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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.
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.