A Classical Theory Of The Free Lunch.

 

A Free Lunch (as a technical term in physics) means the non-conservation of energy, to advantage.
Noether's theorem tells us that conservation of energy is a consequence of the uniformity of time.
Ergo the essential ingrediant of any Free Lunch is Non-uniform Time.
Why didn't YOU think of this.
Non-uniform Time is necessary for the non-conservation of energy.
The zero point energy density of the vacuum is masked by the zero point curvature of space-time.
The matching is dynamic.
Everything in the universe is made from Left Over Vacuum Energy.

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Does your cosmology violate the principal of the conservation of energy but you don't really know what to do about it?
Then why not try the new, improved, Friedmann Lemaitre Robertson Walker Metric with Non-uniform Time.
Non-uniform Time allows conservation of energy to be violated in a controlled and predictable way leaving you safe to enjoy your free lunch.
Professor X writes: "I had a scaler field dumping energy into the vacuum but staying exactly the same. I just couldn't get my head around it. Then I discovered Non-uniform Time. Non-uniform Time produces an extra term in the second Friedmann equation that deals with non-conservation of energy simply and straightforwardly. I'd forgotten that there was a second Friedmann equation."

And this from a student. "I was trying to get the zero point energy density of the vacuum to match the zero point curvature of space time to better than 1 part in a million million million million million million million million million million million million million million million million million million million million without actually being equal. It seemed impossible. But with Non-uniform Time all I needed to do was make the gravitational parameter dependent on the temporal scale factor and I could just sit back and let Unruh radiation do the matching for me, simple.
Non-uniform Time. You can't have a Free Lunch without it.

Re-writing Einstein's Field Equations

${\displaystyle {\text{ }}}$
${\displaystyle {\text{The modified FLRW metric, using spherical coordinates, can be written. }}}$

${\displaystyle -c_{*}^{2}\mathrm {d} \tau ^{2}=-c^{2}(t)\mathrm {d} t^{2}+{a}^{2}(t)\left({\frac {\mathrm {d} r^{2}}{1-kr^{2}}}+r^{2}\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}\theta \,\mathrm {d} \phi ^{2}\right)}$

${\displaystyle {\text{Where:}}}$

${\displaystyle r,\theta (theta),\phi (phi){\text{ are the polar co-ordinates of space.}}}$

${\displaystyle t{\text{ is the co-ordinate of time.}}}$

${\displaystyle a{\text{ is the spatial scale factor and is taken to be a function of time.}}}$

${\displaystyle c{\text{ is the temporal scale factor and is taken to be a function of time.}}}$

${\displaystyle c_{*}{\text{ is a constant and is the limiting value of }}c{\text{.}}}$

${\displaystyle \tau (tau){\text{ is a measure of proper time.}}}$

${\displaystyle k{\text{ is the curvature index, and is taken to belong to the set (-1,0,+1).}}}$

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The cosmological constant and zero point values

${\displaystyle {\text{Einstein's field equations may be written}}}$

${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }}$

${\displaystyle {\text{If we replace the cosmological constant and the gravitational constant with parameters dependant upon }}c{\text{ then we get}}}$

${\displaystyle G_{\mu \nu }+\Lambda (c)g_{\mu \nu }=GRAV(c)T_{\mu \nu }}$

${\displaystyle {\text{alternatively, if we write Einstein's field equations using zero point values we get}}}$

${\displaystyle G_{\mu \nu }+G_{\mu \nu }^{^{_{ZP}}}=GRAV(c)T_{\mu \nu }+GRAV(c)T_{\mu \nu }^{^{_{ZP}}}}$

${\displaystyle {\text{Where }}G_{\mu \nu }^{^{_{ZP}}}=\lambda ^{^{_{ZP}}}g_{\mu \nu }\ {\text{ and }}\ \lambda ^{^{_{ZP}}}{\text{ is the zero point curvature of space time }}}$

${\displaystyle {\text{and where }}T_{\mu \nu }^{^{_{ZP}}}=\rho ^{^{_{ZP}}}g_{\mu \nu }\ {\text{ and }}\ \rho ^{^{_{ZP}}}{\text{ is the zero point energy density of the vacuum.}}}$

