Inflation and Topological Phase Transition Driven by Exotic Smoothness

Abstract

We will discuss a model which describes the cause of inflation by a topological transition. The guiding principle is the choice of an exotic smoothness structure for the space-time. Here we consider a space-time with topology 𝑆³×ℝ. In case of an exotic 𝑆³×ℝ, there is a change in the spatial topology from a 3-sphere to a homology 3-sphere which can carry a hyperbolic structure. From the physical point of view, we will discuss the path integral for the Einstein-Hilbert action with respect to a decomposition of the space-time. The inclusion of the boundary terms produces fermionic contributions to the partition function. The expectation value of an area (with respect to some surface) shows an exponential increase; that is, we obtain inflationary behavior. We will calculate the amount of this increase to be a topological invariant. Then we will describe this transition by an effective model, the Starobinski or 𝑅² model which is consistent with the current measurement of the Planck satellite. The spectral index and other observables are also calculated.

1. Introduction

General relativity (GR) has changed our understanding of space-time. In parallel, the appearance of quantum field theory (QFT) has modified our view of particles, fields, and the measurement process. The usual approach for the unification of QFT and GR, to a quantum gravity, starts with a proposal to quantize GR and its underlying structure, space-time. There is a unique opinion in the community about the relation between geometry and quantum theory: the geometry as used in GR is classical and should emerge from a quantum gravity in the limit (Planck’s constant tends to zero). Most theories went a step further and try to get a space-time from quantum theory. Then, the model of a smooth manifold is not suitable to describe quantum gravity. But, there is no sign for a discrete space-time structure or higher dimensions in current experiments. Hence, quantum gravity based on the concept of a smooth manifold should also able to explain the current problems in the standard cosmological model (ΛCDM) like the appearance of dark energy/matter or the correct form of inflation. But before we are going in this direction we will motivate the usage of the smooth manifold as our basic concept.

When Einstein developed GR, his opinion about the importance of general covariance changed over the years. In 1914, he wrote a joint paper with Grossmann. There, he rejected general covariance by the now famous hole argument. But after a painful year, he again considered general covariance now with the insight that there is no meaning in referring to the space-time point A or the event A, without further specifications. Therefore the measurement of a point without a detailed specification of the whole measurement process is meaningless in GR. The reason is simply the diffeomorphism invariance of GR which has tremendous consequences. Furthermore, GR do not depend on the topology of space-time. All restrictions on the topology of the space-time were formulated using additional physical conditions like causality (see [1]). This ambiguity increases in the 80's when the first examples of exotic smoothness structures in dimension 4 were found. The (smooth) atlas of a smooth 4-manifold 𝑀 is called the smoothness structure (unique up to diffeomorphisms). One would expect that there is only one smooth atlas for 𝑀; all other possibilities can be transformed into each other by a diffeomorphism. But in contrast, the deep results of Freedman [2] on the topology of 4-manifolds combined with Donaldson’s work [3] gave the first examples of nondiffeomorphic smoothness structures on 4-manifolds including the well-known ℝ⁴. Much of the motivation can be found in the FQXI essay [4, 5]. Here we will discuss another property of the exotic smoothness structure: its quantum geometry in the path integral.

Diffeomorphism invariance is the most important property of the Einstein-Hilbert action with far reaching consequences [6]. One of our results is a close relation between geometry and foliation to exotic smoothness [7, 8]. In the particular example of the exotic ℝ⁴, we discussed the exotic smoothness structures as a manifestation of quantum gravity (by using string theory [9, 10]). This exotic ℝ⁴ has some interesting properties as first noted by Brans [11, 12]. More importantly as shown by Sładkowski [13], the exotic ℝ⁴ has a nontrivial curvature in contrast to the flat standard ℝ⁴. It was the first result that an exotic ℝ⁴ can be seen as a source of gravity (or it must contain sources of gravity). Sładkowski [14–16] went further and showed a relation to particle physics also related to quantum gravity. But why there is a relation to quantum gravity? In [17] we presented the first idea to understand this relation which was further extended in [18]. An exotic 4-manifold like 𝑆³×ℝ is also characterized by the property that there is no smoothly embedded 3-sphere but a topological embedded one. This topological 𝑆³ is wildly embedded; that is, the image of the embedding must be triangulated by an infinite polyhedron. In [18], we proved that the (deformation) quantization of a usual (or tame) embedding is a wild embedding which can be seen as a quantum state. But then any exotic 4-manifold can be interpreted as a quantum state of the 4-manifold with standard smoothness structure. From this point of view, the calculation of the path integral in quantum gravity has to include the exotic smoothness structures. Usually it is hopeless to make these calculations. But by using the close relation of exotic smoothness to hyperbolic geometry, one has a chance to calculate geometric expressions like the expectation value of the surface area. In this paper we will show that this expectation value has an inflationary behavior; that is, the area grows exponentially (along the time axis). Therefore quantum gravity (in the sense of exotic smoothness) can be the root of inflation.

2. Space-Time and Smoothness

From the mathematical point of view, the space-time is a smooth 4-manifold endowed with a (smooth) metric as basic variable for general relativity. The existence question for Lorentz structure and causality problems (see Hawking and Ellis [1]) give further restrictions on the 4-manifold: causality implies noncompactness; Lorentz structure needs a nonvanishing normal vector field. Both concepts can be combined in the concept of a global hyperbolic 4-manifold 𝑀 having a Cauchy surface 𝒮 so that 𝑀=𝒮×ℝ.

All these restrictions on the representation of space-time by the manifold concept are clearly motivated by physical questions. Among these properties there is one distinguished element: the smoothness. Usually one starts with a topological 4-manifold 𝑀 and introduces structures on them. Then one has the following ladder of possible structures: Topology⟶piecewise-linear(PL)⟶Smoothness⟶bundles,Lorentz,Spin,etc.⟶metric,geometry,…. (1) We do not want to discuss the first transition, that is, the existence of a triangulation on a topological manifold. But we remark that the existence of a PL structure implies uniquely a smoothness structure in all dimensions smaller than 7 [19]. Here we have to consider the following steps to define a space-time.(i)Fix a topology for the space-time 𝑀.(ii)Fix a smoothness structure, that is, a maximal differentiable atlas 𝒜.(iii)Fix a smooth metric or get one by solving the Einstein equation.The choice of a topology never fixes the space-time uniquely; that is, there are two space-times with the same topology which are not diffeomorphic. The main idea of the paper is the introduction of exotic smoothness structures into space-time. If two manifolds are homeomorphic but nondiffeomorphic, they are exotic to each other. The smoothness structure is called an exotic smoothness structure.

In dimension four there are many examples of compact 4-manifolds with countable infinite nondiffeomorphic smoothness structures and many examples of noncompact 4-manifolds with uncountable infinite many nondiffeomorphic smoothness structures. But in contrast, the number of nondiffeomorphic smoothness structures is finite for any other dimension [19]. As an example, we will consider the space-time 𝑆³×ℝ having uncountable many nondiffeomorphic smoothness structures in the following.

