aa r X i v : . [ a s t r o - ph ] M a r The Spatially Closed Universe
Chan-Gyung Park ∗ Astrophysical Research Center for the Structure and Evolution of the Cosmos, Sejong University, Seoul, 143-747, Korea (Dated: October 29, 2018)The general world model for homogeneous and isotropic universe has been proposed. For thispurpose, we introduce a global and fiducial system of reference (world reference frame) constructedon a 5-dimensional space-time that is embedding the universe, and define the line element as theseparation between two neighboring events that are distinct in space and time, as viewed in the worldreference frame. The effect of cosmic expansion on the measurement of physical distance has beencorrectly included in the new metric, which differs from the Friedmann-Robertson-Walker metricwhere the spatial separation is measured for events on the hypersurface at a constant time while thetemporal separation is measured for events at different time epochs. The Einstein’s field equationswith the new metric imply that closed, flat, and open universes are filled with positive, zero, andnegative energy, respectively. The curvature of the universe is determined by the sign of meanenergy density. We have demonstrated that the flat universe is empty and stationary, equivalentto the Minkowski space-time, and that the universe with positive energy density is always spatiallyclosed and finite. In the closed universe, the proper time of a comoving observer does not elapseuniformly as judged in the world reference frame, in which both cosmic expansion and time-varyinglight speeds cannot exceed the limiting speed of the special relativity. We have also reconstructedcosmic evolution histories of the closed world models that are consistent with recent astronomicalobservations, and derived useful formulas such as energy-momentum relation of particles, redshift,total energy in the universe, cosmic distance and time scales, and so forth. It has also been shownthat the inflation with positive acceleration at the earliest epoch is improbable.
PACS numbers: 04.20.Cv, 98.80.-k, 98.80.Jk
I. INTRODUCTION
The main goal of modern cosmology is to build a cos-mological model that is consistent with astronomical ob-servations. To achieve this goal, tremendous efforts havebeen made both on theories and on observations sincethe general theory of relativity was developed. So far themost successful model of the universe is the Friedmann-Robertson-Walker (FRW) world model [1, 2, 3, 4]. TheFRW world model predicts reasonably well the currentobservations of the cosmic microwave background (CMB)radiation and the large-scale structures in the universe.The precisely determined cosmological parameters of theFRW world model imply that our universe is consistentwith the spatially flat world model dominated by dark en-ergy and cold dark matter (ΛCDM) with adiabatic initialcondition driven by inflation [5, 6].Although the flat FRW world model is currently themost reliable physical world model, one may have the fol-lowing fundamental questions on the nature of the FRWworld model. First, mathematically, if a space-time man-ifold is flat, then the Riemann curvature tensor shouldvanish, and vice versa. However, the Riemann curva-ture tensor of the flat FRW world model does not vanishunless the cosmic expansion speed and acceleration arezeros, which implies that the physical space-time of theflat FRW world is not geometrically flat but curved. Onlyits spatial section at a constant time is flat. ∗ Electronic address: [email protected]
Secondly, the cosmic evolution equations of the FRWworld model can be derived from an application of theNewton’s gravitation and the local energy conservationlaws to the dynamical motion of an expanding spherewith finite mass density [7, 8]. Besides, the New-ton’s gravitation theory has been widely used to mimicthe non-linear clustering of large-scale structures in theuniverse even on the horizon-sized N -body simulations[9, 10]. On large scales, the close connection between theFRW world model and the Newton’s gravitation law isusually attributed to the fact that the linear evolution oflarge-scale density perturbations satisfies the weak grav-itational field condition. Recently, Hwang and Noh [11]show that the relativistic fluid equations perturbed tosecond order in a flat FRW background world coincideexactly with the Newtonian results, and prove that theNewtonian numerical simulation is valid in all cosmolog-ical scales up to the second order. However, one mayhave a different point of view that the Newton’s gravi-tational action at a distance appears to be valid even onthe super-horizon scales in the FRW world just becausethe world model does not reflect the full nature of therelativistic theory of gravitation.Thirdly, according to the FRW world model, the uni-verse at sufficiently early epoch ( z & § . k = − . ± . k = − . ± . , − , − , − ) for the metric tensor g ik , and denote a 4-vector in space-time as p i ( i = 0 , , ,
3) and a 3-vectorin space as p α ( α = 1 , ,
3) or p . The Einstein’s fieldequations are R ik − g ik R = 8 πGT ik + Λ g ik , (1) where R ik = R aiak is the Ricci tensor, R = R ii theRicci scalar, T ik the energy-momentum tensor, G theNewton’s gravitational constant, and Λ the cosmologi-cal constant. The Riemann curvature tensor is givenby R aibk = ∂ b Γ aki − ∂ k Γ abi + Γ abn Γ nki − Γ akn Γ nbi , with theChristoffel symbol Γ aik = g ab ( ∂ i g kb + ∂ k g ib − ∂ b g ik ). Theenergy-momentum tensor for perfect fluid is T ik = ( ε b + P b ) u i u k − P b g ik , (2)where ε b and P b are background energy density and pres-sure of ordinary matter and radiation, and u i is the 4-velocity of a fundamental observer. We assume that thecosmological constant acts like a fluid with effective en-ergy density ε Λ = Λ / πG and pressure P Λ = − ε Λ . Thelimiting speed in the special theory of relativity is set tounity ( c ≡ II. HOW TO DEFINE METRIC FORHOMOGENEOUS AND ISOTROPICUNIVERSES?
The starting point for constructing a physical worldmodel is to define the space-time separation between twoneighboring events, i.e., the line element ds = g ik ( x ) dx i dx k , (3)where g ik ( x ) is the metric tensor which determines allthe geometric properties of space-time in a system of co-ordinates. In (3), the two events are generally distinct inspace and time, separated by dx i = ( dx , dx , dx , dx ).In the special theory of relativity, a separation betweentwo distinct events is given by ds = dt − d r , (4)which is invariant in all inertial reference frames. Themetric g ik = diag(1 , − , − , −
1) is called the Minkowskimetric.The early development in modern cosmology was fo-cused on finding the metric appropriate for the real uni-verse whose space-time structure is inconsistent with theMinkowski metric due to the expansion of the universeand the presence of matter in it. As a pioneer, Ein-stein [27] developed a model of the static closed uni-verse that is spatially homogeneous and isotropic, byadopting a metric with g = 1 and g α = 0 from thestatic condition. Friedmann [1] developed a more gen-eral world model that includes both stationary and non-stationary universes of positive spatial curvature, by as-suming that one can make g = 1 and g α = 0 by anappropriate choice of time coordinate. Weyl [28] pos-tulated that in a cosmological model the world lines ofparticles (e.g., galaxies) form a 3-bundle of nonintersect-ing geodesics orthogonal to a series of space-like hyper-surfaces, which implies that g depends on x = t onlyand g α = 0. Robertson [3] also argued that the lineelement may be expressed as ds = dt + g αβ dx α dx β for a universe where the matter has on the whole thetime-like geodesics x α = const. as world lines, and thecoordinate t can be interpreted as a mean time whichserves to define proper time and simultaneity for the en-tire universe. Walker [4, 29] demonstrated that the lineelement for the 3-dimensional Riemannian space of con-stant curvature, dσ = h αβ dx α dx β , is invariant underall transformations belonging to a group of motions G ,and proposed a metric for the non-stationary space-timemanifold as ds = dt − a ( t ) dσ , where t is a physicaltime of an observer and is invariant under a transforma-tion from one observer to another.The resulting line element for spatially homogeneousand isotropic universe (FRW metric) is concisely writtenas ds = dt − a ( t ) dσ = dt − a ( t ) (cid:2) dχ + S k ( χ )( dθ + sin θdφ ) (cid:3) , (5)where a ( t ) is a cosmic expansion scale factor, dσ is thecomoving-space separation between events, and S k ( χ ) =sinh χ , χ , and sin χ for open ( k = − k = 0), andclosed ( k = +1) spaces, respectively (see Ref. [30] for adetail derivation of the FRW metric). The correspondingcosmic evolution equations known as Friedmann equa-tions are ¨ aa = − πG ε b + 3 P b ) + Λ3 , (6)and (cid:18) ˙ aa (cid:19) = 8 πG ε b − ka + Λ3 . (7)The dot denotes a differentiation with respect to time.We note that previous studies all assumed that g = 1and g α = 0 in the form of line element, where spatialand temporal distances were considered to be separate.In particular, a serious inconsistency is seen in the FRWmetric. The real-space separation a ( t ) dσ is measured forevents on the hypersurface at a constant time (a slice ofsimultaneity). On the other hand, the temporal separa-tion