Evolution of the Cosmological Horizons in a Concordance Universe
EEvolution of the Cosmological Horizonsin a Concordance Universe
Berta Margalef–Bentabol Juan Margalef–Bentabol , Jordi Cepa , [email protected] [email protected] [email protected] Departamento de Astrofísica, Universidad de la Laguna, E-38205 La Laguna, Tenerife, Spain . Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, E-28040 Madrid, Spain. Facultad de Ciencias Físicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain. Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain.
Abstract
The particle and event horizons are widely known and studied concepts, but the study of their properties,in particular their evolution, have only been done so far considering a single state equation in a deceler-ating universe. This paper is the first of two where we study this problem from a general point of view.Specifically, this paper is devoted to the study of the evolution of these cosmological horizons in an accel-erated universe with two state equations, cosmological constant and dust. We have obtained closed-formexpressions for the horizons, which have allowed us to compute their velocities in terms of their respectiverecession velocities that generalize the previous results for one state equation only. With the equations ofstate considered, it is proved that both velocities remain always positive.
Keywords : Physics of the early universe – Dark energy theory – Cosmological simulationsThis is an author-created, un-copyedited version of an article accepted for publication inJournal of Cosmology and Astroparticle Physics. IOP Publishing Ltd/SISSA Medialab srlis not responsible for any errors or omissions in this version of the manuscript or any versionderived from it. The definitive publisher authenticated version is available online at: http://iopscience.iop.org/1475-7516/2012/12/035
In a Robertson-Walker universe, for any observer A we can define two regions in the instantaneous three-dimensional space t = t . The first one is the region defined by the comoving points that have already beenobserved by A (those comoving objects emitted some light in the past and it has already reached us), andthe second one is its complement in the three-dimensional space, i.e. the region that cannot be observedby A at a time t . The boundary between these two regions is the particle horizon at t , that defines theobservable universe for A . Notice that the particle horizon takes into account only the past events withrespect to A . Another horizon could be defined taking into account also the A (cid:48) s future. This horizon isthe event horizon and it is defined as the hyper-surface in space-time which divides all events into twoclasses, those that will be observable by A , and those that are forever outside A (cid:48) s range of observation.This horizon determines a limit in the future observable universe [7].A deep study of the horizons has been made before in [2] where only one state equation was considered.However, it is currently widely accepted that our present Universe is a concordance universe i.e. itis flat and dominated by two state equations (cosmological constant and dust), one of them of negativepressure, that drives the universe into an accelerated expansion. This fact leads us to make a deep studyof more general situations that we summarize in a couple of papers, in this one, we make a deep studytaking into account these two state equations with no curvature, so that we can obtain results applicableto the currently accepted cosmology, and in the second one [5], a complete general study (at least from amathematical point of view) has been made considering countably infinitely many state equations with orwithout curvature. - 1 - a r X i v : . [ a s t r o - ph . C O ] J un or numerical values in a concordance universe, we are going to use the following cosmological parameters:Hubble constant H = 70 . − Mpc − Current density of matter (dark and baryonic) parameter Ω m = 0 . Current density of cosmological constant parameter Ω Λ0 = 0 . This cosmological parameters have been obtained from [6] where a combination of data from WMAP, BAOand SNCONST is considered. Note that as a good first approximation, we are neglecting in this paper theparameter of density of