VII: LECTURE THE EARLY UNIVERSE, FLUCTUATIONS,
AND THE DEVELOPMENT OF STRUCTURE
AST 204 25 Feb 2008
I. The Planck Era and the Notion of Time and Space
First thing today, well discuss the fact that theres an epoch, associated with a density, a
temperature, a mass, and a timescale earlier than which it makes no sense to even discuss
the notions of time and space. This is caleld the Planck time.
The energy of a photon or other highly relativistic ( >> 1) particle is just E = h = hc/.
The Schwarzschild radius associated with a black hole of mass m is rs = 2Gm/c2 in
ordinary units, or, since the energy E associated with a mass m is mc2 , rs = 2GE/c4 .
Thus an energetic particle so energetic that its own self-gravity makes a black hole in which
its wavelength just ts has a wavelength which satises
= rs =
2GE
2Gh
=
,
c4
c3
so this critical wavelength, the Planck length, is
lP = =
2Gh
= 5 1033 cm
c3
The energy, the Planck energy, is
EP =
hc
=
hc5
= 4 1016 erg = 2 1019 GeV
2G
And, of course, an associated temperature, the Planck temperature
TP = EP /k = 3 1032 K.
The associated mass, the Planck mass, is
mP = E/c2 = 4 105 g
There is a density associated with this notion, namely one Planck mass per Planck length
cubed, or
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P = mP /lP = 3 1092 g/cm3
and a time which might be either of two expressionsthe inverse of the frequency associated
with the wavelength lP or the expansion time of the universe associated with P . These
two quantities are the same within a factor of two; the Planck time is usually called
lp /c = 2 1043 sec. The fact that the two notions agree is profound. Can you gure out
why they agree?
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We do not know whether the universe was ever this dense, hot, or rapidly expanding, but
we do know that it does not make sense to discuss conditions even more extreme than this.
In order to study a more extreme condition (smaller length scales, say), one would have to
resolve ner detail, which would involve particles of even shorter wavelength. But Planck
energy particles already cannot propagate because they are within their own event horizon. So the concepts of space and time as a classical arena for events breaks down here, or
probably actually well beforeit makes no sense to talk about the spacetime continuum on
scales shorter than the Planck length and time; the universe is a fundamentally quantum
gravitational entity at this epoch. The notion of even what quantum gravity is is rapidly
evolving at the present time, and the whole eld of fundamental physics is currently doing
a grand experiment of an unprecedented kind today. There is much hope that superstring
theory will somehow produce the Holy Grail of the Theory of Everything, but the motivation now is mathematical beauty rather than any real connection with physics. We will see
(hopefully within your lifetimes, but I suspect not within mine) whether this crusade will
succeed. Perhaps the ination is associated with this epoch, not with the GUT epoch at all
(see Aureliens lectures next week), and with the birthplace of this and possibly/probably
an innity of other universes in a seething timeless quantum gravitational froth, some tiny
uctuation in which spawned the whole thing.
V. A Summary Table, From the Beginning Till Now
We present below a table which summarizes the physical conditions in the universe from
the Planck time to the present, which well use in the next few lectures.
Cosmological Conditions as Functions of Time and Redshift
(sec)
R( )
1+z
T
(K )
kT
(eV )
T
g/cm3
m
g/cm3
R 0 uh
(cm)
rh
(cm)
mh
M
description
4(17)
8(12)
4(11)
10
3(7)
2(11)
2(34)
1(34)
2(36)
1(43)
2/3
1
1500
7000
5(9)
8(12)
1(15)
1(28)
2(45)
5(62)
??
2.7
4000
2(4)
1(10)
1(13)
1(15)
1(28)
??
1(28)
2(32)
2.3(4)
0.3
1.9
1M
1G
100G
1(15)G
??
1(15)G
2(19)G
9(30)
1.2(20)
2.5(18)
5(5)
1.6(18)
4(26)
3(77)
3(77)
3(77)
3(92)
3(30)
1(20)
1(18)
0.2
5(8)
1(15)
1(54)
(106)
2(158)
??
4(28)
1.1(27)
1.7(26)
4(21)
1.4(17)
1.2(5)
1.5(3)
3.5(20)
8(37)
??
4(28)
7(23)
2.4(22)
(11)
1.8(4)
1.2
1.5(25)
1.5(25)
1.5(25)
5(33)
3.8(23)
(18)
3(16)
130
5(12)
3(18)
7(54)
0.1
1(51)
??
present
combination
equal m&r
nuclear reac
quark-gluon
baryogenesis
end ination
mid-ination
beg. ination
Planck era
1/2
eH
??
