Exact solutions for Big Bounce in loop quantum cosmology

Exact solutions for Big Bounce in loop quantum cosmology

Jakub Mielczarek jakubm@poczta.onet.pl Astronomical Observatory, Jagiellonian University, 30-244 Kraków, Orla 171, Poland The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen, Denmark    Tomasz Stachowiak toms@oa.uj.edu.pl Astronomical Observatory, Jagiellonian University, 30-244 Kraków, Orla 171, Poland    Marek Szydłowski uoszydlo@cyf-kr.edu.pl Department of Theoretical Physics, Catholic University of Lublin, Al. Racławickie 14, 20-950 Lublin, Poland Marc Kac Complex Systems Research Centre, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

In this paper we study the flat () cosmological FRW model with holonomy corrections of Loop Quantum Gravity. The considered universe contains a massless scalar field and the cosmological constant . We find analytical solutions for this model in different configurations and investigate its dynamical behaviour in the whole phase space. We show the explicit influence of on the qualitative and quantitative character of solutions. Even in the case of positive the oscillating solutions without the initial and final singularity appear as a generic case for some quantisation schemes.

I Introduction

In recent years Loop Quantum Cosmology (LQC) has inspired realisation of the cosmological scenario in which the initial singularity is replaced by the bounce. In this picture, the Universe is initially in the contracting phase, reaches the minimal, nonzero volume and, thanks to quantum repulsion, evolves toward the expanding phase. Such a scenario has been extensively studied with use of the numerical methods Singh:2006im ; Ashtekar:2006uz . However, as it was shown for example in Stachowiak:2006uh exact solutions for bouncing universe with dust and cosmological constant can be found. The aim of the present paper is to show that analytical solutions can also be obtained for the bouncing models arising from LQC. The main advantage of such exact solutions is that they allow for investigations in whole ranges of the parameter domains.

In this paper we consider the flat FRW model with a free scalar field and with the cosmological constant. Quantum effects are introduced in terms of correction to the classical theory. Generally one considers two types of of quantum correction: correction from inverse volume and holonomy corrections. The leading effect of the volume corrections is the appearance of the super-inflationary phase. The effect of holonomy corrections, on the other hand, is the appearance of a bounce instead of singularity. The aim of this paper is to investigate analytically these effects in a flat FRW model. That is to say, we neglect corrections from inverse volume, these effects however, has been extensively studied elsewhere. Moreover, these two types of corrections are not equally important in the same regimes. The inverse volume corrections are mainly important for small values of the scale factor, whereas holonomy corrections are mainly important for large values of the Hubble parameter. In other words, when the minimal scale factor (during the bounce) is large enough, the effects of inverse volume corrections can be neglected.

The flat FRW model in the Loop Quantum Cosmology has been first investigated in the pioneer works of Bojowald Bojowald:2002gz ; Bojowald:2003md and later improved in the works of Ashtekar, Pawłowski and Singh Ashtekar:2006rx ; Ashtekar:2006uz ; Ashtekar:2006wn . Bojowald’s original description of the quantum universe in currently explored in the number of works and regarded as a parallel line of research Bojowald:2006gr ; Bojowald:2008pu . In the present paper, we restrict ourselves to the flat FRW models arising in the framework proposed by Ashtekar and co-workers. Beside the flat models this approach has also been applied to the FRW models in Ashtekar:2006es ; Vandersloot:2006ws ; Szulc:2006ep ; Szulc:2007uk and Bianchi I in Chiou:2007mg ; Chiou:2007dn . In these models the unambiguity in the choice of the elementary area for the holonomy corrections appear. In the present paper we consider two kind of approaches to this problem: the so called scheme and scheme (for a more detailed description see Appendix A). We find analytical solutions for the considered models in these two schemes.

