temperatureSo as the universe expanded, the temperature dropped. We know that each particle has an energy associated with it through the equation_{past}/ temperature_{now}= scale factor_{now}/ scale factor_{past}= red shift (z) + 1

energy = mass * cSimilarly, each of the fundamental forces has an energy scale associated with it. By dividing each energy by^{2}

temperaturewe find the temperature at which each force becomes effective, and at which each particle condenses from its constituent parts. For instance, above 10_{ freeze out}= energy / k_{B}

1.673 * 10the constituent quarks in the proton are cool enough to be bound together by the strong force:^{-27}kg * c^{2}/ k_{B}= 10^{13}K

event | temperature (K) | scale factor_{now} / scale factor_{then} | time |
---|---|---|---|

strong forces freeze out | 10^{27} | 3.7 * 10^{26} | 10^{-35} s |

weak forces freeze out | 10^{15} | 3.7 * 10^{14} | 10^{-10} s |

protons, neutrons freeze out | 10^{13} | 3.7 * 10^{12} | 0.0001 s |

neutrinos decouple | 3 * 10^{10} | 1.1 * 10^{10} | 1 s |

electrons freeze out | 6 * 10^{9} | 2.2 * 10^{9} | 100 s |

primordial ^{2}H, ^{4}He form | 9 * 10^{8} | 3.3 * 10^{8} | 2-15 minutes |

When the protons and neutrons froze out, the ratio of protons to neutrons was about 6:1 because of the mass difference between the neutron and proton (the neutron is slightly heavier). The large number of neutrons available, as well as the fact that neutron capture occurs faster than proton fusion, caused theWe now take up our history:nucleosynthesisreactions here to be somewhat different from those taking place in the Sun. Once the universe cooled enough to allow Deuterium (^{2}H) to exist, the ratio was 7:1 (from neutron decays) and the following reactions occurred:This sequence lasted about 15 minutes. The final ratio of

^{1}H + n ->^{2}H + γ^{2}H +^{2}H ->^{3}He + n + γ^{3}He + n ->^{4}He + γ^{4}He to^{1}H was about 1:12, so that the universe was about 75% Hydrogen and 25% Helium by mass. A small amount of Deuterium survived; since it does not survive in stars, what Deuterium we observe isprimordial. This is a sensitive indicator of the density of normal matter (not dark) in the universe, since a denser universe would have contained more protons and produced more Deuterium during nucleosynthesis.Heavier elements were not formed because the temperature and density were both dropping very quickly; in stars they do form because the temperature and density increase (slowly).

event | temperature (K) | scale factor_{now} / scale factor_{then} | time |
---|---|---|---|

photons decouple, atoms form | 3000 | 1091 | 377000 years |

first stars | 60 | 10.4 | 10^{9} years |

today | 2.73 | 1 | 1.378 * 10^{10} years |

The scale factors in the tables above were obtained by using TDecouplingmeans that those particles are no longer in thermal equilibrium with their environment. When neutrinos decoupled, the universe became transparent to them; similarly for photons.

The temperatures in our history indicate that the early universe was filled with intense radiation.
Even before the decoupling of photons, the density of matter became greater than the density of radiation (due to red-shifting, as we shall see).
Wide field surveys indicate that the universe is now very close to being a
**homogeneous** "dust" of galaxy clusters.
The pressure from the
radiation has all but vanished. But the Type IA supernovae surveys we discussed previously indicate that the expansion of the universe is
accelerating. We do not know what causes this acceleration, and for now we simply give it the commonly-used name
**dark energy**. So the universe has passed through three distinct phases:

**radiation-dominated phase**(z > 3250),**matter-dominated phase**(3250 > z > 0.37), and now**dark energy-dominated phase**(z < 0.37):

Log-Log plot of temperature vs scale factor.