3. The Path Integral in Exotic 𝑆³×ℝ

For simplicity, we consider general relativity without matter (using the notation of topological QFT). Space-time is a smooth oriented 4-manifold 𝑀 which is noncompact and without boundary. From the formal point of view (no divergences of the metric) one is able to define a boundary 𝜕𝑀 at infinity. The classical theory is the study of the existence and uniqueness of (smooth) metric tensors 𝑔 on 𝑀 that satisfy the Einstein equations subject to suitable boundary conditions. In the first order Hilbert-Palatini formulation, one specifies an SO(1,3)-connection 𝐴 together with a cotetrad field 𝑒 rather than a metric tensor. Fixing 𝐴∣_𝜕𝑀 at the boundary, one can derive first-order field equations in the interior (now called bulk) which are equivalent to the Einstein equations provided that the cotetrad is nondegenerate. The theory is invariant under space-time diffeomorphisms 𝑀→𝑀. In the particular case of the space-time 𝑀=𝑆³×ℝ (topologically), we have to consider a smooth 4-manifolds 𝑀_𝑖,𝑓 as parts of 𝑀 whose boundary 𝜕𝑀_𝑖,𝑓=Σ_𝑖⊔Σ_𝑓 is the disjoint union of two smooth 3-manifolds Σ_𝑖 and Σ_𝑓 to which we associate Hilbert spaces ℋ_𝑗 of 3-geometries, 𝑗=𝑖,𝑓. These contain suitable wave functionals of connections 𝐴∣_Σ𝑗. We denote the connection eigenstates by |𝐴|_Σ𝑗⟩. The path integral, 𝐴|_Σ𝑓𝑇_𝑀𝐴|_Σ𝑖=_{𝐴∣𝜕𝑀𝑖,𝑓}𝑖𝐷𝐴𝐷𝑒expℏ𝑆_𝐸𝐻𝑒,𝐴,𝑀_𝑖,𝑓, (2) is the sum over all connections 𝐴 matching 𝐴∣_{𝜕𝑀𝑖,𝑓} and over all 𝑒. It yields the matrix elements of a linear map 𝑇_𝑀∶ℋ_𝑖→ℋ_𝑓 between states of 3-geometry. Our basic gravitational variables will be cotetrad 𝑒^𝐼_𝑎 and connection 𝐴_𝑎^𝐼𝐽 on space-time 𝑀 with the index 𝑎 to present it as 1-forms and the indices 𝐼,𝐽 for an internal vector space 𝑉 (used for the representation of the symmetry group). Cotetrads 𝑒 are “square-roots” of metrics and the transition from metrics to tetrads is motivated by the fact that tetrads are essential if one is to introduce spinorial matter. 𝑒^𝐼_𝑎 is an isomorphism between the tangent space 𝑇_𝑝(𝑀) at any point 𝑝 and a fixed internal vector space 𝑉 equipped with a metric 𝜂_𝐼𝐽 so that 𝑔_𝑎𝑏=𝑒^𝐼_𝑎𝑒^𝐽_𝑏𝜂_𝐼𝐽. Here we used the action 𝑆_𝐸𝐻𝑒,𝐴,𝑀_𝑖,𝑓,𝜕𝑀_𝑖,𝑓=_{𝑀𝑖,𝑓}𝜖_𝐼𝐽KL𝑒^𝐼∧𝑒^𝐽∧(𝑑𝐴+𝐴∧𝐴)^KL+_{𝜕𝑀𝑖,𝑓}𝜖_𝐼𝐽KL𝑒^𝐼∧𝑒^𝐽∧𝐴^KL, (3) in the notation of [20, 21]. Here the boundary term 𝜖_𝐼𝐽KL(𝑒^𝐼∧𝑒^𝐽∧𝐴^KL) is equal to twice the trace over the extrinsic curvature (or the mean curvature). For fixed boundary data, (2) is a diffeomorphism invariant in the bulk. If Σ_𝑖=Σ_𝑓 are diffeomorphic, we can identify Σ=Σ_𝑖=Σ_𝑓 and ℋ=ℋ_𝑖=ℋ_𝑓; that is, we close the manifold 𝑀_𝑖,𝑓 by identifying the two boundaries to get the closed 4-manifold 𝑀^′. Provided that the trace over ℋ can be defined, the partition function, 𝑍𝑀^′=tr_ℋ𝑇_𝑀=𝑖𝐷𝐴𝐷𝑒expℏ𝑆_𝐸𝐻[𝑒,𝐴,𝑀,𝜕𝑀], (4) where the integral is now unrestricted, is a dimensionless number which depends only on the diffeomorphism class of the smooth manifold 𝑀^′. In case of the manifold 𝑀_𝑖,𝑓, the path integral (as transition amplitude) ⟨𝐴|_Σ𝑓|𝑇_𝑀|𝐴|_Σ𝑖⟩ is the diffeomorphism class of the smooth manifold relative to the boundary. But the diffeomorphism class of the boundary is unique and the value of the path integral depends on the topology of the boundary as well on the diffeomorphism class of the interior of 𝑀_𝑖,𝑓. Therefore we will shortly write Σ_𝑓𝑇_𝑀Σ_𝑖=𝐴|_Σ𝑓𝑇_𝑀𝐴|_Σ𝑖 (5) and consider the sum of manifolds like 𝑀_𝑖,ℎ=𝑀_𝑖,𝑓∪_Σ𝑓𝑀_𝑓,ℎ with the amplitudes Σ_ℎ𝑇_𝑀Σ_𝑖=_{𝐴∣Σ𝑓}Σ_ℎ𝑇_𝑀Σ_𝑓Σ_𝑓𝑇_𝑀Σ_𝑖, (6) where we sum (or integrate) over the connections and frames on Σ_ℎ (see [22]). Then the boundary term 𝑆_𝜕Σ_𝑓=_Σ𝑓𝜖_𝐼𝐽KL𝑒^𝐼∧𝑒^𝐽∧𝐴^KL=_Σ𝑓𝐻√ℎ𝑑³𝑥 (7) is needed where 𝐻 is the mean curvature of Σ_𝑓 corresponding to the metric ℎ at Σ_𝑓 (as restriction of the 4-metric). Therefore we have to divide the path integration into two parts: the contribution by the boundary (boundary integration) and the contribution by the interior (bulk integration).

3.1. Boundary Integration

The boundary Σ of a 4-manifold 𝑀 can be understood as embedding (or at least as immersion). Let 𝜄∶Σ↪𝑀 be an immersion of the 3-manifold Σ into the 4-manifold 𝑀 with the normal vector 𝑁. The spin bundle 𝑆_𝑀 of the 4-manifold splits into two subbundles 𝑆^±_𝑀, where one subbundle, say 𝑆⁺_𝑀, can be related to the spin bundle 𝑆_Σ of the 3-manifold. Then the spin bundles are related by 𝑆_Σ=𝜄^∗𝑆⁺_𝑀 with the same relation 𝜙=𝜄_∗Φ for the spinors (𝜙∈Γ(𝑆_Σ) and Φ∈Γ(𝑆⁺_𝑀)). Let ∇^𝑀_𝑋,∇^Σ_𝑋 be the covariant derivatives in the spin bundles along a vector field 𝑋 as section of the bundle 𝑇Σ. Then we have the formula ∇^𝑀_𝑋(Φ)=∇^Σ_𝑋1𝜙−2∇_𝑋𝑁⋅𝑁⋅𝜙, (8) with the obvious embedding 𝜙↦^𝜙₀=Φ of the spinor spaces. The expression ∇_𝑋𝑁 is the second fundamental form of the immersion, where the trace tr(∇_𝑋𝑁)=2𝐻 is related to the mean curvature 𝐻. Then from (8) one obtains a similar relation between the corresponding Dirac operators 𝐷^𝑀Φ=𝐷^3𝐷𝜙−𝐻𝜙, (9) with the Dirac operator 𝐷^3𝐷 of the 3-manifold Σ. Near the boundary Σ, the 4-manifolds looks like Σ×[0,1] and a spinor Φ on this 4-manifold is a parallel spinor and has to fulfill the following equation: 𝐷^𝑀Φ=0; (10) that is, 𝜙 yields the eigenvalue equation 𝐷^3𝐷𝜙=𝐻𝜙, (11) with the mean curvature 𝐻 of the embedding 𝜄 as eigenvalue. See our previous work [23] for more details.

Now we will use this theory to get rid of the boundary integration. At first we will discuss the deformation of an immersion using a diffeomorphism. Let 𝐼∶Σ↪𝑀 be an immersion of Σ (3-manifold) into 𝑀 (4-manifold). A deformation of an immersion 𝐼^′∶Σ^′↪𝑀^′ is diffeomorphisms 𝑓∶𝑀→𝑀^′ and 𝑔∶Σ→Σ^′ of 𝑀 and Σ, respectively, so that 𝑓∘𝐼=𝐼^′∘𝑔. (12)

