An expanding 4D universe in a 5D Kaluza-Klein cosmology with higher dimensional matter
aa r X i v : . [ g r- q c ] D ec An expanding D universe in a D Kaluza-Klein cosmology with higherdimensional matter
F. Darabi ∗ Department of Physics, Azarbaijan University of Tarbiat Moallem, 53714-161, Tabriz, Iran .Research Institute for Astronomy and Astrophysics of Maragha, 55134-441, Maragha, Iran.
October 22, 2018
Abstract
In the framework of Kaluza-Klein theory, we investigate a (4+1)-dimensional universeconsisting of a (4 + 1) dimensional Robertson-Walker type metric coupled with a (4 + 1)dimensional energy-momentum tensor. The matter part consists of an energy densitytogether with a pressure subject to 4 D part of the (4 + 1) dimensional energy-momentumtensor. The dark part consists of just a dark pressure ¯ p , corresponding to the extra-dimension endowed by a scalar field, with no element of dark energy. It is shown thatthe reduced Einstein field equations are free of 4 D pressure and are just affected byan effective pressure produced by the 4 D energy density and dark pressure. It is thenproposed that the expansion of the universe may be controlled by the equation of statein higher dimension rather than four dimensions. This may account for the emergenceof unexpected current acceleration in the middle of matter dominant era.PACS: 95.36.+x; 98.80.-k; 04.50.CdKey words: Dark pressure; inflation; accelerating universe; non compact Kaluza-Kleincosmology. ∗ e-mail: [email protected] Introduction
The recent distance measurements from the light-curves of several hundred type Ia supernovae[1, 2] and independently from observations of the cosmic microwave background (CMB) bythe WMAP satellite [3] and other CMB experiments [4, 5] suggest strongly that our universeis currently undergoing a period of acceleration. This accelerating expansion is generallybelieved to be driven by an energy source called dark energy. The question of dark energyand the accelerating universe has been therefore the focus of a large amount of activities inrecent years. Dark energy and the accelerating universe have been discussed extensively fromvarious point of views over the past few years [6, 7, 8]. In principle, a natural candidate fordark energy could be a small positive cosmological constant. One approach in this direction isto employ what is known as modified gravity where an arbitrary function of the Ricci scalar isadded to the Einstein-Hilbert action. It has been shown that such a modification may accountfor the late time acceleration and the initial inflationary period in the evolution of the universe[9, 10]. Alternative approaches have also been pursued, a few example of which can be found in[11, 12, 13]. These schemes aim to improve the quintessence approach overcoming the problemof scalar field potential, generating a dynamical source for dark energy as an intrinsic feature.The goal would be to obtain a comprehensive model capable of linking the picture of theearly universe to the one observed today, that is, a model derived from some effective theoryof quantum gravity which, through an inflationary period would result in today acceleratedFriedmann expansion driven by some Ω Λ -term. However, the mechanism responsible for thisacceleration is not well understood and many authors introduce a mysterious cosmic fluid, theso called dark energy, to explain this effect [14].Since in a variety of inflationary models scalar fields have been used in describing thetransition from the quasi-exponential expansion of the early universe to a power law expansion,it is natural to try to understand the present acceleration of the universe by constructingmodels where the matter responsible for such behavior is also represented by a scalar field.Such models are worked out, for example, in Ref [15].Bellini et al , on the other hand, have published extensively on the evolution of the universefrom noncompact vacuum Kaluza-Klein theory [16]. They used the essence of STM (Space-Time-Matter) theory and developed a 5D mechanism to explain, by a single scalar field, theevolution of the universe including inflationary