Alexander's trick

A cosmological horizon is a measure of the distance from which one could possibly retrieve information. This observable constraint is due to various properties of general relativity, the expanding universe, and the physics of Big Bang cosmology. Cosmological horizons set the size and scale of the observable universe. This article will explain a number of these horizons. This article will report distances in units of kiloparsecs (kpc), megaparsecs (Mpc), and gigaparsecs (Gpc).

Particle horizon

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Hubble horizon

One can define a so-called "Hubble Horizon" which shows roughly how far light would travel if space were not expanding. This size is

$\chi =ct$

where $t$ is the lookback time since the Big Bang (otherwise known as the age of the universe) which, according to the Friedmann Equations, is:

$t=\underset{0}{\overset{a}{\int }}\frac{da}{H_0\sqrt{{\mathrm{\Omega }}_R{a}^{-2}+{\mathrm{\Omega }}_m{a}^{-1}+{\mathrm{\Omega }}_k+{\mathrm{\Omega }}_{\mathrm{\Lambda }}{a}^2}}$

where $H_0$ is the Hubble Constant and the $\mathrm{\Omega }$ density parameters are, in order, the density of radiation, matter, curvature, and dark energy scaled to the critical density of the universe.

Today, roughly:

${\chi }_0=\frac{c}{H_0}$,

yielding a Hubble horizon of some 4.2 Gpc. This horizon is not really a physical size, but it is often used as useful length scale as most physical sizes in cosmology can be written in terms of those factors.

Event horizon

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The particle horizon differs from the cosmic event horizon, in that the particle horizon represents the largest comoving distance from which light could have reached the observer by a specific time, while the event horizon is the largest comoving distance from which light emitted now can ever reach the observer in the future. At present, this cosmic event horizon is thought to be at a comoving distance of about 46.6 billion light years.

In general, the proper distance to the event horizon at time $t$ is given by

$d_e(t)=a(t)\underset{t}{\overset{t_{max}}{\int }}\frac{cd{t}^{\prime }}{a(t^{\prime })}$

where $t_{max}$ is the time-coordinate of the end of the universe, which would be infinite in the case of a universe that expands forever.

For our case, assuming that dark energy is due to a cosmological constant, $d_e(t_0)\to \mathrm{\infty }$.

Future horizon

In an accelerating universe, there are events which will be unobservable as $t\to \mathrm{\infty }$ as signals from future events become redshifted to arbitrarily long wavelengths in the exponentially expanding de Sitter space. This sets a limit on the farthest distance that we can possibly see as measured in units of proper distance today. Or, more precisely, there are events that are spatially separated for a certain frame of reference happening simultaneously with the event occurring right now for which no signal will ever reach us, even though we can observe events that occurred at the same location in space that happened in the distant past. While we will continue to receive signals from this location in space, even if we wait an infinite amount of time, a signal that left from that location today will never reach us. Additionally, the signals coming from that location will have less and less energy and be less and less frequent until the location, for all practical purposes, becomes unobservable. In a universe that is dominated by dark energy which is undergoing an exponential expansion of the scale factor, all objects that are gravitationally unbound with respect to the Milky Way will become unobservable, in a futuristic version of Kapteyn's Universe.

Practical horizons

While not technically "horizons" in the sense of an impossibility for observations due to relativity or cosmological solutions, there are practical horizons which include the optical horizon, set at the surface of last scattering. This is the farthest distance that any photon can freely stream. Similarly, there is a "neutrino horizon" set for the farthest distance a neutrino can freely stream and a gravitational wave horizon at the farthest distance that gravitational waves can freely stream. The latter is predicted to be a direct probe of the end of cosmic inflation.

References

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