# Particle horizon

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 particles could have traveled to the observer in the age 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. Due to the expansion of the universe it is not simply the age of the universe times the speed of light (approximately 13.8 billion 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.

In terms of comoving distance, the particle horizon is equal to the conformal time ${\displaystyle \eta }$ that has passed since the Big Bang, times the speed of light ${\displaystyle c}$. In general, the conformal time at a certain time ${\displaystyle t}$ is given by,

${\displaystyle \eta =\int _{0}^{t}{\frac {dt'}{a(t')}}}$
${\displaystyle a(t)H_{p}(t)=a(t)\int _{0}^{t}{\frac {cdt'}{a(t')}}}$
${\displaystyle H_{p}(t_{0})=c\eta _{0}=14.4\ {\rm {Gpc}}=46.9\ {\rm {billion\ light\ years}}}$.
${\displaystyle H(z)=H_{0}{\sqrt {\sum \Omega _{i0}(1+z)^{n_{i}}}}}$
${\displaystyle {\text{The particle horizon }}H_{p}{\text{ exists if and only if }}N>2}$
${\displaystyle {\frac {dH_{p}}{dt}}=H_{p}(z)H(z)+c}$
${\displaystyle H_{p}(t_{\rm {CMB}})=c\eta _{\rm {CMB}}=284\ {\rm {Mpc}}=8.9\times 10^{-3}H_{p}(t_{0})}$.
${\displaystyle a_{\rm {CMB}}H_{p}(t_{\rm {CMB}})=261\ {\rm {kpc}}}$
${\displaystyle f=H_{p}(t_{\rm {CMB}})/H_{p}(t_{0})}$
• Hubble function ${\displaystyle H={\frac {\dot {a}}{a}}}$
• The critical density ${\displaystyle \rho _{c}={\frac {3}{8\pi }}H^{2}}$
• The i-th dimensionless energy density ${\displaystyle \Omega _{i}={\frac {\rho _{i}}{\rho _{c}}}}$
• The dimensionless energy density ${\displaystyle \Omega ={\frac {\rho }{\rho _{c}}}=\sum \Omega _{i}}$
• The redshift ${\displaystyle z}$ given by the formula ${\displaystyle 1+z={\frac {a_{0}}{a(t)}}}$
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