Quantum BPS Cosmology

Quantum cosmology Halliwell ; Kiefer0 ; Kiefer1 in minisuperspace is based on the idea that the Hamiltonian constraint in a gravitating system containing a finite number of variables, such as scale factors and scalar fields, is interpreted as a differential equation of the wave function of the Universe. This equation, the Wheeler–DeWitt equation for a minisuperspace, becomes a hyperbolic second-order partial differential equation by a straightforward construction. This makes it difficult to define a positive-definite probability density. Various solutions have been proposed for this problem in definition of the probability density, including a simple and conventional adoption of the square of the absolute value of the wave function. Furthermore, determining what kind of “initial conditions” to solve the Wheeler–DeWitt equation is the most important when considering the very early universe. For this purpose, we have only to choose plausible initial conditions to solve the equation.²²2Note that it is known that the wave packet solution obtained by the conventional method and interpretation corresponds to the classical solution in the gravitational system KN ; Kiefer2 ; Kiefer3 ; GS ; KKST2 .

Recently, Russo Russo2 ; Russo1 has been studying a class of Einstein gravity theories involving scalar fields in which the classical equations of motion are reduced to first-order ordinary differential equations of time, such as the BPS equations.³³3The same or similar systems have been considered by many authors SB ; BGLM ; BLRR ; SA ; Lidsey ; Taylor from the past to the present, but there has been little mutual citation between them. The first-order differential equation represents the streamline in phase space, and has the advantage that the behavior of the dynamical system can be directly obtained. Although solvable systems are special cases, they often play an important role in the development of all areas of physics. The solutions of BPS type equations may not, of course, represent all possible solutions of the original equations of motion. Solution with restrictions become effective only when the existence of underlying symmetries, such as supersymmetry (or mock symmetry), is presumed. It seems that the model including scalars discussed here includes the possibility of being connected to fundamental theories with supersymmetry such as string theory. Quantum cosmology with supersymmetry has already been actively researched Graham1 ; Graham2 ; OSB ; OPR ; RB ; Moniz , but in this paper we will be careful to analyze the model without specifying the symmetry.

On the other hand, a quantum version of the BPS equation has already been used through studies of partition functions in topological string theory OVV ; CV . Therefore, we will use the quantum version of the BPS equation as an alternative version of the Wheeler–DeWitt equation for the similar model to Russo’s model, and study the solution to the wave function of the Universe obtained from the equation. This could provide a new perspective on plausible choice of initial conditions for restricted wave functions of the Universe. Our goal is to learn something useful about the probability interpretation of such solutions.

Therefore, although the origins of symmetries and restrictions are not discussed in the current simple models, as mentioned earlier, we must choose a solution that is compatible with the appropriate initial conditions of the quantum Universe. We believe that there is some significance in examining whether the wave function of the specious Universe can be obtained from the BPS-like equation.

The structure of this paper is as follows. Section II reviews Russo’s work on classical cosmological equations. In Section III, we consider the BPS-like equations and their quantum version, and find the specific solution of the wave function. The construction of probability density will be discussed in Section IV. In Section V, we examine cosmology of the Dirac–Born–Infeld type model by use of the BPS equation for the model. The final section provides a summary and future prospects.

We use metric signature $(-+\cdots+)$ and units $16\pi G=c=\hbar=1$. $\mu,\nu,\dots=0,1,\cdots,D-1$ are coordinate indices of spacetime, while $i,j=1,2,\cdots,D-1$ are indices for space.

In this section, we briefly review Russo’s model Russo2 ; Russo1 in $D$-dimensional spacetime. The action describing the system of gravitating scalar fields is as follows:

$$S=\int d^{D}x\sqrt{-g}\left[R-\frac{1}{2}g^{\mu\nu}G_{ab}(\phi)\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}-V(\phi)\right]\,,$$

(1)

where $g$ denotes the determinant of the metric tensor $g_{\mu\nu}$, $g^{\mu\nu}$ is the inverse of the metric tensor, $R$ is the scalar curvature, $\phi^{a}$ $(a=1,\cdots,N)$ are the scalar fields, and $V(\phi)$ is the potential of scalar fields $\{\phi^{a}\}$. The metric of the internal space $G_{ab}(\phi)$ is given by a symmetric matrix, $G_{ab}=G_{ba}$ and is generally dependent on scalar fields $\{\phi^{a}\}$. Further, the metric of the $D$ dimensional spacetime is assumed as

$$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-e^{2\gamma\varphi(t)}dt^{2}+e^{2\beta\varphi(t)}\sum_{i=1}^{D-1}(dx^{i})^{2}=-e^{2\gamma\varphi(t)}dt^{2}+a^{2}(t)\sum_{i=1}^{D-1}(dx^{i})^{2}\,,$$

(2)

where $\gamma$ is a constant, and

$$\beta=\left(\sqrt{2(D-1)(D-2)}\right)^{-1}\,,$$

(3)

and $a(t)$ is a scale factor of the flat, homogeneous and isotropic space. Note that this metric becomes the standard Friedmann–Lemaître–Robertson–Walker metric when $\gamma=0$, while the metric becomes a conformally-flat metric when $\gamma=\beta$.

