Re: OT: Dark Matter....does it....matter that is?

Friar Broccoli wrote:
> David Ewan Kahana wrote:
> > Friar Broccoli wrote:
>
> [... savage snipping ...]
>

[snip]

>
> So in summary, there are about 30,000,000,000 CBR cosmic
> background photons, and a similar number of low energy
> (probably now non-relativistic) neutrinos for each and
> every electron and baryon (proton and neutron) in the
> universe.
>
> I'll probably round that down a bit in future discussions.
> No point in exaggerating. Say: 10,000,000,000 neutrinos
> (that will almost certainly never be detectable) for each
> particle of ordinary matter. Wonder what the Intelligent
> Designer intended with all that useless material?
>

Better yet ... why did he bother with at least two extra sets of
basically similar particles: muon, tau lepton, and their two
associated neutrinos in the first place. All that's really needed
for life would seem to be the electron, its neutrino and the up
and down quarks.

As I.I Rabi is reported to have put the question:

`So who ordered the muon?'

So far we've got no really good answer to that question.


> >
> > The neutrinos are very weakly interacting, very light, and
> > it turns out they decouple from everything else at a
> > temperature of T ~ 2-3 MeV: that temperature is still very high
>
> The foregoing was moved here from near the beginning of
> your post. I am interested in knowing the approximate
> density associated with 3 MeV. Specifically, is it less
> than or about the same as the density of a neutron star?
>

The baryon density at weak decoupling would have been far
less than that in a neutron star. In fact, it would have
been far less than that of ordinary matter. This is actually
a critical condition in order for the genesis of the light
elements to work in the hot big bang scenario.

I'll describe this in some detail.

Energy densities at the weak decoupling time are very high,
but temperatures approaching the same level are achieved in
the cores of massive stars towards the end of their
lifetimes. Certainly such temperatures are achieved during
the collapse that leads to type II supernovae.

Here are baryon densities and temperatures typical of
materials we know about either directly or indirectly:

(1) Solid Hydrogen: n = 5.3 x 10^22/cm^3, T =
14 K

(2) Liquid Water: n = 3.3 x 10^22/cm^3, T =
300 K

(3) Core of the Sun: n = 9.6 x 10^25/cm^3, T =
1.5 x 10^6 K

(4) Tokamak (TFTR): n = 3.5 x 10^13/cm^3, T =
5.1 x 10^8 K

(5) Core of Supergiant prior to Collapse: n = 1.0 x 10^30/cm^3, T =
1 - 10 x 10^9K

(6) Neutron Stars: n = 30-45 x 10^37/cm^3, T > 10^10 K
(initial temperatures)

nuclear matter density is near 2.7 x 10^14 g/cm^3 (about
0.17 baryons / fm^3). Thus the baryon density in the core of
a neutron star approaches 10-15 times nuclear matter
density.

> The reason I ask is that I really do not believe in the
> infinite density stuff for the early stages of the Big Bag.
> (I like some type of reverse black hole model - and again I
> don't accept the singularities). When I make an argument,
> I like to be comfortable that I believe all its parts.
>
> In this case, if 3 MeV corresponds to a density that
> exceeds the density that ordinary matter can take, I cannot
> be 100% sure that a neutrino decoupling event ever took
> place. Obviously, it is important that I be sure of that
> since I am talking about the existence of something that is
> probably impossible to measure.
>
>

The baryon density at weak decoupling wouldn't have been all
that high; the energy density on the other hand would have
been very high.

I'll go through this in some detail, using only order of
magnitude estimates. This can all be done, and has been,
using infinitely refined pictures and correct rate equations
and all, but the content is basically the same, and it is
already contained in the much simpler argument.

The reasoning that leads to the result is a little bit
complex.

It's easier to work in reverse starting from the time of
nucleosynthesis. The basic assumptions that allow the
abundances of the light elements to be explained in a hot
big bang scenario are:

(I) The temperature of a black body distribution of freely
propagating particles in the early universe scales as 1/a
where a is the expansion parameter. Richard Tolman showed
that a thermal distribution of relativistic particles
remains thermal in a homogenous isotropic expanding universe
if the only influence is the universal expansion, and this
can be regarded as a consequence of general relativistic
cosmology.

(II) In the critical range of temperatures (T ~ 10 MeV - 0.1
MeV) one is concerned with: the expansion of the universe is
radiation dominated. This means that the energy density of
the universe, thus also the time dependence of the universal
expansion parameter a, are dominated by those particle
species whose energies are much larger than their rest
masses, so that they are relativistic. The energy density of
the universe is then proportional to T^4 and to (1/a
da/dt)^2: and one also gets a very simple expression for the
expansion time in terms of temperature -- the time goes as
1/T^2.

(III) Densities at the time of nucleosynthesis are low
enough that two-body nuclear reactions can be assumed to
dominate the rates. These reactions have cross-sections with
energy dependencies which are the same as those directly
measured in terrestrial experiments at low density and
temperature.

Having solved condition (II) for the expansion time, it then
follows that for a conserved number like baryon number, the
density of the particles associated with that number falls
as 1 / a^3.

