Questions re electroweak de-unification during inflation

[Buzz Bloom]

Before I ask my questions I need some context. Here are two quotes about electroweak unification:
from (1) https://en.wikipedia.org/wiki/Electroweak_interaction ,
and (2) from https://en.wikipedia.org/wiki/Fundamental_interaction .

(1) Mathematically, the unification is accomplished under an SU(2) × U(1) gauge group. The corresponding gauge bosons are the three W bosons of weak isospin from SU(2) (W+, W0, and W−), and the B0 boson of weak hypercharge from U(1), respectively, all of which are massless.

In the Standard Model, the W± and Z0 bosons, and the photon, are produced by the spontaneous symmetry breaking of the electroweak symmetry from SU(2) × U(1)Y to U(1)em, caused by the Higgs mechanism.

(2) Electromagnetism and weak interaction appear to be very different at everyday low energies. They can be modeled using two different theories. However, above unification energy, on the order of 100 GeV, they would merge into a single electroweak force.

Electroweak theory is very important for modern cosmology, particularly on how the universe evolved. This is because shortly after the Big Bang, the temperature was approximately above 10^15 K. Electromagnetic force and weak force were merged into a combined electroweak force.​

I assume that (2) implies that when the average of the mass-energy of particles had an average value of 100 GeV, their temperature was about 10¹⁵ K.
Q1. Is the above assumption correct?

I do not have the skills to calculate this, so I am hopeful someone will supply an answer for me.
Q2. What is an approximate value of the scale factor a (currently a = 1 exactly) and time t (currently t = 13.8 Gy approx) corresponding to 10¹⁵ K?

Since wikipedia gives conflicting information about the timing of inflation and unification, I am making the following guesses:

a) Electroweak unification occurs at a lower temperature than it's Grand Unification with the strong force. Therefore, electroweak de-unification occurred later than de-unification with the strong force.
b) In particular, de-unification with the strong force began when inflation started, and electroweak de-unification ended when inflation ended.
c) The time corresponding to the temperature 10¹⁵ K was when electroweak de-unification started.​

Q3. Are my guesses (a), (b), and (c) likely to be approximately OK?

Let t_s represent the time when electroweak de-unification started. and t_e represent the time when electroweak de-unification ended.

Some questions based on my interpretation of (1)

Q4. The bosons W+ and W- existed since the end of the Planck epoch. Is that correct?

Q5. Before time t_s the Z bosons do not exist, and they do exist after t_e. Is that correct?

Q6. Before time t_s there are no photons. Is that correct? If that is correct, does this mean there is no electrostatic force between charged particles? Or, is this force is carried by a different particle, perhaps B0?

I expect to have some additional questions after getting some answers.

 

[Orodruin]

Q1. Is the above assumption correct?

You can check this yourself, divide 100 GeV with the Boltzmann constant. This will give you the temperature scale. This is not an overly difficult thing to do.

Q2, There are several online applications that will calculate this for you. Alternatively you can just look for a plot of the scale factor vs temperature, there should be plenty of those.

Q3. No. (b) is not correct. The breaking of the interaction symmetries is a priori not related to inflation.Q4-6: The fields that correspond to these particles always exist. The big difference is in how they manifest themselves at different energy scales. Above EW symmetry breaking, you have three SU(2) gague bosons, W1, W2, and W3, and one U(1) gauge boson B coupling to hypercharge. These are all massless and therefore have two degrees of freedom each. In addition you have a hypercharged SU(2) doublet scalar field.

Below EW symmetry breaking, the scalar field takes a vacuum expectation value which breaks the gauge symmetry spontaneously. This results in three of the gauge bosons becoming massive, the W1, W2, and a linear combination of W3 and B. The W1 and W2 can be seen as together forming a single complex vector field rather than two different real ones. The linear combination of the W3 and B which becomes massive is the Z field and there remains an orthogonal linear combination, the photon field A.

In becoming massive, the gauge fields gain a degree of freedom. This degree of freedom is "eaten" from the scalar doublet, which originally had four. The remaining scalar degree of freedom is the one corresponding to the SM Higgs boson.

Therefore, I would not say that Zs and photons did not exist above any scale, the corresponding fields were there but it makes more sense to use a different basis above EW symmetry breaking.

 

[Buzz Bloom]

Hi Orodruin:

I much appreciate your answers to my questions.

You can check this yourself, divide 100 GeV with the Boltzmann constant.

Re Q1: I applogize. I should have thought of that.
When I multiply k = 8.6 ×10^-5 by 10¹⁵ I get 86 GeV. I guess that the quoted value of 100 GeV implies that the rounding to the nearest order of magnitude is a reasonable approximation based what how precise the values of the various parameters/variables involved in this topic can be known.
Q7: Is this a reasonable inference?

Q2, There are several online applications that will calculate this for you. Alternatively you can just look for a plot of the scale factor vs temperature, there should be plenty of those.

I was unable to find an online calculator or a graph covering the time frame and temperatures of interest. I searched on "online calculation of cosmological temperature vs scale factor" and "graph of cosmological temperature vs scale factor".

Assuing a flat universe, I think I can perfrom a caculation of scale vs. temperature if I ignore inflation and the affects due to (a) the formation of distinct protons and neutrons from quarks, (b) particle antiparticle annihilation, (c) nuclear fusion to form He and other nuclei, and (d) the formation of atoms and molecules. I would do this in three steps: (1) starting at t = now with non-zero values for Ω_Λ and Ω_M to a time t' when Ω_Λ is much less than Ω_M; (2) continiuing back in time from t' with non-zero values for Ω_R and Ω_M to a time t" when Ω_R approx. = 1; (3) for t < t", use t ∝ a² and temperature T ∝ 1/a.
Q8: Is this approach ignoring inflation and the four temperature effects I listed likely to to give a reasonable aproximation?

Re Q4-6: I confess that the discussion of particles in terms of gauge fields and group algebras is way over my head.
What I would very much like to undestand is the following.

Q9: What particles (including antiparticles, fermions, and gauge bosons) are present during the electroweak de-uninfication process? Is the Higgs boson also present?

Q10: What interactions among all these particles are possible?

Q11: What are the relative abundances of these particles?

Regards,
Buzz