for an illustration). For $q>-1$, this ensures that the “big bang” singularity
at $H_{0}t=-1/(q+1)$ is avoided, where the scale factor vanishes. For $q<-1$,
one has instead a “big rip” singularity at positive $H_{0}t=1/|q-1|$, where
the scale factor diverges.

The Hubble rate, $H(t)=\dot{a}(t)/a(t)$, for the power-law scale factors is

$H(t)=\frac{H_{0}}{a^{q+1}(t)},$ (174)

and, accordingly, $H_{0}\equiv H(0)$ parametrizes the Hubble rate at time
$t=0$. A positive (negative) value of $H_{0}$ corresponds to an expanding
(contracting) universe, while $H_{0}=0$ corresponds to a static universe.

(a)  (b)

Figure 16:  Power-law class of scale factors as a function of time $t$ (see
eq. (173), upper panel), and as a function of conformal time $\eta$ (see eq.
(176), lower panel).

The relation between cosmic and conformal time reads

$\eta-\eta_{0}=\frac{1}{H_{0}q}\left[1+(q+1)H_{0}t\right]^{q/(q+1)},$ (175)

where the integration constants have been fixed such that $t=0$ corresponds to
$\eta=\eta_{0}+1/(qH_{0})$. The limit $q\to 0$ of eq. 175 can be taken with
$\eta_{\text{i}}\to-1/(qH_{0})$ resulting in a logarithmic form of $\eta(t)$.
Then, the scale factor can also be written as a power-law in conformal time
$\eta$,

$a(\eta)=\begin{cases}\left[qH_{0}(\eta-\eta_{0})\right]^{1/q}&q\neq 0\\\
\exp[H_{0}(\eta-\eta_{0})]&q=0.\end{cases}$ (176)

Note that $\eta=\eta_{0}$ corresponds to the big bang or big rip singularities
for $q>0$ and $q<0$, respectively. Consequently, the allowed regime for the
scale factor is $\eta>\eta_{0}$ in the former case and $\eta<\eta_{0}$ in the
latter.

The scattering potentials corresponding to power-law scale factors (176) are
(for $m_{\phi}=\xi=0$),

$\displaystyle
V_{\text{pl}}(\eta)=\begin{cases}\frac{D-1}{2q}\left[\frac{D-1}{2q}-1\right]\frac{1}{(\eta-\eta_{0})^{2}}&\text{for}\quad
q\neq 0,\\\ \left(\frac{D-1}{2}\right)^{2}H_{0}^{2}&\text{for}\quad
q=0.\end{cases}$ (177)

Having the solution for the case of general $D$ and $q$, we can check whether
there are equivalences between different cases on the level of the mode
functions. "Indeed, the scattering potential (177) for de Sitter space in
$D=3$ spatial dimensions, where $q=-1$, and the scattering potential in $D=2$
spatial dimensions for $q=-1/2$ are equal." However, the curvatures differ
substantially in the two cases. For the exponential expansion in $D=3$ one
finds

$\mathcal{R}(\eta)=6H_{0}^{2}\left[2+\kappa(\eta-\eta_{0})^{2}\right],$ (178)

whereas for the quadratic expansion in $D=2$ one has

$\mathcal{R}(\eta)=\frac{H_{0}^{4}}{8}(\eta-\eta_{0})^{2}\left[8+\kappa(\eta-\eta_{0})^{2}\right].$
(179)

Figure 17: Scattering potential landscape of the power-law class as a function
of conformal time for the case of $D+1=3+1$ spacetime dimensions. For the case
$q=0$, $V(\eta)$ is displayed in units of $H_{0}^{2}$.

The mode equation (23) admits exact analytic solutions in case of the power-
law potential landscape (177) (see Ref. [103, 104]). The case $q\neq 0$ yields
Bessel’s differential equation leading to the mode function

$\displaystyle\psi_{k}^{\pm}(\eta)$
$\displaystyle=\sqrt{\pm\eta}\bigg{[}c_{1}(k)H_{\alpha}^{(1)}\left(\pm\sqrt{-h(k)}\eta\right)$
(180) $\displaystyle\hskip
56.9055pt+c_{2}(k)H_{\alpha}^{(2)}\left(\pm\sqrt{-h(k)}\eta\right)\bigg{]},$

with

$\alpha=\frac{1}{2}\left(\frac{D-1}{q}-1\right),$ (181)

where $+$ and $-$ signs correspond to the cases $q>0$ and $q<0$, respectively.