dt is measured for events at different time epochs. Inother words, spatial and temporal distances are relatedto two different couples of events, in contradiction to thegeneral definition of the line element (3) in which only acouple of events should be involved.Besides, in the FRW metric the geometry of physicalspace separates into the geometry of comoving space andthe expansion history a ( t ). This is the reason why the flatFRW world model is geometrically flat only in the comov-ing space, not in the physical space and time. Therefore,the effect of cosmic expansion on the space-time geome-try has not been correctly reflected in the FRW metric,giving an inaccurate measure of distance between events.To derive the general form of line element for homoge-neous and isotropic universe, let us introduce a 4 + 1Minkowski space-time composed of 4-dimensional Eu-clidean space and 1-dimensional time, and assume that our universe is spatially a 3-dimensional hypersurfacewith uniform curvature embedded in the 4-dimensionalspace. In fact, the hypersurface with negative curva-ture (open space) cannot be embedded in the Euclideanspace. Here we restrict our attention to flat and closedspaces, deferring the discussion about the open space toSec. III C.Throughout this paper, we call a reference frame con-structed on the high dimensional Minkowski space-timeas world reference frame . It is a global system of refer-ence provided with a rigid measuring rod and a numberof clocks to indicate position and time ( world time ) of anevent. Distances between events on the universe will bemeasured based on this fiducial system.Although the world reference frame has been intro-duced for mathematical convenience, it is useful in thatthe space-time coordinates of the frame are independentof the dynamics of the universe. As will be shown later,the proper time as measured by an observer in the uni-verse does not elapse uniformly, being affected by thecosmic expansion. Thus, it is natural to use the worldtime coordinate with uniform lapse for a fair compar-ison of physical phenomena in the expanding universe.The space coordinates of the world reference frame canbe conveniently used to describe the geometry of a 3-dimensional hypersurface embedded in the 4-dimensionalEuclidean space (see Sec. III B).An example of the expanding 1-dimensional flat hy-persurface (with uniform and zero curvature) is shown inFig. 1(a). At initial time t , a hypersurface is given asa straight line, on which there are equally spaced events(open circles) with mutual comoving separation δx , allat rest with respect to the comoving coordinate systemwhose spatial coordinate x is related to the real-spacecoordinate by r = a ( t ) x . After an infinitesimal time δt ,the straight line has been expanded, and the proper sep-aration between neighboring events on the hypersurfacehas increased from a ( t ) δx to a ( t + δt ) δx .We define the line element δs as the space-time sep-aration between two distinct events located at ( t, r ) and( t + δt, r + δr ), which correspond to events 1 and 2 ′ in Fig.1(a) without loss of generality. The spatial separation be-tween events 1 and 2 ′ as measured in the world referenceframe is δr ′ = a ( t + δt )[ x + δx ] − a ( t ) x ≃ ˙ a ( t ) xδt + a ( t ) δx up to the first order of δt and δx . Therefore, we get δs ′ = δt − δr ′ = (1 − ˙ a x ) δt − aaxδtδx − a δx , (8)where the effect of cosmic expansion on the physicalseparation between events has been included explicitly.Note that the line element (8) implies that generally g is a function of both time- and space-coordinates and g α = 0, which violates the Weyl’s postulate unless ˙ a = 0.The line element for non-flat space can be defined anal-ogously. As an example of the closed space, Fig. 1(b)shows an expanding 1-dimensional circle (1-sphere) withradius a ( t ) at two distinct (infinitesimally separated)world times. The expanding circle is a 1-dimensionalhypersurface with uniform and positive curvature em- FIG. 1: Schematic diagrams showing the expansion of (a) flatand (b) closed 1-dimensional homogeneous and isotropic hy-persurfaces. Events denoted as open circles are equally spacedout on two hypersurfaces that are infinitesimally separated by δt . Note that the time axis, which is orthogonal to r - and w -axes, is omitted in (b). The line element is defined as thespace-time separation between two distinct events 1 and 2 ′ (see text). bedded in the 2-dimensional Euclidean space, rw -plane.Events on the circle are equally spaced out with δχ , where χ is a comoving coordinate related to r -coordinate by r = a sin χ . The line element is defined as the space-time separation between distinct events 1 and 2 ′ , writtenconcisely as δs ′ = δt − δl ′ = (1 − ˙ a ) δt − a δχ , (9)where δl ′ = δr ′ + δw ′ is the spatial separation be-tween events 1 and 2 ′ as measured in the world referenceframe. The spatial distances projected on r - and w -axesare given by δr ′ = a ( t + δt ) sin( χ + δχ ) − a ( t ) sin χ and δw ′ = a ( t + δt ) cos( χ + δχ ) − a ( t ) cos χ , respectively.In the second equality of (9), δr ′ and δw ′ have been expanded up to the first order of δt and δχ . From twocases, it is clear that the expansion of space affects bothspace and time intervals in the line element, which is themain difference from the FRW metric. Generally, the lineelement (metric) should reflect the fact that the cosmicexpansion is a dynamical phenomenon.Eq. (8) implies that the FRW line element for the1-dimensional flat space is valid only at a local regionaround an observer at x = 0. The FRW line element forthe closed space also has a similar form to the flat case,i.e., δs = δτ − a δχ , where, to be consistent with (9), δτ should be interpreted as a proper time interval mea-sured by a local observer ( δτ = δt √ − ˙ a ). To such anobserver who may be located between events 1 and 2 (orbetween events 1 ′ and 2 ′ after δt ), the spatial separationbetween neighboring events appears to be aδχ [e.g., thearc length between events 1 and 2 in Fig. 1(b)]. There-fore, both FRW line elements describe the space-timeseparation between events near an observer, which is thelocal nature of the FRW metric. III. METRIC AND EVOLUTION EQUATIONSOF EXPANDING UNIVERSES
In this section, we define the general forms of metricfor homogeneous and isotropic universes of various spa-tial curvature types, and derive the corresponding cosmicevolution equations from the Einstein’s field equations.
A. Flat universe
Suppose that the universe is spatially an expanding3-dimensional flat hypersurface (with uniform and zerocurvature) embedded in a 4-dimensional Euclidean spacewith the Cartesian coordinates ( r , r , r , r ). The em-bedded flat space is infinite, homogeneous, and isotropic,and is Euclidean at an instant of time. To simplify theproblem, let us assume that the hypersurface is orthogo-nal to the r -axis so that the fourth Cartesian coordinatecan be ignored. Then, each event on the flat hypersur-face is labelled by the world time t and the real-spaceposition r defined as r = a ( t ) x , (10)where x is the comoving-space position vectorwith the Cartesian coordinates ( x , x , x ) =( x sin θ cos φ, x sin θ sin φ, x cos θ ) in relation to thecomoving spherical coordinate x α = ( x, θ, φ ) with x = | x | = ( x + x + x ) / .With the help of the differential of (10) d r = ˙ a ( t ) dt x + a ( t ) d x , (11)the line element is defined as the space-time separationbetween events located at ( t, r ) and ( t + dt, r + d r ): ds = dt − d r = (cid:0) − ˙ a x (cid:1) dt − aadt x · d x − a d x = (cid:0) − ˙ a x (cid:1) dt − aaxdtdx − a (cid:2) dx + x ( dθ + sin θdφ ) (cid:3) . (12)The metric tensor in the coordinate system x i =( t, x, θ, φ ) is g ik = − ˙ a x − ˙ aax − ˙ aax − a − a x
00 0 0 − a x sin θ . (13)We calculate the Christoffel symbols and the Riemanncurvature tensor from the metric tensor. The non-zeroChristoffel symbols Γ aik (= Γ aki ) areΓ = ¨ aa x, Γ = Γ = Γ = ˙ aa , Γ = − x, Γ = − x sin θ, Γ = Γ = 1 x , Γ = − sin θ cos θ, Γ = cot θ. (14)One easily verifies that the Riemann curvature tensorvanishes: R aibk = 0 . (15)Thus the Ricci tensor R ik and the Ricci scalar R alsovanish. This demonstrates explicitly that the space-timecurvature of the expanding flat universe is zero.The proper time interval as measured by an observerin arbitrary motion is obtained from (12) as dτ = dt (cid:20) − ˙ a x − aax (cid:18) dxdt (cid:19) − a v (cid:21) / , (16)where av = ( − v α v α ) / is the magnitude of proper 3-velocity, v α = dx α /dt is the 3-velocity in the comovingcoordinate system, and v α = g αβ v β . For an observer whois at rest ( v α = 0) in the comoving coordinate system(hereafter a comoving observer), we get the proper timeinterval dτ = dt (1 − ˙ a x ) / (17)and the 4-velocity u i = dx i dτ = (cid:18) √ − ˙ a x , , , (cid:19) (18)of the observer. Inserting (13) and (18) into (2) gives theenergy-momentum tensor, whose non-zero componentsare T = ε b (1 − ˙ a x ) , T = T = − ε b ˙ aax,T = ε b ˙ a a x + P b a − ˙ a x , T = P b a x ,T = P b a x sin θ. (19) From (13), (15) and (19), the Einstein’s field equations(1) reduce to P b = − ε b = Λ8 πG . (20)The ε b and P b should not be negative because they areenergy density and pressure of ordinary matter and ra-diation, suggesting that ε b = P b = 0 and Λ = 0 . (21)Even if non-zero energies of matter and radiation with anequation of state (20) can exist, the total energy densityand pressure should vanish: ε b + ε Λ = 0 and P b + P Λ = 0 . (22)Therefore, in the domain of classical physics the flat uni-verse is empty, which is consistent with the Einstein’sclaim that the infinite universe has vanishing mean den-sity [24].Eq. (20) does not give any information about the cos-mic expansion history. Actually, the spatial homogeneityand isotropy condition constrains the flat universe to bestationary. In (16), the proper time interval of an ar-bitrary observer depends on the choice of the comovingcoordinate system. The proper time of an observer mov-ing faster at farther distance from the origin goes slower.Only at x = 0 or if ˙ a = 0, it becomes dτ = dt √ − a v ,the same form of the proper time as in the special relativ-ity, irrespective of the choice of the comoving coordinatesystem. Since the dependence of the proper time inter-val on the choice of reference frame is contradictory tothe spatial homogeneity and isotropy condition, the flatuniverse should be stationary ( ˙ a = 0). In conclusion,the flat universe is empty and stationary, and thereforeis equivalent to the Minkowski space-time. B. Closed universe
