A topological model for inflation
In this paper we will discuss a new model for inflation based on topological ideas. For that purpose we will consider the change of the topology of the spatial component seen as compact 3-manifold. We analyzed the topology change by using Morse theory and handle body decomposition of manifolds. For the general case of a topology change of a manifold, we are forced to introduce a scalar field with quadratic potential or double well potential. Unfortunately these cases are ruled out by the CMB results of the Planck misssion. In case of 3-manifolds there is another possibility which uses deep results in differential topology of 4-manifolds. With the help of these results we will show that in case of a fixed homology of the 3-manifolds one will obtain a scalar field potential which is conformally equivalent to the Starobinsky model. The free parameter of the Starobinsky model can be expressed by the topological invariants of the 3-manifold. Furthermore we are able to express the number of e-folds as well as the energy and length scale by the Chern-Simons invariant of the final 3-manifold. We will apply these result to a specific model which was used by us to discuss the appearance of the cosmological constant with an experimentally confirmed value.
pacs:98.80.Jk, 98.80.Cq, 02.40.Sf
- I Introduction
- II The Model
III Inflation in four dimensions
- III.1 Cobordism between 3-manifolds and 4-manifold topology
- III.2 Casson handles or the infinite process of handle-cancellations
- III.3 A geometric interpretation of the Starobinsky model
- III.4 Determine the number of e-folds
- III.5 Determine the Energy scale and the Parameter
- III.6 Reheating and topology
- IV A realistic example
- V Conclusion
- A Connected and boundary-connected sum of manifolds
- B Casson Handles
- C Hyperbolic 3-/4-Manifolds and Mostow-Prasad rigidity
- D Models of Hyperbolic geometry
Because of the influx of observational data111In particular, see the recent results of the Planck satellite in arXiv from 1303.5062 to 1303.5090., recent years have witnessed enormous advances in our understanding of the early universe. To interpret the present data, it is sufficient to work in a regime in which spacetime can be taken to be a smooth continuum as in general relativity, setting aside fundamental questions involving the deep Planck regime. However, for a complete conceptual understanding as well as interpretation of the future, more refined data, these long-standing issues will have to be faced squarely. As an example one may ask, can one show from first principles that the smooth spacetime of general relativity is valid at the onset of inflation? At the same time, this approach has some problems like special initial conditions and free parameters like the amount of increase (number of e-folds). Furthermore, there are many possibilities for a specific model (chaotic or fractal inflation, Starobinsky model etc.). But the impressive results of the PLANCK mission excludes many models PlanckInflation2013 (). Nevertheless, the main questions for inflation remain: what is the scalar field? what is the number of e-folds? which model is realistic? what is the energy scale? etc. In this paper we will focus mainly on the question about the origin of inflation. Today inflation is the main theoretical framework that describes the early Universe and that can account for the present observational data WMAP-7-years (); PlanckCosmoParameters2013 (); PlanckCosmoParameters2015 (). In thirty years of existence Guth1981 (); Linde1982 (), inflation has survived, in contrast with earlier competitors, the tremendous improvement of cosmological data. In particular, the fluctuations of the Cosmic Microwave Background (CMB) had not yet been measured when inflation was invented, whereas they give us today a remarkable picture of the cosmological perturbations in the early Universe. In nearly all known models, the inflation period is caused by one or more scalar field(s)InflationBook (). But the question about the origin of this scalar field remains among other problems (see for instance Penrose1989 ()).
In this paper we will go a different way to explain inflation. Inspired by Wheelers idea of topology change at small scales in quantum gravity, we will consider a spatial topology change. The description of the change using the concept of a cobordism (representing the spacetime) will lead automatically to a scalar field. On general grounds, one can show that there are two kinds of changes: adding a submanifold or change/deform a submanifold. In the first case, we will get the quadratic potential of chaotic inflation whereas in the second case we will obtain the double well potential of topological inflation. But both models were ruled out by the PLANCK mission. Amazingly, only in four dimension there is another possibility of a topology change. This modification is an infinite process of submanifold deformations (arranged along a tree). At the first view, it seems hopeless to calculate something. But in contrast to the two cases above, we are able to determine everything in this model. The potential of the scalar field is which is conformally equivalent to the Starobinsky model. The number of e-folds is determined by a topological invariant of the spatial space. This invariant will be used to get expressions for (free parameter in the Starobinsky model) or the energy scale of inflation. Why is this miracle possible? Mostow-Prasad rigidity (see C) is the cause for this behavior. The infinite process of submanifold deformations implies a hyperbolic geometry for the underlying space. But any deformation of a hyperbolic space must be an isometry. Therefore geometric expressions like volume or curvature are topological invariants. It is the point where geometry and topology meet. In a previous paper AsselmeyerKrol2018a () we discussed a concrete model of topology change in the evolution of the cosmos with two phases. In particular, we obtained a realistic value of the cosmological constant. Here we will use this model to calculate the values of the inflation parameters in the Starobinsky model, i.e. (coupling of the term), energy scale, number of e-folds , the spectral tilt and the tensor-scalar ratio . We will also compare these values with the current measurements. One point remains, how does this model couples to matter (reheating)? In a geometric/topological theory of inflation one also needs a geometric model of matter to explain this coupling. Fortunately, this theory was partly developed in previous work AsselmeyerRose2012 (); AsselmeyerBrans2015 (). By using these ideas, we will explain the coupling between the scalar field and matter. The coupling constant is given by a topological invariant again.