One of the diffeomorphisms (say 𝑓) can be absorbed into the definition of the immersion and we are left with one diffeomorphism 𝑔∈Diff(Σ) to define the deformation of the immersion 𝐼. But as stated above, the immersion is directly given by an integral over the spinor 𝜙 on Σ fulfilling the Dirac equation (11). Therefore we have to discuss the action of the diffeomorphism group Diff(Σ) on the Hilbert space of 𝐿²-spinors fulfilling the Dirac equation. This case was considered in the literature [24]. The spinor space 𝑆_𝑔,𝜎(Σ) on Σ depends on two ingredients: a (Riemannian) metric 𝑔 and a spin structure 𝜎 (labeled by the number of elements in 𝐻¹(Σ,ℤ₂)). Let us consider the group of orientation preserving diffeomorphism Diff⁺(Σ) acting on 𝑔 (by pullback 𝑓^∗𝑔) and on 𝜎 (by a suitable defined pullback 𝑓^∗𝜎). The Hilbert space of 𝐿²-spinors of 𝑆_𝑔,𝜎(Σ) is denoted by 𝐻_𝑔,𝜎. Then according to [24], any 𝑓∈Diff⁺(Σ) leads in exactly two ways to a unitary operator 𝑈 from 𝐻_𝑔,𝜎 to 𝐻_𝑓^∗_𝑔,𝑓^∗_𝜎. The (canonically) defined Dirac operator is equivariant with respect to the action of 𝑈 and the spectrum is invariant under (orientation preserving) diffeomorphisms. In particular we obtain for the boundary term 𝑆_𝜕Σ_𝑓,ℎ=_Σ𝑓𝐻√ℎ𝑑³𝑥=_Σ𝑓𝜙𝐷^3𝐷𝜙𝑑³𝑥, (13) with |𝜙|²=const. (see [25]). But then we can change the integration process from the integration over the metric class ℎ on the 3-manifold Σ_𝑓 with mean curvature to an integration over the spinor 𝜙 on Σ_𝑓. Then we obtain 𝑍Σ_𝑓=𝑖𝐷ℎexpℏ𝑆Σ_𝑓=,ℎ𝐷𝜙𝐷𝑖𝜙expℏ_Σ𝑓𝜙𝐷^3𝐷𝜙𝑑³𝑥=det𝐷^3𝐷𝐷^∗3𝐷𝑒^{𝑖𝜋𝜂(Σ𝑓)/2}, (14) where 𝜂(Σ_𝑓) is the Eta invariant of the Dirac operator at the 3-manifold Σ_𝑓 (here we use a result of Witten; see [26]).

From the physical point of view, we obtain fermions at the boundary. The additional term with the Eta invariant reflects also an important fact. The state space of general relativity is the space of the (Lorentzian) metric tensor up to the group of coordinate transformations. This group of coordinate transformations is not the full diffeomorphism group; it is only one connected component of the diffeomorphism. That is the group of diffeomorphisms connected to the identity. In addition, there is also the (discrete) group of global diffeomorphisms which is in our case detected by the Eta invariant. For 3-manifolds there is a deep relation to the Chern-Simons invariant [27] which will be further studied at our forthcoming work.

3.2. Bulk Integration

Now we will discuss the path integral of the action 𝑆_𝐸𝐻[𝑒,𝐴,𝑀]=_𝑀𝜖_𝐼𝐽KL𝑒^𝐼∧𝑒^𝐽∧(𝑑𝐴+𝐴∧𝐴)^KL (15) in the interior of the 4-manifold 𝑀. The contribution of the boundary was calculated in the previous subsection. In the (formal) path integral (2) we will ignore all problems (ill-definiteness, singularities, etc.) of the path integral approach. Next we have to discuss the measure 𝐷𝑒 of the path integral. Currently there is no rigorous definition of this measure and as usual we assume a product measure.

Then we have two possible parts which are more or less independent from each other:(i)integration 𝐷𝑒_𝐺 over geometries;(ii)integration 𝐷𝑒_𝐷𝑆 over different differential structures parametrized by some structure (see below).

Now we have to consider the following path integral: 𝑍(𝑀)=_{Diffstructures}𝐷𝑒_{𝐷𝑆Geometries}𝐷𝑒_𝐺𝑖expℏ𝑆_𝐸𝐻[𝑒,𝑀], (16) and we have to calculate the influence of the differential structures first. At this level we need an example, an exotic 𝑆³×ℝ.

3.3. Constructing Exotic 𝑆³×ℝ

In [28], Freedman constructed the first example of an exotic 𝑆³×ℝ of special type. There are also uncountable many different exotic ℝ⁴ having an end homeomorphic to 𝑆³×ℝ but not diffeomorphic to it. But Freedman’s first example is not of this type (as an end of an exotic ℝ⁴). Therefore to get an infinite number of different exotic 𝑆³×ℝ one has to see 𝑆³×ℝ as an end of ℝ⁴ also expressible as complement ℝ⁴⧵𝐷⁴ of the 4-disk. A second possibility is the usage of the end-sum technique of Gompf, so that the standard 𝑆³×ℝ can be transformed into an exotic 𝑆³×ℝ by end-sum with an exotic ℝ⁴. Here we will concentrate on the first construction; that is, the exotic 𝑆³×ℝ is an end of an exotic ℝ⁴.

Furthermore we will restrict on a subclass of exotic ℝ⁴ called small exotic ℝ⁴ (exotic ℝ⁴ which can be embedded in a 4-sphere 𝑆⁴). For this class there is an explicit handle decomposition. Small exotic ℝ⁴’s are the result of an anomalous behavior in 4-dimensional topology. In 4-manifold topology [2], a homotopy equivalence between two compact, closed, simply connected 4-manifolds implies a homeomorphism between them (the so-called h cobordism). But Donaldson [29] provided the first smooth counterexample that this homeomorphism is not a diffeomorphism; that is, both manifolds are generally not diffeomorphic to each other. The failure can be localized at some contractible submanifold (Akbulut cork) so that an open neighborhood of this submanifold is a small exotic ℝ⁴. The whole procedure implies that this exotic ℝ⁴ can be embedded in the 4-sphere 𝑆⁴. The idea of the construction is simply given by the fact that every smooth h-cobordism between nondiffeomorphic 4-manifolds can be written as a product cobordism except for a compact contractible sub-h-cobordism 𝑉, the Akbulut cork. An open subset 𝑈⊂𝑉 homeomorphic to [0,1]×ℝ⁴ is the corresponding sub-h-cobordism between two exotic ℝ⁴’s. These exotic ℝ⁴’s are called ribbon ℝ⁴’s. They have the important property of being diffeomorphic to open subsets of the standard ℝ⁴. In [30] Freedman and DeMichelis constructed also a continuous family of small exotic ℝ⁴. Now we are ready to discuss the decomposition of a small exotic ℝ⁴ by Bižaca and Gompf [31] using special pieces, the handles forming a handle body. Every 4-manifold can be decomposed (seen as handle body) using standard pieces such as 𝐷^𝑘×𝐷^4−𝑘, the so-called 𝑘-handle attached along 𝜕𝐷^𝑘×𝐷^4−𝑘 to the boundary 𝑆³=𝜕𝐷⁴ of a 0-handle 𝐷⁰×𝐷⁴=𝐷⁴. The construction of the handle body for the small exotic ℝ⁴, called 𝑅⁴ in the following, can be divided into two parts: 𝑅⁴=𝐴_cork𝐷2CHdecompositionofsmallexoticℝ⁴. (17) The first part is known as the Akbulut cork, a contractible 4-manifold with boundary a homology 3-sphere (a 3-manifold with the same homology as the 3-sphere). The Akbulut cork 𝐴_cork is given by a linking between a 1-handle and a 2-handle of framing 0. The second part is the Casson handle CH which will be considered now.

Let us start with the basic construction of the Casson handle CH. Let 𝑀 be a smooth, compact, simple-connected 4-manifold and 𝑓∶𝐷²→𝑀 a (codimension-2) mapping. By using diffeomorphisms of 𝐷² and 𝑀, one can deform the mapping 𝑓 to get an immersion (i.e., injective differential) generically with only double points (i.e., #|𝑓⁻¹(𝑓(𝑥))|=2) as singularities [32]. But to incorporate the generic location of the disk, one is rather interesting in the mapping of a 2-handle 𝐷²×𝐷² induced by 𝑓×𝑖𝑑∶𝐷²×𝐷²→𝑀 from 𝑓. Then every double point (or self-intersection) of 𝑓(𝐷²) leads to self-plumbings of the 2-handle 𝐷²×𝐷². A self-plumbing is an identification of 𝐷²₀×𝐷² with 𝐷²₁×𝐷², where 𝐷²₀,𝐷²₁⊂𝐷² are disjoint subdisks of the first factor disk. In complex coordinates the plumbing may be written as (𝑧,𝑤)↦(𝑤,𝑧) or (𝑧,𝑤)↦(𝑤,𝑧) creating either a positive or negative (resp.,) double point on the disk 𝐷²×0 (the core). Consider the pair (𝐷²×𝐷²,𝜕𝐷²×𝐷²) and produce finitely many self-plumbings away from the attaching region 𝜕𝐷²×𝐷² to get a kinky handle (𝑘,𝜕⁻𝑘), where 𝜕⁻𝑘 denotes the attaching region of the kinky handle. A kinky handle (𝑘,𝜕⁻𝑘) is a one-stage tower (𝑇₁,𝜕⁻𝑇₁) and an (𝑛+1)-stage tower (𝑇_𝑛+1,𝜕⁻𝑇_𝑛+1) is an 𝑛-stage tower union kinky handles ∪^𝑛_ℓ=1(𝑇_ℓ,𝜕⁻𝑇_ℓ), where two towers are attached along 𝜕⁻𝑇_ℓ. Let 𝑇⁻_𝑛 be (interior𝑇_𝑛)∪𝜕⁻𝑇_𝑛 and the Casson handle, CH=_ℓ=0𝑇⁻_ℓ, (18) is the union of towers (with direct limit topology induced from the inclusions 𝑇_𝑛↪𝑇_𝑛+1).