The homogeneous and isotropic closed space is usu-ally described by the spatially finite hypersphere withuniform and positive curvature. By extending the ex-ample in Fig. 1(b), let us consider our universe as anexpanding 3-sphere of curvature radius a ( t ), embeddedin a 4-dimensional Euclidean space where each point islabelled by the Cartesian coordinates ( x, y, z, w ) and theworld time t . The equation of the 3-sphere in x - y - z - w coordinate system is x + y + z + w = r + w = a , (23)where r is the radial distance in x - y - z coordinate system.The coordinates x , y , z have transformation relationswith the spherical coordinates r , θ , φ as x = r sin θ cos φ , y = r sin θ sin φ , and z = r cos θ .The line element is defined as the space-time distancebetween two infinitesimally separated events located at( t, x, y, z, w ) and ( t + dt, x + dx, y + dy, z + dz, w + dw ): ds = dt − dx − dy − dz − dw = dt − dl , (24)where dl is a spatial separation as measured in the worldreference frame. For two distinct events on the expanding3-sphere, the spatial separation is written as [42] dl = dx + dy + dz + dw = dr + r ( dθ + sin θdφ ) + ( ada − rdr ) a − r = da + a (cid:2) dχ + sin χ ( dθ + sin θdφ ) (cid:3) , (25)where w has been removed by (23) and its differential rdr + wdw = ada = a ˙ adt, (26)and r has been replaced with the comoving coordinate χ by a parametrization r = a sin χ (0 ≤ χ ≤ π ) , (27)and its differential dr = ˙ a sin χdt + a cos χdχ. (28)Therefore, the general form of line element for the non-stationary closed universe is ds = (cid:2) − ˙ a ( t ) (cid:3) dt − a ( t ) (cid:2) dχ + sin χ ( dθ + sin θdφ ) (cid:3) . (29)The metric tensor in the coordinate system x i =( t, χ, θ, φ ) is g ik = diag (cid:2) − ˙ a , − a , − a sin χ, − a sin χ sin θ (cid:3) . (30)We calculate the Christoffel symbols and the Ricci ten-sor from the metric tensor. The non-zero Christoffel sym-bols areΓ = − ˙ a ¨ a − ˙ a , Γ = a ˙ a − ˙ a , Γ = a ˙ a − ˙ a sin χ, Γ = a ˙ a − ˙ a sin χ sin θ, Γ = Γ = Γ = ˙ aa , Γ = − sin χ cos χ, Γ = − sin χ cos χ sin θ, Γ = Γ = cot χ, Γ = − sin θ cos θ, Γ = cot θ. (31)The non-zero components of the Ricci tensor are R = − (cid:18) ¨ aa (cid:19) − ˙ a ,R = a ¨ a (1 − ˙ a ) + 21 − ˙ a ,R = R sin χ, R = R sin χ sin θ, (32)and the Ricci scalar is R = − (cid:18) ¨ aa (cid:19) − ˙ a ) − a (1 − ˙ a ) . (33) From (29), we obtain a proper time interval as mea-sured by an observer in arbitrary motion as dτ = dt (cid:0) − ˙ a − a v (cid:1) / , (34)and thus express the 4-velocity of the observer as u i = dx i dτ = ( γ, γ v ) , (35)where γ = (1 − ˙ a − a v ) − / (36)is a contraction factor. Note that the contraction fac-tor depends on the expansion speed of the universe aswell as the peculiar motion of the observer. For a co-moving observer, the energy-momentum tensor for per-fect fluid is obtained by inserting the 4-velocity u i =(1 / √ − ˙ a , , ,
0) into (2): T ik = diag[ ε b (1 − ˙ a ) , P b a , P b a sin χ, P b a sin χ sin θ ] , (37)where ε b and P b are energy density and pressure as de-fined in the world reference frame. Inserting (30), (32),(33) and (37) into (1), we get evolution equations forhomogeneous and isotropic closed universe. They areconcisely written as1(1 − ˙ a ) ¨ aa = − πG ε b + 3 P b ) + Λ3 (38)and 1 a (1 − ˙ a ) = 8 πG ε b + Λ3 . (39)Combining (38) and (39) gives a continuity equationfor energy density and pressure − (cid:18) ˙ aa (cid:19) ( ε + P ) = ˙ ε, (40)where ε and P are total energy density and pressure ofradiation (R), matter (M), and the cosmological constant(Λ): ε = P I ε I and P = P I P I ( I = R , M , Λ). Note that ε b = ε R + ε M . The continuity equation is equivalent to T i i = 0, with a semicolon denoting a covariant deriva-tive. It is worth noting that the time-time component ofthe metric tensor ( g = 1 − ˙ a ) is positive according tothe sign convention adopted. The positiveness of the left-hand side of (39) suggests that the total energy densityshould be positive in the closed universe ( ε > P I = w I ε I foreach species I , we obtain a solution to (40) as ε I ∝ a − w I ) . Thus the total energy density is written as ε = P I ε I ( a/a ) − w I ) . The subscript 0 denotes thepresent epoch t . Hereafter we call universes dominatedby radiation, matter, and the cosmological constant asradiation-universe (R-u), matter-universe (M-u), and Λ-universe (Λ-u), respectively. The energy density evolvesas ε R ∝ a − in the radiation-universe ( w R = ), ε M ∝ a − in the matter-universe ( w M = 0), and ε Λ = const.in the Λ-universe ( w Λ = − ε Λ = ε Λ0 .Let us define a dimensionless function of redshift z ≡ a /a ( t ) − A ( z ) ≡ a a (1 − ˙ a )= 8 πGa (cid:2) ε R0 (1 + z ) + ε M0 (1 + z ) + ε Λ (cid:3) = a (cid:20) (1 + z ) a + (1 + z ) a + 1 a (cid:21) , (41)where a I = (3 / πGε I ) / . Throughout this paper, afunction of time t will be expressed in terms of redshift z interchangeably. The radius parameter a R0 ( a M0 ) canbe interpreted as a free-fall time or radius for the gravi-tational collapse of a stationary radiation- (matter-) uni-verse with the present radiation (matter) energy density,and a Λ = a Λ0 as the minimum radius of Λ-universe. Byintroducing another dimensionless quantity D I ≡ πGε I a (1 − ˙ a ) = (cid:18) a a I (cid:19) (1 + z ) w I ) A ( z ) , (42)we can rewrite (39) as D R + D M + D Λ = 1 , (43)which holds during the whole history of the universe. The D I can be interpreted as the fraction of energy of species I (see Sec. IV E). C. Open universe
We now consider the geometry of homogeneous andisotropic expanding 3-dimensional space with uniformand negative curvature. Such a negatively curved spacecannot be embedded in a 4-dimensional Euclidean space.At an instant of time, it is a pseudosphere with imagi-nary radius ia ( §
111 of Ref. [31]). We replace a with − a in (23) to obtain an expression analogous to (25) fora spatial separation between two distinct events on theexpanding 3-pseudosphere [43], dl = dx + dy + dz + dw = dr + r ( dθ + sin θdφ ) − ( ada + rdr ) a + r = − da + a (cid:2) dχ + sinh χ ( dθ + sin θdφ ) (cid:3) , (44)where the radial distance r = ( x + y + z ) / in x - y - z coordinate system has been parametrized with thecomoving coordinate χ by r = a sinh χ ( χ ≥ . (45) Therefore, the line element for the non-stationary openuniverse is ds = (cid:2) a ( t ) (cid:3) dt − a ( t ) (cid:2) dχ + sinh χ ( dθ + sin θdφ ) (cid:3) . (46)The non-zero components of the Ricci tensor calcu-lated from the metric (46) are R = − (cid:18) ¨ aa (cid:19)
11 + ˙ a ,R = a ¨ a (1 + ˙ a ) −
21 + ˙ a ,R = R sinh χ, R = R sinh χ sin θ, (47)and the Ricci scalar is R = − (cid:18) ¨ aa (cid:19) a ) + 6 1 a (1 + ˙ a ) . (48)For a comoving observer with 4-velocity u i =(1 / √ a , , , T ik = diag[ ε b (1 + ˙ a ) , P b a ,P b a sinh χ, P b a sinh χ sin θ ] . (49)The resulting evolution equations for the open universeare obtained in the same way as those for the closed uni-verse are obtained. They are1(1 + ˙ a ) ¨ aa = − πG ε b + 3 P b ) + Λ3 (50)and 1 a (1 + ˙ a ) = − πG ε b − Λ3 . (51)Combining (50) and (51) also gives the same continuityequation as (40). Since the time-time component of themetric tensor ( g = 1 + ˙ a ) is always positive, equa-tion (51) suggests that the total energy density shouldbe negative in the open universe ( ε < D. Our universe is spatially closed
In Secs. III A–III C, we have demonstrated that flatuniverse is equivalent to the Minkowski space-time, whichis empty and stationary, and that closed and open uni-verses have positive and negative energy densities, re-spectively. In other words, the curvature of the universeis determined by the sign of mean energy density, not bythe ratio of the energy density to the critical density as inthe FRW world. The open universe is unrealistic becausethe mean density of the universe is known to be positivefrom astronomical observations. Therefore, we concludethat our universe is spatially closed and finite. The spa-tial closure of the universe has been deduced from thepurely theoretical point of view. Our conclusion verifiesthe Einstein’s claim for the finiteness of the universe.
