Semiclassical and quantum features of the Bianchi I cosmology
in the polymer representation

One of the most relevant phenomenological implications of Loop Quantum Gravity (LQG) [1, 2] is certainly the emergence of a Big Bounce in the quantum dynamics of the isotropic Universe [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Despite the cosmological implementation of the formalism of the full theory present some limitations, especially concerning the proper assessment of the $SU(2)$ symmetry (see [14, 15, 16]), the discreteness of the geometrical operator (area and volume) spectra is at the ground of a deep revision in the concept of the early Universe.

Actually, the emergence of a Big Bounce and the corresponding existence of a cut-off for the matter energy density take place in their own evidence mostly on a semiclassical level, where the mean value of localized wave packets follows a revised dynamics in which the singularity is removed (for completeness, see [7] where for the Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe it is obtained an analytically solvable model named sLQC and a full quantum analysis is performed). On this level, the most intriguing open question concerns the specific morphology of this semiclassical turning point in the past of the present Universe. In fact, two possible representations are possible: one imposing a kinematical cut-off on the minimal area element [5] and the other one assigning the same cut-off on a dynamical level, i.e. involving the cosmic scale factor in the discrete spectrum [6, 7]. In the first approach, the turning point in the past depends on the wave packet profile as fixed at a given instant of time and also the critical energy density can approach very large values [17, 2]. In the dynamical reformulation, both the Big Bounce and the critical energy density become intrinsic features of the Universe, namely fixed by the fundamental constants and parameters only [6]. From the point of view of the adopted configurational variables, the difference can be summarized in the use of the pure Ashtekar connection in the kinematical spectrum case or the Universe volume when an intrinsic cut-off comes out. With respect to this dualism of possible representations, see the discussion in [13] where the same question is addressed in Polymer Quantum Mechanics (PQM) by tracing a close parallelism with Loop Quantum Cosmology (LQC). Indeed, the polymer approach [18, 19] is the most simple treatment able to provide insights on the emergence of a bouncing cosmology without entering the subtleties of LQG and LQC, but reliably retaining the same information of the latter.

Here, we apply PQM to the Bianchi I model in the presence of a massless scalar field and we explore the dynamics in different sets of configurational variables by applying both the so-called semiclassical approach and the purely quantum one [19]. We compare the use of the natural Ashtekar connections to the implementation of two different sets of volume-like variables (here dubbed the anisotropic volume-like coordinates as well as the Universe volume plus two anisotropies), following the LQC formulation in [20, 21]. On a semiclassical level we show that, when PQM is applied to the anisotropic Universe dynamics, the Big Bounce as an intrinsic feature is reached only in the pure volume formulation, whereas both in the anisotropic volume-like coordinates and in the Ashtekar connections the resulting bouncing cosmology depends on the initial conditions on the system. This result suggests that it is strictly the use of the Universe volume coordinate that ensures an intrinsic cut-off, whereas the geometrical dimensions of the variables described as discrete on the polymer lattice is only linked to the determination of whether or not the Big Bounce exists (see [22] where it is shown that the Bianchi I and IX models in the Misner variables are still singular in the polymer formulation). This analysis is supported by the derivation of the polymer-modified Friedmann-like equation in the pure volume formulation, from which we can analyze the expression of the total energy density at the Bounce, including the anisotropic contribution to the standard matter energy density term. Moreover, we investigate the dynamical equivalence of the three different formulations in the semiclassical polymer approach, by generalizing the analysis performed in [13]. However, from our study comes out that only the dynamics in the Ashtekar variables and that in the anisotropic volume-like ones can be mapped, by considering the polymer parameter depending on the configurational coordinates (rather than constant) when performing the canonical change of variables.

