Can the Hubble expansion inject energy in AGN and quasar outflows?

Question

I found this article about the behavior of quasar outflows in cosmology and how they can create a magnetic field.

In section 2.1.4., the authors say that when a quasar produces a "wave" or an outflow, the material will be emitted with energy coming from both the quasar itself and the Hubble flow of the expansion of the Universe itself.

Does this really mean that the expansion of the universe can inject energy into these outflows? Are there any other papers acknowledging this?

Then, is energy conserved in this case? Wouldn't the Hubble expansion be adding "new" energy to the system according to the article?

Answer 1

The paper says

At the onset of shell expansion, there are two energy reservoirs: the input energy of the quasar itself, $E_0$, and the kinetic energy of the preexisting Hubble flow within a distance $R_\text{max}$ of the quasar, $(3/10)M_s v^2$.

The latter is the Newtonian kinetic energy of uniformly distributed, uniformly expanding dust:

$$\int \frac12 v^2 dm = \int_0^{R_\text{max}} \frac12 (Hr)^2 (4πr^2ρ\,dr) = \frac25 π ρ H^2 R_\text{max}^5 = \frac3{10}M_s v^2$$

where $M_s = \frac43 π R_\text{max}^3 ρ$ and $v = H R_\text{max}$.

A fraction $f_d$ of that, plus the energy $E_0$ of the explosion, is supposed to equal the kinetic energy of the final thin-shell state, $\frac12 M_s v^2$.

It's not clear to me that this model makes sense, since the distances involved (~1 Mpc comoving) seem too small for uniform Hubble expansion to be a good approximation even at $z=10$. But in any case, the extra energy in the model comes from the motion of surrounding matter, and not from a cosmological expansion force.