The case $q=0$ yields the mode function solutions

$\displaystyle\psi_{k}(\eta)$
$\displaystyle=c_{1}(k)\text{e}^{H_{0}m\eta}+c_{2}(k)\text{e}^{-H_{0}m\eta},$
(182)

with

$m=\left[\left(\frac{D-1}{2}\right)^{2}+\frac{h(k)}{H_{0}^{2}}\right]^{1/2},$
(183)

where the superhorizon modes with
$-h(k)>\left(\tfrac{D-1}{2}\right)^{2}H_{0}^{2}$ decay and correspond to
tunneling modes of the scattering problem, whereas the subhorizon modes
oscillate.

The conformal time bases in eqs. 180 and 182 can be transformed to the cosmic
time bases chosen in our previous work [16], where specific power-law
expansions in $D=2$ spatial dimensions were investigated.

### Bouncing and anti-bouncing spacetimes

Another interesting class of scale factors are of the form

$a(t)=\sqrt{1+st^{2}}.$ (184)

On one hand, the case $s>0$ corresponds to a bounce with $a(t)\to\sqrt{s}|t|$
at asymptotically times $|t|\to\infty$ (several examples are shown in fig.
18). The turn-over point between contraction and expansion is $t=0$. This
model was investigated in [105] as a quintom bounce scenario.

On the other hand, the case $s<0$ and cosmic time restricted to
$\absolutevalue{t}<1/\sqrt{-s}$ (to avoid cosmological singularities), the
scale factor in eq. (184) describes an anti-bounce of a universe that is
expanding after a big bang until the turnover time $t=0$ and, then, contracts
again towards a big crunch.

Figure 18: (Anti-)Bouncing class of scale factors as a function of cosmic time
$t$. for the case $D+1=3+1$

The Hubble rate for the bouncing and anti-bouncing scale factors is

$H(t)=\frac{st}{1+st^{2}},$ (185)

whereas the deceleration parameter (48)

$q(t)=\frac{1}{st^{2}},$ (186)

indicating that the bouncing scenario ($s>0$) is always accelerating, while
the anti-bouncing ($s<0$) is always decelerating.

Here, the relation between conformal and cosmic time is given by

$\eta-\eta_{0}=\begin{cases}\text{arcsinh}(\sqrt{s}t)/\sqrt{s}&\text{for
bounce},\\\ \arcsin(\sqrt{-s}t)/\sqrt{-s}&\text{for anti-bounce},\end{cases}$
(187)

where we fixed the (anti-)bouncing time $\eta(t=0)=\eta_{0}$. With this, we
find a simple form of the scale factor as a function of conformal time,

$a(\eta)=\begin{cases}\cosh[\sqrt{s}(\eta-\eta_{0})]&\text{ for bounce },\\\
\cos[\sqrt{-s}(\eta-\eta_{0})]&\text{ for anti-bounce.}\end{cases}$ (188)

The corresponding scattering potential for the bouncing and anti-bouncing
cosmologies shown in fig. 19 is given by

$V(\eta)=\frac{D-1}{2}\left[s+\frac{D-3}{2}\mathcal{H}^{2}(\eta)\right],$
(189)

with the conformal Hubble rate

$\mathcal{H}(\eta)=\begin{cases}\sqrt{s}\tanh\left[\sqrt{s}(\eta-\eta_{0})\right]&\text{for
bounce},\\\ -\sqrt{-s}\tan\left[\sqrt{-s}(\eta-\eta_{0})\right]&\text{for
anti-bounce},\end{cases}$ (190)

as shown in fig. 20. Interestingly, analytic solutions of the mode equation
also exist for the (anti-)bouncing case. Here, eq. 23 can be written as an
associated Legendre differential equation with the general solution for the
bouncing case

