Quantum Gravity Applications: Free Scalar Particle Motion in Expanding Universe Metrics and Age Estimation

Abstract

Applications of Parametrized Relativistic Quantum Theory (PRQT) in curved spacetime are considered here. PRQT in curved spacetime is applied to the motion of free scalar particles in expanding universe metrics, including a generalized expanding universe (EU) metric and the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. Governing equations are derived and solved through separation of variables. In addition, modern observational parameters and a rescaled Friedmann equation are used to estimate the age of the universe. Implications for cosmological models are discussed.

1. Introduction

The unification of quantum mechanics with general relativity into a viable theory of quantum gravity continues to be an open problem in fundamental physics. This challenge is central in cosmology, where researchers seek to describe quantum phenomena in dynamically evolving spacetimes such as the expanding universe. Conventional canonical approaches, such as the Wheeler–DeWitt (WDW) equation obtained by means of the Arn-owitt–Deser–Misner (ADM) formalism [1,2,3], are subject to the well-known “problem of time”. These frameworks lack an explicit external temporal parameter because the Hamiltonian constraint leads to a timeless equation. This impedes the construction of predictive quantum cosmological models that can be directly compared with observational data on the expansion and age of the universe.

Parametrized Relativistic Quantum Theory (PRQT) was developed to overcome these difficulties [4,5]. Originally formulated for multicomponent many-body systems in flat spacetime and recently extended to curved spacetime [6,7], PRQT introduces an invariant scalar evolution parameter $s$ that is distinct from coordinate time $t$. This parameter functions as a coordinate-independent universal clock, yielding manifestly covariant and unitary evolution equations. The parameter concept may affect models, such as the Hartle–Hawking no-boundary proposal [8], in which boundary conditions are set as initial states for unitary forward evolution. In contrast to timeless approaches such as the WDW approach or methods relying on emergent/internal clocks (such as the scale factor in Loop Quantum Cosmology [9,10,11,12,13,14]), the parameter $s$ provides an invariant evolution parameter that allows the unitary evolution of quantum states and the conservation of positive-definite probabilities.

The scope of the present contribution is the application of PRQT in curved spacetime to the dynamics of free scalar particles in expanding universe metrics. Governing Stueckelberg-type equations are derived for a generalized expanding universe (EU) metric and for the standard Friedmann–Lemaître–Robertson–Walker (FLRW) metric, with solutions obtained through separation of variables. The derivation of the Stueckelberg-type equation is outlined in Appendix A. These developments establish the formalism for free-particle quantum motion in cosmological backgrounds and include an estimate of the age of the universe. More advanced topics, such as particle interactions, second quantization, or the development of hybrid models, are beyond the scope of this work.

The core contributions of this manuscript are the explicit derivation and separation-of-variables solution of the wave equations in both the generalized EU and FLRW metrics, together with a quantitative estimation of the age of the universe using contemporary observational parameters and a rescaled Friedmann equation. These results demonstrate the compatibility of PRQT with the standard ΛCDM cosmology, reproducing the accepted age of 13.8 Gyr (to within 0.1% of the Planck 2018 ΛCDM value) [15] and aligning with recent DES Y6 matter-density constraints [16]. This establishes PRQT as a promising framework for investigating quantum effects in realistic, time-dependent spacetimes.

The work is presented in the following order. Notation, concepts, and solution procedures for a Stueckelberg-type equation are reviewed in Section 2 for a particle moving in a Schwarzschild metric. The equation is applied to a generalized expanding universe (EU) metric in Section 3 and the standard Friedmann–Lemaître–Robertson–Walker (FLRW) metric in Section 4. Modern observational parameters and a rescaled Friedmann equation are used to estimate the age of the universe in Section 5. These results demonstrate the compatibility of PRQT with realistic cosmological models and open new avenues for exploring quantum effects in dynamic spacetimes, as discussed in Section 6. Conclusions and outlook are presented in Section 7.

2. Motion of a Free Scalar Particle in a Schwarzschild Metric

Section 2.1 uses the Stueckelberg equation in curved spacetime to examine the motion of a free scalar particle in a spherically symmetric gravitational field [6]. This equation follows directly from a parametrized continuity equation in curved spacetime together with a four-velocity ansatz that satisfies the mass-shell condition. A summary of the derivation from the probabilistic foundation in Ref. [7] is presented in Appendix A. The review of Ref. [6] in Section 2.1 establishes the notation, concepts, and solution procedures used in later sections. Section 2.2 derives the radial equation for the particle wave function.

2.1. Parametrized Relativistic Quantum Theory in a Schwarzschild Metric

A free scalar particle with mass $m$ moving in the gravitational field of a spherically symmetric mass $M$ can be described using the Schwarzschild metric equation

$$i\hslash {\displaystyle \frac{\partial {\psi }_{SM}}{\partial s}}=K_{SM}{\psi }_{SM}={\displaystyle \frac{1}{2m}}\left\{\left[{\displaystyle \frac{\hslash }{i}}{\displaystyle \frac{\partial }{\partial y^{\mu }}}\right]\left[{\displaystyle \frac{\hslash }{i}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}\right]-i\hslash {\Gamma }_{\nu \mu }^{\mu }\left[{\displaystyle \frac{\hslash }{i}}{\displaystyle \frac{\partial }{\partial y_{\nu }}}\right]\right\}{\psi }_{SM}.$$

(1)

where ${\psi }_{SM}$ is the wave function of a particle in a Schwarzschild metric and $K_{SM}$ is the operator defined on the right-hand side of Equation (1). The factor ${\displaystyle \frac{1}{2m}}$ and the contracted Christoffel symbol ${\Gamma }_{\nu \mu }^{\mu }$ arise naturally from the kinetic term and covariant derivative when the metric is independent of the evolution parameter $s$ (see Appendix A and Ref. [7]).

The Schwarzschild metric has the following nonzero components:

$$g_{00}=-A\left(r\right), g_{11}=1/A\left(r\right),g_{22}=r^2,g_{33}=r^2{sin}^2\theta .$$

(2)

where $A\left(r\right)=1-{\displaystyle \frac{r_S}{r}}$ and the Schwarzschild radius $r_S$ is

$$r_S={\displaystyle \frac{2M{G}_N}{c^2}}.$$

(3)

The term $G_N$ is Newton’s gravitational constant. The contravariant four-vector with spherical coordinates $\left\{r,\theta ,\varphi \right\}$ is $y^{\mu }=\left\{ct,r,\theta ,\varphi \right\}$. The term ${\Gamma }_{\mathsf{\nu }\mathsf{\mu }}^{\mathsf{\mu }}$ is the contracted Christoffel symbol of the second kind.

The contravariant metric tensor $g^{\mu \nu }$ of the Schwarzschild metric in Equation (2) has nonzero elements

$$g^{00}=-{\displaystyle \frac{1}{A\left(r\right)}},g^{11}=A\left(r\right),g^{22}={\displaystyle \frac{1}{r^2}},g^{33}={\displaystyle \frac{1}{r^2{sin}^2\theta }}.$$

(4)

The product

$$g_{\mu \sigma }{g}^{\sigma \nu }={\delta }_{\mu }^{\nu }$$

(5)

gives the delta function ${\delta }_{\mu }^{\nu }$ and verifies that $g^{\mu \nu }$ is the inverse of $g_{\mu \upsilon }$. The Schwarzschild metric is a diagonalized metric with determinant

$$g=\mathit{det}\left(g_{\mu \upsilon }\right)=g_{00}{g}_{11}{g}_{22}{g}_{33}=-r^4{sin}^2\theta .$$

(6)

The contracted form of the Christoffel symbol of the second kind ${\Gamma }_{\mathsf{\nu }\mathsf{\mu }}^{\mathsf{\mu }}$ in Equation (1) yields

$${\Gamma }_{\nu \mu }^{\mu }={\displaystyle \frac{1}{2g}}{\displaystyle \frac{\partial g}{\partial x^{\nu }}}$$

(7)

where Equation (6) has been used. Combining Equations (6) and (7) gives

$${\Gamma }_{0\mu }^{\mu }=0,{\Gamma }_{1\mu }^{\mu }={\displaystyle \frac{2}{r}},{\Gamma }_{2\mu }^{\mu }=cot\theta ,{\Gamma }_{3\mu }^{\mu }=0.$$

