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# The Fate of the Universe

In this quiz we will consider the ultimate fate of the universe. We learned in the quiz on Hubble's Law that the other galaxies in the universe are receding from our own Milky Way. We will answer the question of whether gravity is strong enough to hold the universe together. We will begin, however, with a more human dilemma.

A young adventurer named Jane leaves her solo spaceship to do a repair job while cruising through inter-galactic space. Being a reckless type, she goes outside the ship without being physically attached to it in any way, and without any means of controlling the ship. In a careless moment she slips and drifts out of reach of the craft. Her only hope is that the gravitational attraction between her body and the ship will eventually bring them back together.

Suppose Jane can see her spaceship drifting away from her with speed $$v$$. (In the depths of space, with no nearby reference, it is impossible to tell whether she is drifting away from the ship or if it is drifting away from her).

Is she doomed?

Like in the black holes quiz, an object will never return if it can escape to a separation where its potential energy is larger than $$0.$$

Suppose Jane's velocity is $$\SI[per-mode=symbol]{1}{\centi\meter\per\second}$$ away from the ship when they are separated by $$\SI{1}{\meter}.$$ Will Jane ever fall back to her ship?

Details and Assumptions

• Take Jane's mass to be $$\SI{60}{\kilo\gram}$$ and her ship's to be $$\SI{1000}{\kilo\gram}.$$
• Gravitational potential energy of a mass $$m_1$$ near a mass $$m_2$$ is $U_g = -G m_1 m_2/r$ where $$G=\SI[per-mode=symbol]{6.67e-11}{\meter\cubed\per\kilo\gram\per\second\squared}.$$
• Jane's kinetic energy is $$\frac12 m v^2.$$

Calculate the minimum mass of the spaceship that would allow her to survive this accident, assuming everything else stays the same

Cosmologists have little interest in the travails of reckless space explorers—we concern ourselves with grander things. However, a similar line of reasoning can be used to predict the fate of the universe.

In the Hubble's Law quiz, we observed that nearly every galaxy is moving away from our own. Just like clumsy Jane and her rocket, we can ask whether gravity is strong enough to eventually pull the matter in the universe back together again, or if the fate of the universe is eternal expansion until matter is scattered over such a vast volume that it becomes cold, dark and lifeless.

To address this question, consider an arbitrary sphere taken out of the universe. If the sphere is large enough, what assumption can we make about the volume inside it?

We are going work out the fate of the universe by considering what would happen to just this sphere if it were held in isolation. According to homogeneity, if we find that this sphere will collapse on itself, this suggests that the whole universe will eventually do the same.

Our sphere has radius $$r$$. If the average mass density of the universe is $$\rho$$, what is the mass of our sphere?

You may recognize the graph below from the Hubble's Law quiz.

It tells us the rate of expansion of the universe by showing how the relative speed $$\Delta v$$ of a distance galaxy increases with distance $$r$$ from us.

By taking data from the graph, find the speed $$\Delta v$$ of a galaxy that is a distance $$r$$ from the sphere's center.

Details

• The relationship can be written $$\Delta v = H r$$ where $$H$$ is a constant. Find $$H.$$
• You will need to convert the speed and distance into compatible units to get your answer. $$\SI{1}{Mpc} = \SI{3.1e19}{\kilo\meter}$$

We can now evaluate the total energy $$K+U_g$$ for an object with mass $$m$$ on the surface of our sphere. So far we have obtained equations for the mass of our sphere of space in terms of mass density $$\rho$$ and radius $$r,$$ and also an expression for the velocity of expansion in terms of $$H$$ and radius $$r$$.

It has potential and kinetic energy according to the equations we used earlier for Jane, and the values of the mass of the sphere (equivalent to the spaceship) and the velocity you just calculated.

Which expression below gives the total energy $$(E=K+U_g)$$ of our object on the surface of the sphere?

Details & Assumptions

• Take $$v = Hr$$, where $$H$$ is the quantity you calculated on the previous problem.

Astoundingly, we can look at the term inside the square brackets to determine the overall sign of the energy—without knowing the mass $$m$$ or the radius $$r.$$

When $$E=0$$ the universe will have just enough energy to expand forever. Using your estimate of $$H,$$ find the largest mass density that the universe can have in order to continue expanding.

Details

• If the mass density is larger, then gravity will eventually pull all the matter in the universe back together.

This is a pretty small number for a density. But then, the universe is mostly empty space, so perhaps this is not surprising. Most of the mass in the universe is in the form of hydrogen, which has an atomic mass of $$\SI{1.66e-27}{\kilo\gram}$$. At the universe's critical density, how many Hydrogen atoms, on average, would be in $$\SI{1}{\meter\cubed}$$?

You have determined the value for a critical parameter which should determine the future of our universe. This idea was developed after Edwin Hubble observed the expanding universe, but before astronomers had managed to estimate the mass density of the universe. At the time, three possible scenarios were imagined:

1. An open universe, which would keep expanding forever.
2. A closed universe, which eventually collapses in on itself
3. A flat universe. This looks a lot like an open universe, except that the kinetic and potential energy are perfectly matched - the mass density is at its critical value. If it were any less, the universe would collapse.

In the next quiz, we will set about trying to estimate the amount of mass in the universe, to see if we can determine which path our universe will follow.

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