Knot Universes in Bianchi Type I Cosmology
We investigate the trefoil and figure eight knot universes from Bianchi type I cosmology. In particular, we construct several concrete models describing the knot universes related to the cyclic universe and examine those cosmological features and properties in detail. Finally some examples of unknotted closed curves solutions (spiky and Mobius strip universes) are presented.
Inflation is one of the most important phenomena in modern cosmology and has been confirmed by recent observations on cosmic microwave background (CMB) radiation [1–4]. Furthermore, it is suggested by the cosmological and astronomical observations of Type Ia supernovae [5, 6], CMB radiation [1–4], large scale structure (LSS) [7, 8], baryon acoustic oscillations (BAO) , and weak lensing  that the expansion of the current universe is accelerating. In order to explain the late time cosmic acceleration, we need to introduce so-called dark energy in the framework of general relativity or modify the gravitational theory, which can be regarded as a kind of geometrical dark energy (for reviews on dark energy, see, e.g., [11–16], and for reviews on modified gravity, see, e.g., [17–23]).
It is considered that there happened a Big Bang singularity in the early universe. In addition, at the dark energy dominated stage, the finite-time future singularities will occur [24–70]. There also exists the possibility that a Big Crunch singularity will happen. To avoid such cosmological singularities, there are various proposals such as the cyclic universe [71–80] (in other approach of the cyclic universe, see ), the ekpyrotic scenario [82–85], and the bouncing universe [86–97].
On the other hand, as a related theory to the cyclic universe, the trefoil and figure-eight knot universes have been explored in [98–103]. In the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) and the homogeneous and anisotropic Bianchi-type I cosmologies, the geometrical description of these knot theories corresponds to oscillating solutions of the gravitational field equations. Note that the terms “the trefoil knot universe” and “the figure-eight knot universe” were introduced for the first time in [98–103]. Moreover, the Weierstrass , , and functions and the Jacobian elliptic functions have been applied to solve several issues on astrophysics and cosmology [104–106]. In particular, very recently, by combining the reconstruction method in [17, 18, 66, 107, 108] with the Weierstrass and Jacobian elliptic functions, the equation of state (EoS) for the cyclic universes  and periodic generalizations of Chaplygin gas type models [110–112] for dark energy  have been examined. This procedure can be considered to be a novel approach to cosmological models in order to investigate the properties of dark energy.
In this paper, we explore the cosmological features and properties of the trefoil and figure-eight knot universes from Bianchi-type I cosmology in detail. In particular, we construct several concrete models describing the trefoil and figure-eight knot universes based on Bianchi-type I spacetime. In our previous work [98–103], the models of the knot universes from the homogeneous and isotropic FLRW spacetime were studied. By using the equivalent procedure, as continuous investigations, in this work we explicitly demonstrate that the knot universes can be constructed by Bianchi-type I spacetime. In other words, our purpose is to establish the formalism which can describe the knot universes.
It is significant to emphasize that according to the recent cosmological data analysis [1–4], it is implied that the universe is homogeneous and isotropic. In fact, however, recently the feature of anisotropy of cosmological phenomena such as anisotropic inflation [114, 115] has also been studied in the literature. In such a cosmological sense, it can be regarded as reasonable to consider the anisotropic universe including Bianchi-type I spacetime. The units of the gravitational constant with and being the gravitational constant and the seed of light are used.
The organization of the paper is as follows. In Section 2, we explain the model and derive the basic equations. In Section 3, we investigate the trefoil knot universe. Next, we study the figure-eight knot universe in Section 4. In Section 5 we present some unknotted closed curve solutions of the model. Finally, we give conclusions in Section 6.
2. The Model
In this section we briefly review some basic facts about Einstein's field equation. We start from the standard gravitational action (chosen units are ) where is the Ricci scalar, is the cosmological constant, and is the matter Lagrangian. For a general metric , the line element is The corresponding Einstein field equations are given by where is the Ricci tensor. This equation forms the mathematical basis of the theory of general relativity. In (2.3), is the energy-momentum tensor of the matter field defined as and satisfies the conservation equation where is the covariant derivative which is the relevant operator to smooth a tensor on a differentiable manifold. Equation (2.5) yields the conservations of energy and momentums, corresponding to the independent variables involved. The general Einstein Equation (2.3) is a set of nonlinear partial differential equations. We consider the Bianchi-I metric where we assume that , , are dimensionless (usually we put ). Here the metric potentials , and are functions of alone. This insures that the model is spatially homogeneous. The statistical volume for the anisotropic Bianchi type-I model can be written as The Ricci scalar is where and so on. The nonvanishing components of Einstein tensor are We define as the average scale factor so that the average Hubble parameter may be defined as We write this average Hubble parameter sometimes as where are the directional Hubble parameters in the directions of , , and , respectively. Hence we get the important relations where are integration constants. The other important cosmological quantity is the deceleration parameter , which for our model reads as
Next, we assume that the energy-momentum tensor of fluid has the form Here are the pressures along the axes, recpectively, is the proper density of energy. Then the Einstein equations (with gravitational units, and ) read as where we assumed . For the metric (2.6) these equations take the form In terms of the Hubble parameters this system takes the form Also we can introduce the three EoS parameters as and the deceleration parameters Finally we want to present where is the average pressure. Hence we can calculate the average parameter of the EoS as Let us also present the expression of in terms of . From (2.8) and (2.13) follows Now we want to present the knot and unknotted universe solutions of the system (2.18) or its equivalent (2.19). Consider some examples.
