Editing Particle horizon (section)

== Kinematic model ==
The particle horizon is a distance in a comoving coordinate system, a system that has the expansion of the universe built-in. The expansion is defined by a (dimensionless) [[Scale factor (cosmology)|scale factor]] <math>a(t)</math> set to have a value of one today. The time that light takes to travel a distance {{mvar|dx}} in the comoving coordinate system will be <math>dx=dt/a(t)</math> in units of light years (<math>c=1</math>). The total distance light can travel in the time {{mvar|t}} since the [[Big Bang]] at <math>t=0</math> sums all the incremental distances:<ref name=Dodelson-2003>{{Cite book |last=Dodelson |first=Scott |title=Modern cosmology |date=2003 |publisher=Academic Press |isbn=978-0-12-219141-1 |location=San Diego, Calif}}</ref>{{rp|34}}

:<math display="block">\eta = \int_{0}^{t} \frac{dt'}{a(t')}</math>

The ''comoving horizon'' <math>\eta</math> increases monotonically and thus can be used a time parameter: the particle horizon is equal to the ''conformal time'' <math>\eta</math> that has passed since the [[Big Bang]], times the [[speed of light]] <math>c</math>.<ref name=Dodelson-2003/>{{rp|34}}

By convention, a subscript 0 indicates "today" so that the conformal time today <math>\eta(t_0) = \eta_0 = 1.48 \times 10^{18}\text{ s}</math>. Note that the conformal time is ''not'' the [[age of the universe]] as generally understood. ''That'' age refers instead to a time as defined by the Robertson-Walker form of the cosmological metric, which time is presumed to be measured by a traditional clock and estimated to be around <math>4.35 \times 10^{17}\text{ s}</math>. By contrast <math>\eta_0</math> is the age of the universe as measured by a Marzke-Wheeler "light clock".<ref>{{Cite book |last1=Marzke |first1=R. F. |title=Gravitation and relativity |last2=Wheeler |first2=J. A. |date=1964 |publisher=Benjamin |editor-last=Chiu |editor-first=H. Y. |publication-date=1964 |pages=40–64}}</ref>

The particle horizon recedes constantly as time passes and the conformal time grows. As such, the observed size of the universe always increases.<ref name="books.google.com" /><ref>{{Cite book |last1=Hobson |first1=M. P. |url=https://books.google.com/books?id=xma1QuTJphYC&pg=PA419 |title=General relativity: an introduction for physicists |last2=Efstathiou |first2=George |last3=Lasenby |first3=A. N. |date=2006 |publisher=Cambridge University Press |isbn=978-0-521-82951-9 |location=Cambridge, UK; New York |pages=419– |oclc=ocm61757089}}</ref> Since proper distance at a given time is just comoving distance times the scale factor<ref name="expandingconfusion">{{Cite journal |last1=Davis |first1=Tamara M. |last2=Lineweaver |first2=Charles H. |year=2004 |title=Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe |url=https://www.cambridge.org/core/product/identifier/S132335800000607X/type/journal_article |journal=Publications of the Astronomical Society of Australia |language=en |volume=21 |issue=1 |pages=97–109 |arxiv=astro-ph/0310808 |bibcode=2004PASA...21...97D |doi=10.1071/AS03040 |issn=1323-3580 |s2cid=13068122}}</ref> (with [[comoving distance]] normally defined to be equal to proper distance at the present time, so <math>a(t_0) = 1</math> at present), the proper distance, <math>d_p(t),</math> to the particle horizon at time <math>t</math> is given by<ref name="Giovannini">{{Cite book |last=Giovannini |first=Massimo |url=https://archive.org/details/primeronphysicso0000giov |title=A primer on the physics of the cosmic microwave background |publisher=World Scientific |year=2008 |isbn=978-981-279-142-9 |location=Singapore; Hackensack, NJ |pages=[https://archive.org/details/primeronphysicso0000giov/page/70 70]– |oclc=191658608 |url-access=registration}}</ref>{{rp|417}}

:<math>d_p(t) = a(t) \int_{0}^{t} \frac{c\,dt'}{a(t')}</math>

The value of the distance to the horizon depends on details in <math>a(t)</math>.