Cosmic Recombination occurred when ?

[Widdekind]

According to the Saha equation, applied to a pure hydrogen gas:

 

[math]n \frac{f^2}{1-f} = \mathcal{N}_0 \tau^{\frac{3}{2}} e^{-\frac{1}{\tau}} \equiv \mathcal{N}(\tau)[/math]

where:

 

[math]f \equiv \frac{n_e}{n}[/math]

 

[math]\mathcal{N}_0 \equiv \frac{1}{ \lambda_C^3 } \left( \frac{ 2 \pi \chi }{ m_e c^2 } \right)^{\frac{3}{2}}[/math]

 

[math]\tau \equiv \frac{k_B T}{\chi}[/math]

 

[math]\chi \approx 14 eV[/math]

and where i have attributed ions (protons) two degrees of freedom (g₊ = 2), and neutrals (hydrogens = protons + electrons) four degrees of freedom (g₀ = 4). Ancient Greek scholar Pythagoras gives the ionization fraction:

 

[math]f = \frac{\mathcal{N}(\tau)}{2 n} \left( \sqrt{1 + \frac{4 n}{\mathcal{N}(\tau)}} - 1 \right)[/math]

Cosmically,

 

[math]n \propto (1+z)^3[/math]

 

[math]T \propto (1+z)[/math]

Plugging in the appropriate numbers (perhaps correctly), and plotting the resulting ionization fraction, as a function of redshift, f(z):

 

http://www4a.wolframalpha.com/Calculate/MSP/MSP78261a441cc510d470g50000443f9hf1bad436df?MSPStoreType=image/gif&s=58&w=300&h=189&cdf=RangeControl

plot (Sqrt(1 + 4.5e-24 * z^1.5 * Exp(1/(2.3e-4 * z)) ) - 1)/(2.25e-24 * z^1.5 * Exp(1/(2.3e-4 * z))) from z=1 to 200

shows that (according to these equations) space gas de-ionized over the redshift range z~(100-80). Prof. Abraham Loeb states that "residual electrons [and ions]" existed, in sufficient number density to couple space gas to the CMB, until redshifts z~(200-160). From where arises the "canon" figure of z~1000, for the redshift of Reionization ?

Edited November 24, 2012 by Widdekind

[alpha2cen]

 

Cosmically,

 

[math]n \propto (1+z)^3[/math]

 

[math]T \propto (1+z)[/math]

 

How to drive these relations?

[Widdekind]

The de-ionization of pan-cosmic primordial plasma impacted space sounds, i.e. "Baryon Acoustic Oscillations (BAO)". In the early universe, ionized plasma was supported by radiation pressure from photons. The extra pressure support drastically increased sound speeds (even to trans-luminal levels), and so drastically increased Jeans' wavelengths. After universal expansion diffused radiation, and cooled matter, the de-ionized space gas, oblivious to photons, lost radiation pressure support. Jeans' wavelengths plummeted, from "large" scale wavelengths (~100Mpc = BAO = CMB 1-degree anisotropies), to "small" scale wavelengths (~1kpc = globular star clusters):

 

[math]P = P_R + P_M[/math]

Assuming that matter clumps entrained clumps of photons; and assuming that matter clumps contracted & expanded adiabatically; such that local matter temperature always equaled local radiation temperature, within contracting / expanding clumps; then:

 

[math]P_R = \frac{a T^4}{3}[/math]

 

[math]P_M = P_0 \left( \frac{\rho}{\rho_0} \right)^{5/3}[/math]

 

[math]\frac{T}{T_0} = \left( \frac{\rho}{\rho_0} \right)^{2/3}[/math]

 

[math]P = P_{R,0} \left( \frac{\rho}{\rho_0} \right)^{8/3} + P_{M,0} \left( \frac{\rho}{\rho_0} \right)^{5/3}[/math]

 

