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Recently, an extension of the geodesic equations of motion using the DarkEnergy length scale was proposed. Here, we apply this extension to theanalyzing the motion of test particles at the galactic scale and longer. Acosmological check of the extension is made using the observed rotationalvelocity curves and core sizes of 1393 spiral galaxies. We derive the densityprofile of a model galaxy using this extension, and with it, we calculate$\sigma_8$ to be $0.73_{\pm 0.12}$; this is within experimental error of theWMAP value of $0.761_{-0.048}^{+0.049}$. We then calculate $R_{200}$ to be$206_{\pm 53}$ kpc, which is in reasonable agreement with observations.

Dark Energy and Extending the Geodesic Equations of Motion: Connecting the Galactic and Cosmological Length Scales