The Hamiltonian of the considered model is given by


In Appendix A we show the derivation of this Hamiltonian in the Loop Quantum Gravity setting. The canonical variables for the gravitational field are and for the scalar field . The canonical variables for the gravitational field can be expressed in terms of the standard FRW variables . Where the factor is called Barbero-Immirzi parameter and is a constant of the theory, and is the volume of the fiducial cell. The volume is just a scaling factor and can be chosen arbitrarily in the domain . Since is the more natural variable than here, we present mostly in the figures. is always the positive square root of so the shape of the graphs would be essentially the same when drawn with . The equations of motions can now be derived with the use of the Hamilton equation


where the Poisson bracket is defined as follows


From this definition we can immediately retrieve the elementary brackets


With use of the Hamiltonian (1) and equation (2) we obtain equations of motion for the canonical variables


where . The Hamiltonian constraint implies


The variable corresponds to the dimensionless length of the edge of the elementary loop and can be written in the general form


where and is a constant (this comes from the fact that is positively defined). The choice of and depends on the particular scheme in the holonomy corrections. In particular, boundary values correspond to the cases when is the physical distance (, scheme) and coordinate distance (, scheme). However, the case does not lead to the correct classical limit. When , the classical limit can not be recovered either. Only for negative values of is the classical limit correctly recovered


Strict motivation of the domain of the parameter comes form the investigation of the lattice states Bojowald:2006qu . The number of the lattice blocks is expressed as where is the average length of the lattice edge. This value is connected to the earlier introduced length , namely . During evolution the increase of the total volume is due to the increase of the spin labels on the graph edges or due to the increase of the number of vortices. In this former case the number of the lattice blocks is constant during evolution, . Otherwise, when the spin labels do not change, the number of vortices scales with the volume, . The physical evolution correspond to something in the middle, it means the power index lies in the range . Applying the definition of we see that the considered boundary values translate to the domain of introduced earlier, . More detailed investigation of the considered ambiguities can be found in the papers Bojowald:2008pu ; Bojowald:2007yy and in the Appendix A.

Combining equations (6), (7) and the first one from the set of equations (5) we obtain


where new parameters are defined as follow


Equation (9) is, in fact, a modified Friedmann equation


where the effective constants are expressed as follow




We will study the solutions of the equation (9) for both scheme and scheme.

The organisation of the text is the following. In section II we consider models with . We find solutions of the equations (9) for both scheme and scheme. Next, in section III we add to our considerations a non-vanishing cosmological constant . We carry out an analysis similar to the case without lambda. We find analytical solutions of the equation(9) for scheme. Then, we study the behaviour of this case in scheme. In section V we summarise the results.

Ii Models with

In this section we begin our considerations with the model without . Equation (9) is then simplified to the form


We solve this equation for both scheme and scheme in the present section.

ii.1 scheme  

In the scheme, as is explained in the Appendix A, the is expressed as


where is the area gap. So and .

To solve (20) in the considered scheme we introduce a new dependent variables in the form


With use of the variable the equation (20) takes the form


and has a solution in the form of a second order polynomial


where is a constant of integration. We can choose now , so that the minimum of occurs for . Going back to the canonical variable we obtain a bouncing solution


The main property of this solutions is that never reaches zero value for non vanishing . The minimal value of is given by


We show the solution (25) in the Fig. 1.

Figure 1: Typical solution with in the scheme. The canonical variable is expressed in the units and time in the units.

The dynamical behaviour in this model is simple. For negative times we have a contracting pre-Big Bang Universe. For we have a Big Bang evolution from minimal (26). It is, however, convenient to call this kind of stage Big Bounce rather than Big Bang because of initial singularity avoidance.

ii.2 Scheme  

In the Scheme the is expressed as


So and . To solve (20) in this scheme, we change the time


and introduce a new variable as follows


Applying this new parametrisation to the equation (20) leads to the equation in the form


which has a solution


We can now choose so that the minimal value of occurs for . Now we can go back to the initial parameters and , then


Introducing the variable


we can rewrite integral (33) to the simplest form


and the solution of such integral is given by


where is the hypergeometric function, defined as


where is the Pochhammer symbol defined as follow


This solution is very similar to the one in scheme. However, the time parametrisation is expressed in a more complex way. We show this solution in the Fig. 2.

Figure 2: Typical solution with in the scheme. The canonical variable is expressed in units and time in the units.

In this case minimal value of is expressed as


Iii Models with

In this section, we investigate the general model with non vanishing cosmological constant. It will be useful to write equation (9) in the form


We see that when we perform the multiplication in this equation and define


we recover equation (9). In this and the next section we use equivalently and to simplify notation.

iii.1 scheme  

In this case equation (40) can be rewritten to the form


where the parameters are defined as


To solve equation (42) we re-parametrise the time variable


and introduce a new variable as follows


This change of variables leads to the equation in the form




There are three general types of solutions corresponding to the values of the parameters . We summarise these possibilities in the table below.

