SI definition of a second: "The duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom." If we give the cosmic time (equal to the universe age equal to the proper time of the observer resting in the CMB reference frame) in seconds, we can easily give it in the number of radiation periods from SI definition of a second.

In the same manner we can define a physical, conformal age of the universe. That's the duration of a certain number of the extending CMB radiation periods proportional to the extending peak wavelength of this radiation that passed through a point at which the CMB is isotropic, since its emission. Proportionality factor is the speed of light, because c=λ/T where λ is the extending peak wavelength, and T is the extending wave period.

Conformal time η=∫dη=∫dt/a(t)=47Gy is the conformal age of the universe and I don't question it. I'm proposing a physical definition for it. The inverse of the scale factor 1/a(t) is increasing with time counted backwards, because 0<a(t)≤1 and a(t₀)=1, where t₀ is the present, proper age of the universe. That makes dt/a(t) the equivalent of the wave period extending over time counted backwards. We're integrating over it to sum it up.

Is there something wrong with the proposed, physical definition?

Edited December 13Dec 13 by Jacek

53 minutes ago, Jacek said:

SI definition of a second: "The duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom." If we give the cosmic time (equal to the universe age equal to the proper time of the observer resting in the CMB reference frame) in seconds, we can easily give it in the number of radiation periods from SI definition of a second.

In the same manner we can define a physical, conformal age of the universe. That's the duration of a certain number of the extending CMB radiation periods proportional to the extending peak wavelength of this radiation that passed through a point at which the CMB is isotropic, since its emission. Proportionality factor is the speed of light, because c=λ/T where λ is the extending peak wavelength, and T is the extending wave period.

Conformal time η=∫dη=∫dt/a(t)=47Gy is the conformal age of the universe and I don't question it. I'm proposing a physical definition for it. The inverse of the scale factor 1/a(t) is increasing with time counted backwards, because 0<a(t)≤1 and a(t₀)=1, where t₀ is the present, proper age of the universe. That makes dt/a(t) the equivalent of the wave period extending over time counted backwards. We're integrating over it to sum it up.

Is there something wrong with the proposed, physical definition?

Surely that depends on which twin you ask...

56 minutes ago, dimreepr said:

Surely that depends on which twin you ask...

Which one are you? :)

2 hours ago, Jacek said:

universe. That's the duration of a certain number of the extending CMB radiation periods proportional to the extending peak wavelength of this radiation

Since this wavelength has changed over time, which one are you choosing?

In what way is this age conformal?

13 minutes ago, swansont said:

Since this wavelength has changed over time, which one are you choosing?

I don't choose a single one. I'm adding their changing periods just like I'm adding a changing wavelengths by integrating over cdt/a(t) to get the observable universe radius.

17 minutes ago, swansont said:

In what way is this age conformal?

Conformal age 47Gy is conformal, because it's proportional to the observable universe radius equal to 47Gly, and the proportionality factor is the speed of light. In the same way the extending radiation periods are all proportional to the extending wavelengths. That makes them conformal.

3 minutes ago, Jacek said:

I don't choose a single one. I'm adding their changing periods just like I'm adding a changing wavelengths by integrating over cdt/a(t) to get the observable universe radius.

I don’t see it in your integral

Just now, swansont said:

I don’t see it in your integral

But I see it. If you extrapolate the CMB peak wavelength backwards to the Big Bang, then it will be infinitesimally short, just like the differential dt. That's why I consider cdt/a(t) as the extending wavelength, and dt/a(t) as the extending period, but with time counted backwards.

2 minutes ago, Jacek said:

But I see it. If you extrapolate the CMB peak wavelength backwards to the Big Bang, then it will be infinitesimally short, just like the differential dt. That's why I consider cdt/a(t) as the extending wavelength, and dt/a(t) as the extending period, but with time counted backwards.

I will rephrase. You don’t show it in your integral. Will you do so?

This is a discussion forum. Being coy seems to be the opposite of discussion.

1 minute ago, swansont said:

I will rephrase. You don’t show it in your integral. Will you do so?