Regarding the proper application of PQM, we analyze the evolution of the Bianchi I cosmology only limiting our attention to the choice of the Ashtekar variables (i.e. the privileged set in LQG) and the proper volume ones (i.e. the Universe volume plus two anisotropies). We choose the matter scalar field as a clock before quantizing by means of an Arnowitt-Deser-Misner (ADM) reduction of the variational principle [23]. Actually, the use of this Schrödinger-like approach is justified by the need to avoid all the issues regarding the well-definition of a conserved probability density constructed using the Wheeler-DeWitt equation (in this respect, see [24, 25]). In the Ashtekar variables the evolution of a localized quantum wave packet is studied by following the peaks of the probability density (i.e. the square modulus) and allows to show that there is a good coincidence between the semiclassical trajectories and the behavior of the Universe wave packet. Therefore, we can infer that the results obtained in the semiclassical sector in terms of the Ashtekar variables remain valid in the full quantum picture. In other words, the feature of a non-universal Big Bounce depending on the initial conditions on the wave packets seems to be well-grounded on a quantum level too. Then, in the pure volume representation the quantum mean value of the volume operator is studied, showing a good consistency with the volume semiclassical trajectory. Since the Bianchi I wave packet outlines a clear spreading over time, we also compute the standard deviation of the volume operator, highlighting its non-linear behavior over time. This result points out the need of going beyond the pure semiclassical description of the Big Bounce, in view of its intrinsic quantum nature [26]. Nonetheless, this numerical study allows us to claim the existence of the quasi-classical limit, since the relative error does not exceed the unity for a significant time interval.

The paper is structured as follows. In Sec. II the theory of PQM is introduced, in order to clarify the formalism at the ground of the following cosmological analysis. In Sec. III the main results obtained by describing the FLRW model in the polymer picture are presented, providing the analyses in the Ashtekar variables and in the volume ones into two subsections. In Sec. IV the Hamiltonian formulation of the Bianchi I model in the Ashtekar variables is described, in order to introduce the original part of the paper developed in Secc. V-VIII. In particular, in Secc. V and VI it is solved the semiclassical polymer dynamics of the Bianchi I model in the Ashtekar variables and in the two sets of volume-like variables respectively, whereas in Sec. VII a proposal about the possibility to recover the equivalence between the dynamics in the three sets of variables is investigated. Furthermore, in the Ashtekar set and in the proper volume one a full quantum treatment is introduced in Sec. VIII using a Scrödinger-like formalism. Finally, in Sec. IX some concluding remarks are commented.

As a first step, we introduce the polymer quantization of a system, which is a non-equivalent representation of the quantum mechanics with respect to the standard Schrödinger one (see [19]). This formulation is based on the assumption that one or more variables of the phase space are discretized. Hence, in this approach it is not possible to define $\hat{q}$ and $\hat{p}$ as operators at the same time. However, the power of this alternative quantum approach is the possibility to describe cut-off physics effects through a simple formalism and this is particularly useful in the cosmological setting, where the generalized coordinates are identified with the cosmic scale factors.

First of all, we introduce a set of abstract kets $\ket{\mu}$ with $\mu\in{\rm I\!R}$ and fix the inner product as

$$\langle\mu\ket{\nu}=\delta_{\mu\nu}\,,$$

(1)

being $\delta_{\mu\nu}$ a Kronecker delta. This procedure defines the non-separable Hilbert space $H_{poly}$ where we can define two fundamental operators: the label operator $\hat{\epsilon}$, whose action on the kets is given by $\hat{\epsilon}\ket{\mu}=\mu\ket{\mu}$, and the shift operator $\hat{s}(\lambda)$ ($\lambda\in{\rm I\!R}$), where $\hat{s}(\lambda)=\ket{\mu+\lambda}$. The action of $\hat{s}(\lambda)$ is discontinuous since the kets are orthonormal $\forall\lambda$. Therefore, no Hermitian operator could generate it by exponentiation.

