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1. Integrating Friedmann Equation of Multi-component universe respect to a and t

Try substituting x = 1/a and then use a table of integrals.
2. Mediaeval wormholes

This proves conclusively that wormholes exist.
3. Is the speed of light constant for a co-moving observer?

If you're talking about a "global" comoving frame, then the 1 lyr ruler, which is rigid, will be shrinking by just over 1 mile/day. The photon will always take a year to travel the length of the ruler, even though the ruler will have "shrunk" ~430 miles WRT the global comoving frame during that...
4. Just how dark is Pluto from orbit? How about Saturn?

The OP asked about the relative surface brightness of Earth vs. Saturn vs. Pluto when viewed close. Obviously an object will reduce in brightness if viewed from far away (specifically if the angular size is < 1 minute of arc, it cannot be resolved by the eye, so the farther away it is brought...
5. Problem interpreting Mpc/h in maps of DM

Note that h (\propto H) and a have different time evolution. Therefore a parameter that scales with h does not scale with a, and comoving distance scales with a. I think that the scale in the image should read "44/(1+z) Mpc", noting that 1/(1+z) \propto a.
6. Problem interpreting Mpc/h in maps of DM

Problem interpreting Mpc/h in "maps" of DM In charts of ρ distribution, such as the z = 0 image http://www.mpa-garching.mpg.de/galform/virgo/millennium/seqF_063a_half.jpg (taken from http://www.mpa-garching.mpg.de/galform/virgo/millennium/ ), the distance scale is usually expressed in...
7. Just how dark is Pluto from orbit? How about Saturn?

100 W bulb puts out about 5 lightwatts; zenith sun at earth is about 300 lightwatts/m².
8. Just how dark is Pluto from orbit? How about Saturn?

Pluto direct sun illumination would look like average residential indoor lighting at night, or a typical 100 W tungsten light at 8 feet.
9. Energy in co-moving co-ordinates?

As with most things, one must be careful of definitions. I assume your coordinate system defines v=0 locally as 'at rest with respect to CMB'. However, there are two different ways that lengths (and time intervals) can then be defined. The "normal" way (such as CGS or MKS, for example)...
10. Energy in co-moving co-ordinates?

h has units of (mass)(length2)(time-1). If the time measurement unit is changing, then h must become variable, whether you let l units change in ratio with t units or keep l units constant. Similarly, E has units of ml2t-2, so if you keep c constant by letting l and t units vary in ratio, then E...
11. Energy in co-moving co-ordinates?

No. If you impose a time-dependent time (and/or length) scale, then h is also changing, as is the relative meaning of E.
12. Cosmological constant from first principles

There's a possible problem here: he's saying (I think) that λLF2 is ~1/nμ where n is the # of phase space cells within the Hubble radius (and μ turns out to be ~1.2). However, during the matter era, n \propto ρ-3/4 \propto t3/2, which would make λ variable. This is not allowed in GR. (When I say...
13. Co-moving density should be constant?

In co-moving length units, Planck's constant h would be decreasing, so the photon energy will be decreasing.
14. Understanding expansion

The shrinkage atoms and gravitational orbits, with associated increase in atomic and orbital frequencies (to keep c constant) is a different but usable way to think of expansion. Note that if atoms and gravitational orbits were shrinking (at the same rate), we would not see the shrinking because...
15. Number of Voids in the (observable) Universe

What might be informative here is a plot of observations (or a plot derived from the millennium simulation) which shows the distribution of energy density. The x axis would be space volume, with the least dense on the left ("x"=0) and the most dense on the right ("x"=1 or 100%-- this is a...

For both radiation and matter solutions of ρ(t), with curvature = 0, ρ \propto 1/t2 (although the radiation value of ρ is 9/8 of ρ for a matter-only solution). For radiation, a \propto t1/2, whereas for matter only, a \propto t2/3. For both solutions, the temperature scales as 1/a. Therefore...
17. Big bang and first proton

There is the possibility that there were essentially no protons immediately after the hadron epoch, and that almost all the hadrons were neutrons, of which the majority soon decayed to protons.
18. Characteristic energy units of primordial fluctuations if gaussian

Characteristic energy "units" of primordial fluctuations if gaussian Correct me if wrong, but I think a purely gaussian distribution of the primordial fluctuations could be characterized by a certain unit of energy (which I'll express as mass). If so, then the observed fluctuations are...
19. Is the Milne model compatible with Einstein's field equation?

There may be a simpler way to say this. I think that some boundary conditions must exist such that GR gives a flat space (for all t) and a flat spacetime. This would be an empty space in which da/dt = 0. Normally, Milne has da/dt > 0, I believe.
20. Question on neutron freeze-out

Due to pair production, at around t = 1μ sec, neutrons would have existed (along with antineutrons) in roughly the same number as photons. Then after ~ 10 μ sec, the antineutrons were gone and the neutrons were down to around 1 per billion photons. (There were also proton- antiproton pairs but...
21. Scale factor

a'/a is the fractional expansion per unit time. At present, it's about 2.5x10-18 per second or 1/[14 billion] per year.
22. How the shape of the universe affects density

If I remember right, a deviation of 1 part per billion in density at age 1 second would lead to ~ a factor of 2 now.
23. The phase transition of the universe during inflation

The temperature would have been in the 10 trillion K range (~ a microsecond age) when it was cool enough for protons and neutrons to exist.
24. How to find scale factor at recombination?

athen/anow = 2.725/3000.
25. Scalefactor calculations for special cases using the Friedmann Equation

Also remember that the average energy per blackbody photon is ~2.7kT, which is a little higher than you might guess. Just to give a few clues, with radiation only, you'll find that a \propto t1/2 whereas with matter only, a \propto t2/3. Then \rho \propto k/tn where n is a certain positive...
26. Scalefactor calculations for special cases using the Friedmann Equation

Just remember that the average energy/photon is proportional to 1/a. I'm guessing that you should assume zero curvature, that the density is always at critical, and that the kinetic energy of the matter can be neglected. You'll need to assume some value for the amount of matter per photon (about...
27. Theories for the Universe.

I think that perfect scale invariance of the spectrum is not what is observed and that the degree of deviation from scale invariance is very important. Am I correct in the assertion that if it were perfect scale invariance, then the density deviations over length scales typical of the whole...
28. Light elements abundance in a static toy universe

The meaning of steady state might be a problem. After a very long time, everything might be photons and neutrinos (after black hole evaporation, and after any possible proton decay with the resulting positrons annihilating the existing electrons). I still doubt that the He could ever be over 1%...
29. Light elements abundance in a static toy universe

Then 4He, 3He, and 2H would be much rarer than observed. Most of these isotopes would be inside active stars and would be consumed during the stars' lives. The initial collapse into stars would be different (I'm not sure of the details) because the initial gas would not be composed of a quarter...
30. Light elements abundance in a static toy universe

Are you assuming that one begins with nothing but pure hydrogen, and that any transmutation was stellar in origin ?