It is important to notice that the product is negative in all cases. This property will be useful to solve equation (48). In the Fig. 3 we show values of the roots and as functions of .

Figure 3: Parameters and as functions of . Region 1 corresponds to oscillatory solution, region 2 to bouncing solution. There is no physical solutions in region 3. The parameter is expressed in units of .

Thus, there are two values of cosmological constants where the signs of the roots change, namely and


where we have used the value of the calculated in the work Ashtekar:1997yu . More recent investigation of the black hole entropy indicate however that the value of the Barbero-Immirzi parameter is Domagala:2004jt . In particular, Meissner has calculated Meissner:2004ju . However, this freedom in the choice of does not change the qualitative results. Only the region of the parameter space where the particular kind of motion is allowed can be shifted.

We now perform a change of variables in equation (48) in the form




We also introduce the parameters


and then equation (48) takes the form


The equation (55) is the equation of shifted harmonic oscillator and its solution is


where we set the integration constant to zero. It is now possible to go back to the initial parameters and which are expressed as follows


The integral (58) can be easily solved but solutions depend on the parameters in (57).

We define now an integral


so the expression (58) can now be written as


Solutions of integral (59) depend on the value of parameter and read

In Fig. 4 we show the solution for – or equivalently with parameter .

Figure 4: Parametric solution with . Oscillating curve (green) is and the increasing curve (red) is . The canonical variable is expressed in the units and time in the units.
Figure 5: Solution with . The canonical variable is expressed in units of and time in .

In Fig. 5 we show the solution for which corresponds to .

Figure 6: Parametric solution with . Top curve (green) shows and bottom curve (red) is . The canonical variable is expressed in units of and time in .
Figure 7: Solution with . The canonical variable is expressed in units of and time in .

iii.2 Scheme  

In this section we study the last case in which and . Equation (40) can then be written in the form




and the polynomial’s coefficients are expressed as


The discriminant of polynomial (62) is


Inserting the values of parameters listed above we obtain


For , or


polynomial (62) has three real roots. When relation (69) is fulfilled and , oscillatory solutions occur. For equation (61) has bouncing type solutions. This can be seen when we redefine equation (61) to the point particle form. This approach is useful for qualitative analysis and will be fully used in the next section. However, here we will use it to distinguish between different types of solutions.

Introducing a new time variable


we can rewrite equation (61) to the form


where the potential function


Equation (71) has the form of Hamiltonian constraint for a point particle in a potential well. We see that for potential (72) has only one extremum (for ) and only a bouncing solution is possible. For potential (72) has a minimum for . In this case physical solutions correspond to the condition and the energy of the imagined particle in the potential well is greater than the minimum of well. The particle then oscillates between the boundaries of the potential. The condition is equivalent to relation (69) calculated from the discriminant.

Upon introducing the parameters


and the variable


then equation (61) takes on the form of the Weierstrass equation


Detailed analysis and plots of solutions of this equation for different values of parameters can be found in the appendix to the article Dabrowski:2004hx . The solution of equation (76) is the Weierstrass -function




So the parametric solution for the parameter is


The time variable can be expressed as the integral


In the Fig. 8 and 9 we show an exemplary parametric bounce solution with and a possible oscillatory solution with .

Figure 8: Parametric solution with . The canonical variable is expressed in units of , is dimensionless.
Figure 9: Parametric solution with and relation (69) satisfied. The canonical variable is expressed in units of and is dimensionless.

In general the solution is expressible as an explicit function of time, by means of the so called Abelian functions. However, we chose not to employ them here as the appropriate formulae are not as clear, and the commonly used numerical packages do not allow for their direct plotting. The fact of existence of such solutions allows us to assume that the above integral is well defined, and so is the solution itself.

Iv Qualitative methods of differential equations in study evolutional paths

The main advantage of using qualitative methods of differential equations (dynamical systems methods) is the investigation of all solutions for all admissible initial conditions. We demonstrate that dynamics of the model can be reduced to the form of two dimensional autonomous dynamical system. In our case the phase space is . First we can find the solutions corresponding to vanishing of the right hand sides of the system which are called critical points. The information about their stability and character is contained in the linearisation matrix around a given critical point. In the considered case, the dynamical system is of the Newtonian type. For such a system, the characteristic equation which determines the eigenvalues of the linearisation matrix at the critical point is of the form where is a potential function and , being the scale factor. As it is well known for dynamical systems of the Newtonian type, only two types of critical points are admissible. If the diagram of the potential function is upper convex then eigenvalues are real and of opposite signs, and the corresponding critical point is of the saddle type. In the opposite case, if then the eigenvalues are purely imaginary and conjugate. The corresponding critical point is of the centre type.