Ii The Model
The main idea of our model can be summarized by a simple assumption: during the cosmic evolution (i.e. directly after the Big Bang) the spatial component (space) undergoes a topology change. This assumption is mainly motivated by all approaches to quantum gravity. Notable are first ideas by Wheeler Wheeler62 (). But topology changes are able to produce singularities and causal discontinuities as shown in deWitteAnderson1986 (). In many cases one can circumvent these problems as discussed in Dowker1997 (); DowkerGarcia:1998 (); DowkerGarciaSurya:2000 (). Here, we will implicitly assume that the topology changes is causal continuous.
At first we have to discuss the description of a spatial topology change. Let and be two compact closed 3-manifolds so that is changed to . Now there is a spacetime with called a cobordism. For now we have to face the question to characterize the topology of the cobordism. It is obvious that two diffeomorphic 3-manifolds will generate a trivial cobordism . Interestingly, it is not true if there is a counterexample to the smooth Poincare conjecture in dimension 4. Then this cobordism between diffeomorphic 3-manifold can be also non-trivial (i.e. non-product). In general, the difference between two manifolds is expressed in a complicated topological structure of the interior. The prominent example is a cobordism between two disjoint circles and one circle, the so-called trouser. There, the non-triviality of the cobordism is given by the appearance of a ’singular’ point, the crotch. Fortunately, this behavior can be generalized to all other cases too. To understand this solution we have to introduce Morse theory and handlebody decomposition of manifolds. By using these methods we will show that a topology change requires a scalar field including an interaction potential (related to the so-called Morse function).
ii.1 Morse theory and handles
In Morse theory one analyzed the (differential-)topology of a manifold by using a (twice-)differentiable function . The main idea is the usage of this function to generate a diffeomorphism via the gradient equation
in a coordinate system. Away from the fix point , the solution of this differential equation is the desired diffeomorphism. This behavior breaks down at the fix points. The fix points of this equation are the critical points of . Now one has to assume that these critical points are isolated and that the matrix of second derivatives has maximal rank (non-degenerated critical points). This function is called a Morse function. Then, the function in a neighborhood of an isolated, non-degenerated point (i.e. , ) looks generically like
where the number is called the index of the critical point. In the physics point of view, the Morse function is a scalar field over the manifold. The Morse function (2) at the critical point is a quadratic form w.r.t the coordinates. This form is invariant by the action of the group . If one interpret this group as isometry group then one can determine the geometry in the neighborhood of the critical point. But the the Morse function reflects the topological properties of the underlying space, i.e. the analytic properties of are connected with the topology of . For that purpose we will define the level set of , i.e.
Now consider two sets and for . If there is no critical point in the compact set then and are homotopy equivalent (and also topologically equivalent, at least in dimension smaller than 5). If the compact set contains a critical point of index then and are related by the attachment of a handle ( is the disc), i.e. . Therefore, the topology of is encoded in the critical values of the Morse function. But let us give two words of warning: firstly this approach gives only the number handles but not the detailed attachment of the handles and secondly the number of handles as induced by the Morse function can be larger as the minimal number of handles used to decompose the manifold. Both facts are expressed in the Morse relations: let be the number of critical points of index and the th Betti number (= the rank of the homology group with values in ). Then, one has
where is the Euler characteristics of . So, Morse theory extracted only the homological properties of the manifold but not the whole topological information.
ii.2 Cobordism, handles and Cerf theory
Now we will discuss the cobordism between two different 3-manifolds and . In case of 3-manifolds, the word ’different’ means non-diffeomorphic which agrees with non-homeomorphic (see Moi:52 ()). What is the structure of the cobordism for different 3-manifolds? The answer can be simply expressed that there are one or more handles in the interior of the cobordism. A proof can be found in Mil:65 () and we will discuss a simple example now. Let us consider a pant or trouser like above, i.e. a cobordism between two disjoint circles and one circle (see Fig. 1).
