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.


Introduction
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]. Furthermore, it is suggested by the cosmological and astronomical observations of Type Ia Supernovae [2], CMB radiation [1], large scale structure (LSS) [3], baryon acoustic oscillations (BAO) [4], and weak lensing [5] 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., [6]- [11], and for reviews on modified gravity, see, e.g., [12]- [17]).
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 [18]- [23]. 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 [24]- [26] (in other approach of the cyclic universe, see [27]), the ekpyrotic scenario [28], and the bouncing universe [29].
On the other hand, as a related theory to the cyclic universe, the trefoil and figure-eight knot universes have been explored in Ref. [30]. 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 Ref. [30]. Moreover, the Weierstrass ℘(t), ζ(t) and σ(t) functions and the Jacobian elliptic functions have been applied to solve several issues on astrophysics and cosmology [31]. In particular, very recently, by combining the reconstruction method in Refs. [12,22,33,34] with the Weierstrass and Jacobian elliptic functions, the equation of state (EoS) for the cyclic universes [35] and periodic generalizations of Chaplygin gas type models [36]- [38] for dark energy [39] have been examined. This procedure can be considered to 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 [30], 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], 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 [46] 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 8πG = c = 1 with G and c being the gravitational constant and the seed of light are used.
The organization of the paper is as follows. In Sec. II, we explain the model and derive the basic equations. In Sec. III, we investigate the trefoil knot universe. Next, we study the figure-eight knot universe in Sec. IV. In Sec. V we present some unknotted closed curve solutions of the model. Finally, we give conclusions in Sec. VI.

The model
In this section we briefly review some basic facts about the Einstein's field equation. We start from the standard gravitational action (chosen units are c = 8πG = 1) where R is the Ricci scalar, Λ is the cosmological constant and L m is the matter Lagrangian. For a general metric g µν , the line element is The corresponding Einstein field equations are given by where R µν is the Ricci tensor. This equation forms the mathematical basis of the theory of general relativity. In (2.3), T µν is the energy-momentum tensor of the matter field defined as 4) and satisfies the conservation equation where ∇ µ is the covariant derivative which is the relevant operator to smooth a tensor on a differentiable manifold. Eq.(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 non-linear partial differential equations. We consider the Bianchi -I metric Here the metric potentials A, B and C are functions of τ = t 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 We define a = (ABC) 1 3 as the average scale factor so that the average Hubble parameter may be defined as (2.14) 3 We write this average Hubble parameter H sometimes as 15) where 2.16) are the directional Hubble parameters in the directions of x 1 , x 2 and x 3 respectively. Hence we get the important relations where A 0 , B 0 , C 0 are integration constants. The other important cosmological quantity is the deceleration parameter q, which for our model reads as Next, we assume that the energy-momentum tensor of fluid has the form Here p i are the pressures along the x i axes recpectively, ρ is the proper density of energy. Then the Einstein equations (with gravitational units, 8πG = 1 and c = 1) read as where we assumed Λ = 0. For the metric (2.6) these equations take the forṁ In terms of the Hubble parameters this system takes the form Also we can introduce the three EoS parameters as 2.29) and the deceleration parameters Finally we want present the equation where is the average pressure. Hence we can calculate the average papameter of the EoS as Let us also we present the expression of R in terms of H i . From (2.8) and (2.16) follows Now we want present the knot and unknotted universe solutions of the system (2.21)- (2.24)

Example 3.
Now we present a new kind of the trefoil knot universes. Let the system (2.21)-(2.24) has the solution (3.44) where cn(t) ≡ cn(t, k) and sn(t) ≡ sn(t, k) are the Jacobian elliptic functions which are doubly periodic functions, k is the elliptic modulus. Fig. 8 shows the knotted closed curve corresponding to the solution (3.42)-(3.44) with (3.16). Substituting the formulas (3.42)-(3.44) into the system (2.21)-(2.24) we get the corresponding expressions for ρ and p i that gives us the parametric EoS. This parametric EoS reads as where 54) ( 3.57) The evolution of the volume for (3.57) is presented in Fig.9 The scalar curvature has the form Figure 9: The evolution of the volume of the trefoil knot universe with respect to the cosmic time τ for Eq.(3.57) In Fig.10 we plot the evolution of the R with respect of the cosmic time τ .

Other unknotted models of the universe
In this section we would like to present some unknotted but closed curve solutions of the Einstein equation for the Bianchi I type metric.
As an examples we consider the spiky and Mobious strip universe solutions.

Spiky universe solutions
Our aim in this subsection is to present some unknotted closed curve solutions namely the spiky universe solutions.
In Fig.25 we plot the evolution of the R with respect of the cosmic time τ . In this example, we Figure 25: The evolution of the R with respect of the cosmic time τ for Eq. (5.17) have shown that the Einstein equations admit the spike-like solution. We can show this solution describes the accelerated and decelerated expansion phases of the universe.
( 5.107) In Fig.34 we plot the evolution of the R with respect of the cosmic time τ .

Integrable models
The system (2.21)-(2.24) contents 4 equations for 7 unknown functions. This means we can add 3 new additional equations. It gives us in particular to construct integrable Bianchi models. In this subsection we present two examples of such integrable models.

Euler top equation
Let us we assume that A = A 1 , B = A 2 , C = A 3 obey the Euler top equation. The simple Euler top equation reads asȦ

Heisenberg ferromagnet equation
Our second example is the Heisenberg ferromagnet equation (HFE). Here we assume that the variables A = S 1 , B = S 2 , C = S 3 satisfy the equation iS t + 1 ω [S, W ] = 0, iW x + ω[S, W ] = 0, where S = S i σ i , W = W i σ i and σ i are Pauli's matrices. It the principal chiral type equation with the U = −iλS, V = − iλ ω(λ+ω) W.. Note this principal chiral type equation is the particular case of the following (2+1)-dimensional M-XCIX equation [47] (see also [48])

Conclusion
In the present paper, we have constructed several concrete models describing the trefoil and figureeight knot universes from Bianchi-type I cosmology and examined the cosmological features and properties in detail.
To realize the cyclic universes, it is necessary to a non-canonical scalar field with non well-defined vacuum in the context of the quantum field theory or extend gravity, e.g., with adding higher order derivative terms and f (R) gravity [25]. Indeed, however, these modified gravity theories have to satisfy the tests on the solar system scale as well as cosmological constraints so that those can be alternative gravitational theories to general relativity. The significant cosmological consequence of our approach is that we have shown the possibility to obtain the knot universes related to the cyclic universes from Bianchi-type I spacetime within general relativity.
Furthermore, recently it has been pointed out that the asymmetry of the EoS for the universe can lead to cosmological hysteresis [26]. On the other hand, Bianchi-type I spacetime describes the spatially anisotropic cosmology and hence the EoS for the universe has the asymmetry in the oscillating process through the expanding and contracting behaviors. Accordingly, it is considered that in the constructed models of the knot universes cosmological hysteresis could occur. The observation of this phenomenon in our models is one of our future works on the knot universes.
Finally, it should be remarked that by summarizing the results of our previous [30] and this works, the knot universes describing the cyclic universes can be realized from the homogeneous and isotropic FLRW spacetime as well as the homogeneous and anisotropic Bianchi-type I cosmology. In these series of works, the formulations of model construction method of the knot universes have been established. Thus, it can be expected that the presented formalism is useful to realize the universes with other features from both the isotropic and anisotropic spacetimes.
Finally we would like to note that all solutions presented above describe the accelerated and decelerated expansion phases of the universe.