Assuming that all scalar fields are also spatially uniform, that is, a function of time only, $\phi^{a}=\phi^{a}(t)$, the effective Lagrangian on these variables is given by⁴⁴4The Gibbons–Hawking–York term York ; GH is implicitly taken into account, as a standard approach in Einstein gravity.

$$L=-\frac{1}{2}e^{\delta\varphi}\dot{\varphi}^{2}+\frac{1}{2}e^{\delta\varphi}G_{ab}(\phi)\dot{\phi}^{a}\dot{\phi}^{b}-e^{(2\gamma+\delta)\varphi}V(\phi)\,,$$

(4)

where $\delta=(D-1)\beta-\gamma$ and the dot $(\dot{~{}})$ denotes the time derivative.

The equations of motion derived from the Lagrangian (4) are

$$\frac{d}{dt}\left[e^{\delta\varphi}G_{ab}\dot{\phi}^{b}\right]=-e^{(2\gamma+\delta)\varphi}\partial_{a}V+\frac{e^{\delta\varphi}}{2}(\partial_{a}G_{bc})\dot{\phi}^{b}\dot{\phi}^{c}\,,$$

(5)

and

$$\frac{d}{dt}(-e^{\delta\varphi}\dot{\varphi})=-\frac{1}{2}\delta\,e^{\delta\varphi}\dot{\varphi}^{2}+\frac{1}{2}\delta\,e^{\delta\varphi}G_{cd}\dot{\phi}^{c}\dot{\phi}^{d}-(2\gamma+\delta)e^{(2\gamma+\delta)\varphi}V\,,$$

(6)

where $\partial_{a}=\frac{\partial}{\partial\phi^{a}}$.

These equations are satisfied by the following two simultaneous equations including the first-order derivative:

$$e^{\delta\varphi}\dot{\varphi}=\epsilon\partial_{\varphi}\mathcal{W}\,,\quad e^{\delta\varphi}\dot{\phi}^{a}=-\epsilon G^{ab}\partial_{b}\mathcal{W}\,,$$

(7)

where the constant $\epsilon$ satisfies $\epsilon^{2}=1$, $\partial_{\varphi}=\frac{\partial}{\partial\varphi}$, and $G^{ab}$ is the inverse matrix of $G_{ab}$, if the scalar potential takes the following form:

	$\displaystyle V(\phi)$	$\displaystyle=$	$\displaystyle e^{-2(\gamma+\delta)\varphi}\left[\frac{1}{2}(\partial_{\varphi}\mathcal{W})^{2}-\frac{1}{2}G^{ab}(\phi)\partial_{a}\mathcal{W}\partial_{b}\mathcal{W}\right]$		(8)
		$\displaystyle=$	$\displaystyle\frac{1}{2}\alpha^{2}({W(\phi}))^{2}-\frac{1}{2}G^{ab}(\phi)\partial_{a}{W(\phi)}\partial_{b}{W(\phi)}\,,$

where

$$\mathcal{W}=e^{(\gamma+\delta)\varphi}W(\phi)=e^{(D-1)\beta\varphi}W(\phi)=e^{\alpha\varphi}W(\phi)=a^{D-1}W(\phi)\,,$$

(9)

and

$$\alpha=(D-1)\beta=\sqrt{\frac{D-1}{2(D-2)}}\,.$$

(10)

The potential $V(\phi)$ is mostly determined by $W(\phi)$, so we call $W(\phi)$ prepotential.

We find that the Lagrangian (4) can be rewritten in the form

	$\displaystyle L$	$\displaystyle=$	$\displaystyle-\frac{1}{2}e^{-\delta\varphi}(e^{\delta\varphi}\dot{\varphi}-\epsilon\partial_{\varphi}\mathcal{W})^{2}+\frac{1}{2}e^{-\delta\varphi}G_{cd}\left[e^{\delta\varphi}\dot{\phi}^{c}+\epsilon G^{ce}\partial_{e}\mathcal{W}\right]\left[e^{\delta\varphi}\dot{\phi}^{d}+\epsilon G^{de}\partial_{e}\mathcal{W}\right]$		(11)
			$\displaystyle-\epsilon(\dot{\varphi}\partial_{\varphi}\mathcal{W}+\dot{\phi}^{a}\partial_{a}\mathcal{W})\,,$

so, it is apparent that the equations (7) represent a stationary point of the action $S=\int Ldt$.