A circle is the union of one 0-handle () and one 1-handle () glued together along and the boundary of the 0-handle , i.e. along the end-points of the two intervals . Now lets go from two disjoint circles on one side of the cobordism to one circle of the other side of the cobordism. The two disjoint circles are build from two 0-handles and two 1-handles whereas the one circle is decomposed by one 0-handle and one 1-handle. Therefore the pair of one 0-handle and one 1-handle was destroyed. Each process of this kind will produce a handle in the interior of the cobordism. In this case it is a 1-handle (the crotch of the pant). The critical point of this handle represents the topology change. It is the critical point of the corresponding Morse function for the cobordism. Of course the process can be reversed (cobordism classes are forming a group). Then a 0-/1-handle pair appears and one circles splits into two disjoint circles but the main observation is the same: A topology change produces an additional handle in the interior of the cobordism. For the discussion later, we have to make an important remark. The 1-handle in the interior of the cobordism is a saddle and the critical point is a saddle point. The corresponding Morse function is given by . Now we are interested in the geometry of the cobordism. The group fixes the function by the usual action. The group is the group of hyperbolic rotations preserving the area and orientation of a unit hyperbola. It is a subgroup of the Möbius group, the isometry group of the 2-dimensional hyperbolic space. Because of the saddle point, the interior of the cobordism is a saddle surface having a hyperbolic geometry (with negative curvature). This observation can be generalized to any saddle point of a cobordism (or to any handle for with the dimension of the cobordism). In dimension four, one obtains hyperbolic geometries for 1- and 3-handles and the geometry for 2-handles so2-2-symmetry () for the isometry group inside of the cobordism. Then two geodesics will be separated exponentially after passing the critical point of the handle. This behavior explains also the appearance of an inflationary phase after a topology change which will be discussed later.
The process of ’killing’ the 0-/1-handle pair is visualized in Fig. 2.
There, one can also find an analytic expression for this process. For completeness, we remark that the theory behind this description is called Cerf theory Cer:70 (). In general, the modification of any handle structure can be simplified to one process of this kind. The idea of Cerf theory can be simply expressed by considering the function , i.e. a one-parameter family of Morse functions at the boundary of the cobordism. Central point of Cerf theory is the existence of two generic singularities (with vanishing first derivatives). The first kind is the Morse singularity: and the two other cases and . For these two cases , there are resolutions as one-parameter families: and (with the parameter ). Interestingly, the case can be reduced to the case. The one-parameter family is visualized in Fig. 2 for (from left to right). Now the cancellation of a handle pair is described by the function
and we will use this function to describe the change in topology as the effect of handle canceling.
One example is the simplification of a handle decomposition. As explained in the previous subsection, this handle decomposition of the cobordism can be non-uniquely given. An example is the following picture Fig. 3.
A cobordism between two circles (usually the trivial cylinder ) is decomposed by an extra pair of one 1-handle and one 2-handle. But as the figure indicated, there is a flow (determined by the corresponding Morse function, see (1)) from one critical point (1-handle) to the other critical point (2-handle). This flow is a diffeomorphism which can be used to cancel both critical points, see Mil:63 () and Mil:65 () for the details. For the successful canceling, one needs an implicit assumption which will be explained later.
There is another example where canceling pairs appear, the so-called homology cobordism. This example will become important later. It is motivated by the following question: how does a simple-connected, 4-dimensional cobordism with trivial homology look like? In general one would expect that the corresponding 3-manifolds have to be simply connected too. But let us consider a disk with boundary . is simply connected in contrast to the boundary. So, every non-contractable curve in a 3-manifold can be transformed to a contractable curve inside of the cobordism by attaching a disk or better a 2-handle . The corresponding change of the cobordism can be changed by the attachment of a 3-handle which cancels the 2-handle. A simple argument using the Mayer-Vietoris sequence shows that a cobordism which looks like is a cobordism between two homology 3-spheres , i.e. 3-manifolds with the same homology like the 3-sphere. One call this cobordism, a homology cobordism. A sequence of these homology cobordism will look like but inside of this cobordism you have an ongoing topology change. Later on we will construct the Starobinsky model from this cobordism.
ii.3 The physics view on cobordism
In Wit:82a (), Witten presented a physics view on Morse theory using supersymmetric quantum mechanics. This work was cited by many followers and it was the beginning of the field of topological quantum field theory. Part of this work will be used to describe the cobordism and its handle decomposition. As explained above, a cobordism between different manifolds and contains at least one handle, say handle . This handle is embedded in the cobordism by using a map
to visualizing the attachment of the handle. It can be locally modeled by a map
where is a chart of the cobordism representing the attachment of the handle. In principle, this local description is enough to understand the adding of a handle to the interior of the cobordism (and representing the non-triviality of the cobordism). So, if we are choosing the map
then we have an equivalent description given by a set of four scalar fields . Importantly, this description can be generalized to all dimensions expressing the topology change. In the special case of 3-manifolds we will later present another description using a valued scalar field.