radiation, Ω r , although it should be taken into account when considering timesnear the beginning of the Universe, when radiation dominates over dust, as indeed we did in [5].This paper could be summarized as follows: in section one, we briefly introduce some basic cosmologicalconcepts and establish the nomenclature to be used throughout the paper. In sections two and three,we define the formal concepts of particle and event horizons, and derive their integral expressions atany cosmological time. In section four we study the evolution of the horizons, obtaining more suitableexpressions for the horizons at every time. Section five is devoted to gathering all the results obtained,discussion and conclusions. In this last section, some relevant graphics about the horizons and theirderivatives in the at concordance universe are included. Throughout this paper, we use geometrical unitswhere c = G = 1 . In order to introduce the particle and event horizons, we need to define the proper distance D p , which isthe distance between two simultaneous events at a cosmological time t measured by an inertial observer.Considering a homogenous and isotropic universe, we can write the Robertson-Walker metric, where a ( t ) is the scale factor (albeit with units of longitude in our system) and k the sign of the curvature: ds = − dt + a ( t ) (cid:18) − kr dr + r d Ω (cid:19) (2.1)Isotropy of the Robertson-Walker universe allows us to consider any direction, i.e. θ, φ constant and so d Ω = 0 , and homogeneity allows us to consider that the observer is at r = 0 . Therefore by the definitionof the proper distance (where dt = 0 ) we have for a given time t : D p ( R ) = (cid:90) D p ds = a ( t ) (cid:90) R dr √ − kr (2.2)Notice that R is the radial comoving coordinate of the measured point. On the other hand, if we considerlight rays ( ds = 0) , we have, from (2.1) (the minus sign coming from the fact that we are considering thelight coming towards us): dta = − dr √ − kr −→ (cid:90) t o t e dt (cid:48) a ( t (cid:48) ) = − (cid:90) r e dr √ − kr = (cid:90) r e dr √ − kr (2.3)Where the e subscript stands for emission and the o subscript for observation. Notice that under ourassumptions r o = 0 . Therefore, using the last two equations, the proper distance from r = 0 to r = R at agiven time t can be expressed as a distance measured using light as follows: D p ( t e ) = a ( t ) (cid:90) tt e dt (cid:48) a ( t (cid:48) ) (2.4)Notice that we have exchanged the variables R and t e using the biunivocal correspondence given by (2.3).So to speak, the previous formula represents the distance covered by the light between two points ofthe space-time, but considering an expanding universe since the a factor accounts for this expansion. Inparticular, if a is constant, then we are just measuring one cathetus using the another one and the factthat the speed of light is the same for all inertial frames (hence the angle formed is always the same) aswe can see in figure 1. - 2 - t e D p ( R ) ≡ D p ( t e ) t − t e c = D p ( t e ) = Z tt e dt ′ = a ( t ) Z tt e dt ′ a ( t ′ )45 o r e = R r = 045 o Figure 1 – Representation of the measurement of the proper distance with light when a ( t ) ≡ a . The
Hubble parameter H is defined as: H ( t ) = 1 a dadt (2.5)whose value at t = t is the Hubble constant H . The product of H times the proper distance, hasdimensions of velocity and is known as the recession velocity (of a point located at comoving coordinate R at time t ): v r ≡ HD p ( R ) (2.6)which physically is the instantaneous velocity of an object at a proper distance D p with respect to aninertial observer.We are considering the universe as a perfect fluid with density ρ and pressure p , but it can also beapproximated as composed by separate constituents of density ρ i and partial pressure p i , where all togetheradd up to ρ and p respectively. Furthermore, the i -th quantities are related by the linear state equation p i = w i ρ i . Now using second Friedmann equation [3, chap.3] (where the dot stands for time derivation): ˙ ρ = − H ( ρ + p ) = − w ) ρ ˙ aa (2.7)and considering the i -th state equation only, we can obtain the i -th density ρ i in terms of the redshift z : ρ i = ρ i (1 + z ) w i ) (2.8)Now we define the following time dependent magnitudes: • Critical density as ρ c = 38 π H • Dimensionless energy density as