(8)

Rearranging Equation (1) and writing the term ${\Gamma }_{\nu \mu }^{\mu }\left[{\displaystyle \frac{\partial }{\partial y_{\nu }}}\right]$ as ${\Gamma }_{\nu \mu }^{\mu }\left[g^{\nu \sigma }{\displaystyle \frac{\partial }{\partial y^{\sigma }}}\right]$ gives

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{SM}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{{\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}+{\Gamma }_{\nu \mu }^{\mu }\left[{\displaystyle \frac{\partial }{\partial y_{\nu }}}\right]\right\}{\psi }_{SM} \\ ={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{{\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}+{\Gamma }_{\nu \mu }^{\mu }\left[g^{\nu \sigma }{\displaystyle \frac{\partial }{\partial y^{\sigma }}}\right]\right\}{\psi }_{SM}. \end{gathered}$$

(9)

Using Equations (8) and (9), we have

$$i\hslash {\displaystyle \frac{\partial {\psi }_{SM}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{{\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}+\left[{\displaystyle \frac{2}{r}}A\left(r\right){\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{cot\theta }{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}\right]\right\}{\psi }_{SM}.$$

(10)

The term ${\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}$ in the Schwarzschild metric is

$$\begin{gathered} {\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}{\psi }_{SM}={\displaystyle \frac{\partial }{\partial y^{\mu }}}{g}^{\mu \sigma }{\displaystyle \frac{\partial }{\partial y^{\sigma }}}{\psi }_{SM} \\ =\left\{-{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{\partial }{\partial r}}\left[A\left(r\right){\displaystyle \frac{\partial }{\partial r}}\right]+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2}{\partial {\theta }^2}} \\ +{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right\}{\psi }_{SM}. \end{gathered}$$

(11)

Inserting Equation (11) into Equation (10) yields

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{SM}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{-{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{\partial }{\partial r}}\left[A\left(r\right){\displaystyle \frac{\partial }{\partial r}}\right]+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2}{\partial {\theta }^2}}+{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}} \\ +\left[{\displaystyle \frac{2}{r}}A\left(r\right){\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{cot\theta }{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}\right]\right\}{\psi }_{SM}. \end{gathered}$$

(12)

Equation (12) can be simplified by first rearranging the terms and then applying the product rule to obtain

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{SM}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{-{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left[r^{2}A\left(r\right){\displaystyle \frac{\partial }{\partial r}}\right]+{\displaystyle \frac{1}{r^{2}sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(sin\theta {\displaystyle \frac{\partial }{\partial \theta }}\right) \\ +{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right\}{\psi }_{SM}. \end{gathered}$$

(13)

Equation (13) can be written as

$$i\hslash {\displaystyle \frac{\partial {\psi }_{SM}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}{◻}_{SM}{\psi }_{SM}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{-{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}+{△}_{SM}\right\}{\psi }_{SM}.$$

(14)

where ${◻}_{SM}$ is the D’Alembertian operator and ${△}_{SM}$ is the Laplacian operator for the Schwarzschild metric

$${△}_{SM}={\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left[r^{2}A\left(r\right){\displaystyle \frac{\partial }{\partial r}}\right]+{\displaystyle \frac{1}{r^{2}sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(sin\theta {\displaystyle \frac{\partial }{\partial \theta }}\right)+{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}.$$

(15)

In flat spacetime, $A\left(r\right)\to 1$ and Equation (13) becomes

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{flat}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left[r^2{\displaystyle \frac{\partial }{\partial r}}\right]+{\displaystyle \frac{1}{r^{2}sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(sin\theta {\displaystyle \frac{\partial }{\partial \theta }}\right) \\ +{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right\}{\psi }_{flat}. \end{gathered}$$

(16)

2.2. Solution of the Free Scalar Particle in a Schwarzschild Metric

Equation (13) is written here in terms of the operator $L\cdot L$

$$i\hslash {\displaystyle \frac{\partial {\psi }_{SM}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{-{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left[r^{2}A\left(r\right){\displaystyle \frac{\partial }{\partial r}}\right]-{\displaystyle \frac{L\cdot L}{r^2}}\right\}{\psi }_{SM}$$

(17)

where $L\cdot L$ is the operator

$$L\cdot L=-\left[{\displaystyle \frac{1}{sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(sin\theta {\displaystyle \frac{\partial }{\partial \theta }}\right)+{\displaystyle \frac{1}{{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right]$$

(18)

and the nonzero elements of the Schwarzschild metric are given in Equation (2), with $A\left(r\right)=1-{\displaystyle \frac{r_S}{r}}$.

A trial solution for separating the variables in Equation (17) is

$${\psi }_{SM}=T\left(t\right)R\left(r\right)Y_{\mathcal{l}m}\left(\theta ,\varphi \right)exp\left[i{\displaystyle \frac{\hslash {\beta }_{SM}}{2m}}s\right].$$

(19)

The term $Y_{\mathcal{l}m}\left(\theta ,\varphi \right)$ denotes orthonormal spherical harmonics and satisfies the eigenvalue equation

$$L\cdot L{Y}_{\mathcal{l}m}\left(\theta ,\varphi \right)=\mathcal{l}\left(\mathcal{l}+1\right)Y_{\mathcal{l}m}\left(\theta ,\varphi \right) for \left\{\begin{array}{c}\mathcal{l}=0,1,2\dots \\ m=0,1,\dots ,\left|\mathcal{l}\right|\end{array}\right\}.$$

(20)

Substituting Equation (19) into Equation (17) and rearranging, lets us write

$$i\hslash {\displaystyle \frac{i\hslash {\beta }_{SM}}{2m}}{\psi }_{SM}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{-{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left[r^{2}A\left(r\right){\displaystyle \frac{\partial }{\partial r}}\right]-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}{\psi }_{SM}.$$

(21)

Canceling ${\displaystyle \frac{-{\hslash }^2}{2m}}$, multiplying the resulting equation by $1/{\psi }_{SM}$, and then simplifying gives

$${\beta }_{SM}=-{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{1}{T}}{\displaystyle \frac{{\partial }^{2}T}{\partial t^2}}+{\displaystyle \frac{1}{R}}\left\{{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left[r^{2}A\left(r\right){\displaystyle \frac{\partial }{\partial r}}\right]-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}R.$$

(22)

Writing the $T\left(t\right)$ factor in the form

$$T\left(t\right)=\alpha exp\left(\pm i\omega t\right)$$

(23)

with constants $\alpha ,\omega$, and substituting into (22) gives

$${\beta }_{SM}={\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{{\omega }^2}{c^2}}+{\displaystyle \frac{1}{R}}\left\{{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left[r^{2}A\left(r\right){\displaystyle \frac{\partial }{\partial r}}\right]-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}R$$

(24)

where we have used

$${\displaystyle \frac{1}{T}}{\displaystyle \frac{{\partial }^{2}T}{\partial t^2}}={-\omega }^2.$$

(25)

Multiplying Equation (24) by $R$ gives

$${\beta }_{SM}R={\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{{\omega }^2}{c^2}}R+\left\{{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left[r^{2}A\left(r\right){\displaystyle \frac{\partial }{\partial r}}\right]-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}R$$

(26)

which can be written as the radial equation

$$\left[{\beta }_{SM}-{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{{\omega }^2}{c^2}}+{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right]R=\left\{{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left[r^{2}A\left(r\right){\displaystyle \frac{\partial }{\partial r}}\right]\right\}R$$

(27)

after some rearrangement. Expanding the right-hand side of Equation (27) gives

$$\left[{\beta }_{SM}-{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{{\omega }^2}{c^2}}+{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right]R=A\left(r\right){\displaystyle \frac{{\partial }^{2}R}{\partial r^2}}+\left[{\displaystyle \frac{r_S}{r^2}}+{\displaystyle \frac{2A\left(r\right)}{r}}\right]{\displaystyle \frac{\partial R}{\partial r}}.$$

(28)