This is a discussion forum. Being coy seems to be the opposite of discussion.

I can't show it in "my" integral, because it's not mine. It's the integral of the FLRW metric equation for null geodesic. I'm just giving you my interpretation of cdt/a(t) as the extending wavelength, and dt/a(t) as the extending period over time counted backwards.

Ps. I've reached the maximum number of posts I can make per day. That's a limitation for a new accounts on this forum. @swansont I will gladly answer you tomorrow.

Edited December 13Dec 13 by Jacek

Just now, Jacek said:

I can't show it in "my" integral, because it's not mine. It's the integral of the FLRW metric equation for null geodesic. I'm just giving you my interpretation of cdt/a(t) as the extending wavelength, and dt/a(t) as the extending period.

Which you should be able to show mathematically.

The next question is the limits on the integral. You imply it starts at t=0 (or it could be something quite small, like 10^-43 s) but the CMB didn’t exist until later. How do you reconcile this?

21 hours ago, swansont said:

The next question is the limits on the integral. You imply it starts at t=0 (or it could be something quite small, like 10^-43 s) but the CMB didn’t exist until later. How do you reconcile this?

In my understanding, the background radiation was already there before the recombination, when the hydrogen atoms were created, and some of it was absorbed by them in the process of their creation. However, the electrons in newly formed atoms were generally in the excited state from which they immediately transitioned to the ground state, emitting the photons that were added back to the background radiation. Is this correct?

If it is, then the idea of extrapolating it back to the Big Bang seems sensible to me the more if the radiation coexisting with the plasma before the recombination was the black body radiation.

If it's not, then I'm left with the purely mathematical extrapolation, but I don't see how it could be incorrect, since the radiation's redshift directly corresponds to the scale factor describing the equal expansion of the universe and the radiation filling it.

21 hours ago, swansont said:

Which you should be able to show mathematically.

What I can show is trivial.

"According to the FLRW metric which is used to model the expanding universe, if at present time we receive light from a distant object with a redshift of z, then the scale factor at the time the object originally emitted that light is a(t)=1/(z+1)."

[imath]1+z=1/a(t)[/imath]

I'm aware that

[imath]dz/dt = -\dot{a}/a^2 = -(1/a)(\dot{a}/a)[/imath]

[imath]\dot{a}/a = H(t) = H(z(t)) = H(z)[/imath]

[imath]dt = -(1+z)^{-1}H(z)^{-1}dz[/imath]

[imath]dt/a(t) = -dz/H(z)[/imath]

But I'm not using it; dz is the differential of dimentionless redshift, and I need dimentional, temporal dt and spatial cdt.

It's enough for me that 1/a(t)=1+z(t), so dt/a(t)=(1+z(t))dt. Now, if we interpret dt as the initial period and cdt as the initial wavelength at - let's say - the Planck time, then z(t)+1 is the redshift+1 equal to the expansion of the wavelength, period and the universe itself.

Ps. What's the deal with latex?

Edited December 14Dec 14 by Jacek

24 minutes ago, Jacek said:

In my understanding, the background radiation was already there before the recombination, when the hydrogen atoms were created, and some of it was absorbed by them in the process of their creation. However, the electrons in newly formed atoms were generally in the excited state from which they immediately transitioned to the ground state, emitting the photons that were added back to the background radiation. Is this correct?

Recombination/ionization is not a thermal effect; the photons emitted would have a thermal variation reflecting the temperature of the (primarily) hydrogen at that time. But the energy levels do not have a thermal distribution. The nominal ionization energy is still 13.6 eV. That would “reset” any thermal distribution you might have beforehand.

6 minutes ago, swansont said:

Recombination/ionization is not a thermal effect; the photons emitted would have a thermal variation reflecting the temperature of the (primarily) hydrogen at that time.

So 3000K was a temperature of hydrogen back then?

Edited December 14Dec 14 by Jacek

8 minutes ago, Jacek said:

So 3000K was a temperature of hydrogen at that time?

Yes