Editing Particle horizon (section)

==Evolution of the particle horizon==

In this section we consider the [[Friedmann–Lemaître–Robertson–Walker metric|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 <math>\rho_i</math>, partial pressure <math>p_i</math> and [[Equation of state (cosmology)|state equation]] <math>p_i=\omega_i \rho_i</math>, such that they add up to the total density  <math>\rho</math> and total pressure <math>p</math>.<ref>{{Cite journal |last1=Margalef-Bentabol |first1=Berta |last2=Margalef-Bentabol |first2=Juan |last3=Cepa |first3=Jordi |date=2012-12-21 |title=Evolution of the cosmological horizons in a concordance universe |journal=Journal of Cosmology and Astroparticle Physics |volume=2012 |issue=12 |pages=035 |arxiv=1302.1609 |bibcode=2012JCAP...12..035M |doi=10.1088/1475-7516/2012/12/035 |issn=1475-7516 |s2cid=119704554}}</ref> Let us now define the following functions:

* Hubble function   <math>H=\frac{\dot a}{a}</math>
* The critical density   <math>\rho_c=\frac{3}{8\pi G}H^2</math>
* The ''i''-th dimensionless energy density   <math>\Omega_i=\frac{\rho_i}{\rho_c}</math>
* The dimensionless energy density <math>\Omega=\frac \rho {\rho_c}=\sum \Omega_i</math>
* The redshift <math>z</math> given by the formula   <math>1+z=\frac{a_0}{a(t)}</math>

Any function with a zero subscript denote the function evaluated at the present time <math>t_0</math> (or equivalently <math>z=0</math>). The last term can be taken to be <math>1</math> including the curvature state equation.<ref name="Evolution2">{{Cite journal |last1=Margalef-Bentabol |first1=Berta |last2=Margalef-Bentabol |first2=Juan |last3=Cepa |first3=Jordi |date=February 2013 |title=Evolution of the cosmological horizons in a universe with countably infinitely many state equations |journal=Journal of Cosmology and Astroparticle Physics |series=015 |volume=2013 |issue=2 |pages=015 |arxiv=1302.2186 |bibcode=2013JCAP...02..015M |doi=10.1088/1475-7516/2013/02/015 |issn=1475-7516 |s2cid=119614479}}</ref> It can be proved that the Hubble function is given by

:<math> H(z)=H_0\sqrt{\sum \Omega_{i0}(1+z)^{n_i}}</math>

where the dilution exponent <math>n_i=3(1+\omega_i)</math>. Notice that the addition ranges over all possible partial constituents and in particular there can be countably infinitely many. With this notation we have:<ref name="Evolution2" />

:<math> \text{The particle horizon } d_p \text{ exists if and only if } N>2</math>

where <math>N</math> is the largest <math>n_i</math> (possibly infinite). The evolution of the particle horizon for an expanding universe (<math>\dot{a}>0</math>) is:<ref name="Evolution2" />

:<math> \frac{d}{dt}d_p=d_p(z)H(z)+c</math>

where <math>c</math> is the speed of light and can be taken to be <math>1</math> ([[natural units]]). Notice that the derivative is made with respect to the FLRW-time <math>t</math>, while the functions are evaluated at the redshift <math>z</math> which are related as stated before. We have an analogous but slightly different result for [[event horizon]].