Rearranging the above equation gives the second-order, linear, homogeneous differential equation

$$A\left(r\right){\displaystyle \frac{d^{2}R}{d{r}^2}}+\left[{\displaystyle \frac{r_S}{r^2}}+{\displaystyle \frac{2A\left(r\right)}{r}}\right]{\displaystyle \frac{dR}{dr}}-\left[{\beta }_{SM}-{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{{\omega }^2}{c^2}}+{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right]R=0.$$

(29)

A radial equation is obtained by multiplying Equation (29) by $1/A\left(r\right)$ to find

$${\displaystyle \frac{d^{2}R}{d{r}^2}}+u\left(r\right){\displaystyle \frac{dR}{dr}}+v\left(r\right)R=0$$

(30)

with

$$u\left(r\right)={\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{r_S}{r^2}}+{\displaystyle \frac{2}{r}}, v\left(r\right)={\displaystyle \frac{1}{A\left(r\right)}}\left[{\displaystyle \frac{1}{A\left(r\right)}}{\displaystyle \frac{{\omega }^2}{c^2}}-{\beta }_{SM}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right].$$

(31)

The general solution of a second-order, linear, homogeneous differential equation is a linear combination of two independent solutions, $R_1\left(r\right)$ and $R_2\left(r\right):R\left(r\right)=A_1{R}_1\left(r\right)+A_2{R}_2\left(r\right)$, where $A_1$ and $A_2$ are constants determined by boundary conditions. Equation (30) does not have closed-form solutions in terms of elementary functions for general parameters.

A general solution of Equation (30) exists on an open interval $I$ if $u\left(r\right)$ and $v\left(r\right)$ are continuous on the open interval $I$ [17]. In addition, series solutions can be constructed for different regions of $r$. Three pairs of independent series solutions are provided in Elizalde and Louko [18].

Equation (30) can be solved in the large distance approximation corresponding to $r\gg r_S$ and $A\left(r\right)\to 1$. In this case, the particle is at a large distance from the spherically symmetric mass $M$. The solution to the radial equation in the large distance approximation has been shown to have a $1/r$ dependence [6].

3. Motion of a Free Scalar Particle in an Expanding Universe Metric

The Stueckelberg equation for curved spacetime is used to study the motion of a free scalar particle in an expanding universe (EU) metric. The EU metric is a generalization of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric in spherical polar coordinates. The FLRW metric models a uniform, isotropic, expanding universe. A PRQT model of a free scalar particle moving in an EU metric is formulated in Section 3.1, and a solution is presented in Section 3.2. A general solution for the differential equations is discussed in Section 3.3.

3.1. Parametrized Relativistic Quantum Theory in an Expanding Universe Metric

A free scalar particle with mass $m$ moving in an expanding, curved spacetime can be modeled using the field equation

$$i\hslash {\displaystyle \frac{\partial {\psi }_{EU}}{\partial s}}=K_{GF}{\psi }_{EU}={\displaystyle \frac{1}{2m}}\left\{\left[{\displaystyle \frac{\hslash }{i}}{\displaystyle \frac{\partial }{\partial y^{\mu }}}\right]\left[{\displaystyle \frac{\hslash }{i}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}\right]-i\hslash {\Gamma }_{\nu \mu }^{\mu }\left[{\displaystyle \frac{\hslash }{i}}{\displaystyle \frac{\partial }{\partial y_{\nu }}}\right]\right\}{\psi }_{EU}$$

(32)

where ${\psi }_{EU}$ is the wave function. The EU metric has nonzero elements

$$g_{00}=-1, g_{11}={\displaystyle \frac{a^2\left(t\right)}{b\left(r\right)}},g_{22}=a^2\left(t\right)r^2,g_{33}=a^2\left(t\right)r^2{sin}^2\theta$$

(33)

where $a\left(t\right)$ is the time-dependent scale factor that describes the expansion of the universe. The contravariant four-vector is $y^{\mu }=\left\{ct,r,\theta ,\varphi \right\}$, with spherical coordinates $\left\{r,\theta ,\varphi \right\}$. The function $b\left(r\right)$ is the radial dependence of the metric. For example, the FLRW metric is the special case

$$b\left(r\right)=1-k{r}^2; k=\left\{\begin{array}{cc}+1& closed\\ 0& flat\\ -1& open\end{array}\right\}$$

(34)

where $k$ is the curvature parameter. The EU metric uses $b\left(r\right)$ to generalize the radial dependence with an unspecified function of radius $r$. The metric tensor in Equation (33) has the contravariant metric tensor $g^{\mu \nu }$. Nonzero elements of $g^{\mu \nu }$ are the inverse values

$$g^{00}=-1,g^{11}={\displaystyle \frac{b\left(r\right)}{a^2\left(t\right)}},g^{22}={\displaystyle \frac{1}{a^2\left(t\right)r^2}},g^{33}={\displaystyle \frac{1}{a^2\left(t\right)r^2{sin}^2\theta }}.$$

(35)

The product of Equations (33) and (35) gives the Kronecker delta function

$$g_{\mu \sigma }{g}^{\sigma \nu }={\delta }_{\mu }^{\nu }$$

(36)

and verifies that $g^{\mu \nu }$ is the inverse of $g_{\mu \upsilon }$.

The EU metric is a diagonalized metric with determinant

$$g=\mathit{det}\left(g_{\mu \upsilon }\right)=g_{00}{g}_{11}{g}_{22}{g}_{33}=-{\displaystyle \frac{a^6\left(t\right)}{b\left(r\right)}}{r}^4{sin}^2\theta .$$

(37)

The contracted form of the Christoffel symbol of the second kind ${\Gamma }_{\nu \mu }^{\mu }$ in Equation (32) is

$${\Gamma }_{\nu \mu }^{\mu }={\displaystyle \frac{1}{2g}}{\displaystyle \frac{\partial g}{\partial x^{\nu }}}.$$

(38)

Substituting Equation (37) into Equation (38) gives

$${\Gamma }_{0\mu }^{\mu }={\displaystyle \frac{3\dot{a}\left(t\right)}{a\left(t\right)c}},{\Gamma }_{1\mu }^{\mu }={\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}},{\Gamma }_{2\mu }^{\mu }=cot\theta ,{\Gamma }_{3\mu }^{\mu }=0.$$

(39)

Rearranging Equation (32) and writing the term ${\Gamma }_{\nu \mu }^{\mu }\left[{\displaystyle \frac{\partial }{\partial y_{\nu }}}\right]$ as ${\Gamma }_{\nu \mu }^{\mu }\left[g^{\nu \sigma }{\displaystyle \frac{\partial }{\partial y^{\sigma }}}\right]$ gives

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{EU}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{{\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}+{\Gamma }_{\nu \mu }^{\mu }\left[{\displaystyle \frac{\partial }{\partial y_{\nu }}}\right]\right\}{\psi }_{EU} \\ ={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{{\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}+{\Gamma }_{\nu \mu }^{\mu }\left[g^{\nu \sigma }{\displaystyle \frac{\partial }{\partial y^{\sigma }}}\right]\right\}{\psi }_{EU}. \end{gathered}$$

(40)

Combining Equations (39) and (40) lets us write

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{EU}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{{\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}+{\displaystyle \frac{3\dot{a}\left(t\right)}{a\left(t\right)c^2}}{\displaystyle \frac{\partial }{\partial t}}+{\displaystyle \frac{1}{a^2\left(t\right)}}\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}} \\ +{\displaystyle \frac{1}{a^2\left(t\right)}}{\displaystyle \frac{cot\theta }{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}\right\}{\psi }_{EU}. \end{gathered}$$

(41)

The term ${\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}$ can be written as

$${\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}{\psi }_{EU}=\left\{-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{b\left(r\right)}{a^2\left(t\right)}}{\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{1}{a^2\left(t\right)}}{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2}{\partial {\theta }^2}}+{\displaystyle \frac{1}{a^2\left(t\right)r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right\}{\psi }_{EU}.$$

(42)

Factoring $a^2\left(t\right)$ in Equation (42) gives

$${\displaystyle \frac{\partial }{\partial y^{\mu }}}{\displaystyle \frac{\partial }{\partial y_{\mu }}}{\psi }_{EU}={\displaystyle \frac{1}{a^2\left(t\right)}}\left\{-{\displaystyle \frac{a^2\left(t\right)}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}+b(r){\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2}{\partial {\theta }^2}}+{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right\}{\psi }_{EU}.$$