${\displaystyle {\text{Subtracting the third equation from the second we get}}}$

${\displaystyle \Lambda (c)g_{\mu \nu }-G_{\mu \nu }^{^{_{ZP}}}=-GRAV(c)T_{\mu \nu }^{^{_{ZP}}}}$
${\displaystyle \Lambda (c)g_{\mu \nu }=G_{\mu \nu }^{^{_{ZP}}}-GRAV(c)T_{\mu \nu }^{^{_{ZP}}}}$
${\displaystyle \Lambda (c)g_{\mu \nu }=\lambda ^{^{_{ZP}}}g_{\mu \nu }-GRAV(c)\rho ^{^{_{ZP}}}g_{\mu \nu }}$
${\displaystyle \Lambda (c)=\lambda ^{^{_{ZP}}}-GRAV(c)\rho ^{^{_{ZP}}}}$

${\displaystyle c_{*}{\text{ is the limiting value of }}c{\text{ such that }}\Lambda (c_{*})=0{\text{ so therefore }}}$

${\displaystyle \lambda ^{^{_{ZP}}}=GRAV(c_{*})\rho ^{^{_{ZP}}}}$

${\displaystyle \Lambda (c)=GRAV(c_{*})\rho ^{^{_{ZP}}}-GRAV(c)\rho ^{^{_{ZP}}}}$

${\displaystyle {\frac {\Lambda (c)}{GRAV(c_{*})}}=\rho ^{^{_{ZP}}}-{\frac {GRAV(c)}{GRAV(c_{*})}}\rho ^{^{_{ZP}}}}$


${\displaystyle {\frac {-\Lambda (c)}{GRAV(c_{*})}}{\text{ is the Left Over Vacuum Energy.}}}$

${\displaystyle {\mathbf {Everything}}\ {\mathbf {in}}\ {\mathbf {the}}\ {\mathbf {universe}}\ {\mathbf {is}}\ {\mathbf {made}}\ {\mathbf {from}}\ {\mathbf {Left}}\ {\mathbf {Over}}\ {\mathbf {Vacuum}}\ {\mathbf {Energy.}}\ }$



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A Classical Theory Of Emergent Time

Re-calculating The Friedmann Equations

${\displaystyle \lambda ^{^{_{ZP}}}=GRAV(c_{*}).\rho ^{^{_{ZP}}}}$


${\displaystyle G_{\mu \nu }+G_{\mu \nu }^{^{_{ZP}}}=GRAV(c).T_{\mu \nu }^{^{_{ZP}}}+GRAV(c).T_{\mu \nu }}$


${\displaystyle {\frac {G_{\mu \nu }}{g_{\mu \nu }}}+{\frac {G_{\mu \nu }^{^{_{ZP}}}}{g_{\mu \nu }}}=GRAV(c){\frac {T_{\mu \nu }^{^{_{ZP}}}}{g_{\mu \nu }}}+GRAV(c){\frac {T_{\mu \nu }}{g_{\mu \nu }}}}$

${\displaystyle {\frac {G_{\mu \nu }}{g_{\mu \nu }}}+\lambda ^{^{_{ZP}}}=GRAV(c)\rho ^{^{_{ZP}}}+GRAV(c){\frac {T_{\mu \nu }}{g_{\mu \nu }}}}$

${\displaystyle {\frac {G_{\mu \nu }}{g_{\mu \nu }}}=GRAV(c)\rho ^{^{_{ZP}}}-\lambda ^{^{_{ZP}}}+GRAV(c){\frac {T_{\mu \nu }}{g_{\mu \nu }}}}$

${\displaystyle {\frac {G_{\mu \nu }}{g_{\mu \nu }}}=GRAV(c)\rho ^{^{_{ZP}}}-GRAV(c_{*})\rho ^{^{_{ZP}}}+GRAV(c){\frac {T_{\mu \nu }}{g_{\mu \nu }}}}$

${\displaystyle {\frac {G_{\mu \nu }}{g_{\mu \nu }}}=\left(GRAV(c)-GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+GRAV(c){\frac {T_{\mu \nu }}{g_{\mu \nu }}}}$


${\displaystyle {\frac {G_{tt}}{g_{tt}}}=\left(GRAV(c)-GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+GRAV(c){\frac {T_{tt}}{g_{tt}}}}$

${\displaystyle 3{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}c^{2}}}=\left(GRAV(c)-GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+GRAV(c)\rho }$