Ω = ρρ c • Dimensionless i -th energy density as Ω i = ρ i ρ c All this quantities, when referred to the current time, are denoted with a zero subindex. If we now substitutein the definition of Ω i , the expressions of ρ c , Ω i and ρ i , we obtain: Ω i ( z ) = Ω i H (1 + z ) w i ) H ( z ) (2.9)Finally, adding all the Ω i (where the index i ranges over all possible state equations) leads to: H ( z ) = H (cid:118)(cid:117)(cid:117)(cid:116) n (cid:88) i =1 Ω i Ω (1 + z ) w i ) (2.10) - 3 - Particle Horizon
As the age of the Universe and the light velocity have finite values, there exists a particle horizon H p , thatrepresents the longest distance from which we can retrieve information from the past, so it defines the pastobservable universe. Then the particle horizon H p at the current moment is given by the proper distancemeasured by the light coming from t = 0 (the origin of the universe, where a hot big bang is assumed,compatible with the dominant equation of state) to t : H p = lim t e → D p ( t e ) = (cid:90) t dt (cid:48) a ( t (cid:48) ) (3.1)For a rigorous discussion of the convergence of this integral see [5], in this paper we will just work out ourparticular case. It is well known that the scale factor a is related with the redshift z by: z = a a (3.2)Combining equations (2.4), (2.5), the derivative of (3.2), and taking into account that z ( t ) = 0 , it followsthat the proper distance can be written as D p ( z e ) = − (cid:90) z ( t o ) z ( t e ) dyH ( y ) = (cid:90) z e dyH ( y ) (3.3)Particularizing equation (2.10) to a flat universe ( Ω = 1 ) with non-relativistic matter ( w m = 0 ) andcosmological constant ( w Λ = − ) [3, chap.3], we have: H ( z ) = H (cid:112) Ω m (1 + z ) + Ω Λ0 (3.4)Applying (3.1), (3.3), (3.4), and using the boundary condition that z → ∞ when t → , we obtain theexpression of the particle horizon at the current cosmic time: H p = 1 H (cid:90) ∞ dy (cid:112) Ω m (1 + y ) + Ω Λ0 (3.5) We have so far obtained the particle horizon for the current cosmic time (corresponding to z = 0 ). If wewant to know the expressions of this horizon at any cosmic time, or equivalently at any other z (cid:48) , we have toreplace in (3.5) the constant parameters H , Ω m and Ω Λ0 (corresponding to z = 0 ) with the parameterscorresponding to z (cid:48) that we denote H ( z (cid:48) ) , Ω m ( z (cid:48) ) and Ω Λ ( z (cid:48) ) . Where we recall from equation (2.9) that Ω m ( z (cid:48) ) and Ω Λ ( z (cid:48) ) are the dimensionless energy densities given by: Ω Λ ( z (cid:48) ) = Ω Λ0 H H ( z (cid:48) ) Ω m ( z (cid:48) ) = Ω m H H ( z (cid:48) ) (1 + z (cid:48) ) (3.6)Therefore, we obtain the particle horizon for any redshift z (where the prime is omitted in order to simplifythe notation): H p ( z ) = 1 H ( z ) (cid:90) ∞ dy (cid:112) Ω m ( z )(1 + y ) + Ω Λ ( z ) (3.7) The event horizon represents the barrier between the future events that can be observed, and those thatcannot. It sets up a limit in the future observable universe, since in the future the observer will be able toobtain information only from events which happen inside their event horizon. According to its definitionthe event horizon can be expressed as: H e = lim t →∞ ( − D p ) (4.1)Proceeding as in section 3.1, but now taking into account that z → − as t → ∞ , we obtain that the eventhorizon is: H e = 1 H (cid:90) − dy (cid:112) Ω m (1 + y ) + Ω Λ0 (4.2) - 4 - .2 At any cosmic time Now proceeding as in section 3.2, the event horizon at any z is: H e ( z ) = 1 H ( z ) (cid:90) − dy (cid:112) Ω m ( z )(1 + y ) + Ω Λ ( z ) (4.3) In this section we are going to obtain the expressions of particle and event horizons (as well as its derivatives)using hypergeometric functions. We would like to recall that not only is this approach interesting becausewe obtain analytical expressions for the horizons, but it is also a great advantage since these functions havebeen widely used and analysed.
The theory of hypergeometric functions can be found in many books devoted to advanced calculus. Forgeneral theory see for example [1] or [4, chap. 9], the latter is the one we are following here. In what follows,we are going to establish the minimum amount of theory that we need for our purposes.
