(a) Constant curvature spacetime, with spacetime sectional curvature K: \begin{align} \newcommand{\e}{{\rm e}} \newcommand{\CC}{{\rm CC}} \displaystyle \CC^m_{K} = \left\{\begin{array}{@{}ll@{}} \{(m,K,\cosh(\sqrt{K}t)) & \mid K>0, ~ I=\mathbb{R} \}, \nonumber \\ \{(m,0,1) & \mid K=0, ~ I=\mathbb{R} \}, \nonumber \\ \{(m,K,\cos(\sqrt{-K}t)) & \mid K<0, ~ I=\mathbb{R} \}. \end{array}\right. \nonumber \end{align} \tag

(b) Einstein static universe, with spatial sectional curvature K\ne 0:

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ESU}{{\rm ESU}} \displaystyle \ESU^m_{K} = \{(m,K,1) \mid m>1, ~ I=\mathbb{R} \}. \nonumber \end{align} \tag{ 2 }

(c) Spatially flat constant scalar curvature spacetime, with spacetime scalar curvature m(m+1)K and such that \frac{f^{\prime2}}{f^2}(I) = J:

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\CS}{{\rm CS}} \newcommand{\CSC}{{\rm CSC}} \displaystyle \CSC^{m,0}_{K,J} = &\left\{(m,\alpha,f) \mid m>1, ~ f^\prime\ne 0,\right. \nonumber \\ &\left.\left(\frac{f^{\prime\prime}}{f} - \frac{f^{\prime2}}{f^2}\right) + \frac{(m+1)}{2} \left(\frac{f^{\prime2}}{f^2} - K\right) = 0 \right\}. \nonumber \end{align} \tag{ 3 }

(d) Generic constant scalar curvature spacetime, with spacetime scalar curvature m(m+1)K, normalized radiation density constant Ω and such that \frac{\alpha}{f^2}(I) = J:

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\CS}{{\rm CS}} \newcommand{\CSC}{{\rm CSC}} \displaystyle \CSC^m_{K,\Omega,J} = \left\{(m,\alpha,f) \mid m>1, ~ \alpha \ne 0, ~ f^\prime \ne 0,\right. \nonumber \\ \left. \frac{f^{\prime2}}{f^2} + \frac{\alpha}{f^2} = K + \Omega \frac{\vert \alpha\vert ^{(m+1)/2}}{f^{m+1}} \right\}. \nonumber \end{align} \tag{ 4 }

(e) Spatially flat FLRW spacetime with normalized pressure function P defined on an open interval J, with 0 < \frac{f^{\prime2}}{f^2}(I) = J and

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\del}{\partial} \displaystyle P(u) \left[\del_u P(u) - \frac{1}{2\kappa}\right] \ne 0 \nonumber \end{align} \tag{ 5 } everywhere on J:

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\FLRW}{{\rm FLRW}} \displaystyle \FLRW^{m,0}_{P,J} = \left\{(m,0,f) \mid \left(\frac{f^{\prime\prime}}{f} - \frac{f^{\prime2}}{f^2}\right)+\frac{m}{2}\frac{f'^2}{f^2} = -\kappa P\left((\,f^\prime/f)^2\right) \right\}. \nonumber \end{align} \tag{ 6 }

(f) Generic FLRW spacetime with normalized energy function E defined on an open interval J, with 0 \not\in \frac{\alpha}{f^2}(I) = J and

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\del}{\partial} \displaystyle \del_u \left[u \del_u E(u) - \frac{(m+1)}{2} \right] \ne 0 \nonumber \end{align} \tag{ 7 } everywhere on J:

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\FLRW}{{\rm FLRW}} \displaystyle \FLRW^m_{E,J} = \left\{(m,\alpha,f) \mid m>1, ~ \alpha \ne 0, ~ \frac{f^{\prime2}}{f^2} + \frac{\alpha}{f^2} = \kappa E(\alpha/f^2) \right\}. \nonumber \end{align} \tag{ 8 }

(a) Constant scalar, with scalar value Φ, on a constant curvature spacetime with scalar curvature K:

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\CS}{{\rm CS}} \newcommand{\CC}{{\rm CC}} \displaystyle \CC^m_K \CS_\Phi = \{(m,\alpha,f,\Phi) \mid (m,\alpha,f) \in \CC^m_K \}. \nonumber \end{align} \tag{ 9 }

