Particle horizon
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The particle horizon (also called the cosmological horizon, the comoving horizon (in Dodelson's text), or the cosmic light horizon) is the maximum distance from which light from
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
s could have traveled to the
observer An observer is one who engages in observation or in watching an experiment. Observer may also refer to: Computer science and information theory * In information theory, any system which receives information from an object * State observer in co ...
in the
age of the universe In physical cosmology, the age of the universe is the time elapsed since the Big Bang. Astronomers have derived two different measurements of the age of the universe: a measurement based on direct observations of an early state of the universe, ...
. Much like the concept of a terrestrial horizon, it represents the boundary between the observable and the unobservable regions of the universe, so its distance at the present epoch defines the size of the
observable universe The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these obj ...
. Due to the expansion of the universe, it is not simply the
age of the universe In physical cosmology, the age of the universe is the time elapsed since the Big Bang. Astronomers have derived two different measurements of the age of the universe: a measurement based on direct observations of an early state of the universe, ...
times the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
(approximately 13.8 billion light-years), but rather the speed of light times the conformal time. The existence, properties, and significance of a cosmological horizon depend on the particular
cosmological model Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
.


Conformal time and the particle horizon

In terms of
comoving distance In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects. ''Proper distance'' roughly corresponds to where a distant object would be at a spec ...
, the particle horizon is equal to the conformal time \eta that has passed since the
Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
, times the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
c. In general, the conformal time at a certain time t is given by :\eta = \int_^ \frac, where a(t) is the
scale factor In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
of the
Friedmann–Lemaître–Robertson–Walker metric The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe tha ...
, and we have taken the Big Bang to be at t=0. By convention, a subscript 0 indicates "today" so that the conformal time today \eta(t_0) = \eta_0 = 1.48 \times 10^\text. Note that the conformal time is ''not'' the
age of the universe In physical cosmology, the age of the universe is the time elapsed since the Big Bang. Astronomers have derived two different measurements of the age of the universe: a measurement based on direct observations of an early state of the universe, ...
, which is estimated around 4.35 \times 10^\text. Rather, the conformal time is the amount of time it would take a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
to travel from where we are located to the furthest observable distance, provided the universe ceased expanding. As such, \eta_0 is not a physically meaningful time (this much time has not yet actually passed); though, as we will see, the particle horizon with which it is associated is a conceptually meaningful distance. The particle horizon recedes constantly as time passes and the conformal time grows. As such, the observed size of the universe always increases. Since proper distance at a given time is just comoving distance times the scale factor (with
comoving distance In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects. ''Proper distance'' roughly corresponds to where a distant object would be at a spec ...
normally defined to be equal to proper distance at the present time, so a(t_0) = 1 at present), the proper distance to the particle horizon at time t is given by :a(t) H_p(t) = a(t) \int_^ \frac and for today t = t_0 : H_p(t_0) = c\eta_0 = 14.4\text = 46.9\text.


Evolution of the particle horizon

In this section we consider the FLRW cosmological model. In that context, the universe can be approximated as composed by non-interacting constituents, each one being a perfect fluid with density \rho_i, partial pressure p_i and
state equation In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or interna ...
p_i=\omega_i \rho_i, such that they add up to the total density \rho and total pressure p. Let us now define the following functions: * Hubble function H=\frac * The critical density \rho_c=\fracH^2 * The ''i''-th dimensionless energy density \Omega_i=\frac * The dimensionless energy density \Omega=\frac \rho =\sum \Omega_i * The redshift z given by the formula 1+z=\frac Any function with a zero subscript denote the function evaluated at the present time t_0 (or equivalently z=0). The last term can be taken to be 1 including the curvature state equation. It can be proved that the Hubble function is given by : H(z)=H_0\sqrt where the dilution exponent n_i=3(1+\omega_i). Notice that the addition ranges over all possible partial constituents and in particular there can be countably infinitely many. With this notation we have: : \text H_p \text N>2 where N is the largest n_i (possibly infinite). The evolution of the particle horizon for an expanding universe (\dot>0) is: : \frac=H_p(z)H(z)+c where c is the speed of light and can be taken to be 1 (
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a Coherence (units of measurement), coherent unit of a quantity. For e ...
). Notice that the derivative is made with respect to the FLRW-time t, while the functions are evaluated at the redshift z which are related as stated before. We have an analogous but slightly different result for
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s. In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact obj ...
.


Horizon problem

The concept of a particle horizon can be used to illustrate the famous horizon problem, which is an unresolved issue associated with the
Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
model. Extrapolating back to the time of recombination when the
cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
(CMB) was emitted, we obtain a particle horizon of about which corresponds to a proper size at that time of: Since we observe the CMB to be emitted essentially from our particle horizon (284\text \ll 14.4\text), our expectation is that parts of the
cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
(CMB) that are separated by about a fraction of a
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
across the sky of (an
angular size The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is ...
of \theta \sim 1.7^\circ) should be out of
causal contact Two entities are in causal contact if there may be an event that has affected both in a causal way. Every object of mass in space, for instance, exerts a field force on all other objects of mass, according to Newton's law of universal gravita ...
with each other. That the entire CMB is in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
and approximates a
blackbody A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
so well is therefore not explained by the standard explanations about the way the
expansion of the universe The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not exp ...
proceeds. The most popular resolution to this problem is
cosmic inflation In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from  seconds after the conjectured Big Bang singularity ...
.


See also

*
Cosmological horizon 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. Co ...
*
Observable universe The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these obj ...


References

{{reflist Physical cosmology