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### Further Maths

• A-Level Physics

# Cosmology

Updated: Aug 22

Instead of using metres and kilometres, distances are measured in far larger units:

• Astronomical units, AU: the mean distance between the Earth and the Sun. ~150 million km

• Light-Years, ly: the distance and electromagnetic wave travels in one year. ~ 9.5x1015 m

• Parsecs, pc: the length of the adjacent side of a right-angled triangle with angle 1 arcsecond and opposite side length 1 parsec. ~ 3.1x1016 m

The angle is measured in arcseconds because of stellar parallax – objects further away appear to be moving more slowly than object closer to earth, and nearby stars appear to move differently depending on where in the earth’s orbit we are. Parallax in arcseconds, p, can be calculated for a known distance, d, measured in parsecs:

p = 1/d parallax (in arcseconds) = 1 / distance (in parsecs)
d = 1/p distance (in parsecs) = 1 / parallax (in arcseconds)

1 second of arc = (1/3600)°

## The Cosmological Principal

The Cosmological Principal states that, on a large scale, the universe is:

• homogeneous (every part is the same as another)

• isotropic (looks the same in every direction)

• and the laws of physics apply everywhere.

We apply this principle when investigating the universe, allowing us to use standard mechanics to solve problems on a cosmological scale.

## Doppler Shift & Hubble's Law

The Doppler Shift is the way in which frequency and wavelength change with the motion of the wave's source, relative to you. This happens because the waves bunch together in front of the source and stretch out behind it. The extent of bunching and stretching depends on the sources velocity.

• Red shift occurs when a light source moves away from us, as the wavelengths become longer and the frequencies become lower, so the light shifts towards the red end of the spectrum.

• Blue shift occurs when a light source moves towards us, as the opposite happens and the light shifts towards the blue end of the spectrum.

• The same principal applies to sound waves: as the source moves towards you, the frequency increases and wavelength decreases; as it moves away the frequency decreases and wavelength increases. For example, a high speed train approaching, passing, and disappearing from a level crossing.

The amount of shift can be calculated with the Doppler Equation:

Δλ/λ ≈ Δf/f ≈ v/c

Δλ and Δf represent the difference between observed and emitted wavelength and frequency. v is the velocity of the source, and c the speed of light, 3 E8 m/s.

### Hubble's Law

Until the early 20th century, it was believed the universe was infinite in both space and time – Steady State Theory. However, Edwin Hubble noticed that the spectra from far away galaxies all show red shift, implying they are moving away from us, and that the universe is expanding.

The amount of galactic red shift gives the recessional velocity, and Hubble’s Law shows that recessional velocity is directly proportional to the distance of receding galaxies:

v ≈ H(0) x d recessional velocity ≈ the Hubble constant x distance

The Hubble constant, H(0), can be taken as roughly 65-80 km s-1 Mpc-1 (velocity is in kms-1  and distance in Mpc). There is not yet a universally accepted accurate reading.

• To find the Hubble constant in SI units (s-1), velocity must be in ms-1 and distance in m.

## The Big Bang Theory

If the universe is expanding and cooling down (conservation of energy principal), then back in time, it must have been smaller and hotter. At time t=0, it must have been at a point of singularity (infinitely dense and infinitely hot). This is the Big Bang Theory.

Cosmic Microwave Background Radiation is the evidence for the Big Bang: if there were a point of singularity from which very rapid expansion happened, there would be a great deal of gamma radiation produced. This would still be observable today, and it is, but as microwave radiation, because as the universe has expanded, the wavelength has been stretched and lost energy. This was picked up accidentally by Penzias and Wilson in the 60’s.

The age of the universe can be estimated as 1/the Hubble constant. However, this assumes that the universe has been expanding at the same rate for all of time (which is unlikely), and since we do not have an accurate value for the Hubble constant, we can only roughly estimate the universe’s age to be about 13.8 bn years.

It is difficult to establish how large the universe is, as we can only see the ‘observable universe’ – a sphere around the earth with a radius of 13.8 billion light years (as that is how far light can have travelled in time). Taking into account the expansion of the universe, it is probably around 46-47 billion ly in radius.

## The Evolution of the Universe

This is the current theory of the evolution of the universe. The times given is the age of the universe at each point, so the amount of time after the big bang.

• 0 s (Big Bang)

• Time and space are formed, the universe is at a point of singularity (infinitely dense and hot).

• 10^-35 s

• Universe is expanding rapidly, known as inflation. There is no matter yet, and temepratures are around 10^28 K.

• 10^-6 s

• First fundamental particles (quarks, leptons etc.) form. We do not know exactly how this happens, but we know it involves the Higgs Boson.

• 10^-3 s

• Quarks combing to become hadrons (protons and neutrons) through pair production.

• 1 s

• Matter production stops, temperature dropped to about 10^9 K.

• 100 s

• Protons and neutrons fuse to form light elements like deuterium (hydrogen-2) and helium

• 380,000 years

• Universe is cold enough for heavier atoms to form as the nuclei take on electrons. The microwave background radiation originates from this time.

• 30 million years

• Stars begin to form, and the fusion that occurs in them form the first heavy elements (anything after lithium).

• 200 million years

• The Milky Way forms, as clouds of hydrogen and stars are pulled together by gravitational forces.

• 9 billion years

• Our solar system forms out of the supernova of a dead larger star

• 10 billion years The earth forms

• 11 billion years Primitive life on earth begins

• 13.8 billion years (today)

• Temperature has dropped to about 2.7K, diameter of universe is probably around 46 - 47 billion light years.

## Dark Energy & Matter

In the 30’s, Fritz Zwicky calculated the mass of cluster galaxies based on their velocity and luminosity. The two measurements gave vastly different results, with velocity giving a far more massive result. This implies the existence of mass that cannot be seen – dark matter.

In the 70’s, Vera Rubin observed that stars at the edges of galaxies were moving faster than they should be given their mass, again, implying the existence of dark matter.

Since all things with mass are attracted to one another through gravity, we would expect the expansion of the universe to be slowing down. However, in the 90’s it was discovered that the expansion is, in fact, speeding up. This acceleration is only possible if there is some undetectable energy that spans all the universe – dark energy.

Today, it is estimated that there is approximately five times as much dark matter as there is ordinary matter, and that this makes up around a quarter of the entire universe. It is also believed that around 70% of the universe is dark energy. This leaves only 5% of the universe to be ordinary, detectable matter.

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