The stars are not enough

Knowing the critical density, we can see how the visible matter compares to expectations ($\Omega_{\rm m,0} \approx 0.3$, $\Omega_{\rm bary,0} \approx 0.05$). But that requires inferring the amount of mass behind the light we see, which depends on what it is (a star, cold or hot gas, etc).

Stars with different masses give off different amounts of light, and if the number of stars of each type can be identified or inferred, then the total amount of mass in stars can be estimated. For example,

Thus each type of star has its own mass-to-light ratio, or $M/L$, usually expressed in units of $M_\odot / L_{\odot, V}$. For galaxies actively forming stars, the average ratio is $\approx 0.3 M_\odot / L_{\odot, V}$, while for quiescent galaxies (that haven't formed stars in a billion years or so, and thus have no bright, high mass stars left — they've all gone supernova) the ratio is more like $8 M_\odot / L_{\odot, V}$. Since there are both types of galaxies in the local universe the value falls in between these, given a density parameter in stars of $\Omega_{*,0} \approx 0.005$. This falls significantly below the expected amount of baryons in the universe — we believe these atoms are in a hard to detect gaseous phase just outside of and in between galaxies: we call this the "missing baryon problem." In galaxy clusters, however, this gas is so hot it emits X-rays that are more easily detected, and we can estimate that the mass in gas is about $10\times$ higher in gas than stars, consistent with $\Omega_{\rm bary,0}$.

Evidence for Dark Matter

Galaxies

Under Newtonian gravity, the orbital speed of stars or gas in a galaxy should be found by equating its circular acceleration $a=v^2/R$ with the acceleration from the gravitational force $a = G M(<R) / R^2$. As when deriving the Friedmann equation, we assuming the mass contributing to gravity is only interior, with no contribution exterior (so $M(<R)$ is the total mass enclosed by a sphere of radius $R$. Therefore, measuring the velocity as a function of radius allows us to infer the total amount of matter:

Beyond a radius where you can detect stars and gas, $M(<R)$ becomes a constant and we expect $v \propto R^{1/2}$$v \propto R^{-1/2}$ (otherwise known as a Keplerian velocity curve — which is what the velocities of planets in the solar system follow). Observations beyond this radius, however, show flat rotation curves, which means there must be some additional, but dark (or really, invisible), matter causing $M(<R) \propto R$.

Inverting the velocity equation and putting it in terms of useful units:

The mass-to-light ratio inferred using this mass estimate depends on the radius of the dark matter halo (i.e., how far out the dark matter extends). Since this is, almost by definition, hard to determine, total mass estimates will also generally be fairly uncertain. For our own galaxy, and perhaps galaxies in general, $\left<M/L_V\right> \approx 200~{\rm M}_\odot/{\rm L}_{\odot,V}$, much larger than what we expect from stars.

Clusters of Galaxies

For any collection of gravitationally bound objects, eventually they will reach an equilibrium state in which object energies get distributed in the system in such a way that the average kinetic energy $K$ of an object directly relates to the average gravitational potential energy $W$. This is called the virial theorem and can be written as

Here, $M$ is the total mass of the system, $\left<v^2\right>$ is the average squared velocity, $r_h$ is the half-mass radius (radius within which half the mass is contained), and $\alpha$ is a numerical factor of order 1 that depends on how the objects are distributed. If you don't know the exact object distribution, setting $\alpha=1$ should get you a good estimate to with a factor of a few (but just realize it's more of an order of magnitude estimate). Again, the equation can be inverted to estimate the total mass:

For a cluster of galaxies, we can't directly estimate $\left<v^2\right>$ because that requires knowing the full 3D space velocity of the galaxies, and these galaxies are too far away to estimate transverse velocities from proper motions. However, we can measure radial (or line-of-sight) velocities from the Doppler shift, and if we assume the velocities are symmetric in all directions, we can estimate this from the radial motions alone:

The root-mean-square of the radial velocities is called the velocity dispersion $\sigma_r$. In the second equation, we assume that the dispersion in the 2 other spatial dimensions is the same, hence the factor of 3. The mass-to-light ratio of galaxy clusters, using this method to estimate the mass, is then $\left<M/L_V\right> \sim 400~{\rm M}_\odot/{\rm L}_{\odot,V}$, comparable to what is found in individual galaxies.

Another method to find the mass is to use the thermodynamical properties of the hot, X-ray emitting gas in between the galaxies in clusters. If this gas is in hydrostatic equilibrium with the gravitational potential, the gradient of its pressure will balance the gravitational force at that radius, and the total mass can be estimated from the temperature and density profiles of the observed gas:

Gravitational Lensing

Another indirect method for detecting dark matter (or the total amount of matter, whether dark or not) is gravitational lensing, where light follows geodesics curved by the presence of massive objects (of any mass really, but the light must pass closer to less massive objects for an effect to be seen). The angle $\alpha$ a light ray is bent is given by

where $b$ is the "impact parameter" or closest distance the light comes to the lensing object. If a lensed object is directly (or nearly directly) behind the lensing object, there's a particular value of $b$ that causes the rays to get focused directly to us, and that can cause multiple images (or a continuous ring) to be seen. This impact parameter is called the Einstein radius $\theta_E$.

We can use this radius to search for compact clumps of dark matter (called MACHOs), which cause the total amount of light detected from the background object to temporarily increase (as the direct light and extra light from $\theta_E$ are added together):

Except for interferometric radio telescopes, this radius is too small to be resolved (optical telescopes on the ground can typically resolve stars to ~1"). While many objects have been detected with this method, they are generally consistent with being white dwarfs, the leftover cores of evolved low mass stars, and there are not enough of them to account for the amount of dark matter we infer in any case.

We can also see gravitational lensing effects in galaxy clusters, where the Einstein radii tend to be more easily observable:

The masses inferred from lensing and from hydrostatic equilibrium in clusters generally agree, again suggesting the presence of some unseen, exotic form of matter beyond the standard model of particle physics.