29. Cosmology29. CosmologyGoalsGoals:1. Examine a simple cosmological model to

gain insights into the parameters of interest.

2. Note how the observed cosmological parameters depend upon the type of universe considered: flat, closed, or open.

3. Add complications to the simple model to make it more similar to reality, then examine what observations tell us about the applicability of the adopted models.

Observational BackgroundThe rectangular area below lies upwards from the Galactic plane, thereby sampling the universe.

A close-up of the region of sky where the Hubble Space Telescope Ultra Deep Field image lies.

Every “fuzzy” object in the

image is a distant galaxy.

Foreground stars in our own Galaxy

have associated diffraction

spikes because they are

point-like images.

Star

Redshift-distance relation (again).

The inverse of the Hubble constant (which is the slope of the Hubble relation) has units of time and is called the Hubble Time. It is an estimate of the age of the universe (backwards extrapolation),

provided that the expansion began at some point in the past and has been continuing at the same

rate ever since.

for H0 = 71 km/s/Mpc, and 15 billion years forH0 = 65 km/s/Mpc.

The Milky Way’s globular clusters are all less than 14 billion years old.

yearsbillion8.13)km/s/Mpc(

secondsbillion9781

00

HH

Schematics of the meanings for “homogeneous” and “isotropic.”

A Simple Pressureless “Dust” Model of the UniverseChapter 29 begins with the presentation of a simple model to illustrate the basic cosmological parameters, an expanding universe filled with pressureless “dust” of uniform density, ρ(t), with an arbitrary point as the origin. Such a universe is isotropic and homogeneous, and expands about an arbitrary point chosen as the origin.

Let r(t) be the radius of a thin spherical shell of mass m at time t. The shell expands along with the universe with recessional velocity v(t) = dr(t)/dt. As the shell expands, its kinetic energy K decreases and its gravitational potential U increases, but the total energy E remains the same. The total energy of the shell can be expressed in terms of two constants, k and , such that E = −½mkc22. The constant k has units of (length)−2, while (“varpi”) can be thought of as the present radius of the shell, in other words: r(t0) = .

The pressureless “dust” model of the universe.

Conservation of energy for the mass shell implies that:

where Mr is the mass interior to the shell:

Although the radius and density of the shell constantly change, the combination r3(t)ρ(t) remains constant because the total mass of dust in the shell does not change as the universe model expands. Removal of m and substituting for Mr leads to:

Note that k > 0 represents a closed universe, k < 0 an open universe, and k = 0 a flat universe.

The cosmological principle requires that the expansion is the same for all shells, which implies that the radius of a particular shell identified by is:

Here r(t) is the co-ordinate distance while is the co-moving co-ordinate. R(t) is a dimensionless scale factor.Thus, R(t0) = 1 corresponds to r(t0) = . The scale factor is equal to Remitted/Robserved. which from previously means that the scale factor and redshift are related by:

The comment that r3(t)ρ(t) remains constant means that the product R3ρ remains constant for all shells, and, since R(t0) = 1, implies:

where ρ0 is the density of the dust-filled universe at present. From above we know that:

The evolution of this pressureless “dust” universe is described by the time behaviour of the scale factor R(t). The Hubble parameter, H(t), can be expressed in terms of the scale factor as:

But v(t) is the time derivative of r(t):Therefore:

The conservation of energy equation therefore becomes:

or:

The left side of the equation applies to all shells while the right side of the equation contains only constants. It can be rewritten as:

For a flat, closed universe, k = 0, so the density ρ is the critical density ρc required for closure:

We can evaluate the parameter for our present universe using:

That yields the present value for the critical density:

about 11h2 hydrogen atoms per cubic metre.

Protons and neutrons are baryons, so hydrogen ions and element nuclei in stars are a form of baryonic (quark-based) matter. Electrons are leptons, but when bound to atomic nuclei are counted as a component of baryonic matter, along with mesons. Non-baryonic matter, such as so-called dark matter, is generally of unknown composition. One of the questions of cosmology is how much of the universe is baryonic and how much non-baryonic.The ratio of the measured density of the universe to the critical density is denoted Ω, the density parameter:

with a present value of:

Some estimates of interest:

From the previous equation for density ratios:

or:

From previous relations it follows that:

And, at t = t0:

If Ω0 > 1, then k > 0 and the universe is closed. But if Ω0 < 1, then k < 0 and the universe is open. Most cosmologists appear to want Ω0 = 1 (k = 0) in order to have a flat universe.

Equating some of the previously-derived equations yields:

With the previous two equations one obtains:

And:

According to the equations, at very early times as R → 0 and z → ∞ the sign of Ω − 1 does not change and Ω → 1 no matter what its present value. By inference, the very early universe was essentially flat, regardless of its present nature.