But now we will consider the case of a manifold. It is not an accident that scalar fields are describing a topology change. Critical points of a scalar field (the Morse function) express the topology of the underlying space, at least partly. Here, in the case of a topology change we have to consider the change of a scalar field. This change leads to the appearance of one or more handles in the interior of the cobordism whose location is a tupel of four scalar fields to describe the embedding. A vector field is not suitable because there is no ’direction’ in an embedding. But we are able to simplify this description. The embedding can be chosen in such a manner that the flow to the critical point of the handle is normal to the boundary. The Morse function for the handle is given by in the coordinate system . The normal direction in the cobordism is expressed by a coordinate seen as a (locally) non-vanishing vector field (as section of the normal bundle of the boundary ). The case of a cobordism with more than one critical point (or handle) is more interesting. This case includes also the situation to simplify the cobordism (see Fig. 3). InWit:82a (), this situation was considered. The Morse function inside of the cobordism generates the ’potential energy’ of the problem to be . Then the flow from one critical point to another critical point can be described as a tunneling path. These paths are the paths of steepest descent (leading from one critical point to another critical point) expressed as solutions of the equation (1) now written as
with respect to a metric of . Here, the variable the path of steepest descent. But as shown in Mil:65 (), one can order the handles so that the coordinate of the cobordism cane be identified with this parameter which will be done now. As Witten pointed out in the paper Wit:82a (), the relevant action is given by
Now, we identify the coordinates of the handle with one direction in the coordinate system of the cobordism, say , and using a Lorentz transformation at the same time then no direction is preferred (a standard argument). Finally we obtain the action of the nonlinear sigma model
and the path of steepest descent is given by a choice of a function (as embedding of the curve). This action can be also used to describe the canceling process of a handle pair. Both handles agreed in nearly all directions except one direction. It is the direction with coordinate . The handle is given by the Morse function whereas the handle is determined by the Morse function . The difference between both Morse functions is concentrated at the direction: the function for the handle and for the handle. Both handles are connected along this direction and the canceling of both handles has its origin in this connection. In the above mentioned Cerf theory Cer:70 (), this handle pair is described by one function
with one parameter . The main result of Cerf theory states that this expression is unique (or better generic) up to diffeomorphisms. The canceling is described schematically in Fig. 2 for the parameter , and . But then we need only one scalar field in the action (4) and the function is given by the bracket term in the expression (5). Finally we obtain the action
for . But as explained above, one has the freedom to embed the handle pair into the coordinate system of the cobordism. For the curve connecting the two handles, one can rewrite the total derivative
and in a small neighborhood we can use the usual relation between the four velocities. Instead to integrate only along the curve, we will consider a field of handle pairs on the cobordism . Then can be interpreted as a kind of density for handle pairs in . Then we have to integrate over the whole cobordism to obtain the action
where we set . The argumentation can be generalized to other cases like the appearance of a handle as described by the function leading to the general action
As remarked above, we looked for the generic cases like and but higher powers are also possible. In combination with the Einstein-Hilbert action we obtain the two generic models
of chaotic inflation and topological inflation.
By using the model of a topology change, we are able to reproduce two known inflationary models. The advantage of this model is the natural appearance of the scalar field which is associated to the topology change. Unfortunately, both models were ruled out by recent results of the Planck mission PlanckInflation2013 (). Only potentials like with or long-tailed expressions like are possible. But functions like cannot be generated by Morse functions and one cannot reproduce this model by topological methods. Interestingly, only in dimension four there is the possibility to obtain long-tailed expressions. There is a simple reason why this is possible: in dimension four a pair of handles like or handle pairs cannot be canceled smoothly but topologically using an infinite process. In the next section we will describe this process which will lead to the Starobinsky model which is one of the favored models of the Planck mission.
Iii Inflation in four dimensions
In this section we will specialize to 4D spacetime. Here, the cancellation of a handle pair is described by an infinite process i.e. one needs a special handle known as Casson handle. Casson handles are parametrized by all trees. In principle, this fact is the reason for the exponential potential leading to Starobinsky inflation.
iii.1 Cobordism between 3-manifolds and 4-manifold topology
In the previous section we described the general case of a topology change. Implicitly we assumed that the canceling of handles is always possible. But in dimension 4, there is a problem which is at the heart of all problems in 4-dimensional topology. As an example let us consider a pair of one 2-handle and one 1-handle . The 1-handle and 2-handle cancel each other if the attaching sphere of the 2-handle meets the belt sphere of the 1-handle tranversally in one point. To understand the problem, we have to consider the attachment of handles. A handle is attached to via the boundary by the map . Then a 2-handle is attached to by an embedding of , the solid torus. But this map is equivalent to , i.e. the attaching of a 2-handle is determined by a knot. There is also an additional number, the framing, which describes how a parallel copy of the knots wind around the knot. The attachment of 1- and 3-handles are easier to describe. In case of a 1-handle, Akbulut AkbKir:79 () found another amazing description: a 1-handle is a removed 2-handle with fixed framing (see GomSti:1999 () section 5.4 for the details). An example of a non-canceling pair of one 1-handle and one 2-handle is visualized in Fig. 4.