Given those conditions, Gamow (and Bethe and Alpher)
reasoned:

(1) If the temperature in the early universe was above 2-3
MeV (3 x 10^10 K) then complex nuclei basically could not be
formed out of a sea of neutrons and protons. None could be
formed, because the sea of very hot thermal photons would
have sufficient energy to dissociate large nuclei by
electromagnetic interactions, tearing one proton away at a
time. So nucleosynthesis had to have occurred at lower
temperatures.

(2) Neutron and proton numbers would have been kept near
thermal equilibrium ratios at such high temperatures (10^10
K) by fast two-body reactions like: e- + p -> n + nu. As the
temperature dropped below 1 MeV, such reactions would begin
to go out of equilibrium, and all neutrons would then begin
to decay into protons on a time scale of about 10 minutes,
thus limiting the overall time available for nucleosynthesis
(all lone protons left over at the end of the neutron decays
will become hydrogen).

(3) Above about T = 10^9 K the radiative capture reaction p
+ n -> deuteron + photon would be fast enough to prevent
deuterons from existing more than ephemerally: they would
not have time to collide with anything else before breaking
up. So still, no complex nuclei could possibly form.

(4) Below that temperature, the rate of photodissociation of
deuterons (photon + d -> p + n) drops, and it becomes just
possible for deuterons to begin to build up.

(5) The deuteron reactions are the key initial step for
nucleosynthesis. So the probability for a neutron to undergo
neutron capture in the above scenario had to be high enough
at the initiation of nucleosynthesis that heavier nuclei
were also formed in significant numbers. But the probability
also couldn't be too high, because some deuterons were still
left over.

Condition (II) gives an expansion time, or age of the
universe, at nucleosynthesis of about 100-200 seconds, and
condition (5) requires:

<sigma v> * n * t ~= 1.

Here: n = baryon density, t = expansion time, sigma =
radiative capture cross-section, v = neutron proton relative
velocity, <> = thermal average.

An estimate of the thermal average of the product of the
cross-section for neutron+proton radiative capture to
deuterons multiplied by the relative velocity is not hard to
make, and solving the constraint here leads to a baryon
density of:

n = 10^18/cm^3; for a temperature of T = 10^9K as suggested
by (3) and (4).

To get the weak decoupling density and temperature one then
extrapolates the densities and temperatures backwards to see
when the reaction: (nu) + (anti-nu) <--> (e+) + (e-) would
have been in equilibrium.

A theoretical number for this particular thermal reaction
rate is not hard to estimate.

This reaction can't be measured directly in an accelerator
because neutrinos are so hard to detect. But the reaction is
nevertheless a direct consequence of the existence of
neutral currents and of the modern theory of the weak
interactions. This theory is very well tested, certainly
including the existence of the neutral currents: and
basically, the theory would have to be very wrong if the
cross-section weren't what is theoretically predicted.

The neutrino densities can be directly related to the photon
densities by a consideration of what happens as the universe
passes through the temperature of T = 0.511 MeV (the
electron mass). At temperatures higher than this, photons
are kept in equilibrium with a sea of electron-positron
pairs, through the reaction photon + photon -> (e+) + (e-)
and below this temperature the pairs all annihilate into
photons, transferring all of their entropy to the
photons, whose number and temperature are then directly
related to the number of neutrinos above T=0.511 MeV.
That then fixes the temperature and neutrino density
at weak decoupling (by a similar argument to Gamow's for
the deuteron reaction) as:

n(neutrinos) ~= 30 x 10^31 / cm^3

and

T ~= 3 x 10^10 K

We then see from consideration (I) that the expansion
parameter a would have increased by a factor 30 between
the weak decoupling time and the initial condensation
of deuterons. This leads to a baryon density a factor of
2.7 x 10^4 greater than that at the beginning of
nucleosynthesis, or:

n = 2.7 x 10^22/cm^3, and T = 3 x 10^10 K

So while the baryon density at weak decoupling is comparable
to that of solid hydrogen or liquid water: that is to the
density of quite ordinary matter, the energy densities at
that time are only obtained in systems such as supergiant
stars at the end of their life-cycles and in
supernovae explosions. But we do definitely have
such systems to study, so we can glean a great
deal of knowledge about the conditions indirectly.

What we do know from the theory of supernovae explosions:
what we have to say to make theory consistent with the mere
experimental fact that supernovae do actually happen, though
this isn't direct knowledge by any means, is certainly
generally consistent with the theory that's used for
neutrino cross-sections in big bang models of nucleosynthesis.

If the (nu)+(anti-nu) <--> (e+) + (e-) cross-section
discussed were very different from what theory predicts,
then supernovae would almost certainly not happen. Neutrinos
are the only matter which transports at all inside the
collapsing core of a high mass star.

So on the whole I believe one can regard the weak decoupling
as an all but certain consequence of the hot big bang. One
is certainly not making really wild extrapolations to reach
that conclusion.

If you want to push temperatures into the multi-GeV range
and/or above, and baryon densities to many times nuclear matter
density, things certainly become theoretically less well
understood, and direct tests of theoretical assumptions by
experiments are far harder to come by.

That will change to some extent as the new accelerators come
on line, though there will always be areas where theory has
to come in.



> In closing, thank you for a wonderful and clear post. I
> would also like to thank the folks who asked supplementary
> questions. For example, I wasn't at all sure about those
> non relativistic neutrinos until it was directly addressed.
>

Cheers!

David

.

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