(b) Constant energy scalar, with energy density \rho > 0 and J = \phi(I), on an Einstein static universe with spatial sectional curvature K = \frac{2}{m(m-1)} \kappa \rho, or equivalently with cosmological constant \Lambda = \frac{(m-1)}{m} \kappa \rho:

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\CES}{{\rm CES}} \newcommand{\ESU}{{\rm ESU}} \displaystyle \ESU^m_{K} \CES_{\rho,J} = \{(m,K,1, \sqrt{2\rho/m} t) \mid I = J/\sqrt{2\rho/m} \}. \nonumber \end{align} \tag{ 10 }

(c) Spatially flat massless minimally-coupled scalar spacetime, with cosmological constant Λ, J = \phi(I) and J^\prime = \frac{f^\prime}{f}(I) \not\ni 0 and \frac{2\Lambda/\kappa}{m(m-1)} < \frac{1}{\kappa} (J^\prime){\hspace{0pt}}^2:

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\MMS}{{\rm MMS}} \displaystyle \textstyle \MMS^{m,0}_{\Lambda,J,J^\prime} = &\left\{(m,0,f,\phi) \mid \phi^\prime < 0, ~ \frac{f^\prime}{f} \ne 0, \right. \nonumber \\ &\textstyle \left. \frac{f^{\prime2}}{f^2} = \frac{\kappa \phi^{\prime2} + 2\Lambda}{m(m-1)}, ~ \left(\frac{f^{\prime\prime}}{f} - \frac{f^{\prime2}}{f^2}\right) + m\frac{f^{\prime2}}{f^2} = \frac{2\Lambda}{(m-1)} \right\}. \nonumber \end{align} \tag{ 11 }

(d) Generic massless minimally-coupled scalar spacetime, with cosmological constant Λ, normalized scalar energy constant \Omega > 0, J = \phi(I) and J^\prime = \frac{f^\prime}{f}(I) \not\ni 0:

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\MMS}{{\rm MMS}} \displaystyle \textstyle \MMS^m_{\Lambda,\Omega,J,J^\prime} =& \left\{(m,\alpha,f,\phi) \mid \alpha \ne 0, ~ \frac{f^\prime}{f} \ne 0, \right. \nonumber \\ &\textstyle \left. \phi^\prime = -\sqrt{\Omega} \frac{\vert \alpha\vert ^{\frac{m}{2}}}{f^m}, ~ \frac{f^{\prime2}}{f^2} + \frac{\alpha}{f^2} = \frac{2\Lambda + \kappa \Omega \vert \alpha\vert ^m/f^{2m}}{m(m-1)} \right\}. \nonumber \end{align} \tag{ 12 }

(e) Spatially flat nonlinear Klein-Gordon spacetime, with non-constant scalar self-coupling potential V\colon J\to \mathbb{R}, with J = \phi(I), and expansion profile \Xi\colon J \to \mathbb{R}, satisfying \Xi(u) \ne 0, \newcommand{\del}{\partial} \frac{1}{\kappa}\del_u \Xi(u) > 0 and \mathfrak{H}_V(\Xi) = 0 in the notation of (, 18: #cqgaa9f61eqn019):

\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\NKG}{{\rm NKG}} \newcommand{\del}{\partial} \displaystyle \textstyle \NKG^{m,0}_{V,\Xi,J} = \left\{(m,0,f,\phi) \mid \frac{f^\prime}{f} = \Xi(\phi), \phi^\prime = -\frac{(m-1)}{\kappa} \del_\phi \Xi(\phi) \right\}. \nonumber \end{align} \tag{ 13 }

(f) Generic nonlinear Klein-Gordon spacetime, with non-constant scalar potential V\colon J \to \mathbb{R}, with J = \phi(I), and expansion profile (\Pi, \Xi) \colon J\to \mathbb{R}^2, satisfying \Pi < 0, \Xi \ne 0, \kappa\frac{\Pi^2+V}{m(m-1)} \ne \Xi^2 and \mathfrak{G}_V(\Pi, \Xi) = 0 in the notation of (, 20: #cqgaa9f61eqn021):