In this example, the attaching sphere of the 2-handle (the knot without the dot) meets the belt sphere of the 1-handle (the knot with the dot) twice. Usually, a curve meeting a second curve twice can be separated. This process is called the Whitney trick. For the realization of this process in a controlled manner, one needs an embedded disk. But it is known that the Whitney trick fails because the disk contains self-intersections (it is immersed in contrast to embedded), see Asselmeyer2007 (). Interestingly, it is possible to realize the Whitney trick topologically. But then one needs an infinite process as shown by Freedman Fre:82 (). Now we will describe this process to use it for the Starobinsky model.
iii.2 Casson handles or the infinite process of handle-cancellations
In dimension 4, the process of handle canceling can be an infinite process. One can understand the reason simply. Two disks in a 4-manifold intersect in a point. At the same time, disks in a 4-manifold can admit self-intersections, i.e. we obtain an immersed disk (in contrast to an embedded disk with no self-intersections). As explained above, one can cancel the self-intersections but one needs another disk admitting self-intersections again. As Freedman Fre:82 () showed, one needs infinitely many disks (or stages) to cancel the self-intersection topologically. In this process, an immersed disk can admit more than one self-intersection and therefore needs more than one disk for its canceling. Thus, one obtains a tree of disks, called a Casson handle (see the appendix and Cas:73 (); Fre:79 (); Fre:82 (); GomSti:1999 ()). Here we have also a special situation: an infinite object, the Casson handle , has to be embedded into a compact 4-manifold, the cobordism . Furthermore, the tree can be exponentially large (see Biz:94 ()). But then we have to find an embedding of this tree into a compact submanifold. For simplicity we can assume a disk of radius . The infinite tree has a root and a continuum of leaves at infinity. If we put the root of the tree in the middle of the disk then the leaves have to be at the boundary of the disk. The corresponding disk has to admit a special geometry to reflect this properties, it is the Poincare hyperbolic disk with metric
The boundary of the disk (i.e. ) represents the point at infinity. The (scalar) curvature is negative, i.e. the disk carries a hyperbolic metric. This metric can be simply transformed into
by using , . The tree of the Casson handle is embedded along a fixed angle , i.e. . As mentioned above, the root of the tree is located at the center of the disk. Then the whole tree is located between where is containing the leaves of the tree. By using , we obtained for the metric
A Morse function on the disk is given by
and a cancelling pair (by using Cerf theory) can be expressed by
with the deformation parameter . For one has the canceling pair of two handles and for both handles are disappeared. The cancellation point is given by or by . For the following argumentation, we will start with a Morse function and deform it to a pair of two handles by choosing
One handle is at the middle of the disk () and the canceling handle is located at infinity (after adding the whole Casson handle), i.e. at the boundary of the disk . One can also construct the previous Morse function directly from this data. For that purpose , we have to choose the first derivative to be
with respect to the metric (7)
In our philosophy, we have to use the coordinate instead of which is equivalent to transform the problem back into the Euclidean space. Then we will obtain the scalar field and the new potential in these coordinates
leading to the action
of the Starobinsky model written as scalar field action. Here we take the opportunity to scale the potential by the free parameter , i.e. by scaling the function . The classical Starobinsky model can be constructed after performing a conformal transformation
with the scalar curvature and . Then one obtains
the usual Starobinsky model. It is one of the few models which agrees with the results of the Planck mission. But what is the meaning of this conformal transformation? Is it possible to determine the free parameter and what is its meaning? What is the real geometric background of this model? All these question have to be addressed to get a full derivation of the model. Therefore we will start with the model (12) to obtain (15).
iii.3 A geometric interpretation of the Starobinsky model
The infinite process of handle-cancellation in dimension four was used to construct the scalar field model (12) which is conformally equivalent to the Starobinsky model. Before we start we have to give an overview about the model leading to this action. We considered a topology change of a 3-manifold into another 3-manifold represented by a cobordism, i.e. by a 4-manifold with boundary . In this process, one changed the handle structure of into the handle structure of . Some handles will be canceled and some other handles are created. For the special case of 3-manifolds, one can consider a scalar field where the variation of this field gives the topology change. Analytically one has to consider a one parameter family of functions for this creation/annihilation process. But in dimension four, there are problems to realize this process. One needs a complicated infinite tree-like structure (Casson handle) to manage this process which has to be embedded into the compact cobordism. The embedding can be realized by using the hyperbolic metric of the Poincare hyperbolic disk. At the end we obtained an analytic expression (see the potential (11)) for the corresponding handle structure of this Casson handle. Now we will understand the geometric origin of this potential.