${\displaystyle 3{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}=\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+c^{2}.GRAV(c)\rho }$


${\displaystyle 3{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}-c^{2}.GRAV(c)\rho =\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}}$












${\displaystyle {\frac {G_{rr}}{g_{rr}}}={\frac {G_{\theta \theta }}{g_{\theta \theta }}}={\frac {G_{\phi \phi }}{g_{\phi \phi }}}=\left(GRAV(c)-GRAV(c_{*})\right)\rho ^{^{_{ZP}}}-GRAV(c).p}$
${\displaystyle {\frac {G_{rr}}{g_{rr}}}={\frac {G_{\theta \theta }}{g_{\theta \theta }}}={\frac {G_{\phi \phi }}{g_{\phi \phi }}}=2{\frac {\ddot {a}}{ac^{2}}}+{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}c^{2}}}-2{\frac {{\dot {a}}{\dot {c}}}{ac^{3}}}=\left(GRAV(c)-GRAV(c_{*})\right)\rho ^{^{_{ZP}}}-GRAV(c).p}$
${\displaystyle {\frac {G_{rr}}{g_{rr}}}={\frac {G_{\theta \theta }}{g_{\theta \theta }}}={\frac {G_{\phi \phi }}{g_{\phi \phi }}}=2{\frac {\ddot {a}}{a}}+{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}-2{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}=\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}-c^{2}.GRAV(c).p}$
${\displaystyle 2{\frac {\ddot {a}}{a}}+{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}-2{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}=\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}-c^{2}.GRAV(c).p}$
${\displaystyle 2{\frac {\ddot {a}}{a}}+{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}-2{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}+c^{2}.GRAV(c).p=\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}}$
${\displaystyle 2{\frac {\ddot {a}}{a}}+{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}-2{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}+c^{2}.GRAV(c).p=3{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}-c^{2}.GRAV(c)\rho }$


${\displaystyle 2{\frac {\ddot {a}}{a}}+{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}-2{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}-3{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}=-c^{2}.GRAV(c).p-c^{2}.GRAV(c)\rho }$


${\displaystyle 2{\frac {\ddot {a}}{a}}-2{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}-2{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}=-c^{2}.GRAV(c).p-c^{2}.GRAV(c)\rho }$


${\displaystyle {\frac {\ddot {a}}{a}}-{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}-{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}=-{\frac {1}{2}}c^{2}.GRAV(c).p-{\frac {1}{2}}c^{2}.GRAV(c)\rho }$


${\displaystyle {\frac {\ddot {a}}{a}}-{\frac {{\dot {a}}^{2}}{a^{2}}}-{\frac {kc^{2}}{a^{2}}}-{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}=-{\frac {1}{2}}c^{2}.GRAV(c).p-{\frac {1}{2}}c^{2}.GRAV(c)\rho }$


${\displaystyle {\frac {\ddot {a}}{a}}-{\frac {{\dot {a}}^{2}}{a^{2}}}={\frac {kc^{2}}{a^{2}}}+{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}-{\frac {1}{2}}c^{2}.GRAV(c).p-{\frac {1}{2}}c^{2}.GRAV(c)\rho }$




${\displaystyle H={\frac {\dot {a}}{a}}}$
${\displaystyle {\dot {H}}={\frac {\ddot {a}}{a}}-{\frac {{\dot {a}}^{2}}{a^{2}}}}$
${\displaystyle {\dot {H}}={\frac {kc^{2}}{a^{2}}}+{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}-{\frac {1}{2}}c^{2}.GRAV(c).p-{\frac {1}{2}}c^{2}.GRAV(c)\rho }$

${\displaystyle 3{\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}=\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+c^{2}.GRAV(c)\rho }$


${\displaystyle {\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}={\frac {1}{3}}\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+{\frac {1}{3}}c^{2}.GRAV(c)\rho }$


${\displaystyle {\frac {{\dot {a}}^{2}}{a^{2}}}=-{\frac {kc^{2}}{a^{2}}}+{\frac {1}{3}}\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+{\frac {1}{3}}c^{2}.GRAV(c)\rho }$


${\displaystyle H^{2}=-{\frac {kc^{2}}{a^{2}}}+{\frac {1}{3}}\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+{\frac {1}{3}}c^{2}.GRAV(c)\rho }$





