Dark energy and 3-manifold topology
aa r X i v : . [ g r- q c ] N ov Dark energy and 3-manifold topology ∗ Torsten Asselmeyer-Maluga † German Aerospace Center, Rutherfordstr. 2, 12489 Berlin, Germany andHelge Ros´e ‡ Fraunhofer FIRST, Kekulestr. 7, 12489 Berlin, GermanyWe show that the differential-geometric description of matter by differ-ential structures of spacetime leads to a unifying model of the three typesof energy in the cosmos: matter, dark matter and dark energy. Using thismodel we are able to calculate the ratio of dark energy to the total energyof the cosmos.PACS numbers: 04.20.Gz,98.80.JK,02.40.Vh
1. Introduction
For centuries it has been our firm conviction that matter and energyof the same kind as is surrounding us also constitute the rest of the world.Thorough examinations of supernovae [15, 12] and of cosmic background ra-diation [2, 8], however, have replaced this conviction by the insight that theglobal structure of the cosmos is dominated at 95 % by an energy form thathas hitherto been entirely unknown. About two thirds of this energy formconsist in “dark energy”, and one third in “dark matter”. This is the mostradical revolution in our understanding of the cosmos after Kopernikus. Inthe last years, great effort has been invested to understand these unknownforms of energy [14, 13, 17]. Many explanations of dark energy assume thatbesides spacetime geometry and baryonic matter, there is an additional en-tity that acts as source of the dark energy. For instance, particle-theoretic ∗ Presented at the conference “Matter to the deepest” Ustron 2007 † [email protected] ‡ rose@first.fhg.de (1) dark-energy-3MF printed on November 8, 2018 models attribute this role to the vacuum energy [24, 20, 3], or introduceadditional global scalar fields [21, 16].A first evaluation of the WMAP data favors a Poincar´e sphere as topol-ogy of the cosmos [11]. That means that the cosmos is a closed 3-manifoldhaving the same homology as the 3-sphere. Furthermore one knows thatthis 3-manifold has a positive curvature. In this paper we support this resultand show that the dark energy can be explained by the curvature of twoPoincar´e spheres. Details of the calculation can be found in the expandedversion of the paper [1].
2. Basic Model
Our model based on the fact from general relativity theory that everyform of energy is related to the curvature of the spatial 3-manifold, i.e.matter must be interpreted by curved 3-manifolds. Then Einstein’s equationis the dynamical equation for the evolution of 3-geometries. Thus we canstate our basic assumptions:
Basic assumptions : The 4-manifold of all possible spacetime events is acompact, closed 4-manifold M which is differentiable and simply connected.The cosmos is an embedded 3-manifold Σ which is compact and closed.The energy density of any kind of matter is described by the curvature ofthe associated submanifold of Σ.Before we study the implications we will motivate these assumptions.Compactness of the 4-manifolds means that every serie of spacetime eventsconverges to an event belonging to the same 4-manifold. The manifold isclosed i.e. it has no boundary or in any neighborhood of an arbitrary pointthere is always inner points of the manifold. The assumptions of compact-ness and closedness can be interpreted that all points of the manifold areinner points and any spacetime event must be part of the manifold. Thatassumption seems natural from our knowledge of space and time. The as-sumption of simple-connectness is more delicate. It means that any closed(time-like) curve is contractable i.e. any time circle contradicting causalitycan be shrunk to a point. In the following we will study the implications viadifferential topology of the assumptions leading to very strong restrictionson the possible 4- and 3-manifolds. The 4-manifold can be determined by the following argumentation. Ifevery kind of energy is described by the curvature of some submanifold of the Compactness is not contradictory to the possible infiniteness of proper time for worldlines. Curves in manifolds can have infinite length like Peanos curve. ark-energy-3MF printed on November 8, 2018 spatial 3-manifold Σ then there are no source terms in Einstein’s equation.Every kind of energy including matter must be given by geometry. ThenEinstein’s equation R µν = 0 is the statement that the 4-manifold mustbe Ricci-flat. But there is only one Ricci-flat compact, simply-connected4-manifold, the K3-surface [23] which is defined by { ( z , z , z , z ) ∈ ICP | z + z + z + z = 0 } . Thus we determine the topology of the 4-manifold by our basic assumptions.The second restriction, which will determine the structure of the 3-manifold,is the choice of a particular differential structure. It is a well-known result [7]that the differential or smooth structure of a compact, simply-connected 4-manifold is determined by a contractable submanifold, the Akbulut cork A .The boundary of the Akbulut cork is a 3-manifold which is a so-called homology 3-sphere , i.e. a 3-manifold with the same homology as the 3-sphere S . Thus we assume the cosmos Σ is a homology 3-sphere . In case of the K3-surface we know the structure of the Akbulut cork andits boundary Σ = Σ(2 , ,
7) [7] the Brieskorn sphereΣ(2 , ,
7) = { ( x, y, z ) ∈ IC | | x | + | y | + | z | = 1 , x + y + z = 0 } . From the structure theory of 3-manifolds [9, 19] we know that there are only three kinds of 3-manifolds that can form Σ. In the particular case we obtainΣ = K K K N S S /I ∗ S /I ∗ . (1)with S /I ∗ rep-resents the Poincar´e sphere which forms the global structure of the cosmos[11]. Now we identify the pieces with1. negatively curved pieces K i (matter, radiation)2. positively curved 3-spheres S (dark matter)3. two positively curved Poincar´e spheres S /I ∗ (dark energy),The details can be found in the expanded paper [1]. This remarkable factmotivates the following Conjecture:
The three types of 3-manifolds that constitute the cosmosas a homology 3-sphere, correspond to the three kinds of matter: baryonicmatter, dark matter, and dark energy.
