Particle horizon

The particle horizon at cosmic time $t$ is the greatest comoving distance from which a light signal could have reached an observer since the initial time:Regions separated by more than twice this distance cannot have exchanged signals by time $t$.

The radial null integral from the Big Bang is ${\chi }_p=c{\int }_0^{t}d{t}^{\prime }/R(t^{\prime })=c{t}^{1-p}/[A(1-p)]$. Multiplication by the scale factor gives the proper radial horizon distance. It is $3ct$ for a flat matter-only model and $2ct$ for a flat radiation-only model. A finite initial time different from zero or an earlier expansion era changes the result.

Divide the particle horizon in a power-law universe at emission by the angular diameter distance in a flat power-law universe. The result is the small angular extent of one radial horizon length. Matter-only and radiation-only histories have powers $1/3$ and $1/2$, respectively. These are idealized single-era histories; a real recombination horizon and distance depend on different portions of the expansion history.

The proper radial particle horizon is the distance travelled by a null signal since the initial boundary, measured on the current time slice. A spherical patch with that radius has twice that physical diameter. Dividing one or the other by an angular diameter distance therefore differs by a factor of two. A question using horizon size should state which linear extent is intended.

In a spatially flat FLRW metric, the supremum of the comoving radial distance from which a signal emitted at the initial time can eventually reach a fixed comoving observer is the total conformal time interval. At an intermediate time $t$, this equals its comoving particle horizon plus the comoving radius of its cosmological event horizon, when both integrals converge. This is a visibility bound for emission arbitrarily close to the initial boundary; it does not assert that a transparent universe or a source existed there.

Use $c=1$, $a_0=1$, positive pressureless matter and positive cosmological constant, with ${\Omega }_m+{\Omega }_{\Lambda }=1$ and no radiation. The Friedmann equation gives ${\chi }_{\max }=H_0^{-1}{\int }_0^{\infty }dy/({\Omega }_{\Lambda }+{\Omega }_m{y}^3{)}^{1/2}$. Set $y=({\Omega }_{\Lambda }/{\Omega }_m{)}^{1/3}u$, then $v=u^3$. The remaining integral is $\frac{1}{3}{\int }_0^{\infty }{v}^{-2/3}(1+v{)}^{-1/2}dv=\frac{1}{3}B(1/3,1/6)$, proving the formula by the beta function integral. Its two endpoints converge for positive densities, but the lifetime diverges in either pure-fluid limit.

The comoving particle horizon is the particle horizon expressed in comoving coordinates, assuming a finite initial cosmic-time boundary at zero. In a spatially flat FLRW metric, it is the distance a photon could have travelled since that boundary. The corresponding proper distance in cosmology at time $t$ is $a(t)r_p(t)$. It differs from the future-directed cosmological event horizon and from the instantaneous comoving Hubble radius.

For an Einstein-de Sitter universe normalized by $a(t_0)=1$, $a=(t/t_0{)}^{2/3}$. Its current comoving particle horizon is $3c{t}_0$. A photon emitted inward at that location at $t_0$ covers $3c{t}_0[(t/t_0{)}^{1/3}-1]$ in comoving distance by time $t$. Equating these distances gives arrival at $8t_0$, a travel time of $7t_0$.

Neglecting curvature and dark energy, integration of the Friedmann equation for radiation in cosmology and pressureless matter gives the displayed comoving particle horizon. At matter-radiation equality, $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$, this is $2(\sqrt{2}-1)$ times the radiation-only extrapolation $c{a}_{\mathrm{eq}}/(H_0\sqrt{{\Omega }_{r,0}})$. Its associated inverse length differs by an order-one factor from the matter-radiation equality scale $a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$.

In a spatially flat Friedmann-Lemaître-Robertson-Walker metric with initial conformal time zero, two comoving worldlines at separation $d$ can exchange a one-way light signal by time $\eta$ only if $d\le \eta$. Their past light cones overlap if $d\le 2\eta$: a common event halfway between them can influence both. These are different statements. For opposite emission events observed at conformal time ${\eta }_0$, their separation is $2({\eta }_0-{\eta }_e)$, so the respective thresholds are ${\eta }_e\ge 2{\eta }_0/3$ and ${\eta }_e\ge {\eta }_0/2$.

For a radiation-filled expanding universe with $0<{\Omega }_0<1$ and no cosmological constant, set $x=a/a_0$. The particle horizon radial proper distance is $H_0^{-1}{\int }_0^1[{\Omega }_0+(1-{\Omega }_0)x^2{]}^{-1/2}dx$. An inverse-hyperbolic-sine antiderivative gives the formula. The spatially flat limit is $d_{h0}=H_0^{-1}$; radial proper distance is distinct from the area radius of the corresponding horizon sphere.

A horizon-length patch on the matter-radiation equality surface has physical size $a_{\mathrm{eq}}{\tau }_{\mathrm{eq}}$ in a flat universe with conformal time measured from the Big Bang. Dividing by its angular diameter distance $a_{\mathrm{eq}}({\tau }_0-{\tau }_{\mathrm{eq}})$ gives its small angular size. For the radiation-matter scale factor in conformal time, this is $(\sqrt{2}-1)\sqrt{{\Omega }_{r,0}}/[1-\sqrt{2{\Omega }_{r,0}}]$. A patch diameter is twice the angle associated with one horizon radius. The particle horizon is an integral over past evolution and must not be silently replaced by the instantaneous Hubble radius.