Thoughts on an analogy: baking raisin bread is often used to picture an expanding universe. No matter which raisin represents the Sun, all raisins appear to increase their distances with time, at a rate proportional to their

distance. But where is the edge?

The expansion of a flat (Ω0 = 1), one-component universe of pressureless dust as function of time is found by further combination of the equations:

Taking the square root, rearranging, and integrating yields:

Solving gives:

and:

or: for Ω0 = 1.

tH = 1/H0 is referred to as the Hubble time.

If Ω0 ≠ 1 the density is not equal to the critical density. For Ω0 > 1 the universe is closed and the solutions become:

For Ω0 < 1 the universe is open and the solutions become:

The variable x parameterizes the solutions. The behaviour of a closed universe for Ω0 = 2 is shown in the appended figure. The “bounce” that occurs after contraction of the universe is a mathematical artifact and does not necessarily imply an oscillating universe.

Recall the hyperbolic trigonometric functions:

so Ropen increases monotonically with t. The appended figure displays the solution for Ω0 = 0.5. When Ω0 ≤ 1 the model universe continues to expand forever.

Because of deceleration, the age of the universe must be less than the Hubble time.

Age of the Pressureless “Dust” Universe.Keep in mind that all such mathematical solutions are extrapolations only and do not refer to “the age of the universe.” Recall:

and

Therefore: for Ω0 = 1 (flat).

for Ω0 > 1 (closed).

for Ω0 < 1 (open).

In the limit as z → ∞ all equations reduce to:

where higher order terms are neglected for Ω0 ≠ 1. Because the early universe was flat to a close approximation, highly precise observations are necessary to establish if the universe is flat, closed, or open. The current age of the universe t0 may be found by setting z = 0 in the equation, giving:

for Ω0 = 1 (flat).

Note that a Hubble constant of H0 = 50 km/s/Mpc yields a Hubble time of tH = 1/H0 = 20 109 years and an age for the present universe of 13.3 109 years, consistent with the derived maximum ages of globular clusters. The solutions for Ω0 ≠ 1 are more complicated, namely:

namely:

for Ω0 > 1 (closed).

for Ω0 < 1 (open).

Solutions for the age of the universe in such models are summarized in the following figure.

Note that a Hubble constant of H0 = 71 km/s/Mpc produces conflicts with the ages of globular clusters.

Lookback Time

The lookback time tL is defined as how far back in time one looks when viewing an object of redshift z. That is simply the difference between the present age of the universe and its age at time t(z), i.e.:

Solutions depend upon the value of Ω0. The results are:

for Ω0 = 1 (flat).

Solutions for for Ω0 > 1 (closed) and Ω0 < 1 (open) are more complex. See appended figure for details.

Example 29.1.2If the redshift of quasar SDSS 1030+0524 is z = 6.28, for a flat universe of pressureless dust the lookback time for the quasar is:

or 0.6327266. But the age of a flat universe is t0 = 2tH/3. Therefore:

Thus, the light from SDSS 1030+0524 was emitted when only ~5% of the history of the universe had unfolded. The universe at that time was also smaller by a factor of ~7 since the scale factor was:

The Inclusion of PressureThe inclusion of pressure begins with the original differential equation:

with Einstein’s relation Erest = mc2 and the mass density ρ broadened in definition to include relativistic particles of equivalent mass density, where mass now is Erest/c2. It is also useful to describe the conservation of mass using the relation described earlier:

Generalization of the first equation is done using the 1st law of thermodynamics, where conservation of internal energy U, work done W, and heat Q is written as:

Thermodynamics are used to generalize the pressureless “dust” universe to incorporate pressure-producing components. It is useful to first consider that the entire universe has the same temperature, so there is no heat flow, i.e. dQ = 0. Therefore, any change in internal energy must be produced by work done by the “fluid” component of the universe:

With V = 4/3 πr3 one obtains:

Define the internal energy per unit volume u as:

to obtain:

Now substitute for u the equivalent mass density:

which generates the relationship:

Finally, with:one obtains the fluid equation:

In a universe of pressureless dust, P = 0, so R3ρ is constant. With the equations noted previously it is possible to derive the acceleration equation:

Acceleration equation:

which is an illustration of Birkhoff’s theorem. Note that the effect of pressure (P > 0) is to slow down the expansion.The acceleration equation and the fluid equation contain three unknowns: ρ, P, and R. A solution requires a third relationship for the parameters, in particular the equation of state:

where w is a constant of proportionality. Inserting the last into the fluid equation produces:

where ρ0 is the present value of the equivalent mass density.