Equation (40) can be written in the form


where or equivalently as


where we have made the following time reparametrisation


Now we are able to write the Hamiltonian constraint in the form analogous to the particle of the unit mass moving in the one dimensional potential well


where the potential function


As we can see, the constant plays the role of the total energy of the fictitious particle. The domain admissible for motion in the configuration space is determined by the condition .

A dynamical system of the Hamiltonian type has the following form


where the prime denotes differentiation with respect to a new re-parametrised time variable which is a monotonous function of the original, cosmological time.

The structure of the phase plane is organised by the number and location of critical points. In our case, critical points in the finite domain are located only on the line and the second coordinate is determined form the equation which is


The number of critical points depends on the value of and . We can distinguish two cases:

  • for :

    : ;

    : and

  • for :

    : ;

    : degenerate .

The full analysis of the behaviour of trajectories requires investigation also at the infinity. To this aim we introduce radial coordinates on the phase plane for compactification of the plane by adjoining the circle at infinity , .

The phase portraits for both cases are shown at Figs. 10 and 11.

For the general case of we can write the parametric equation of the boundary of the physically admissible region in the phase space, namely


where the dot denotes differentiation with respect to cosmological time . This equation greatly simplifies for the special case , and we receive the value of the Hubble parameter at the boundary


a) b)

Figure 10: The phase space diagram for the case and a) and b) . The physical domain admissible for motion is shaded. Note that for the case (a) the boundary of admissible for motion is bounded by a homoclinic orbit and all solutions in this area are oscillating without initial and final singularities. All trajectories situated in physical region posses the minimum value of scale factor.

a) b)

Figure 11: The phase space diagram for the case and a) and b) . The physical domain admissible for motion is shaded. In the second case there is no physically allowed region for which . This case is distinguished by behaviour at infinity when Hubble function is finite like for the de Sitter solution (see formula (90)). This state is the global attractor in the future. All solutions are of the bouncing type.

In this special case and we can integrate eq.(1) for and choosing the integration constant equal to zero


which gives the maximal value o parameter


In the same case with and choosing integration constant equal to zero we have


For and in the case we obtain the de Sitter solution


These solutions represent the lines on the boundaries of the physically admissible domains.

V Summary

We have studied dynamics and analytical solutions of the flat Friedmann-Robertson-Walker cosmological model with a free scalar field and the cosmological constant, modified by the holonomy corrections of Loop Quantum Gravity.

We performed calculations in two setups called scheme and scheme, explained in the appendix A. We have explored whole range and whole allowed range. In the case of scheme resulting solutions are of oscillating type for and bouncing type for . For there are no physical solutions. In the case of scheme for bouncing solutions occur. When both and relation (69) are fulfilled, oscillatory behaviour occurs. Otherwise, bouncing solutions appear. In all considered cases with the initial singularity is avoided.

We have investigated the evolutional paths of the model, from the point of view of qualitative methods of dynamical systems of differential equations. We found that in the special case of the boundary trajectory approaches the de Sitter state; and demonstrate that in the case of positive cosmological constant there are two types of dynamical behaviours in the finite domain. For the case of there appear oscillating solutions without the initial and final singularities, and that they change into bouncing for .

The results of this paper can give helpful background dynamics to study variety of physical phenomenas during the bounce epoch in Loop Quantum Cosmology. For example, the interesting question of the fluctuations like gravitational wavesBojowald:2007cd ; Mielczarek:2007zy ; Mielczarek:2007wc or scalar perturbations Bojowald:2006tm during this period. We tried to show that numerical calculations can be “shifted” one step further, since the basic model is explicitly solvable, and can be treated as starting ground for more complex problems, like the above, which cannot be solved analytically.

Authors are grateful to Orest Hrycyna for discussion. This work was supported in part by the Marie Curie Actions Transfer of Knowledge project COCOS (contract MTKD-CT-2004-517186).

Appendix A Flat FRW model and holonomy corrections in Loop Quantum Gravity

In this appendix, we derive the form of the Hamiltonian (1) considered in the paper.

The FRW spacetime metric can be written as


where is the lapse function and the spatial part of the metric is expressed as