$\psi_{k}^{\text{b}}(\eta)=c_{1}(k)\,P_{l}^{m}\left(\frac{\mathcal{H}(\eta)}{\sqrt{s}}\right)+c_{2}(k)\,Q_{l}^{m}\left(\frac{\mathcal{H}(\eta)}{\sqrt{s}}\right),$
(191)

with

$\displaystyle l=\frac{D-3}{2}\quad\text{and}\quad
m=\left[\frac{(D-1)^{2}}{4}+\frac{h(k)}{s}\right]^{1/2},$ (192)

and for the anti-bouncing case one finds

$\displaystyle\psi_{k}^{\text{ab}}(\eta)$
$\displaystyle=c_{1}(k)\,P_{l}^{m}\left(\text{i}\frac{\mathcal{H}(\eta)}{\sqrt{-s}}\right)+c_{2}(k)\,Q_{l}^{m}\left(\text{i}\frac{\mathcal{H}(\eta)}{\sqrt{-s}}\right),$
(193)

with

$\displaystyle l=\frac{D-3}{2}\quad\text{and}\quad
m=\left[\frac{(D-1)^{2}}{4}-\frac{h(k)}{s}\right]^{1/2}.$ (194)

Finally, note that for $D=3$, the potential eq. 189 becomes constant, i.e. we
have $V_{\text{b/ab}}=s$, with $s<0$ for the anti-bounce and $s>0$ for the
bounce. On the level of the mode functions, this reveals an interesting
duality between the bouncing scenario (184) in $D=3$ dimensions and the linear
expansion (cf. (173) with $q=0$). However, to admit scattering processes, the
potential landscape (and thus also the analogue cosmology) needs to have
transitions between different epochs. These appear naturally in an analogue
quantum field simulation, as we showed in section III.

Figure 19: Scattering potential of the (anti-)bouncing class as a function of
conformal time for the case of $D=2$ spatial dimensions. Figure 20: The
conformal Hubble rate for the (anti-)bouncing class of scale factors. For the
anti-bouncing class, the Hubble rate falls from positive infinity towards
negative infinity, whereas it is bounded for the bouncing class such that the
argument of the Legendre functions in eq. 191 lies within $(-1,1)$.

### Scattering amplitudes of bouncing and anti-bouncing spacetimes

Let us consider the (anti)-bouncing scale-factors (184) as the dynamical
evolution in region II and compute the scattering amplitudes in region III. We
make an Ansatz for the mode functions for a bouncing potential (189)

$\psi_{k}(\eta)=\begin{cases}c_{k}\text{e}^{-\text{i}\omega_{k}\eta}&\eta<\eta_{\text{i}},\\\\[5.0pt]
f_{k}\,P_{l}^{m}\left(\frac{\mathcal{H}(\eta)}{\sqrt{s}}\right)+g_{k}\,Q_{l}^{m}\left(\frac{\mathcal{H}(\eta)}{\sqrt{s}}\right)&\eta\in[\eta_{\text{i}},\eta_{\text{f}}],\\\\[5.0pt]
a_{k}\text{e}^{-\text{i}\omega_{k}\eta}+b_{k}\text{e}^{\text{i}\omega_{k}\eta}&\eta>\eta_{\text{f}},\end{cases}$
(195)

with $\omega_{k}=\sqrt{E_{k}}$. The boundary conditions at
$\eta=\eta_{\text{i}}$ are

$\displaystyle c_{k}\text{e}^{-\text{i}\omega_{k}\eta_{\text{i}}}$
$\displaystyle=f_{k}\,P_{l}^{m}\left(\frac{\mathcal{H}_{\text{i}}}{\sqrt{s}}\right)+g_{k}\,Q_{l}^{m}\left(\frac{\mathcal{H}_{\text{i}}}{\sqrt{s}}\right),$
(196)
$\displaystyle-\text{i}\omega_{k}c_{k}\text{e}^{-\text{i}\omega_{k}\eta_{\text{i}}}$
$\displaystyle=f_{k}\Bigg{[}(l+1)\mathcal{H}_{\text{i}}P_{l}^{m}\left(\frac{\mathcal{H}_{\text{i}}}{\sqrt{s}}\right)-\sqrt{s}(l-m+1)\,P_{l+1}^{m}\left(\frac{\mathcal{H}_{\text{i}}}{\sqrt{s}}\right)\Bigg{]}$
$\displaystyle\quad\qquad+g_{k}\Bigg{[}(l+1)\mathcal{H}_{\text{i}}Q_{l}^{m}\left(\frac{\mathcal{H}_{\text{i}}}{\sqrt{s}}\right)-\sqrt{s}(l-m+1)\,Q_{l+1}^{m}\left(\frac{\mathcal{H}_{\text{i}}}{\sqrt{s}}\right)\Bigg{]},$