Definition 5.1. F has the following (integral) definition: F ( α, β ; γ ; x ) = Γ( γ )Γ( β )Γ( γ − β ) (cid:90) t β − (1 − t ) γ − β − (1 − tx ) α dt Re ( γ ) > Re ( β ) > | arg (1 − x ) | < π (5.1)where Γ( z ) is the gamma function, defined by: Γ( z ) = (cid:90) ∞ e − t t z − dt, Re ( z ) > (5.2)and by analytic continuation for every z ∈ C apart from z = 0 , − , − . . . where simple poles appears. Remarks. Γ( z + 1) = z Γ( z ) for every z ∈ C apart from z = 0 , − , − . . . and Γ(1) = 1 .5.3 Since no distinction between hypergeometric functions p F q is necessary, from now on, we are goingto write F ≡ F .5.4 The function F , as a function of x , is defined in the whole complex plane cut along [1 , ∞ ) , thoughwe would only need real negative values.5.5 The function F can be extended for every α , β ∈ C and for every γ (cid:54) = 0 , − , − . . . through recursiveequations, in fact, F ( α, β ; γ ; x ) / Γ( γ ) is an entire function of the parameters α, β, γ . For more detailssee [4, sect. 9.4].5.6 When we restrict F , considered as a function of x with fixed real parameters, to real values of x , weobtain a real-valued function F : ( −∞ , → R .5.7 Applying the properties of the beta function B ( x, y ) B ( x, y ) = (cid:90) t x − (1 − t ) y − dt, Re ( x ) , Re ( y ) > we can rewrite Γ( γ )Γ( β )Γ( γ − β ) = 1 B ( β, γ − β ) .Now we are going to state some useful properties (always considering that the third argument is differentfrom , − , − . . . and the fourth one verifies | arg (1 − x ) | < π ) that can be found in [1] or in [4, sect. 9.2–9.8]. Properties. F ( α, β ; γ ; 0) = 1 ∂F∂x ( α, β ; γ ; x ) = − γ − x (cid:20) F ( α, β ; γ ; x ) − F ( α, β ; γ − , x ) (cid:21) - 5 - .10 lim x → − F ( α, β ; γ ; x ) = B ( β, γ − β − α ) B ( β, γ − β ) Re ( γ − β − α ) > F ( α, β ; β ; x ) = (1 − x ) − α F ( α, β ; γ ; x ) = 1(1 − x ) α F (cid:18) α, γ − β ; γ ; xx − (cid:19) lim x →∞ F ( α, β ; γ ; − x ) = 0 α, β > α − β / ∈ Z The last expression follows by one of the analytic extensions of F given in [4, sect. 9.5] together withproperty 5.8. Now we are going to obtain some particular values of F that we will need. Lemmas. (cid:90) ∞ ds √ s + A = 2 F (cid:18) ,
16 ; 76 ; − A (cid:19) if A > (cid:90) ds √ s + A = 1 √ A F (cid:18) ,
13 ; 43 ; − A (cid:19) if A > ∂F∂x ( α, β ; β + 1; x ) = − βx (cid:20) F ( α, β ; β + 1; x ) − − x ) α (cid:21) lim x → − F (cid:18) ,
1; 1 + c ; x (cid:19) = ∞ if < c < Proof. s = t − / leads tothe left hand side of the equation.5.15 Analogous to the previous case, but now making the change of variable s = t / .5.16 This statement follows immediately from properties 5.9 and 5.11.5.17 Notice first that property 5.10 cannot be applied as the conditions on the parameters do not hold.The result follows applying property 5.8 to the analytical extension in | z − | < cut along [1 , ∞ ) that can be found in [4], equations (9.5.7) or (9.5.10). The hypothesis of this equations are satisfiedas c ∈ (cid:0) , (cid:1) and x ∈ R .It is important to note that in the third equation, we have managed to established the derivative of F (withone restriction in the parameters) in terms of the F itself. Finally, we introduce the auxiliary functions A, B and gather some straightforward computations