Given the wide field surveys and observations of the CMBR, we need the most general metric which is homogeneous and **isotropic**
(the same in all directions).
This is the **Friedmann-Robertson-Walker (FRW) metric**,
and it is described by two parameters. The first is "k", the **curvature constant**; if we
choose a time "t" and take all the points in the spacetime which have the same value of t (called a **spatial section**),

- k = 1 means the section is positively curved:
- cross sections in a fixed direction are circles;
- the universe is closed like a sphere;
- the sum of the interior angles in a triangle is more than 180 degrees;

- k = 0 means the section is not curved; it is
**flat space**:- cross sections in a fixed direction are straight lines;
- the sum of the interior angles in a triangle is 180 degrees;

- k = -1 means the section is negatively curved:
- cross sections in a fixed direction are hyperbolas;
- the neighborhood of every point in the universe is shaped like a saddle;
- the sum of the interior angles in a triangle is less than 180 degrees:

Surfaces of constant curvature, and their triangles.(Strictly speaking, what we have said about k, and the images above, are relevant to 2-dimensional surfaces. You have to use your imagination to see how all this applies to 3 dimensions, but with a little practice it can be done.)

For nonpositive k, the universe can be either finite (although obviously*very*large!) or infinite. A finite universe need not have an edge: a flat**closed universe**(finite, without boundary) could be like a cube, but with identifications: walking "out" one side is the same as walking "in" the opposite side. Similarly, a negatively curved space can have identifications, but these are much harder to visualize. An infinite universe is called an**open universe**(without boundary). If k is positive, the universe cannot be open.

The other parameter is a function "a(t)"; this is the scale factor we mentioned earlier.
It measures the "size" of the universe as a function of time. The **Big Bang**
occurred when a(t) was zero, and the expansion of the universe means that a(t) increases as a function of time. The
Hubble Parameter is defined as the rate of change of a(t) divided by a(t). The rate of change of a(t) is
denoted a'(t), and it must be positive as long as the universe is expanding:

H(t) = a'(t) / a(t)Our matter/energy expression must be able to describe our three act history of the universe: radiation-dominated, matter-dominated and dark energy-dominated. This can be done using a

p = w ρwhere "w" is a constant:

- if w=0, the equation describes dust;
- if w=1/3 it describes radiation, and
- if w=-1 it describes the Cosmological Constant.

If the dark energy is not due to a Cosmological Constant,
it is usually given the name **quintessence**, and it has a different equation of state. Current data is consistent
with a Cosmological Constant, and we will assume that simple scenario in the following.

With these parameterizations, Einstein's Equations reduce to two simple equations:

a'and^{2}(t) (ρ(t) / ρ_{c}- 1) = k c^{2},

a''(t) = - (4 π G / 3 cwhere we have shown explicitly the dependence of the density and pressure on the time (since density and pressure both depend on volume and therefore on the scale factor, which is a function of time). ρ^{2}) a(t) (3 p(t) + ρ(t))= - (3 w + 1) (4 π G / 3 c^{2}) a(t) ρ(t)

3 Hequivalent to about 5.5 protons per cubic meter.^{2}c^{2}/ 8 π G= 8.2487 * 10^{-10}J / m^{3},

- If ρ = ρ
_{c}, k must equal zero. - If ρ > ρ
_{c}, k must be positive, and - if ρ < ρ
_{c}, k must be negative.

Einstein's Equations are supplemented by a **conservation equation** which guarantees that energy is conserved.
With our simple equation of state, the conservation equation has the solution

ρ(t) is proportional to 1 / aThis tells us something we already knew: for matter, the density drops as the volume increases, and the radiation pressure drops as a^{3(w + 1)}(t).