For that purpose, we have to consider the potential (11). We derived it for the Casson handle embedded in the Poincare hyperbolic disk. A quick look at the defining formula (9) for the scalar field will give us the defining area: is between leading to . But the final potential told us more: outside of the Poincare hyperbolic disk, the scalar field can admit negative values and the potential increased exponentially. At this point we have to remember on the interpretation of : it is directly the deformation of the 3-manifold into . Obviously, this deformation will lead to a deformation of the metric at the 3-manifold as well. Now we will consider the metric (7)
by using the coordinates . Along the coordinate we have the exponentially crowing tree and along we have also an exponential increase given by or large positive . For , one has an exponential increase of the metric (see (8)) induced by the hyperbolic metric used to embed the tree of the Casson handle.
The embedding of the tree will mimic also the embedding of the whole Casson handle. The Casson handle (see Appendix) is homeomorphic to (see Fre:82 ()) and therefore we will need a four-dimensional version of the Poincare hyperbolic disk, the Poincare hyperbolic 4-ball with metric
with the angle coordinates (a tupel of 3 angles) and the radius . Interestingly the calculation remained the same because the expression (16) qualitatively agreed with (8). The interesting part is independent of the dimension (see hyperbolic-geometry ()). But there is an important difference: now we have a four-dimensional hyperbolic submanifold admitting Mostow rigidity or Mostow-Prasad rigidity. Mostow rigidity is a powerful property. As shown by Mostow Mos:68 (), every hyperbolic manifold with finite volume has this property: Every diffeomorphism (especially every conformal transformation) of a hyperbolic manifold with finite volume is induced by an isometry. See C for more information. Therefore one cannot scale a hyperbolic 3- and 4-manifold with finite volume. The volume and the curvature (or the Chern-Simons invariant) are topological invariants. Now one may ask that the embedding of the Casson handle is rather artificial then generic. But there is a second argument. In the appendix we worked out how the canceling 1-/2-handle pair with a Casson handle attached looks like. Especially we will show that the corresponding sequence of 3-manifolds is a sequence of hyperbolic 3-manifolds of finite volume. Now we are able to argue similarly: one has
and one will get an exponential increase by along all directions. This discussion showed that the scalar field can be interpreted as the deformation of the cobordism metric via a conformal transformation
for all positive values . Which geometrical expression determines this conformal transformation? The field is directly related to the radius of the hyperbolic disk via (9). In a very small neighborhood of , one has approximately an Euclidean metric (vanishing curvature ). For large values , the curvature of the curves inside the disk increases (relative to the background metric). The negative values correspond to the area outside of the hyperbolic disk. Here a curve passing the hyperbolic disk will be changed according to the negative curvature of the disk. Then we obtain the simple relation
with the strictly increasing function (i.e. for and for ). The simplest function is the linear function, i.e.
and the positivity of the exponential function implied
with . But with the discussion above, we can interpret the Starobinsky model geometrically. The potential of the scalar action is given by
or in terms of the curvature
This expression is flat for positive curvatures reaching slowly the
value . But for negative values (), it
grows rapidly. As discussed above, the positive value is related to
the curvature of a curve in the interior of the hyperbolic disk. The
limit values corresponds to the curvature of the whole disk which
is needed to embeds the whole tree of the Casson handle (see above).
But negative values (or the contraction of the metric) are leading
to a strongly increasing potential or the contraction of the disk
(containing the embedded tree) is impossible. This behavior goes over
to the 4-dimensional case (the embedding of the whole Casson handle
into the Poincare hyperbolic 4-ball). Then we can state:
The Starobinsky model is the simplest realization of Mostow rigidity, i.e. there is a 4-dimensional hyperbolic submanifold of curvature which cannot be contracted.
This submanifold is also the cause for inflation: it is the reaction of the incompressibility of the submanifold. But Mostow rigidity has a great advantage: geometric expression are topological invariants. For this reason we should be able to determine the free parameter by the topological invariants of the cobordism.
