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The CMB rest frame is the frame in which we calculate the age of the universe, so it's definitely an interesting idea to think about how our elapsed time might differ. However, for 370 km/s motion, the effect of relativistic time dilation is about one part in a million, and the gravitational time dilation (due to the Milky Way's gravitational potential) is of similar order.

That's correct that the orbits within the extended galactic mass distribution resemble orbits about a point mass in two dimensions (or an infinite line mass in 3D). In both cases the gravitational potential is logarithmic with respect to distance. That's coincidental, and I'm not familiar with any mathematical tricks that take advantage of the correspondence.

The central gravitational force and the centrifugal force cancel exactly. The same principle applies for any orbit/free fall motion, for example our orbit about the sun, or a spacecraft's orbit about the earth.

The Milky Way and Andromeda are moving toward each other at about 110 km/s, according to this article. They're expected to collide in about 4 billion years (keeping in mind that they accelerate as they approach).

The difference is that you are thinking about orbits fully outside the gravitating body (the star, planet, or moon). In contrast, objects in a galaxy are orbiting inside an extended mass distribution. This means more distant objects feel the gravitational influence of more mass below them.

> Actually, I've just looked up the moon's orbital velocity at 1km/s and low earth orbit as 7km/s so that's the complete opposite of what the simulation implies, which definitely requires prograde burns to increase apoapsis.

Both are correct. You have to speed up to get to a higher orbit, and yet that results in you moving slower on average! As I noted in another comment, that is very interesting because it means gravitational systems have a negative heat capacity (adding energy cools them).

The opposite: since we are currently moving (with respect to the CMB frame) slower than the galaxy by about 180 km/s, in half a galactic year we should be faster than the galaxy by about the same, achieving about 730 km/s.

Yes, you have to speed up to get to a higher orbit -- and paradoxically, that still results in you moving slower, on average! This is an extremely interesting feature of gravitational systems; for example, it means they have a negative heat capacity (adding energy cools them).

> The CMB is light that was emitted, but not until about 380k years after the big bang. Before that matter was so dense in the universe that any light that was emitted was reabsorbed by other matter.

Since all photons are identical, there is no way to objectively say whether photons were absorbed/reemitted or scattered. Regardless, last scattering at ~370000 years did not cause any change in the energy density in photons (which continued to drop as a^-4 as usual), which is why for the purpose of this discussion, it is reasonable to ignore it.

> Around 24k years after the big bang, there was more matter than energy in the universe.

Matter is energy, but it's closer to 52k years that the energy density of matter began to exceed the energy density of radiation.

> The CMB doesn't dominate the energy in the universe. It's about 10 orders of magnitude lower than the average matter/energy density of the universe.

No, just 4 orders of magnitude today, as I mentioned.

The temperature variation in the CMB corresponds extremely precisely to what is expected if the CMB is isotropic (the same in all directions) in some frame and we are moving with respect to that frame. If you've seen the usual pictures of CMB temperature variations, the "dipole" temperature variation due to our motion (example) is about 10 times more extreme than those temperature variations, and those nice pictures are only obtained after subtracting it off.

Put another way, we can say purely by analyzing the CMB that there is a reference frame in which it is about the same in all directions, and we are moving at 370 km/s with respect to that frame.

Toward the orange patch in this figure, which is where the CMB is hottest. That's a Mollweide projection of the whole sky, where the Milky Way disk is the noisy horizontal red line. The galactic center is in the middle. Our orbital velocity is toward some point on the left side.

You might have misheard or been misinformed -- the impact of the extended galactic mass distribution (including dark matter) is that the orbital velocity remains approximately uniform over a wide range of radii (see again figure 16 of this review article). The orbital period does not.

Orbital periods are only uniform near the very centers of some galaxies (not ours, and mostly dwarf galaxies). That's actually a challenge to the standard dark matter picture (the core-cusp problem) because it requires that the system's density be uniform in the relevant region, which is not what dark matter simulations predict. But there are lots of proposed solutions to this.

> Hang on, if all motion is relative, why can't I pick an object moving in the direction I would like to travel at 99% of the speed of light, the travel at the speed of light relative to that, thereby traveling at 199% speed of light relative to my starting location.

See relativistic velocity addition

Most of the Milky Way galaxy, including the Sun, orbits at around 200-250 km/s; see e.g. figure 16 of this review article. Note that this implies the galaxy cannot be rotating rigidly. Objects closer to the center have shorter orbital periods.

That's about 1/1400 the speed of light, so the Sun and Earth cover about that fraction of a light year in one year, with respect to the galaxy.

Of course, all motion is relative. Why choose the Milky Way galaxy as a reference? Actually, there is a fairly natural "rest frame" for the local universe, and that's the rest frame of the cosmic microwave background (CMB). An observer in that frame finds the CMB to be equally hot in every direction. We do not, so we infer that the Sun is moving at about 370 km/s (1/800 the speed of light) with respect to the CMB rest frame.

Interestingly, that motion is anti-aligned with our motion about the galaxy, which means the Milky Way itself is moving at about 550 km/s with respect to the CMB. See table 3 of this article for more velocity comparisons; LSR is the local standard of rest, referring to the average motion of nearby stars; GC is the Milky Way galactic center; CMB is the cosmic microwave background; and LG is the local group containing the Milky Way, Andromeda, and many smaller galaxies.