Now, we suppose that the configurational coordinate $q$ has a discrete character, i.e. the position is defined on a lattice having a given spacing. The projection of the states in the $p$-polarization is ($\hbar=1$)

$$\psi_{\mu}(p)=\langle p|\mu\rangle=e^{ip\mu}\,.$$

(2)

By applying the shift operator on these states as

$$\hat{s}(\lambda)\psi_{\mu}(p)=e^{i\lambda p}e^{i\mu p}=\psi_{(\mu+\lambda)}(p)\,,$$

(3)

we can conclude that the operator $\hat{p}$ cannot be defined in a rigorous fashion, whereas the action of $\hat{q}$ reads as

$$\hat{q}\psi_{\mu}(p)=-i\partial_{p}e^{i\mu p}=\mu\psi_{\mu}(p)\,,$$

(4)

representing exactly that one of the label operator $\hat{\epsilon}$.

On a dynamical level we have to face the problem of defining an approximated version of the $\hat{p}$ operator, since the Hamiltonian $\mathcal{H}$ is a function of both $(q,p)$ as

$$\mathcal{H}(q,p)={p^{2}\over 2m}+V(q)\,,$$

(5)

where we have considered the case of a non-relativistic particle of mass $m$ in a potential $V(q)$. In order to overcome this problem, we introduce a regular graph

$$\gamma_{\mu_{0}}=\{q\in{\rm I\!R}\,|\,q=n\mu_{0},\forall n\in\mathbb{Z}\}\,,$$

(6)

i.e. a numerable set of equidistant points whose spacing is given by the scale $\mu_{0}$. In this way, we can restrict the action of the shift operator $e^{i\lambda p}$ by imposing $\lambda=n\mu_{0}$ in order to remain in the lattice. Then, we can use it to approximate any function of $p$ since

$$p\approx\frac{1}{\mu_{0}}\sin\left(\mu_{0}p\right)=\frac{1}{2i\mu_{0}}\left(e^{i\mu_{0}p}-e^{-i\mu_{0}p}\right)\,.$$

(7)

We notice that this approximation is good for $\mu_{0}p\ll 1$. Under this hypothesis one derives

$$\hat{p}_{\mu_{0}}\ket{\mu_{n}}=-{i\over 2\mu_{0}}(\ket{\mu_{n+1}}-\ket{\mu_{n-1}})\,.$$

(8)

Thus, it is possible to define a regularized operator $\hat{p}_{\mu_{0}}$ that depends on the scale $\mu_{0}$ and a modified version of the Hamiltonian as

$$\hat{\mathcal{H}}_{\mu_{0}}:={\hat{p}^{2}_{\mu_{0}}\over 2m}+V(\hat{q})\,.$$

(9)

In what follows we will apply this same picture to configurational variables describing the Bianchi I cosmology. Each independent variable will be associated to the representation traced above for the polymer framework.

In this section we provide a brief analysis of the homogeneous and isotropic Universe as described by the FLRW model. The ADM line element of the FLRW Universe is given by

$$ds^{2}=-N(t)^{2}dt^{2}+a^{2}(t)\Big{(}{dr^{2}\over 1-Kr^{2}}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta d\phi^{2}\Big{)}\,,$$

(10)

where $N(t)$ is the lapse function, $a(t)$ the cosmic scale factor and $K=0,\pm 1$ the signature of the spatial curvature. The factor $a(t)$ is the only degree of freedom available to define the dynamical properties of the Universe. Substituting the metric (10) in the Einstein-Hilbert action we obtain

	$\displaystyle S_{\mathrm{RW}}$	$\displaystyle=\int_{t_{1}}^{t_{2}}dt\left(p_{a}\dot{a}-N\mathcal{H}_{\mathrm{RW}}\right)$		(11)
		$\displaystyle=\int_{t_{1}}^{t_{2}}dt\left[p_{a}\dot{a}-N\left(-\frac{\kappa}{24\pi^{2}}\frac{p_{a}^{2}}{a}-\frac{6\pi^{2}K}{\kappa}a+2\pi^{2}\rho a^{3}\right)\right],$

where $\rho=\rho(a)$ is the matter energy density and $\kappa=8\pi G$ is the Einstein constant (we have set the speed of light equal to one).