Based on the Lagrangian (4), the conjugate momenta of $\varphi$ and $\phi^{a}$ are expressed by

$$\Pi_{\varphi}=\frac{\partial L}{\partial\dot{\varphi}}=-e^{\delta\varphi}\dot{\varphi}\,,\quad\Pi_{a}=\frac{\partial L}{\partial\dot{\phi}^{a}}=e^{\delta\varphi}G_{ab}(\phi)\dot{\phi}^{b}\,,$$

(12)

respectively, and the Hamiltonian is expressed by

$$\mathcal{H}=-\frac{1}{2}e^{-\delta\varphi}\Pi_{\varphi}^{2}+\frac{1}{2}e^{-\delta\varphi}G^{ab}(\phi)\Pi_{a}\Pi_{b}+e^{(2\alpha-\delta)\varphi}V(\phi)\,.$$

(13)

Therefore, if the BPS-like equations (7) are substituted in the Hamiltonian, we find that $\mathcal{H}=0$ holds classically. The Hamiltonian constraint is automatically satisfied for the solutions of (7) in this case as a gravitating system.⁵⁵5As is well known, the Hamiltonian constraint is derived by, replacing $t\rightarrow Nt$ in the action $S=\int Ldt$ and regarding that the variation $\frac{\delta S}{\delta N}$ vanishes Halliwell ; Kiefer0 ; Kiefer1 .

The BPS-like equations (7) can directly represent the flow of solutions in phase space, so they are very convenient for examining the relationship between initial conditions and subsequent evolution.

According to the paper OVV , the BPS equation can be utilized for quantization. Letting the wave function be $\Phi(\varphi,\phi)$, the quantization is given by substitution of momenta

$$\Pi_{\varphi}\rightarrow-i\frac{\partial}{\partial\varphi}\,,\quad\Pi_{a}\rightarrow-i\frac{\partial}{\partial\phi^{a}}\,,$$

(14)

as operators on the wave function Halliwell ; Kiefer0 ; Kiefer1 , and then, the resulting equations from (7) are

$$\left[-i\partial_{\varphi}+\epsilon\partial_{\varphi}\mathcal{W}\right]\Phi(\varphi,\phi)=0\,,\quad\left[-i\partial_{a}+\epsilon\partial_{a}\mathcal{W}\right]\Phi(\varphi,\phi)=0\,.$$

(15)

As a simple solution to these equations,

$$\Phi(\varphi,\phi)=e^{-i\epsilon\mathcal{W}(\varphi,\phi)}\,,$$

(16)

can be considered.

Let us take a look at the differences between the commonly used Wheeler–DeWitt equation and the quantum BPS equations. The Wheeler–DeWitt equation is derived by replacing (14) in the Hamiltonian constraint $\mathcal{H}=0$, and can be read as

$$\left[\partial_{\varphi}^{2}-G^{ab}(\phi)\partial_{a}\partial_{b}+2e^{2\alpha\varphi}V(\phi)\right]\Phi=0\,,$$

(17)

up to operator orderings. Note that the Wheeler–DeWitt equation is independent of the choice of the constant $\delta$, in turn the choice of $\gamma$.

As is clear from the form of the solution (16) of the BPS equations, this solution is a solution of the Hamilton–Jacobi equation for the Wheeler–DeWitt equation (17),

$$\Phi=e^{iS}\,,\quad-(\partial_{\varphi}S)^{2}+G^{ab}(\phi)\partial_{a}S\partial_{b}S+2e^{2\alpha\varphi}V(\phi)=0\,,$$

(18)

and its simple solution is $S=-\epsilon\mathcal{W}$. Therefore, the semiclassical wave function (or the WKB wave function) (16) corresponds to a specific solution representing a certain classical trajectory. By the way, as a general matter, it is natural that the difference obtained by quantizing classical equations can always appear, because there are ambiguities which depends on the operator-ordering prescription.⁶⁶6A similar attitude has been adopted in work on minisuperspace quantum cosmology from the early-days study Halliwell to the most recent study Godet . The discrepancy from the specific ordering may modify the solution for the wave function multiplicatively, so we can neglect the possible correction which could also come from the one-loop effect of the same order of the Planck constant $\hbar$.

In general, the wave function of the Universe, which is given as the solution of the Wheeler–DeWitt equation, is written by an ensemble of semi-classical universes D37 ; D50

$$\Phi=\sum_{k}e^{iS_{k}(c)}\chi_{k}(c,q)\,,$$

(19)

where the sum is taken over the classical trajectory, $S_{k}(c)$ is the classical solution of the Hamilton–Jacobi equation, and $\chi_{k}(c,q)$ is the contribution of the quantum variables $q$. Thus, taking only $S=-\epsilon\mathcal{W}$ as the solution corresponds to choosing a particular classical solution for the wave function.

The choice of Hawking’s no-boundary condition HH ; Hawking or Vilenkin’s tunneling wave function D33 ; D37 are conjectures that attempt to select a solution representing the initial state of the Universe among the general solutions. By specifying the BPS solution, we are proposing a new conjecture for the problem of the initial condition regarding the wave function of the Universe in minisuperspace quantum cosmology. It would be nice if it could be connected to the classical inflationary solution for the Universe.