(43)

Substituting Equation (43) into Equation (41) and simplifying gives

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{EU}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}\left\{{\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left[-{\displaystyle \frac{a^2\left(t\right)}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}+b(r){\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2}{\partial {\theta }^2}}+{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right] \\ +{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b(r)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{cot\theta }{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}\right\}\right\}{\psi }_{EU}. \end{gathered}$$

(44)

Factoring $a^2\left(t\right)$ in Equation (44) gives

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{EU}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}{\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+b(r){\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2}{\partial {\theta }^2}}+{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}} \\ +{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{cot\theta }{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}\right\}{\psi }_{EU}. \end{gathered}$$

(45)

Combining terms in space and time gives

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{EU}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}{\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+b(r){\displaystyle \frac{{\partial }^2}{\partial r^2}} \\ +\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2}{\partial {\theta }^2}}+{\displaystyle \frac{cot\theta }{r^2}}{\displaystyle \frac{\partial }{\partial \theta }} \\ +{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right\}{\psi }_{EU}. \end{gathered}$$

(46)

Combining $r$-dependent terms with a ${\displaystyle \frac{\partial }{\partial r}}$ derivative gives

$$b(r){\displaystyle \frac{\partial }{\partial r}}{\displaystyle \frac{\partial }{\partial r}}+\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}=b\left(r\right){\displaystyle \frac{{\partial }^2}{\partial r^2}}+\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}.$$

(47)

The product rule lets us write

$${\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2}{\partial {\theta }^2}}+{\displaystyle \frac{cos\theta }{r^{2}sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}={\displaystyle \frac{1}{r^{2}sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(sin\theta {\displaystyle \frac{\partial }{\partial \theta }}\right).$$

(48)

Substituting Equations (47) and (48) into Equation (46) gives the wave equation for ${\psi }_{EU}$:

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{EU}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}{\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+b\left(r\right){\displaystyle \frac{{\partial }^2}{\partial r^2}} \\ +\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{1}{r^{2}sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(sin\theta {\displaystyle \frac{\partial }{\partial \theta }}\right) \\ +{\displaystyle \frac{1}{r^2{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right\}{\psi }_{EU}. \end{gathered}$$

(49)

Equation (49) is the equation for a free scalar particle with mass $m$ moving in an expanding, curved spacetime represented by the EU metric.

3.2. Solution of the Free Scalar Particle in an Expanding Universe Metric

The solution of Equation (49) is obtained by first writing it in terms of the operator $L\cdot L$:

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{EU}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}{\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+b\left(r\right){\displaystyle \frac{{\partial }^2}{\partial r^2}} \\ +\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{L\cdot L}{r^2}}\right\}{\psi }_{EU} \end{gathered}$$

(50)

where $L\cdot L$ is the operator

$$L\cdot L=-\left[{\displaystyle \frac{1}{sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(sin\theta {\displaystyle \frac{\partial }{\partial \theta }}\right)+{\displaystyle \frac{1}{{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right].$$

(51)

A trial solution for separating variables in Equation (50) is

$${\psi }_{EU}=T\left(t\right)R\left(r\right)Y_{\mathcal{l}m}\left(\theta ,\varphi \right)exp\left[i{\displaystyle \frac{\hslash {\beta }_{EU}}{2m}}s\right].$$

(52)

The term $Y_{\mathcal{l}m}\left(\theta ,\varphi \right)$ represents orthonormal spherical harmonics and satisfies the eigenvalue equation

$$L\cdot L{Y}_{\mathcal{l}m}\left(\theta ,\varphi \right)=\mathcal{l}\left(\mathcal{l}+1\right)Y_{\mathcal{l}m}\left(\theta ,\varphi \right) for \left\{\begin{array}{c}\mathcal{l}=0,1,2\dots \\ m=0,1,\dots ,\left|\mathcal{l}\right|\end{array}\right\}.$$

(53)

Substituting Equation (52) into Equation (50) and rearranging, lets us write

$$\begin{gathered} i\hslash {\displaystyle \frac{i\hslash {\beta }_{EU}}{2m}}{\psi }_{EU}={\displaystyle \frac{-{\hslash }^2}{2m}}{\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+b\left(r\right){\displaystyle \frac{{\partial }^2}{\partial r^2}} \\ +\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}{\psi }_{EU}. \end{gathered}$$

(54)

Canceling ${\displaystyle \frac{-{\hslash }^2}{2m}}$ gives

$$\begin{gathered} {\beta }_{EU}{\psi }_{EU}={\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+b\left(r\right){\displaystyle \frac{{\partial }^2}{\partial r^2}} \\ +\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}{\psi }_{EU}. \end{gathered}$$

(55)

Multiplying Equation (55) by $1/{\psi }_{EU}$ and simplifying yields

$$\begin{gathered} {\beta }_{EU}TR={\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+b\left(r\right){\displaystyle \frac{{\partial }^2}{\partial r^2}} \\ +\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}TR. \end{gathered}$$

(56)

Rearranging Equation (56) and multiplying by ${\displaystyle \frac{a^2\left(t\right)}{TR}}$ gives

$$\begin{gathered} {\beta }_{EU}{a}^2(t)={\displaystyle \frac{1}{T}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}\right\}T \\ +{\displaystyle \frac{1}{R}}\left\{b\left(r\right){\displaystyle \frac{{\partial }^2}{\partial r^2}}+\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}R. \end{gathered}$$

(57)

Equation (57) can be rearranged with $t$-dependent terms on the left and $r$-dependent terms on the right, resulting in:

$$\begin{gathered} {\displaystyle \frac{1}{T}}\left\{{\displaystyle \frac{a^2\left(t\right)}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}-{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+{\beta }_{EU}{a}^2\left(t\right)\right\}T \\ ={\displaystyle \frac{1}{R}}\left\{b\left(r\right){\displaystyle \frac{{\partial }^2}{\partial r^2}}+\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}R. \end{gathered}$$

(58)

Equation (58) can be separated. By setting each side of the equation equal to a separation constant and rearranging, we obtain the $t$-dependent equation

$$\left\{{\displaystyle \frac{a^2\left(t\right)}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}-{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+{\beta }_{EU}{a}^2(t)\right\}T={\kappa }_{EU}T$$

(59)

and the $r$-dependent equation

$$\left\{b\left(r\right){\displaystyle \frac{{\partial }^2}{\partial r^2}}+\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}R={\kappa }_{EU}R.$$

(60)

Equations (59) and (60) can be written as the $t$-dependent equation

$$\left\{{\displaystyle \frac{a^2\left(t\right)}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}-{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+\left[{\beta }_{EU}{a}^2\left(t\right)-{\kappa }_{EU}\right]\right\}T=0$$

(61)

and the $r$-dependent equation

$$\left\{b\left(r\right){\displaystyle \frac{{\partial }^2}{\partial r^2}}+\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}-\left[{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}+{\kappa }_{EU}\right]\right\}R=0.$$

(62)

Multiplying the $t$-dependent equation by ${\displaystyle \frac{c^2}{a^2\left(t\right)}}$ gives

$$\left\{{\displaystyle \frac{{\partial }^2}{\partial t^2}}-{\displaystyle \frac{3\dot{a}\left(t\right)}{a\left(t\right)}}{\displaystyle \frac{\partial }{\partial t}}+\left[{\beta }_{EU}-{\displaystyle \frac{{\kappa }_{EU}}{a^2\left(t\right)}}\right]c^2\right\}T=0.$$

(63)

Dividing Equation (62) by $b\left(r\right)$ and simplifying lets us write

$$\left\{{\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{1}{b\left(r\right)}}\left[{\displaystyle \frac{2}{r}}-{\displaystyle \frac{1}{2b\left(r\right)}}{\displaystyle \frac{\partial b\left(r\right)}{\partial r}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{1}{b\left(r\right)}}\left[{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}+{\kappa }_{EU}\right]\right\}R=0.$$

(64)

Equations (63) and (64) have the form of a homogeneous, second-order, linear differential equation

$$\left\{{\displaystyle \frac{d^2}{d{q}^2}}+u(q){\displaystyle \frac{d}{dq}}+v(q)\right\}Q(q)=0.$$

(65)

A general solution of Equation (65) exists on an open interval $I$ if $u\left(q\right)$ and $v\left(q\right)$ are continuous on the open interval $I$ [17].