The numbers in parentheses are powers of 10, so that, for example, 1(43) = 1 1043 .
The columns in the table are mostly self-explanatory, but briey are as follows: is the
2
cosmic time at the epoch in question, Rlaw the relation between the scale factor and .
1+ z the redshift factor, T the temperature in K , kT the typical thermal energy in eVlater
MeV and GeV; T the total mass-energy density in g cm3 .
The column headed m is the density of rest mass Actually, this is a swindle for early
times, because what it is is simply the present rest mass density multiplied by (1 + z )3 ,
and as such is just a tracer for the mass today. Clearly early on at energies where protons
and neutrons do not exist, we cannot easily calculate the rest mass density, and during
ination, if the identities of protons and neutrons were carried along, the densities would
become ridiculous. These entries are noted by the presence of a (*). Note that this simple
bookkeeping calculation assumes that the baryons, were they conserved, continue to carry
one proton mass, which is probably wrongin fact, they may carry no mass at all, because
the mechanism which gives particles mass may do so only at relatively low energies. But
this calculation illustrates graphically that baryon number cannot be conserved in ination.
R0 uh is the comoving size of the particle horizon computed as if there were no ination
referred to the present universe. At and prior to the end of the ination era, we just
follow an incoming light ray which is at that horizon radius at the end of ination, which
illustrates how immense the eective horizon becomes during ination. Note that the
physical size of the horizon remains at the event horizon during this time, but the redshift
factors become so large that the comoving distances become huge. About 55 efolds are
required for the inated particle horizon to encompass the entire observable universe today,
and after about 80, roughly the supposed number, the horizon is 7 orders of magnitude
larger than the present observable universe. Thus any point in the universe at the end of
ination had been able to receive information and send information to a volume which is
now immensely larger than the present observable universe.
The adjacent column, the physical size of the horizon at the relevant period, is just the
comoving one divided by (1 + z ).
The column mh is the rest mass, again referred to the present universe, within the horizon
just the present rest mass density within the comoving radius R0 uh .
We have talked about most of the phenomena identied in the table: The present, recombination, the equality of radiation and mass density, the era of nuclear reactions. Above
a temperature corresponding to something like 1 GeV (though it is somewhat uncertain)
baryons and mesons no longer exist and are replaced by a plasma of free quarks and gluons;
at somewhat higher energies, above about 100 Gev, the electroweak symmetry is restored,
and at something like these energies the processes which lead to baryogenesis probably
freeze out and create the baryon number we see today.
Before this, at least according to our present hazy understanding, not very much interesting happens over a very large range in temperature and expansion until the energies
of the GUT era are reached. There are only the electroweak and strong forces, only the
fundamental leptons, quarks, any decaying heavy remnants of the GUT era, the gauge
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bosons responsible for the forces, including photons, Ws, Zs, and gluons. Then we hit the
GUT era and ination, prior to which there may (or may not) have been a more-or-less
ordinary expansion era from the Planck time.
The numbers in this table are mostly, especially in the columns pertaining to the earliest
times, very uncertain. The assumptions made in calculating the entries are that H0 is
about 70km/s/Mpc, that 0 is unity but is made up currently of m = 0.3 and = 0.7;
that baryogenesis is associated with the decay or interaction of weakly interacting particles
of mass about 100 GeV, that ination is associated with the GUT energy scale at about
1015 GeV, and that the universe undergoes about 80 efolds of ination.
III. Small Perturbations: The Linear Regime
We will see that uctuations which arose in the ination era reappear across the particle
horizon after ination ends; uctuations which enter the horizon before the end of radiation
dominance at z 2500 do not grow appreciably until radiation dominance ends, though
uctuations in the baryonic uid are coupled to the radiation and participate in acoustic
wave motion both before the era of matter domination and after until recombination. (We
will follow common practice and call the time and redshift of the switchover from radiation
to matter dominance eq and zeq , though the epoch has no good name that one does not
stumble over.) We believe, however, that most of the matter is dark matter which does
not couple to the radiation at all, and so uctuations in it can begin to grow immediately
after eq . The initial icker-noise power spectrum is modied on large scales, though it is
essentially preserved in the dark matter on smaller scales. The table above shows that the
horizon at the turnover is on comoving scales of about 2 1026 cm, about 70 Mpc, and
that the horizon encloses a total mass or order 3 1016 solar masses. The largest bound
objects we know in the universe are the great clusters, with masses of a few times 1015 solar
masses; galaxies have masses three orders of magnitude smaller. So we are interested in
small enough regions that we can proceed with purely Newtonian physics. We believe that
the uctuations grow to become the clusters of galaxies and galaxies we see in the universe
today, and in this lecture we will see how they grow and how they can be characterized.