${\displaystyle {\dot {H}}={\frac {\ddot {a}}{a}}-{\frac {{\dot {a}}^{2}}{a^{2}}}={\frac {kc^{2}}{a^{2}}}+{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}-{\frac {1}{2}}c^{2}.GRAV(c).p-{\frac {1}{2}}c^{2}.GRAV(c)\rho }$

${\displaystyle {\frac {\ddot {a}}{a}}={\frac {{\dot {a}}^{2}}{a^{2}}}+{\frac {kc^{2}}{a^{2}}}+{\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}-{\frac {1}{2}}c^{2}.GRAV(c).p-{\frac {1}{2}}c^{2}.GRAV(c)\rho }$

${\displaystyle {\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}={\frac {1}{3}}\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+{\frac {1}{3}}c^{2}.GRAV(c)\rho }$


${\displaystyle {\frac {\ddot {a}}{a}}={\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}-{\frac {1}{2}}c^{2}.GRAV(c).p-{\frac {1}{2}}c^{2}.GRAV(c)\rho +{\frac {1}{3}}\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+{\frac {1}{3}}c^{2}.GRAV(c)\rho }$



${\displaystyle {\frac {\ddot {a}}{a}}={\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}-{\frac {1}{2}}c^{2}.GRAV(c).p-{\frac {1}{6}}c^{2}.GRAV(c)\rho +{\frac {1}{3}}\left(c^{2}.GRAV(c)-c^{2}.GRAV(c_{*})\right)\rho ^{^{_{ZP}}}}$

A Possible Beginning

${\displaystyle {\text{For no or negligible curvature}}}$

${\displaystyle {\dot {H}}={\frac {\dot {a}}{a}}{\frac {\dot {c}}{c}}-{\frac {1}{2}}c^{2}GRAV(c).p_{max}-{\frac {1}{2}}c^{2}GRAV(c)\rho _{max}}$

${\displaystyle H^{2}={\frac {1}{3}}\left(c^{2}.GRAV(c)-c^{2}GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+{\frac {1}{3}}c^{2}GRAV(c)\rho _{max}}$

${\displaystyle {\text{In the simplest case of dependence }}c^{2}GRAV(c){\text{ would be constant and would be equal to }}c_{*}^{2}GRAV(c_{*})}$

${\displaystyle {\dot {H}}=H{\frac {\dot {c}}{c}}-{\frac {1}{2}}c_{*}^{2}GRAV(c_{*}).p_{max}-{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\rho _{max}}$

${\displaystyle H^{2}={\frac {1}{3}}\left(c_{*}^{2}.GRAV(c_{*})-c^{2}GRAV(c_{*})\right)\rho ^{^{_{ZP}}}+{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\rho _{max}}$

${\displaystyle H^{2}={\frac {1}{3}}GRAV(c_{*})\left(c_{*}^{2}\rho ^{^{_{ZP}}}+c_{*}^{2}\rho _{max}-c^{2}\rho ^{^{_{ZP}}}\right)}$

${\displaystyle H^{2}={\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)\left(1-{\frac {\rho ^{^{_{ZP}}}}{\rho ^{^{_{ZP}}}+\rho _{max}}}{\frac {c^{2}}{c_{*}^{2}}}\right)}$

${\displaystyle {\text{define}}\ \ \psi \ {\text{such that}}\ \ {\frac {\rho ^{^{_{ZP}}}}{\rho ^{^{_{ZP}}}+\rho _{max}}}{\frac {c^{2}}{c_{*}^{2}}}=\sin ^{2}{(\psi )}}$


${\displaystyle H^{2}={\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)\cos ^{2}{(\psi )}}$

${\displaystyle H={\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\cos {(\psi )}}$

${\displaystyle {\dot {H}}=-{\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\sin {(\psi )}{\frac {d{\psi }}{dt}}}$


${\displaystyle {\dot {H}}=H{\frac {\dot {c}}{c}}-{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)}$

${\displaystyle -{\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\sin {(\psi )}{\frac {d{\psi }}{dt}}={\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\cos {(\psi )}{\frac {\dot {c}}{c}}-{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)}$

${\displaystyle -{\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\sin {(\psi )}{\frac {d{\psi }}{dt}}={\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\cos {(\psi )}{\frac {\cos(\psi )}{\sin(\psi )}}{\frac {d{\psi }}{dt}}-{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)}$

