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Composition and Mass of observable universe
Idris Qasim
The size, complexity and composition of the universe have bewildered mankind
for millenniums. Only in the late 20th
century have we been able to unravel the
mysteries, understand, explain and confidently speak about it. Cosmology moved
from the realm of meta-physics to precision physics. We understand and speak
with a high degree of certainty about the origin, age, dimension and composition
of the universe.
In this paper I will compute the mass and composition of the observable
universe explain the theory behind the numbers and provide supporting evidence
(observational facts) .
Edwin Hubble meticulously measured the velocities and distances of
galaxies and discovered the galaxies moving away from one another with the
farther galaxies moving faster. He captured this observation in the Hubble’s Law
V = H0d where V is velocity of a Galaxy, d the distance and H0 the Hubble’s
constant. The Hubble constant also evolves with time, H0 is its current value.
General Relativity tells us that matter (or energy) curves space around it.
So the total amount of matter in the universe determines its curvature. Friedmann
equations connect the Mass of the universe to its curvature.Consider a test
particle on the surface of an expanding sphere of radius R. Newton’s equation of
motion gives us
(1)

Multiply both-sides by (2)
= 2R => = 2 (3)
( )’ = - (4)
’ (5)
’ (6)
Integrating both sides (7)
where is the density (8)
Multiply by (9)
Since H = (10)
Let = => where is the critical density (11)
Substituting (11) in (10) (12)
re-arranging (13)

Multiply (13) by (14)
Substituting (11) (15)
The friedmann equation (15) connects there quantities, the density , the space
curvature and the expansion rate H of the universe. The nature of the
curvature depends upon the density
Then the universe has a negative curvature like a saddle.
Then the universe has a positive curvature like a balloon.
Then the universe has a 0 curvature and is flat as a sheet.
Experiments (BOOMERANG, COBE, WMAP etc) on CMB have proven
the universe to be flat with Ω = / c = 1. So the density of the universe equal to
its critical density. is = = .
The current radius R of the observable universe is given by
(16)
Given the radius and density we can derive Mass of universe as
M = (17)
M = (18)
M = (19)

Using the current values of H from the WMAP satellite we compute M =~ 24x1053
kg.
Once the amount of matter (or density) of the universe is known, one can
begin inventory of what constitutes the universe. It is customary to express the
current density as a fraction of the critical density Ω = / c . At a high level we
can consider this density is comprised densities of matter (both luminous and
non-luminous), radiation (photons) and anything else (we shall elaborate more
later on this) .
The sum of matter (electrons, baryons, neutrinos etc), radiation (photons) and
everything else is equal to the
(20)
(21)
Ω (22)
Where , is the density of matter and radiation respectively.
The density of the radiation has been measured by calculating the
number of photons in a cubic centimeter, then converting the energy of these
photons to mass using E = mc2
. This energy is mostly due to CMP and cosmic
neutrinos. WMAP assessed this to be =~ 8x10-5
which is very low compared to
the . Using the current CMB temperature of 2.70
K in Bose-Einstein energy
and Fermi-Dirac distribution equations a similar low value for can be derived.
Most of the observed radiation is from CMB. Although stars are intense sources
of radiation, they contribute a tiny fraction because they are miniscule compared
to the size of the universe. The CMB is very weak and cold, but the total CMB

contains more energy than has been emitted by all the stars and galaxies that
have ever existed.
Variety of techniques is used to determine the density of matter. Some are
highlighted below.
1. Direct Observation: Using the mass-to-light ratio.
2. Measuring the amount of light absorbed quantifies the amount of
hydrogen light encounters along its way pass a galaxy.
3. Computing the baryon contents of universe from the anisotropies of the
CMB.
4. Gravitational Lensing.
5. Inferring the baryonic content from the abundance of the light elements
in the universe. Models for big bang Nucleosynthesis predict certain
amount of light elements in the universe and these predictions match
very well with the observation when as shown in Figure 1
BBN and Abundance of lighter elements

Figure 1 BBN and Abundance of lighter elements
A consensus emerges on . So normal matter atoms constitute only
4% of the observable universe.
Galaxies rotate and the speed of rotation depends upon the distribution of
mass inside the galaxies. Gravity keeps stars bound to galaxies as they rotate.
The orbital velocity of any object (star or planet) is given by . One would
expect the orbital velocities to decrease for stars on the fringes of galaxies. But
orbital velocities measured and plotted by Vera Rubin and others show orbital
velocities flattening out near the peripheries of the galaxies. Further given the
observable mass of a galaxy, the velocity of stars at the peripheries is too large
for them to be in an orbit within the galaxy. The star should have flown away from

the galaxy. So there is some additional un-seen matter that gravitationally binds
these fast moving stars to the galaxies. This un-seen matter is called dark matter.
Figure 2 Rotation Curve of Galaxy NGC 3198
The curve indicates that the mass of galaxies is not concentrated in the
center but has a mass distributed in a halo surrounding the galaxy. One of the
many models for the mass distribution is where r - radius, a - halo
length, the density. There are other models too. All the luminous content
including the inter-galactic gas does not account for the unseen matter. The
velocity rise appears to flatten out around 50Kpc and one can infer that the dark
matter is 5-7 times more than the luminous matter thereby making .
Dark matter thus interacts gravitationally, but does not interact
electromagnetically (no light emission/absorption), holds matter in galaxies
together. Evidence for Dark Matter include
1. Rotational curves described above
2. Inter galactic gases are too hot for gravity to hold them in galaxy
clusters. Without dark matter these gases would have evaporated.