that we will need to obtain the derivative of bothhorizons. A ≡ Ω Λ ( z )Ω m ( z ) = Ω Λ0 Ω m z ) −→ dAdz = − A z (5.3) B ≡ A = Ω m Ω Λ0 (1 + z ) −→ dBdz = 3 B z (5.4) H ( z ) = H √ Ω Λ0 √ B + 1 (5.5) dzdt = − a a dadt = − a a H = − (1 + z ) H ( z ) (5.6)Where the last expression is obtained applying first equation (3.2), then equation (2.5) and finally equation(3.2) again. - 6 - .2 Expressing the particle horizon through hypergeometric functions The expression of the particle horizon (3.7) can be expressed as follows, where z should verify z ∈ ( − , ∞ ) according to equation (3.2): H p ( z ) = 1 H ( z ) (cid:90) ∞ dy (cid:112) Ω m ( z )(1 + y ) + Ω Λ ( z ) s = y +1 = 1 H ( z ) (cid:112) Ω m ( z ) (cid:90) ∞ ds (cid:112) s + A ( z ) . == 2 H ( z ) (cid:112) Ω m ( z ) F (cid:18) ,
16 ; 76 ; − A ( z ) (cid:19) eq. (3.6) = 2 H (cid:112) Ω m (1 + z ) F (cid:18) ,
16 ; 76 ; − A ( z ) (cid:19) Finally, by definition of A (5.3): H p ( z ) = 2 (cid:112) A ( z ) H √ Ω Λ0 F (cid:18) ,
16 ; 76 ; − A ( z ) (cid:19) (5.7) Now we are going to obtain the derivative of H p . Notice that in the expression of H p , the parameters of F verify γ = β + 1 , so we can use lemma 5.16 and obtain F with the same parameters. For that reason andin order to simplify the notation, we omit the parameters and we also omit the argument z in the A and B function. dH p dz = 1 H √ Ω m (cid:20) dA/dz √ A F ( − A ) + 2 √ A (cid:18) − dAdz (cid:19) ∂F∂x ( − A ) (cid:21) . == 1 H √ Ω m (cid:20) − √ A z F ( − A ) + √ A z (cid:18) F ( − A ) − √ A (cid:19)(cid:21) B =1 /A == − z ) 2 √ AH √ Ω m F ( − A ) − z ) 1 H √ Ω m √ B eq. (5.7) , (5.5) == − H p ( z )1 + z − z ) H ( z ) Finally, applying the chain rule and equation (5.6), we obtain: dH p dt = H p ( z ) H ( z ) + 1 (5.8)Note that H p ( z ) H ( z ) represents the recession velocity of the particle horizon (2.6). Its physical meaningis explained in subsection 6.5, where all the conclusions are provided. Analogously for H e we have: H e ( z ) = 1 H ( z ) (cid:90) − dy (cid:112) Ω m ( z )(1 + y ) + Ω Λ ( z ) s =1+ y = 1 H ( z ) (cid:112) Ω m ( z ) (cid:90) ds √ s + A . == 1 H ( z ) (cid:112) Ω m ( z ) 1 √ A F (cid:18) ,
13 ; 43 ; − A (cid:19) = 1 H ( z ) (cid:112) Ω Λ ( z ) F (cid:18) ,
13 ; 43 ; − A (cid:19) Where in the last equality, the definition of A (5.3) is used. Now using equation (3.6) we obtain thefollowing expression: H e ( z ) = 1 H √ Ω Λ0 F (cid:18) ,
13 ; 43 ; − B ( z ) (cid:19) (5.9) - 7 - .5 Obtaining the derivative of the event horizon We are again in the hypothesis of lemma 5.16, and then it is meaningful to omit the parameters. dH e dz = 1 H √ Ω Λ0 (cid:18) − dBdz (cid:19) ∂F∂x ( − B ) . == − z ) 1 H √ Ω Λ0 F ( − B ) + 11 + z H √ Ω Λ0 √ B eq. (5.9) , (5.5) == − H e ( z )1 + z + 1(1 + z ) H ( z ) Finally, applying the chain rule and equation (5.6), we obtain: dH e dt = H e ( z ) H ( z ) − (5.10)Where now H e ( z ) H ( z ) is the recession velocity of the event horizon. Its physical implications will beexplained in subsection 6.5. Below we show some important values of the horizons, summarized in table 1, and the required computa-tions to obtain them. z t Ω m ( z ) Ω Λ ( z ) H p ( z ) H e ( z ) dH p dt dH e dt Origin of the universe ∞ c ∞ Current time t .
278 0 .
