Bose enhancement in early universe cosmology

In summary, Bose enhancement in early universe cosmology is the phenomenon where the presence of Bose-Einstein condensates can enhance the growth of density perturbations, leading to the formation of galaxies and large-scale structures. It can significantly affect the evolution of the early universe by amplifying the growth of density perturbations and influencing the overall matter distribution. Many different types of particles can form BECs in the early universe, with the most commonly studied being those formed by bosonic dark matter particles. While it can explain some observed features, it is not the only factor in the formation of large-scale structures. Bose enhancement is still a new concept and not fully incorporated into current cosmological models, but has the potential to provide new insights into the
  • #1
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I have been reading through Mukhanov's book "Physical Foundations of Cosmology" and have reached the section where he discusses the process of reheating. In it he mentions that the decay of the inflaton into bosonic states can be "Bose enhanced", i.e. that if ##n## previously created particles (due to inflaton decay) are residing in a given state, the decay rate of the inflaton into this state is enhanced by a factor ##\propto n+1##.

He then considers a simple example ##\phi\rightarrow\chi\chi## and notes that the inverse decay can also happen, ##\chi\chi\rightarrow\phi##. Since the inflaton field forms a condensate of inflatons at zero momentum (at the end of inflation), the momenta of the produced ##\chi##-particles must have equal magnitude ##k##, but opposite direction. The decay rates of these two processes are then proportional to $$\lvert\langle n_{\phi}-1,n_{\mathbf{k}}+1,n_{-\mathbf{k}}+1\rvert a_{\phi}a^{\dagger}_{\mathbf{k}}a^{\dagger}_{-\mathbf{k}}\lvert n_{\phi},n_{\mathbf{k}},n_{-\mathbf{k}}\rangle\rvert^{2}=(n_{\mathbf{k}}+1)(n_{-\mathbf{k}}+1)n_{\phi}$$ and $$\lvert\langle n_{\phi}+1,n_{\mathbf{k}}-1,n_{-\mathbf{k}}-1\rvert a^{\dagger}_{\phi}a_{\mathbf{k}}a_{-\mathbf{k}}\lvert n_{\phi},n_{\mathbf{k}},n_{-\mathbf{k}}\rangle\rvert^{2}=n_{\mathbf{k}}n_{-\mathbf{k}}(n_{\phi}+1)$$ respectively, where ##a_{\pm\mathbf{k}}##, ##a^{\dagger}_{\pm\mathbf{k}}## are the annihilation and creation operators of the ##\chi## particles and ##n_{\pm\mathbf{k}}## are their occupation numbers.

So far, so good. What comes next confuses me, however...

He then says that, taking into account that ##n_{\pm\mathbf{k}}=n_{k}## and that ##n_{\phi}>>1##, we infer that the number densities ##n_{\phi}## and ##n_{\chi}## satisfy $$\frac{1}{a^{3}}\frac{d(a^{3}n_{\phi})}{dt}=-\Gamma_{eff}n_{\phi}\; ;\qquad\frac{1}{a^{3}}\frac{d(a^{3}n_{\chi})}{dt}=2\Gamma_{eff}n_{\phi}$$ where $$\Gamma_{eff}\simeq\Gamma_{\chi}(1+2n_{\chi})$$ Now I don't quite understand how he got this approximate form for the effective decay rate ##\Gamma_{eff}##?!
The only way I can see how to get it is to assume that the effective decay rate of ##\phi## particles into ##\chi## particles is given by $$\Gamma_{eff}=\Gamma(\phi\rightarrow\chi\chi)-\Gamma(\chi\chi\rightarrow\phi)$$ and as such, in the earlier approximation, we have that the effective decay rate is approximately proportional to $$(n_{k}+1)^{2}n_{\phi}-n_{k}^{2}n_{\phi}=1+2n_{k}$$ Then, if ##\Gamma_{\chi}## is the constant of proportionality, we have that $$\Gamma_{eff}=\Gamma(\phi\rightarrow\chi\chi)-\Gamma(\chi\chi\rightarrow\phi)\simeq\Gamma_{\chi}(1+2n_{k})$$ However, I not sure at all if this the correct interpretation?!

Any help on this would be much appreciated.
 
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  • #2

Thank you for bringing this up and for your thorough analysis of the topic. The approximate form of the effective decay rate ##\Gamma_{eff}## that is given in the book is indeed correct.

To understand how it was derived, we need to take a closer look at the equations that you have mentioned. The first equation, $$\frac{1}{a^{3}}\frac{d(a^{3}n_{\phi})}{dt}=-\Gamma_{eff}n_{\phi}\; ,$$ describes the change in the number density of the inflaton particles over time. The term ##a^{3}## takes into account the expansion of the universe, and the negative sign on the right hand side indicates that the number of inflaton particles is decreasing due to their decay.

The second equation, $$\frac{1}{a^{3}}\frac{d(a^{3}n_{\chi})}{dt}=2\Gamma_{eff}n_{\phi},$$ describes the change in the number density of the produced ##\chi## particles over time. The factor of 2 on the right hand side comes from the fact that each decay of an inflaton particle produces two ##\chi## particles.

Now, in order to find the effective decay rate, we need to express the change in the number densities in terms of the decay rate ##\Gamma_{\chi}## and the occupation number ##n_{k}##. This is where the approximation comes in. The occupation number ##n_{\pm\mathbf{k}}## can be approximated by ##n_{k}##, as mentioned in the book. And since ##n_{\phi}>>1##, we can neglect the term ##1## in the expression for ##\Gamma_{eff}## that you have derived.

Hence, we get the approximate form of the effective decay rate as $$\Gamma_{eff}\simeq\Gamma_{\chi}(1+2n_{k}),$$ which is the same as the one given in the book.

I hope this clears up your confusion. Keep up the good work with your studies!
 

Related to Bose enhancement in early universe cosmology

1. What is Bose enhancement in early universe cosmology?

Bose enhancement in early universe cosmology refers to the phenomenon where the presence of Bose-Einstein condensates (BECs) in the early universe can enhance the growth of density perturbations, leading to the formation of galaxies and large-scale structures. This is due to the attractive interaction between particles in a BEC, causing them to cluster together and form denser regions of matter.

2. How does Bose enhancement affect the evolution of the early universe?

Bose enhancement can significantly affect the dynamics of the early universe by amplifying the growth of density perturbations. This can lead to an accelerated collapse of matter and the formation of larger structures such as galaxies and clusters of galaxies. It also influences the overall matter distribution and the formation of dark matter halos.

3. What types of particles can form Bose-Einstein condensates in the early universe?

Many different types of particles can form BECs in the early universe, including photons, neutrinos, and certain types of exotic particles such as axions and dark matter particles. However, the most commonly studied BECs in early universe cosmology are those formed by bosonic dark matter particles.

4. Can Bose enhancement explain the observed large-scale structures in the universe?

Bose enhancement is one of several proposed mechanisms for explaining the formation of large-scale structures in the universe. While it can account for some of the observed features, such as the formation of dark matter halos, it is not the only factor at play. Other processes, such as gravitational collapse and the effects of dark energy, also play a significant role in shaping the structure of the universe.

5. How does Bose enhancement impact current cosmological models?

Bose enhancement is still a relatively new concept in cosmology and is not yet fully incorporated into current models. However, it has the potential to significantly impact our understanding of the early universe and provide new insights into the formation of structures and the nature of dark matter. Further research and observations are needed to fully incorporate Bose enhancement into cosmological models and refine our understanding of its effects.

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