${\displaystyle {\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)={\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\cos {(\psi )}{\frac {\cos(\psi )}{\sin(\psi )}}{\frac {d{\psi }}{dt}}+{\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\sin {(\psi )}{\frac {d{\psi }}{dt}}}$

${\displaystyle {\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)={\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\csc {(\psi )}{\frac {d{\psi }}{dt}}}$

${\displaystyle {\frac {dt}{d{\psi }}}={\frac {\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}{{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)}}\csc {(\psi )}}$

${\displaystyle \int \csc {(\psi )}\,d\psi =-\ln {\left|{\frac {1+\cos {(\psi )}}{\sin {(\psi )}}}\right|}+C}$

${\displaystyle c_{*}d\tau =cdt}$
${\displaystyle {\frac {c}{c_{*}}}={\sqrt {\frac {\rho ^{^{_{ZP}}}+\rho _{max}}{\rho ^{^{_{ZP}}}}}}\sin {(\psi )}}$



${\displaystyle {\frac {d\tau }{d{\psi }}}={\frac {d\tau }{dt}}{\frac {dt}{d{\psi }}}={\frac {c}{c_{*}}}{\frac {dt}{d{\psi }}}={\sqrt {\frac {\rho ^{^{_{ZP}}}+\rho _{max}}{\rho ^{^{_{ZP}}}}}}\sin {(\psi )}{\frac {\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}{{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)}}\csc {(\psi )}={\frac {1}{\sqrt {{\frac {3}{4}}c_{*}^{2}GRAV(c_{*})\rho ^{^{_{ZP}}}}}}{\frac {\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}{\left(p_{max}+\rho _{max}\right)}}}$

${\displaystyle \int _{0}^{\tau }d\tau =\int _{0}^{\frac {\pi }{2}}{\frac {1}{\sqrt {{\frac {3}{4}}c_{*}^{2}GRAV(c_{*})\rho ^{^{_{ZP}}}}}}{\frac {\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}{\left(p_{max}+\rho _{max}\right)}}d\psi }$

${\displaystyle \tau ={\frac {\frac {\pi }{2}}{\sqrt {{\frac {3}{4}}c_{*}^{2}GRAV(c_{*})\rho ^{^{_{ZP}}}}}}{\frac {\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}{\left(p_{max}+\rho _{max}\right)}}}$

Co-ordinate Curvature

${\displaystyle {\frac {\dot {a}}{a}}=H={\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\cos {(\psi )}}$

${\displaystyle {\frac {d{\psi }}{dt}}={\frac {{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)}{\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}}\sin {(\psi )}}$
${\displaystyle {\frac {\dot {c}}{c}}={\frac {\cos(\psi )}{\sin(\psi )}}{\frac {d{\psi }}{dt}}}$
${\displaystyle {\frac {\dot {c}}{c}}={\frac {\cos(\psi )}{\sin(\psi )}}{\frac {{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)}{\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}}\sin {(\psi )}}$

${\displaystyle {\frac {\dot {c}}{c}}={\frac {{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)}{\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}}\cos(\psi )}$

${\displaystyle {\frac {{\dot {c}}/c}{{\dot {a}}/a}}={\frac {{\frac {{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)}{\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}}\cos(\psi )}{{\sqrt {{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}\cos {(\psi )}}}}$

${\displaystyle {\frac {{\dot {c}}/c}{{\dot {a}}/a}}={\frac {{\frac {1}{2}}c_{*}^{2}GRAV(c_{*})\left(p_{max}+\rho _{max}\right)}{{\frac {1}{3}}c_{*}^{2}GRAV(c_{*})\left(\rho ^{^{_{ZP}}}+\rho _{max}\right)}}}$
${\displaystyle {\frac {{\dot {c}}/c}{{\dot {a}}/a}}={\frac {3}{2}}{\frac {p_{max}+\rho _{max}}{\rho ^{^{_{ZP}}}+\rho _{max}}}}$

${\displaystyle {\frac {{\frac {d}{dt}}\ln {c}}{{\frac {d}{dt}}\ln {a}}}={\frac {3}{2}}{\frac {p_{max}+\rho _{max}}{\rho ^{^{_{ZP}}}+\rho _{max}}}}$