Observations of the universe are limited by the physical properties of light. Terrestrial observations are limited by the fact that light propagates in straight lines, while the surface of Earth is curved. The distance one can see can be increased by observing from a higher vantage point, but there is a limit: an observer on an infinitely tall platform could only see one quarter of the way around Earth's circumference. Astronomical observations are limited by the fact that light propagates with a finite velocity, c. The only objects that can be seen are those whose emitted light has had enough time, since the beginning of the Universe, to reach Earth today. In both cases, the physical limit to vision is called the horizon.
The mathematics of the terrestrial horizon is straightforward; knowing the observer's height above the surface, and the radius of Earth, simple trigonometry will suffice to calculate the horizon distance. The mathematics of the cosmological horizon is also straightforward, but far from simple. Exactly how far into the Universe can be seen depends upon a large number of cosmological parameters. The rate at which the Universe is expanding, the rate at which that expansion is slowing down (due to the mutual gravitational attraction between matter), the curvature of the Universe (whether it is negative, zero, or positive), and the matter content of the Universe (whether it is dominated by massive or massless particles), all enter into the calculation. Unlike the terrestrial horizon distance, the cosmological horizon distance is a dynamical quantity; it evolves in time, and does not expand at the same rate as the Universe expands. The ratio of the horizon volume to the Universe volume (that fraction of the Universe that can be in causal contact) changes as the Universe evolves, with important cosmological implications.
The calculation of the horizon distance simplifies if the Universe is flat, and its matter content is dominated by massive particles (matter-dominated). A universe whose content is primarily radiation, massless photons, is said to be radiation-dominated. Certainly the luminous matter, what can be seen with telescopes, is composed of massive particles. This may constitute as little as ten percent of the entire mass content, however, and it remains an open question as to whether the dark matter is massive or massless. In this case, the horizon distance at any time, d(t), is given by d(t)=3ct, where t is the time since the big bang.
In the hot big bang model, a flat, matter dominated universe grows slower than the rate at which the horizon distance grows. If the big bang is considered as happening backwards, the horizon distance shrinks faster than does the Universe. What constitutes the observable Universe today was, in the past, composed of many regions which were causally disconnected. There had not been enough time since the bang for light signals to propagate between these regions. This problem with the standard big bang model helped lead to the theory of cosmic inflation
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