TABLE I: Possible ranges of total energy density in the FRWand new world models.Curvature Type FRW World a New Worldclosed ε ≥ / πGa ε > ε ≥ ε = 0open − / πGa ≤ ε < ε < a Equality signs for the FRW world models correspond to cases ofstationary universes ( ˙ a = 0, ¨ a = 0). E. The Friedmann equations
In the non-flat universes, the proper time interval of acomoving observer is related to the world time by dτ = (cid:2) − k ˙ a ( t ) (cid:3) / dt, (52)where k = +1 for closed and − dτ = dt ).From (52), we obtain relations between world- andproper-time derivatives of the curvature radius:1 a (cid:18) dadτ (cid:19) + ka = ka (1 − k ˙ a ) (53)and 1 a (cid:18) d adτ (cid:19) = 1(1 − k ˙ a ) ¨ aa , (54)where the proper time τ acts as the time of non-flat FRWworld models. As will be shown in Sec. IV E [Eq. (92)],energy density and pressure defined in the world refer-ence frame are equivalent to those measured by the co-moving observer. Therefore, Friedmann equations (6)and (7) for non-flat universes can be derived from thenew cosmic evolution equations [Eqs. (38) and (39) forclosed and (50) and (51) for open universes] by the time-parametrization (52). This, along with the local natureof the FRW metric as discussed in Sec. II, implies thatthe Friedmann equations describe the evolution of thelocal universe around a comoving observer.Table I lists possible ranges of total energy density inthe FRW and new world models. In the new world mod-els, the energy density is strictly positive, zero, and nega-tive for closed, flat, and open universes, respectively. Onthe other hand, the FRW world models have rather com-plicated ranges of energy density. From (53), one findsthat ( da/dτ ) ≥ ≤ ( da/dτ ) < da/dτ ) suggests that the closed FRW world model havepositive energy density larger than or equal to 3 / πGa ,and that the open model accommodate only negative en-ergy density not smaller than − / πGa . Note that thepositive energy density appears to be allowable in the open FRW world model if the constraint on ( da/dτ ) isnot imposed.The Friedmann equations for the flat universe ( k = 0)cannot be derived with any world-proper time relation,but can be obtained by neglecting the curvature term − k/a in (7) for the closed model, resulting in ε ≥ t with uniform lapse, while the latter has a local time τ whose lapse depends on the cosmic expansion speed. Anevent on the expanding 3-sphere may be labelled by co-ordinates ( t, x, y, z, w ) or ( t, a, χ, θ, φ ). By omitting thecurvature radius a that is a function of t , we regard thecoordinate system x i = ( t, χ, θ, φ ) as equivalent to theworld reference frame. The comoving observer’s frameis a system of physical space and proper time coordi-nates adopted by an observer like us whose comovingcoordinate x α = ( χ, θ, φ ) is fixed during the expansion ofthe universe. Actually, the comoving observer’s frame isequivalent to the locally inertial frame. IV. PHYSICAL AND ASTRONOMICALASPECTS OF THE EXPANDING CLOSEDUNIVERSE
In this section, we investigate interesting properties ofthe expanding closed universe, such as time-varying lightspeed, cosmic expansion history, energy-momentum rela-tion of particles, redshift, and cosmic distance and timescales. All the quantities, not otherwise specified, aredefined and expressed in the world reference frame, andsecondarily in the comoving observer’s frame. In the lat-ter frame, all the physical quantities and their evolutionare the same as those in the closed FRW world.
A. Time-varying light speed and cosmic expansionspeed
In the special theory of relativity, the speed of light isconstant and equal to the limiting speed ( c = 1), whichapplies to the Minkowski space-time, or equivalently tothe flat universe. In the expanding closed universe, how-ever, the light speed is less than or equal to the limitingspeed. From the photon’s geodesic equation ( ds = 0),one can express the speed of light as η ( t ) ≡ (1 − ˙ a ) / = (1 + z ) A ( z ) . (55)It should be noted that the light speed varies with time,depending on the cosmic expansion speed ˙ a and satisfying η + ˙ a = 1. Both speeds cannot exceed the limiting speed(0 ≤ η ≤ − ≤ ˙ a ≤ A ( z ) evolves as (1 + z ) in the radiation-dominated era[Eq. (41)]. Therefore, we expect ˙ a = 1 and η = 0 at thebeginning of the universe.On the other hand, the comoving observer always mea-sures the speed of light as unity because the observer’sproper time interval varies in the same way as the world-frame light speed does [Eqs. (52) and (55)]. Besides, thecosmic expansion speed in the comoving observer’s framehas no limit ( da/dτ = ˙ a/ √ − ˙ a < ∞ ).In the open universe, the speed of light is η = (1 +˙ a ) / ≥
1: the light propagates faster than the limitingspeed. There is no upper limit on the cosmic expansionand the light speeds in the world reference frame. How-ever, the comoving observer perceives that the speed oflight is always unity and that the cosmic expansion speedis bounded to unity ( da/dτ = ˙ a/ √ a < B. Cosmic evolution history
The evolution of homogeneous and isotropic universeis described by the evolution of physical quantities duringthe history of the universe. The most important quantityis the curvature radius a . Although it is not easy to getthe general solution to (39), there exist analytic solutionsfor special cases of the universe dominated by energy ofthe single species. For radiation-universe ( ε M = 0, ε Λ =0), the curvature radius is given by a ( t ) = b R sin( t/b R ) (0 ≤ t ≤ πb R ) , (56)where b R = (cid:0) πGε R0 a / (cid:1) / = ( a /a R0 ) a is the max-imum curvature radius of the radiation-universe. At t = 0, the universe expands with the maximum speedand zero acceleration ( ˙ a = 1, ¨ a = 0). The positive accel-eration is not allowable in the radiation-universe.For matter-universe ( ε R = 0, ε Λ = 0), the solution forthe curvature radius is a ( t ) = b M − b M ( t − b M ) (0 ≤ t ≤ b M ) , (57)where b M = 8 πGε M0 a / a /a M0 ) a is the maximumcurvature radius of the matter-universe. The initial con-dition a (0) = 0 has been assumed. The cosmic expansionacceleration is negatively constant in the matter-universe(¨ a = − / b M ). Due to the negative acceleration, the ex-panding universe containing only radiation and matter isbound to contract into the single point. If the universe does not contain the ordinary matterand radiation but is dominated by the cosmological con-stant or dark energy (Λ-universe), Eq. (39) becomes dadt = ± r − a = ± r − (cid:16) a Λ a (cid:17) , (58)where a Λ ≡ (3 / Λ) / = (3 / πGε Λ ) / is the (minimum)radius of the Λ-universe at initial time t i . For the ex-panding Λ-universe, we get a ( t ) = (cid:2) a + ( t − t i ) (cid:3) / ( t ≥ t i ) . (59)The expansion speed and acceleration of the Λ-universeare ˙ a ( t ) = ( t − t i ) / [ a + ( t − t i ) ] / and ¨ a ( t ) = a / [ a +( t − t i ) ] / , which go over asymptotically into unity andzero, respectively, as t goes to infinity. Starting with a ( t i ) = a Λ , ˙ a ( t i ) = 0, and ¨ a ( t i ) = a − , the Λ-universeexpands eternally.It is interesting to consider a universe dominated byenergy of a hypothetical species with an equation of state w H = − [33]. This universe (H-universe) has a simpleexpansion history a ( t ) = t [1 − ( a H0 /a ) ] / ( t ≥ , (60)where a H0 = (3 / πGε H0 ) / . The important property ofthe H-universe is that a ∝ t and ¨ a = 0. The universeexpands with the constant speed. The energy densityof the hypothetical species evolves as ε H ∝ a − . Foran extreme case of a H0 = 0 ( ε H0 = ∞ ), the H-universeexpands with the limiting speed ( a = t ).To reconstruct the evolution history of the closed uni-verse, we have adopted two world models that are con-sistent with the recent astronomical observations. Themodel parameters, which are listed in Table II, are basedon a non-flat ΛCDM FRW world model that best fitswith the WMAP CMB data only (Model I; H = 55 kms − Mpc − , Ω M = 0 . Λ = 0 . § . H = 71 km s − Mpc − ,Ω M = 0 . Λ = 0 . H = 100 h kms − Mpc − is the Hubble constant, and Ω I is the cur-rent density parameter of the FRW world model. TheHubble constant of Model I is quite lower than the pop-ular value of Model II, but is allowable because the lowHubble constant has been reported from observations ofCepheids plus SNIa ( H = 62 . ± . ± . − Mpc − ;[20]) and of eclipsing binaries ( H = 61 km s − Mpc − ;[35]).Using the FRW model parameters as input, we calcu-late radius parameters a R0 , a M0 , and a Λ from a formula a I = (3 / πGε I ) / = H − Ω − / I . The radiation energydensity has been calculated from ε R0 = π k T / ~ (see Sec. IV F) with the CMB temperature T cmb = 2 . a has been obtained from the relation H = a − ( da/dτ ) = (cid:0) a − + a − + a − − a − (cid:1) / . (61)0 TABLE II: Cosmological parameters of the two closed world models.Parameters Symbols Model I a Model II b InputCMB temperature T cmb .