3. The Trefoil Knot Universe
Our aim in this section is to construct the simplest examples of the knot universes, namely, the trefoil knot universes. Consider the following examples.
3.1. Example 1
Let us assume that our universe is filled by the fluid with the following parametric EoS: where Substituting these expressions for the pressures and density of energy into the system (2.18), we obtain the following solution: where are some real constants. We see that this solution describes the trefoil knot. In fact the solution (3.3) is the parametric equation of the trefoil knot. In Figure 1 we plot the trefoil knot for (3.3), where we assume and the initial conditions are . The Hubble parameters for the solution (3.3) with (3.4) read as In Figure 2 we plot the evolution of for thr solution (3.5) with (3.4). It is interesting to study the evolution of the volume of the trefoil knot universe. For our case it is given by In Figure 3 we plot the evolution of the volume of the trefoil knot universe with respect to the cosmic time for (3.6) with (3.4). To get , we must consider , if exactly for example, as . But below for simplicity we take the case (3.4). The other interesting quantity is the scalar curvature. For the trefoil knot solution (3.3), it has the form
In Figure 4 we plot the evolution of the with respect of the cosmic time .
So we have shown that the universe can live in the trefoil knot orbit according to the solution (3.3). It is interesting to note that this trefoil knot solution admits infinite number accelerated and decelerated expansion phases of the universe. To show this, as an example let us consider the solution for from (3.3) that is . In this case we have so that (accelerating phase) as and (decelerating phase) as with the transion points as , where is integer that is .
3.2. Example 2
Now we consider the following parametric EoS: where Substituting these expressions for the pressures and density of energy into the system (3.8), we obtain the following solution: We see that this solution again describes the trefoil knot but for the "coordinates" . Note that the scale factors we can recovered from (2.14). We get where are some real constants. In Figure 5 we plot the evolution of accordingly to (3.11) and for the initial conditions , where we assume that . For this example, the volume of the universe is given by The evolution of the volume for (3.12) is presented in Figure 6 for and for the intial condition .
The scalar curvature has the form In Figure 7 we plot the evolution of the with respect of the cosmic time . Finally we conclude that the Einstein equations for the Bianchi I type metric admit the trefoil knot solution of the form (3.10) or (3.11). These solutions describe the accelerated and decelerated phases of the expansion of the universe.
3.3. Example 3
Now we present a new kind of the trefoil knot universes. Let the system (2.18) has the solution where and are the Jacobian elliptic functions which are doubly periodic functions, and is the elliptic modulus. Figure 8 shows the knotted closed curve corresponding to the solution (3.14) with (3.4). Substituting (3.14) into the system (2.18) we get the corresponding expressions for and that gives us the parametric EoS. This parametric EoS reads as where The volume of the universe for the solution (3.14) with (3.4) looks like The evolution of the volume for (3.17) is presented in Figure 9 The scalar curvature has the form In Figure 10 we plot the evolution of the with respect of the cosmic time .
3.4. Example 4
Our fourth example is given by Which again the knotted closed curve in Figure 8 but for the "coordinates" . Note that the corresponding parametric EoS looks like where In Figure 11 we plot the evolution of for (3.20). The scalar curvature has the form In Figure 12 we plot the evolution of the with respect of the cosmic time .
4. The Figure-Eight Knot Universe
Our aim in this section is to demonstrate some examples of the figure-eight knot universes for the Bianchi type I metric (2.6). We give some particular figure-eight knot universe models.
4.1. Example 1
Again, let us assume that our universe is filled by the fluid with the following parametric EoS: where
Substituting these expressions for the pressuries and the density of energy into the system (2.18), we obtain the following its solution [98–103]: This solution is nothing but the parametric equation of the figure-eight knot as we can see from Figure 13, where we assume that and the initial conditions have the form . And for that reason in [98–103] we called such models as the figure-eight knot universes. Note that the “coordinates” with (3.4) satisfy the equation where . Let us calculate the volume of the universe. For our case it is given by where we used (3.4). In Figure 14 we present the evolution of the volume for the solution (4.3) with (3.4). The scalar curvature has the form In Figure 15 we plot the evolution of the with respect of the cosmic time . So we found the figure-eight knot solution of the Einstein equations which again describe the accelerated and decelerated expansion phases of the universe.
4.2. Example 2
Now we consider the system (2.19). Its solution is given by Then the coorresponding scale factors read as For this solution the parametric EoS looks like where In Figure 16 we plot the EoS (4.9). For this example, the evolution of the volume of the universe is given by The evolution of the volume is presented in Figure 17 for and for the intial condition . The scalar curvature has the form In Figure 18 we plot the evolution of the with respect of the cosmic time . Again we have shown that the Einstein equations admit the figure-eight knot solution and it again describe the accelerated and decelerated expansion phases of the universe.
4.3. Example 3
Now we present the figure-eight knot universe induced by the Jacobian elliptic functions. Let the system (2.18) have the solution Note that and are the doubly periodic Jacobian elliptic functions. Figure 19 shows the knotted closed curve corresponding to the solution (4.13) with (3.4). Substituting the formulas (4.13) into the system (2.18) we get the corresponding expressions for and that gives us the parametric EoS. The evolution of the volume of the universe for (3.4) reads as
The scalar curvature has the form