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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 light from 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,[1] so its distance at the present epoch defines the size of the observable universe.[2] 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 light-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.

Conformal time and the particle horizon[edit]

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

η=∫0tdt′a(t′),{displaystyle eta =int _{0}^{t}{frac {dt'}{a(t')}},}

where a(t){displaystyle a(t)} is the scale factor of the Friedmann–Lemaître–Robertson–Walker metric, and we have taken the Big Bang to be at t=0{displaystyle t=0}. By convention, a subscript 0 indicates 'today' so that the conformal time today η(t0)=η0=1.48×1018 s{displaystyle eta (t_{0})=eta _{0}=1.48times 10^{18}{text{ s}}}. Note that the conformal time is not the age of the universe. Rather, the conformal time is the amount of time it would take a photon to travel from where we are located to the furthest observable distance provided the universe ceased expanding. As such, η0{displaystyle eta _{0}} is not a physically meaningful time (this much time has not yet actually passed), though, as we will see, the particle horizon with which it is associated is a conceptually meaningful distance.

The particle horizon recedes constantly as time passes and the conformal time grows. As such, the observed size of the universe always increases.[1][3] Since proper distance at a given time is just comoving distance times the scale factor[4] (with comoving distance normally defined to be equal to proper distance at the present time, so a(t0)=1{displaystyle a(t_{0})=1} at present), the proper distance to the particle horizon at time t{displaystyle t} is given by[5]

a(t)Hp(t)=a(t)∫0tcdt′a(t′){displaystyle a(t)H_{p}(t)=a(t)int _{0}^{t}{frac {c,dt'}{a(t')}}}

and for today t=t0{displaystyle t=t_{0}}

Hp(t0)=cη0=14.4 Gpc=46.9 billion light years.{displaystyle H_{p}(t_{0})=ceta _{0}=14.4{text{ Gpc}}=46.9{text{ billion light years}}.}

Evolution of the particle horizon[edit]

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 ρi{displaystyle rho _{i}}, partial pressure pi{displaystyle p_{i}} and state equation pi=ωiρi{displaystyle p_{i}=omega _{i}rho _{i}}, such that they add up to the total density ρ{displaystyle rho } and total pressure p{displaystyle p}.[6] Let us now define the following functions:

  • Hubble function H=a˙a{displaystyle H={frac {dot {a}}{a}}}
  • The critical density ρc=38πGH2{displaystyle rho _{c}={frac {3}{8pi G}}H^{2}}
  • The i-th dimensionless energy density Ωi=ρiρc{displaystyle Omega _{i}={frac {rho _{i}}{rho _{c}}}}
  • The dimensionless energy density Ω=ρρc=∑Ωi{displaystyle Omega ={frac {rho }{rho _{c}}}=sum Omega _{i}}
  • The redshift z{displaystyle z} given by the formula 1+z=a0a(t){displaystyle 1+z={frac {a_{0}}{a(t)}}}

Any function with a zero subscript denote the function evaluated at the present time t0{displaystyle t_{0}} (or equivalently z=0{displaystyle z=0}). The last term can be taken to be 1{displaystyle 1} including the curvature state equation.[7] It can be proved that the Hubble function is given by

H(z)=H0∑Ωi0(1+z)ni{displaystyle H(z)=H_{0}{sqrt {sum Omega _{i0}(1+z)^{n_{i}}}}}

where ni=3(1+ωi){displaystyle n_{i}=3(1+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:[7]

The particle horizon Hp exists if and only if N>2{displaystyle {text{The particle horizon }}H_{p}{text{ exists if and only if }}N>2}

where N{displaystyle N} is the largest ni{displaystyle n_{i}} (possibly infinite). The evolution of the particle horizon for an expanding universe (a˙>0{displaystyle {dot {a}}>0}) is:[7]

dHpdt=Hp(z)H(z)+c{displaystyle {frac {dH_{p}}{dt}}=H_{p}(z)H(z)+c}

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

Horizon problem[edit]

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

Hp(tCMB)=cηCMB=284 Mpc=8.9×10−3Hp(t0){displaystyle H_{p}(t_{text{CMB}})=ceta _{text{CMB}}=284{text{ Mpc}}=8.9times 10^{-3}H_{p}(t_{0})}

which corresponds to a proper size at that time of:

aCMBHp(tCMB)=261 kpc{displaystyle a_{text{CMB}}H_{p}(t_{text{CMB}})=261{text{ kpc}}}

Since we observe the CMB to be emitted essentially from our particle horizon (284 Mpc≪14.4 Gpc{displaystyle 284{text{ Mpc}}ll 14.4{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

f=Hp(tCMB)Hp(t0){displaystyle f={frac {H_{p}(t_{text{CMB}})}{H_{p}(t_{0})}}}

(an angular size of θ∼1.7∘{displaystyle theta sim 1.7^{circ }})[8] 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.