725 K 2 .
725 KMatter density Ω M .
415 0 . Λ .
630 0 . h .
55 0 . a η /c .
208 0 . a /c .
978 0 . a . × − Mpc − . × − Mpc − Cosmic age (in world time) t . τ . . a R0 . . a M0 a Λ b R b M . V . × Mpc . × Mpc Total radiation energy E R0 . × erg 1 . × ergTotal matter energy E M0 . × erg 1 . × ergTotal dark energy E Λ0 . × erg 2 . × ergFraction of radiation energy D R0 . × − . × − Fraction of matter energy D M0 .
397 0 . D Λ0 .
603 0 . a Based on parameters of a non-flat FRW world model that best fits with the WMAP 3-year data only [5]. b Based on parameters of a non-flat FRW world model that jointly fits with the CMB, SNIa, γ -ray bursts, theshape parameter Γ = Ω M h , Hubble constant, matter density (Ω M h ), and big-bang nucleosynthesis data [34].The Hubble constant is an average of recent measurement values. The basic parameters characterizing the closed worldmodel are the curvature radius of the universe ( a ) andthe radius parameters ( a R0 , a M0 , a Λ ) at the present time.It is interesting to note that converting parameters ofthe flat FRW world model into those of the new closedmodel always gives limiting values of a = ∞ , ˙ a = 1,and ¨ a = 0. The flat FRW world is a limiting case of theclosed universe with infinite curvature radius.The evolution histories of the two closed world mod-els are summarized in Figs. 2–5 below, where we havealso plotted analytic solutions for radiation, matter, andΛ-universes ( t i = 0 is assumed for Λ-u). All the numer-ical values given in the text are based on Model I. Theevolution of curvature radius of the closed world modelis shown in Fig. 2, where we have performed a numericalintegration of (39) to obtain a ( t ). Note that the solu-tion of H-universe with infinite energy density greatlyapproximates the evolution of curvature radius of ouruniverse, which differs from a = t by maximally about2% at t ≃
100 Gyr, as shown in the small panel.Fig. 3 shows the relation between the proper timeof a comoving observer and the world time, obtained byintegrating (52). The total elapsed time until the presenttime, the age of the universe, is denoted as a star at t = 85 . τ = 15 . τ - t relations expected in radiation, matter, and Λ-universesare written in analytic forms as τ = b R [1 − cos( t/b R )] (R-u) , (62) τ = b M " (cid:18) t − b M b M (cid:19) s − (cid:18) t − b M b M (cid:19) + π (cid:18) t − b M b M (cid:19) (M-u) , (63)and τ = a Λ ln t − t i a Λ + s (cid:18) t − t i a Λ (cid:19) (Λ-u) , (64)respectively. Inserting (62) into (56) gives a ( τ ) =[ τ (2 b R − τ )] / , which goes over into a ∝ τ / if τ ≪ b R ,the behavior of a scale factor in the radiation-dominatedFRW universe.Fig. 4 shows the time-variation of cosmic expansionspeed (top) and speed of light (bottom). At the present1 FIG. 2: Evolution of curvature radius a over the world time t (thick solid curve; Model I). For comparison, the curvatureradius of radiation, matter, and Λ-universes are shown as dot-ted (R), dashed (M), and long-dashed (Λ) curves, respectively.The small panel shows the fractional difference of the curva-ture radius of our universe relative to a = t (∆ a/a ≡ [ a − t ] /t in unit of percent; thick and thin solid curves for Model I andII, respectively). The stars denote quantities at the presenttime t = 85 . z ≈ t /t − time, the universe is expanding faster than the light bya factor of 4 . a = 0 .
978 and η = 0 .
208 (denotedas stars). The speed of light in radiation, matter, andΛ-universes are written as η ( t ) = sin( t/b R ) = a/b R (R-u) , (65) η ( t ) = s − (cid:18) t − b M b M (cid:19) = ( a/b M ) / (M-u) , (66)and η ( t ) = a Λ [ a + ( t − t i ) ] / = a Λ /a (Λ-u) , (67)respectively. The behavior of time-varying light speedimplies that photons are frozen ( η = 0) when the universeexpands with the maximum speed, e.g., at the beginningor far in the future of the universe.Fig. 5 shows the history of cosmic expansion accelera-tion calculated from¨ a = − ε + 3 P )16 πGε a , (68)which has been obtained by combining (38) and (39).In the radiation-dominated era, the expansion acceler-ation had gradually decreased from zero, and became FIG. 3: The proper time of a comoving observer τ versus theworld time t (thick and thin solid curves for Model I and II,respectively). The τ - t relations expected in radiation, matter,and Λ-universes are shown as dotted (R), dashed (M), andlong-dashed (Λ) curves, respectively. The star indicates t =85 . τ = 15 . a and(bottom) speed of light η over the world time t (in unit ofthe limiting speed of the special relativity; thick and thinsolid curves for Model I and II, respectively). The relation η + ˙ a = 1 holds. The speed of light in radiation, matter,and Λ-universes are shown as dotted (R), dashed (M), andlong-dashed (Λ) curves. The stars denote quantities at thepresent time. FIG. 5: Evolution of cosmic expansion acceleration ¨ a in unitof Mpc − (thick and thin solid curves for Model I and II,respectively). The acceleration in radiation, matter, and Λ-universes are shown as dotted (R), dashed (M), and long-dashed (Λ) curves, respectively. The star denotes the accel-eration at the present time. negatively constant during the matter-dominated era( t = 0 . t = 58 . τ = 10 . z = 0 . t = 93 . τ = 17 . z = − . C. Energy-momentum relation of particles
Now we define energy and momentum of a free particlewith rest mass m in the closed universe. Here, we meanthe rest mass by the intrinsic mass of the particle thatis independent of its peculiar motion and the dynamicsof the universe. Thus m is the mass as measured in alocally inertial frame comoving with the particle. It isalso equivalent to the rest mass in the stationary universe.The action for the free material particle moving along atrajectory with end points A and B has the form ( § S = − m Z BA ds = Z t B t A Ldt, (69) where the Lagrangian L = − m (cid:0) − ˙ a − a v (cid:1) / (70)goes over into − m + ma v / av ≪ a = 0. The motion of the particle is determined from theprinciple of least action, δS = − mδ R ds = 0 (e.g., §
87 ofRef. [31]), resulting in the geodesic equation d x i ds + Γ ikl dx k ds dx l ds = 0 . (71)From the Lagrangian, we calculate energy and momen-tum of the material particle. The 3-momentum of theparticle is obtained from p α = ∂L/∂v α , with individ-ual components p = mγv a , p = mγv a sin χ , and p = mγv a sin χ sin θ . The energy of the particle isgiven by E p = p α v α − L = m (1 − ˙ a )(1 − ˙ a − a v ) / , (72)and the relativistic mass by m r = E p /η = mγ. (73)Both energy and mass of a particle are tightly relatedto the expansion speed of the universe. For a comovingparticle with a fixed comoving coordinate ( v α = 0), theenergy and the relativistic mass become E p = m (1 − ˙ a ) / and m r = m (1 − ˙ a ) − / , respectively.Let us define the 4-momentum vector of a particle as p i = ( mγ, mγv n ) = (cid:18) E p − ˙ a , pa n (cid:19) , (74)where p is the magnitude of the proper momentum de-fined as p = ( − p α p α ) / = mγav and n is a unitvector indicating the direction of motion of the parti-cle. According to this definition, the particle’s energyis the time component of the covariant 4-momentum p k = mu k = ( E p , − ap n ). From the square of the 4-momentum p i p i = m , (75)we obtain an energy-momentum relation E = (cid:0) p + m (cid:1) (1 − ˙ a ) , (76)which goes over into E = p + m in the limit of ˙ a = 0.One important expectation from (76) is that the energyof a material particle vanishes when ˙ a = 1, e.g., at thebeginning of the universe.The equation of motion of a particle with small pecu-liar velocity ( v α ≪
1) is obtained from the space compo-nent of (71) as dv α dt + (cid:18) ˙ a ¨ a − ˙ a + 2 ˙ aa (cid:19) v α = 0 , (77)3where any quadratic of v α has been dropped. The solu-tion to this equation av α ∝ √ − ˙ a a (78)shows how the proper peculiar velocity of a particleevolves as a result of the cosmic expansion.From (74) and (76), the energy and the 4-momentumof a massless photon are E γ = p γ (1 − ˙ a ) / , (79)and p i = (cid:18) p γ √ − ˙ a , p γ a n (cid:19) , (80)where p γ is the photon’s proper spatial momentum.The photon’s energy and spatial momentum are usu-ally expressed as photon’s frequency and inverse wave-length multiplied by the Planck constant ( E γ = hν and p γ = h/λ ). Therefore, Eq. (79) is equivalent to νλ = η = (1 − ˙ a ) / . (81)A comoving observer measures frequency and wavelengthof the same photon as ν c = ν (1 − ˙ a ) − / and λ c = λ . Thus, ν c λ c = 1 in the locally inertial frame. Thesubscript c denotes a quantity measured by the comovingobserver. D. Doppler shift and cosmological redshift ofphotons
