Canonical quantization of anisotropic Bianchi I cosmology from scalar vector tensor Brans Dicke gravity

TL;DR

This paper performs canonical quantization of an anisotropic Bianchi I cosmology within a scalar-vector-tensor Brans Dicke gravity framework, revealing quantum conditions and implications for early universe inflation.

Contribution

It introduces a novel quantization approach for anisotropic cosmology using scalar-vector-tensor Brans Dicke gravity, deriving new quantum conditions and insights into inflation.

Findings

Wave functional described by Hermit polynomials

Quantization condition on ADM mass related to quantum numbers

Inflationary expansion can originate from big bang without naked singularity

Abstract

We applied a generalized scalar-vector-tensor Brans Dicke gravity model to study canonical quantization of an anisotropic Bianchi I cosmological model. Regarding an anisotropic Harmonic Oscillator potential we show that the corresponding Wheeler de Witt wave functional of the system is described by Hermit polynomials. We obtained a quantization condition on the ADM mass of the cosmological system which raises versus the quantum numbers of the Hermit polynomials. Our calculations show that the inflationary expansion of the universe can be originate from the big bang with no naked singularity due to the uncertainty principle.

Equations

$$I=\frac{1}{16\pi }\int d^{4}x\sqrt{-g}\left\{\varphi R-\frac{\omega }{\varphi }{g}^{\mu \nu }{\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi \right\}$$

$$+\frac{1}{16\pi }\int d^{4}x\sqrt{-g}\{\zeta (x^{\nu })(g^{\mu \nu }{N}_{\mu }{N}_{\nu }+1)+2\varphi F_{\mu \nu }{F}^{\mu \nu }+U(\varphi ,N_{\mu })$$

$$-\varphi N_{\mu }{N}^{\nu }(2F^{\mu \lambda }{\Omega }_{\nu \lambda }+F^{\mu \lambda }{F}_{\nu \lambda }+{\Omega }^{\mu \lambda }{\Omega }_{\nu \lambda }-2R_{\nu }^{\mu }+\frac{2\omega }{{\varphi }^2}{\nabla }^{\mu }\varphi {\nabla }_{\nu }\varphi )\},$$

$$F_{\mu \nu }=2({\nabla }_{\mu }{N}_{\nu }-{\nabla }_{\nu }{N}_{\mu }),{\Omega }_{\mu \nu }=2({\nabla }_{\mu }{N}_{\nu }+{\nabla }_{\nu }{N}_{\mu })$$

$$g_{\mu \nu }{N}^{\mu }{N}^{\nu }=-1.$$

$$d{s}^2=-d{t}^2+e^{2a(t)}\{e^{-4b(t)}d{x}^2+e^{2b(t)}(d{y}^2+d{z}^2)\}$$

$$N_{\mu}(t)=\left(\begin{array}[]{c}N_{t}\\ N_{x}\\ N_{y}\\ N_{z}\\ \end{array}\right)=\left(\begin{array}[]{c}\cosh\alpha\\ e^{a-2b}\sinh\alpha\cos\beta\\ e^{a+b}\sinh\alpha\sin\beta\cos\gamma\\ e^{a+b}\sinh\alpha\sin\beta\sin\gamma\\ \end{array}\right)$$

$$I=\frac{1}{16\pi}\int dxdydz\int dte^{3a}\bigg{[}-\omega\frac{\dot{\phi}^{2}}{\phi}-6\phi\dot{a}^{2}+6\phi\dot{b}^{2}-U(a,b,\phi)\bigg{]}$$

$$dt=F(t)d\tau ,F=\frac{e^{3a}\varphi }{16\pi }$$

$$\varphi =\frac{e^{Z(\tau )}}{G}$$

$$\mathcal{L}=-\omega Z^{\prime 2}-6a^{\prime 2}+6b^{\prime 2}-V(a,b,Z)$$

$$V=\frac{F^{2}U}{\varphi }$$

$${\Pi }_a=\frac{\partial \mathcal{L}}{\partial a^{\prime }},{\Pi }_b=\frac{\partial \mathcal{L}}{\partial b^{\prime }},{\Pi }_Z=\frac{\partial \mathcal{L}}{\partial Z^{\prime }}$$

$$\mathcal{H}=a^{\prime }{\Pi }_a+b^{\prime }{\Pi }_b+Z^{\prime }{\Pi }_Z-\mathcal{L}$$

$$\mathcal{H}=-\frac{{\Pi }_a^2}{24}+\frac{{\Pi }_b^2}{24}-\frac{{\Pi }_Z^2}{4\omega }+V(a,b,Z)$$