As the reader has already noticed, in the system we are currently dealing with, we are considering a flat universe, which naturally requires a different treatment from the quantum cosmology represented by Hawking and Vilenkin in the models with a homogeneous space of positive curvature. We will explain how to think about wave functions in our system and the case of specific potentials $V(\phi)$ and prepotentials $W(\phi)$ in later sections.

For an interpretation of the wave function, we would like to obtain conserved quantity for constructing probability density. Taking into account the covariance of the minisuperspace, i.e., by the so-called Laplace–Beltrami ordering, we write down a more sophisticated Wheeler–DeWitt equation

$$\left[e^{-p\varphi}\partial_{\varphi}e^{p\varphi}\partial_{\varphi}-\frac{1}{\sqrt{G}}\partial_{a}\sqrt{G}G^{ab}\partial_{b}+2e^{2\alpha\varphi}V(\phi)\right]\Phi=0\,,$$

(20)

where $G\equiv\det G_{ab}$, and the constant $p$ represents an ordering ambiguity.

The conservation equation created from the second-order partial differential equation D33 ; D37 ; D39 ; HGC ,

$$\partial_{\varphi}\rho+\nabla_{a}J^{a}=0\,,$$

(21)

where

$$\rho\equiv\frac{i}{2}\epsilon e^{p\varphi}(\Phi^{*}\partial_{\varphi}\Phi-\partial_{\varphi}\Phi^{*}\Phi)\,,\quad J^{a}\equiv-\frac{i}{2}\epsilon e^{p\varphi}(\Phi^{*}\nabla^{a}\Phi-\nabla^{a}\Phi^{*}\Phi)\,,$$

(22)

where $\nabla_{a}$ is a covariant derivative in metric $G_{ab}$. Thus, we conclude that

$$\frac{d}{d\varphi}\int\rho\,\sqrt{G}\,d^{N}\phi=0\,,$$

(23)

where $d^{N}\phi\equiv\prod_{a=1}^{N}d\phi^{a}$, and the probability measure can be taken as D33 ; D37 ; D39 ; HGC

$$dP=\rho\sqrt{G}\,d^{N}\phi\,.$$

(24)

Now, substituting the solution $\Phi=e^{-i\epsilon\mathcal{W}}$ into the above definitions, we find that

$$\alpha e^{p\varphi}\sqrt{G(\phi)}\,\mathcal{W}=\alpha e^{(\alpha+p)\varphi}\sqrt{G(\phi)}\,W(\phi)$$

(25)

expresses the probability density for $\{\phi^{a}\}$ at a fixed $\varphi$. As is often thought, $\varphi$ is defined as time, or the increase of a volume element $a^{D-1}=e^{\alpha\varphi}$ indicates time development of the Universe. Furthermore, if the constant $p$ takes the value $-\alpha$, the probability density is proportional to

$$\sqrt{G(\phi)}\,W(\phi)\,,$$

(26)

which is independent of $\varphi$.

This becomes a positive definite value only if the prepotential $W(\phi)$ is chosen as a positive-definite function. That is, the resolution of the problem of the positivity of the probability measure is set aside in the present analysis.⁷⁷7Note that the change the sign of $\epsilon$ is equivalent to the change $W\rightarrow-W$.

We propose that the initial value for scalar fields are determined by the maximum of the positive prepotential $W(\phi)$,⁸⁸8Note that this means the maximum expansion rate, since $\frac{\dot{a}}{a}$ is proportional to $W(\phi)$, which is seen from (7). whereas nothing is said about the initial condition on the scale of the Universe. This is natural in a sense, since in the present analysis, the volume element is treated as time. Also, since we are dealing with a model in which the curvature of the space at any time slice is zero, the singularity problem does not appear seriously. We also pay attention to the fact that the classical action becomes $\int_{t_{i}}^{t_{f}}Ldt=-\epsilon(\mathcal{W}|_{t_{f}}-\mathcal{W}|_{t_{i}})$, if the equations (7) hold. Therefore, the extremum of the classical action corresponds to the maximum probability density. In this case, the factor $\sqrt{G}$ simply arises from the Jacobian of the scalars.

Here, we give an example with a single scalar field (i.e., $G_{ab}\rightarrow 1$). Suppose that

$$W(\phi)=w^{2}(\cosh m\phi)^{-b}\,,$$

(27)

where the constants $w$, $m$ and $b$ takes positive values. In this case, the most probable value of $\phi$ in the quantum Universe is near $\phi=0$.

The classical trajectory from (7) satisfies

$$\frac{d\phi}{d\varphi}=\frac{\dot{\phi}}{\dot{\varphi}}=-\frac{W^{\prime}(\phi)}{\alpha W(\phi)}=\frac{bm}{\alpha}\tanh m\phi\,,$$

(28)

where $W^{\prime}(\phi)=\frac{dW}{d\phi}$. The solution of this equation is found to be

$$e^{\alpha\varphi}=a^{D-1}=C(\sinh m\phi)^{\frac{\alpha^{2}}{bm^{2}}}\,,$$

(29)

where $C$ is a constant.⁹⁹9The constant $C$ is not physical, since it can be absorbed into the rescaling of spatial coordinates. One can see that the scale factor becomes $a\approx 0$ when $\phi\rightarrow 0$. So, in this case, the BPS equations also give the initial condition $a\approx 0$.