The stretch of photon’s wavelength is induced bythe receding motion of an observer from a light source(Doppler shift), or by the cosmic expansion (cosmologi-cal redshift).First, let us consider the Doppler shift. For simplicity,the cosmic expansion speed is assumed to be fixed. Sup-pose that an observer with 4-velocity u i = ( γ, γv n ) ismoving away from a light source emitting photons with 4-momentum p i = ( p em / √ − ˙ a , p em n /a ), and is receiv-ing photons from the source. The observed momentumof a photon is given by the inner product of u i and p i : p ob = p i u i = √ − ˙ a − av cos θ √ − ˙ a − a v p em , (82)where p em and p ob are the proper spatial momenta ofemitted and observed photons, respectively, and n · n =cos θ ( θ = 0 for receding and θ = π for approachingobservers). The ratio of momenta (or energies) of ob-served to emitted photons for the longitudinal Dopplereffect ( θ = 0) is p ob p em = E ob E em = s c χ − vc χ + v , (83) where c χ ≡ η/a is the light speed in the comoving co-ordinate system. The ratio for the transversal Dopplereffect ( θ = π/
2) is p ob p em = E ob E em = 1 q − v /c χ . (84)The two formulas for the Doppler effect are similar inform to those in the special relativity.Next, for the cosmological redshift, let us suppose thatphotons, emitted at world time t from a light source atcomoving coordinate χ , have arrived at the origin at t .Using the photon’s geodesic equation and assuming thatphotons have traveled radially by the symmetry of space,we get Z χ dχ ′ = − Z tt √ − ˙ a a dt ′ = Z t t c χ ( t ′ ) dt ′ = Z t + δt t + δt c χ ( t ′ ) dt ′ , (85)where the minus sign in front of the second integral in-dicates that photons have propagated from the distantsource to the origin. The δt and δt are the world timeintervals during which a photon’s wave crest propagatesby the amount of its wavelength at the points of emissionand observation, respectively (i.e., ν = δt − ). The thirdequality holds because the integral does not change afterthe infinitesimal time intervals δt and δt .Manipulating (85) gives the frequency ratio of emittedto observed photons, νν = δt δt = c χ ( t ) c χ ( t )= a a (cid:18) − ˙ a − ˙ a (cid:19) / = (1 + z ) A (0) A ( z ) ≡ r ( z ) . (86)The cosmological time dilation function r ( z ) is useful forcomparing physical quantities at past and present epochs.The variation of r ( z ) is shown in Fig. 6. The correspond-ing function in the FRW world model, 1+ z , has a similarvalue to r ( z ) only at low redshift ( z . r ( z ) is almost constant duringthe radiation-dominated era with the maximum value of r ( ∞ ) = [( ε R0 + ε M0 + ε Λ ) /ε R0 ] / = 113 . (87)From (81) and (86), the photon’s frequency and wave-length vary as ν ∝ √ − ˙ a a and λ ∝ a. (88)Because the wavelength of a photon increases in pro-portional to a , the redshift is equal to the fractionaldifference between wavelengths at the points of obser-vation and emission of the photon: z = a /a − FIG. 6: Variation of r ( z ) along with redshift (thick solidcurve; Model I). The upper bound value of r ( z ) at infiniteredshift is 113. For comparison, the cosmological time dila-tion factor 1+ z in the FRW world model is shown as a dashedcurve. ( λ ob − λ em ) /λ em . Besides, the photon’s energy and spa-tial momentum vary as E γ ∝ √ − ˙ a a and p γ ∝ a . (89)From (56), (57), and (59), one finds that E γ = const.(R-u), E γ ∝ a − / (M-u), and E γ ∝ a − (Λ-u). Inthe comoving observer’s frame, both photon’s energy andspatial momentum always vary as a − . E. Total energy in the universe
The conservation of energy in the classical physics isclosely related to the invariance of physical laws undera time-translation (Noether’s theorem), which applies tothe physics in the Minkowski space-time. In general rel-ativity there is not necessarily a time coordinate withthe translation-symmetry, so the conservation of energyis not generally expected. However, in an asymptoticallyflat region or in a locally inertial frame, it is possible todefine the conserved energy. For this reason, it has beenusually said that there is not a global but a local energyconservation law.Let us estimate the total energy in the universe basedon the definition of energy in Sec. IV C. First, we needto define the volume element. The 4-dimensional volumeelement is given by dV = √− gdx dx dx dx = √− gdtdχdθdφ, (90) FIG. 7: Variation of total radiation ( E R ), matter ( E M ), anddark ( E Λ ) energies in unit of 10 erg over the world time t (thick solid curves; Model I), together with that of totalenergy E = E R + E M + E Λ (long dashed curve). Thin solidcurves represent the variation of the corresponding energyfraction parameters ( D R , D M , D Λ ; D R + D M + D Λ = 1) witha dimensionless unit. The vertical dotted line indicates t . where g is the determinant of the metric tensor g ik . Weobtain the 4-volume of the universe from V ( t ) = R dV =2 π R t (1 − ˙ a ) / a dt ′ . The proper 3-volume of the uni-verse is the time-derivative of V : V ( t ) = dV dt = 2 π (1 − ˙ a ) / a = 2 π a (1 + z ) A ( z ) . (91)The factor (1 − ˙ a ) / appears as a natural contractioneffect due to the expansion of the universe. At the presenttime, V = 6 . × Mpc . One can verify that the 3-volume of the universe evolves as V ∝ a (R-u), V ∝ a / (M-u), and V ∝ a (Λ-u). Note that V c = dV /dτ ∝ a in the comoving observer’s frame and thus V = (1 − ˙ a ) / V c .If there are N comoving particles with rest mass m inthe universe, then the matter energy density is ε M = N E p V = N m (1 − ˙ a ) / (1 − ˙ a ) / V c = N mV c = ε Mc . (92)Therefore, matter energy densities both in the world ref-erence and the comoving observer’s frames are equivalentto each other, which also applies to the radiation energydensity if m is replaced with p γ in (92). The matterenergy density is related to the matter density ρ M by ε M = ρ M η because ρ M = N m r /V = N m/V c (1 − ˙ a ) = ε M / (1 − ˙ a ).5The total energy of each species I is calculated from E I = ε I V = 2 π a ε I (1 + z ) w I A ( z ) . (93)The present values of total radiation, matter, and darkenergies ( E R0 , E M0 , E Λ0 ) are listed in Table II. In radia-tion, matter, and Λ-universes, the total energy evolves as E R = const., E M ∝ a / , and E Λ ∝ a , respectively. Onthe other hand, E Rc ∝ a − , E Mc = const., and E Λc ∝ a in the comoving observer’s frame, implying that the radi-ation energy is infinite at the initial time, and the matterenergy is always constant.Fig. 7 shows the variation of total radiation, mat-ter, and dark energies during the history of the universe(thick solid curves), together with that of energy fractionparameters ( D I ; thin solid curves). The total radiationenergy remained constant in the radiation-dominated era( t . − Gyr), and thereafter has decreased. The totalmatter energy was zero at t = 0, arrived at the maxi-mum value at t = 58 . τ = 10 . M = E M0 (1 − ˙ a ) − / ≈ M ⊙ , which corresponds toabout 10 galaxies with a typical mass of 10 M ⊙ .From the definition of energy fraction parameters [Eqs.(42) and (43)], one finds that εV c χ is a conserved quan-tity such that ε ( t ) V ( t ) c χ ( t ) = ε V c χ = const. , (94)where ε is the total energy density. Thus, the ratio ofpresent to past total energies in the universe is obtainedas E E ( t ) = ε V ε ( t ) V ( t ) = c χ ( t ) c χ ( t ) = r ( z ) , (95)where E = P I E I (long dashed curve in Fig. 7) and E =1 . × erg. The initial amount of total energy in theuniverse is E (0) = E /r ( ∞ ) = 9 . × erg. Since onlythe radiation contributes to the total energy at t = 0, thesame value is obtained from E (0) = E R (0) = E R0 r ( ∞ )with the help of E R ( t ) = E R0 r ( z ) deduced from (86).According to the definition of energy in this paper,the total energy is not conserved in the expanding closeduniverse, but increases with time. Especially, the totalenergy is finite at the beginning of the universe. F. Energy density, pressure, and temperature inthermal equilibrium