$$V(a,b,Z)=\frac{1}{2}{K}_a{a}^2-\frac{1}{2}{K}_b{b}^2+\frac{1}{2}{K}_Z{Z}^2.$$

$${\hat{\Pi }}_a=-i\frac{\delta }{\delta a},{\hat{\Pi }}_b=-i\frac{\delta }{\delta b},{\hat{\Pi }}_Z=-i\frac{\delta }{\delta Z},$$

$$\bigg{\{}\frac{1}{24}\frac{\delta^{2}}{\delta a^{2}}-\frac{1}{24}\frac{\delta^{2}}{\delta b^{2}}+\frac{1}{4\omega}\frac{\delta^{2}}{\delta Z^{2}}-\frac{K_{a}}{2}a^{2}+\frac{K_{b}}{2}b^{2}-\frac{K_{Z}}{2}Z^{2}\bigg{\}}\Psi=0.$$

$$\bigg{[}\frac{1}{24}\frac{\delta^{2}}{\delta a^{2}}+\epsilon_{a}-\frac{K_{a}}{2}a^{2}\bigg{]}A(a)=0$$

$$\bigg{[}\frac{1}{24}\frac{\delta^{2}}{\delta b^{2}}+\epsilon_{b}-\frac{K_{b}}{2}b^{2}\bigg{]}B(b)=0$$

$$\bigg{[}\frac{1}{4\omega}\frac{\delta^{2}}{\delta Z^{2}}+\epsilon_{Z}-\frac{K_{Z}}{2}Z^{2}\bigg{]}C(Z)=0$$

$${ϵ}_a+{ϵ}_Z={ϵ}_b.$$

$$\epsilon_{a}=\bigg{(}N_{a}+\frac{1}{2}\bigg{)}\sqrt{\frac{K_{a}}{12}},~{}~{}~{}\epsilon_{b}=\bigg{(}N_{b}+\frac{1}{2}\bigg{)}\sqrt{\frac{K_{b}}{12}},~{}~{}~{}\epsilon_{Z}=\bigg{(}N_{Z}+\frac{1}{2}\bigg{)}\sqrt{\frac{K_{Z}}{2\omega}}$$

$$A_{N_a}(a)=\frac{H_{N_a}({\lambda }_{a}a)e^{-{\lambda }_a{a}^2/2}}{\sqrt{2^{N_a}{N}_a!\sqrt{\pi }}},{\lambda }_a=(12K_X{)}^{\frac{1}{4}}$$

$$B_{N_b}(b)=\frac{H_{N_b}({\lambda }_{b}b)e^{-{\lambda }_b{b}^2/2}}{\sqrt{2^{N_b}{N}_b!\sqrt{\pi }}},{\lambda }_b=(12K_b{)}^{\frac{1}{4}}$$

$$C_{N_Z}(Z)=\frac{H_{N_Z}({\lambda }_{Z}Z)e^{-{\lambda }_Z{Z}^2/2}}{\sqrt{2^{N_Z}{N}_Z!\sqrt{\pi }}},{\lambda }_Z=(2\omega K_Z{)}^{\frac{1}{4}}$$

$$\Psi (X,Y,X)={\Sigma }_{N_a=0}^{\infty }{\Sigma }_{N_b=0}^{\infty }{\Sigma }_{N_Z=0}^{\infty }{D}_{N_a{N}_b{N}_Z}{\Psi }_{N_a{N}_b{N}_Z}(a,b,Z)$$

$$\bigg{(}N_{a}+\frac{1}{2}\bigg{)}\sqrt{\frac{K_{a}}{K_{b}}}+\bigg{(}N_{Z}+\frac{1}{2}\bigg{)}\sqrt{\frac{6}{\omega}}\sqrt{\frac{K_{Z}}{K_{b}}}=\bigg{(}N_{b}+\frac{1}{2}\bigg{)}.$$

$$\mathcal{M}={ϵ}_b-({ϵ}_a+{ϵ}_Z)$$

$$\mathcal{M}_{N_{a}N_{b}N_{Z}}=\bigg{(}N_{b}+\frac{1}{2}\bigg{)}\sqrt{\frac{K_{b}}{12}}-\bigg{(}N_{a}+\frac{1}{2}\bigg{)}\sqrt{\frac{K_{a}}{12}}-\bigg{(}N_{Z}+\frac{1}{2}\bigg{)}\sqrt{\frac{K_{Z}}{2\omega}}$$