expansion and the present day observed ac-celerated expansion. The STM theory is proposed by Wesson and his collaborators, which isdesigned to explain the origin of matter in terms of the geometry of the bulk space in whichour 4 D world is embedded, for reviews see [17]. More precisely, in STM theory, our world isa hypersurface embedded in a five-dimensional Ricci-flat ( R AB = 0) manifold where all thematter in our world can be thought of as being manifestations of the geometrical properties ofthe higher dimensional space according to G αβ = 8 πGT αβ , provided an appropriate definitionis made for the energy-momentum tensor of matter in terms of the extra part of the geometry.Physically, the picture behind this interpretation is that curvature in (4 + 1) space induceseffective properties of matter in (3 + 1) space-time. The fact that such an embedding canbe done is supported by Campbell’s theorem [18] which states that any analytical solution ofthe Einstein field equations in N dimensions can be locally embedded in a Ricci-flat manifoldin ( N + 1) dimensions. Since the matter is induced from the extra dimension, this theory is2lso called the induced matter theory. The sort of cosmologies stemming from STM theory isstudied in [19, 20, 21].Another higher dimensional work has already been done with a multi-dimensional compact Kaluza-Klein cosmological model in which the scale factor of the compact space evolves as aninverse power of the radius of the observable universe [22]. The Friedmann-Robertson-Walkerequations of standard four-dimensional cosmology were obtained where the pressure in the4 D universe was an effective pressure, expressed in terms of the components of the higherdimensional energy-momentum tensor, capable of being negative to explain the accelerationof our present universe.In this work, motivated by the work done in the compact model [22] and interested inits non-compact version, a 5 D non-compact Kaluza-Klein cosmological model is introducedwhich is not Ricci flat, but is extended to couple with a higher dimensional energy momentumtensor. In the present non-compact model, it is shown that a dark pressure along the higherdimensional sector together with the 4 D energy density may induce an effective pressure infour dimensional universe so that the reduced field equations on 4 D universe are free of 4 D pressure and are just affected by the effective pressure. The main point of this paper is toshow the possibility that perhaps 4 D pressure does not directly control the dynamics of theuniverse, rather the cosmological eras including inflation, deceleration and current accelerationare just happened due to either the evolution in equation of state along higher dimension oran interplay between equations of state in 4 D universe and along higher dimension. Moreover,it is appealing to consider the current acceleration of the universe as a result of a new phasewhich is started recently along extra dimension. In this way, the emergence of unexpectedacceleration in the middle of matter dominated era is easily justified because in this modelthe real dynamics of the universe is controlled not by 4 D physics but through the higherdimensional physics. In other words, the unexpected emergence of current acceleration maybe related to a higher dimensional effect which is hidden for 4 D observers. We start with the 5 D line element dS = g AB dx A dx B , (1)in which A and B run over both the space-time coordinates α, β and one non compact extradimension indicated by 4. The space-time part of the metric g αβ = g αβ ( x α ) is assumed todefine the Robertson-Walker line element ds = dt − a ( t ) (cid:18) dr (1 − kr ) + r ( dθ + sin θdφ ) (cid:19) , (2)where k takes the values +1 , , − g α = 0 , g = ǫ Φ ( x α ) , where ǫ = 1 and the signature of the higher dimensional part of the metric is left general.This choice has been made because any fully covariant 5 D theory has five coordinate degreesof freedom which can lead to considerable algebraic simplification, without loss of generality.3he extra dimensional independence of the scalar field and the 4 D metric may pose anambiguity