In order to measure the density of the Universe, it is necessary to sample a region that is larger than the scale on which the Universe becomes approximately homogeneous. The volume of this region must then be measured, as well as the mass of the matter it contains. The ratio of mass to volume then gives the density.

The volume of the surveyed region obviously depends on the
distance of the objects at the edge of the region.
The volume of a sphere of radius *R* is given by
*V = (4*pi/3)R ^{3}*. The radius is the distance

The mass of an object can be derived from the orbital velocities of the particles within the object. Consider a mass on a string that you swing around. As you make the mass go faster, the pull you need to exert on the string gets larger and larger. If you replace the string by a central mass, this means that the central mass must be larger if the orbital velocities are larger. When applied to clusters of galaxies we find a mass to luminosity ratio that implies a density of the Universe that is about 0.3 times the critical density.

A moderately technical further discussion of these idea follows below, with links to more detailed derivations.

The mass of the objects in the surveyed region is determined using the virial theorem or one of its variants, which states that

vwhere^{2}= GM/R

M = R*vIf the object has a measured flux^{2}/G = theta*cz*v^{2}/GH_{o}

L = 4*pi*F*(cz/Hand thus the mass-to-luminosity ratio of the object is_{o})^{2}

M/L = (H_{o}/cz)*theta*v^{2}/(4*pi*F*G)

If *N* objects of this mass are seen in the surveyed volume,
then the derived density of the Universe is given by

rho = N*M/V = N*(3/4*pi)*<theta*cz>/(czNote that_{max})^{3}*(v^{2}/G)H_{o}^{2}

We can also compute the luminosity density of the sampled region:

N*L/V = N*<F*(cz)The luminosity density of the Universe is about 190 million solar luminosities per cubic Megaparsec (Mpc) at visual wavelengths for^{2}>*H_{o}/[3*(cz_{max})^{3}]

If we use the virial theorem on galaxies instead of clusters of galaxies
then we get a mass-to-luminosity ratio that is about 30.
Thus the mass-to-luminosity ratio appears to vary with the size of the region
measured, from 3 in the solar neighborhood to 30 in galaxies to 300
in clusters of galaxies. Is there a possibility that for even larger objects
the ratio could reach the critical value of 700? For such large regions
we cannot use the virial theorem because these regions are still expanding
with the Hubble flow.
However, we can compute the gravitational acceleration due to the large
density contrasts in the nearby superclusters. The density contrast,
*d(rho)/rho*, can be measured by counting galaxies. The gravitational
acceleration is proportional to *d(rho)* which is the measured
density contrast times the unknown density. The gravitational
acceleration times the age of the Universe gives our peculiar velocity
relative to the CMB, which can be
determined from the dipole
anisotropy of the CMB.
Different groups have reached different conclusions
about whether the resulting *Omega* could reach the critical value
of 1. But it definitely appears that the
dark matter
fraction increases
with the size of objects at least up to clusters of galaxies (1 Mpc radius).

We can use the observed abundances of hydrogren, helium and lithium isotopes
to estimate the total density of baryons in the Universe. This gives
a value of 4.4% of the critical density for *H _{o} = 71*.
Since the

One proposed way out of the need for dark matter is to modify the law of gravity. This is done in the Modification Of Newtonian Dynamics [MOND] theory of Milgrom. There are many papers about MOND on the astro-ph preprint server, just in the 1993-1998 time period: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. Many papers continue to published on MOND. While the MOND alternative to non-baryonic dark matter is not widely accepted, it is not suppressed by the establishment, as proved by these citations.

MOND proposes that the acceleration of gravity *g* is not always
given by the Newtonian formula *g _{N} = GM/r^{2}*.
If

Aguirre, Schaye & Quataert (2001) find that while MOND works well on galaxy scales, it fails for Lyman alpha clouds and clusters of galaxies. The basic problem with MOND on cluster scales is that a point mass in MOND gives the observed isothermal profile in clusters, but most of the mass in clusters is due to a diffuse cloud of hot gas. Thus the observed gas density profile generates a gravity field under MOND that is not consistent with the observed gas temperature and density profile. Sanders (2002), written by a long time supporter of MOND, recognizes this problem and finds that dark matter is needed in clusters even under MOND.

A long standing problem with MOND was that it did not have a relativistic
version. Bekenstein & Sanders (2005) have proposed
a solution to this
problem. But
Zhao *et al.* (2005) find that this model requires different values
for the universal constant a_{o} in different gravitationally
lensing clusters of galaxies, so the problem of not having a consistent
relativistic version of MOND remains.

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© 1998-2016 Edward L. Wright. Last modified 09 Mar 2016