Full text

1 Introduction

The standard cosmology has a great success in explaining the observations of the cosmic microwave background radiation (CMBR) temperature [1, 2, 3, 4]. This model is based on the validity of the cosmological principle (the spatial homogeneity and isotropy but not in the time direction) and the Einstein‘s general relativity explain most large-scale observations with unprecedented accuracy. However, several directional anomalies have been reported in various large-scale observations (see [5] and references therein). They have no place in standard cosmology and are not being studied. In fact origin of these anomalies do not understood and so there are two different proposals to understand them as follows: a) Perhaps they are originated from cosmological effects which should be described via alternative gravity theories instead of the Einstein‘s general theory of relativity. b) Other possibility which arises these directional anomalies can be systematic errors or contaminations of measuring instruments and etc., which should be exclude from the future data analysis. In the latter case one usually accept validity of the standard cosmological $\Lambda CDM$ model while in the former proposal one use an alternative gravity model instead of the Einstein‘s general theory of relativity. Zhao and Santos, provided full review about these proposals in ref. [6] where the directional anomalies predict a preferred axis $"Axis~{}of~{}Evil"$ in large scale of the Universe. However, the general covariance principal leads us to believe that these anomalies have cosmological origin and it can be described by some alternative gravity models where the cosmological principals should be violated (see ([5]) and reference therein). To describe the above mentioned anomalies the anisotropic Bianchi cosmological models are applicable [7, 8, 9] for anisotropic cosmological constant and dark energy. As an alternative gravity model we consider scalar-vector-tensor gravity model [10, 11] which is made from generalization of the well known Jordan-Brans-Dicke scalar tensor gravity [12] by transforming the background metric as $g_{\mu\nu}\to g_{\mu\nu}+2N_{\mu}N_{\nu}$. $N_{\mu}$ is dynamical four vector field which can be called as four velocity of a preferred reference frame. Several classical and quantum applications of this model are studied previously for FLRW cosmology which are addressed in references of the work [13]. As an application of the gravity model for anisotropic cosmological models we applied it to study Bianchi I classical cosmology in ref. [5] where the corresponding Freedmann equations read to anisotropic inflationary expanding universe and they satisfied observational data successfully. In the present work we want to resolve the naked singularity of the anisotropic Bianchi I cosmological model by applying the canonical quantization approach. To do so we use self-interaction anisotropic three dimensional Harmonic Oscillator potential to calculate Hamiltonian operator of the system. Then we obtain Wheeler de Witt wave functional of the anisotropic Bianchi I background metric versus the Hermit polynomials. Outlook of this work is to present a quantization condition on the energy density of the system with no naked singularity where the system is stable at high energy quantum level. Organization of the paper is as follows.

In section 2, we introduce the scalar-vector-tensor gravity model [10, 11] under consideration. In section 3, we use the Bianchi I type of the background metric to obtain exact form of Hamiltonian density. In section 4 we solve Wheeler de Witt wave equation of the system and obtain a quantization condition on the energy density of the Bianchi I quantum cosmological system. The obtained Wheeler de Witt wave solution is described versus the quantum anisotropic harmonic Oscillator (Hermit polynomials) for three dimensional anisotropic Oscillator potential. In section 5 we discuss about the quantization of the ADM energy of the system. In section 6 we discuss about the big bang naked singularity of the expanding universe which how can it is removed in the quantum perspective of the system. Section 7 denotes to concluding remark.

2 The gravity model

Let us start with the Brans-Dicke scalar-vector tensor gravity [10, 11]

[TABLE]

[TABLE]

[TABLE]

where $g$ is absolute value of determinant of the metric tensor $g_{\mu\nu}$ with signature as (-,+,+,+), $\phi$ is the Brans-Dicke scalar field and $\omega$ is the Brans-Dicke adjustable coupling constant. The tensor fields $F_{\mu\nu}$ and $\Omega_{\mu\nu}$ are defined versus the time like vector field $N_{\mu}$ as follows.

[TABLE]

The above action is written in units $c=G=\hbar=1$ and the undetermined Lagrange multiplier $\zeta(x^{\nu})$ controls $N_{\mu}$ to be an unit time-like vector field. $\phi$ describes inverse of variable Newton’s gravitational coupling parameter and its dimension is $(lenght)^{-2}$ in units $c=G=\hbar=1$. Present limits of dimensionless BD parameter $\omega$ based on time-delay experiments [14, 15, 16, 17] requires $\omega\geq 4\times 10^{4}.$ Varying the above action functional with respect to $\zeta$ we obtain time-like condition on the vector field $N_{\mu}$ as follows.

[TABLE]

In the next section we apply the above mentioned action functional to study the anisotropic Bianchi I cosmological model.

3 Bianchi I quantum cosmology

Spatially homogenous but anisotropic dynamical flat universe is defined by the Bianchi I metric which from point of view of free falling (comoving) observer is defined by the following line element [18].

