that a universal speed limit exists, namely the velocity of light c. We know this is true observationally as the redshift diverges as v -> c.

This assumption allows us to define the Hubble radius R(t) by

c = H(t) R(t).

Then the Hubble radius, R_0, at the present time t_0 is given by:

c = H_0 R_0

where H_0 is the present Hubble parameter value.

As we have:

R(t) = R_0 a(t)

R(t) = (c / H_0) * a(t)

Therefore

a'(t) / a(t) = c / R(t)

a'(t)/ a(t) = H_0 / a(t)

So that finally

a(t) = H_0 * t

or equivalently

R(t) = c * t

Thus we seem to have derived a scale factor with a linear time dependence using only two facts:

1. Space is expanding uniformly.

2. A cosmological speed limit, namely c, exists.

Is this correct?

I expect I'm also assuming that space is flat.

This very simple scale factor time dependence, the so-called coasting cosmology, is actually close to what is observed.

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