between the compact or non-compactness of the present model. In the compacttheory the cylindrical condition is imposed to account for this independence and justify the non-observability of extra dimension. In non-compact theory, however, one may account for thisindependence within a different framework in which the non-observability of extra dimensionsis justified in a different way. Therefore, the answer to the question of whether a higherdimensional model in which all variables are independent of extra dimension is compact ornon-compact depends on the way by which the model justifies the non-observability of extradimensions. In the present model we aim to follow the non-compact model where a reasonablejustification will be given for the non-observability of extra dimension.Unlike the noncompact vacuum Kaluza-Klein theory, we will assume the fully covariant 5 D non-vacuum Einstein equation G AB = 8 πGT AB , (3)where G AB and T AB are the 5 D Einstein tensor and energy-momentum tensor, respectively.Note that the 5 D gravitational constant has been fixed to be the same value as the 4 D one .In the following we use the geometric reduction from 5 D to 4 D as appeared in [23]ˆ R αβ = R αβ + ∂ Γ αβ − ∂ β Γ α + Γ λαβ Γ λ + Γ αβ Γ D D − Γ αλ Γ λβ − Γ Dα Γ βD , (4)where ˆ denotes the 4 D part of the 5 D quantities. Evaluating the Christoffel symbols for themetric g AB gives ˆ R αβ = R αβ − ∇ α ∇ β ΦΦ . (5)In the same way we obtain R = − ǫ Φ (cid:3) Φ . (6)We now construct the space-time components of the Einstein tensor G AB = R AB − g AB R (5) . In so doing, we first obtain the 5 D Ricci scalar R (5) as R (5) = g AB R AB = ˆ g αβ ˆ R αβ + g R = g αβ ( R αβ − ∇ α ∇ β ΦΦ ) + ǫ Φ ( − ǫ Φ (cid:3) Φ)= R − (cid:3) Φ , (7)where the α R is the 4 D Ricci scalar. The space-time components of theEinstein tensor is written ˆ G αβ = ˆ R αβ − ˆ g αβ R (5) . Substituting ˆ R αβ and R (5) into the space-timecomponents of the Einstein tensor givesˆ G αβ = G αβ + 1Φ ( g αβ (cid:3) Φ − ∇ α ∇ β Φ) . (8) In compact Kaluza-Klein theory one may define a 5 D gravitational constant G (5) which is reduced to 4 D one as G = G (5) R dy where R dy is the volume of extra compact dimension. In non-compact theory, however, wedo not require a 5 D gravitational constant because there is no finite volume of extra dimension, and assumingthe gravitational constant as G will result in the correct 4 D Einstein equations.
4n the same way, the 4-4 component is written G = R − g R (5) , and substituting R , R (5) into this component of the Einstein tensor gives G = − ǫR Φ . (9)We now consider the 5 D energy-momentum tensor without specifying its nature or origin . Theform of energy-momentum tensor is dictated by Einstein’s equations and by the symmetriesof the metric (2). Therefore, we may assume a perfect fluid with nonvanishing elements T αβ = ( ρ + p ) u α u β − pg αβ , (10) T = − ¯ pg = − ǫ ¯ p Φ , (11)where ρ and p are the conventional density and pressure of perfect fluid in the 4 D standardcosmology and ¯ p acts as a pressure living along the higher dimensional sector. Notice thatthe perfect fluid is isotropic on the 3 D geometry and anisotropic regarding the 5 th dimension .The field equations (3) are to be viewed as constraints on the simultaneous geometric andphysical choices of G AB and T AB components, respectively.Substituting the energy-momentum components (10), (11) in front of the 4 D and extradimensional part of Einstein tensors (8) and (9), respectively, we obtain the field equations G αβ = 8 πG [( ρ + p ) u α u β − pg αβ ] + 1Φ [ ∇ α ∇ β Φ − (cid:3) Φ g αβ ] , (12)and R = 16 πG ¯ p. (13)By evaluating the g αβ trace of Eq.(12) and combining with Eq.(13) we obtain (cid:3) Φ = 13 (8 πG ( ρ − p ) + 16 πG ¯ p )Φ . (14)This equation infers the following scalar field potential V (Φ) = −