The main idea of the construction above is very simple: an immersed disk (disk with self-intersections) can be deformed into an embedded disk (disk without self-intersections) by sliding one part of the disk along another (embedded) disk to kill the self-intersections. Unfortunately the other disk can be immersed only. But the immersion can be deformed to an embedding by a disk again and so forth. In the limit of this process one “shifts the self-intersections into infinity” and obtains the standard open 2-handle (𝐷²×ℝ²,𝜕𝐷²×ℝ²). In the proof of Freedman [2], the main complications come from the lack of control about this process.

A Casson handle is specified up to (orientation preserving) diffeomorphism (of pairs) by a labeled finitely branching tree with base-point *, having all edge paths infinitely extendable away from *. Each edge should be given a label + or −. Here is the construction: tree →CH. Each vertex corresponds to a kinky handle; the self-plumbing number of that kinky handle equals the number of branches leaving the vertex. The sign on each branch corresponds to the sign of the associated self-plumbing. The whole process generates a tree with infinitely many levels. In principle, every tree with a finite number of branches per level realizes a corresponding Casson handle. Each building block of a Casson handle, the “kinky” handle with 𝑛 kinks, is diffeomorphic to the 𝑛-times boundary connected sum ♮_𝑛(𝑆¹×𝐷³) (see Appendix A) with two attaching regions. The number of end connected sums is exactly the number of self-intersections of the immersed two handle. One region is a tubular neighborhood of band sums of Whitehead links connected with the previous block. The other region is a disjoint union of the standard open subsets 𝑆¹×𝐷² in #_𝑛𝑆¹×𝑆²=𝜕(♮_𝑛𝑆¹×𝐷³) (this is connected with the next block).

For the construction of an exotic 𝑆³×ℝ, denoted by 𝑆³×_𝜃ℝ, we consider the complement 𝑅⁴⧵𝐷⁴ or the decomposition: 𝑆³×_𝜃ℝ=𝑅⁴⧵𝐷⁴=𝐴_cork⧵𝐷⁴_𝐷2CH. (19)

The first part 𝐴_cork⧵𝐷⁴ contains a cobordism between the 3-sphere 𝑆³ and the boundary of the Akbulut cork 𝜕𝐴_cork (a homology 3-sphere). The complement ℝ⁴⧵𝐷⁴ is conformally equivalent to 𝑆³×ℝ. Equivalently, the complement 𝑅⁴⧵𝐷⁴ is diffeomorphic to an exotic 𝑆³×ℝ. But the exoticness is not confined to a compact subset but concentrated at infinity (for instance at +∞). In our case we choose a decomposition like 𝑆³×_𝜃ℝ=𝑀𝑆³,𝜕𝐴_cork𝐷2CH, (20) where 𝑀(𝑆³,𝜕𝐴_cork) is a cobordism between 𝑆³ and 𝜕𝐴_cork. For the Casson handle we need another representation obtained by using Morse theory (see [33]). Every kinky handle (𝑘,𝜕⁻𝑘) is given by 𝑛 pairs of 1-/2-handle pairs, where 𝑛 is the number of kinks (or self-intersections). These handles are given by the level sets of the Morse functions 𝑓₁=𝑥²+𝑦²+𝑧²−𝑡²𝑓forthe1-handle,₂=𝑥²+𝑦²−𝑧²−𝑡²forthe2-handle, (21) that is, by the sets 𝐿(𝑓_𝑖,𝐶)={(𝑥,𝑦,𝑧,𝑡)∣𝑓_𝑖(𝑥,𝑦,𝑧,𝑡)=𝐶=const.} for 𝑖=1,2. Now we represent the Casson handle by the union of all 𝑛-stage towers CH=_{levelℓoftree𝒯}int𝑇_ℓ∪𝜕⁻𝑇_ℓ, (22) arranged along the tree 𝒯. But every tower 𝑇_ℓ is given by the union of pairs (𝑓₁,𝑓₂). But what is the geometry of 𝑇_ℓ (and better of int(𝑇_ℓ))? Every level set 𝐿(𝑓₁,𝐶) and 𝐿(𝑓₂,𝐶) is a hyperbolic 3-manifold (i.e., with negative curvature) and the union of all level sets is a hyperbolic 4-manifold. A central point in our argumentation is Mostow rigidity, a central property of all hyperbolic 3-manifolds (or higher) with finite volume explained in the next subsection.

3.4. The Hyperbolic Geometry of 𝐶𝐻

The central element in the Casson handle is a pair of 1- and 2-handles representing a kinky handle. As we argued above this pair admits a hyperbolic geometry (or it is a hyperbolic 3-manifold) having negative scalar curvature. A 3-manifold admits a hyperbolic structure in the interior if there is a diffeomorphism to ℍ³/Γ, where Γ is a discrete subgroup Γ⊂SO(3,1) of the Lorentz group and we have a representation of the fundamental group 𝜋₁(𝑀) into SO(3,1) (the isometry group of the hyperbolic space ℍ³). One property of hyperbolic 3- and 4-manifolds is central: Mostow rigidity. As shown by Mostow [34], every hyperbolic 𝑛-manifold 𝑛>2 with finite volume has this property: Every diffeomorphism (especially every conformal transformation) of a hyperbolic 𝑛-manifold with finite volume is induced by an isometry. Therefore one cannot scale a hyperbolic 3-manifold with finite volume. Then the volume vol() and the curvature are topological invariants but for later usages we combine the curvature and the volume into the Chern-Simons invariant CS(). But more is true: in a hyperbolic 3-manifold there are special surfaces which cannot be contracted, called incompressible surface. A properly embedded connected surface 𝑆⊂𝑁 in a 3-manifold 𝑁 is called 2-sided if its normal bundle is trivial and 1-sided if its normal bundle is nontrivial. The “sides” of 𝑆 then correspond to the components of the complement of 𝑆 in a tubular neighborhood 𝑆×[0,1]⊂𝑁. A 2-sided connected surface 𝑆 other than 𝑆² or 𝐷² is called incompressible if for each disk 𝐷⊂𝑁 with 𝐷∩𝑆=𝜕𝐷 there is a disk 𝐷^′⊂𝑆 with 𝜕𝐷^′=𝜕𝐷; that is, the boundary of the disk 𝐷 can be contracted in the surface 𝑆. The boundary of a 3-manifold is an incompressible surface. More importantly, this surface can be detected in the fundamental group 𝜋₁(𝑁) of the 3-manifold, that is; there is an injective homomorphism 𝜋₁(𝑆)→𝜋₁(𝑁). The consequence of all properties is the following conclusion.

The tower 𝑇_ℓ has a hyperbolic geometry (with finite volume) and therefore fixed size; that is, it cannot be scaled by any diffeomorphism or conformal transformation. Then we obtain an invariant decomposition of the Casson handle into towers arranged with respect to a tree. Secondly, inside of every tower 𝑇_ℓ there is (at least one) an incompressible surface also of fixed size.

In case of the tower 𝑇_ℓ, one knows two incompressible surfaces, the two tori coming from the complement of the Whitehead link (with two components) used in the construction.