[math]C_S^2 \equiv \frac{\partial P}{\partial \rho} |_{\rho=\rho_0} = \frac{1}{\rho_0} \left( \frac{8}{3} P_{R,0} + \frac{5}{3} P_{M,0} \right)[/math]

 

[math] = c^2 \left( \frac{8}{9} \frac{u_R}{u_M} + \frac{5 k_B T}{3 \bar{m} c^2} \right) [/math]

 

[math] \approx c^2 \left( 10^{-4.5} + 10^{-8} \right) \times (1+z)[/math]

(ionized)

[math] \approx c^2 \left( 0 + 10^{-8.5} \right) \times (1+z)[/math]

(neutral)

De-ionization drastically reduced the sound speed, by about two orders of magnitude, so reducing the Jeans' wavelength, by a similar amount (from ~100Mpc to ~1Mpc ?). But, how coupled was matter to energy? The Thompson cross section is so small, that the mean free photon path, through space plasma, is enormous:

 

[math]\lambda_0 = \frac{1}{n_{e,0} \sigma_{T,0}} \approx 10^{27.5} m \approx 10^{11.5} ly \approx 10^{11} pc = 100 Gpc[/math]

 

[math]\lambda \approx 100 Gpc \times \left( 1+z \right)^{-3}[/math]

assuming full ionization. The Jeans wavelength (L) is approximately equal to the sound-speed multiplied by the free-fall time [math]\left( C_S / \sqrt{G \rho} \right) \propto (1+z)^{-1}[/math]. At redshifts z~(1000-100):

 

[math]\lambda \approx 0.1 - 100 Kpc[/math]

[math]L \approx 1 - 10 Kpc[/math]

So, photon mean-free paths, through the ionized space plasma, began to become comparable to the Jeans' wavelengths, by the epoch of de-ionization (z~100). And so, radiation would have ceased supporting space plasma, against gravity collapse, at long wavelengths of space sound. At redshifts z~(100-0):

 

[math]\lambda \rightarrow \infty[/math]

[math]L \rightarrow 0.1 - 10 Kpc[/math]

And so, loss of radiation pressure support, drastically reduced the capacity of space gas to propagate low frequency "bass notes". Space sounds with wavelengths longer than (0.1-10)Kpc began collapsing into clumps, instead of transmitting pressure perturbations (according to these calculations). Such size scales, resemble large star clusters, and small proto-galaxies^[1].

^[1]

Collecting & calculating correct numerical factors, L ~ 20 Kpc / (1+z).

Cp.

http://www.astronomy.ohio-state.edu/~dhw/A825/notes6.pdf.

Edited November 29, 2012 by Widdekind

[Widdekind]

the posted equation for sound speed was wrong:

 

[math]P_M = P_0 \left( \frac{\rho}{\rho_0} \right)^{5/3}[/math]

[math]\frac{T}{T_0} = \left( \frac{\rho}{\rho_0} \right)^{2/3}[/math]

[math]P_R = \frac{a T^4}{3}[/math]

 

[math]\boxed{P = P_{R,0} \left( \frac{\rho}{\rho_0} \right)^{8/3} + P_{M,0} \left( \frac{\rho}{\rho_0} \right)^{5/3}}[/math]

 

 

[math]U_M = \rho c^2 + \frac{3}{2} P_M[/math]

 

[math]\boxed{ U = U_{R,0} \left( \frac{\rho}{\rho_0} \right)^{8/3} + U_{M,0} \left( \frac{\rho}{\rho_0} \right) + \frac{3}{2} P_{M,0} \left( \frac{\rho}{\rho_0} \right)^{5/3} }[/math]

[math]C_S^2 \equiv c^2 \frac{\partial P}{\partial U} |_{U=u_0}[/math]

 

[math] = c^2 \frac{\partial P}{\partial \rho} / \frac{\partial U}{\partial \rho} |_{\rho=\rho_0}[/math]

 

