According to Newtonian gravity, the rotational velocity $v(r)$ of an object is related to the mass enclosed by its orbit (integral mass): \[ v^2(r) = \frac{G M(r)}{r}. \] Rotation curves plot orbital velocity of a star vs. its distance from the center of the galaxy. Rubin et al. (1978) measured the rotation curves of various galaxies and found that the rotation curve at the edge of the luminous galaxy does not decline as expected $v(r) \propto 1/\sqrt{r}$. Instead, the curve is fairly flat or even increase (See Fig. 1 and Fig. 2). If Newtonian gravity is correct in these systems, there must be some invisible mass extended over the luminous part to provide gravity. These are the classical evidences of existence of dark matter.

The dark matter distribution can be inferred from the rotation curve. The flatness of rotation curves in spiral galaxies indicate an integral mass $M(r) \propto r^{1+\delta}$. It has been shown by N-body simulation that there exists universal dark matter halo profile, \[

\rho(r) = \rho_0 \left( \frac{r_0}{r} \right)^\gamma \left[ \frac{1 + \left( \frac{r_0}{a} \right)^\alpha}{1 + \left( \frac{r}{a} \right)^\alpha} \right]^{\frac{\beta - \gamma}{\alpha}} \] Some widely used dark matter halo profiles are, Kravtsov et al. (1998)$ (\alpha, \beta, \gamma) = (2, 3, 0.4), a = r_0 = 10$ kpc, Navarro et al. (1995) $(\alpha, \beta, \gamma) = (1, 3, 1), a = r_0 = 20$ kpc, Moore et al. (1999) $(\alpha, \beta, \gamma) = (1.5, 3, 1.5), a = r_0 = 28$ kpc and Bergstrom et al. (1998) $(\alpha, \beta, \gamma) = (2, 2, 0), a = r_0 = 3.5$ kpc

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Fig. 2: rotation curves of several spiral galaxies. By Rubin et al. (1978) |

\rho(r) = \rho_0 \left( \frac{r_0}{r} \right)^\gamma \left[ \frac{1 + \left( \frac{r_0}{a} \right)^\alpha}{1 + \left( \frac{r}{a} \right)^\alpha} \right]^{\frac{\beta - \gamma}{\alpha}} \] Some widely used dark matter halo profiles are, Kravtsov et al. (1998)$ (\alpha, \beta, \gamma) = (2, 3, 0.4), a = r_0 = 10$ kpc, Navarro et al. (1995) $(\alpha, \beta, \gamma) = (1, 3, 1), a = r_0 = 20$ kpc, Moore et al. (1999) $(\alpha, \beta, \gamma) = (1.5, 3, 1.5), a = r_0 = 28$ kpc and Bergstrom et al. (1998) $(\alpha, \beta, \gamma) = (2, 2, 0), a = r_0 = 3.5$ kpc

#### Dark Matter Distribution

In summary, evidences of existence of dark matter can be found in galaxies, clusters and from cosmological probes. Cosmological observations suggest overall density fraction of invisible mass $\Omega_{DM} \simeq 0.23 $. Bahcall et al. (1995) collected the mass-to-light ratio on different scales (See Fig. 3). Their study shows mass-to-light ratio of dark matter on large scale structures is consistent with the cosmological constraints. This conclusion also suggests dark matter from cosmology is the same thing with the dark matter in galaxy and clusters.Fig. 3 |

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