Note that for the Cosmological Constant, w = -1, so ρ is a constant:It is customary to use the ratio ρ / ρit does not depend on a or t.

a'So if Ω = 1, k must be zero; if Ω < 1, k must equal -1, and if Ω > 1, k must equal 1.^{2}(t) (Ω - 1) = k c^{2}

When the universe cooled so much that there was insufficient energy to ionize Hydrogen atoms, the universe became transparent to photons.
Before then, it consisted of a dense **plasma** (electrically charged fluid) containing electrons, baryons,
and photons. There was also **dark matter**: massive particles (not yet understood as part of the
Standard Model) which do not exchange
photons, and so do not interact electromagnetically. Dark matter may come in two forms: **Cold Dark Matter (CDM)**
or **Hot Dark Matter (HDM)**; HDM travels at speeds
close to that of light, while CDM is non-relativistic (speeds ≤ 50 km/s).
There was also dark energy, but its influence appears to have been negligible
in the early universe.

- Since the baryons are roughly 2000 times more massive than electrons, they were the fundamental source of
**inertia**(resistance to acceleration). - Baryons and dark matter were the fundamental sources of gravitational attraction.
- Photons were the fundamental source of pressure, which was therefore experienced by the electrons and baryons but not the dark matter.
- Neutrinos contributed to the radiation energy density because they are nearly massless and move at relativistic speeds, but because they are electrically neutral they did not contribute to the pressure: they simply passed through the plasma unimpeded.

Because of their differing physical effects, the contributions of radiation, baryons, HDM, CDM and the dark energy to Ω are distinguished by subscripts:

Ω = Ωwhere "vac" denotes the dark energy contribution. Sometimes "Λ" is used instead of "vac" when we are particularly interested in the cosmological constant. In addition, we sometimes write_{rad}+ Ω_{b}+ Ω_{HDM}+ Ω_{CDM}+ Ω_{vac}

Ωto denote the contribution of matter to Ω._{M}= Ω_{b}+ Ω_{HDM}+ Ω_{CDM}

Solutions of the Friedmann Equation for radiation (in red), dust (in green) and Cosmological Constant (in blue), for each value of k.

The graph on the right shows the solutions for a Cosmological Constant; these solutions are exponential and must be fitted to the appropriate solution from the graph on the left at the appropriate time. From these graphs we see that for smaller values of k, the universe expands more rapidly, and that radiation tends to make the universe expand more rapidly than dust.

The universe of course has all three components: matter, radiation and Cosmological Constant. When all three are present, it is quite difficult to solve the Friedmann Equation analytically, but it can be solved numerically. In order to do so, we must have values for the various Ωs and k.

While our equation of state above implies that we can have either matter or radiation, but not both, the solutions to the conservation equation are valid even when both are present. Equating the energy densities as functions of the scale factor allows us to find the red shift at the transition from the radiation-dominated era to the matter-dominated era:

ρUsing Ω_{radiation}≡ ρ_{matter}Ω

_{rad}/ (a_{rad→M})^{4}= Ω_{M}/ (a_{rad→M})^{3}a

_{rad→M}= Ω_{rad}/ Ω_{M}z

_{rad→M}= Ω_{M}/ Ω_{rad}- 1

The energy density from photons alone can be found from the black body distribution asWe can perform a similar calculation, using ρρ_{γ}= 8 π^{5}(k_{B}* T_{CMBR})^{4}/ (15 (h c)^{3})= 5.06 * 10^{-5}ρ_{c}

Portfolio Exercise:Verify the computations of Ω_{rad}and ρ_{γ}, and compute Ω_{&Lambda}. For Ω_{&Lambda}, use z_{M→Λ}= 0.374397.

For our numerical solution, we have assumed values for H_{0}, z_{rad→M}, Ω_{M},
Ω_{Λ} and k consistent with the
WMAP7+BAO+SNSALT data set for the ΛCDM model.
The numerical solution for the scale factor is shown below in blue; the left plot shows the past, and the right plot shows the future:

The scale factor up to the present (left) and in the future (right).