A last parameter can be introduced to describe the acceleration of the universal expansion, namely the deceleration parameter q(t):

It can also be shown, for a pressureless “dust” universe:

Thus, for the present time in such a universe, q0 = 0.5 for a flat universe, q0 > 0.5 for a closed universe, and q0 < 0.5 for an open universe.

The Cosmic Microwave Background

A new observational parameter was added to the field of cosmology in 1965 with the discovery, by Arno Penzias and Robert Wilson working at the Bell Laboratories in New Jersey, of the 3 K cosmic microwave background radiation, or CMB. The existence of the CMB has subsequently been confirmed by observations with the Cosmic Background Explorer (COBE) satellite. It is usually thought of as the residual glow of the Big Bang, greatly redshifted from 3000 K to 3 K.

It may be noted that a variety of alternative explanations have been proposed to explain the glow, but most appear to have weak points. There is even one pertaining to steady-state cosmology, although the predictions of most variants of steady-state cosmology are contradicted by the abundances of helium (He) and the light elements lithium, beryllium, and boron (Li, Be, B).

Penzias and Wilson with the radio horn used to discover the 3K microwave background

radiation.

Origin of the 3K

microwave backgroun

d

The 3K background revealed by the COBE satellite, displays a “Doppler shift” of 370.6 ±0.4 km s−1 relative to

the Hubble flow, attributed to mass asymmetry in the early universe when matter separated from radiation.

The “Doppler effect” seen in COBE measurements of the 3K background.

Note that the direction of motion is mainly towards the Virgo cluster.

The 3K microwave background matches the radiation from a black body with T = 2.728 K. Note that WMAP

finds a best value of [T0]WMAP = 2.725 ± 0.002 K.

The 2.728 K background is the constant faint glow from the universe when T = 3000 K, now redshifted by z ≈ 1000.

The net motion of the Earth relative to the Hubble flow needs to be adjusted for the motion of the Sun about the Galactic centre and the motion of the Galaxy in the Local Group. Both of those adjustments have likely been done erroneously. Thus, the resulting motion of the Local Group relative to the Hubble flow of 627 km s−1 may be slightly in error. No one has yet generated an improved value (likely through lack of knowledge about the LSR velocity). The effect on the temperature of the CMB is given by (Problem 29.21):

or:

The 3K microwave background with the Doppler shift removed, as recorded by WMAP.

The wavelength of peak radiative output of black bodies varies according to Wien’s Law as:

And the peak wavelength must stretch with the expansion of the universe according to the scale factor:

The radiation temperature of the universe must therefore decrease as the universe expands according to:

And the radiation density of the universe must vary as:

whereas the matter density of the universe must vary as:

It follows that, even if the radiation density now is much less than the matter density, it must have dominated the matter density in the distant past (see figure).

The remaining CMB radiation is surprisingly isotropic, but does have hotter and cooler areas about 1° or less in diameter. They are usually attributed to convective bubbles in the early universe. They deviate from the CMB by only one part in 105.

The Sunyaev-Zel’dovich Effect, resulting from inverse Compton scattering of low energy CMB photons by ionized intracluster gas of T 108 K, was discussed earlier. It can result in an apparent temperature decrease in the temperature T0 of the CMB.

The effective temperature decrease is given by:

where τ is the optical depth. Typical values for ΔT/T0 are of order 10−4.

Two-Component Model of the Universe.In the pressureless “dust” universe, the expansion was slowed by self-gravity of the dust. In a Big Bang universe, however, the relativistic equivalence between mass and energy is important, since the early universe was dominated by CMB photons and neutrinos with their dynamical influence. Their effect is negligible in the present universe.In such a universe, the equation of state:

determines how different particles are included, either as matter (wm = 0), relativistic particles (photons, neutrinos), wrad =1/3), or even something more exotic, such as the hypothesized dark energy (wde = −1). For example, an electron gas at T > 6 109 K has kT > mec2, implying that it is in the relativistic regime described by a constant of proportionality w =1/3.

The velocity equation under such a situation becomes:

where the mass density of the CMB is given by:

where a is the radiation constant, which can be rewritten in terms of the number of degrees of freedom g for the particle, namely the number of possible spin states nspin and the possible existence of an antiparticle (nant = 1 or 2), i.e.:

For photons, with only two possible polarization states and being their own antiparticle, grad = 2.

The cooling of the black body radiation follows from the energy density equation:

and the fluid equation:

Since wrad = 1/3:

and

since R0 = 1. The above equation becomes:

So:

indicating that the temperature of the universe is related inversely to the scale factor R, as noted previously.

