Origin of motion and relative speed of bodies in the universe Charged particles can hit the earth at relativistic speeds. But it
seems that all large bodies have fairly low relative speed. Of course,
speed can increase considerably when a body orbits close to a massive
object, but then it will not travel very far, and it can be discounted
by averaging the speed over a long enough time (maybe a year).
The Earth cruises around the Sun at 30 km/s, the sun cruises at
200 km/s and the Milky Way at 600 km/s. Not that much.
I have two somewhat opposite questions:


*

*What do we know of the relative speeds of massive bodies in the
universe, small or large (not particles)?  I am not sure whether it was
the proper way to state the question. Also, do relative speeds get very
high if we correct out the part due to universe expansion? Are there
braking phenomena?

*On the other hand, what initiated that motion? If the initial soup
had been homogenous, the coalescence of randomly moving particles
should have produced structures essentially at rest (ahem: where is
the energy going?) with respect to each other. There were small variations
in temperature, but how should that create speed differences? Or did
it cause large scale streams that coalesced into moving bodies?
Even if some speed is due to contraction of large scale rotating structures, it does require initial momentum to exist.
To put it together, do we have measured speed statistics in conformance with
universe evolution models? What does it say about speed?
Sorry if the questions are not well stated; that is my best. A
reference to a paper for non-specialists would do too.

Added after 3 weeks, considering the answer by @BenCrowell and the
comment of @ChrisWhite (thank you both).
Please, forget the part about cosmological speed comparisons.
This is my own answer to my question; I leave it as the question, since
it raises other issues.  The first point is: does my answer make
sense? It is only guesswork on my part. I wrote this answer because, though really helpful, Ben Crowell's answer did not really answer the heart of my initial question.
Though I suggested it in my initial question, I realize even more now
that the problem is only the origin of momentum, probably only angular momentum. This was confirmed by
the answer of Ben Crowell regarding the fact that structures were
essentially at rest to start with.
Probably my main error was to think that there might be another "source of
speed and momentum", and my awkward attempt to discount observed very high
speeds as going nowhere because in tight orbit.
I do not see how large scale momentum could come from some form of
accretion of spontaneously emerging angular momentum quanta. I would
think they would balance on average without any visible macroscopic
effect, outside very special and anisotropic places like a black hole
horizon (and even then).  I feel the same regarding linear momentum
quanta (if such a thing exists).
My guess is that high scale momentum arises from momentum exchange
between large collapsing structures. It is well known that celestial
structure, such as planets and satellites, can exchange angular
momentum. Though I no longer try to follow the mathematical analysis,
I also read that much of the exchange can be mediated by tidal
effects. But tidal effect should be even stronger when it is between
structures that are not yet collapsed and are hence very deformable.
Angular Momentum exchange is not necessarily one structure slowing while
another is speeding up. It is vectorial and may be two structures both
speeding up in opposite directions, provided extra energy gets in from
somewhere, such as potential energy from collapse.
So it would be the case that, as they are collapsing, structures are being
deformed under the gravitational influence of other structures, so
that instead of converging to some global center of gravity, some subparts
collapse separately around their own centers of gravity, and
rotate around the main center of gravity.
Is it actually in that way that the originally small variations of
temperature (density?) created the various structures of the universe?
Observed momentum arising from such momentum exchanges, there is no
reason it should have the same orientation in all subparts of a
larger structure, and this is indeed observed in the solar system.
(see Can the axis of rotation of a celestial body point in any arbitrary direction?)
What about other structures in the universe?
The variations in speed (earth in solar system, sun in galaxy, etc.) are obviously just the consequence of variations in mass and size
of the structures, very massive structure permitting higher speed
as observed. The speed formula being $v = \sqrt{\frac{G\cdot M} r}$, and the mass
most likely increasing like the cube of the (initial) structure size.
Hence the "average speed" in a structure should grow more or less like
the initial linear size of the structure (assuming an almost uniform
initial density).  As reported in the initial question above, this is not
quite what is observed.  Where do I err?
Could it be that very large structures collapse less than smaller ones
because of the angular momentum of the subpart?  This could reduce the
influence of the total mass on individual subparts, and increase the
orbital radius, thus reducing the speed.
Is there any way to measure the degree of collapse as a function of
initial size of a structure, and some notion of reference speed of
substructures?
Licence CC BY-SA 3.0 from the author.
 A: The early universe was a hot, high-friction environment in which solid objects couldn't form, and even if one had, it would have been kept at rest relative to the Hubble flow because of friction.
Much, much later, stars, solar systems, and other structures began to form by gravitational collapse. There is a kind of scale-invariance in this collapse. What I mean by that is the following. Take a uniform, spherical cloud of gas and dust of radius $r$. Calculate the gravitational acceleration at the edge of the cloud, and from that find the time it takes for the cloud to collapse by some fixed fraction of $r$, say $r/10$. This time turns out not to depend on $r$. Because of this, all the different levels of the hierarchical structure of the universe formed more or less at the same time -- it wasn't top-down or bottom-up.
In such a collapse, the matter accelerates due to gravity, and by conservation of energy the final speeds depend on the final size of the system like $r^{-1/2}$. Therefore if you want to see very rapidly moving objects, you want to look at things that have collapsed to very small sizes, such as neutron stars or matter falling into or orbiting around black holes. These objects have motion at speeds comparable to the speed of light.
Because of scale-invariance, this argument about $v\propto r^{-1/2}$ is basically not coupled at all to cosmological structure.

Also, do relative speeds get very high if we correct out the part due to universe expansion ?

This is a little subtle, but really we can't correct out this effect -- we can't even define what it is. That is, general relativity doesn't provide any well defined, unique way of describing the motion of one object from another object far away. You may hear people talking about the speed at which cosmologically distant galaxies are receding from us, but that's either (a) sloppy popularizations, or (b) people who have in mind a particular and somewhat arbitrary definition. The arbitrary definition is that you place a chain of rulers stretching from object A to cosmologically distant object B, and let each ruler be at rest relative to the cosmic microwave background. If you do use this definition, there are objects that we can observe and that are receding from us faster than the speed of light (Davis 2004), which should make it clear that this definition doesn't mean anything dynamically.

where is the energy going?

Energy isn't conserved in cosmology: Total energy of the Universe
Davis and Lineweaver, Publications of the Astronomical Society of Australia, 21 (2004) 97, msowww.anu.edu.au/~charley/papers/DavisLineweaver04.pdf
A: "on the other hand, what initiated that motion?"
time? surly given enough time everything would spin as there are more ways to spin than not spin.
i would think there is only 1 way for something not to spin but an infinate amount of directions, angles and speeds at which to spin at(depending on how accurate you can measure).
is it not the same as why wires or string is always tangled up? there is only 1 way for it not to be tangled up and an infinate number of ways it can be.