Up to the present day, the numerical scale factor is approximately

a(left, in red), but in the distant future it increases exponentially:_{past}(t) = (t / t_{ now})^{2/3}

a(right, in red). a_{future}(t) = 4/3 e^{c t √ (Λ / 3)}

There's a caveat here: for the past, we are using the solution for the matter-dominated universe, and in the early stages of its evolution, the universe was dominated by radiation. The scale factor was essentially linear in t during that era, but we can't match the solutions easily because we have no distance information from those early times. So we will pretend this value is accurate, but remember that this is only an approximation.Recalling our discussion of the Hubble relation, we know that the expansion speed of an object located at a distance r(t) is

vsince r(t) = r(t_{exp}= H(t) * r(t)= a'(t) * r(t) / a(t)= a'(t) * r(t

_{now}),

H(a) = 2 / (3 t_{now}a^{3/2})

The Hubble parameter / H0, from past to present as a function of scale factor.

But using a(t) = a_{future}(t) implies that in the future, the Hubble parameter approaches a constant: c √ (Λ/3),
or approximately 11.85 km/s / Mpc.

We have seen images of proto-galaxies at z = 1 and beyond. Let us assume that an extraterrestrial named Bob lives on
an early-developed planet in one of those protogalaxies (at z=1), and is not moving with respect to it.
Using a_{past}(t), it is possible to compute the current distance to Bob,
even though the light he emits now will not be seen by us for a long time. Our equation for v_{exp} is actually a differential equation:

dr(t)/dt = a'(t)/a(t) * r(t)If we consider a light ray traveling toward us from the past, this equation describes the path of the ray as affected by the expansion of the universe:

dr(t)/dt = a'The expansion of the universe essentially modifies the speed of light, which is only guaranteed to be constant in_{past}(t)/a_{past}(t) * r(t) - c

rwhere b_{past}(t) = b_{past}* t^{2/3}- 3 c t,

Since Bob was at z = 1 when he emitted light that we are just seeing now at z = 0 (a = 1), we can find the time since the emission by setting

aand solving for t_{past}(t_{then}) = 1 / (z + 1)

With Bob's current distance from us, we can now compute Bob's distance from us at any time, and his recession velocity at any time as well. Here are plots of his recession velocity (with c = 1), in the past (on the left) and in the future (on the right). The horizontal scale is the scale factor, and the red line corresponds to Bob, while the blue line corresponds to an object midway between us:

Expansion speeds of Bob (red) and an object between us (blue), vs scale factor (c = 1).

We can see that Bob's recession velocity is *greater than the speed of light* for a < 0.26 (in the past) and a > 6.8 (in the future). We also
see that the recession velocity of objects closer than Bob dropped below the speed of light earlier than Bob's did, and will become greater
than the speed of light later than Bob's will. The opposite is true of objects further from us than Bob; *the length of time an object's recession
velocity is below the speed of light varies inversely with its distance from us*.

None of this contradicts what we have said previously about nothing being able to move faster than the speed of light. Bob isIf we plot Bob's position at various times in his history (he is aat rest: his velocity relative to his neighbors is zero. The distance between events is increasing faster than the speed of light because the universe is expanding there at that rate.

r(t) for Bob; the black line is the edge of the light cone (past on the left, future on the right).

Once we have r_{now}, we can find the path of a light ray from Bob at any time by setting r_{past}(t_{then}) equal to
r_{now}*a_{then},
and solving for a new b_{past}. There will be a new b_{past} for every light ray emitted by Bob at a different time.

If we consider a light ray traveling toward us from Bob at some time in the future, the solution is

rwhere b_{future}(t) = b_{future}* e^{c t √ (Λ / 3)}+ √ (3 / &Lambda),

These computations help us to realize what it means to study GeneralBob's light cones are defined by a "left edge" and a "right edge". The left edge is just rRelativity!

dr(t)/dt = a'with new solutions_{past}(t)/a_{past}(t) * r(t) + c,

rand_{past}(t) = b_{past}* t^{2/3}+ 3 c t

rNote that these equations have different values for b_{future}(t) = b_{future}* e^{c t √ (Λ / 3)}- √ (3 / &Lambda).

