Exact solutions for Big Bounce in loop quantum cosmology
Abstract
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 superinflationary 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 coworkers. 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
(1) 
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 BarberoImmirzi 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
(2) 
where the Poisson bracket is defined as follows
(3)  
From this definition we can immediately retrieve the elementary brackets
(4) 
With use of the Hamiltonian (1) and equation (2) we obtain equations of motion for the canonical variables
(5) 
where . The Hamiltonian constraint implies
(6) 
The variable corresponds to the dimensionless length of the edge of the elementary loop and can be written in the general form
(7) 
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
(8) 
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
(9)  
where new parameters are defined as follow
(10)  
(11)  
(12)  
(13)  
(14) 
Equation (9) is, in fact, a modified Friedmann equation
(15) 
where the effective constants are expressed as follow
(16)  
(17) 
and
(18)  
(19) 
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 nonvanishing 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
(20) 
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
(21) 
where is the area gap. So and .
To solve (20) in the considered scheme we introduce a new dependent variables in the form
(22) 
With use of the variable the equation (20) takes the form
(23) 
and has a solution in the form of a second order polynomial
(24) 
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
(25) 
The main property of this solutions is that never reaches zero value for non vanishing . The minimal value of is given by
(26) 
The dynamical behaviour in this model is simple. For negative times we have a contracting preBig 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
(27) 
So and . To solve (20) in this scheme, we change the time
(28) 
and introduce a new variable as follows
(29) 
Applying this new parametrisation to the equation (20) leads to the equation in the form
(30) 
which has a solution
(31) 
We can now choose so that the minimal value of occurs for . Now we can go back to the initial parameters and , then
(32)  
(33) 
Introducing the variable
(34) 
we can rewrite integral (33) to the simplest form
(35) 
and the solution of such integral is given by
(36)  
where is the hypergeometric function, defined as
(37) 
where is the Pochhammer symbol defined as follow
(38) 
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.
In this case minimal value of is expressed as
(39) 
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
(40)  
We see that when we perform the multiplication in this equation and define
(41) 
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
(42) 
where the parameters are defined as
(43)  
(44)  
(45) 
To solve equation (42) we reparametrise the time variable
(46) 
and introduce a new variable as follows
(47) 
This change of variables leads to the equation in the form
(48) 
where
(49)  
(50) 
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 .
Thus, there are two values of cosmological constants where the signs of the roots change, namely and
(51) 
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 BarberoImmirzi 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
(52) 
and
(53) 
We also introduce the parameters
(54) 
and then equation (48) takes the form
(55) 
The equation (55) is the equation of shifted harmonic oscillator and its solution is
(56) 
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
(57)  
(58) 
The integral (58) can be easily solved but solutions depend on the parameters in (57).
We define now an integral
(59) 
so the expression (58) can now be written as
(60) 
Solutions of integral (59) depend on the value of parameter and read
In Fig. 4 we show the solution for – or equivalently with parameter .
In Fig. 5 we show the solution for which corresponds to .
iii.2 Scheme
In this section we study the last case in which and . Equation (40) can then be written in the form
(61) 
where
(62) 
and the polynomial’s coefficients are expressed as
(63)  
(64)  
(65)  
(66) 
The discriminant of polynomial (62) is
(67) 
Inserting the values of parameters listed above we obtain
(68) 
For , or
(69) 
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
(70) 
we can rewrite equation (61) to the form
(71) 
where the potential function
(72) 
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
(73)  
(74) 
and the variable
(75) 
then equation (61) takes on the form of the Weierstrass equation
(76) 
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
(77) 
where
(78)  
(79) 
So the parametric solution for the parameter is
(80) 
The time variable can be expressed as the integral
(81) 
In the Fig. 8 and 9 we show an exemplary parametric bounce solution with and a possible oscillatory solution with .
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
(82) 
where or equivalently as
(83) 
where we have made the following time reparametrisation
(84) 
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
(85) 
where the potential function
(86) 
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
(87) 
where the prime denotes differentiation with respect to a new reparametrised 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
(88) 
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 , .
For the general case of we can write the parametric equation of the boundary of the physically admissible region in the phase space, namely
(89) 
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
(90) 
In this special case and we can integrate eq.(1) for and choosing the integration constant equal to zero
(91) 
which gives the maximal value o parameter
(92) 
In the same case with and choosing integration constant equal to zero we have
(93) 
For and in the case we obtain the de Sitter solution
(94) 
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 FriedmannRobertsonWalker 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.
Acknowledgements.
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 MTKDCT2004517186).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
(95) 
where is the lapse function and the spatial part of the metric is expressed as