We solve the BPS-like equations (7), using the standard cosmic time in the Friedmann–Lemaître–Robertson–Walker metric, obtained by choosing $\gamma=0\,(\leftrightarrow\delta=\alpha)$ in the metric (2). If $\gamma=0$, the equations (7) for a single scalar field becomes

$$\dot{\varphi}=\alpha W(\phi)\,,\quad\dot{\phi}=-W^{\prime}(\phi)\,,$$

(30)

and then, the expansion rate is

$$H=\frac{\dot{a}}{a}=\alpha\beta W(\phi)\,.$$

(31)

In the early time in the Universe, when $m\phi\ll 1$, we find

$$e^{\alpha\varphi}=a^{D-1}\approx Ce^{\alpha^{2}w^{2}(t-t_{0})}\,,\quad\phi\approx m^{-1}e^{b\,m^{2}w^{2}(t-t_{0})}\,,$$

(32)

where $t_{0}$ is a constant. This shows the de Sitter expansion, of which expansion rate is $H=\frac{\alpha^{2}w^{2}}{D-1}=\frac{w^{2}}{2(D-2)}$.

In the late time, when $m\phi\gg 1$, we find

$$e^{\alpha\varphi}=a^{D-1}\approx C[b^{2}m^{2}w^{2}(t-t_{1})]^{\frac{\alpha^{2}}{b^{2}m^{2}}}\,,\quad\phi\approx\frac{1}{bm}\ln[2^{b}b^{2}m^{2}w^{2}(t-t_{1})]\,,$$

(33)

where $t_{1}$ is a constant. For $m\phi\gg 1$, the potential takes the asymptotic form

$$V(\phi)=\frac{1}{2}\frac{w^{4}}{(\cosh m\phi)^{2b}}\left(\alpha^{2}-b^{2}m^{2}\tanh^{2}m\phi\right)\approx\frac{1}{2}2^{2b}w^{4}(\alpha^{2}-b^{2}m^{2})e^{-2bm\phi}\quad(\mbox{for~{}}b^{2}m^{2}\neq\alpha^{2})\,.$$

(34)

The cosmological model of a scalar field with an exponential potential has been studied by many authors until Russo0 ; Neupane ; ENO . They found that the potential $e^{-\lambda\phi}$ leads to $e^{\alpha\varphi}\sim t^{4\alpha^{2}/\lambda^{2}}$ and $\phi\sim\frac{2}{\lambda}\ln t$. The asymptotic solution (33) coincides with their solution. Interestingly, the positivity of the potential at $m\phi\gg 1$ corresponds to a condition of accelerating expansion, $\frac{\alpha^{2}}{b^{2}m^{2}}>1$, i.e. the power-law inflation. In an exceptional case with $b^{2}m^{2}=\alpha^{2}$, we find

$$V(\phi)=\frac{1}{2}\frac{\alpha^{2}w^{4}}{(\cosh m\phi)^{2b+2}}\approx\frac{1}{2}2^{2b+2}\alpha^{2}w^{4}e^{-2(b+1)m\phi}\quad(\mbox{for~{}}b^{2}m^{2}=\alpha^{2})\,,$$

(35)

asymptotically. In this case, the relation between the coefficient of the exponent of the asymptotic potential and the exponent in the scale factor differs from that in Russo0 ; Neupane ; ENO .

Incidentally, the slow-roll parameters in Russo2 are expressed by the prepotential $W(\phi)$, as

	$\displaystyle\varepsilon_{H}$	$\displaystyle=$	$\displaystyle-\frac{\dot{H}}{H^{2}}=\frac{\dot{\phi}^{2}}{2(D-2)H^{2}}=2(D-2)\frac{W^{\prime}(\phi)^{2}}{W(\phi)^{2}}\,,\quad$		(36)
	$\displaystyle\eta_{H}$	$\displaystyle=$	$\displaystyle-\frac{\ddot{H}}{2H\dot{H}}=-\frac{\ddot{\phi}}{H\dot{\phi}}=2(D-2)\frac{W^{\prime\prime}(\phi)}{W(\phi)}\,.$		(37)

For the present example, $W(\phi)=w^{2}[\cosh(m\phi)]^{-b}$, we find

$$\varepsilon_{H}=2(D-2)b^{2}m^{2}\tanh^{2}m\phi\,,\quad\eta_{H}=2(D-2)bm^{2}[(b+1)\tanh^{2}m\phi-1]\,.$$

(38)

Thus, the slow-roll condition $\varepsilon_{H}\ll 1$ is satisfied if $m\phi\ll 1$ and/or $bm\ll 1$.

In summary, this example shows that the Universe could experience the nearly de Sitter phase and the subsequent power-law inflation phase.