and at $\eta=\eta_{\text{f}}$

$\displaystyle
f_{k}\,P_{l}^{m}\left(\frac{\mathcal{H}_{\text{f}}}{\sqrt{s}}\right)+g_{k}\,Q_{l}^{m}\left(\frac{\mathcal{H}_{\text{f}}}{\sqrt{s}}\right)=a_{k}\text{e}^{-\text{i}\omega_{k}\eta_{\text{f}}}+b_{k}\text{e}^{\text{i}\omega_{k}\eta_{\text{f}}},$
(197) $\displaystyle
f_{k}\Bigg{[}(l+1)\mathcal{H}_{\text{f}}P_{l}^{m}\left(\frac{\mathcal{H}_{\text{f}}}{\sqrt{s}}\right)-\sqrt{s}(l-m+1)\,P_{l+1}^{m}\left(\frac{\mathcal{H}_{\text{f}}}{\sqrt{s}}\right)\Bigg{]}+g_{k}\Bigg{[}(l+1)\mathcal{H}_{\text{f}}Q_{l}^{m}\left(\frac{\mathcal{H}_{\text{f}}}{\sqrt{s}}\right)-\sqrt{s}(l-m+1)\,Q_{l+1}^{m}\left(\frac{\mathcal{H}_{\text{f}}}{\sqrt{s}}\right)\Bigg{]}$
$\displaystyle=-\text{i}\omega_{k}a_{k}\text{e}^{-\text{i}\omega_{k}\eta_{\text{f}}}+\text{i}\omega_{k}b_{k}\text{e}^{\text{i}\omega_{k}\eta_{\text{f}}}.$

Then, we find for the amplitudes

$\begin{split}a_{k}&=\frac{c_{k}e^{\text{i}\omega_{k}(\eta_{\text{f}}-\eta_{\text{i}})}}{2\text{i}\omega_{k}\sqrt{s}}\frac{\Gamma(l-m+1)}{\Gamma(l+m+1)}\\\
&\times\Bigg{\\{}\left[\left(\text{i}\omega_{k}-(l+1)\mathcal{H}_{\text{f}}\right)P_{l}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{f}}}}{\sqrt{s}}\right)+\sqrt{s}(l-m+1)P_{l+1}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{f}}}}{\sqrt{s}}\right)\right]\\\
&\times\left[\left(\text{i}\omega_{k}+(l+1)\mathcal{H}_{\text{i}}\right)P_{l}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{i}}}}{\sqrt{s}}\right)-\sqrt{s}(l-m+1)P_{l+1}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{i}}}}{\sqrt{s}}\right)\right]\\\
&-\left[\left(\text{i}\omega_{k}-(l+1)\mathcal{H}_{\text{f}}\right)Q_{l}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{f}}}}{\sqrt{s}}\right)+\sqrt{s}(l-m+1)Q_{l+1}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{f}}}}{\sqrt{s}}\right)\right]\\\
&\times\left[\left(\text{i}\omega_{k}+(l+1)\mathcal{H}_{\text{i}}\right)Q_{l}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{i}}}}{\sqrt{s}}\right)-\sqrt{s}(l-m+1)Q_{l+1}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{i}}}}{\sqrt{s}}\right)\right]\Bigg{\\}},\\\
b_{k}=&\frac{c_{k}e^{-\text{i}\omega_{k}(\eta_{\text{f}}+\eta_{\text{i}})}}{2\text{i}\omega_{k}\sqrt{s}}\frac{\Gamma(l-m+1)}{\Gamma(l+m+1)}\\\
&\times\Bigg{\\{}\left[\left(\text{i}\omega_{k}+(l+1)\mathcal{H}_{\text{f}}\right)P_{l}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{f}}}}{\sqrt{s}}\right)-\sqrt{s}(l-m+1)P_{l+1}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{f}}}}{\sqrt{s}}\right)\right]\\\
&\times\left[\left(\text{i}\omega_{k}+(l+1)\mathcal{H}_{\text{i}}\right)P_{l}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{i}}}}{\sqrt{s}}\right)-\sqrt{s}(l-m+1)P_{l+1}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{i}}}}{\sqrt{s}}\right)\right]\\\
&-\left[\left(\text{i}\omega_{k}+(l+1)\mathcal{H}_{\text{f}}\right)Q_{l}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{f}}}}{\sqrt{s}}\right)-\sqrt{s}(l-m+1)Q_{l+1}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{f}}}}{\sqrt{s}}\right)\right]\\\
&\times\left[\left(\text{i}\omega_{k}+(l+1)\mathcal{H}_{\text{i}}\right)Q_{l}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{i}}}}{\sqrt{s}}\right)-\sqrt{s}(l-m+1)Q_{l+1}^{m}\left(\frac{\mathcal{\mathcal{H}_{\text{i}}}}{\sqrt{s}}\right)\right]\Bigg{\\}}.\end{split}$
(198)

