Classical and quantum cosmology of K-essentially modified \(R^2\) and pure \(R^p\) gravity

Abstract

We present a gravitational action with a modified higher order term of a combination of scalar curvature and Lagrangian density of a scalar field. This type of models has been considered first by Cruz-Dombriz et al. The classical and quantum cosmologies governed by the modified action are studied. Models described by a positive-definite action and a pure arbitrary-powered scalar curvature action without the standard Einstein–Hilbert term are also investigated. We show some particular cases in which exact solutions can be obtained.

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Notes

  1. 1.

    Here, we adopt the normalization of the spatial volume by considering the space as \(S^3\) \((k=1)\). Even for other values of k, we can still use this convention because the normalization can be absorbed into the redefinition of \(\alpha \) and \(\beta \).

  2. 2.

    In comparison with [17, 47], absence of small parameter here is an artifact; the replacement \(r\rightarrow r/\sqrt{\lambda }\) bring about the parameter \(\lambda \). Thus, we conserve the present expressions here.

  3. 3.

    Indeed, we find that \(\frac{\partial ^2}{\partial x^2}-\frac{\partial ^2}{\partial y^2}= \frac{1}{r}\frac{\partial }{\partial r}r\frac{\partial }{\partial r}-\frac{1}{r^2}\frac{\partial ^2}{\partial \theta ^2}\) by the coordinate transformation (A7).

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Appendix A: Solvable pure \(R^2\) gravity in different variables and the factor ordering

Appendix A: Solvable pure \(R^2\) gravity in different variables and the factor ordering

A different factor ordering gives a different solution of the WDW equation, at least in a certain region of variables. In this “Appendix A”, we review the solvable model of pure \(R^2\) gravity in four dimensions as an example, and considered the factor ordering when the different variables are used.

For the flat four-dimensional spacetime, the pure \(R^2\) gravity, i.e., in absence of the additional scalar field, is known to be solvable in the minisuperspace formalism [53, 54]. In such a case, the effective Lagrangian becomes

$$\begin{aligned} L'(a,{{\dot{a}}},Q,{\dot{Q}})=-2\left( \frac{Q}{a}\right) ^{\cdot }{{\dot{a}}} -\frac{1}{36\pi ^2\beta }Q^2, \end{aligned}$$
(A1)

since the term which comes from the spatial curvature is also absent. If we use new variables \(\sigma \equiv a^3\) and \(\tau \equiv Q/a\), the Lagrangian can be written as

$$\begin{aligned} L'(\sigma ,{\dot{\sigma }},\tau ,{\dot{\tau }})=-\frac{2}{3\sigma ^{2/3}} {\dot{\sigma }}{\dot{\tau }} -\frac{\sigma ^{2/3}\tau ^2}{36\pi ^2\beta }, \end{aligned}$$
(A2)

and the Hamiltonian is given by

$$\begin{aligned} H(\sigma ,\tau ,P_\sigma ,P_\tau )=-\frac{3\sigma ^{2/3}}{2}\left( P_{\sigma }P_{\tau } -\frac{\tau ^2}{54\pi ^2\beta }\right) . \end{aligned}$$
(A3)

Therefore, the WDW equation reads

$$\begin{aligned} \left( \frac{\partial }{\partial \sigma }\frac{\partial }{\partial \tau }+ \frac{\tau ^2}{54\pi ^2\beta }\right) \Psi (\sigma ,\tau )=0. \end{aligned}$$
(A4)

This equation can be exactly in the form

$$\begin{aligned} \Psi (\sigma ,\tau )=\int _{-\infty }^\infty A(\lambda )\exp \left[ -\lambda \sigma +\frac{\tau ^3}{162\pi ^2\beta \lambda } \right] d\lambda , \end{aligned}$$
(A5)

where \(A(\lambda )\) indicates the amplitude [53, 54].

Here, we consider the Fourier transform of the elementary wave solution. That is

$$\begin{aligned} w=\int _{-\infty }^\infty \exp \left[ -\lambda \sigma +\frac{\tau ^3}{162\pi ^2\beta \lambda } \right] e^{-il\theta }d\theta . \end{aligned}$$
(A6)

The definitions of the variables are the same as in the previous sections, i.e.,

$$\begin{aligned} x=\frac{Q}{a}-a=r\sinh \theta ,\quad y=a+\frac{Q}{a}=r\cosh \theta . \end{aligned}$$
(A7)

Thus, we find

$$\begin{aligned} \sigma =a^3=\frac{r^3}{8}e^{-3\theta },\quad \tau =\frac{Q}{a}=\frac{r}{2}e^{\theta }. \end{aligned}$$
(A8)

Then, we obtain

$$\begin{aligned} w= & {} \int _{-\infty }^\infty \exp \left[ -\lambda \frac{r^3}{8}e^{-3\theta } +\frac{r^3e^{3\theta }}{8\cdot 162\pi ^2\beta \lambda } -il\theta \right] d\theta \nonumber \\= & {} \frac{1}{3}(-i9\sqrt{2\beta }\pi \lambda )^{-il/3}\int _{-\infty }^\infty \exp \left[ -i\frac{l}{3}t-i\frac{r^3}{36\sqrt{2\beta }\pi }\cosh t \right] dt \nonumber \\\propto & {} K_{il/3}(ir^3/(36\sqrt{2\beta }\pi )). \end{aligned}$$
(A9)

Because \(K_{il/3}(ir^3/(36\sqrt{2\beta }\pi ))\) is expressed by a linear combination of \(J_{\pm il/3}(r^3/(36\sqrt{2\beta }\pi ))\), general solution can be written by

$$\begin{aligned} \Psi= & {} \int \int \left[ A(l,m) J_{i\frac{l}{3}}(r^3/(36\sqrt{2\beta }\pi ))\right. \nonumber \\&\left. + B(l,m) J_{-i\frac{l}{3}}(r^3/(36\sqrt{2\beta }\pi ))\right] e^{il\theta } dl. \end{aligned}$$
(A10)

This is the form of the general explicit solution of the following equation:

$$\begin{aligned} \left[ \frac{1}{r}\frac{\partial }{\partial r}r\frac{\partial }{\partial r}-\frac{1}{r^2}\frac{\partial ^2}{\partial \theta ^2}+\frac{r^4}{288\pi ^2\beta } \right] \Psi (r,\theta )=0. \end{aligned}$$
(A11)

We conclude that the exact solution obtained by Schmidt [53, 54] is equivalent of the solution of (A11), where the parameter of ordering s equals to one,Footnote 3 whereas the authors of [17, 47] chose the different factor ordering (\(s=0\)) in the ‘kinetic’ term (in the Starobinsky model).

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Kan, N., Shiraishi, K. & Yashiki, M. Classical and quantum cosmology of K-essentially modified \(R^2\) and pure \(R^p\) gravity. Gen Relativ Gravit 51, 90 (2019). https://doi.org/10.1007/s10714-019-2573-7

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Keywords

  • Modified gravity
  • Inflation
  • Quantum cosmology
  • Exact solutions