3.5. The Path Integral of the Exotic 𝑆³×ℝ

Now we will discuss the path integral using the decomposition 𝑆³×_𝜃ℝ=𝑀𝑆³,𝜕𝐴_{cork𝐷2levelℓoftree𝒯}int𝑇_ℓ∪𝜕⁻𝑇_ℓ, (23) and we remark that the construction of the cobordism 𝑀(𝑆³,𝜕𝐴_cork) requires the usage of a Casson handle again, denoted by 𝑀(𝑆³,𝜕𝐴_cork)∪CH_cork. Therefore we have to clarify the role of the Casson handle. In the previous Section 3.4, we discussed the strong connection between geometry and topology for hyperbolic manifolds. The topology of 𝑆³×_𝜃ℝ is rather trivial but the smoothness structure (and therefore the differential topology) can be very complicate.

As stated above, the boundary terms can be factorized from the terms in the interior. Formally we obtain 𝑍𝑆³×_𝜃ℝ=_ℓ𝑍𝜕⁻𝑇_ℓ𝑍𝜕𝐴_cork𝑍𝑆³×_ℓ𝑍int𝑇_ℓ𝑍CH_cork, (24) and for an expectation value of the observable 𝒪𝑆³×_𝜃ℝ|𝒪|𝑆³×_𝜃ℝ, (25) but for the following we have to discuss it more fully. To understand the time-like evolution of a disk (or a surface), we have to describe a disk inside of a Casson handle as pioneered by Bižaca [35]. With the same arguments, one can also describe the modification of the 3-sphere into homology 3-spheres Σ. But then we obtain (formally) an infinite sequence of homology 3-spheres Σ₁→Σ₂→⋯ with amplitudes 𝑍𝑆³×_𝜃ℝ=Σ₁𝑇_𝑀Σ₂Σ₂𝑇_𝑀Σ₃⋯, (26) including the boundary terms. Every spatial section Σ_𝑛 can be seen as an element of the phase space in quantum gravity. Therefore this change of transitions is a topological phase transition which will be further investigated in our work.

The choice of the boundary term has a kind of arbitrariness. We can choose the decomposition much finer to get more boundary terms. Therefore the path integral (14) must be extended away from the boundary. We will discuss this extension also in our forthcoming work.

Before we go ahead we have to discuss the foliation structure of 𝑆³×_𝜃ℝ or the appearance of different time variables. As stated above, our space-time has the topology of 𝑆³×ℝ with equal slices parametrized by a topological time 𝑡_TOP; that is, 𝑆³×_𝜃ℝ_TOP=𝑝,𝑡_TOP∣𝑝∈𝑆³,𝑡_TOP=𝑆∈ℝ₃×𝑡_TOP∣𝑡_TOP∈{−∞,…,+∞}, (27) defined by the topological embedding 𝑆³↪𝑆³×_𝜃ℝ. It is the defining property of exotic smoothness that 𝑆³ inside of 𝑆³×_𝜃ℝ is only a topological 3-sphere; that is, it is wildly embedded and so only represented by an infinite polyhedron. There is another possibility to introduce 𝑡_TOP which will point us to the smooth case. For that purpose we define a map 𝐹∶𝑆³×ℝ→ℝ by (𝑥,𝑡)↦𝑡 so that 𝑡_TOP=𝐹(𝑝) for 𝑝∈𝑆³×ℝ. In contrast one also has the smooth time 𝑡_Diff which we have to define now. Locally it is the smooth (physical coordinate) time. We know also that the exotic 𝑆³×_𝜃ℝ is composed by a sequence Σ₁→Σ₂→Σ₃→⋯ of homology 3-spheres or better by a sequence 𝑀(Σ₁,Σ₂)∪_Σ2𝑀(Σ₂,Σ₃)∪_Σ3… of (homology) cobordism between the homology 3-spheres. All sequences are ordered and so it is enough to analyze one cobordism 𝑀(Σ₁,Σ₂). Every cobordism between two homology 3-spheres Σ₁ and Σ₂ is characterized by the existence of a finite number of 1-/2-handle pairs (or dually 2-/3-handle pairs). Now we define a smooth map 𝐹_cob∶𝑀(Σ₁,Σ₂)→[0,1] which must be a Morse function (i.e., it has isolated critical points) [33]. The number of critical points 𝑁 of 𝐹 is even, say 𝑁=2𝑘, where 𝑘 is the number of 1-/2-handle pairs. These critical points are also denoted as naked singularities in GR (but of bounded curvature). Like in the case of topological time 𝑡_TOP we introduce the smooth time by 𝑡_Diff=𝐹_cob(𝑝) for all 𝑝∈𝑀(Σ₁,Σ₂)⊂𝑆³×_𝜃ℝ. The extension of 𝑡_Diff to the whole 𝑆³×_𝜃ℝ by the Morse function 𝐹∶𝑆³×_𝜃ℝ→ℝ is straightforward 𝑡_Diff=𝐹(𝑝) for all 𝑝∈𝑆³×_𝜃ℝ. The cobordism 𝑀(Σ₁,Σ₂) is part of the exotic 𝑆³×_𝜃ℝ and can be embedded to make it 𝑆³×[0,1] topologically. Therefore function 𝐹_cob is a continuous function which is strictly increasing on future directed causal curves, so it is a time function (see [36, 37]). But there is also another method to construct 𝑡_Diff by using codimension-1 foliations. In [7] we uncovered a strong relation between codimension-1 foliations (also used to construct a Lorentz structure on a manifold) and exotic smoothness structures for a small exotic ℝ⁴. The coordinate of this codimension-1 submanifold is also the smooth time 𝑡_Diff. This approach will be more fully discussed in our forthcoming work.

4. The Expectation Value of the Area and Inflation

In Section 3.4 we described the hyperbolic geometry originated in the exotic smoothness structure of 𝑆³×_𝜃ℝ. Because of this hyperbolic geometry, there are incompressible surfaces inside of the hyperbolic manifold as the smallest possible units of geometry. Then Mostow rigidity determines the behavior of this incompressible surface. At first we will concentrate on the first cobordism 𝑀(𝑆³,𝜕𝐴_cork) between 𝑆³ and the boundary 𝜕𝐴_cork of the Akbulut cork. The area of a surface is given by 𝐴(𝑒,𝑆)=_𝑆𝑑²𝜎𝐸^𝑎𝐸^𝑏𝑛_𝑎𝑛_𝑏, (28) with the normal vector 𝑛_𝑎 and the densitized frame 𝐸^𝑎=det(𝑒)𝑒^𝑎. The expectation value of the area 𝐴, 𝑆³|𝐴(𝑒,𝑆)|𝜕𝐴_cork=1𝑍𝑀𝑆³,𝜕𝐴_cork×()𝑖𝐷𝑒𝐴𝑒,𝑆expℏ𝑆_𝐸𝐻𝑒,𝑀𝑆³,𝜕𝐴_cork, (29) depends essentially on the hyperbolic geometry. As argued above, this cobordism has a hyperbolic geometry but in the simplest case, the boundary of the Akbulut cork is the homology 3-sphere 𝜕𝐴_cork=Σ(2,5,7), a Brieskorn homology 3-sphere. Now we study the area of a surface where one direction is along the time axis. Then we obtain a decomposition of the surface into a sum of small surfaces so that every small surface lies in one component of the cobordism. Remember, that the cobordism 𝑀(𝑆³,𝜕𝐴_cork) is decomposed into the trivial cobordism 𝑆³×[0,1] and a Casson handle CH=∪_ℓ𝑇_ℓ. Then the decomposition of the surface 𝑆=∪_ℓ𝑆_ℓ, (30) corresponds to the decomposition of the expectation value of the area 𝐴_ℓ𝑒,𝑆_ℓ=_𝑆ℓ𝑑²𝜎𝐸^𝑎𝐸^𝑏𝑛_𝑎𝑛_𝑏, (31) so that 𝜕⁻𝑇_ℓ𝐴_ℓ′𝑒,𝑆_ℓ′𝜕⁻𝑇_ℓ+1=𝜕⁻𝑇_ℓ𝐴_ℓ𝑒,𝑆_ℓ𝜕⁻𝑇_ℓ+1𝛿_ℓℓ′,𝑆³|𝐴(𝑒,𝑆)|𝜕𝐴_cork=_ℓ𝜕⁻𝑇_ℓ𝐴_ℓ𝑒,𝑆_ℓ𝜕⁻𝑇_ℓ+1. (32) The initial value for ℓ=0 is the expectation value 𝜕⁻𝑇₀𝐴₀𝑒,𝑆₀𝜕⁻𝑇₁=𝑎²₀, (33) where 𝑎₀ is the radius of the 3-sphere 𝑆³. But because of the hyperbolic geometry (with constant curvature because of Mostow rigidity) every further level scales this expectation value by a constant factor. Therefore, to calculate the expectation value, we have to study the scaling behavior.