3. Distorted images of background galaxies behind a cluster due to weak
lensing
Figure 3 Abell 2218 image

4. Collision of enormous cluster of galaxies captured by the Chandra
observatory shows normal matter(hot gas shown in pink) wrenched
away from the dark matter (blue)
Figure 4 Bullet cluster showing dark matter
While existence of dark matter has been demonstrated, its constituent
components are unknown. It may be a new form of matter. Some potential
candidates for this new form of matter are
- WIMPs (Weak interactive massive particle)
- MACHOs (Massive compact halo objects)
- Neutrinos
The standard candle technique has been used for ages to measure
distances, but this method suffers from calibration and standardization problems.
One is hard pressed to determine the absolute luminosity of the candle. The
Type Ia supernovae offer an effective alternative. These are bright explosions
that occur when a white dwarf upon acquiring sufficient mass from a companion
star crosses the Chandrasekhar limit and explodes. Its intrinsic luminosity is well
understood both theoretically and empirically, hence serving as excellent

standard candle. The shift in the peak luminosity of the SNe over weeks is a
great indication of the rate of expansion of the universe. Further Goobar &
Perlmutter (1995) showed the possibility of separating the relative contributions
of the Ωm to the changes in expansion rate by studying supernovae red shifts.
Two independent teams initially studied the SNe Ia and discovered the
Hubble curve having a slight bent upwards indicating an accelerating universe
pointing to existence of energy that is driving this acceleration against the pull of
gravity. Their results captured in the famous charts reproduced in Figure 5
Hubble charts showing an accelerating universe.

Figure 5 Hubble charts showing an accelerating universe
This hitherto unknown energy is called dark energy and it acts like a repulsive
gravity causing the universe to accelerate. Examining (22) we see
(21)
Ω (22)
Ω (23)
Dark energy contributes to the and thus accounts for 71% of the
observable universe.
Friedmann equation and GR can be analyzed to understand the
theoretical origins of the acceleration.
(24)
(25)

is the factor contributing to the dark energy. As the universe expands ∝ a-4
in
a radiation dominated universe and ∝ a-3
in a matter dominated universe.
(Figure 6 Mass density over timeFigure 6 Mass density over time). In either
case the RHS in (25) will decrease assuming is κ constant. So we would expect
the universe to be decelerating. But since the SNe Ia data shows an
accelerating universe, Feiedmann’s equation has to be modified to include a
dark-energy term
(26)
The mysterious phantom dark-energy is causing the expansion to
accelerate, is smoothly distributed and varies very slowly (if at all) with time.
One candidate for the dark energy component is the energy of Vacuum. It
is possible that the dark energy is property of space itself. Another theory called
Super symmetric proposes that for every normal matter particle there is a
corresponding s-particle (electrons vs selectron). Or maybe we need a new
theory of gravity that goes beyond general relativity. Currently there is no
comprehensive and compelling theory that explains dark energy. Turner et.al
succinctly summarized and assessed these theories as
- Cosmological constant: simple, but no underlying physics.
- Vacuum energy: Mathematically equivalent to a cosmological
constant w = −1 is consistent with all data, but all attempts to
estimate its size are at best orders of magnitude too large.
- Scalar Fields: Temporary period of cosmic acceleration,w varies
between−1 and 1 (and could also be < −1), possibly related to

inflation, but does not address the cosmological constant problem
and may lead to new long-range forces.
- New gravitational physics: Cosmic acceleration could be a clue to
going beyond GR, but no self-consistent model has been put forth.
- Old gravitational physics: It may be possible to ﬁnd an
inhomogeneous solution that is observationally viable, but such
solutions do not yet seem compelling
How did the composition of the universe evolve over time? As the
universe expanded it cooled. Going backwards in time, reverses the expansion,
the red shift of the CMP becomes blue shift and the CMB gets hotter (E =hc/λ).
Closer to the birth (big bang), the radiation is so got that it breaks matter and the
universe becomes plasma of fundamental particles and photons. The universe
thus evolved to its current form (matter dominated) from hot, dense plasma
(radiation dominated). As it expanded and cooled the densities of matter and
radiation changed too. In a radiation dominated universe ∝ i.e.
wavelength stretches, the density is derived as
∝ (26)
∝ (27)
While in a matter dominated universe the density is and the density is
∝ (28)

Since the radius of the universe is function of time a plot of the density over time
looks like
Figure 6 Mass density over time
In conclusion we can say that the universe started its life in a radiation
dominated era as a hot dense radiation, as it expanded and cooled matter
formed, as it continued expansion the density of radiation fell faster than the
density of matter and universe entered a matter dominated era. We are now at
an interesting juncture where the matter density is same as the dark energy
density (Cosmic Coincidence problem) and the future era will be one in which the
dark energy will dominate. The matter of the universe is spread among luminous
matter, non-luminous (dark matter), radiation energy and dark energy with
Luminous matter = 1%

Dark matter = 24%
Dark energy = 71%
The pie in Figure 7 Composition of the Universe shows just that.
Figure 7 Composition of the Universe

References
Michael Rauch et al. 1997 ApJ 489 7
Burles, S., et al. 2001, Astrophys. J. 552, L1.
Schindler S. 2001, Space Science Reviews 100: 299-309
Gentile, SAlucci, Klein, Vergani, Kalberla, 2008, ApJ, 484, 637
Gentile, G., Salucci, P., Klein, U., Vergani, D. and Kalberla, P. (2004), Monthly Notices of the
Royal Astronomical Society, 351: 903–922.
Michael S. Turner 2002 ApJ 576
Frieman,Turner, Huterer 2008. REVIEWS Annu. Rev. Astro. Astrophys. 2008.46:385-432
Perlmutter S, Aldering G, Goldhaber G, Knop RA, Nugent P, et al. 1999. Astrophys. J. 517:565
Riess AG, Filippenko AV, Challis P, Clocchiatti A, Diercks A, et al. 1998. Astron. J. 116:1009

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