We describe the evolution of energy density, pressure,and temperature of gas in the early universe (see [37] fordetails). Our discussion is restricted to the relativisticgas particles in thermal equilibrium. The particles are assumed to have low rest mass compared to their kineticenergy.The number of particles of species I per unit spatialvolume dV per unit momentum volume dW can be ex-pressed as dN I = g I f I ( x i , p i )(2 π ~ ) dV dW, (96)where g I is the spin-degeneracy of the particle, f I is theparticle distribution function for species I , equivalent tothe mean number of particles occupying a given quantumstate, and ~ = h/ π . Assuming zero chemical potential,we can write f I = [exp( E p /k B T I ) ± − , where plus andminus signs are for fermions (f) and bosons (b), respec-tively, k B is the Boltzmann constant, and T I is the ther-modynamic temperature of species I . The proper spatialand momentum volume elements in the world referenceframe are given by dV = √− gdχdθdφ = (1 − ˙ a ) / dV c ,dW = 1 √− g dp χ dp θ dp φ = (1 − ˙ a ) − / dW c , (97)where dV c and dW c are proper volume elements in thecomoving observer’s frame. Note that dV dW = dV c dW c .The momentum volume element is written as dW =(1 − ˙ a ) − / πp dp for isotropic gas particles with propermomentum p .The energy density of the relativistic gas is obtainedby integrating over the momentum space the particle’senergy multiplied with its distribution function: ε I = g I (2 π ~ ) Z E p f I dW = g I (2 π ~ ) Z ∞ πp dpe E p /k B T I ±
1= 4 πg I k T I (2 π ~ ) (1 − ˙ a ) Z ∞ x e x ± dx = 7 π g I k T I ~ (1 − ˙ a ) (f) and π g I k T I ~ (1 − ˙ a ) (b) , (98)where x ≡ E p /k B T I and E p ≃ p (1 − ˙ a ) / for the rela-tivistic gas.The pressure of the relativistic gas is obtained in asimilar way: P I = g I (2 π ~ ) Z p v f I dW = g I π ~ ) Z ∞ πp dpe E p /k B T I ± ε I , (99)where p and v are proper momentum and velocity inone direction: v = ( p /E p )(1 − ˙ a ) and p = p / w I = and the energy densityvarying as ε I ∝ a − . Therefore, from (98) the thermo-dynamic temperature of the relativistic gas evolves as T I ∝ √ − ˙ a a . (100)6In the comoving observer’s frame, T I c ∝ a − . From aformula of the entropy density σ I = ( ε I + P I ) /k B T I , onecan verifies that the total entropy S I = σ I V of the rela-tivistic gas is constant.For photons, the quantity x = hν/k B T R is invariantduring the cosmic expansion history because the pho-ton’s frequency varies in the same way as the tempera-ture does. Since x is also frame-independent ( x = x c ),we have T R = T Rc ( ν/ν c ) = T Rc (1 − ˙ a ) / . The presentCMB temperature in the world reference frame is T R0 = T Rc0 (1 − ˙ a ) / = 0 .
57 K ( T Rc0 = T cmb ). From (100), theratio of past to present radiation temperatures is T R T R0 = a a (cid:18) − ˙ a − ˙ a (cid:19) / = r ( z ) , (101)which enables us to estimate the radiation temperatureat the past epoch. For example, at the beginning of theuniverse T R (0) = T R0 r ( ∞ ) = 64 . r ( z )implies that T R = const. in the radiation-universe.The epoch of radiation-matter equality is determinedfrom the condition ε M = ε R :1 + z eq = ε M0 ε R0 = (cid:18) a R0 a M0 (cid:19) = 5081 . (102)At this epoch ( t eq = 1 . × − Gyr, τ eq = 29700 yr),the size of the universe was a eq = 5 .
05 Mpc and theradiation temperature was T R, eq = T R0 r ( z eq ) = 45 . T Rc, eq = T Rc0 (1 + z eq ) = 13850 K.To summarize, as judged in the world reference frame,the early universe was cold and all the physical pro-cesses in it were extremely slow. Especially, the universestarted from a regular (non-singular) point in the sensethat physical quantities have finite values at the initialtime. The singular nature of the FRW universe comesfrom the fact that the flow of the proper time was frozen( dτ = 0) at t = 0. G. Cosmic distance and time scales
Lastly, we consider cosmic distance and time scales inthe closed world model. As the most popular distancemeasure, the coordinate distance ( d C ) to a galaxy at red-shift z is obtained by integrating the photon’s geodesicequation ( dχ = dt √ − ˙ a /a ), d C ( z ) = a χ C ( z ) = a Z t t √ − ˙ a a dt ′ = Z z a dz ′ [ A ( z ′ ) − (1 + z ′ ) ] / . (103)Here χ C ( z ) is the comoving coordinate distance. Forsufficiently large a , Eq. (103) goes over into d C ( z ) ≈ R z a dz ′ /A ( z ′ ), which is equivalent to the coordinate dis-tance in the flat FRW world model. The coordinate dis-tance in the matter-universe has an analytic form d C ( z ) = 2 a arctan "(cid:18) a a M0 (cid:19) (1 + z ) − / − a arctan "(cid:18) a a M0 (cid:19) − / (M-u) . (104)We can derive other astronomical distances based onluminosity and angular size of distant sources. For theluminosity distance, let us imagine that a light sourceat redshift z has intrinsic bolometric luminosity L c asmeasured at the source. Since both photon’s energy andarrival rate vary in proportion to a − in the locally in-ertial frame, the flux of the light source as measured bythe present comoving observer can be written as L c πd = L c πa sin χ C (cid:18) aa (cid:19) , (105)where d L is the luminosity distance to the source [44], d L ( z ) = a (1 + z ) sin χ C ( z ) . (106)The angular size distance ( d A ) to a galaxy with physicalsize r g and angular size θ g is given by d A ( z ) = r g θ g = √ g θ g θ g = a sin χ C ( z )1 + z . (107)The recession velocity ( v rec = z ) of a galaxy has alsobeen used as a distance measure in the local universe.By integrating dz = − da ( a /a ) from the definition ofredshift, we get z = Z t t (cid:16) a a (cid:17) (cid:18) ˙ aa (cid:19) dt ′ = a Z t t H ( τ ′ ) √ − ˙ a a dt ′ , (108)where H ( τ ) = a − ( da/dτ ) = a − [ A ( z ) − (1 + z ) ] / is the Hubble parameter. Since the Hubble parameterremains almost constant during the recent epoch ( z . . v rec ≈ H d C ( z ), which is the Hubble’s law.The age of the universe or the lookback time have beenused as a measure of cosmic time scales. The age of theuniverse measured in world time is calculated from t ( z ) = Z t dt ′ = Z ∞ z a dz ′ (1 + z ′ ) s A ( z ′ ) A ( z ′ ) − (1 + z ′ ) , (109)while the age measured in proper time of a comovingobserver (us) is obtained by integrating (52): τ ( z ) = Z t p − ˙ a dt ′ = Z ∞ z a dz ′ (1 + z ′ ) 1[ A ( z ′ ) − (1 + z ′ ) ] / , (110)7 FIG. 8: (top) Coordinate ( d C ), luminosity ( d L ), and an-gular size ( d A ) distances in the closed world model (solidcurves; Model I). The coordinate distance to the big-bang is d C ( ∞ ) = 15 . t ; top curve) and in proper time of a comoving observer( τ ; bottom curve). The dashed curves are the correspondingdistances and age for the flat FRW world model that best fitswith the WMAP CMB data (Ω M h = 0 . h = 0 . The latter is equivalent to the age of the FRW universe.Note that t ≥ τ (see Fig. 3 for τ - t relation). At thepresent time, t = 85 . τ = 15 . t − t ( z ) or τ − τ ( z ).Fig. 8 compares coordinate, luminosity, angular sizedistances (top) and cosmic ages (bottom) as a functionof redshift in the closed world model with Model I pa-rameters of Table II. Also shown are the correspondingdistances and age for the flat FRW world model that bestfits with the WMAP CMB data (Ω M h = 0 . h = 0 . z & V. INFLATION