I used Dodelson's Modern Cosmology originally. However:

• There's a recent, very concise review article on the growth of structure.
• For the expansion history, the Wikipedia article on Friedmann equations has most of the relevant mathematics.
• The thermal history is more complicated. I hinted at some of its features when I mentioned 40% neutrinos and that photon domination only begins at a time of a few minutes (that's about when the electron-positron phase transition ends). I frequently refer to this article for details about the thermal history.

The overall energy density of light today is about one ten thousandth (10^(-4)) that of matter or dark energy. Thus, its contribution to the overall spacetime curvature is negligible.

That was not always the case, though. Despite all of the light emitted in galaxies, the cosmic microwave background (CMB) still dominates the energy density in radiation (see for example the first figure in this article). This is not light that was "emitted" per se; rather it is left over from a time when the universe was much hotter and denser. Between a few minutes and roughly 50000 years, this light dominated the energy density of the universe. (In particular, the energy density was about 60% photons and 40% neutrinos.)

The gravitational influence of this radiation (that is, its influence on spacetime curvature) led to a different cosmic expansion history. When radiation dominates, cosmic expansion decelerates more efficiently than when matter dominates (the size scales as time^(1/2) rather than time^(2/3)).

Another major impact of radiation domination is that structure growth is suppressed. Matter can cluster, so for example, any small density excess tends to pull in surrounding matter, becoming even denser. That's what is meant by the growth of structure. However, radiation cannot cluster, so when radiation dominates, structures grow much less efficiently.

> to detect it at all ever, it has to be less homogeneous than, say, the cosmic microwave background

That's not accurate: a uniform distribution exerts a nontrivial gravitational effect (and indeed this is something Newton got wrong, although with careful treatment, Newtonian gravity correctly predicts it).

Ordinary matter loses its kinetic energy through inelastic collisions, which allows it to gravitationally condense into star systems. Our best understanding of dark matter is that it is not able to cool in this way. Consequently, star systems are extremely overdense in ordinary matter but much less overdense in dark matter.

In particular, the local density of dark matter is expected to be about 0.4 GeV/cm^(3), i.e. less than a proton mass per cubic centimeter. The mass of dark matter that is relevant to the earth's orbit is then about 10^-17 times as massive as the sun. So dark matter affects the earth's orbit at the 10^-17 level, which is unfortunately not measurable.

You can make almost anything a black hole if you compress it small enough. If you compressed the earth down to about a centimeter, it would become a black hole. For a 10^20 gram asteroid, the relevant size is under a nanometer.

The first problem is that typical dark matter particles are moving at ~300 km/s with respect to the earth. But even if one particle was very fortunate and fell toward the earth from essentially zero relative velocity, the problem is conservation of energy. The particle would gain speed as it fell, pass through the earth, and then lose the same amount of speed on the way out of the system, escaping earth's influence again.

In principle a particle could be temporarily trapped in the earth's influence via an interaction with the moon, so that it would transfer its energy to the moon. However this still leaves it on an orbit that takes it at least as far as the moon, and it would eventually be ejected by another interaction with the moon. (This sometimes happens with solar system objects.)

It could -- that's the self-interacting dark matter idea -- but not at a level that is important in large galactic halos. In fact, I've seen recent suggestions that in the context of self-interacting dark matter, observations favor a velocity-dependent interaction strength that scales as v^(-4), i.e. decreases very strongly with velocity. This would make interactions irrelevant in clusters and large galaxies (which have very high velocity dispersions) and most relevant toward the centers of the smallest galaxies.

Not necessarily electromagnetic, but we think dark matter should have some nongravitational interaction with ordinary matter in order to be created in the first place. Although, purely gravitational production is also possible...

To be clear, we don't directly measure dark matter velocities; there are several theoretical steps along the chain. See this reply. I'm not sure of a good pedagogical source right now, but here's an academic one (see e.g. section 4).

Yeah, sqrt{ mean[ (v-mean(v))^2 ] }

Good question! It's indirect. For example, we can measure the circular orbit velocities of stars. Dark matter must have the same circular orbit velocity (i.e. velocity of a dark matter particle that is on a circular orbit), because it's subject to the same gravity. That's not yet the velocity dispersion, but it's close.

Next, we have observational evidence (e.g. rotation curves, lensing) that dark matter halos extend much farther than galaxies. This suggests that unlike the ordinary matter, the dark matter cannot efficiently cool -- otherwise it would condense into the galaxies as well. In fact the dark matter halos around large galaxies are consistent with what we find in simulations of dark matter that only interacts gravitationally. This suggests that dark matter is effectively collisionless in this context.

Since dark matter halos form by nearly isotropic collapse and accretion -- dark matter comes in from all directions -- their net angular momentum is small. Thus they should have very little net rotational motion and almost all random motion. This is also what the same simulations tell us.

The specific number "270 km/s" was a quick estimate I made by taking the isothermal sphere model, which is a good approximation for galactic halos over a pretty wide range of radii, and noting that its velocity dispersion is sqrt(3/2) its circular velocity. The local circular velocity is known to be 220 km/s, so that yields 270 km/s.