Thus we obtain a unified approach for all observed kinds of energy den-sities. The global structure of the cosmos can thus be derived from the dark-energy-3MF printed on November 8, 2018 differential geometry of spacetime itself, without additional entities, and itis possible to compare the observed energy densities with the curvatures ofthe three types of Σ. In the next section we will calculate the ratio of theenergy densities of the dark energy and the total energy by using a resultof Witten [22].
3. Calculation of the dark energy density
The investigation of the global characteristics of the differential structureof spacetime led us to the result that the cosmos is a Brieskorn sphereΣ = Σ(2 , ,
7) split up into three types of piecesΣ = K K K N S S /I ∗ S /I ∗ . Let us now suppose a 4-manifold M with an Akbulut cork bounded by Σ.The metric g µν of M is given by the Einstein equation R µν − g µν R = Λ g µν + 8 πGc · T µν (2)with the cosmological constant Λ and the energy-momentum tensor T µν .The cosmological constant Λ = 8 πGc ρ D (3)corresponds to the energy density ρ D = E D /vol (Σ) of the dark energy. Inthe previous section we showed that one can identify the dark energy withthe curvature of two Poincar´e spheres Σ D = S /I ∗ S /I ∗ . Let R be thescalar curvature of the cosmos Σ. Inserting the Robertson-Walker-metric ds = c dt − a ( t ) h ik dx i dx j (4)with Σ × [0 ,
1] and the scaling function a ( t ) in (2), one gets the Friedmanequation (cid:18) ˙ a ( t ) c · a ( t ) (cid:19) + ka ( t ) = 8 πGc ρ πGc ( ρ + ρ D )3with curvature k = 0 , ±
1. Then the scalar curvature of the cosmos Σ is R = 3 k/a ( t ) and the Hubble constant is given by ˙ a ( t ) /a ( t ) = H . Thus,the relation of the total density ρ and scalar curvature R of the cosmos isgiven by ρ = c πG R + 3 H c πG . (5) ark-energy-3MF printed on November 8, 2018 Using homogeneity and isotropy of the matter distribution we get by inte-gration ρ = c πG R Σ R √ hd xvol (Σ) + ρ C , ρ C = 3 H c πG (6)with the critical density ρ C . Replacing Σ by Σ D we obtain in an analogway the dark energy density ρ D = c πG R Σ D ( R D + R C ) √ hd xvol (Σ D ) , R C = 3 H c . (7)The main step of the calculation is to solve the integral, i.e. the Einstein-Hilbert action of the dark energy defined on Σ D S EH (Σ D ) = Z Σ D ( R D + R C ) √ hd x . Witten has discussed the 3-dimensional Einstein-Hilbert action in more de-tail [22]. He was able to derive the important result that S EH is related toa pure topological property – the Chern-Simons invariant of the manifold.Then one gets the simple relation between the Chern-Simons invariant of a SU (2) connection A and the Einstein-Hilbert action (see [1] for details) S EH (Σ D ) = 16 π (1 − R C / · CS ( A, Σ D ) . Using this result we are able to calculate the ratio of the energy density ofdark matter (7) and the total density (6) of the cosmos Σ yielding ρ D ρ = CS ( A, Σ D ) CS ( A, Σ) · vol (Σ) vol (Σ D ) . With the dark energy part Σ D = S /I ∗ S /I ∗ by using CS ( A, M M ) = CS ( A, M ) + CS ( A, M ) we get ρ D ρ = vol (Σ) vol (Σ D ) · CS ( A, S /I ∗ ) CS ( A, Σ) . (8)By homogeneity, the density ρ | Σ D = E D /vol (Σ D ) restricted to the subsetΣ D has to be equal to the energy density ρ = E/vol (Σ) on the wholemanifold Σ, i.e. E D /vol (Σ D ) = E/vol (Σ). With