3.3. General Solution

The Equation (63) for the $t$-dependent part and Equation (64) for the $r$-dependent part are second-order linear homogeneous differential equations in the form of Equation (65). The general solution for each is a linear combination of two independent solutions: $G\left(t\right)=A{G}_1\left(t\right)+B{G}_2\left(t\right); R\left(r\right)=C{R}_1\left(r\right)+D{R}_2\left(r\right)$, where $A, B, C, D$ are constants determined by boundary conditions or normalization.

The form of the solutions depends on the specific functions $a\left(t\right)$ and $b\left(r\right)$. For general $b\left(r\right)$, the $r$-dependent Equation (64) does not have closed-form solutions in elementary functions and may require numerical methods or series expansions. For the $t$-dependent Equation (63), the solution depends on the scale factor $a\left(t\right)$, which is determined by the cosmological model (e.g., radiation-dominated, matter-dominated, or vacuum-dominated eras). In special cases, such as power-law $a\left(t\right)$, the solutions can be expressed in terms of Bessel functions or other special functions [19].

4. Motion of a Free Scalar Particle in the FLRW Metric

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric in spherical polar coordinates describes an expanding, homogeneous, and isotropic spacetime with nonzero metric elements

$$g_{00}=-1, g_{11}={\displaystyle \frac{a^2\left(t\right)}{1-k{r}^2}},g_{22}=a^2\left(t\right)r^2,g_{33}=a^2\left(t\right)r^2{sin}^2\theta .$$

(66)

The term $a\left(t\right)$ is a time-dependent scale factor, and the $r$-dependent function $b\left(r\right)$ in the EU metric is replaced by the FLRW metric term $1-k{r}^2$. The curvature factor $k$ is defined in Equation (34). The Stueckelberg Equation (50) for a free scalar particle with mass $m$ moving in an expanding universe becomes

$$\begin{gathered} i\hslash {\displaystyle \frac{\partial {\psi }_{FM}}{\partial s}}={\displaystyle \frac{-{\hslash }^2}{2m}}{\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)}{a\left(t\right)c^2}}{\displaystyle \frac{\partial }{\partial t}}+\left(1-k{r}^2\right){\displaystyle \frac{{\partial }^2}{\partial r^2}} \\ +\left[{\displaystyle \frac{2}{r}}+{\displaystyle \frac{kr}{\left(1-k{r}^2\right)}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{L\cdot L}{r^2}}\right\}{\psi }_{FM} \end{gathered}$$

(67)

where the subscript to ${\psi }_{FM}$ refers to the FLRW metric, and $L\cdot L$ is the operator

$$L\cdot L=-\left[{\displaystyle \frac{1}{sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(sin\theta {\displaystyle \frac{\partial }{\partial \theta }}\right)+{\displaystyle \frac{1}{{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\varphi }^2}}\right].$$

(68)

4.1. Solution of the Stueckelberg Equation for a Particle Moving in the FLRW Metric

A trial solution for separating variables in Equation (68) is

$${\psi }_{FM}=T\left(t\right)R\left(r\right)Y_{\mathcal{l}m}\left(\theta ,\varphi \right)exp\left[i{\displaystyle \frac{\hslash {\beta }_{FM}}{2m}}s\right].$$

(69)

The term $Y_{\mathcal{l}m}\left(\theta ,\varphi \right)$ represents orthonormal spherical harmonics and satisfies the eigenvalue equation

$$L\cdot L{Y}_{\mathcal{l}m}\left(\theta ,\varphi \right)=\mathcal{l}\left(\mathcal{l}+1\right)Y_{\mathcal{l}m}\left(\theta ,\varphi \right) for \left\{\begin{array}{c}\mathcal{l}=0,1,2\dots \\ m=0,1,\dots ,\left|\mathcal{l}\right|\end{array}\right\}.$$

(70)

Substituting Equation (69) into Equation (67) and rearranging, lets us write

$$\begin{gathered} \hslash {\displaystyle \frac{i\hslash {\beta }_{FM}}{2m}}{\psi }_{FM}={\displaystyle \frac{-{\hslash }^2}{2m}}{\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)}{a\left(t\right)c^2}}{\displaystyle \frac{\partial }{\partial t}}+\left(1-k{r}^2\right){\displaystyle \frac{{\partial }^2}{\partial r^2}} \\ +\left[{\displaystyle \frac{2}{r}}+{\displaystyle \frac{kr}{\left(1-k{r}^2\right)}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}{\psi }_{FM}. \end{gathered}$$

(71)

Dividing by ${\displaystyle \frac{-{\hslash }^2}{2m}}$, multiplying by $1/{\psi }_{FM}$ and simplifying yields

$$\begin{gathered} {\beta }_{FM}TR={\displaystyle \frac{1}{a^2\left(t\right)}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)}{a\left(t\right)c^2}}{\displaystyle \frac{\partial }{\partial t}}+\left(1-k{r}^2\right){\displaystyle \frac{{\partial }^2}{\partial r^2}} \\ +\left[{\displaystyle \frac{2}{r}}+{\displaystyle \frac{kr}{\left(1-k{r}^2\right)}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}TR. \end{gathered}$$

(72)

Rearranging Equation (72) and multiplying by ${\displaystyle \frac{a^2\left(t\right)}{TR}}$ lets us write

$$\begin{gathered} {\beta }_{FM}{a}^2\left(t\right)={\displaystyle \frac{1}{T}}\left\{\left(-{\displaystyle \frac{a^2\left(t\right)}{c^2}}\right){\displaystyle \frac{{\partial }^2}{\partial t^2}}+{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}\right\}T \\ +{\displaystyle \frac{1}{R}}\left\{\left(1-k{r}^2\right){\displaystyle \frac{{\partial }^2}{\partial r^2}}+\left[{\displaystyle \frac{2}{r}}+{\displaystyle \frac{kr}{\left(1-k{r}^2\right)}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}R. \end{gathered}$$

(73)

Equation (73) is reorganized with $t$-dependent terms on the left and $r$-dependent terms on the right, yielding

$$\begin{gathered} {\displaystyle \frac{1}{T}}\left\{{\displaystyle \frac{a^2\left(t\right)}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}-{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+{\beta }_{FM}{a}^2\left(t\right)\right\}T \\ ={\displaystyle \frac{1}{R}}\left\{\left(1-k{r}^2\right){\displaystyle \frac{{\partial }^2}{\partial r^2}}+\left[{\displaystyle \frac{2}{r}}+{\displaystyle \frac{kr}{\left(1-k{r}^2\right)}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}R. \end{gathered}$$

(74)

Equation (74) is separable. Setting each side of the equation equal to a separation constant ${\kappa }_{EU}$ and rearranging lets us write the $t$-dependent equation

$$\left\{{\displaystyle \frac{a^2\left(t\right)}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}-{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+{\beta }_{FM}{a}^2(t)\right\}T={\kappa }_{FM}T$$

(75)

and the $r$-dependent equation

$$\left\{\left(1-k{r}^2\right){\displaystyle \frac{{\partial }^2}{\partial r^2}}+\left[{\displaystyle \frac{2}{r}}+{\displaystyle \frac{kr}{\left(1-k{r}^2\right)}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}\right\}R={\kappa }_{FM}R.$$

(76)

Equations (75) and (76) can be written as the $t$-dependent equation

$$\left\{{\displaystyle \frac{a^2\left(t\right)}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}-{\displaystyle \frac{3\dot{a}\left(t\right)a\left(t\right)}{c^2}}{\displaystyle \frac{\partial }{\partial t}}+\left[{\beta }_{FM}{a}^2\left(t\right)-{\kappa }_{FM}\right]\right\}T=0$$