[math]\boxed{ C_S^2 = \frac{c^2}{3} \left( \frac{1 + \frac{5}{8} \frac{P_M}{P_R} }{1 + \frac{3}{8} \frac{U_M}{U_R} + \frac{5}{16} \frac{P_M}{P_R} } \right) }[/math]

 

In the limit of radiation dominance [math]\left( U_R \gg U_M \right)[/math], the sound speed approaches the canonical value [math]C_S^2 \longrightarrow \frac{c^2}{3}[/math]. In the Classical limit of matter dominance [math]\left( \rho c^2 \gg \frac{3}{2} P, U_R \rightarrow 0 \right)[/math], the sound speed approaches the corresponding canonical value [math]C_S^2 \longrightarrow \frac{5}{3} \frac{P}{\rho} [/math].

 

During the epoch of Recombination ("De-ionization"), radiation & matter shared a common temperature [math]T \propto (1+z)[/math]. Thus, the ratio of matter-to-radiation Pressure was independent of redshift:

 

[math]\frac{P_M}{P_R} = \left( \frac{\rho_0 k_B T_0}{\bar{m}} \times (1+z)^4 \right) / \left( \frac{1}{3} a T_0^4 \times (1+z)^4 \right) \approx 3 \times 10^{-8}[/math]

 

Given a baryon-to-photon ratio of 3e-8, the pressure per quanta (baryon, photon) is approximately equal, suggesting some sort of "equipartition of energy".

 

During De-ionization, the ratio of matter-to-radiation Density increased with decreasing redshift:

 

[math]\frac{U_M}{U_R} = \frac{ \rho_0 c^2 \times (1+z)^3 }{a T_0^4 \times (1+z)^4} \approx \frac{22,000}{1+z}[/math]

 

where critical density has been assumed. During De-ionization ( z~(1000-100) ), the sound speed, in the coupled radiation-matter fluid, so calculated, would have varied as:

 

[math]C_S^2 \approx \frac{c^2}{3} \times \frac{8}{3} \frac{1+z}{22,000}[/math]

 

[math]\beta_S \approx \frac{\sqrt{1+z}}{150} \approx 0.2 - 0.07[/math]

 

The Jeans' wavelength would have increased with decreasing redshift:

 

[math]\lambda_J \approx \frac{C_S}{\sqrt{G \rho}} \approx \frac{c/150}{\sqrt{H_0^2 \Omega_0}} \times \frac{1}{1+z} \approx \frac{D_0}{150} \times \frac{1}{1+z} \approx \frac{100 Mpc}{1+z}[/math]

 

which would have been of order ~(0.1-1) Mpc. Including more correct numerical factors, ~(0.2-2) Mpc. At present epoch, such size scales would have stretched to ~200 Mpc. Such size scales, as calculated, accord closely with canonical estimates, of the Jeans' wavelength, at the epoch of De-ionization, and last scattering of CMB photons, accounting for CMB one-degree anisotropies; and for observed "Baryon Accoustic Oscillations" of the large scale structuring of the spatial distribution of galaxies, on ~500 Mly scales. The above corrections seemingly show that this amended analysis is consistent with canonical calculations, in common cosmology textbooks. Indeed, perturbations the size of the Jeans' wavelength, as calculated, would be about one-degree on the sky, (largely) independent of redshift:

 

[math]\theta \approx \frac{\lambda_J}{D_A} \approx \frac{200 Mpc}{1+z} \times \frac{1+z}{2 D_0 \left(1 - (1+z)^{-1/2} \right) } \approx \frac{0.2 Gpc}{2 \times 14 Gpc} \approx \frac{1}{2}^{\circ}[/math]

 

i understand, that perhaps the term "relic Baryon-Photon Accoustic Oscillations" might be more accurate, in that those "large" scale perturbations, corresponding at present epoch to Voids & Super-Clusters, were imprinted before De-ionization, in the coupled matter-energy fluid.

Edited December 11, 2012 by Widdekind