Since perturbations in the matter cannot grow before matter dominance, we can assume as
the initial conditions, which we take at the beginning of matter dominance, the that power
spectrum is whatever emerged from ination, and that the Hubble constant is uniform,
since the radiation, which up until the initial time provided essentially all the energy
density, had time to smooth itself out on the length scales we are considering.
Consider for the sake of argument a tophat perturbation in which the matter density is
dierent from the critical density in the universe by some factor (1 + + ) in some spherical
region of initial radius ri , and is the average density in the universe outside this region.
Now if m is dierent from unity, whether or not the total is dierent from unity, the
mean density is dierent from the critical density as well. Let (1 + e ) be the ratio of the
mean matter density to the critical density c (e stands for external); note that e is
probably negative.
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Now when we were developing the description for the expanding universe, we wrote a
Newtonian equation which was the direct analog of the Friedman equation for the radius
of a shell in an expanding universe, just balancing the kinetic and potential energies:
8Gr2
= 2
(1)
3
We begin, we argue, with the same value of H = r/r for all the shells in our model, inside
the perturbation and out. Let us think about the behavior of a perturbation with the
density just such as to make = 0. It is clear that in this case = c and r = C 2/3 ,
because this is just part of a critical-density universe. If the external density is less than
c , it will expand faster and leave the little tophat alone.
r2
Now consider a density in the tophat greater than the critical density. Then is negative,
and the material in the tophat will reach some maximum radius at some point in the
future, turn around, and collapse again. How does the density excess behave early in the
expansion? We can perturb Equation 1 to follow the small dierence in behavior from the
critical case, which we can solve exactly, but it is messy and we do not need to; we know
the answer already. The cosmological relations you derived in your problem sets say that
at early times
1 0
1 m
(1 + z )1
m (1 + z )
and furthermore that any contribution to the density in the present universe in the form of
a cosmological constant is completely negligible early in the universe. The perturbation is
a little piece of a higher-density universeit does not know about the exterior, remember,
and
c
=
= +
1 m = 1
c
c
The shell in question is not expanding, to be sure, exactly the way the universe is, but
when the perturbation is still small, it is certainly expanding approximately the way the
universe is, so we can neglect for a while the dierence in the factor (1+z) inside and out;
so we get, for no work,
+ (z ) (1 + z )1 R
Of course, the same proportionality applies to e , so whether we refer the density perturbation to the critical density or the mean density, the contrast R. This is a very
important result.
Think again about the energetics. Let us write the energy equation (1) in the following
way for the initial conditions:
8Gc r2
8G( c )r2
= 2.
3
3
Here we have just added and subtracted the term 8Gc r2 /3. But the rst two terms just
cancel (this is the denition of the critical density), and leave us with (dividing by 2):
r2
4G( c )r2
= .
3
5
We can rephrase this in the suggestive way
Gm
=
r
where m is just the excess mass in the perturbation interior to the shell at radius r
compared to the mass that would be contained if the shell contained just the critical
density. Clearly
m = m +
where m is the total mass inside the shell. Then m goes like 2/3 as well, and this can be
thought of protably and interchangeably in two very dierent ways. The excess mass is
growing because the density contrast is growing because the shell is expanding somewhat
less rapidly than the external universe. This is correct and precise. The excess mass is
growing because the density contrast is growing because matter is slowly moving inward
with respect to a shell which is expanding at the same rate as the external universe is
expanding. This is also correct, but expresses a rather dierent point of view.
We have restricted ourselves here to spherical perturbations; clearly one does not need to
restrict oneself to uniform ones. Everything we have said applies to a shell in a general
spherical perturbation, with all the densities and density contrasts replaced with the mean
density inside the shellwhat matters, after all, is just how much mass there is within the
shell. Rather more surprising is that these results are not conned to spherical perturbations while the perturbations are very smallif one has some general density excess eld
(u) = ((u) )/, then
( u) R
This result is true as long as << 1 but is crudely correct as long as < 1i.e. , as
long as the perturbation in the density is not much larger than the density itself. This
approximation is the linear approximation and the realm in which it is valid is called the
linear regime.
IV. Spherical Perturbations in the Nonlinear Regime
Now if we are going to make a galaxy or a cluster or any other kind of bound structure,
we are clearly interested in perturbations with negative . We are interested in structures
with enormous density contrast, and furthermore structures which are bound. Remember
that most of the matter is dark matter, which does not radiate or otherwise interact with
either itself or other matter except gravitationally, which means that it cannot lose energy.