iii.4 Determine the number of e-folds
In the usual models of inflation, the number of e-folds is a free parameter which will be choose to be . But topology in combination with Mostow rigidity should determine this value by purely topological methods. In AsselmeyerKrol2014 () we described the way to get this value in principle. Here we will adapt the derivation of the formula to the case in this paper. For that purpose we we will state two deep mathematical results, the details can be found in AsselmeyerKrol2018a (). Above we introduced a hyperbolic disk (Poincare disk) to embed the infinite tree. This deep result can be expressed a different manner: there is no freedom or we have to choose the hyperbolic metric (which is up to isometries given by (8)). The second deep result is the representation of the topology change as an infinite chain of 3-manifolds where each spatial 3-manifold admits a a homogenous metric of constant negative curvature. Equivalently this change is given by an infinite chain of cobordisms
representing the chain of changes. This chain of cobordisms is also embedded in the spacetime. As shown in AsselmeyerKrol2018a (), is a model for an end of an exotic or a model of an exotic . As shown in this paper, the embedded admits a hyperbolic geometry. This hyperbolic geometry of the cobordism is best expressed by the metric
also called the Friedmann-Robertson-Walker metric (FRW metric) with the scaling function for the (spatial) 3-manifold (denoted as in the following). As explained above, the spatial 3-manifold admits (at least for the pieces) a homogenous metric of constant curvature. Now we have the following situation: each spatial 3-manifold admits a hyperbolic metric and the whole process (given as infinite chain of cobordisms) admits also a hyperbolic metric of constant negative scalar curvature which is realized by the equation , i.e. is the so-called cosmological constant. Then one obtains the equation
having the solutions for , for and for all with exponential behavior. At first we will consider this equation for constant topology, i.e. for the spacetime . But as explained above and see AsselmeyerKrol2018a (), the embedding and every admits a hyperbolic structure. Now taking Mostow-Prasad rigidity seriously, the scaling function must be constant, or . Therefore we will get
so that CC is related to the curvature of the 3D space. By using , we are able to define a scaling parameter for every . By Mostow-Prasad rigidity, is also constant, . But the change increases the volumes of , , by adding specific 3-manifolds (i.e. complements of the Whitehead links). Therefore we have the strange situation that the spatial space changes by the addition of new (topologically non-trivial) spaces. To illustrate the amount of the change, we have to consider the embedding directly. It is given by the embedding of the Casson handle as represented by the corresponding infinite tree . As explained above, this tree must be embedded into the hyperbolic space. For the tree, it is enough to use a 2D model, i.e. the hyperbolic space . There are many isometric models of (see the appendix D for two models). Above we used the Poincare disk model but now we will use the half-plane model with the hyperbolic metric
to simplify the calculations. The infinite tree must be embedded along the axis and we set . The tree , as the representative for the Casson handle, can be seen as metric space instead of a simplicial tree. In case of a simplicial tree, one is only interested in the structure given by the number of levels and branches. The tree as a metric space (so-called tree) has the property that any two points are joined by a unique arc isometric to an interval in . Then the embedding of is given by the identification of the coordinate with the coordinate of the tree representing the distance from the root. This coordinate is a real number and we can build the new distance function after the embedding as
But as discussed above, the tree grows with respect to a time parameter so that we need to introduce an independent time scale . From the physics point of view, the time scale describes the partition of the tree into slices. This main ideas was used in AsselmeyerKrol2018a () to get the relation between the number of e-folds and a topological invariant (Chern-Simons invariant) of the 3-manifold (as result of the change). In the following we will describe only the main points in the derivation of the formula (see AsselmeyerKrol2018a () for the details):
The growing of the tree with respect to the hyperbolic structure is given by
This equation agrees with the Friedman equation for a (flat) deSitter space, i.e. the current model of our universe with a CC. This equation can be formally integrated yielding the expression
One important invariant of a cobordism is the signature , i.e. the number of positive minus the number of negative eigenvalues of the intersection form. Using the Hirzebruch signature theorem, it is given by the first Pontryagin class
with the curvature 2-form of the tangent bundle . By Stokes theorem, this expression is given by the difference
of two boundary integrals where
is known as Chern-Simons invariant of a 3-manifold .
Using ideas of Witten Wit:89.2 (); Wit:89.3 (); Wit:91.2 () we will interpret the connection as connection. Note that is the Lorentz group by Wigner-Inönü contraction or the isometry group of the hyperbolic geometry. For that purpose we choose
with the length and 1-form with values in the Lie algebra so that the generators fulfill the commutation relations
with pairings , . This choice was discussed in Wise2010 () in the context of Cartan geometry.
For vanishing torsion , we obtain
and finally the relation
From (23) it follows that
Then we identify with the time and using (25) we will obtain the expression
where the extra factor (equals ) is the normalization of the curvature integral. This normalization of the curvature changes the absolute value of the curvature into
and we choose the scaling factor by the relation to the volume .