[TABLE]

where ${x,y,z}$ are cartesian coordinates of the comoving observer and $t$ is cosmic time. In the above metric equation we assume that the spatial part has cylindrical symmetry for which $e^{a(t)}$ is an global isotropic scale factor and $b(t)$ represents a deviation from the isotropy. Substituting (4) into the equation (3) we obtain

[TABLE]

where $(\alpha,\beta,\gamma)$ are angular constant parameters of the vector field $N_{\mu}$ which makes as fixed its direction at the 4D anisotropic space time (4). Substituting (4) and (5) into the action functional (2) with some simple calculations, one can show that the action functional read

[TABLE]

where dot $\dot{}$ denotes to ‘cosmic‘ time derivative $\frac{d}{dt}$ and we used the ansatz $\alpha=0$ to eliminate frictional terms $\ddot{a}$ and $\ddot{b}$ in the action functional (6) (see ref.[5] for more detail). In general, if an action functional contains time derivative of velocity of the dynamical fields then there will be some frictional forces which cause that the extremum point of the action functional does not fixed. The latter kind of dynamical systems are not closed and so stable. They behave usually as chaotic dynamical systems. Hence we should eliminate these acceleration terms of the dynamical fields to fix extremum points of the system. Usually one obtained an effective action functional by integrating by part and removing divergence-less counterpart of the action functional. Here we can eliminate the frictional terms $\ddot{a}$ and $\ddot{b}$ without to use an effective action functional instead of the action functional (6) just by setting $\alpha=0$. This restrict us to choose a particular direction for the time-like dynamical vector field (5) where the lagrangian of the system has not friction terms $\ddot{a}$ and $\ddot{b}$.

When we study canonical quantization of a mini-super-space quantum cosmology, then there the local coordinates do not have an important role but the dynamical fields themselves play an important role. Hence it will be useful to apply a conformal frame with the following conformal time $\tau(t)$ in what follows.

[TABLE]

where $F$ is called as lapse (red shift) function in the ADM formalism of the decomposition of the background metric. Substituting (7) and

[TABLE]

into the action functional (6) we obtain $I=\int dxdydz\int\mathcal{L}d\tau$ where $G$ is Newton‘s coupling constant and $\mathcal{L}$ is the Lagrangian density of the system from point of view of the conformal frame.

[TABLE]

where $\prime$ denotes to derivative with respect to the conformal time $\tau$ and

[TABLE]

is a suitable super-potential. Calculating canonical momenta of the fields $(a,b,Z)$ as

[TABLE]

and applying definition of the Hamiltonian density

[TABLE]

one can infer

[TABLE]

where we see that signature of the de Witt superspace metric is Lorentzian form $(-,+,-).$ In the next section we see that the canonical momentum operator of the anisotropic counterpart of the metric field $b$ behaves as a time-evolution parameter of the system in the canonical quantum cosmology of the minisuperspace de Witt metric. However we obtain now Wheeler de Witt probability wave solution of the cosmological system under a quantization condition on the ADM energy of the system.

4 Canonical quantization

To study quantum stability of the Bianchi I cosmology we should first fix the potential $V(a,b,Z)$ where we choose anisotropic harmonic Oscillator potential defined on the mini-super-space de Witt metric as follows.

[TABLE]

Substituting the Dirac‘s canonical quantization condition for the momentum operators as

[TABLE]

and the potential form (14) into the Hamiltonian density (13) we obtain the corresponding Wheeler de Witt wave equation $\hat{\mathcal{H}}\Psi(a,b,X)=0$ as follows.

[TABLE]

Negativity sign of the differential operator $\frac{\delta^{2}}{\delta b^{2}}$ in the above equation shows that the anisotropy field $Y$ can behave as the time evolution parameter in the three dimensional de Witt superspace metric. In other word the Kinetic terms in the above differential equation is similar to a Klein Gordon operator defined on the mini-super-space de Witt metric. To solve the equation (16) one can apply the standard method of separation of variables as follows. We assume the Wheeler de Witt wave solution to be separable as $\Psi(a,b,Z)=A(a)B(b)C(Z)$ and substitute it into the equation (16), then we can obtain the following differential equations for the fields $A(a),B(b),C(Z)$ respectively as follows.

[TABLE]

[TABLE]

[TABLE]

where the constants of separation of variables $\epsilon_{a,b,Z}$ satisfy the following relation.

[TABLE]

One can show that the equations (17), (18) and (19) can be described by the well known Hermit polynomials if we set the following quantization conditions on the parameters $\epsilon_{X,Y,Z}.$

[TABLE]

in which $N_{\{a,b,Z\}}$ can take quantized values $0,1,2,3,\cdots.$ In the latter case we will obtain normalized form of the solutions $A(a),B(b)$ and $C(Z)$ as follows.