A comment should be placed here on the “classicalization”. The initial fluctuation of the scalar field and the scale factor in quantum cosmology should produce corresponding classical quantities which obey the classical equation of motion in classical cosmology. The validity of this classicalization needs to be explained theoretically. Here, we consider the model with the simplest assumption that the initial state is described by $\phi\approx 0$ and $a\approx 0$, including fluctuations, and that the system rapidly transitions to classical system.

Finally in this section, we would like to give a brief comment on the Euclidean treatment of the BPS equations. The authors of OVV suggested the use of the imaginary time and found the Hartle–Hawking state in their theory. The use of the imaginary time yields

$$\Phi\propto e^{\mp\mathcal{W}}\,.$$

(39)

In this case, we should consider the commonly used probability density, $|\Phi|^{2}$. Also in this case, the extremum of the prepotential $W(\phi)$ gives the most probable initial value of the scalar fields.

The BPS-like equation for gravitating Dirac–Born–Infeld type scalar field theory was found in ST ; AST . The action for the gravitating Dirac–Born–Infeld scalar fields is

	$\displaystyle S$	$\displaystyle=$	$\displaystyle\int d^{D}x\sqrt{-g}\left[R-\frac{1}{f(\phi)}\sqrt{1+f(\phi)g^{\mu\nu}G_{ab}(\phi)\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}}-V(\phi)\right]$		(40)
		$\displaystyle=$	$\displaystyle\int d^{D}x\sqrt{-g}\left[R-\frac{1}{f(\phi)}\left\{\sqrt{1+f(\phi)g^{\mu\nu}G_{ab}(\phi)\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}}-1\right\}-U(\phi)\right]\,,$

where $f(\phi)$ is a function of the scalar fields $\{\phi^{a}\}$ and $U(\phi)=V(\phi)+\frac{1}{f(\phi)}$. Notice that the kinetic term reduces to the one in the previous canonical scalar model in the limit of $f\rightarrow 0$. Using the same ansatze for the metric and scalar fields as in Section II, we find the effective Lagrangian

$$L=-\frac{1}{2}e^{\delta\varphi}\dot{\varphi}^{2}-\frac{e^{(\delta+2\gamma)\varphi}}{f(\phi)}\sqrt{1-{e^{-2\gamma\varphi}}f(\phi)G_{ab}(\phi)\dot{\phi}^{a}\dot{\phi}^{b}}-e^{(\delta+2\gamma)\varphi}V(\phi)\,.$$

(41)

The equations of motion obtained from this Lagrangian are

			$\displaystyle\frac{d}{dt}\left[\left(\sqrt{1-{e^{-2\gamma\varphi}}fG_{cd}\dot{\phi}^{c}\dot{\phi}^{d}}\right)^{-1}e^{\delta\varphi}G_{ab}\dot{\phi}^{b}\right]$		(42)
		$\displaystyle=$	$\displaystyle-e^{(2\gamma+\delta)\varphi}\partial_{a}V+\frac{e^{\delta\varphi}}{2}\left(\sqrt{1-{e^{-2\gamma\varphi}}fG_{cd}\dot{\phi}^{c}\dot{\phi}^{d}}\right)^{-1}(\partial_{a}G_{be})\dot{\phi}^{b}\dot{\phi}^{e}$
			$\displaystyle+e^{(2\gamma+\delta)\varphi}\frac{\partial_{a}f}{2f^{2}}\left[\sqrt{1-{e^{-2\gamma\varphi}}fG_{cd}\dot{\phi}^{c}\dot{\phi}^{d}}+\left(\sqrt{1-{e^{-2\gamma\varphi}}fG_{cd}\dot{\phi}^{c}\dot{\phi}^{d}}\right)^{-1}\right]\,,$

	$\displaystyle\frac{d}{dt}(-e^{\delta\varphi}\dot{\varphi})$	$\displaystyle=$	$\displaystyle-\frac{\delta}{2}e^{\delta\varphi}\dot{\varphi}^{2}-(\gamma+\delta)\frac{e^{(2\gamma+\delta)\varphi}}{f}\sqrt{1-{e^{-2\gamma\varphi}}fG_{cd}\dot{\phi}^{c}\dot{\phi}^{d}}$		(43)
			$\displaystyle-\gamma\frac{e^{(2\gamma+\delta)\varphi}}{f}\left(\sqrt{1-{e^{-2\gamma\varphi}}fG_{cd}\dot{\phi}^{c}\dot{\phi}^{d}}\right)^{-1}-(2\gamma+\delta)e^{(2\gamma+\delta)\varphi}V\,.$

These equations are solved by

$$e^{\delta\varphi}\dot{\varphi}=\epsilon\partial_{\varphi}\mathcal{W}\,,\quad\left(\sqrt{1-{e^{-2\gamma\varphi}}fG_{cd}\dot{\phi}^{c}\dot{\phi}^{d}}\right)^{-1}e^{\delta\varphi}\dot{\phi}^{a}=-\epsilon G^{ab}\partial_{b}\mathcal{W}\,,$$