Consider a cobordism 𝑀(Σ₀,Σ₁) between the homology 3-spheres Σ₀,Σ₁. As shown by Witten [38–40], the action, _Σ0,1³𝑅√ℎ𝑑³𝑥=𝐿⋅CSΣ_0,1, (34) for every 3-manifold (in particular for Σ₀ and Σ₁ denoted by Σ_0,1) is related to the Chern-Simons action CS(Σ_0,1) (defined in Appendix B). The scaling factor 𝐿 is related to the volume by 𝐿=³vol(Σ_0,1) and we obtain formally 𝐿⋅CSΣ_0,1,𝐴=𝐿³⋅CSΣ_0,1𝐿²=_Σ0,1CSΣ_0,1𝐿²√ℎ𝑑³𝑥, (35) by using 𝐿³=volΣ_0,1=_Σ0,1√ℎ𝑑³𝑥. (36) Together with ³𝑅=3𝑘𝑎², (37) one can compare the kernels of the integrals of (34) and (35) to get for a fixed time 3𝑘𝑎²=CSΣ_0,1𝐿². (38) This gives the scaling factor 𝑎𝜗=²𝐿²=3CSΣ_0,1, (39) where we set 𝑘=1 in the following. The hyperbolic geometry of the cobordism is best expressed by the metric 𝑑𝑠²=𝑑𝑡²−𝑎(𝑡)²ℎ_𝑖𝑘𝑑𝑥^𝑖𝑑𝑥^𝑘, (40) also called the Friedmann-Robertson-Walker metric (FRW metric) with the scaling function 𝑎(𝑡) for the (spatial) 3-manifold. But Mostow rigidity enforces us to choose 𝑎̇𝑎²=1𝐿², (41) in the length scale 𝐿 of the hyperbolic structure. But why is it possible to choose the FRW metric? At first we state that the FRW metric is not sensitive to the topology of the space-time. One needs only a space-time which admits a slicing with respect to a smooth time 𝑡_Diff and a metric of constant curvature for every spatial slice. Then for the cobordism 𝑀(Σ₁,Σ₂) between Σ₁ and Σ₂ we have two cases: the curvature parameter 𝑘(Σ₁) of Σ₁ (say 𝑘(Σ₁)=+1) jumps to the value 𝑘(Σ₂) of Σ₂ (say 𝑘(Σ₂)=−1) or both curvatures remain constant. The second case is the usual one. Each homology 3-sphere Σ₁,Σ₂ has the same geometry (or geometric structure in the sense of Thurston [41]) which is hyperbolic in most case. The first case is more complicated. Here we need the smooth function to represent the jump in the curvature parameter 𝑘. Let us choose the function 𝑘∶ℝ→ℝ𝑡⟼+10≤𝑡1−2⋅exp−𝜆⋅𝑡⁻²𝑡>0, (42) which is smooth and the parameter 𝜆 determines the slope of this function. Furthermore the metric (40) is also the metric of a hyperbolic space (which has to fulfill Mostow rigidity because the cobordism 𝑀(Σ₁,Σ₂) is compact).

In the following we will switch to quadratic expressions because we will determine the expectation value of the area. Then we obtain 𝑑𝑎²=𝑎²𝐿²𝑑𝑡²=𝜗𝑑𝑡², (43) with respect to the scale 𝜗. By using the tree of the Casson handle, we obtain a countable infinite sum of contributions for (43). Before we start we will clarify the geometry of the Casson handle. The discussion of the Morse functions above uncovers the hyperbolic geometry of the Casson handle (see also Section 3.4). Therefore the tree corresponding to the Casson handle must be interpreted as a metric tree with hyperbolic structure in ℍ² and metric 𝑑𝑠²=(𝑑𝑥²+𝑑𝑦²)/𝑦². The embedding of the Casson handle in the cobordism is given by the following rules. (i)The direction of the increasing levels 𝑛→𝑛+1 is identified with 𝑑𝑦² and 𝑑𝑥² is the number of edges for a fixed level with scaling parameter 𝜗.(ii)The contribution of every level in the tree is determined by the previous level best expressed in the scaling parameter 𝜗.(iii)An immersed disk at level 𝑛 needs at least one disk to resolve the self-intersection point. This disk forms the level 𝑛+1 but this disk is connected to the previous disk. So we obtain for 𝑑𝑎²∣_𝑛+1 at level 𝑛+1𝑑𝑎²∣_𝑛+1∼𝜗⋅𝑑𝑎²_𝑛 (44)up to a constant.

By using the metric 𝑑𝑠²=(𝑑𝑥²+𝑑𝑦²)/𝑦² with the embedding (𝑦²→𝑛+1, 𝑑𝑥²→𝜗) we obtain for the change 𝑑𝑥²/𝑦² along the 𝑥-direction (i.e., for a fixed 𝑦) 𝜗/(𝑛+1). This change determines the scaling from the level 𝑛 to 𝑛+1; that is, 𝑑𝑎²_𝑛+1=𝜗𝑛+1⋅𝑑𝑎²_𝑛=𝜗^𝑛+1(𝑛+1)!⋅𝑑𝑎²₀; (45) and after the whole summation (as substitute for an integral for the discrete values) we obtain for the relative scaling 𝑎²=^∞_𝑛=0𝑑𝑎²_𝑛=𝑎²₀⋅^∞_𝑛=01𝜗𝑛!^𝑛=𝑎²₀⋅exp(𝜗)=𝑎²₀⋅𝑙_scale (46) with 𝑑𝑎²|₀=𝑎²₀. With this result in mind, we consider the expectation value where we use the constant scalar curvature (Mostow rigidity). By using the normalization, many terms are neglected (like the boundary terms): 𝑆³|𝐴(𝑒,𝑆)|𝜕𝐴_cork=_ℓ𝑍𝜕⁻𝑇_ℓ𝑍𝜕𝐴_cork𝑍𝑆³×^∞_𝑛=0𝜕⁻𝑇_𝑛𝐴_𝑛𝑒,𝑆_𝑛𝜕⁻𝑇_𝑛+1×_ℓ𝑍𝜕⁻𝑇_ℓ𝑍𝜕𝐴_cork𝑍𝑆³⁻¹=^∞_𝑛=0𝜕⁻𝑇_𝑛𝐴_𝑛𝑒,𝑆_𝑛𝜕⁻𝑇_𝑛+1. (47) Finally we obtain for the area 𝑎²₀ for the first level ℓ=0, 𝑆³|𝐴(𝑒,𝑆)|𝜕𝐴_cork=^∞_𝑛=0𝜕⁻𝑇_𝑛𝐴_𝑛𝑒,𝑆_𝑛𝜕⁻𝑇_𝑛+1=𝑎²₀⋅^∞_𝑛=013𝑛!CS𝜕𝐴_cork^𝑛=𝑎²₀3⋅expCS𝜕𝐴_cork, (48) with the radius 𝑎₀ of Σ₀ and arrive at 𝑎 for Σ₁. From the physical point of view we obtain an exponential increase of the area; that is, we get an inflationary behavior. This derivation can be also extended to the next Casson handle but we have to determine the 3-manifold in which 𝜕𝐴_cork can change. It will be done below.