Neutrino DecouplingThe equation of state for a neutrino universe is described by:

where Tν is the temperature of the neutrino “gas.” With a expressed as before, one obtains:

since there is an antiparticle equivalent for each of the three types of neutrinos: electron, muon, and tau.

For T > 3.5 1010 K, Tbb Tν = T, but as the universe expanded and cooled there was a neutrino decoupling in which the neutrinos expanded and cooled at their own rate.

It can be shown that the neutrino temperature is related to the temperature of the CMB photons by:

The total neutrino energy density is therefore given by:

and the total energy density for relativistic particles, photons and neutrinos, is given by:

where:

is the effective number of degrees of freedom for relativistic particles.

The equivalent mass density of relativistic particles is also defined by:

which appears to be valid back in time to the end of electron-positron annihilation, ~1.3 s after the Big Bang. For earlier times T was higher and g* was also larger. The solution to the velocity equation changes from:

to:

where:

is the density parameter for matter (baryonic and otherwise),

and the density parameter for relativisitic matter is:

For T0 = 2.725 K (WMAP) and H = 71 km/s/Mpc, Ωrel,0 = 8.24 10−5, a relatively small number compared to the similar value from WMAP for matter, namely Ωm,0 = 0.27.

Given that wrel = 1/3 for relativistic particles, the relation:

as before, produces:

whereas the scale factor for matter varies as:

It follows that, in the very early universe when R → 0, there must have been an era when radiation, including all relativistic particles not just photons and neutrinos, dominated the universe. The transition from a radiation era to the present matter era occurred when ρrel = ρm, or when Ωrel = Ωm. That occurred when:

WMAP derived a value of Rr,m = 3.05 10−4, which corresponds to a redshift of:

giving zr,m = 3270 for the WMAP values, compared with a WMAP value of zr,m = 3233 ±200.

Since RT = T0, the temperature of the universe at that point was:

or Tr,m = 8920 K with the WMAP results.

Recall the velocity equation simplified with the previous results:

For a flat early universe, k = 0, so the equation can be solved as follows:

leading to an expression for the age of the universe in terms of the scale factor R:

where:

The transition from a radiation-dominated to a matter-dominated universe is derived using R/Rr,m = 1, which, with the WMAP values, is 5.52 104 years. Deep in the radiation era when R → 0 it is possible to show that:

and:

At the other extreme where R >> Rr,m:

So:

and, with tH = 1/H0 for the Hubble time:

The smaller Ωm,0 becomes, the closer t(z) is to the Hubble time for z = 0.

Since the temperature varies as 1/D(t), as described previously as a function of time:

Whereas, since g* = 2 for radiation, the time dependence for the relativistic equivalent matter density for radiation is described by:

It follows that the radiation density dominated the matter density at some point in the past when the temperature was much larger than at present.

In other words...

Big Bang NucleosynthesisThe process that produced the lightest elements in the early universe is referred to as Big Bang nucleosynthesis, described by the reactions:

The reactions occur either way because the mass difference between a proton and a neutron amounts to:

whereas the characteristic thermal energy of particles at 1013 K is ~86 MeV. The Boltzmann equation gives the number density ratio as:

At 1013 K the conversions between neutrons and protons proceeded nearly equally in both directions, and according to the Boltzmann equation that was also true around 1012 K. But just above 1010 K the characteristic thermal energy of the photons, kT, fell below the 1.022 MeV energy needed to create electron-positron pairs via the reaction γ → e− + e+ and the neutrino energies had become too small for them to participate in the reactions cited above. At that point the number density of neutrons to protons became frozen at its present value of nn/np = 0.223.

By then there were 1000 protons for every 223 neutrons, with no new neutrons being created. Beta decay therefore continued to create protons at the characteristic half life of 614 seconds. But the production of deuterium via:

could not proceed until temperatures dropped to ~109 K.

The elimination of “antimatter” in the early universe.

Below 109 K a number of reactions could result in the production of helium:

These reactions differ from the PP chain operating in low-mass stars. It was also possible to generate lithium via:

Because the helium nucleus is ~4 times more massive than a proton, the mass fraction of helium-4 in the universe should have been about:

with an observed value of ~0.23-0.24.

Calculations:

Predicted versus Observed element abundances for a Big Bang Model

Origin of the CMBThe CMB originated only when the time scale for the average scattering of photons by electrons reached the magnitude of the Hubble time, i.e.:

Only then did the photons become decoupled from electrons = time of decoupling. At that point it becomes possible to identify a “surface” of last scattering. The formation of neutral atoms at the moment of recombination can be described by the Saha equation:

For a composition of pure hydrogen, ZI = 2 and ZII = 1, and one can specify the fraction of ionized hydrogen atoms by:

or:

For ionized hydrogen there is one free electron for every proton, so the number density of free electrons is:

where ρb is the density of baryonic matter.

