The Big Bang, Critical Density and Dark Matter

The Big Bang

Hubble’s law suggests that at some time in the past, all the matter in the universe was far more concentrated today. It was then blown apart on an immense explosive called big bang, giving all observable matter more or less the velocities that we observe today.

According to Hubble law, the matter at a distance away from us is travelling with speed \(v = H_0r\).

The time needed to travel a distance r is given by, \( t = \frac rv \)

\begin{align*} \therefore t &= \frac {r}{H_0r} = \frac {1}{H_0} \\ &= 4.3 \times 10^{17} \: s\\ &= 1.4 \times 10^{10} \: \text {years} \\\end{align*}

By this hypothesis, the big bang occurred about 14 billion years ago. It assumes that all speeds are constant after the big bang; that is it neglects any changes in the expansion rate due to gravitational attraction or other effects.

The events that went on during succeeding time after the big bang are summarised as follows:

1. 14 billion years ago initial expansion began. This event was referred to be singularity because at this time the volume of the universe was zero and density of mass-energy was infinite.
2. At t = 10^-43 s the temperature of the universe was about 10³² K, the average energy per particle was about 10¹⁹ The entire universe was much smaller than a proton.
3. At t = 10^-35 s the temperature had decreased to about 10²⁷ K and average energy to about 10¹⁴, The universe has undergone a tremendously rapid inflation increasing the size by a factor of about 10³⁰.
4. At t = 10^-32 s the universe was a mixture of quarks, leptons and mediating bosons.
5. At t = 10^-6 s the temperature was about 10¹³ K and the typical energies were 1GeV. At this time, quarks began to bind together to form nucleons and antinucleons.
6. At t = 1 min, the universe has now cooled enough so that protons and neutrons in colliding, can stick together to form the low mass nuclei H², He³, He⁴, and Li⁷ the predicted relative abundance of these nuclides are just what we observe in the universe today.
7. At t = 300,000 years, the temperature was about 10⁴ K and electrons can stick together to bear nucleus when they collide forming atoms.

Atoms of hydrogen and helium under the influence of gravity begin to clump together, starting the formation of galaxies and stars. Early supernovas spewed out the various elements heavier than helium that later became incorporated in stars and in their satellite planets.

Critical Density

The average density of matter in the universe determines whether the universe continues to expand indefinitely or not. The particular density needed just to stop the expansion of the universe is called critical density.

Expression for critical density

Consider a large spherical volume of the universe with the radius R and mass M containing many galaxies as shown in the figure. Let the mass of our universe m and is located at the surface of the sphere. According to the cosmological principle, the average distribution of the matter within the sphere is uniform. Let r be the density of matter inside this sphere.

The total energy E of the galaxy is the sum of its kinetic and potential energies, that is

\begin{align*} E &= \frac 12mv^2 + \frac {(-GMm)}{R} \\ &= \frac 12mv^2 - \frac {GMm}{R}\dots (i) \\ \end{align*}

If E is positive, our galaxy has enough energy to escape from the gravitational attraction of the mass inside the sphere, in this case the universe should keep expanding forever. If E is negative, our galaxy cannot escape and the universe should eventually pull back together. The cross over between these two cases occurs when E = 0, and r = r_c.

\begin{align*} \frac 12mv^2 &= \frac {GMm}{R} \dots (ii) \\ \text {But}\: M &= \frac 43 \pi R^3 \rho _c \dots (iii) \\ \end{align*}

If v be the speed of our galaxy relative to the centre of the sphere, then by Hubble’s law,

\begin{align*} v &= H_0R \dots (iv) \\ \text {From} \: (ii), (iii) \: \text {and} (iv) \: \text {we get}, \\ \frac 12m(H_0R)^2 &= \frac {Gm}{R} \left ( \frac 43 \pi R^3 \rho \right ) \\ \text {or,} \: \rho_c &= \frac {3H_0^2}{8\pi G} \dots (v) \\ \text {putting}\: H_0 &= 2.3 \times 10^{-18} \: s^{-s}, \: G = 6.677 \times 10^{-11} \: kg m^2s^{-2} \: \text {in equation} \: (v) \: \text {we get} \\ \rho _c &= \frac {3(2.3 \times 10^{-18})^2}{8\pi(6.67 \times 10^{-11})kg\:m^{-3} } \\ &= 6.3 \times 10^{27} \: kg\: m^{-3}\end{align*}

The mass of a hydrogen atom is 1.67 ´ 10^-27 kg so this density is equivalent to about four hydrogen atom per cubic meter.

Dark Matter

Dark matter is the non-luminous material distributed throughout the universe that cannot be directly detected by observing any form of electromagnetic radiation but whose existence is suggested by gravitational effects on the visible matter. According to present observations of a structure larger than the galaxy, big bang cosmology, dark matter and dark energy account for the vast majority of the mass in the observable universe.

The galaxies near the Milky Way appear to be rotating faster than the rotation rate expected from the amount of visible matter that appears to be in these galaxies. Many astronomers believe that 96% of the matter in a typical galaxy is invisible. Some astronomers argue that the cluster galaxies are bound together from billions of years by the gravity due to the presence of enough mass which include up to 96 % dark matter and energy (72% dark energy, 24% dark matter).

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.