To achieve a complete canonical description of the dynamics for the isotropic Universe, we consider the variations of the action above with respect to the lapse function $N$ and the conjugate variables $(a,p_{a})$. In particular, the variation of (11) with respect to $N$ provides the Hamiltonian constraint for the FLRW model

$$\mathcal{H}_{RW}=\frac{p_{a}^{2}}{a^{4}}+\frac{144\pi^{4}}{\kappa^{2}a^{2}}K-\frac{48\pi^{4}}{\kappa}\rho=0\,,$$

(12)

which coincides with the Friedmann equation

$$H^{2}=\bigg{(}{\dot{a}\over a}\bigg{)}^{2}={\kappa\rho\over 3}-{K\over a^{2}}\,.$$

(13)

Finally, we notice that the dynamics of the isotropic Universe in the Hamiltonian formulation resembles that one of a one-dimensional pinpoint particle, with generalized coordinate $a$ and momentum $p_{a}$.

The aim of this section is to introduce some general features of the classical dynamics of the Bianchi I cosmological model as expressed in terms of the Ashtekar variables (see [20]), before introducing the original analysis performed in the paper in the next section. The importance of studying this model resides in the legitimacy of considering more general cosmological models near the singularity with respect to the highly symmetric isotropic Universe.

In particular, the Bianchi I model represents the simplest homogeneous but anisotropic geometry that reduces to the flat FLRW model in the isotropic limit. Its line element reads as

$\displaystyle{ds^{2}=-N(t)^{2}dt^{2}+a_{1}^{2}dx_{1}^{2}+a_{2}^{2}dx_{2}^{2}+a_{3}^{2}dx_{3}^{2}}\,,$

(27)

where $a_{1},a_{2},a_{3}$ are the three independent scale factors, one for each direction. The phase space of the Bianchi I model in the Ashtekar variables is six-dimensional and it is expressed through the canonical couple $(c_{i},p_{j})$ defined as

$$p_{i}=|\epsilon_{ijk}a_{j}a_{k}|\text{sign}(a_{i})\,,\hskip 14.22636ptc_{i}=\gamma\dot{a}_{i}\,,$$

(28)

with $i=1,2,3$ and $\{c_{i},p_{j}\}=\kappa\gamma\delta_{ij}$. Starting from the metric (27), we find the structure of the Bianchi I Hamiltonian constraint in the Ashtekar variables when a massless scalar field is considered, i.e.

$$\mathcal{H}=-{1\over{\kappa\gamma^{2}V}}(c_{1}p_{1}c_{2}p_{2}+c_{1}p_{1}c_{3}p_{3}+c_{2}p_{2}c_{3}p_{3})+{p^{2}_{\phi}\over 2V}=0\,,$$

(29)

with $V=\sqrt{p_{1}p_{2}p_{3}}$. This constraint reduces to that one of the isotropic model (15) if the isotropy condition is imposed.

The classical dynamics of the system is clearly completed by the following Hamiltonian equations:

$$\begin{cases}\begin{aligned} &\dot{p_{i}}=Nk\gamma{{\partial\mathcal{H}}\over{\partial c_{i}}}=-{{p_{i}}\over\gamma p_{\phi}}\big{(}c_{j}p_{j}+c_{k}p_{k}\big{)}\\ &\dot{c_{i}}=-Nk\gamma{{\partial\mathcal{H}}\over{\partial p_{i}}}={c_{i}\over{\gamma p_{\phi}}}\big{(}c_{j}p_{j}+c_{k}p_{k}\big{)}\end{aligned}\end{cases}$$

(30)

for $i,j,k=1,2,3$, $i\neq j\neq k$. After imposing the time gauge $\dot{\phi}=1=:N{{\partial\mathcal{H}}\over{\partial p_{\phi}}}\Rightarrow N={V\over p_{\phi}}\,,$ we can solve equations (LABEL:AshBI) and the scalar constraint (29) by assigning proper initial conditions. In particular, the initial value problem must satisfy the Hamiltonian constraint, so that the solution can be numerically provided.