[TABLE]

[TABLE]

[TABLE]

in which $H_{N_{a}},H_{N_{b}},H_{N_{Z}}$ are Hermit polynomials. Multiplying the above solutions we obtain $\Psi_{N_{a}N_{b}N_{Z}}=A_{N_{a}}(a)B_{N_{b}}(b)C_{N_{Z}}(Z)$ which in fact describes quantum fluctuations of the metric field of the Bianchi I cosmology defined by (4) when the cosmological system is in quantum state $(N_{a},N_{b},N_{Z})$ with corresponding eigne energy $(\epsilon_{N_{a}},\epsilon_{N_{b}},\epsilon_{N_{Z}})$. Now we can show that general solution of the Wheeler de Witt wave equation (16) by expanding it versus the eigne functionals such that

[TABLE]

where the coefficient $D_{N_{X}N_{Y}N_{Z}}$ describes probability of the quantum Bianchi I cosmology which takes the eigne state $\Psi_{N_{X}N_{Y}N_{Z}}.$ This can be determined by regarding the initial condition of the system and orthogonal condition on the Hermit polynomials. However no one does not know about initial condition of ”physical cosmology” but we can predicts some physical statements about our obtained solutions in what follows. Substituting (21) into the condition (20) we can obtain allowable eigne states of the system as follows.

[TABLE]

5 ADM energy and mass

In the previous section we assume the ADM mass of the cosmological system has a zero value. If it is not permissible then we should solve the extended Wheeler de Witt wave equation as $\hat{\mathcal{H}}\Psi=\mathcal{M}\Psi$ in which $\mathcal{M}$ is called to be the ADM mass of the system. In the latter case the equation (20) should be extended to the following form.

[TABLE]

which by substituting the quantization conditions (21) we obtain a quantization condition on the ADM mass of the Bianchi I cosmological model such that

[TABLE]

In fact ADM energy is a special way to define the energy in general relativity, which is only applicable to some special geometries of spacetime that asymptotically approach a well-defined metric tensor at infinity. If the background metric approaches to Minkowski space asymptotically then the Neother‘s theorem implies that the ADM energy or mass should be invariant because of time independence of the Minkowski metric. According to general relativity, the conservation law for the total energy does not hold in more general. For instance for time-dependent background metrics it will be violated. For example, it is completely violated in physical cosmology. In fact cosmic inflation in particular is able to produce energy or mass from ”nothing” because the vacuum energy density is roughly constant, but the volume of the Universe grows exponentially. Here we can modeled the violation of the ADM energy in the anisotropic Bianchi I inflationary cosmology by quantization approach. Here we show that the raising the ADM energy in the Bianchi I cosmology can be described by quantum fluctuations of the mass parameters of the fields $(a,b,Z)$ which we called with $K_{a},K_{b},K_{Z}$ respectively.

6 Big Bang naked singularity

The background metric (4) shows that the naked singularity lies on the particular hypersurface $(a,b)\to(-\infty,-\infty)$ at the classical cosmological feature where the big bang originated. Applying $(a,b)\to(-\infty,-\infty)$ one can infer that for naked singularity we have $A(-\infty)=0=B(-\infty).$ This reads

[TABLE]

Physical interpretation of the above result can be described as follows: There is a zero probability where the big bang originates from a naked singularity. In other words the big bang originates from an anisotropic quantum Oscillation of the quantum matter fields $(a,b,Z)$ at small scales of the space time. This happened because of the uncertainty relations between the fields and the corresponding canonical momenta as $\Delta X_{i}\Delta\Pi_{X_{i}}\equiv 1$ where $X_{i}=(a,b,Z).$

7 Concluding remarks

In this paper we choose scalar vector tensor Brans Dicke gravity to study anisotropic Bianchi I cosmology in the canonical quantization approach. We solved Wheeler de Witt wave equation and obtained its solutions versus the Hermit polynomials for an anisotropic harmonic Oscillator potential. We obtained a quantization condition on the ADM boundary energy of the system. Mathematical calculations predict that the inflation of the universe can be originated from a big bang state without a naked singularity. Furthermore anisotropic counterpart of the metric field can play as a time evolution parameter on the mini-super-space because of Lorentzian signature of the de Witt metric. As a future work one can study backreaction of the Hawking thermal radiation of the quantum matter (the vector and the Brans Dicke scalar) fields on the solutions of the Wheeler de Witt equations and so the corresponding eigne energies (quantized ADM energy) of the system which does not considered here.