722 2 . cH √ Ω Λ0 . cH √ Ω Λ0 . c . c Future − ∞ ∞ cH √ Ω Λ0 ∞ Table 1 – Some important values of the horizons and their velocities in physical units. Notice thatthe numerical values appearing on the current time values, depend on both Ω Λ0 and Ω m . • H p ( ∞ ) – Notice that A → when z → ∞ and use property 5.8 in (5.7). • H e ( ∞ ) – Notice that B → ∞ when z → ∞ and use property 5.13 in (5.9). • dH p dt ( ∞ ) – Again, A → when z → ∞ , then apply property 5.12 to eq. (5.8), which leads to: dH p dt = 2 F (cid:18) ,
1; 76 ; AA + 1 (cid:19) + 1 −→ F (cid:18) ,
1; 76 ; 0 (cid:19) + 1 . = 3 • dH e dt ( ∞ ) – Again, B → ∞ when z → ∞ , then apply property 5.12 to eq. (5.10), which leads to: dH e dt = F (cid:18) ,
1; 1 + 13 ; BB + 1 (cid:19) − . −−−−−−−→ ∞• H p ( − – Notice that A → ∞ when z → − and apply property 5.12 to eq. (5.7), which leads to: H p = 2 H √ Ω Λ0 (cid:114) AA + 1 F (cid:18) ,
1; 1 + 16 ; AA + 1 (cid:19) . −−−−−−−→ ∞ - 8 - H e ( − – Notice that B → when z → − and use property 5.8 in (5.9). • To determine the values of the derivatives for z = − , just substitute H ( −
1) = H √ Ω Λ0 in equations(5.8) and (5.10), and use the values H p ( − and H e ( − we have just obtained above. • Finally, we obtain the values at the current cosmic time through numerical calculus using (5.7), (5.9),(5.8) and (5.10).
It is clear from equation (5.8) that the velocity of the particle horizon is always positive, but from equation(5.10) we cannot conclude the same for the event horizon. Let us see that with a little more effort weare able to obtain some information about the behaviour of the horizon velocities, including the fact thatthey are indeed always positive. In order to do that, we compute the second derivative which is quitestraightforward using all the previous computations: d H p dt = H (cid:18) − B B ) H p H + 1 (cid:19) d H e dt = H (cid:18) − B B ) H e H − (cid:19) If we equal both equations to zero, we obtain (where the correspondent parameters have been omitted):For H p : F ( − A ) = √ A − A For H e : F ( − B ) = 2 √ B − B From the definition of F , it is easy to prove applying elementary inequalities rules, that when x ∈ ( −∞ , with positive parameters the function F ( x ) is strictly positive and increasing. Then on one hand, we havethat F ( − x ) is positive but decreasing when x ∈ ( − , ∞ ) . Computing the derivative with respect to A ofthe right hand side of first equation, we obtain on the other hand, that it is increasing where it is defined,takes positive values in the interval I = ( − , / and negative for larger values. Gathering these two factswe have that if there exists a solution A for the first equation, it must be unique and A ∈ I . Clearlywhen A = 0 , the right hand side is and from property 5.8 F (0) = 1 , so the unique solution is A = 0 .Analogously for the second equation (but now I = ( − , that does not affect the argument) we obtainthat B = 0 is the unique solution for the second equation.Therefore, from the definition of A and B , we have that the acceleration of H p is zero only at the originof the universe z → ∞ and the acceleration of H e is zero only at the far future z = − , and for the rest ofthe values of z , both accelerations are positive.As at the origin of the universe both velocities were positive, we conclude they are always nonnegative. Infact, they can only vanish for the limits z → ∞ and z → − , and according to table 1, it only happens atthe far future for the event horizon speed. We have included on the appendix (page 12) some figures thatshow the behaviour of both horizons and their derivative. In this section we are going to prove that in fact we are generalizing the previous results obtained takinginto account just one state equation such as the ones stated in [2]. In general, it can be proved that for aunique state equation the following results are obtained for any cosmic time (e.g. using (3.1), (3.3), (2.10)and (2.9) for just one state equation, and (5.6) to compute their derivatives): H ∗ p ( z ) = 2 H ∗ ( z )(1 + 3 w ) dH ∗ p dt = 3(1 + w )1 + 3 w if w > − / H ∗ e ( z ) = − H ∗ ( z )(1 + 3 w ) dH ∗ e dt = − w )1 + 3 w if w < − / Where the ∗ stand for the fact that we are taking into account just one state equation, in contrast withthe equations obtained in this paper taking into account two state equations. • If it dominates the state equation of matter, we have w = 0 , then there only exists the particle horizonwhose equations, as proved in [2], are: H ∗ p ( z ) = 2 H ∗ ( z ) dH ∗ p dt = 3 A and B , that by definition are both positive when z ranges over ( − , ∞ ) , would play the role of the x . - 9 - n this case Ω m ( z ) = Ω m = 1 and Ω Λ ( z ) = Ω Λ0 = 0 , and therefore H ∗ ( z ) = H (cid:112) (1 + z ) . Nowsubstituting the A function in