How to Integrate the Boltzmann Equation Over Heavy Momenta in Cosmology?

[alejandrito29]

I need integrate "over heavy momenta" the following equation
\sigma v>=
$$<\sigma v>= \frac{1}{n^{(0)}_1n^{(0}}_2 \int \frac{d^3p_1}{(2Pi)^3 2 E_1} \int \frac{d^3p_2}{(2Pi)^3 2 E_2} \int \frac{d^3p_3 }{(2Pi)^3 2 E_3} \int \frac{d^3p_4}{(2Pi)^3 2 E_4} e^{-(E_1+E_2)/T} \\ \cdot (2 Pi)^4 \delta^3 (p_1+p_2-p_3-p_4) \delta (E_1+E_2-E_3-E_4) M^2$$

where $$n+ \nu \rightarrow p+ e^-$$

I need to find that_

$$n_{\nu}^{(0)}< \sigma v>= \frac{Pi}{4m^2}\int \frac{d^3p_\nu}{(2Pi)^3 2 p_\nu} e^{-p_\nu/T}\int \frac{d^3p_e}{(2Pi)^3 2 p_e} \delta (Q-p_\nu-p_e) M^2$$

where $$Q=m_n-m_p$$

I tried very ways, for example using $$\delta^3 (p_1+p_2+p_3+p_4)= \delta(p_1)\delta(p_2)\delta(p_3)\delta(p_4)$$ but i have very problems with the integration and with the propierties of Dirac Delta...

Help please

 

[clamtrox]

I tried very ways, for example using $$\delta^3 (p_1+p_2+p_3+p_4)= \delta(p_1)\delta(p_2)\delta(p_3)\delta(p_4)$$ but i have very problems with the integration and with the propierties of Dirac Delta...

Help please

You are confusing your notation here. p_n are not different components of a 4-vector, they are 3-vectors corresponding to different particle species.


Now, let's start from the beginning. What are E_p, E_n, E_v and E_e? Replace them in your equation, and you should only have particle momenta left. Then consider integrating.

 

[alejandrito29]

You are confusing your notation here. p_n are not different components of a 4-vector, they are 3-vectors corresponding to different particle species.


Now, let's start from the beginning. What are E_p, E_n, E_v and E_e? Replace them in your equation, and you should only have particle momenta left. Then consider integrating.

i think that if the particle is heavym (proton and neutron)

$$E=\sqrt{m^2+p^2} = m\sqrt{1+\frac{p^2}{m^2}} \approx m$$

$$\delta^3 (p_1+p_2-p_3-p_4) \delta (m_n+E_\nu-m_p-E_e) M^2$$

$$\int \frac{d^3p_e}{(2Pi)^3 2 E_e} e^{-(E_n+E_\nu)/T} \delta^3 (p_n+p_\nu-p_p-p_e) \delta (Q+E_\nu-E_e) M^2$$

but ¿how i use de Dirac Delta? ¿what is the propierty?

 

[DEvens]

Here is the wiki page on the Dirac Delta.

http://en.wikipedia.org/wiki/Dirac_delta_function

 

[alejandrito29]

Here is the wiki page on the Dirac Delta.

http://en.wikipedia.org/wiki/Dirac_delta_function

thanks, I had already read, but in this links there is not a propierty for my problem

 

[clamtrox]

That's good! And also for the relativistic particles, E_e = p_e and same for neutrino.

Now you notice that your integrand does not depend on the heavy momenta at all. So let's look at the standard delta function identity
$$\int d^3 x \delta^3(x-a) = 1$$ if a is in the integral domain. Use this to get rid of one of the integrals (it's really that easy!).

Then you're left with another integral over a heavy momentum, but the integrand still doesn't depend on the integral variable. This integral looks a lot like a phase space number density, so maybe you can find a nice identity for that one somewhere from your material.

 

[alejandrito29]

That's good! And also for the relativistic particles, E_e = p_e and same for neutrino.

Now you notice that your integrand does not depend on the heavy momenta at all. So let's look at the standard delta function identity
$$\int d^3 x \delta^3(x-a) = 1$$ if a is in the integral domain. Use this to get rid of one of the integrals (it's really that easy!).

Then you're left with another integral over a heavy momentum, but the integrand still doesn't depend on the integral variable. This integral looks a lot like a phase space number density, so maybe you can find a nice identity for that one somewhere from your material.

very thanks, but if E_e = p_e then my equation is


$$\int \frac{d^3p_e}{(2Pi)^3 2 p_e} e^{-(E_n+E_\nu)/T} \delta^3 (p_n+p_\nu-p_p-p_e) \delta (Q+p_\nu-p_e) M^2$$

but, the integrand $$\frac{1}{p_e}$$ itself depends on the heavy momentum p_e..

then $$\int \frac{d^3p_e}{(2Pi)^3 2 p_e} \delta^3 (p_n+p_\nu-p_p-p_e) \neq 1$$

¿ how i use $$\int d^3 x \delta^3(x-a) = 1$$ ?

 

[clamtrox]

... but, the integrand $$\frac{1}{p_e}$$ itself depends on the heavy momentum p_e..

Assume that proton and neutron are heavy, p << m, and that electron and neutrino are light, p >> m.

 

[alejandrito29]

thank, but my problem is the integral of last post,

 

[andrien]

I need integrate "over heavy momenta" the following equation
\sigma v>=
$$<\sigma v>= \frac{1}{n^{(0)}_1n^{(0}}_2 \int \frac{d^3p_1}{(2Pi)^3 2 E_1} \int \frac{d^3p_2}{(2Pi)^3 2 E_2} \int \frac{d^3p_3 }{(2Pi)^3 2 E_3} \int \frac{d^3p_4}{(2Pi)^3 2 E_4} e^{-(E_1+E_2)/T} \\ \cdot (2 Pi)^4 \delta^3 (p_1+p_2-p_3-p_4) \delta (E_1+E_2-E_3-E_4) M^2$$

where $$n+ \nu \rightarrow p+ e^-$$

I need to find that_

$$n_{\nu}^{(0)}< \sigma v>= \frac{Pi}{4m^2}\int \frac{d^3p_\nu}{(2Pi)^3 2 p_\nu} e^{-p_\nu/T}\int \frac{d^3p_e}{(2Pi)^3 2 p_e} \delta (Q-p_\nu-p_e) M^2$$

where $$Q=m_n-m_p$$

I tried very ways, for example using $$\delta^3 (p_1+p_2+p_3+p_4)= \delta(p_1)\delta(p_2)\delta(p_3)\delta(p_4)$$ but i have very problems with the integration and with the propierties of Dirac Delta...

First of all, i have problems here regarding the reaction.It should be probably
n→p+e- +v^-(however it does not seem from the delta function you have written).Anyway,you are probably calculating cross section,right?So why are there 4 phase space integrals.There should be only two for your reaction and three for mine but there are four?the way to handle the space δ is to integrate over anyone of the three variable,which will left only two momentum integral(according to me) then you have to move forward to time delta function with relation E=√p²+m².

 

[clamtrox]

thank, but my problem is the integral of last post,

You are not asked to calculate that integral in this problem.