I want to calculate the following integral:
$$I_1 = \int_{\eta_0}^0d\eta'\int d^3x'~a^4(\eta')~G(x,x')~\frac{\lambda~\delta^3\left(\vec{x}'\right)}{a^3(\eta')}~,$$
where $$G(x,x') = -\frac{\theta(\eta - \eta')}{2\pi}\left[\frac{\delta\left[(\eta-\eta')^2 - (\vec{x}-\vec{x}')^2\right]}{a(\eta)a(\eta')} + \frac{H^2}{2}\theta(\eta-\eta' - |\vec{x} - \vec{x}'|)\right]$$
If I consider the first term, I can do the following:
$$T_1 = - \frac{1}{2\pi~a(\eta)}\int_{\eta_0}^0d\eta'\theta(\eta - \eta')\int d^3x' ~\delta^3(\vec{x}')\delta\left[(\eta- \eta')^2 - (\vec{x}-\vec{x}')^2\right]~.$$
Now, I can write the one dimensional delta function as $\delta(f(\eta'))$ where
$$f(\eta') = \eta'^2 - 2 \eta~\eta' + \eta^2 - (\vec{x}-\vec{x}')^2.$$ Then, using the property of delta functions $\delta(f(y)) = \sum_i\frac{\delta(y-y_i)}{f'(y_i)}$, I can evaluate this term to be
$$T_1 = -\frac{1}{4\pi~a(\eta)}\int_{\eta_0}^0\theta(\eta - \eta')\left\{\frac{\delta(\eta'-\eta - |\vec{x}|)}{\vec{x} - \vec{x}\cdot \vec{v}} - \frac{\delta(\eta'-\eta + |\vec{x}|)}{\vec{x} + \vec{x}\cdot \vec{v}}\right\}$$
with $\vec{v} = \frac{d\vec{x}}{d\eta'}$. But now, I obtain this product of delta function and the step function which I am not sure how to integrate. The result obtained so far also does not look very similar to what I'm supposed to get which is given in Eqn 19 of this paper. How do I proceed?