B Matter density right after the decoupling

[janeczek]

Once photons decoupled from matter, they traveled freely through the universe without interacting with matter and constitute what is observed today as cosmic microwave background radiation (in that sense, the cosmic background radiation is infrared and some red black-body radiation emitted when the universe was at a temperature of some 3000 K, redshifted by a factor of 1100 from the visible spectrum to the microwave spectrum).

https://en.wikipedia.org/wiki/Recombination_(cosmology)

Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum, and not the medium through which the light propagates with the speed lower than $({\epsilon_0\mu_0})^{-1/2}$? I'm asking this in context of the calculation of the observable universe radius, where the time integral of the inverse of the scale factor is multiplied by the constant speed of light $c$.

 

[PeterDonis]

Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum

The Wikipedia page you linked to gives a formula for the number density of hydrogen as a function of the redshift $z$. From that you can calculate what the matter density would be. What do you get?

 

[janeczek]

Is $Q_H$ the energy of hydrogen ionization?

 

[PeterDonis]

Is $Q_H$ the energy of hydrogen ionization?

Yes. But you don't need to know that to do the calculation I described in post #2.

 

[janeczek]

Right... So the formula for the total density of hydrogen is $(1+z)^3\cdot 1.6\,m^{-3}$.

So for $z=0$ it's 1.6 hydrogen atoms per cubic meter... Right after the decoupling or today? More like today I guess.

So for $z=1100$ it's $1101^3 \cdot 1.6$ hydrogen atoms per cubic meter... That gives ~$2.1\cdot 10^9$ atoms per cubic meter.

 

[Jaime Rudas]

https://en.wikipedia.org/wiki/Recombination_(cosmology)

Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum, and not the medium through which the light propagates with the speed lower than $({\epsilon_0\mu_0})^{-1/2}$? I'm asking this in context of the calculation of the observable universe radius, where the time integral of the inverse of the scale factor is multiplied by the constant speed of light $c$.

I consider the radius of the observable universe to be, by definition, the greatest distance that anything could reach during the age of the universe propagating at speed $c$, that is, without taking into account that, on average, the speed of light in a given non-empty medium may be less than $c$.

 

[PeterDonis]

Right... So the formula for the total density of hydrogen is $(1+z)^3\cdot 1.6\,m^{-3}$.

Yes.

So for $z=0$ it's 1.6 hydrogen atoms per cubic meter... Right after the decoupling or today? More like today I guess.

Yes, $z = 0$ means now.

So for $z=1100$ it's $1101^3 \cdot 1.6$ hydrogen atoms per cubic meter... That gives ~$2.1\cdot 10^9$ atoms per cubic meter.

Yes. How does that compare to, say, the number density of particles per cubic meter in air at the surface of the Earth today?

 

[janeczek]

Number density of air is $2.5\cdot 10^{25}$ molecules per cubic meter... It's larger by 16 orders of magnitude.

Now that makes me wonder how could light be "trapped" in such "low" density plasma before the recombination...

 

[Jaime Rudas]

I consider the radius of the observable universe to be, by definition, the greatest distance that anything could reach during the age of the universe propagating at speed $c$, that is, without taking into account that, on average, the speed of light in a given non-empty medium may be less than $c$.

On the other hand, the margin of error in calculating the age of the universe is on the order of hundreds of millions of years, so what happened in the first 380,000 years is irrelevant for calculating the observable universe radius.

 

[PeterDonis]

Number density of air is $2.5\cdot 10^{25}$ molecules per cubic meter... It's larger by 16 orders of magnitude.

Yes.

Now that makes me wonder how could light be "trapped" in such "low" density plasma before the recombination...

Because plasma doesn't work like air. Even at such extremely low density, it interacts strongly with light.