We can solve these equations for the light rays passing through any event on Bob's world line, and so reconstruct his light cones at several interesting times:

Notice first that these light cones are *not* at 45 degree angles: they are warped by the expansion of the universe, whose expansion
speeds *depend on position*. The red dots outline Bob's world line, and the ends of the axes correspond to a = 1/101, a = 1/11, a = 1 and a = 10
(except for the last plot). The horizontal black line on the future plots (a > 0) represents the present.

As we discussed previously, the blue area below an event on a world line is the past light cone
of that event, and the blue area above it is its future light cone.
The world line of events taking place at the origin is simply the vertical axis, on the left edge of each frame.
The green regions are where light cones from our world line and Bob's overlap.
A green region below an event indicates a point in our mutual
**causal past**: a prior event there is visible from the two later events.
A green region above an event indicates a point in our
mutual **causal future**: an event at one will eventually be visible there.
The horizontal axis represents the time of the
Big Bang; *at that time, the entire universe is the point at the origin*.

It appears that Bob's events at a = 1/1001 are **causally disconnected** from those at the origin. But we must be cautious
about over-interpreting this graph; the resolution is insufficient to determine whether the light cones overlap at very early times.
However, by a = 1/11 it is clear that events at both locations are **causally connected**: they have a common past.

Just after a = 1/4, the incoming light ray from Bob tilts to the left of vertical: Bob's recession speed has dropped below that of light. Because we see Bob currently at a = 1/2, that plot shows the incoming ray arriving at the origin at the top of the frame: the present. Just before a = 7, the incoming light ray from Bob tilts back to the right of vertical as Bob is again receeding from us faster than light. But by the time the universe is 15 times its current size, Bob's light cone will no longer share a future with ours.

In general, the further away something is, the later we first see it, and the sooner it will move beyond our particle horizon.
We do not know if anything exists outside our current particle horizon, but depending on how large the universe actually is, there may be
portions *that we will never see*. Of course nothing is "visible" until after a = 1/11: before then, there were no stars, and the universe
was filled with an infrared glow that eventually red shifted to become the CMBR.

Putting all this together, we see the universe expanding so rapidly during its early history that more distant regions evolved independently. In the future, the accelerating expansion will move the galactic clusters out of each others' future light cones, and each will eventually grow cold surrounded only by the void of the then much colder CMBR. This of course will not happen until the universe is many times older than it is now.

Perhaps we should take some consolation in realizing that we live at a point in the life of the universe when it is *possible* to understand its
evolution. In the far future, these notions would not ever be testable; much less would it occur to us to test them.

When we solved Einstein's Equations, we usedco-moving coordinates: coordinates whichmove withthe point in question. So in our solution, any point in the universe could just as well be the origin, and the horizontal distance in the light-cone diagrams is simply the separation between two points which are "riding along" with the expansion of spacetime.Note that it is only on the scale of clusters of galaxies that the universe appears homogeneous and isotropic; clearly, the universe looks different to us on Earth than it does if we were, say, on the moon. So when we say that spacetime is expanding, we mean that the distances between galactic clusters are growing with the scale factor, not that the distance between your nose and your eyes is changing. Itmight be, but the Friedmann Equation has nothing to say about that: it simply does not apply to scales smaller than megaparsecs. If the distance between your nose and your eyeswerechanging, and it is, for example, 1 cm now, the current value of the Hubble Parameter tells us that the rate of separation would be1 cm * 69.9 km/s / Mpcor about 1/100,000= 10^{-2}m * 2.265 * 10^{-18}/ s= 2.265 * 10

^{-20}m/s,^{th}of the diameter of an atomic nucleus each second. In your lifetime, the expansion of the universe would change that distance about the radius of an atom. So even if the Friedmann Equationdidimply that the distance between atoms is changing with the expansion of the universe (which it does not!), it would be undetectable on such small scales.

©2014, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

Please send comments or suggestions to the author.