Friedmann equation - show Big Bang happened given conditions

In summary: You can see this by the following: Consider a function a(t) such that a'(t)>0 for all t. Then if we go back in time, a(0)>a(t) for all t, so a(t) decreases for all t. If we go back far enough, we eventually get to a time t1 where a(t1)=0. So a(t) decreases for all t>t1.
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binbagsss
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Homework Statement



Use Friedmann equations to show that if ##\dot{a} > 0##, ##k<0## and ##\rho>0## then there exists a ##t*## in the past where ##a(t*)=0##

Homework Equations


[/B]
Friedmann :

##(\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2}##

The Attempt at a Solution


[/B]
re-arrange as:

##\dot{a}^2=1+\frac{8\pi G}{3}a^2\rho##

where I have used ##k<0## can be/is standard to set to ##k=-1##

Then since ##\rho>0##, and ##\dot{a}##>0 implies I should take the positive square root of this , this implies that ##\dot{a}>1## for all time.

Now I do not follow the next part of my solution which says:

Thus ##a(t)>0## is decreasing at a rate that is always greater than ##1## and there was necessarily a finite time ##t*## in the past where ##a(t*)=0##

How have we concluded a decrease, do we not need ##\ddot{a}## to make this conclusion? Can someone please explain where this comes from?

Many thanks in advance
 
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  • #2
I don't think you can just set k=-1, although it doesn't change the conclusion here. It is not necessary to do that, all you need is k<0 in the following argument.

If the first derivative is positive, then the original function decreases if we go back in time. This does not depend on the second derivative.
 
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Related to Friedmann equation - show Big Bang happened given conditions

1. What is the Friedmann equation and how does it relate to the Big Bang theory?

The Friedmann equation is a mathematical equation that describes the expansion of the universe in the context of the Big Bang theory. It relates the rate of expansion of the universe to the matter and energy content of the universe.

2. How does the Friedmann equation show that the Big Bang happened?

The Friedmann equation is derived from Einstein's theory of general relativity, which states that the universe is expanding. When the Friedmann equation is solved, it shows that the universe must have started from a singularity, or a single point of infinite density and temperature, which is a key aspect of the Big Bang theory.

3. What conditions are necessary for the Friedmann equation to show that the Big Bang occurred?

The Friedmann equation requires certain parameters, such as the density and pressure of matter and energy, to be plugged in for it to accurately describe the expansion of the universe. These parameters are based on observations and measurements of the universe, and when they are used in the equation, it shows that the universe began with a Big Bang.

4. Can the Friedmann equation be used to prove the Big Bang theory?

While the Friedmann equation is a fundamental part of the Big Bang theory and provides evidence for its occurrence, it is not the only piece of evidence. Other observations, such as the cosmic microwave background radiation and the abundance of light elements, also support the Big Bang theory.

5. Are there any alternative explanations for the expansion of the universe besides the Big Bang theory and the Friedmann equation?

There are currently no widely accepted alternative explanations for the expansion of the universe. The Big Bang theory, supported by the Friedmann equation, is the most widely accepted explanation among scientists. However, there are ongoing studies and research being conducted to further understand and explain the expansion of the universe.

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