Bose enhancement in early universe cosmology

["Don't panic!"]

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.

 

[AndyW]

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!