5. An Effective Theory

Now will ask for an effective theory where the influence of the exotic smoothness structure is contained in some moduli (or some field). As explained above, the main characteristics is given by a change of the (spatial) 4-manifold (but without changing the homology). Therefore let us describe this change (the so-called homology cobordism) between two homology 3-spheres Σ₀ and Σ₁. The situation can be described by a diagram Σ₁^ΨΣ⟶ℝ,𝜙↓↻↕𝑖𝑑,_0𝜓⟶ℝ, (49) which commutes. The two functions 𝜓 and Ψ are the Morse function of Σ₀ and Σ₁, respectively, with Ψ=𝜓∘𝜙. The Morse function over Σ_0,1 is a function Σ_0,1→ℝ having only isolated, nondegenerated, critical points (i.e., with vanishing first derivatives at these points). A homology 3-sphere has two critical points (located at the two poles). The Morse function looks like ±‖𝑥‖² at these critical points. The transition 𝑦=𝜙(𝑥) represented by the (homology) cobordism 𝑀(Σ₀,Σ₁) maps the Morse function 𝜓(𝑦)=‖𝑦‖² on Σ₀ to the Morse function Ψ(𝑥)=‖𝜙(𝑥)‖² on Σ₁. The function −‖𝜙‖² represents also the critical point of the cobordism 𝑀(Σ₀,Σ₁). But as we learned above, this cobordism has a hyperbolic geometry and we have to interpret the function ‖𝜙(𝑥)‖² not as Euclidean form but change it to the hyperbolic geometry so that −𝜙²=−𝜙²₁+𝜙²₂+𝜙²₃⟶−𝑒^−2𝜙11+𝜙²₂+𝜙²₃; (50) that is, we have a preferred direction represented by a single scalar field 𝜙₁∶Σ₁→ℝ. Therefore, the transition Σ₀→Σ₁ is represented by a single scalar field 𝜙₁∶Σ₁→ℝ and we identify this field as the moduli. Finally we interpret this Morse function in the interior of the cobordism 𝑀(Σ₀,Σ₁) as the potential (shifted away from the point 0 ) of the scalar field 𝜙 with Lagrangian 𝐿=𝑅+𝜕_𝜇𝜙²−𝜌21−exp−𝜆𝜙², (51) with two free constants 𝜌 and 𝜆. For the value 𝜆=2/3 and 𝜌=3𝑀² we obtain the Starobinski model [42] (by a conformal transformation using 𝜙 and a redefinition of the scalar field [43]) 1𝐿=𝑅+6𝑀²𝑅², (52) with the mass scale 𝑀≪𝑀_𝑃 much smaller than the Planck mass. From our discussion above, the appearance of this model is not totally surprising. It favors a surface to be incompressible (which is compatible with the properties of hyperbolic manifolds). In the next section we will determine this mass scale.

6. A Cosmological Model Compared to the Planck Satellite Results

In this section we will go a step further and discuss the path integral for 𝑆³×ℝ, where we sum over all smoothness structures. Furthermore we will assume that 𝑆³×ℝ is the end of a small exotic ℝ⁴. But then we have to discuss the parametrization of all Casson handles. As discussed by Freedman [2], all Casson handles can be parametrized by a dual tree where the vertices are 5-stage towers (with three extra conditions). We refer to [2] or to [44] for the details of the well-known construction. This tree has one root from which two 5-stage towers branch. Every tower has an attaching circle of any framing. Using Bizacas technique [35], we obtain an attaching of a 5-tower along the sum 𝑃#𝑃 of two Poincare spheres 𝑃 (for the two towers). Therefore for the universal case, we obtain two transitions 𝑆⟶^3cork𝜕𝐴⟶_cork^tower𝑃#𝑃, (53) with the scaling behavior, 𝑎=𝑎₀3⋅exp2⋅CS𝜕𝐴_cork+32⋅CS(𝑃#𝑃). (54)

It can be expressed by the expectation value 𝑆³|𝐴(𝑒,𝑆)|𝑃#𝑃=𝑎²₀3⋅expCS𝜕𝐴_cork+3CS(𝑃#𝑃), (55) for the transition 𝑆³→𝑃#𝑃. It is important to note that this expectation value is the sum over all smoothness structures of 𝑆³×ℝ and we obtain also 𝑆³×ℝ|𝐴(𝑒,𝑆)|𝑆³×ℝ=_{diffstructures}𝑆³×_𝜃ℝ|𝐴(𝑒,𝑆)|𝑆³×_𝜃ℝ=𝑎²₀3⋅expCS𝜕𝐴_cork+3(),CS𝑃#𝑃 (56) with 𝑎²₀ as the size of the 3-sphere 𝑆³ at −∞. With the argumentation above, the smoothness structure has a kind of universality so that the two transitions above are generic.

In our model (using the exotic smoothness structure), we obtain two inflationary phases. In the first phase we have a transition 𝑆³⟶𝜕𝐴_cork, (57) and for the simplest case 𝜕𝐴_cork=Σ(2,5,7), a Brieskorn homology 3-sphere. Now we will assume that the 3-sphere has Planck-size 𝑎₀=𝐿_𝑃=ℎ𝐺𝑐³; (58) then we obtain for the size 𝑎=𝐿_𝑃3⋅exp(Σ().2⋅CS2,5,7 (59) We can use the method of Fintushel and Stern [45–47] to calculate the Chern-Simons invariants for the Brieskorn spheres. The calculation can be found in Appendix C. Note that the relation (34) is only true for the Levi-Civita connection. Then the Chern-Simons invariant is uniquely defined to be the minimum, denoted by 𝜏() (see (B.4)). Then we obtain for the invariant (C.7) so that 𝐿_𝑃⋅exp1403≈7.5⋅10⁻¹⁵𝑚 (60) is the size of the cosmos at the end of the first inflationary phase. This size can be related to an energy scale by using it as Compton length and one obtains 165 MeV, comparable to the energy scale of the QCD. For the two inflationary transitions 𝑆³⟶Σ(2,5,7)⟶𝑃#𝑃 (61) one obtains the size 𝑎=𝐿_𝑃⋅exp1403+90≈9.14⋅10²⁴𝑚≈10⁹𝐿𝑗. (62) As explained above, the effective theory is the Starobinsky model. This model is in very good agreement with results of the Planck satellite [48] with the two main observables:  𝑛_𝑠∼0.96 spectral index for scalar perturbations, 𝑟∼0.004 tensor-to-scalar ratio,but one parameter of the model is open, the energy scale 𝑀 in Planck units. In our model it is related to the second derivative of the Morse function, which is the curvature of the critical point. In our paper [49], we determined also the energy scale of the inflation by using a simple argument to incorporate only the first 3 levels of the Casson handle. For the scale 3𝜗=(Σ(,2⋅CS2,5,7)) (63) of the first transition, we obtain the scaling of the Planck energy (associated with the Planck-sized 3-sphere at the beginning) 𝐸_Inflation=𝐸_Planck1+𝜗+𝜗²/2+𝜗³,/6 (64) with the relative scaling 𝐸𝛼=^Inflation𝐸_Planck=11+𝜗+𝜗²/2+𝜗³/6≈5.5325⋅10⁻⁵, (65) leading to the energy scale of the inflation 𝐸_Inflation≈6.7547⋅10¹⁴𝐺𝑒𝑉, (66) by using 𝐸_Planck≈1.2209⋅10¹⁹𝐺𝑒𝑉. We remark that the relative scaling 𝛼≈5.5325⋅10⁻⁵ (in Planck units) above is an energy scale for the potential in the effective theory (52). Therefore we have to identify 𝑀=𝛼 in the parameter 1/6𝑀² of the Starobinsky model (in very good agreement with the measurements). Now we can go a step further and discuss the appearance of the cosmological constant.

Again we can use the hyperbolic geometry to state that the curvature is negative and we have the Mostow rigidity; that is, the scalar curvature of the 4-manifold has a constant value, the cosmological constant Λ. If we assume that the 3-sphere has the size of the Planck length (as above), then we obtain 1Λ=𝐿²_𝑃−3⋅exp(Σ(−3CS2,5,7))().CS𝑃#𝑃 (67) With the values of the Chern-Simons invariants (C.7), we obtain the value Λ⋅𝐿²_𝑃=exp−2803−180≈5⋅10⁻¹¹⁸, (68) in Planck units. In cosmology, one usually relate the cosmological constant to the Hubble constant 𝐻₀ (expressing the critical density) leading to the length scale 𝐿²_𝑐=𝑐²3𝐻²₀. (69) The corresponding variable is denoted by Ω_Λ and we obtain Ω_Λ=𝑐⁵3ℎ𝐺𝐻²₀−3⋅exp(Σ(−3CS2,5,7))(),CS𝑃#𝑃 (70) in units of the critical density. This formula is in very good agreement with the WMAP results; that is, by using the value for the Hubble constant 𝐻₀=74km,𝑠⋅Mpc (71) we are able to calculate the dark energy density to be Ω_Λ=0.729, (72) agreeing with the WMAP results. But it differs from the Planck results [50] of the Hubble constant 𝐻_0Planck=68km,𝑠⋅Mpc (73) for which we obtain Ω_Λ≈0.88, (74) in contrast with the measured value of the dark energy Ω_ΛPlanck=0.683. (75) But there is another possibility for the size of the 3-sphere at the beginning and everything depends on this choice. But we can use the entropy formula of a Black hole in Loop quantum gravity 𝑆=𝐴⋅𝛾₀4⋅𝛾⋅𝐿²_𝑃⋅2𝜋, (76) with 𝛾₀=ln(2)𝜋⋅√3, (77) according to [51] with the Immirzi parameter 𝛾, where the extra factor 2𝜋 is given by a different definition of 𝐿_𝑃 replacing ℎ by ℏ. In the original approach of Ashtekar in Loop quantum gravity one usually set 𝛾=1. If we take it seriously then we obtain a reduction of the length in (70) 1𝐿²_𝑃⟶1𝐿²_𝑃⋅2⋅ln(2)√3=2𝜋𝛾₀𝐿²_𝑃1≈0.80037⋅𝐿²_𝑃, (78) or the new closed formula Ω_Λ=𝑐⁵3ℏ𝐺𝐻²₀⋅𝛾₀−3⋅exp(Σ(−3CS2,5,7))(),CS𝑃#𝑃 (79) correcting the value Ω_Λ≈0.88 to Ω_Λ≈0.704. (80) But 𝛾₀ depends on the gauge group and if one uses the value [52] 𝛾₀=ln(3)𝜋⋅√8, (81) agreeing also with calculations in the spin foam models [53], then one gets a better fit Ω_Λ≈0.6836, (82) which is in good agreement with the measurements.

