CMBR isotropy, then --> space isotropy, today ?

[Widdekind]

Now (z=0) -- According to D.Maoz' Astrophysics in a Nutshell, our cosmos, today, is isotropically smooth, at large size scales, >100 Mpc.

 

Then (z ~ 1100) -- Judging from the "CMBR Power Spectrum", which shows a peak at an angular size scale of (slightly less than) 1 degree, our cosmos, nearly 14 Gya, was isotropically smooth, at large-for-back-then size scales, >1 degree.

 

QUESTION -- should not these size-scales correspond? Should not "1 degree then" somehow correlate, to "100 Mpc today" ? Naively using online dark-energy-incorporating cosmological calculators (and cf. PF), the proper distance to z~1100 is ~45 Gly, at which absolute distance, ~1 degree represents ~600 Mly (200 Mpc).

[Widdekind]

If our space-time fabric was static (i.e., Minkowskian), then light-rays emitted, towards a 'target', from the two lateral limbs of some astronomical object (e.g., towards earth, from the side-limbs, of some CMBR anisotropy), would converge straight in towards that target, like the straight laser beams in games & movies:

 

However, our space-time fabric is 'stretching'. Thus, when some astronomical object 'lets photons fly', 'loosing light' out into the Hubble Flow, the expansion, of the space-time fabric, between those photons, actually forces those photons apart, laterally-side-to-side, even as they stream towards their mutual distant target. Again, if light, from the extreme opposite edges, of some galaxy, is 'let loose' towards earth, then, even as the emitting galaxy, as a gravitationally self-bound object, 'secedes' from the Hubble Flow, its loosed light will stream out into that Hubble Flow, and start to spread apart, side-to-side. Thus, in illustration, "lasers", shot from the "wing-tips" of an "x-wing fighter" would, if they behaved cosmologically, "fan out" far to the sides of the shooting space-ship, before finally, eventually, reconverging towards their cosmologically-distant target:

 

It is for this reason, that the actual observed angular diameters, of distant objects, seen in earth's skies, are actually much much greater, than they would be, in simple, static (i.e., Minkowskian) space-time:

 

[math]\theta = (1 + z) \, \frac{L}{R(t_0) \, w} \approx (1 + z) \, \frac{L}{D_P(t_0)}[/math]

(Carroll & Ostlie.

Intro. Mod. Astrophys

., p.1274)

In particular, then, CMBR anisotropies, of [math]\approx 1^{\circ}[/math], do not correspond to ~200 Mpc spatial size scales, as I naively assumed; but, instead, given that the Last-Scattering-Surface resides out at a radius of z~1100, those angular size scales, having been magnified by a factor of one thousand, actually only correspond, to physical size scales, of ~200 Kpc. Thus, CMBR anisotropies represent (pre-)galactic-scaled 'clumps' of matter.

 

At even smaller angular, and physical, size scales, Silk damping seemingly smoothed out any primordial anisotropies. Thus, entering into its matter-dominated epoch, our cosmos was 'seeded', essentially exclusively, with galactic-scale clumps -- with little 'power', on larger size scales, due to global isotropy & uniformity; and with little 'power', on smaller size scales, due to said smoothing. The gradual emergence, of the < 100 Mpc anisotropies, seen today, can probably, then, be attributed to Hierarchical Structure Formation, over the past 14 Gyr.

[csmyth3025]

Now (z=0) -- According to D.Maoz' Astrophysics in a Nutshell, our cosmos, today, is isotropically smooth, at large size scales, >100 Mpc.

 

Then (z ~ 1100) -- Judging from the "CMBR Power Spectrum", which shows a peak at an angular size scale of (slightly less than) 1 degree, our cosmos, nearly 14 Gya, was isotropically smooth, at large-for-back-then size scales, >1 degree.

 

QUESTION -- should not these size-scales correspond? Should not "1 degree then" somehow correlate, to "100 Mpc today" ? Naively using online dark-energy-incorporating cosmological calculators (and cf. PF), the proper distance to z~1100 is ~45 Gly, at which absolute distance, ~1 degree represents ~600 Mly (200 Mpc).

Perhaps my math is too simple for this calculation, but if one projects a one degree segment on a circle with a radius of 45 Gly, it seems to me that it would represent:

 

[LaTeX]\left(\frac{2\times pi\times45\,Gly}{360} \right)=\,\sim758.4\,Mly[/LaTeX]

 

-and-

 

[LaTeX]758.4\,Mly\times\left(\frac{1\,Mpc}{3.262\,Mly}\right)=\,\sim240.77\,Mpc[/LaTeX]

 

This figure is only somewhat larger than your calculation quoted above, but I would be interested to know if there's an error in the way I calculated it.

 

As you noted in your post #2, this is the current co-moving arc length of one degree at a distance of 45 Gly. As I understand it, the distance to the source of the CMB photons we're seeing today was originally about 42 million light years away from our spot in the universe at the time of recombination. This would make these arc segments ~224.7 Kpc in length - which is roughly the estimated distance between the Milky Way galaxy and the Leo I dwarf galaxy in our local group.

 

Chris

[Widdekind]

Chris' calculations look quite correct. What supposedly happened was, 14 Gya, back at redshift z~1100, CMBR anisotropies "fired" photons, towards the earth. These photons have been in flight, for 14 Gyr, crossing the cosmos, to our earth-based detectors, 'here-and-now'. They have been 'inbound', on radial trajectories, fixed to the comoving coordinate grid, as defined back at z~1100. And, that coordinate grid has been expanding, with the Hubble Flow, the entire time. Thus, after 14 Gyr of Hubble Expansion, those photons are 'inbound', on sight-lines, now spread laterally, side-to-side, ~1100x wider, in angular diameter, than those CMBR anisotropies would look, at their present "true proper distance" of 45 Gly, in flat static Minkowskian space-time. Again, if you had some sort of 'super-luminal true sight'; and if those CMBR anisotropies persisted to the present epoch; then those ~200 kpc anisotropies, would today span only 1/1100^th of a degree -- to wit, ~3" of arc. Expanding space-time "magnifies", or "lenses", cosmologically distant objects. (This site discusses CMBR angular size scales, seemingly sans the red-shift magnification effect.)

 

Edited July 28, 2011 by Widdekind

[Widdekind]

According to A.Loeb's How Did the First Stars & Galaxies Form ?, the CMB Temperature fluctuations are of order O(10^-5), indicating that density fluctuations were of a similar scale, back at z~1100. Now, our sun; and, our galaxy, both have density fluctuations R_S/R ~ 10^-5. So, theoretically, fully-formed stars, or galaxies, as seen today, could account for the ancient anisotropies, billions of years ago, at the CMB LSS. What, then, is the 'problem' with finding sufficient 'seeds' for structure formation ?