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[tex]

\frac{1}{R}\frac{{{d^2}R}}{{d{t^2}}} = - \frac{{4\pi G}}{3}\left[ {{\rho _m} + {\rho _{rad}} + {\rho _\Lambda } + \frac{{3({P_m} + {P_{rad}} + {P_\Lambda })}}{{{c^2}}}} \right]

[/tex]

In the Friedmann equation, ρ is the mass density.

===

https://en.wikipedia.org/wiki/Friedmann_equations

They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density ρ and pressure p.

~~~

ρ and p are the volumetric mass density (and not the volumetric energy density) and the pressure,

===

In the case of matter, pressure P=0,

In the case of radiation, pressure P=(1/3)ρc^2,

In the case of cosmological constant, pressure P=-ρc^2.

When we call the

[tex]{\rho _m} + \frac{{3{P_m}}}{{{c^2}}} = {\rho _m} + 0 = {\rho _m}[/tex]

[tex]{\rho _\Lambda } + \frac{{3{P_\Lambda }}}{{{c^2}}} = {\rho _\Lambda } + \frac{{3( - {\rho _\Lambda }{c^2})}}{{{c^2}}} = - 2{\rho _\Lambda }[/tex]

[tex]{\rho _{rad}} + \frac{{3{P_{rad}}}}{{{c^2}}} = {\rho _{rad}} + \frac{{3(\frac{1}{3}{\rho _{rad}}{c^2})}}{{{c^2}}} = 2{\rho _{rad}}[/tex]

or

[tex]{\rho _{rad}} + \frac{{3{P_{rad}}}}{{{c^2}}} = 0 + \frac{{3(\frac{1}{3}{\rho _{rad}}{c^2})}}{{{c^2}}} = {\rho _{rad}}[/tex]

\frac{1}{R}\frac{{{d^2}R}}{{d{t^2}}} = - \frac{{4\pi G}}{3}\left[ {{\rho _m} + {\rho _{rad}} + {\rho _\Lambda } + \frac{{3({P_m} + {P_{rad}} + {P_\Lambda })}}{{{c^2}}}} \right]

[/tex]

In the Friedmann equation, ρ is the mass density.

===

https://en.wikipedia.org/wiki/Friedmann_equations

They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density ρ and pressure p.

~~~

ρ and p are the volumetric mass density (and not the volumetric energy density) and the pressure,

===

In the case of matter, pressure P=0,

In the case of radiation, pressure P=(1/3)ρc^2,

In the case of cosmological constant, pressure P=-ρc^2.

When we call the

**energy densities**of matter, radiation, and dark energy (ρ_m)c^2, (ρ_rad)c^2, (ρ_lambda)c^2, what form does this take when entering the equation?[tex]{\rho _m} + \frac{{3{P_m}}}{{{c^2}}} = {\rho _m} + 0 = {\rho _m}[/tex]

[tex]{\rho _\Lambda } + \frac{{3{P_\Lambda }}}{{{c^2}}} = {\rho _\Lambda } + \frac{{3( - {\rho _\Lambda }{c^2})}}{{{c^2}}} = - 2{\rho _\Lambda }[/tex]

**In the case of radiation, which one is correct? 2ρ_rad or 1ρ_rad?**[tex]{\rho _{rad}} + \frac{{3{P_{rad}}}}{{{c^2}}} = {\rho _{rad}} + \frac{{3(\frac{1}{3}{\rho _{rad}}{c^2})}}{{{c^2}}} = 2{\rho _{rad}}[/tex]

or

[tex]{\rho _{rad}} + \frac{{3{P_{rad}}}}{{{c^2}}} = 0 + \frac{{3(\frac{1}{3}{\rho _{rad}}{c^2})}}{{{c^2}}} = {\rho _{rad}}[/tex]

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