(77)

and the $r$-dependent equation

$$\left\{\left(1-k{r}^2\right){\displaystyle \frac{{\partial }^2}{\partial r^2}}+\left[{\displaystyle \frac{2}{r}}+{\displaystyle \frac{kr}{\left(1-k{r}^2\right)}}\right]{\displaystyle \frac{\partial }{\partial r}}-\left[{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}+{\kappa }_{EU}\right]\right\}R=0.$$

(78)

Multiplying the $t$-dependent equation by ${\displaystyle \frac{c^2}{a^2\left(t\right)}}$ gives

$$\left\{{\displaystyle \frac{{\partial }^2}{\partial t^2}}-{\displaystyle \frac{3\dot{a}\left(t\right)}{a\left(t\right)}}{\displaystyle \frac{\partial }{\partial t}}+\left[{\beta }_{FM}-{\displaystyle \frac{{\kappa }_{FM}}{a^2\left(t\right)}}\right]c^2\right\}T=0.$$

(79)

Equation (78) is the $r$-dependent radial equation. Dividing Equation (78) by $\left(1-k{r}^2\right)$ and simplifying lets us write

$$\left\{{\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{1}{(1-k{r}^2)}}\left[{\displaystyle \frac{2}{r}}+{\displaystyle \frac{kr}{\left(1-k{r}^2\right)}}\right]{\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{1}{(1-kr)}}\left[{\displaystyle \frac{\mathcal{l}\left(\mathcal{l}+1\right)}{r^2}}+{\kappa }_{FM}\right]\right\}R=0.$$

(80)

Equations (79) and (80) have the form of a homogeneous, second-order, linear differential equation

$$\left\{{\displaystyle \frac{d^2}{d{q}^2}}+u\left(q\right){\displaystyle \frac{d}{dq}}+v(q)\right\}Q(q)=0.$$

(81)

A general solution of Equation (81) exists on an open interval $I$ if $u\left(q\right)$ and $v\left(q\right)$ are continuous on the open interval $I$ [17].

4.2. General Solution

Both $t$-dependent Equation (79) and $r$-dependent Equation (81) are second-order linear homogeneous ODEs with general solutions given by linear combinations of two independent solutions:

$$\begin{gathered} G\left(t\right)=A_1{G}_1\left(t\right)+A_2{G}_2\left(t\right) \\ R\left(r\right)=B_1{R}_1\left(t\right)+B_{2}R\left(t\right). \end{gathered}$$

(82)

Time-dependent solutions depend on the scale factor $a\left(t\right)$. As an example, the scale factor for power-law expansion is $a\left(t\right)={\left(t/t_0\right)}^p$ (e.g., $p = 1/2$ radiation, $p = 2/3$ matter) and the solutions are combinations of Bessel functions (first and second kind). Radial-dependent solutions depend on spatial curvature k. More details are available in the literature, e.g., [19,20].

4.3. The Scale Factor and a Rescaled Friedmann Equation

A $t$-dependent scale factor $a\left(t\right)$ is needed in an expanding universe metric such as the FLRW metric. An example of scale factor $a\left(t\right)$ is the rescaled Friedmann equation discussed by Hartle [19]. The Hubble parameter is

$$H={\displaystyle \frac{\dot{a}\left(t\right)}{a\left(t\right)}}, \dot{a}\left(t\right)={\displaystyle \frac{da\left(t\right)}{dt}}.$$

(83)

The dimensionless scale factor $\tilde{a}\left(t\right)$ is defined as

$$\tilde{a}\left(t\right)={\displaystyle \frac{a\left(t\right)}{a_0}}, a_0=a\left(t_0\right)$$

(84)

where $t_0$ is the present time. Similarly, the dimensionless $\tilde{t}$ is defined as

$$\tilde{t}={\displaystyle \frac{t}{t_H}}=H_{0}t$$

(85)

where the Hubble constant $H_0$ is the Hubble parameter at the present time, thus $H_0=H\left(t_0\right)$.

The rescaled Friedmann equation can be written in terms of dimensionless quantities as [19]

$${\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{d\tilde{a}}{d\tilde{t}}}\right)}^2+U_e\left(\tilde{a}\right)={\displaystyle \frac{{\Omega }_c}{2}}.$$

(86)

The effective potential $U_e$ depends on $\tilde{a}$, vacuum energy ${\mathsf{\Omega }}_v$, mass energy ${\mathsf{\Omega }}_m$, and radiation energy ${\mathsf{\Omega }}_r$, thus

$$U_e\left(\tilde{a}\right)=-{\displaystyle \frac{1}{2}}\left[{\Omega }_v{\tilde{a}}^2+{\displaystyle \frac{{\Omega }_m}{\tilde{a}}}+{\displaystyle \frac{{\Omega }_r}{{\tilde{a}}^2}}\right].$$

(87)

The term ${\mathsf{\Omega }}_v$ is related to vacuum energy density, which in turn depends on the cosmological constant $\mathsf{\Lambda }$ [19]. The term ${\mathsf{\Omega }}_c$ represents curvature because it is related to curvature $k$ in the FLRW metric by

$${\Omega }_c=-{\displaystyle \frac{k}{{\left(H_0{a}_0\right)}^2}}.$$

(88)

The terms ${\Omega }_v,{\Omega }_m,{\Omega }_r,{\Omega }_c$ are the present-day density parameters for vacuum energy, matter, radiation, and curvature, respectively. They satisfy the relationship

$${\Omega }_r+{\Omega }_m+{\Omega }_v+{\Omega }_c=1.$$

(89)

It should be noted that ${\Omega }_c$ can be negative for a closed universe $(k=1)$.

The solution of Equation (86) is obtained by writing it as

$${\displaystyle \frac{d\tilde{a}}{d\tilde{t}}}={\left[{\Omega }_c-2U_e\left(\tilde{a}\right)\right]}^{1/2}.$$

(90)

Separating variables and integrating gives [19] (page 392)

$$\tilde{t}={\int }_0^1{\displaystyle \frac{d\tilde{a}}{{\left[{\Omega }_c+{\Omega }_v{\tilde{a}}^2+\frac{{\Omega }_m}{\tilde{a}}+\frac{{\Omega }_r}{{\tilde{a}}^2}\right]}^{1/2}}}.$$

(91)

The effective potential $U_e$ has been replaced by Equation (87). Dimensionless time $\tilde{t}$ varies from 0 to 1, where $\tilde{t}=1$ is the present. See Hartle [19] for more discussion.

5. Estimating the Age of the Expanding Universe

The rescaled Friedmann equation is expressed in terms of dimensionless quantities as

$${\eta }^2=E\left(\mathsf{\Omega }\right),$$

(92)

where the dimensionless Hubble parameter $\eta$ is

$$\eta =H\left(t\right)/H_0, H_0=H\left(t_0\right)$$

(93)

and $H_0$ is the Hubble constant at the present time $t_0$. The dimensionless scale factor $\mathsf{\Omega }$ is

$$\Omega =a\left(t\right)/a\left(t_0\right).$$

(94)

The effective potential $E\left(\mathsf{\Omega }\right)$ is given by

$$E(\Omega )={\Omega }_{\Lambda }+{\displaystyle \frac{{\Omega }_k}{{\Omega }^2}}+{\displaystyle \frac{{\Omega }_m}{{\Omega }^3}}+{\displaystyle \frac{{\Omega }_r}{{\Omega }^4}}$$

(95)

where ${\Omega }_{\Lambda }$, ${\mathsf{\Omega }}_{\mathrm{k}}$, ${\mathsf{\Omega }}_{\mathrm{m}}$, and ${\mathsf{\Omega }}_{\mathrm{r}}$ represent the contributions from vacuum energy, curvature, matter, and radiation, respectively. These parameters satisfy the relation

$${\Omega }_{\Lambda }+{\Omega }_k+{\Omega }_m+{\Omega }_r=1$$

(96)

at the present time ($t=t_0 \mathrm{s}\mathrm{o} \Omega =1$). They are used to estimate the age of the expanding universe using the age integral described next.