See also[edit]

References[edit]

  1. ^ abEdward Robert Harrison (2000). Cosmology: the science of the universe. Cambridge University Press. pp. 447–. ISBN978-0-521-66148-5. Retrieved 1 May 2011.
  2. ^Andrew R. Liddle; David Hilary Lyth (13 April 2000). Cosmological inflation and large-scale structure. Cambridge University Press. pp. 24–. ISBN978-0-521-57598-0. Retrieved 1 May 2011.
  3. ^Michael Paul Hobson; George Efstathiou; Anthony N. Lasenby (2006). General relativity: an introduction for physicists. Cambridge University Press. pp. 419–. ISBN978-0-521-82951-9. Retrieved 1 May 2011.
  4. ^Davis, Tamara M.; Charles H. Lineweaver (2004). 'Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the universe'. Publications of the Astronomical Society of Australia. 21 (1): 97. arXiv:astro-ph/0310808. Bibcode:2004PASA...21...97D. doi:10.1071/AS03040.
  5. ^Massimo Giovannini (2008). A primer on the physics of the cosmic microwave background. World Scientific. pp. 70–. ISBN978-981-279-142-9. Retrieved 1 May 2011.
  6. ^Berta Margalef-Bentabol; Juan Margalef-Bentabol; Jordi Cepa (21 December 2012). 'Evolution of the cosmological horizons in a concordance universe'. Journal of Cosmology and Astroparticle Physics. 2012 (12): 035. arXiv:1302.1609. Bibcode:2012JCAP...12..035M. doi:10.1088/1475-7516/2012/12/035.
  7. ^ abcBerta Margalef-Bentabol; Juan Margalef-Bentabol; Jordi Cepa (8 February 2013). 'Evolution of the cosmological horizons in a universe with countably infinitely many state equations'. Journal of Cosmology and Astroparticle Physics. 015. 2013 (2): 015. arXiv:1302.2186. Bibcode:2013JCAP...02..015M. doi:10.1088/1475-7516/2013/02/015.
  8. ^'Understanding the Cosmic Microwave Background Temperature Power Spectrum'(PDF). Retrieved 5 November 2015.
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MILAN (AP) — Portugal and France lost their perfect records in this season’s UEFA Nations League after playing out a 0-0 draw against each other on Sunday, and Belgium saw its 13-match unbeaten run ended by a 2-1 defeat against England.

Two of the sport’s best forwards were kept quiet at a chilly Stade de France.

Portugal star Cristiano Ronaldo had a good chance in the second minute of injury time, but goalkeeper Hugo Lloris palmed away his angled shot from the left. France’s Kylian Mbappe was substituted near the end after having failed to make an impression against the Nations League champion’s resolute defense.

Portugal stays top in Group 3 of League A, ahead of France on goal difference, with both teams on seven points.

Third-place Croatia has three points after beating Sweden 2-1. Substitute Andrej Kramarić scored the winner in that match, nine minutes after coming off the bench.

ENGLAND COMEBACK

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England came from behind to beat Belgium and move top in Group 2 of League A.

Marcus Rashford’s penalty canceled out Romelu Lukaku’s spot kick in the first half before Mason Mount’s looping, deflected shot clinched England’s first victory over a top-ranked team in nine years.

England moved a point ahead of second-place Belgium and three above Denmark which beat Iceland 3-0.

STALEMATES

Italy was held to a 0-0 draw against Poland but remained top of its group after the Netherlands drew by the same score against Bosnia-Herzegovina.

The match was supposed to pit two of Europe’s top goalscorers against each other in Robert Lewandowski and Ciro Immobile. However, Italy coach Roberto Mancini surprisingly left Immobile on the bench.

Italy is a point above the Netherlands and Poland in Group 1 of League A. It plays the Netherlands on Wednesday.

HAALAND HAT TRICK

Erling Haaland netted a hat trick to help Norway thrash Romania 4-0 and assuage its European Championship playoff disappointment.

Alexander Sørloth scored Norway’s other goal. Haaland already has six goals in the competition.

Norway lost a Euro 2020 playoff to Serbia on Thursday.

Norway is level on six points with leader Austria in Group 1 of League B.

Austria beat host Northern Ireland 1-0.

OTHER MATCHES

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Finland beat Bulgaria 2-0 in Group 4 of League B and Ireland drew 0-0 against Wales.

Also in League B, it was: Israel 1, Czech Republic 2; Scotland 1, Slovakia 0; Russia 1, Turkey 1; and Serbia 0, Hungary 1.

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There were four draws in League C: Armenia 2, Georgia 2; Estonia 3, North Macedonia 3; Kazakhstan 0, Albania 0; and Lithuania 2, Belarus 2.

Also, Greece beat Moldova 2-0 and Slovenia won 1-0 in Kosovo.

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