equation (5.7), we have (omitting once again the parameters): H p ( z ) = 2 H (cid:112) Ω m (1 + z ) F (cid:18) − Ω Λ0 Ω m z ) (cid:19) = 2 H (cid:112) (1 + z ) F (0) . = 2 H ∗ ( z ) dH p dt = H ( z ) H p ( z ) + 1 = H ∗ ( z ) 2 H ∗ ( z ) + 1 = 3 • If it dominates the state equation of cosmological constant (i.e. w = − ) then there only exists theevent horizon whose equations are: H e = 1 H ∗ ( z ) dH e dt = 0 In this case Ω m ( z ) = Ω m = 0 and Ω Λ ( z ) = Ω Λ0 = 1 , and therefore H ∗ ( z ) = H . Then, equation(5.9) leads to: H e ( z ) = 1 H √ Ω Λ0 F (cid:18) − Ω m Ω Λ0 (1 + z ) (cid:19) = 1 H F (0) . = 1 H = 1 H ∗ ( z ) dH e dt = H ( z ) H e ( z ) − H ∗ ( z ) 1 H ∗ ( z ) − So we conclude that indeed, our results generalize the previous ones obtained in [2].
The cosmic microwave background radiation that we measure today comes from a spherical surface calledthe surface of last scattering . In fact, it is not a surface but it has some thickness. In the currentStandard Model, the surface of last scattering is at redshift z ls = 1 089 with a thickness of ∆ z = 195 [8].The proper distance to the mean value of this surface is: D p = 1 H (cid:90) dy (cid:112) Ω m (1 + y ) + Ω Λ0 = 14 086 . Mpc
The values of particle and event horizons at the current cosmic time are according to table 1 (with thenumerical constant we mentioned at the beginning of the paper): H p = 14 577 . Mpc H e = 4 823 . Mpc
We notice that from the integral definition of the particle horizon (3.5), the distance to the surface of lastscattering is always smaller than the particle horizon, which agrees with the experimental data. The lightwhich is reaching us at the current moment from objects located over the surface of last scattering wasemitted at a time corresponding to a redshift of z ls . As time goes by, the distance to the surface will behigher, but always less than the particle horizon, so in the future we get information emitted from thesurface of last scattering.On the other hand the distance to the last scattering surface is greater than the event horizon, which meansthat the light emitted at the present moment from this surface, is never going to reach us. It does notmean that in the future we are not going to see background radiation, but that the radiation we can seeat the current time and which we will see in the future is the one inside the present event horizon. Thiscan be generalized to any object in the universe, and drives to the conclusion that in an ever accelerateduniverse (i.e. dominated by a negative pressure) with several state equations, where the initial expansionphases were dominated by positive pressure terms, although the particle horizon guarantees that an objectcan be seen, the event horizon prevents to see photons beyond a time given by the instantaneous eventhorizon at that time. Summarizing, the particle horizon defines the events that could be observed at agiven cosmic time, while the event horizon defines the events that will be observed in the future. In this paper, we have first obtained analytical expressions for the particle and the event horizons consider-ing two state equations (cosmological constant and dust as accepted in the concordance cosmology). Theyhave allowed us to compute their first derivative, obtaining these extremely simple equations: dH p dt = H p ( z ) H ( z ) + 1 dH e dt = H e ( z ) H ( z ) − - 10 - here H p H and H e H are the recession velocities of the particle and event horizons respectively. Theseexpressions generalize the previous stated results [2] derived for just one state equation. Notice that allthese results and values are independent of the current values of the involved constants.The equations of the velocity of the horizons have a remarkable physical meaning. As the recession velocity H p H is the instantaneous velocity of an object located at the distance of the particle horizon H p , fromthe first equation we deduce that the instantaneous velocity of the surface of the horizon particle is fasterby the speed of light c (in our units c = 1 ), than the one of the objects over the particle horizon, thenmore and more objects are entering into the particle horizon and they will never get out. Analogously, thesecond equation stands that the recession speed of the event horizon is slower by c than that of the objectsover the event horizon, and then more and more objects are disappearing from the event horizon and theywill never get into again.Finally, computing the second derivatives of both horizons we have proved that under our assumptions,their velocities remain positive throughout the history of the universe. Acknowledgements
This work was partially supported by the Spanish Ministry of Economy and Competitiveness (MINECO)under the grant AYA2011-29517-C03-01.