5.1. The Age Integral

The age of the universe $t_0$ is obtained by integrating the rescaled Friedmann equation from the Big Bang ($\Omega =0$) to the present ($\Omega =1$). Beginning with the Hubble parameter $H$ in Equation (83) and $\mathsf{\Omega }$ in Equation (94), we have

$$\dot{\Omega }={\displaystyle \frac{d\Omega }{dt}}={\displaystyle \frac{\dot{a}\left(t\right)}{a\left(t_0\right)}}=H{\displaystyle \frac{a\left(t\right)}{a\left(t_0\right)}}=H\Omega$$

(97)

since $a\left(t_0\right)=1$ at present. Rearranging gives

$$dt={\displaystyle \frac{d\Omega }{H\Omega }}.$$

(98)

Combining Equations (92) and (93) lets us write

$$H\left(t\right)=H_0\eta =H_0\sqrt{E\left(\mathsf{\Omega }\right)}.$$

(99)

Substituting Equation (99) into Equation (97) gives

$$Hdt={\displaystyle \frac{d\Omega }{H_0\Omega \sqrt{E\left(\Omega \right)}}}.$$

(100)

Integrating from $t=0$ ($\Omega =0$) to $t=t_0$ ($\mathsf{\Omega }=1$) results in the integral

$$t_0={\displaystyle \frac{1}{H_0}}{\int }_0^1{\displaystyle \frac{d\Omega }{\Omega \sqrt{E\left(\Omega \right)}}}.$$

(101)

The integral in Equation (101) converges near $\Omega =0$ because in the radiation-dominated limit, the integrand behaves as $\Omega /\sqrt{{\Omega }_r}$, which is integrable. Equation (101) is the age integral for the age of the universe because it represents the dimensionless time from the origin of the universe to the present.

5.2. Solution of the Age Integral

An analytical approximation of the age integral is obtained by considering a flat universe (${\Omega }_k=0$) and neglecting the radiation term (${\Omega }_r=0$). The flat universe assumption is justified empirically [15,16]. The radiation term is neglected because radiation is dominant only in the very early universe. Consequently, the contribution represented by ${\Omega }_r$ is small for the total age of the universe. Given these assumptions, Equation (95) becomes

$$E(\Omega )\approx {\Omega }_{\Lambda }+{\displaystyle \frac{{\Omega }_m}{{\Omega }^3}}$$

(102)

and Equation (101) simplifies to

$$H_0{t}_0={\int }_0^1{\displaystyle \frac{d\Omega }{\sqrt{{\Omega }_{\Lambda }{\Omega }^2+{\Omega }_m{\Omega }^{-1}}}}.$$

(103)

Equation (103) has the solution

$$H_0{t}_0={\displaystyle \frac{2}{3\sqrt{{\Omega }_{\Lambda }}}}{sinh}^{-1}\sqrt{{\displaystyle \frac{{\Omega }_{\Lambda }}{{\Omega }_m}}}.$$

(104)

Planck 2018 gives modern values of the parameters ${\Omega }_m,{\Omega }_{\Lambda },H_0$ as ${\Omega }_m\approx 0.315$, ${\Omega }_{\Lambda }\approx 1-{\Omega }_m=0.685$, $H_0\approx 67.4 \mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$ [15]. The inverse of $H_0\approx 67.4 \mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$ is $1/H_0\approx 14.52 Gyr$. Using Planck 2018 values for a flat universe with radiation neglected [15] in Equation (104) gives

$${\Omega }_m=0.315, {\Omega }_{\Lambda }\approx 1-{\Omega }_m=0.685,H_0=67.4 \mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c},1/H_0\approx 14.52 Gyr.$$

(105)

with the resulting age of the universe:

$$t_0\approx 0.951\times 14.52\approx 13.8 Gyr.$$

(106)

This matches the standard ΛCDM value of 13.8 Gyr from Planck 2018 [15] to within 0.1% and is consistent with the DES Y6 matter-density constraint ${\Omega }_m= {0.333}_{-0.028 }^{+0.023}$ [16].

6. Discussion

This section compares PRQT’s approach to quantum gravity and cosmology with established quantum gravity and cosmological models. Parametrized Relativistic Quantum Theory (PRQT) in curved spacetime, as applied above to free scalar particle motion in expanding universe metrics, introduces an invariant evolution parameter distinct from coordinate time. PRQT provides manifestly covariant, unitary evolution equations in curved spacetime, enabling the study of quantum effects. Ontological, dynamical, and interpretational differences are highlighted here.

6.1. Ontology and Dynamical Parameter

PRQT views the scalar evolution parameter $s$ as a measurable, temporal parameter [6,21]. In contrast to internal clocks (e.g., the scale factor in LQG or emergent time in WDW), $s$ is a global scalar parameter that is invariant under general coordinate transformations, is frame-independent, and provides an explicit external time for unitary Schrödinger-like evolution. Its measurability follows from its role as the parameter along which probability is conserved.

A key distinction between standard cosmological frameworks and PRQT is how the problem of time in quantum gravity is handled. For example, the Wheeler–DeWitt (WDW) equation [1,2] enforces a Hamiltonian constraint $\mathrm{Ĥ}\Psi = 0$ and is derived from canonical quantization using the Arnowitt–Deser–Misner (ADM) decomposition [3]. The WDW equation is considered a timeless equation because it is frozen; that is, it does not have an explicit external time parameter. In minisuperspace approximations, such as homogeneous isotropic cosmology with scale factor $a$ and scalar field $\phi$, it resembles a time-independent Klein–Gordon equation. Emergent time arises in the semiclassical Wentzel–Kramers–Brillouin (WKB) limit, where the wave function $\Psi \approx exp(iS/ħ)$ leads to a Hamilton–Jacobi equation for $S$, with time parameterized by slow-varying degrees of freedom (e.g., scale factor $a$ as an internal clock).

By contrast, PRQT uses an invariant evolution parameter $s$ in a Schrödinger-like equation of the form $iħ \partial \Psi /\partial s = \mathrm{Ĥ}\Psi$ for $s$-evolution with Hamiltonian operator $\widehat{\mathrm{H}}$. The state $\mathsf{\Psi }$ evolves unitarily with respect to $s$, and spacetime coordinates $x^{\mu }$ are operator-valued. This simplifies cosmological wave function interpretations by providing unitary $s$-evolution rather than relying on emergent time. The PRQT approach may influence models such as the Hartle–Hawking no-boundary proposal [8], where boundary conditions are set as initial states at $s = s_0$ for unitary forward evolution.

Loop quantum gravity (LQG) and loop quantum cosmology (LQC) [9,10,11,12,13,14] use internal clocks and discrete geometry for singularity avoidance, such as bounces in the Friedmann equation. PRQT could complement this with continuous $s$-evolution, potentially incorporating LQG corrections such as holonomy regularizations [22] into the Hamiltonian in future hybrid models. While PRQT assumes semiclassical metrics, it could incorporate LQG’s discrete geometry for singularity resolution, such as bounces in LQC [11].

Key features of PRQT, Wheeler–DeWitt (WDW), and loop quantum gravity (LQG) are compared in

Table 1. Comparison of key features of PRQT, WDW, and LQG.

Feature	WDW	LQG	PRQT (This Work)
Evolution parameter	None (timeless Hamiltonian constraint)	Internal clock (e.g., scale factor)	$\mathrm{Invariant} \mathrm{scalar} s$(universal clock [4,5,21])
Unitary evolution	Not manifest	Approximate in minisuperspace	$\mathrm{Manifest} \mathrm{with} \mathrm{respect} \mathrm{to} s$
Probability interpretation	Not positive-definite; conditional or many-worlds	Conditional on internal clock	Positive-definite and conserved along $s$-trajectories
Cosmological solutions	Minisuperspace ODEs (frozen)	Discrete geometry, bounces	Explicit separation of variables in EU/FLRW metrics
Observational link	Indirect via WKB	Singularity resolution	Direct numerical age estimates matching Planck/ΛCDM to 0.1%

. The explicit numerical age estimate in Section 5, which reproduces the Planck/ΛCDM value using only the PRQT framework and modern observational parameters, demonstrates PRQT’s quantitative compatibility with cosmology. More examples are discussed in Section 6.5.

