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If we take a slice of the universe at a fixed time, the 3D space may be (top) a flat, Euclidean space, in which the angles of a triangle add up to 180°, (middle) a space of negative curvature (hyperbolic space), in which the angles add to <180°, or (bottom) a space of positive curvature (spherical space), in which the angles of a triangle add to >180°.

Einstein’s General Relativity

Einstein put forward his General Theory of Relativity in 1915. The main idea was that gravity is nothing more than the consequence of the bending of space-time around a massive body. So a mass distorts the space (and time) around it and a test particle then has to modify its orbit to follow the geometry of space. The geometry of space-time is curved by matter. In most situations, where the gravitational field is not too strong, Einstein’s theory gives the same result as Newton’s inverse-square law of gravitation.

In 1917 Einstein tried to apply his new theory to the whole universe. To do this he made a very dramatic assumption: the universe is (a) homogeneous (everyone sees the same picture) and (b) isotropic (the universe looks the same in every direction). Einstein used this unlikely assumption, the cosmological principle, to derive a static model of the universe in which the gravitational attraction between galaxies, which would tend to pull them all together, is balanced by a new force, the cosmological repulsion.

Einstein’s inspired guess that the universe must be very simple (homogeneous and isotropic) is confirmed to very high accuracy today. However, his static model of the universe soon ran into trouble with Hubble’s discovery that the universe is expanding. 


The Friedmann cosmological models

The Russian mathematician and meteorologist Alexander Friedmann had already shown in 1922 that expanding universe models are what would be expected according to Einstein’s General Theory of Relativity, assuming the cosmological principle. Friedmann explored a whole class of models, including those with and without the cosmological repulsion.

The fate of the universe is determined by its composition. With just matter (including dark matter) the universe expands out from the Big Bang, gravity slowing down the expansion. Depending on the average density of matter in the universe, the universe may either reach a maximum extent and then fall back together in a “Big Crunch”, or it may just keep on expanding for ever. There is a critical density that divides these two possibilities.

The fate of the universe is also connected, for a universe with matter only, with the spatial curvature of the universe. The critical density universe is spatially flat, while the higher density universe has positive spatial curvature and the lower density universe has negative spatial curvature.

In a universe with dark energy (the cosmological repulsion, chapter 9), the deceleration of the expansion due to gravity is eventually reversed and the universe begins an accelerated expansion. The observed universe seems to be just moving into this accelerated phase at the present epoch. Modern observations (see chapter 12) suggest that the universe is close to being spatially flat.

The fate of the universe. With no cosmological repulsion, the expansion either keeps on going for ever, or reaches a maximum and recollapses to a Big Crunch. The cosmological repulsion generates an exponential acceleration of the expansion.


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