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Numerical Solution of the Friedmann Equation for the LCDM (Lambda Cold Dark Matter) Model

Faculty Commons material developed by Brian Woodahl - Published May 15, 2024

In undergraduate cosmology, an important equation of big bang theory, is the Friedmann Equation [for example see Eq 4.18 of Ryden's "Introduction to Cosmology" 2nd Ed]. It gives us the scale or size, a(t), of the universe as a function of time, t, based upon four physical parameters: Hubble's constant, and three ratios of the present density to the critical density for radiation, matter, and vacuum energy of the universe.
The relationship between the time and the scale factor, a, is obtained by performing the numerical integration of [Eq 5.83 of Ryden]:
![](images/FriedmannSolution/equation1.png "")
where omega_r,o is the current value of the radiation ratio, omega_m,o is the current value of matter ratio, and omega_L,o is the current value of the vacuum energy ratio, and 1 - omega_o is the curvature = 1 - omega_r,o - omega_m,o - omega_L,o, and finally Ho = the Hubble constant.
Using Mathematica, we can easily and quickly obtain the result (plot) of a(t) versus t for typical values of the LCDM model:
omega_r,o = 0.00008
omega_m,o = 0.3
omega_L,o = 0.7
Ho = 70 km/s/megaparsec.
Please now refer to the attached Mathematica Notebook, "LCDM Friedmann.nb". The steps are fairly self-explanatory. We can quickly obtain the plot of a(t) versus t:
![](images/FriedmannSolution/graphic2.png "")

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