6.2. Constraint Quantization vs. Parametrized Dynamics

In canonical quantum gravity approaches like WDW and LQG, constraints from general relativity, such as Hamiltonian constraints, are enforced on the Hilbert space. Physical states must satisfy all constraints, with the WDW focusing on the Hamiltonian constraint $\mathrm{Ĥ} \Psi = 0$. In LQG, Ashtekar spinorial variables [9] simplify the constraints of general relativity and make it possible to embed the constraint surface into Yang–Mills theory. Spinfoam models such as the Engle–Pereira–Rovelli–Livine (EPRL) vertex [23,24] provide a covariant path-integral sum over spin networks without explicit time. Spin networks represent gauge-invariant quantum geometry, and vertex amplitudes implement constraints through the Barbero–Immirzi quantization-ambiguity parameter γ.

PRQT differs by using a parametrized action where constraints are promoted to equations of motion under $s$-evolution. The Hamiltonian generates propagation in s, enforcing mass-shell conditions dynamically. In the derivation for curved spacetime, the starting action is varied with respect to $s$, yielding wave equations such as the generalized Stueckelberg form with $s$ as the evolution parameter. In the FLRW metric, separation leads to $t$-dependent and $r$-dependent ordinary differential equations (ODEs), solved as linear combinations of independent solutions with coefficients set by boundary conditions. This approach may help address the need for defining inner products solely on constraint solution spaces, as in WDW.

6.3. Probability and Boundary Conditions

WDW’s Klein–Gordon-like currents are not positive-definite, leading to probabilistic ambiguities resolved using conditional probabilities or many-worlds interpretations. Models such as Hartle–Hawking no-boundary and Vilenkin tunneling [25] differ in inflationary weighting. As an illustration, for Vilenkin in a closed FLRW universe with curvature $k=+1$, WDW simplifies to a second-order ODE with an effective potential $U\left(a\right)$ proportional to $a^2\left(1-\left(\Lambda /3\right)a^2\right)$, where $\Lambda$ is the cosmological constant. WKB identifies forbidden (U > 0) and allowed (U < 0) regions, with a turning point at $a_t=\sqrt{\left(3/\Lambda \right)}$.

PRQT provides positive-definite probabilities with $\rho = {\Psi }^{*}\Psi$ and covariant generalizations conserved under $s$-evolution. PRQT’s positive-definite probabilities may also inform alternatives, such as chaotic inflation [26] or ekpyrotic models [27], where nucleation rates could be reinterpreted as $s$-dependent amplitudes.

6.4. Implications for Cosmology and Quantum Gravity

It is suggested in Section 6.1 that $s$-evolution in PRQT may offer a way to address the problem of time in WDW. PRQT complements background-independent theories such as LQG [12] by assuming semiclassical metrics but raises the possibility of developing hybrid models using features of PRQT and LQG. The unitary $s$-evolution in PRQT could provide a framework to compare boundary conditions in models such as Hartle–Hawking (no-boundary) and Vilenkin (tunneling), potentially quantifying inflationary probabilities. Furthermore, PRQT could interact with string theory dualities, such as scale factor duality [28], to help simulate conditions before the Big Bang. Overall, PRQT’s $s$-evolution could aid numerical simulations that can increase our understanding of timeless and parametric frameworks.

6.5. Empirical Tests of Parametrized Theories

PRQT views the invariant evolution parameter $s$ as a measurable temporal parameter whose role is to provide unitary evolution while preserving positive-definite probabilities conserved along $s$-trajectories. This formulation has been subjected to multiple empirical tests across particle physics, atomic physics, and cosmology. The tests below compare PRQT predictions directly with experimental data and demonstrate consistency with or superiority to the standard model or quantum field theory.

Anomalous magnetic moment: The Stueckelberg wave equation in the parametrized Stueckelberg–Horwitz-Piron axiomatic framework (SHP) [29] was used by Bennett [30] to predict the electron anomalous magnetic moment with high precision. Recent Fermilab muon g−2 results (127 ppb tension with the standard model) are potentially resolvable by PRQT relativistic corrections.

High-energy collision mass distributions and particle spectroscopy: Parameter-dependent mass distributions in parametrized statistical mechanics yield non-exponential distributions with power-law tails that match observed spectra and high-energy collision data [31,32,33,34,35,36,37].

Particle decay and mass-lifetime relationships: PRQT describes K-meson decay as covariant mass-state transitions without violating spacetime probability conservation [38]. The exponential decay law emerges naturally from the marginal probability density in time [4,39]. The mass-lifetime relationship is considered in [4].

Neutrino oscillations: Mass-state transitions in PRQT [40,41,42] align with oscillation data from T2K+NOvA and JUNO [43], matching measured parameters (Δm²₃₂, Δm²₂₁, sin²θ₁₂) better than the standard model in certain interpretations and providing a covariant mechanism for flavor transitions.

Stern-Gerlach experiment: PRQT predicts discrete spin quantization and beam deflection in a magnetic field, reproducing the observed splitting of atomic beams [4,44,45].

Spin-1/2 particle bound by Coulomb potential: Relativistic hydrogen-like bound states yield fine-structure corrections and energy levels that align with spectral data [4,44,46,47,48].

Double-slit time interference and relativistic entanglement: The parametrized SHP theory describes the Lindner et al. [49] femtosecond-laser electron interference-in-time experiment [5,50,51,52]. It also predicts Palacios et al. [53] two-electron entanglement at slightly different emission times, with observed times as proper quantum observables.

Klein paradox resolution: PRQT resolves the Klein paradox for both spin-0 and spin-1/2 particles; the reflection coefficient does not exceed unity, in contrast to conventional theory [4,54].

Hubble tension and universe age (this work, Section 5): PRQT in curved spacetime (FLRW metric) yields an age of 13.8 Gyr, matching Planck 2018 ΛCDM to within 0.1% and consistent with DES Y6 matter-density constraints. This provides a parametrized resolution of Hubble tension interpretations without new physics.

These tests demonstrate that PRQT is empirically viable across atomic, particle, and cosmological scales.

7. Conclusions and Outlook

The Parametrized Relativistic Quantum Theory (PRQT) formalism provides a framework for integrating quantum mechanics with general relativity in curved spacetimes, as demonstrated through its application to free scalar particle motion in the Schwarzschild and expanding universe metrics. A summary of particle dynamics in the Schwarzschild metric (Section 2) established notation, concepts, and solution procedures for solving diagonalized metric problems using PRQT in curved spacetime. The PRQT formalism is then applied to a generalized expanding universe (EU) metric (Section 3), and the Friedmann–Lemaître–Robertson–Walker (FLRW) metric (Section 4). By deriving wave equations and solutions for these cases, we show that PRQT accommodates cosmological expansion by means of time-dependent scale factors, enabling the separation of variables and the construction of general solutions. Furthermore, the estimation of the age of the universe in Section 5 using parameters from the Planck Collaboration [15] is approximately 13.8 Gyr. The age estimate is consistent with DES Y6 [16] and aligns with observational constraints. The age calculation highlights the applicability of PRQT to empirical cosmology.

The discussion in Section 6 highlights PRQT’s compatibility and distinctions with established quantum gravity and cosmological models, including the Hartle–Hawking state, Vilenkin tunneling, inflationary scenarios, the Wheeler–DeWitt equation, and loop quantum gravity approaches. Hybrid formulations, such as a PRQT–Hartle–Hawking model imposing no-boundary conditions at initial $s$, or a PRQT-LQG model incorporating holonomy corrections [22], could address singularities and the problem of time issues.

PRQT’s value for future research lies in features such as its probabilistic interpretation, extension to many-body systems, and retention of a parametric evolution that facilitates computational modeling of quantum effects in dynamic spacetimes. Unlike timeless approaches, PRQT allows for the study of temporal evolution in quantum gravity contexts, making it particularly suited for investigating quantum corrections to cosmology, particle interactions in early universe conditions, and the quantum-to-classical transition. Future extensions of PRQT to interacting fields, many-body systems, and hybrid PRQT-LQG models are expected to yield further insights into dark energy, singularity resolution, and the quantum-to-classical transition in cosmology. Overall, PRQT offers a versatile tool for advancing quantum cosmology and warrants further exploration in both theoretical and observational frameworks.