(44)

provided that

	$\displaystyle V(\phi)$	$\displaystyle=$	$\displaystyle\frac{1}{2}e^{-2\alpha\varphi}(\partial_{\varphi}\mathcal{W})^{2}-\frac{1}{f}\sqrt{1+{e^{-2\alpha\varphi}}fG^{ab}\partial_{a}\mathcal{W}\partial_{b}\mathcal{W}}$		(45)
		$\displaystyle=$	$\displaystyle\frac{1}{2}\alpha^{2}({W(\phi}))^{2}-\frac{1}{f(\phi)}\sqrt{1+f(\phi)G^{ab}(\phi)\partial_{a}{W(\phi)}\partial_{b}{W(\phi)}}\,,$

where

$$\mathcal{W}=e^{\alpha\varphi}W(\phi)=e^{(D-1)\beta\varphi}W(\phi)\,.$$

(46)

Note that

$$\sqrt{1-{e^{-2\gamma\varphi}}fG_{ab}\dot{\phi}^{a}\dot{\phi}^{b}}\sqrt{1+{e^{2\alpha\varphi}}fG^{ab}\partial_{a}\mathcal{W}\partial_{b}\mathcal{W}}=1\,,$$

(47)

if the equations (44) hold, and notice that $\partial_{a}G^{bc}=-G^{bd}G^{ce}\partial_{a}G_{de}$.

Then, the Lagrangian (41) can be rewritten by

	$\displaystyle L$	$\displaystyle=$	$\displaystyle-\frac{1}{2}e^{-\delta\varphi}(e^{\delta\varphi}\dot{\varphi}-\epsilon\partial_{\varphi}\mathcal{W})^{2}$		(48)
			$\displaystyle+\frac{1}{2}e^{-\delta\varphi}\sqrt{1-{e^{-2\gamma\varphi}}fG_{ab}\dot{\phi}^{a}\dot{\phi}^{b}}\,G_{cd}\left[\left(\sqrt{1-{e^{-2\gamma\varphi}}fG_{ab}\dot{\phi}^{a}\dot{\phi}^{b}}\right)^{-1}e^{\delta\varphi}\dot{\phi}^{c}+\epsilon G^{ce}\partial_{e}\mathcal{W}\right]$
			$\displaystyle\qquad\qquad\qquad\times\left[\left(\sqrt{1-{e^{-2\gamma\varphi}}fG_{ab}\dot{\phi}^{a}\dot{\phi}^{b}}\right)^{-1}e^{\delta\varphi}\dot{\phi}^{d}+\epsilon G^{de}\partial_{e}\mathcal{W}\right]$
			$\displaystyle-\frac{1}{2}\frac{e^{(2\gamma+\delta)\varphi}}{f}\left(\sqrt{1-{e^{-2\gamma\varphi}}fG_{ab}\dot{\phi}^{a}\dot{\phi}^{b}}\right)^{-1}$
			$\displaystyle\qquad\qquad\qquad\times\left[\sqrt{1-{e^{-2\gamma\varphi}}fG_{ab}\dot{\phi}^{a}\dot{\phi}^{b}}\sqrt{1+{e^{-2\alpha\varphi}}fG^{ab}\partial_{a}\mathcal{W}\partial_{b}\mathcal{W}}-1\right]^{2}$
			$\displaystyle-\epsilon(\dot{\varphi}\partial_{\varphi}\mathcal{W}+\dot{\phi}^{a}\partial_{a}\mathcal{W})\,.$

Obviously, the limit $f\rightarrow 0$ yields the Russo’s canonical model reviewed in Section II.

The conjugate momenta in the present system are

$$\Pi_{\varphi}=-e^{\delta\varphi}\dot{\varphi}\,,\quad\Pi_{a}=\frac{e^{\delta\varphi}G_{ab}\dot{\phi}^{b}}{\sqrt{1-{e^{-2\gamma\varphi}}fG_{cd}\dot{\phi}^{c}\dot{\phi}^{d}}}\,,$$

(49)

and then the Hamiltonian can be found as

$$\mathcal{H}=-\frac{1}{2}e^{-\delta\varphi}\Pi_{\varphi}^{2}+\frac{e^{(2\alpha-\delta)\varphi}}{f}\sqrt{1+{e^{-2\alpha\varphi}}fG^{ab}\Pi_{a}\Pi_{b}}+e^{(2\alpha-\delta)\varphi}V(\phi)\,.$$

(50)

The classical Hamiltonian constraint $\mathcal{H}=0$ is satisfied by taking (44) with (49).