16 (8 πG ( ρ − p ) + 16 πG ¯ p )Φ , (15)whose minimum occurs at Φ = 0, for which the equations (12) reduce to describe a usual4 D FRW universe filled with ordinary matter ρ and p . In other words, our conventional 4 D Since this model is supposed to describe the radiation and matter dominated eras, it seems that this fifthdimensional pressure component would be a dark property of conventional matter, including standard modelfields (see refs in [24] for some derivations of the matter contribution in Kaluza-Klein cosmology). The same choice has been made in [22] with the components of the higher dimensional energy-momentumtensor as T ij = diag [ − ρ, p, p, p, p d , ..., p d ] where ρ, p and p d are the density, pressure on the 3 D geometry andpressure along the extra dimensions, respectively. The α R α = 0 , which is an identity with no useful information. (cid:3) Φ = 13 (8 πG ( ρ − p ) + 16 πG ¯ p ) , (16)1Φ ∇ α ∇ β Φ = 13 (8 πG ( ρ − p ) + 16 πG ¯ p ) u α u β . (17)Putting the above replacements into Eq.(12) leads to G αβ = 8 πG [( ρ + ˜ p ) u α u β − ˜ pg αβ ] , (18)where ˜ p = 13 ( ρ + 2¯ p ) . (19)This energy-momentum tensor effectively describes a perfect fluid with density ρ and pressure˜ p . The four dimensional field equations lead to Friedmann equation3 ˙ a + ka = 8 πGρ, (20)and 2 a ¨ a + ˙ a + ka = − πG ˜ p. (21)Differentiating (20) and combining with (21) we obtain the conservation equation ddt ( ρa ) + ˜ p ddt ( a ) = 0 . (22)The equations (20) and (21) can be used to derive the acceleration equation¨ aa = − πG ρ + 3˜ p ) = − πG ρ + ¯ p ) . (23)The acceleration or deceleration of the universe depends on the negative or positive values ofthe quantity ( ρ + ¯ p ).From extra dimensional equation (13) ( or 4-dimensional Eqs.(19), (20) and (21) ) we obtain − k + ˙ a + ¨ aa ) a = 16 πG ¯ p. (24)Using power law behaviors for the scale factor and dark pressure as a ( t ) = a t α and ¯ p ( t ) = ¯ p t β in the above equation, provided k = 0 in agreement with observational constraints, we obtain β = − D universe we may assume the scalar field tobe just a function of time, then the scalar field equation (14) reads as the following form¨Φ + 3 ˙ aa ˙ φ − πG ρ − p ) + 2¯ p )Φ = 0 . (25)6ssuming Φ( t ) = Φ t γ and ρ ( t ) = ρ t δ ( ρ >
0) together with the equations of state formatter pressure p = ωρ and dark pressure ¯ p = Ω ρ we continue to calculate the requiredparameters for inflation, deceleration and then acceleration of the universe . In doing so, werewrite the acceleration equation (23), scalar field equation (25) and conservation equation(22), respectively, in which the above assumptions are included as α ( α −
1) + 8 πG ρ (1 + Ω) = 0 , (26) γ ( γ −
1) + 3 αγ − πG ρ ((1 − ω ) + 2Ω) = 0 , (27)2 ρ [(2 + Ω) α −
1] = 0 , (28)where δ = − t α − inthe conservation equation. The demand for acceleration ¨ a > ρ ( t ) = ρ t δ and ¯ p = Ω ρ , requires ρ (1 + Ω) < < − α > α = which together with the condition α > − < Ω < −
1. Now, one may recognize two options as follows.The first option is to attribute an intrinsic evolution to the parameter Ω along the higherdimension so that it can produce the 4 D expansion evolution in agreement with standardmodel including early inflation and subsequent deceleration, and also current acceleration ofthe universe. Ignoring the phenomenology of the evolution of the parameter Ω, we may require Ω & − f or inflationΩ > − f or decelerationΩ . − f or acceleration . (29)The first case corresponds to highly accelerated universe due to a large α >>>
1. This canbe relevant for the inflationary era if one equate the power law with exponential behavior.The second case corresponds to a deceleration α <
1, and the third case represents an smallacceleration α &
1. In this option, there is no specific relation between the physical phasealong extra dimension, namely Ω, and the ones defined in 4 D universe by ω .The second option is to assume a typical relation between the parameters Ω and ω asΩ = f ( ω ) so that Ω & − f or ω = − > − f or ω = Ω . − f or ω = 0 . (30)The case ω = − α >>>
1. The case ω = corresponds to the radiation dominant era and shows a deceleration α <
