See my previous answer, the expansion is uniform hence exponential.
Your "hence" demands some mathematics! AG
Why uniform expansion implies exponential growth
Uniform expansion does not necessarily mean that the sphere grows linearly.
In fact, uniform expansion implies that the proportion of growth remains
constant at every moment, which is the definition of exponential growth. If
the distance between the points increases proportionally to the current
distance, then we obtain exponential expansion.
You keep doing the same thing; asserting a result without proving it.
Please start by *defining *"uniform expansion", mathematically. AG
Defining "Uniform Expansion"
Mathematically, uniform expansion means:
At each moment, the distance between two points increases by a rate
proportional to the current distance.
*This is presumably based on the geometry of an expanding sphere. Have you
proven your above comment anywhere? I surmise that your final result will
show that Hubble's law can be derived exclusively from the geometry of an
expanding sphere. Do you agree? AG *
We express this as the following differential equation:
d/dt [d(t)] = k * d(t)
where:
d(t) is the distance between two points at time t,
d/dt [d(t)] is the rate of change of the distance,
k is a constant proportionality factor (the expansion rate).
*Please use standard mathematical notation, so your d(t) should be writen
as ds. Then your differential equation needs to be rewritten, as ds/dt =
k*s (I think), which is what Brent wrote earlier. AG*
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This equation states that the rate of change of the distance is
proportional to the current distance.
Solving the Differential Equation
Rewriting the equation:
d/dt [d(t)] = k * d(t)
We can solve this by separating variables:
1 / d(t) * d(d(t)) = k * dt
Integrating both sides:
ln(d(t)) = k * t + C
Exponentiating both sides:
d(t) = e^(k * t + C)
This simplifies to:
d(t) = e^C * e^(k * t)
Let d_0 = e^C represent the initial distance between the points at t = 0.
The solution becomes:
d(t) = d_0 * e^(k * t)
Exponential Growth from Uniform Expansion
The solution shows that uniform expansion — where the rate of change is
proportional to the current distance — leads to exponential growth. The
distance between the points increases exponentially over time:
d(t) = d_0 * e^(k * t)
Summary:
"Uniform expansion" means distances grow proportionally to their current
size.
This leads to the exponential formula d(t) = d_0 * e^(k * t), where k is a
constant.
As a result, distances increase exponentially over time.
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