cosmology - If we were able to double the mass of the ordinary matter in the universe it will recollapse?

What will happen if we suppose to be able to add mass at the universe? For example doubling the mass of the ordinary matter?

Answer

This going to be a rather approximate answer because it involves lots of estimated quantities like the current density of matter and the value of the cosmological constant.

The second Friedmann equation tells us:

$$ \frac{\ddot{a}}{a} = \frac{-4\pi G}{3}\left( \rho + \frac{3p}{c^2} \right) + \frac{\Lambda c^2}{3} $$

It's conventional to take $a = 1$ at the current time, and I'll assume the pressure of the stuff in the universe is negligable, so the equation simplifies to:

$$ \ddot{a} = \frac{-4\pi G}{3} \rho + \frac{\Lambda c^2}{3} $$

The value of $\ddot{a}$ is negative if the expansion of the universe is slowing, and positive if the expansion of the universe is accelerating. From a quick look at the equation it should be obvious that the density of matter (baryonic and dark), $\rho$, makes a negative contribution to $\ddot{a}$, while the dark energy makes a positive contribution to $\ddot{a}$. To answer your question we need to feed in the current values for $\rho$ and $\Lambda$ and see how the two terms in the equation compare.

The current density of baryonic matter is about 0.25 hydrogen atoms per cubic metre, and the density of all matter (baryonic + dark) is about 1.6 hydrogen atoms per cubic metre. Let's take the density of all matter and use it to calculate the first term in the equation above. The result is:

$$ \frac{4\pi G}{3} \rho = 7.5 \times 10^{-37} s^{-2} $$

Wikipedia assures me that the current value of the cosmological constant is $10^{-52}$ m$^{-2}$, and this makes the second term:

$$ \frac{\Lambda c^2}{3} = 3 \times 10^{-36} s^{-2} $$

So we end up concluding that $\ddot{a}$ is positive i.e. the expansion of the universe is accelerating, just as the supernovae observations have confirmed.

Now you've asked what happens if we double the amount of matter i.e. double $\rho$. Normal (i.e. baryonic) matter is only about one sixth of the total matter, so doubling the amount of normal matter only increases the first term to $8.7 \times 10^{-37} s^{-2}$, and $\ddot{a}$ is still positive so the expansion still accelerates. Doubling the amount of all matter doubles the first term to $1.5 \times 10^{-36} s^{-2}$ and $\ddot{a}$ is still positive so the expansion still accelerates. Doubling the amount of matter makes no difference to the future fate of the universe.

Let me end with a caveat: the figures I've used above are very rough so I wouldn't place too much emphasis on the exact values. However the overall conclusion still applies i.e. that the effect of the cosmological constant currently dominates. I estimate you'd need to increase the total matter content of the universe by about a factor of 7 for matter to become dominant again.