The Wheeler–DeWitt equation $\mathcal{H}\Phi=0$ is obtained by replacing the momenta with the operators. The resulting equation is highly nonlinear differential equation in the present case. Nevertheless, the Hamilton–Jacobi equation obtained by setting $\Phi=e^{iS}$ and assuming the WKB approximation $|\partial_{a}\partial_{b}S|\ll|\partial_{a}S|^{2}$ becomes

$$-\frac{1}{2}(\partial_{\varphi}S)^{2}+\frac{e^{2\alpha\varphi}}{f}\sqrt{1+{e^{-2\alpha\varphi}}fG^{ab}\partial_{a}S\partial_{b}S}+e^{2\alpha\varphi}V(\phi)=0\,.$$

(51)

Thus, the solution of the quantum BPS equations

$$\left[-i\partial_{\varphi}+\epsilon\partial_{\varphi}\mathcal{W}\right]\Phi(\varphi,\phi)=0\,,\quad\left[-i\partial_{a}+\epsilon\partial_{a}\mathcal{W}\right]\Phi(\varphi,\phi)=0\,,$$

(52)

which is the same as in the Section II, that is,

$$\Phi=e^{-i\epsilon\mathcal{W}}\,,$$

(53)

is also the solution of the Hamilton–Jacobi equation (51).

At last, unlike for the model in Section II, it is difficult to write down the exact conservation equation with the present wave function. However, under the WKB approximation already adopted, the probability density should take the same form,

$$\sqrt{G(\phi)}\,W(\phi)\,.$$

(54)

Incidentally, the Wheeler–DeWitt equation for a similar model was studied in Psinas , which also included an analysis of the lowest order of scalar derivatives.

That aside, we note that we find the same probability measure in the Dirac–Born–Infeld type model as in the canonical scalar model, which is recovered by sending $f$ to zero. Further, it has been noted that the classical value of $e^{iS}$ is proportional to $e^{-i\epsilon\mathcal{W}}$, regardless of the function $f$. Thus, as in the canonical case, with $f\rightarrow 0$, the maximum probability density is taken to be given by the extremum of the prepotential.

Now, let us consider the similar example as in the previous section. Apparently, the de Sitter phase and the power-law behavior can be found by considering the asymptotic form of $W(\phi)$ and for an appropriate function $f(\phi)$. In the following, for simplicity, we here assume $b=1$ and $f$ takes a constant value. That is,

$$W(\phi)=w^{2}(\cosh m\phi)^{-1}\,.$$

(55)

The equations (44) and (47) lead to

$$\sqrt{1+f(W^{\prime}(\phi))^{2}}\frac{d\phi}{d\varphi}=-\frac{W^{\prime}(\phi)}{\alpha W(\phi)}\,,$$

(56)

and substitution of (55) yields

$$\sqrt{1+fm^{2}w^{2}\frac{\sinh^{2}m\phi}{\cosh^{4}m\phi}}\frac{d\phi}{d\varphi}=\frac{m}{\alpha}\tanh m\phi\,.$$

(57)

The solution of this equation is

	$\displaystyle e^{\alpha\varphi}$	$\displaystyle=$	$\displaystyle C(\sinh m\phi)^{\frac{\alpha^{2}}{m^{2}}}$		(58)
			$\displaystyle\cdot\left[\frac{2\cosh^{2}m\phi+fm^{2}w^{2}+2\sqrt{\cosh^{4}m\phi+fm^{2}w^{2}\sinh^{2}m\phi}}{2\cosh^{2}m\phi+fm^{2}w^{2}\sinh^{2}m\phi+2\sqrt{\cosh^{4}m\phi+fm^{2}w^{2}\sinh^{2}m\phi}}\right]^{\frac{\alpha^{2}}{2m^{2}}}$
			$\displaystyle\cdot\exp\left[-\frac{\sqrt{fm^{2}w^{2}}\alpha^{2}}{2m^{2}}\arctan\frac{\sqrt{fm^{2}w^{2}}(1-\sinh^{2}m\phi)}{2\sqrt{\cosh^{4}m\phi+fm^{2}w^{2}\sinh^{2}m\phi}}\right]\quad(\mbox{for~{}}f>0)\,,$

	$\displaystyle e^{\alpha\varphi}$	$\displaystyle=$	$\displaystyle C(\sinh m\phi)^{\frac{\alpha^{2}}{m^{2}}}$		(59)
			$\displaystyle\cdot\left[\frac{2\cosh^{2}m\phi+fm^{2}w^{2}+2\sqrt{\cosh^{4}m\phi+fm^{2}w^{2}\sinh^{2}m\phi}}{2\cosh^{2}m\phi+fm^{2}w^{2}\sinh^{2}m\phi+2\sqrt{\cosh^{4}m\phi+fm^{2}w^{2}\sinh^{2}m\phi}}\right]^{\frac{\alpha^{2}}{2m^{2}}}$
			$\displaystyle\cdot\exp\left[\frac{\sqrt{|f|m^{2}w^{2}}\alpha^{2}}{2m^{2}}{\arctan}\mbox{h}\frac{\sqrt{|f|m^{2}w^{2}}(1-\sinh^{2}m\phi)}{2\sqrt{\cosh^{4}m\phi+fm^{2}w^{2}\sinh^{2}m\phi}}\right]\quad(\mbox{for~{}}f<0)\,,$