Andrew Conway

(See the section on black holes for an introduction to relativity.)

- Einstein's General Theory of Relativity is needed to build cosmological models.
- The Universe is regarded as a 4 dimensional space-time: 3 space dimensions plus time.
- The distribution of mass and energy throughout the Universe determines the shape of space-time.
- The shape of space-time determines how mass and energy will move.

*Mass tells space-time how to curve, and space-time tells mass how to move.*

John Wheeler

- A homogenous Universe is one that has the same properties everywhere, e.g. the number of galaxies in a given volume of space.
- The Universe appears isotropic if it appears the same everywhere.
- If we look at the largest scales, e.g. by counting the numbers of high redshift galaxies in different parts of the sky, we find that the Universe is isotropic.
- Cosmologists are only interested if the Universe is currently isotropic and homogeneous on the very largest distance scales.

- Solving Einstein's equations from General Relativity requires assumptions.
- The cosmological principle states that the Universe is isotropic and homogeneous.
- If we suppose we are not located in a special place in the Universe, then isotropy makes homogeneity seem reasonable.
- Cosmologists charactertise the Universe in terms of a density and pressure.
- Other quantities, such as temperature, can be derived from knowledge of density and pressure.
- The pressure of the Universe is taken as zero. The distance between stars and galaxies are such that there are no frequent collisions that you'd find in a gas like air.
- Remember that density here refers to both mass and energy.

Source: Andrew Conway CC-BY SA 3.0

s^{2}=x^{2}+y^{2}

- The square of the distance between two points is equal to the sum of squares of the x and y co-ordinates.
- This is a simple example of a metric - a formula for calculating distance.
- In this case we are calculating distance on a 2 dimensional flat surface.
- The shortest distance between two points in flat space is a straight line.

- Consider two cities A and B separated by λ in longitude with latitudes L
_{A}and L_{B}. - Pythagoras's theorem cannot be applied - squaring the differences in longitude and latitude an adding them will not give the distance between A and B.
- The shortest distance between A and B is not a straight line but the arc of a great circle - a circle with the same diameter as the sphere.
- Notice that we are still considering a 2D surface, but it is curved.
- If A and B are really close together then Pythagoras becomes a useful approximation - space is locally flat.

The distance squared ds^{2} between two points is given by

ds^{2} = dx^{2} + dy^{2} + dz^{2}

- dx is the difference in x - likewise for dy and dz.
- The d means that the differences are very small and have to be added up using calculus (using integrals).
- The metric is Pythagoras's theorem applied locally.

ds^{2} = dx^{2} + dy^{2} + dz^{2} - c^{2}dt^{2}

- Notice that a time difference dt is now included and is multiplied by the the speed of light squared and has a minus sign.
- This metric tells you the "distance" between two events.
- An event is defined by a location in space (x,y,z) and time t.
- Events associated with a signal travelling at the speed of light will have ds=0.
- Two events in which one event caused the other must have ds
^{2}<0 - this is equivalent to saying nothing can travel faster than light. - This is the metric used in Special Relativity.

ds^{2} = R^{2}(dx^{2} + dy^{2} + dz^{2}) - c^{2}dt^{2}

- This is the metric used in cosmology.
- R is the scale factor and can vary across time and space.
- For example, if R increases with time t, then the Universe is expanding.
- If R is a function of x, y or z then space-time will be curved.

Recall from before, that if λ_{0} is the stationary wavelength and λ is the observed (shifted) wavelength, then the redshift is given by

z = (λ - λ_{0}) / λ_{0}

The correct explanation for this is not the Doppler effect due to the source's motion, but because the scale factor has changed, which tells us that:

z = (R - R_{0}) / R_{0}

where R is the scale factor at observation and R is the scale factor as the light was emitted.

- Einstein's equations can be solved to relate the density of the Universe ρ, the scale factor R and Hubble's constant H.
- Hubble's constant turns out to be equal to the rate of change of R divided by R.
- And a critical density can be defined in terms of H and the constant G, from Newton's law of gravitations, as follows:

ρ_{crit} = 3 H^{2} / 8 π G

ρ < ρ_{crit}

- The Universe expands forever, though the expansion slows.
- Space-time is curved, and the Universe is open and infinite.

ρ > ρ_{crit}

- The Universe's expansion slows and until it eventually starts to contract.
- Space-time is curved, and the Universe is closed and finite.

ρ = ρ_{crit}

- The Universe expands forever, though the expansion slows, getting closer to halting.
- Space-time is flat, and the Universe is open and infinite.

- In Einstein's original description of the Universe using general relativity, he included a constant Λ (capital lambda) - the cosmological constant.
- It allowed him to describe the Universe as being a steady state - this was before Hubble's work and evidence of an expanding Universe.
- The cosmological constant can be intepreted as causing a repulsion in the Universe.
- Einstein later referred to it as his greatest blunder.
- So far we have assumed Λ=0.

- For many years, observational efforts in cosmology were directed at determining whether the density was above or below the critical density.
- Best estimates suggested we live in a flat Universe, near critical density.
- Efforts were also made to measure the deceleration of the expansion.
- But then, in 1998, observations of distant type Ia supernova revealed that the expansion of the Universe was
*accelerating*. - This can be described in the models if Λ>0, but there is no clear physical interpretation of what might cause this repulsion; for this, and other reasons, we are forced to consider that Universe is mainly composed of dark energy, distinct from dark matter.