5.5.3 Cosmology

So in this section we'll use different unit for distance (as apposed to the meter). This is because the distances are so vast. The new units are as follows:

• Astronomical unit (AU)
• This is the average distance from the Earth to the Sun. It is most often used to express the average distance between the Sun and other planets in our Solar System.
• It is equal to 1.50 x 10¹¹ m
• Light-year (ly)
• This is the distance travelled by light in a vacuum in a time of one year. It is often used when expressing distances to stars or other galaxies.
• It is equal to 3.00 x 10⁸ x (365 x 24 x 60 x 60) = 9.46 x 10¹⁵ m
• Parsec (pc)
• So for this one, we need to be aware that in cosmology we measure angles in arcseconds and arcminutes. One arcsecond is 1/3600th of a degree (1 arcminute is 1/60th of a degree).
• The parsec is defined as the distance at which a radius of one AU subtends an angle of one arcsecond.
• The value of 1pc can be determined by tan(1/3600) = 1AU/1pc. This means that 1pc = 1AU/tan(1/3600). This equals 3.1 x 10¹⁶ m.
• It is important to realise that at a distance of n parsecs, the angle subtended by a radius of 1AU = 1/n arcseconds.

There is a special technique used to determine the distance to stars that are relatively close to Earth (like, less than 100 pc). This is known as stellar parallax. Parallax is the shift in position of a relatively close star against the backdrop of much more distant stars as Earth orbits the Sun. If p (parallax angle) is measured in arcseconds, the distance to the nearby star in parsecs is given by the following equation:

d = 1/p

This technique can't be used to measure the distance between stars bigger than 100pc from the Earth because as d increases the parallax angle decreases. Eventually becoming too small to measure accurately, even with the most advanced astronomical techniques.

The Doppler effect
The Doppler effect is used to determine the speed of moving objects.

When a wave source moves relative to an observer, the frequency and wavelength of the waves received by the observer change compared with what would be observed without relative motion. Originally, two points equidistant from the source would receive waves at the same frequency and wavelength as they were emitted from the source. When the source moves closer to one point, the waves received by this point will be compressed. They have a shorter wavelength and a higher frequency (therefore, a shorter period).

How fast the wave source moves relative to the observer affects the size of the observed shift in wavelength and frequency. For electromagnetic waves we can use the Doppler equation. The equation shows that the faster a source moves, the greater the observed change in wavelength and frequency:

Δλ/λ ≈ Δf/f ≈ v/c

NOTE: The Doppler equation can only be used for galaxies with speed far less than the speed of light.

As we know already, one technique to analyse starlight involves looking at the absorption spectra from stars. The Doppler effect can be used to determine the relative velocity of a distant galaxy. Any difference in the observed wavelengths of the absorption lines must be caused by the relative motion between the galaxy and the Earth.

NOTE: if the galaxy is moving towards the Earth the absorption lines will be blue-shifted (they move toward the blue end of the spectrum because the wavelength appears shorter). If the galaxy is moving away from the Earth the absorption lines will be red-shifted as the wavelength appears stretched.

Using data from the absorption spectra of many distant galaxies, Hubble made two key observations:

1. A confirmation that earlier observations that the light from the vast majority of galaxies was red shifted (they had a relative velocities away from the Earth)
2. He found that in general the further away the galaxy was the greater the observed red shift and so the faster the galaxy was moving

From these observations Hubble formulated his law (Hubble's Law): The recessional speed (v) of a galaxy is almost directly proportional to its distance (d) from the Earth.

This means that a graph of recessional speed against distance for all galaxies will produce a straight line graph through the origin. The gradient is a constant of proportionality - the Hubble constant (Ho). It's (current) value is 2.2  × 10^-18 km s^-1 Mpc^-1. From Hubble's law it can be derived that:

v ≈ Ho x d

Hubble's law has been very useful in determining key evidence for the Big Bang theory and the model of the expanding Universe (following the Big Bang). This model is the accepted explanation of the observation that the light from nearly all the galaxies we can see is red-shifted. The further two points are apart the faster their relative motion (the more red-shifted their spectra).

The cosmological principle is the assumption that (when viewed on a large enough scale) the Universe is homogenous and isotropic and the laws of physics are universal:

• Homogenous means the matter is distributed uniformly across the Universe (ie the density of the universe is uniform
• Isotropic means that the Universe looks the same in all directions to every observer (ie there is no end to the Universe)
• The laws of physics can be applied across the Universe (meaning that theories/models tested on Earth can be applied to everything within the Universe).

The Big Bang
Hubble's law and the microwave background radiation are two key pieces of evidence for the Big Bang theory. Hubble's law shows that space is expanding as the galaxies are receding from each other because space itself is expanding in all dimensions.

The existence of microwave background radiation is the second piece of evidence for the Big Bang. Microwave background radiation can only be explained by the Big Bang and the expansion of space. It's existence can be explained in two ways:

• When the Universe was young and extremely hot space was saturated with high-energy gamma photons. The expansion meant that space itself was stretched over time. The expansion stretched the wavelength of these high-energy photons so we now observe this primordial electromagnetic radiation as microwaves.
• The Universe was extremely hot and dense when it was young. Expansions over billions of years reduced it's temperature to about 2.7 K. The Universe can be treated as a black-body radiator. At this temperature the peak wavelength would correspond to about 1mm (in the microwave region of the spectrum).

We can estimate the age of the Universe by assuming that it has expanded uniformly over time since the Big Bang. This actually isn't the case lol. Results from recent observations show that the expansion of the Universe is accelerating. Nonetheless, this assumption will give a crude indication the Universe's age.

So, Hubble's law shows galaxies are receding from each other. If a galaxy at a distance 'd' is moving away at a constant speed 'v' then a time (d/v) must have elapsed since it was next to our galaxy. This time is roughly the age of the Universe. The ratio d/v is equal to 1/Ho meaning that:

age of the Universe 't' ≈ 1/Ho

NOTE: This gives the age of the Universe to be 4.5 x 10¹⁷ s (≈14 billion years).

As I mentioned above, it is now known that the Universe appears to be expanding at an increasing rate. The most widely accepted theory includes the concept of dark energy. It is suggested that this hypothetical form of energy fills all of space and tends to accelerate the expansion of the Universe. The two most significant discoveries that have changed our understanding of the Universe is the discovery of dark energy and dark matter.

We need energy to accelerate things. The term 'dark energy' was coined to describe a hypothetical form of energy that permeates all space. It is estimated that dark energy makes up around 68% of our Universe.

In the 1970s astronomers studying the Doppler shift in light from galaxies found that the velocity of the stars in the galaxies did not behave as predicted. It was expected that their velocity would decrease as the distance from the centre of the galaxy increases. This effect is observed in other gravitational systems where most of the mass is in the centre (e.g the moons of Jupiter). The observations can be explained if the mass of the galaxy is not concentrated in the centre. We currently think that there must be another type of matter which we cannot see. This dark matter is spread throughout the galaxy, explaining the observations. The Universe must be made up of 27% of this matter (according to calculations).

All we know about dark matter is we know it cannot be seen directly with telescopes and it neither emits not absorbs light.

The rest of the universe is made up of a small percentage of ordinary matter.