E/E D = ρ/ρ D it yields to ρ/ρ D = vol (Σ) /vol (Σ D ). Inserting in (8) we obtain the dark energy fraction ρ D ρ = s CS ( A, S /I ∗ ) CS ( A, Σ) (9) dark-energy-3MF printed on November 8, 2018 which is a purely topological invariant .To give an explicit expression we need the Chern-Simons invariants ofthe Poincar´e sphere S /I ∗ and the Brieskorn sphere Σ = Σ(2 , , minimal Chern-Simons invariant τ (Σ) of a homology 3-sphere τ ( S /I ∗ ) = 1120 , (10) τ (Σ(2 , , . (11)This invariant corresponds to the self-dual or anti-self-dual solutions of a SU (2) gauge theory on Σ × IR . This minimization principle permits anunique determination of the Chern-Simons invariants and we obtain for thedark energy fraction ρ D ρ = s τ ( S /I ∗ ) τ (Σ(2 , , ρ D ρ = vuut (2 · )( ) = r ≈ . . (13)Inserting the observed total energy density ρ obs = (1 . ± . · ρ C fromthe WMAP data [18] we obtain for the dark energy density Ω D = ρ D ρ C = 0 . ± . , (14)with the critical density ρ C = 3 H c / πG and the Hubble-constant H .Our calculated value (14) fits very well with the currently observed data. Inparticular, the fit CMB+2dFGRS+BBN ([4] Table 1) yields (Ω D ) obs = 0 . obs = 1 . D = p /
27 Ω obs = 0 . H ) obs = (72 ± kms /M pc we obtainfor the cosmological constantΛ = q /
27 3( H ) obs Ω obs c ≈ (1 . ± . · − m − We would like to emphasize that our approach deeply requires a positivecurvature of the cosmos, i.e. Ω >
1, because our proposed topology ofthe cosmos – the Brieskorn sphere – is a closed 3-manifold with positivecurvature. This provides a strong possibility for falsification and should bedeterminable by future observations. ark-energy-3MF printed on November 8, 2018
4. Discussion
In the paper we show how to calculate the ratio of the dark energydensity to the total energy density of the cosmos. Although we don’t use aquantum-gravitational argumentation for the calculation, the appearance ofthe Chern-Simons invariant indicates a possible relation to quantum gravity.The exponential of the Chern-Simons invariant (Kodama state) is a wavefunction in Loop quantum gravity which can be used for cosmology. In afollow-up paper we shall deal with this important battery of questions inmore detail. REFERENCES [1] T. Asselmeyer-Maluga and H. Ros´e. Calculation of the cosmological constantby unifying matter and dark energy. available as gr-qc/0609004.[2] C.L. Bennett et al.
Astrophys. J. Suppl. , :1, 2003.[3] S.M. Carroll. Living Rev. Relativity , :1, 2001.[4] G. Efstathiou et al. Mon.Not.Roy.Astron.Soc. , :L29, 2002.[5] R. Fintushel and R.J. Stern. Proc. London Math. Soc. , :109, 1990.[6] D.S. Freed and R.E. Gompf. Comm. Math. Phys. , :79, 1991.[7] R.E. Gompf and A.I. Stipsicz. . AmericanMathematical Society, 1999.[8] G. Hinshaw et al. Astrophys. J. Suppl. , :135, 2003.[9] W. Jaco and P. Shalen. Seifert fibered spaces in 3-manifolds , volume of Mem. Amer. Math. Soc.
AMS, 1979.[10] P. Kirk and E. Klassen.
Math. Ann. , :343, 1990.[11] J.-P. Luminet et al. Nature , :593, 2003.[12] S. Nobili et al. Astronomy and Astrophysics , :789, 2005.[13] T. Padmanabhan. Phys. Rept. , :235, 2003.[14] P.J.E. Peebles and B. Ratra. Rev. Mod. Phys. , :559, 2003.[15] S. Perlmutter et al. Astrophys. J. , :565, 1999.[16] B. Ratra and P.J.E. Peebles. Phys. Rev. D , :3406, 1988.[17] V. Sahni. Lect. Notes Phys. , :141, 2004.[18] D.N. Spergel et al. Astrophys. J. Suppl. , :175, 2003.[19] W. Thurston. Three-Dimensional Geometry and Topology . Princeton Univer-sity Press, Princeton, first edition, 1997.[20] S. Weinberg.
Rev.Mod.Phys. , :1, 1989.[21] C. Wetterich. Nucl. Phys. , B302 :668, 1988.[22] E. Witten.
Nucl. Phys. , B311 :46, 1988/89.[23] S.T. Yau.
Proc. Nat. Acad. Sci. USA , :1748, 1977.[24] Y.B. Zel’dovich. JETP letters ,6