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*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.