The FRW world model has been criticized because oftwo shortcomings, namely, flatness and horizon problems.As shown in Sec. III, the universe with positive energydensity is always spatially closed. If the present universeis traced back to the past, it would become a more curvedhypersurface, a 3-sphere with smaller curvature radius.Therefore, the closed world model proposed in this paperis free from the flatness problem. Next, let us consider thehorizon problem, which is stated as follows. The observedCMB temperature fluctuations separated by more thana degree are similar to each other over the whole sky. Inthe FRW world model, such an angle corresponds to adistance where the causal contact was impossible on thelast scattering surface. The large-scale uniformity of theCMB anisotropy suggests that the observed regions musthave been in causal contact in the past.The inflation paradigm has offered a reasonable solu-tion to the puzzle of the large-scale homogeneity of theobservable universe by proposing that there was a periodof rapid expansion of the universe with positive accelera-tion [38]. According to the inflation theory, the inflationtakes place due to the presence of a scalar field φ , whoseenergy density and pressure are given by ε φ = ˙ φ + V ( φ )and P φ = ˙ φ − V ( φ ), respectively, where V ( φ ) is a po-tential of the scalar field. Here we assume that the dotover φ is the world-time derivative. From (38), for a uni-verse dominated by the scalar field, the condition for thepositive expansion acceleration is˙ φ < V ( φ ) . (111)A distance scale of causally connected region (so calledhorizon size) is usually quantified by the Hubble radiusand the particle horizon size. In the FRW world model,one important implication of the positive expansion ac-celeration is that the comoving Hubble radius decreaseswith time, i.e., d ( aH ) − /dτ <
0. The comoving Hubbleradius is defined as the comoving distance at which therecession velocity as defined in the world reference frameis equal to the speed of light ( ˙ aχ H = η ): χ H ( t ) = √ − ˙ a ˙ a = 1 aH ( τ ) = 1 + z [ A ( z ) − (1 + z ) ] / . (112)The comoving particle horizon size is the comoving dis-tance a photon has travelled during the age of the uni-verse: χ P ( t ) = Z t √ − ˙ a a dt ′ = Z ∞ z dz ′ [ A ( z ′ ) − (1 + z ′ ) ] / . (113)The proper Hubble radius and particle horizon size aregiven by d H ( t ) = a ( t ) χ H ( t ) = 1 /H ( τ ) and d P ( t ) = a ( t ) χ P ( t ), respectively.Fig. 9 compares proper (top) and comoving (bottom)horizon sizes as a function of world time. It is important8 FIG. 9: Time-variation of proper (top) and comoving (bot-tom) horizon sizes in the closed universe (Model I), namely,the Hubble radius ( d H , χ H ) and the particle horizon size ( d P , χ P ). For comparison, the evolution of curvature radius a ( t )is shown as dashed curve in the top panel. The proper Hub-ble radius has an asymptotic maximum d H ( ∞ ) = a Λ = 6867Mpc. The comoving horizon size is shown in unit of degree,with an asymptotic maximum χ P ( ∞ ) = 49 . ◦
4. The verticaldotted lines indicate the present epoch. to note that comoving horizon sizes were zeros at the be-ginning of the universe, and then have increased until therecent epoch. As shown in Sec. IV B, the universe wasexpanding with the limiting speed and zero accelerationat the initial time (Figs. 4 and 5). Therefore, the positiveexpansion acceleration or the decrease in the comovingHubble radius is not allowable at the early stage of theclosed universe. The decrease is only possible in laterΛ-dominated universe ( t &
100 Gyr; Fig. 9, bottom).Actually, the scalar field φ is not essential for drivingthe rapid expansion of the universe. Even if the scalarfield is dominant, the condition (111) is not satisfied. Wecan only expect that ˙ φ = V ( φ ) from the constraint ofzero acceleration, obtaining an equation of state P φ = − ε φ . (114)Thus the curvature radius increases as a ∝ t in the uni-verse dominated by the scalar field. Further constrainingthe universe to expand with the limiting speed demandsthat the energy density of the scalar field should be in- finite, as in the extreme case of H-universe (Sec. IV B).However, the radiation-universe with finite total energyprovides a far simpler expansion history a ≃ t for t ≪ b R [Eq. (56)], which demonstrates the sufficiency of the radi-ation in driving the rapid expansion and the needlessnessof the scalar field. In conclusion, it is improbable thatthe inflation with positive acceleration occurred in theearly universe.If the universe expands with the limiting speed, the pe-culiar velocity of a particle vanishes as implied by (34):the matter and radiation were frozen with zero propa-gation speed at the earliest epoch. Besides, informationat one region could not be easily transferred to otherregions due to the small horizon size. Thus, physical in-formation sharing through the causal contact during theexpansion of the universe is not an efficient way to ex-plain for the large-scale homogeneity of the universe. Thespherically symmetric and uniform distribution of super-nova remnants (e.g., Tycho’s supernova 1572; [39]) drivenby a strong shock into the ambient interstellar mediumshows that the large-scale homogeneity can be generatedfrom the ballistic explosion at the single point, withoutthe causal contact during the expansion. At the begin-ning of the universe, everything was on the single pointso that every information such as temperature and en-ergy density could be shared in full and uniform contact.Therefore, if the initial condition was properly set at thecreation of the universe, e.g., by the quantum processesat t . t p (Planck time), which is out of the scope ofthe classical physics, the observed uniformity of densitydistributions at super-horizon scales may be explained. VI. CONCLUSION
In this paper, the general world model for homoge-neous and isotropic universe has been proposed. By in-troducing the world reference frame as a global and fidu-cial system of reference, we have defined the line ele-ment so that the effect of cosmic expansion on the phys-ical space-time separation can be correctly included inthe metric. With this framework, we have demonstratedtheoretically that the flat universe is equivalent to theMinkowski space-time and that the universe with posi-tive energy density is always spatially closed and finite.The open universe is unrealistic because it cannot accom-modate positive energy density. Therefore, in the worldof ordinary materials, only the spatially closed universeis possible to exist.The naturalness of the finite world with positive en-ergy density comes from the Mach’s principle that themotion of a mass particle depends on the mass distribu-tion of the entire world. The principle is consistent onlywith the finite world because the dynamics of a referenceframe cannot be defined in the infinite, empty world. Theclosed world model satisfies the Mach’s principle and sup-ports Einstein’s perspective on the physical universe.We have reconstructed evolution histories of the closed9world models that are consistent with the recent as-tronomical observations, based on the nearly flat FRWworld models (Model I and II; Sec. IV). The present cur-vature radius of the universe is a = 25 . a = 40 . a = 1, ¨ a = 0).From the local nature of the FRW metric (Sec. II)and of the proper time of a comoving observer (Sec.III E), it is clear that the FRW world model describesthe local universe as observed by the comoving observer.Since the Newton’s gravitation law can be derived fromthe Einstein’s field equations in the weak field and thesmall velocity limits, the gravitational action at a dis-tance usually holds at a local region of space on scalesfar smaller than the Hubble horizon size (e.g., [40]). Theproper Hubble radius d H (Fig. 9, top) may provide areasonable estimate of the characteristic distance scalewhere the Newton’s gravity applies. The cosmic struc-tures simulated by the Newton’s gravity-based N -body method will significantly deviate from the real structureson scales comparable to d H . The variation of the comov-ing Hubble radius χ H = d H /a also implies that in thepast (future) the Newtonian dynamics was (will be) ap-plicable on smaller region of space compared to the sizeof the universe (Fig. 9, bottom).In this paper, the history of the universe has been ten-tatively reconstructed based on cosmological parametersof non-flat FRW world models. The more general cos-mological perturbation theory and parameter estimationare essential for accurate reconstruction of the cosmichistory. Acknowledgments
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