The galaxy is bound now and must have been bound for all of its existence if most of its
mass cannot get rid of any energy.
So consider a shell in a perturbation which has positive + and hence negative . Its energy,
we have seen, is Gm/ri initially, and this is conserved. At some point in its history, it
will reach some maximum radius, and then collapse. We can easily calculate how big it
will get. At its maximum radius, its energy is all potential, and is Gm/rmax . So
Gm/ri = Gm/rmax ,
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and we obtain the exquisitely simple result
rmax
1
m
= +.
=
ri
m
if a shell at the initial time has interior to it a 1 percent density excess, it will expand a
factor of 100 before turning around. If we do a little work, we can also calculate how long
it takes to reach maximum expansion; twice this is the collapse time, the time it takes the
structure it is forming to reach high density, and, well form. We will solve the equation
of motion, which we can write
dr
d
2
1
1
r
rmax
= 2Gm
;
clearly, as we saw above, = Gm/rmax . Then we write
2Gm
3
rmax
1/2
dr/rmax
d =
rmax /r 1
dq
=
1/q 1
qdq
=
,
1q
where we have here let q = r/rmax . If we let q = sin2 (/2), (dont worry about where
the /2 comes fromit is convenient later but is not necessary) (1 q ) is cos(/2) and
dq = sin(/2) cos(/2)d, so
2Gm
3
rmax
1/2
d = sin2 (/2)d
=
1 cos
d.
2
Notice that the term (1 cos )/2 = sin2 (/2) is just r/rmax again. We can integrate this
trivially, and get the parametric solution
1/2
3
rmax
=
8Gm
r
1 cos
.
=
rmax
2
( sin )
For the geometry acionados among you, this is the pair of equations which describe the
development of a cycloid, the locus of a point on a circular hoop of unit radius as it rolls
along the x-axis, starting at x = 0, y = 0, and ending at x = 2 , y = 0; is the angle
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through which the hoop has rolled. In this identication, x is 8Gm/rmax ; y is 2r/rmax .
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Look at the behavior of r/rmax . It is zero for = 0 and 1 for = , so = corresponds
to maximum expansion. Then it is zero again for = 2 , which is fully collapsed. If we
let
3
3
tc = 2 rmax /(8Gm) = rmax /(2Gm)
then the expression for is
=
tc
2
( sin);
when is , we are at maximum expansion, and = tc /2; when is 2 , we have collapsed
and = tc . The quantity tc is the collapse time. With a little manipulation you can also
relate the collapse time to the initial conditions:
tc =
+ 3/2
( )
Hi
where Hi is the Hubble constant at the initial conditions. You will be asked to derive this
in the next problem set.
We have done a lot of manipulation; where has it gotten us? We see that a positive
perturbation is like a little piece of a closed universe, which reaches some maximum radius
and then collapses. If we draw a spacetime diagram, the positive part of the perturbation
separates out from the expanding universe and collapses to form a bound object.
We might well ask what the interface looks likedo we leave a vacuum at the (admittedly
unrealistic) sharp edge of the tophat? The answer is no. Remember that the maximum
radius and collapse time depend on the average density inside a shell. A shell just outside
the tophat feels the same average density as the last shell in the tophat, so it collapses
in the same time. As we go away from the edge of the tophat the average density clearly
tends to the mean density in the universe, and the collapse time gets longer and longer;
if the mean density of the universe is less than the critical density, there is a last bound
shell. Let us see how this goes:
The average density inside a shell is
av =
3m
3,
4ri
but m is just the mass belonging to the mean density in the universe, 4 r3 /3, plus the
excess mass in the tophat, which we can call M :
3
M = 4Ri c ( + e )/3
Here everything refers to conditions at the initial epoch, and capital R and M to the whole
tophat, i.e. at its edge. Then clearly, outside the tophat,
av = +
3M
3,
4ri
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ri > Ri
If we subtract the critical density and divide by it, we can write the much more intuitive
expression:
3
Ri
,
ri > Ri ,
+ (ri ) = e + ( + e )
3
ri
= + ,
ri < Ri .
here + with no argument is the density excess with respect to the critical density inside
the tophat. Thus we see that if e is negative, there will be a radius ri,last ,
ri,last = Ri
e
+
e
1/3
which has energy zero. All shells inside of this eventually fall into the object being formed,
and can represent an amount of mass large compared to the mass in the original tophat,
especially if the original tophat is a large-amplitude perturbation.
In the next lecture we will investigate the energetics of the object being formed, and see
how the present properties of galaxies and clusters connect to the initial conditions.
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