1. Finally, the case ω = 0 corresponds to the matter dominant era and shows an smallacceleration α & As we discussed earlier, the fifth dimensional pressure component could be a dark property of conventionalmatter ρ , through a dark parameter Ω, according to ¯ p = Ω ρ . onclusion A (4 + 1)-dimensional universe consisting of a (4 + 1) dimensional metric of Robertson-Walkertype coupled with a (4 + 1) dimensional energy-momentum tensor in the framework of non-compact Kaluza-Klein theory is studied. In the matter part, there is energy density ρ togetherwith pressure p subject to 4 D part of the (4 + 1) dimensional energy-momentum tensor, anda dark pressure ¯ p corresponding to the extra-dimensional part endowed by a scalar field. Aparticular (anisotropic) equation of state in 5 D is used for the purpose of realizing the 4 D expansion in agreement with observations. This is done by introducing two parameters ω andΩ which may be either independent or related as Ω = f ( ω ). The physics of ω is well knownbut that of the parameter Ω needs more careful investigation based on effective higher di-mensional theories like string theory or Brane theory. The reduced 4 D and extra-dimensionalcomponents of 5 D Einstein equations together with different equations of state for pressure p and dark pressure ¯ p may lead to a 4 D universe which represents early inflation, subsequentdeceleration and current acceleration. In other words, all eras of cosmic expansion may beexplained by a single simple mechanism.The important point of the present model is that the reduced Einstein field equations arefree of 4 D pressure and are just affected by an effective pressure produced by the 4 D energydensity and dark pressure along the extra dimension. This provides an opportunity to considerthe expansion of the universe as a higher dimensional effect and so justify the unexpected currentacceleration in the middle of matter dominant era , along this line of thought. Moreover, thereis no longer “coincidence problem” within this model. This is because, in the present modelthere is no element of “dark energy” at all and we have just one energy density ρ associatedwith ordinary matter. So, there is no notion of coincidental domination of dark energy overmatter densities to trigger the acceleration at the present status of the universe. In fact, adark pressure with different negative values along the 5 th dimension by itself may produceexpanding universe including inflation, deceleration and acceleration without involving withthe coincidence problem. These stages of the 4 D universe may occur as well because ofnegative, positive and zero values of the four dimensional pressure, respectively, which leadsto a competition between energy density ρ and dark pressure ¯ p in the acceleration equation(23). For the same reason that there is no element of dark energy in this model, the apparent phantom like equation of state for dark pressure Ω < − unbounded from below dark energy or vacuum instability [25].The above results are independent of the signature ǫ by which the higher dimension takespart in the 5D metric. Moreover, the role played by the scalar field along the 5 th coordinatein the 5 D metric is very impressed by the role of scale factor over the 4 D universe. At earlyuniverse during the inflationary era the scalar field is highly suppressed and the 5 th coordinateis basically ignored in 5 D line element. At radiation dominant era the scalar field is much lesssuppressed and the 5 th coordinate becomes considerable in 5 D line element. Finally, at matterdominant era the scalar field and its possible fluctuations starts to be super-suppressed andthe observable effect of 5 th coordinate becomes vanishing in 5 D line element at t ≃ Sec ,leaving an effective 4 D universe in agreement with observations.A clear similarity is seen between the results of our 5 D non compact model and that ofmulti-dimensional compact one [22]. Both of these models predict an effective 4 D pressure,8xpressed in terms of the components of the higher dimensional energy-momentum tensor,capable of being negative to explain the acceleration of our present universe. Moreover, bothhigher dimensional metrics dynamically evolves towards an effective four-dimensional one. Acknowledgment
This work has been financially supported by the Research Institute for Astronomy and Astro-physics of Maragha (RIAAM).
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