7. Conclusion

The strong relation between hyperbolic geometry (of the space-time) and exotic smoothness is one of the main results in this paper. Then using Mostow rigidity, geometric observables like area and volume or curvature are topological invariants which agree with the expectation values of these observables (calculated via the path integral). We compared the results with the recent results of the Planck satellite and found a good agreement. In particular as a direct result of the hyperbolic geometry, the inflation can be effectively described by the Starobinski model. Furthermore we also obtained a cosmological model which produces a realistic cosmological constant.

Appendices

A. Connected and Boundary Connected Sum of Manifolds

Now we will define the connected sum # and the boundary connected sum ♮ of manifolds. Let 𝑀,𝑁 be two 𝑛-manifolds with boundaries 𝜕𝑀,𝜕𝑁. The connected sum 𝑀#𝑁 is the procedure of cutting out a disk 𝐷^𝑛 from the interior int(𝑀)⧵𝐷^𝑛 and int(𝑁)⧵𝐷^𝑛 with the boundaries 𝑆^𝑛−1⊔𝜕𝑀 and 𝑆^𝑛−1⊔𝜕𝑁, respectively, and gluing them together along the common boundary component 𝑆^𝑛−1. The boundary 𝜕(𝑀#𝑁)=𝜕𝑀⊔𝜕𝑁 is the disjoint sum of the boundaries 𝜕𝑀,𝜕𝑁. The boundary connected sum 𝑀♮𝑁 is the procedure of cutting out a disk 𝐷^𝑛−1 from the boundary 𝜕𝑀⧵𝐷^𝑛−1 and 𝜕𝑁⧵𝐷^𝑛−1 and gluing them together along 𝑆^𝑛−2 of the boundary. Then the boundary of this sum 𝑀♮𝑁 is the connected sum 𝜕(𝑀♮𝑁)=𝜕𝑀#𝜕𝑁 of the boundaries 𝜕𝑀,𝜕𝑁.

B. Chern-Simons Invariant

Let 𝑃 be a principal 𝐺 bundle over the 4-manifold 𝑀 with 𝜕𝑀≠0. Furthermore let 𝐴 be a connection in 𝑃 with the curvature, 𝐹_𝐴=𝑑𝐴+𝐴∧𝐴, (B.1) and Chern class, 𝐶₂=18𝜋²_𝑀tr𝐹_𝐴∧𝐹_𝐴, (B.2) for the classification of the bundle 𝑃. By using the Stokes theorem we obtain tr𝐹_𝑀𝐴∧𝐹_𝐴=_𝜕𝑀2tr𝐴∧𝑑𝐴+3𝐴∧𝐴∧𝐴, (B.3) with the Chern-Simons invariant ()=1CS𝜕𝑀,𝐴8𝜋²_𝜕𝑀2tr𝐴∧𝑑𝐴+3𝐴∧𝐴∧𝐴. (B.4) Now we consider the gauge transformation 𝐴→𝑔⁻¹𝐴𝑔+𝑔⁻¹𝑑𝑔 and obtain CS𝜕𝑀,𝑔⁻¹𝐴𝑔+𝑔⁻¹𝑑𝑔=CS(𝜕𝑀,𝐴)+𝑘, (B.5) with the winding number 1𝑘=24𝜋²_𝜕𝑀𝑔⁻¹𝑑𝑔³∈ℤ, (B.6) of the map 𝑔∶𝑀→𝐺. Thus the expression, CS(𝜕𝑀,𝐴)mod1, (B.7) is an invariant, the Chern-Simons invariant. Now we will calculate this invariant. For that purpose we consider the functional (B.4) and its first variation vanishes 𝛿CS(𝜕𝑀,𝐴)=0, (B.8) because of the topological invariance. Then one obtains the equation 𝑑𝐴+𝐴∧𝐴=0; (B.9) that is, the extrema of the functional are the connections of vanishing curvature. The set of these connections up to gauge transformations is equal to the set of homomorphisms 𝜋₁(𝜕𝑀)→SU(2) up to conjugation. Thus the calculation of the Chern-Simons invariant reduces to the representation theory of the fundamental group into SU(2). In [45] the authors define a further invariant 𝜏(Σ)=minCS(𝛼)∣𝛼∶𝜋₁(Σ)⟶SU(2), (B.10) for the 3-manifold Σ. This invariants fulfills the relation 𝜏(Σ)=18𝜋²_Σ×ℝtr𝐹_𝐴∧𝐹_𝐴, (B.11) which is the minimum of the Yang-Mills action 18𝜋²_Σ×ℝtr𝐹_𝐴∧𝐹_𝐴≤18𝜋²_Σ×ℝtr𝐹_𝐴∧∗𝐹_𝐴, (B.12) that is, the solutions of the equation 𝐹_𝐴=±∗𝐹_𝐴. Thus the invariant 𝜏(Σ) of Σ corresponds to the self-dual and anti-self-dual solutions on Σ×ℝ, respectively. Or the invariant 𝜏(Σ) is the Chern-Simons invariant for the Levi-Civita connection.

C. Chern-Simons Invariant of Brieskorn Spheres

In [45–47] an algorithm for the calculation of the Chern-Simons invariant for the Brieskorn sphere Σ(𝑝,𝑞,𝑟) is presented. According to that result, a representation 𝛼∶𝜋₁(Σ(𝑝,𝑞,𝑟)→SU(2) is determined by a triple of 3 numbers ⟨𝑘,𝑙,𝑚⟩ with 0<𝑘<𝑝,0<𝑙<𝑞,0<𝑚<𝑟 and the further relations 𝑙𝑞+𝑚𝑟𝑘<1𝑙mod2=𝑚mod2,𝑝+𝑙𝑞+𝑚𝑟𝑘>1,𝑝−𝑙𝑞+𝑚𝑟𝑘<1,𝑝+𝑙𝑞−𝑚𝑟<1. (C.1) Then the Chern-Simons invariant is given by (𝛼)=𝑒CS²4⋅𝑝⋅𝑞⋅𝑟mod1, (C.2) with 𝑒=𝑘⋅𝑞⋅𝑟+𝑙⋅𝑝⋅𝑟+𝑚⋅𝑝⋅𝑞. (C.3) Now we consider the Poincaré sphere 𝑃 with 𝑝=2,𝑞=3,𝑟=5. Then we obtain ⟨⟩11,1,1CS=,⟨⟩1201,1,3CS=49,120 (C.4) and for the Brieskorn sphere Σ(2,5,7)⟨1,1,3⟩CS=81,⟨⟩92801,3,1CS=,⟨⟩2801,2,2CS=169,⟨⟩2801,2,4CS=249.280 (C.5) In [45] the authors define a further invariant 𝜏(Σ)=minCS(𝛼)∣𝛼∶𝜋₁(Σ)⟶SU(2), (C.6) for a homology 3-sphere Σ. For 𝑃 and Σ(2,5,7) one obtains 𝜏(𝑃)=1(Σ(=9120,𝜏2,5,7)).280 (C.7) And we are done.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors acknowledged all critical remarks of the referees increasing the readability of the paper.