Actually, once the constants of motion

$${c_{i}p_{i}}=\mathcal{K}_{i}\,,\hskip 14.22636ptp_{\phi}=\mathcal{K}_{\phi}$$

(31)

have been identified ($i=1,2,3$), the six-equations system (36) decouples as follows:

$$\begin{cases}\begin{aligned} &{{dp_{i}}\over{d\phi}}=-{{p_{i}}\over\gamma p_{\phi}}(\mathcal{K}_{j}+\mathcal{K}_{k})\\ &{{dc_{i}}\over{d\phi}}={c_{i}\over{\gamma p_{\phi}}}(\mathcal{K}_{j}+\mathcal{K}_{k})\end{aligned}\end{cases}$$

(32)

for $i\neq j\neq k$ and $\eqref{Bsystemsolv}$ is made analytically solvable. In particular, Fig. 3 shows that the Universe volume $V=\sqrt{p_{1}p_{2}p_{3}}$ follows the classical singular behavior also in the Ashtekar variables.

In this section we present the original part of this paper by investigating the dynamics of the Bianchi I model in terms of the Ashtekar variables when the polymer paradigm is implemented. We recall that this analysis is expected to provide significantly insight on the behavior of the quantum expectation values in PQM as well in LQC dynamics.

As seen for the FLRW case in Sec. III, we proceed by imposing the polymer substitution for the configurational variables in (29)

$$c_{i}\to{1\over\mu_{i}}sin({\mu_{i}c_{i}})\,,$$

(33)

so the polymer Hamiltonian takes the form

$\displaystyle\mathcal{H}_{poly}=$

$\displaystyle-{1\over{\kappa\gamma^{2}V}}\sum_{i\neq j}{{\sin(\mu_{i}c_{i})p_{i}\sin(\mu_{j}c_{j})p_{j}}\over\mu_{i}\mu_{j}}+{p^{2}_{\phi}\over 2V}=0\,,$

(34)

where $i,j=1,2,3$ and $V=\sqrt{p_{1}p_{2}p_{3}}$. Similarly, $\phi$ represents our internal time, so $N$ is fixed by the gauge

$$\dot{\phi}:=N{{\partial\mathcal{H}_{poly}}\over{\partial p_{\phi}}}=1\Rightarrow N={{\sqrt{p_{1}p_{2}p_{3}}}\over p_{\phi}}\,,$$

(35)

in a way that the equations of motion in the polymer representation are the following:

$$\begin{cases}\begin{aligned} \small&{{dp_{i}}\over{d\phi}}=-{{p_{i}\cos(\mu_{i}c_{i})}\over\gamma p_{\phi}}\Big{[}{p_{j}\over{\mu_{j}}}\sin(\mu_{j}c_{j})+{p_{k}\over{\mu_{k}}}\sin(\mu_{k}c_{k})\Big{]}\\ \small&{{dc_{i}}\over{d\phi}}={\sin(\mu_{i}c_{i})\over{\gamma\mu_{i}p_{\phi}}}\Big{[}{p_{j}\over{\mu_{j}}}\sin(\mu_{j}c_{j})+{p_{k}\over{\mu_{k}}}\sin(\mu_{k}c_{k})\Big{]}\end{aligned}\end{cases}$$

(36)

for $i,j,k=1,2,3$, $i\neq j\neq k$. It is possible to solve this system by establishing the initial conditions on the variables $(c_{i},p_{i})$ that must satisfy the Hamiltonian constraint (34). In this respect, we make the choice¹¹1The considered initial conditions on $c_{i}$ restrict the solutions to those which are synchronized, due to the need of having a simultaneous Bounce in all the three directional scale factors.

