# Cosmology

Cosmology is the study of the origins and history of the universe. It differentiates from Astrophysics in the questions the two disciplines attempt to answer. Astrophysics is concerned with the evolution of stars and the physics of the observable universe as it is. Cosmology is focused on attempting to explain how it all began and if there will be an 'end'...

The big questions which concern cosmologists include: Olber's Paradox, the Big Bang (and the Big Crunch... or Fizzle?), the expansion of the universe, cosmic background radiation, the open and closed universe models, evolution of the universe, critical density, dark matter and dark energy.

Isaac Newton believed in an infinite universe, with no beginning (or presumably end), and which was static. Newton's universe has been uniform (a fairly even distribution of matter throughout) and isotropic (looks much the same in all directions) at all times. People accepted this model, even though it failed to answer an enigmatic question. This question was reformulated in 1823 by the German astronomer, Heinrich Olbers (1758–1840), and has since been known as Olbers' Paradox:

A simple seeming question with a complex answer: Why is the sky dark at night?

This question bamboozled the greatest minds of physics, Newton included, for centuries. Who solved it?: Einstein (as usual!). If the universe is infinite and static, then everywhere we look there should be a star or a galaxy, and therefore the night sky should be ablaze with light (and energy frying our skin). So why is most of the night sky black?

Mathematically, the apparent brightness \$b\$ (or received energy per area per second) from a star with luminosity \$L\$ is: \$b = L/{4πd^2}\$.

The number of stars in a shell of stars, thickness \$t\$, at distance \$d\$ is number density x volume = \$4πd^2nt\$.

The received energy is therefore \$b = L/{4πd^2} ⋅ 4πd^2nt = Lnt\$.

Notice that \$d\$ is not present - i.e. the energy received from a shell is independent of the distance to the shell, and \$Lnt\$ is a constant. Since in an infinite universe there would be an infinite number of shells, irrespective of their thickness, the energy received would be infinite.

Clearly the night sky is not infinitely bright. Olbers' Paradox demonstrates that something is wrong: either the universe is not infinite, or it is not static, in space or time. Or all of these.

#### The Cosmological Principle

The Cosmological Principle combines two principles:

• The Homogeneity Principle
• On a large scale, the universe appears to be uniform. Any part of the sky is identical to any other part of the sky, in terms of the very distant objects.

• The Isotropy Principle
• The universe looks and behaves the same to any observer anywhere in the universe, looking in any direction. The is no special alignment of the universe's mass and energy, no accumulation along any axis or location. There is nowhere in the universe which is 'on the outer regions looking in towards the centre'. the same physical laws apply throughout the universe.

The cosmological principle implies that the universe has no centre or edge. How can this be?

One proposed explanation is that interstellar medium, for example dust, could be absorbing the radiation. However, this explanation does not stand up to scrutiny, because the energy does not disappear when absorbed, but would be re-radiated at exactly the level it is absorbed.

### The Solution to Olbers' Paradox

The radiation received by any observer, anywhere in the universe, is finite and small, for these two reasons:

1. There is a finite number of stars, and these all have a finite lifespan.
2. The universe is finite and expanding at an accelerating rate (which, incidently, we do not understand), and has a finite life (since the Big Bang, 13.6 billion years ago, give or take a day). Therefore, some galaxies are so far away (more than 13.6 billion light-years), the light has not had time to reach us yet.

Another factor is the shift in frequency of visible light due to the high speeds distant galaxies are receding from us at. This is called the red-shift.

## Is the Universe Static or Expanding?

In 1929, Edwin Hubble announced to the world the astonishing finding that the universe not only contains more than our galaxy, but that there are billions of other galaxies. And not only that, these galaxies are in nearly every case travelling away from us at high speed.

This revelation had been predicted by Georges Lemaître (1894 - 1966), who had presented the idea ten years previously to Albert Einstein, known for his insight and support for radical new ideas. However, Einstein himself had fallen into the trap of presuming the universe was static. He had included a 'fudge factor' in his General Relativity Theory in 1915, the cosmological constant, in order to force his equations from causing the universe to expand. He called this his 'greatest blunder'. He compounded the blunder by rejecting Lemaitre's thesis.

However, Einstein redeemed himself, and Lemaître, by immediately recognising the importance of Hubble's observations, and realised that his equations without the fudge described the universe perfectly.

## The Big Bang

Slowly, the radically new idea that the universe had an origin, and was currently expanding at a great rate of knots, began to accumulate supporters. Einstein publically endorsed the idea, and acknowledged Lemaître's contribution, in 1931, after being convinced by Edwin Hubble's measurements of the distance to M31 (the Andromeda Galaxy), the Red-Shift measurement of the speed of recession (how fast the galaxies are moving away from us), and Hubble's Law, that states that the speed of a galaxy is proportional to its distance from us.

In a 1948 paper (the famous Alpha, Bethe, Gamov paper), entitled The Origin of Chemical elements, by Ralph Alpha and George Gamov, presented a theoretical basis for the Big Bang. It describes the primordial universe up to 3 minutes after creation. In this 'primordial soup', it was so hot there were only protons, electrons, and neutrons. Only after the universe had expanded enough to cool down enough, could these sub-atomic particles fuse to form nuclei. This explains Cecilia Payne's discovery that 98% of the universe consists of hydrogen and helium.