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Special functions , Cambridge University Press (2000).[2] E. Harrison,
Hubble spheres and particle horizons , Astrophys. J. (1991) 60.[3] E.W. Kolb and M.S. Turner,
The early universe , West View Press (1990).[4] N.N. Lebedev and R.A. Silverman,
Special functions & their applications , Courier Dover Publications(1972).[5] B. Margalef–Bentabol, J. Margalef–Bentabol and J. Cepa, Evolution of the cosmological horizons withcountably infinitely many state equations , JCAP (2013) 015 [arXiv:1302.2186].[6] NASA Mission Data (2011) WMAP Cosmological Parameters. Model: lcdm+sz+lens+iso1.[7] W. Rindler, Visual horizons in world-models , Mon. Not. Roy. Astr. Soc. (1956) 662.[8] WMAP collaboration, D. Spergel et al.,
First year Wilkinson microwave anisotropy probe (WMAP)observations: Determination of cosmological parameters , Astrophys. J. Suppl . (2003) 175[arXiv:astro-ph/0302209]. - 11 - Graphical Behaviour of the Horizons
In this appendix we include the graphics of the horizons and their velocities obtained with numericalcalculus. As the hypergeometric functions are really well implemented in almost all mathematical programs,obtaining these graphics is extremely easy.
A.1 Particle horizon −1 −0.8 −0.6 −0.4 −0.2 000.511.52 x 10 Future Particle Horizon z - Redshift H p ( M p c ) Past Particle Horizon z - Redshift H p ( M p c ) Figure 2 – Particle Horizon. At the origin of the Universe (right hand side of the graphic on theright), the particle horizon tends to zero, then it increases rapidly with time tending to infinite at thefar future (left hand side of the other graphic on the left). The dot in both graphics represents theconnecting point between them and corresponds to the current time z = 0 . −1 −0.8 −0.6 −0.4 −0.2 0050100150200 Future Speed of the Particle Horizon z - Redshift d H p d t ( u n i t o f c ) Past Speed of the Particle Horizon z - Redshift d H p d t ( u n i t o f c ) Figure 3 – Velocity of the Particle Horizon. At the origin of the Universe it tends to the constantvalue c (cid:39) . · − Mpc / s and increases rapidly to infinite, just as the particle horizon itself. - 12 - .2 Event horizon −1 −0.8 −0.6 −0.4 −0.2 0480048504900495050005050 Future Event Horizon z - Redshift H e ( M p c ) Past Event Horizon z - Redshift H e ( M p c ) Figure 4 – Event Horizon. At the origin of the Universe, the event horizon tends to zero, then itincreases with time, tending at the far future to the constant value c (cid:0) H √ Ω Λ0 (cid:1) − (cid:39) .
08 Mpc with no velocity i.e. with zero slope. −1 −0.8 −0.6 −0.4 −0.2 000.050.10.150.2
Future Speed of the Event Horizon z - Redshift d H e d t ( u n i t o f c ) Past Speed of the Event Horizon z - Redshift d H e d t ( u n i t o f c ) Figure 5 – Velocity of the Event Horizon. At the origin of the Universe it tends to infinite anddecrease to zero at the far future with no acceleration (again with zero slope).– Velocity of the Event Horizon. At the origin of the Universe it tends to infinite anddecrease to zero at the far future with no acceleration (again with zero slope).