particle horizon

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 $a(t)$ 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 $dx=dt/a(t)$ in units of light years ($c=1$). The total distance light can travel in the time {{mvar|t}} since the Big Bang at $t=0$ sums all the incremental distances:{{Cite book |last=Dodelson |first=Scott |title=Modern cosmology |date=2003 |publisher=Academic Press |isbn=978-0-12-219141-1 |location=San Diego, Calif}}{{rp|34}}

:$$\backslash eta\; =\; \backslash int\_\{0\}^\{t\}\; \backslash frac\{dt\text{'}\}\{a(t\text{'})\}$$

The comoving horizon $\backslash eta$ increases monotonically and thus can be used a time parameter: the particle horizon is equal to the conformal time $\backslash eta$ that has passed since the Big Bang, times the speed of light $c$.{{rp|34}}

By convention, a subscript 0 indicates "today" so that the conformal time today $\backslash eta(t\_0)\; =\; \backslash eta\_0\; =\; 1.48\; \backslash times\; 10^\{18\}\backslash text\{\; s\}$. 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 $4.35\; \backslash times\; 10^\{17\}\backslash text\{\; s\}$. By contrast $\backslash eta\_0$ is the age of the universe as measured by a Marzke-Wheeler "light clock".{{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}}

The particle horizon recedes constantly as time passes and the conformal time grows. As such, the observed size of the universe always increases.{{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}} Since proper distance at a given time is just comoving distance times the scale factor{{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}} (with comoving distance normally defined to be equal to proper distance at the present time, so $a(t\_0)\; =\; 1$ at present), the proper distance, $d\_p(t),$ to the particle horizon at time $t$ is given by{{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}}{{rp|417}}

:$d\_p(t)\; =\; a(t)\; \backslash int\_\{0\}^\{t\}\; \backslash frac\{c\backslash ,dt\text{'}\}\{a(t\text{'})\}$

The value of the distance to the horizon depends on details in $a(t)$.

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 $\backslash rho\_i$, partial pressure $p\_i$ and state equation $p\_i=\backslash omega\_i\; \backslash rho\_i$, such that they add up to the total density $\backslash rho$ and total pressure $p$.{{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}} Let us now define the following functions:

• Hubble function $H=\backslash frac\{\backslash dot\; a\}\{a\}$
• The critical density $\backslash rho\_c=\backslash frac\{3\}\{8\backslash pi\; G\}H^2$
• The i-th dimensionless energy density $\backslash Omega\_i=\backslash frac\{\backslash rho\_i\}\{\backslash rho\_c\}$
• The dimensionless energy density $\backslash Omega=\backslash frac\; \backslash rho\; \{\backslash rho\_c\}=\backslash sum\; \backslash Omega\_i$
• The redshift $z$ given by the formula $1+z=\backslash frac\{a\_0\}\{a(t)\}$

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.{{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}} It can be proved that the Hubble function is given by

:$H(z)=H\_0\backslash sqrt\{\backslash sum\; \backslash Omega\_\{i0\}(1+z)^\{n\_i\}\}$

where the dilution exponent $n\_i=3(1+\backslash omega\_i)$. Notice that the addition ranges over all possible partial constituents and in particular there can be countably infinitely many. With this notation we have:

:$\backslash text\{The\; particle\; horizon\; \}\; d\_p\; \backslash text\{\; exists\; if\; and\; only\; if\; \}\; N>2$

where $N$ is the largest $n\_i$ (possibly infinite). The evolution of the particle horizon for an expanding universe ($\backslash dot\{a\}>0$) is:

:$\backslash frac\{d\}\{dt\}d\_p=d\_p(z)H(z)+c$

where $c$ is the speed of light and can be taken to be $1$ (natural units). 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.

{{main|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 model. Extrapolating back to the time of recombination when the cosmic microwave background (CMB) was emitted, we obtain a particle horizon of about

{{block indent|$H\_p(t\_\backslash text\{CMB\})\; =\; c\backslash eta\_\backslash text\{CMB\}\; =\; 284\backslash text\{\; Mpc\}\; =\; 8.9\; \backslash times\; 10^\{-3\}\; H\_p(t\_0)$}}

which corresponds to a proper size at that time of:

{{block indent|$a\_\backslash text\{CMB\}H\_p(t\_\backslash text\{CMB\})\; =\; 261\backslash text\{\; kpc\}$}}

Since we observe the CMB to be emitted essentially from our particle horizon ($284\backslash text\{\; Mpc\}\; \backslash ll\; 14.4\backslash text\{\; Gpc\}$), our expectation is that parts of the cosmic microwave background (CMB) that are separated by about a fraction of a great circle across the sky of

{{block indent|$f\; =\; \backslash frac\{H\_p(t\_\backslash text\{CMB\})\}\{H\_p(t\_0)\}$}}

(an angular size of $\backslash theta\; \backslash sim\; 1.7^\backslash circ$){{Cite web |last=Tojero |first=Rita |date=March 16, 2006 |title=Understanding the Cosmic Microwave Background Temperature Power Spectrum |url=http://www.roe.ac.uk/ifa/postgrad/pedagogy/2006_tojeiro.pdf |access-date=5 November 2015 |website=Royal Observatory, Edinburgh}} should be out of causal contact with each other. That the entire CMB is in thermal equilibrium and approximates a blackbody so well is therefore not explained by the standard explanations about the way the expansion of the universe proceeds. The most popular resolution to this problem is cosmic inflation.

• Cosmological horizon
• Observable universe

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Category:Physical cosmology