The equation can be rewritten as:

With the Saha equation and the black-body temperature at recombination, we find:

For T0 = 2.725 K and χI = 13.6 eV, the equation is solved with f = 0.5, when the scale factor was R 7.25 10−4

(z 1380), corresponding to T 3760 K.

WMAP generated a value of [zdec]WMAP = 1089 ±1 at decoupling, with a corresponding temperature of Tdec = T0(1 + zdec) = 2970 K, smaller than our estimate of 3760 K. According to WMAP:

Additional parameters obtained by WMAP are:

for the time scale over which decoupling occurred, and:

corresponding to the thickness in redshift space of the surface of last scattering.But keep in mind that many of these conclusions are tied to how the CMB is interpretted.

More later.

The section on Relativistic Cosmology introduces terms such as the Robertson-Walker metric for curved spacetime, which is not covered here.

Friedmann Equation and Cosmological ConstantThe solution of Einstein’s field equations for an isotropic, homogeneous universe leads to a differential equation for the scale factor of:

known as the Friedmann equation from its solution by Friedmann in 1922, derived independently by Lemaître in 1927.When Einstein realized that his field equations could not replicate a static universe, he modified his equations by adding an ad hoc term, the cosmological constant Λ. With that term added, the general differential equation is:

It represents an equation with an additional term as if a potential energy term was added of the form:

The conservation of mechanical energy applied to an expanding shell of mass m becomes:

arising from a new potential of the form:

Such a term is included in solutions that attempt to incorporate the “effects of dark energy.”

A cartoon illustration of the cosmological constant.

In such a situation the scale factor varies with time as:

The temporal development of the universe according to the Big Bang model (logarithmic units).

In the Big Bang model of cosmology, the formation of galaxies like the Milky Way occurred only after several previous stages in which the first stars were formed and

galaxy mergers occurred.

Observational CosmologyLuminosity distance is what is derived for the distance of a cosmological object from the inverse square law of light propagation:

as distinct from the proper distance from redshift:

for z << 1. The surface area of a sphere with the Robertson-Walker metric is:

and the energy of each photon Ephoton = hc/λ is reduced by 1 + z for objects at cosmological redshifts. The radiant flux at the surface of the sphere is therefore:

The luminosity distance is therefore:

Since:

it follows that:

where: for Ώ0 = 1.

so dL(z) approximates to:

for z << 1.

The redshift-magnitude relation comes about from using the luminosity distance with the distance modulus relation and H0 = 100h km/s/Mpc:

As well, the luminosity distance gives:

If the last term is expanded in terms of a Taylor series about z = 0, and higher order terms are neglected, then:

Then:

That is where the results of Francis Farley (2009) and Nielsen et al. (2015, below) are relevant.

The residuals provide no convincing evidence for cosmic acceleration from Type Ia supernovae.

How the “scale factor” changes with time

An example of the concept of inflation in cosmology.

Sample QuestionsSample Questions1. For a one-component universe of pressureless dust show that:

What happens as z increases?

Answer: From:

and:

we find by substitution for H2:

That simplifies to:

as desired.

As z → ∞, we find that:

So Ω → 1.In other words, the very early universe was essentially flat, regardless of its present nature, as noted previously.

2. Calculate the magnitude of the variation in the temperature of the CMB arising from the Sun’s peculiar velocity.Answer. The Sun’s peculiar velocity relative to the Hubble flow is estimated to be 370.6 ±0.4 km/s. From the relation:

and θ = 0º and Trest = 2.725 K one obtains:

which gives:

a value (0.003 K) amounting to only a thousandth of the measured mean temperature of the CMB.

3. The inverse square law for light does not necessarily provide a solution to Olber’s paradox. For example, consider a uniform distribution of stars with n stars per unit volume, each of luminosity L. Consider two, thin, spherical shells of stars of radii r1 and r2 centred on Earth each of thickness Δr. Show that the same energy flux reaches Earth from each shell.

Answer. The number of stars in a thin spherical shell of radius r1 and thickness Δr for n stars per unit volume is:

The radiant flux received from a single star of luminosity L is, according to the inverse square law:

The total flux received from the shell is therefore:

The result is independent of the value of r, so an identical value applies to the total flux received from the shell of radius r2.

Thus, for an infinite number of shells, each would provide the same amount of radiant flux to Earth, thereby making the sky infinitely bright, which is Olber’s paradox.