		$\displaystyle c_{i}(0)={\pi\over 2\mu_{i}}\,,\quad p_{1}(0)=\bar{p}_{1}\,,\quad p_{2}(0)=\bar{p}_{2}\,,$		(37)
		$\displaystyle p_{3}(0)=\bar{p}_{3}=\frac{(p_{\phi}^{2}\kappa\gamma^{2}\mu_{1}\mu_{2}-2\bar{p_{1}}\bar{p}_{2})\mu_{3}}{2(\mu_{2}\bar{p}_{1}+\mu_{1}\bar{p}_{2})}\,.$

Moreover, it can be easily seen that the momentum conjugate to the scalar field is a first integral, since the variable $\phi$ is cyclic in (34), and other constants of motion can be obtained by combining the Hamilton equations (36). So, in analogy with (31) we get

$${p_{i}\sin(\mu_{i}c_{i})\over{\mu_{i}}}=\mathcal{K}_{i}\,,\hskip 14.22636ptp_{\phi}=\mathcal{K}_{\phi}\,,$$

(38)

where the considered values of $\mathcal{K}_{i}$ and $\mathcal{K}_{\phi}$ depends on the initial conditions (37). As already mentioned, identifying these first integrals makes possible to transform the six-equations system showed in (36) in the three closed systems

$$\begin{cases}\begin{aligned} \small&{{dp_{i}}\over{d\phi}}=-{{p_{i}\cos(\mu_{i}c_{i})}\over\gamma p_{\phi}}\Big{[}\mathcal{K}_{j}+\mathcal{K}_{k}\Big{]}\\ \small&{{dc_{i}}\over{d\phi}}={\sin(\mu_{i}c_{i})\over{\gamma\mu_{i}p_{\phi}}}\Big{[}\mathcal{K}_{j}+\mathcal{K}_{k}\Big{]}\end{aligned}\end{cases}$$

(39)

Thanks to this procedure, the equations of motion can be solved analytically, leading to the following solutions ($i\neq j,i,j=1,2$):

		$\displaystyle c_{i}(\phi)=\frac{2}{\mu_{i}}\text{arccot}\big{[}\text{exp}\big{(}-\frac{\bar{p}_{3}/\mu_{3}+\bar{p}_{j}/\mu_{j}}{\gamma p_{\phi}}\phi\big{)}\big{]}\,,$		(40)
		$\displaystyle p_{i}(\phi)=\bar{p}_{i}\cosh\Big{[}\frac{\bar{p}_{3}/\mu_{3}+\bar{p}_{j}/\mu_{j}}{\gamma p_{\phi}}\phi\Big{]}\,;$
		$\displaystyle c_{3}(\phi)=\frac{2}{\mu_{3}}\text{arccot}\big{[}\text{exp}\big{(}-\frac{\bar{p}_{1}/\mu_{1}+\bar{p}_{2}/\mu_{2}}{\gamma p_{\phi}}\phi\big{)}\big{]}\,,$
		$\displaystyle p_{3}(\phi)=\bar{p}_{3}\cosh\Big{[}\frac{(\bar{p}_{1}/\mu_{1}+\bar{p}_{2}/\mu_{2})}{\gamma p_{\phi}}\phi\Big{]}\,.$

Using (40), we can obtain the Universe volume behavior $V(\phi)=\sqrt{|p_{1}(\phi)p_{2}(\phi)p_{3}(\phi)|}$ that is shown in Fig. 4. The resulting trajectory highlights that a semiclassical Big Bounce replaces the classical Big Bang thanks to the regularizing polymer effects, which are expected to become dominant near the Planckian region.

In this section we study the dynamics of the Bianchi I Universe for a new choice of variables, in complete analogy with the analysis performed for the FLRW model in Sec. III.1.2. More specifically, the anisotropic character of the Bianchi I model leads to the possibility of taking into account two different sets of volume-like variables, that coincide in the case of the isotropic model. Then, we will compare the obtained results.