Then we will obtain formally
in agreement with the normalization above. Let us note that Mostow-Prasad rigidity enforces us to choose a rescaled formula
with the hyperbolic volume (as a topological invariant). The volume of all other 3-manifolds can be arbitrarily scaled. In case of hyperbolic 3-manifolds, the scalar curvature is negative but above we used the absolute value in the calculation. Therefore we have to modify (26), i.e. we have to use the absolute value of the curvature and of the Chern-Simons invariant . By (26) and (28) using
a simple integration (22) gives the following exponential behavior
For the following, we will introduce the shortening
In principle, the value is the number of e-folds but above (27) we used another normalization of the curvature. But the curvature is related to by . Therefore we have to correct the number of e-folds by the logarithm of the normalization and we will obtain
for the number e-folds. This result determines the number of e-folds and connect it with topological information of the final 3-manifold. Interestingly, this result is independent of the embedding and of the particular Casson handle. As shown in the following, using this result we are able to determine also the other parameters like the energy scale or the parameter in the Starobinsky model.
iii.5 Determine the Energy scale and the Parameter
Starting point is the formula
or the th level will contribute by . To calculate the energy scale, we need the argumentation that the energy scale as represented by an energy change is related to a time change by . Therefore we are enforced to determine the shortest time change. But this change must agree with the number of levels in the tree of the Casson handle where the topology change appears. The Casson handle is designed to produce a (flat) disk with no self-intersections. As explained above, this disk will be used to cancel additional self-intersections and at the end it will lead to the topology change. Therefore we have to ask how many levels are necessary to get the first disk with no self-intersections. In Fre:88 () Freeman answered this question: three levels are needed! Now it seems natural that the shortest time scale will be assumed to be the Planck time . Then we will get
for the shortest time interval of the topology change. Finally we will obtain for the energy scale
of the inflation. In subsection III.3, we gave a geometrical interpretation of the parameter as the radius of a non-contractable core. Following the argumentation above, then this core has to consist of at least three levels. Furthermore, has to be expressed as energy in Planck units. But then using (32) we will obtain
and as we will see below, will be of order below the Planck energy. Finally, the spectral tilt and the tensor-scalar ratio can be determined to be
with the topological invariant .
iii.6 Reheating and topology
Now we will discuss the Einstein-Hilbert action for the sequences of cobordism following our work AsselmeyerBrans2015 (). Let us start with the (Euclidean) Einstein-Hilbert action functional
of the 4-manifold and fix the Ricci-flat metric as solution of the vacuum field equations of the exotic 4-manifold. As discussed above, we consider a sequences of cobordism
and one has to consider the Einstein-Hilbert action functional for every cobordism . In general, for a manifold with boundary one has the expression (see GibHaw1977 ())
and for the cobordism, one obtains
where is the mean curvature of the boundary with metric . In the following we will discuss the boundary term, i.e. we reduce the problem to the discussion of the action
(see also Ashtekar08 (); Ashtekar08a () for this boundary term) along the boundary (a 3-manifold). Now we will show that the action (34) over a 3-manifold is equivalent to the Dirac action of a spinor over . For completeness we present the discussion from AsselmeyerBrans2015 (). At first let us consider the general case of an embedding of a 3-manifold into a 4-manifold. Let be an embedding of the 3-manifold into the 4-manifold with the normal vector . A small neighborhood of looks like . Furthermore we identify and ( is an embedding). Every 3-manifold admits a spin structure with a spin bundle, i.e. a principal bundle (spin bundle) as a lift of the frame bundle (principal bundle associated to the tangent bundle). There is a (complex) vector bundle associated to the spin bundle (by a representation of the spin group), called spinor bundle . A section in the spinor bundle is called a spinor field (or a spinor). In case of a 4-manifold, we have to assume the existence of a spin structure. But for a manifold like , there is no restriction, i.e. there is always a spin structure and a spinor bundle . In general, the unitary representation of the spin group in dimensions is -dimensional. From the representational point of view, a spinor in 4 dimensions is a pair of spinors in dimension 3. Therefore, the spinor bundle of the 4-manifold splits into two sub-bundles where one subbundle, say can be related to the spinor bundle of the 3-manifold. Then the spinor bundles are related by with the same relation for the spinors ( and ). Let be the covariant derivatives in the spinor bundles along a vector field as section of the bundle . Then we have the formula
with the embedding of the spinor spaces from the relation . Here we remark that of course there are two possible embeddings. For later use we will use the left-handed version. The expression is the second fundamental form of the embedding where the trace is related to the mean curvature . Then from (35) one obtains the following relation between the corresponding Dirac operators
with the Dirac operator on the 3-manifold . This relation (as well as (35)) is only true for the small neighborhood where the normal vector points is parallel to the vector defined by the coordinates of the interval in . In AsselmeyerRose2012 (), we extend the spinor representation of an immersed surface into the 3-space to the immersion of a 3-manifold into a 4-manifold according to the work in Friedrich1998 (). Then the spinor defines directly the embedding (via an integral representation) of the 3-manifold. Then the restricted spinor is parallel transported along the normal vector and is constant along the normal direction (reflecting the product structure of ). But then the spinor has to fulfill
in i.e. is a parallel spinor. Finally we get