Ralph Alpha went on to predict that at this point, light was first radiated out into the universe: and therefore, from the edges of the observable universe (the part of the universe travelling at nearly the speed of light), this light from the Big Bang would only just be reaching us now, after 13.8 billion years! And this light would be very 'stretched', due to the redshift - so far in fact, its frequency would be in the microwave bandwidth. He predicted that this radiation could be detected, and this 'background radiation' would be proof of the Big Bang.

In 1964, two American researchers, Arno Penzias (b. 1933) and Robert Wilson (b. 1936) detected this Cosmic Microwave Background (CMB) Radiation, and won themselves the Nobel Prize. They (accidentally) discovered that every part of the sky contained a signal (isotropic = equal from every direction) which was a black-body spectrum corresponding to a temperature of 2.7K. This signal was the afterglow of the enormous amount of energy released at the birth of the universe.

This was the nail in the coffin of the Static Universe.

### Evidence for the Big Bang

The Big Bang occurred 13.8 billion years ago. At that time the whole universe was infinitesimally small, and experienced a rapid expansion. There are three main pieces of evidence for this scenario:

1. The Expansion of the Universe
2. Hubble's redshift technique and Cepheid object distance measurements have been consistently verified by ever more precise telescopes. The only change has been the discovery that the universe is accelerating in its expansion, which was a surprising and yet unexplained phenomenon. In the light of this, doubt must be cast on the precision of the date 3.8 billion years since the Big Bang.

4. The 1964 discovery by Penzias and Wilson is strong evidence, since it correlates so closely to the Gamov/Alpher prediction 16 years previously of the temperature of the universe has a background of 2.7K.

5. Helium Abundance
6. The Big Bang model, as described by Gamov, and confirmed by Cecilia Payne's spectrographic observations of the Sun's constituency, predicts the formation of helium from hydrogen (single protons) through a process of fusion after the cooling in the initial stages of the universe's expansion. Since hydrogen and helium make up more than 98% of the universe, and helium makes up 25%, this is evidence that matter was created as described by the Big Bang model.

## Expansion of the Universe

Every time we walk outside, we are being bathed in radiation which began its journey at the beginning of the universe, in the immediate aftermath of the Big Bang!

So, if there was a Big Bang, how do we answer the question: Where does all the matter come from?

Ideas are based on the standard model of physics, which includes the Higg's Field. Super-symmetry is a model which says for every particle there is an anti-particle.

The scale factor (function \$R(t)\$) of the universe is given by: \$x(t) = R(t)x_0\$, where \$x\$ is the distance between two galaxies at time \$t\$ after time zero (at \$x_0\$). \$R\$ is therefore the radius of the universe, and is a scalar, relying on the assumption of homogeneity and isotropy of the universe.

A closed universe has \$R\$ start at zero, expand to a maximum value, then contract to zero again.

An open universe has \$R\$ start at zero, but expands forever (The Big Fizzle).

A flat universe has \$R\$ expand forever, but at a decreasing rate.

Each of these options for \$R\$ gives a different age to the universe. Which is correct depends on the ratio of the density of the universe, \$ρ\$, to the critical density \$ρ_c\$. This is the density at which there is sufficient mass for gravity to be strong enough to overcome the momentum of the expansion, and cause a reversal of the vectors of the galaxies, to end in a Big Crunch.

If \$ρ < ρ_c\$, the universe is open, and \$R\$ will increase forever (hyperbolic, or saddle, geometry).

If \$ρ = ρ_c\$, the universe is flat, and \$R\$ will increase forever but a decreasing rate till it stalls (Euclidean, or flat, geometry).

If \$ρ > ρ_c\$, the universe is closed, and \$R\$ will reverse and begin to decrease at a certain point (spherical, or spots on the surface of a balloon, geometry).

## Dark Matter

Hubble's Law is \$v = Hd\$, where \$v\$ is the velocity of a distant galaxy, \$d\$ is the distance to the galaxy, and \$H = 72\$ km s\$^{-1}\$ Mpc\$^{-1}\$ is the Hubble's constant.

A sphere of gas, radius \$r\$, expanding at speed \$v\$, has total energy of \$E = 1/2mv^2 - {GMm}/r\$, where \$M\$ is the mass of the cloud of gas, and \$m\$ is a mass at the surface of the sphere. The mass of the cloud can be expressed in terms of the density \$ρ\$, as \$M = ρ4/3πr^3\$.

Using Hubble's Law, \$E = 1/2mr^2(H^2 - {8πρG}/3)\$.

The critical density is the density for which the energy will be zero at infinity: \$ρ_c = {3H^2}/{8πG} ≈ 10^{-26} \$ kg m\$^{-3}\$. The mass of a proton is \$1.673 × 10^{-27}\$ kg, so the critical density is about 6 protons per cubic metre of space.

The difficulty lies in estimating the mass of a galaxy: there is a great deal of matter which cannot be seen because it is too cold to radiate. Also, the universe is known to be bathed in neutrinos, and the mass of neutrinos is not yet known, and may not be zero. Other particles that may add mass are WIMPS (non-baryonic, weakly interacting massive particles), and MACHOS (massive compact halo objects).

## Dark Energy

In 1998, it was discovered that the universe is expanding faster than expected according to the standard Big Bang Theory. In fact, it seems to be accelerating - a phenomenon that has no explanation as yet. The explanation that is currently being investigated is that 73% of the universe consists of dark energy, which is pushing the universe apart, and 27% matter, of which 85% is dark matter. How the dark energy causes the universe to expand with an acceleration in the opposite direction to the effect of gravity, is entirely unknown.

LIGO Interferometer, Hanford, Washington State
Artist's impression of the event which triggered the gravitational wave detected in 2016 by LIGO

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