We make an similar Ansatz for the mode functions for an anti-bouncing
potential (189)

$\psi_{k}(\eta)=\begin{cases}c_{k}\text{e}^{-\text{i}\omega_{k}\eta}&\eta<\eta_{\text{i}},\\\\[5.0pt]
f_{k}\,P_{l}^{m}\left(\text{i}\frac{\mathcal{H}(\eta)}{\sqrt{-s}}\right)+g_{k}\,Q_{l}^{m}\left(\text{i}\frac{\mathcal{H}(\eta)}{\sqrt{-s}}\right)&\eta\in[\eta_{\text{i}},\eta_{\text{f}}],\\\\[5.0pt]
a_{k}\text{e}^{-\text{i}\omega_{k}\eta}+b_{k}\text{e}^{\text{i}\omega_{k}\eta}&\eta_{\text{f}}<\eta,\end{cases}$
(199)

with $\omega_{k}$ and $\mathcal{H}(\eta)$ as above. The boundary conditions at
$\eta=\eta_{\text{i}}$ are

$\displaystyle c_{k}\text{e}^{-\text{i}\omega_{k}\eta_{\text{i}}}$
$\displaystyle=f_{k}\,P_{l}^{m}\left(\frac{\text{i}\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)+g_{k}\,Q_{l}^{m}\left(\frac{\text{i}\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right),$
(200)
$\displaystyle\left(\frac{\mathcal{H}_{\text{i}}}{2}-\text{i}\omega_{k}\right)c_{k}\text{e}^{-\text{i}\omega_{k}\eta_{\text{i}}}$
$\displaystyle=f_{k}\Bigg{[}(l+1)\mathcal{H}_{\text{i}}P_{l}^{m}\left(\frac{\text{i}\mathcal{H}_{\text{i}}}{\sqrt{\sqrt{-s}}}\right)+\text{i}\sqrt{-s}(l-m+1)\,P_{l+1}^{m}\left(\frac{\text{i}\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)\Bigg{]}$
$\displaystyle\hskip
28.45274pt+g_{k}\Bigg{[}(l+1)\mathcal{H}_{\text{i}}Q_{l}^{m}\left(\frac{\text{i}\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)\,Q_{l+1}^{m}\left(\frac{\text{i}\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)\Bigg{]}.$

As the scattering potential for the anti-bouncing case has singularities, the
boundary conditions at $\eta=\eta_{\text{f}}$

$\displaystyle
f_{k}\,P_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)+g_{k}\,Q_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)=a_{k}\text{e}^{-\text{i}\omega_{k}\eta_{\text{f}}}+b_{k}\text{e}^{\text{i}\omega_{k}\eta_{\text{f}}},$
(201) $\displaystyle
f_{k}\Bigg{[}(l+1)\mathcal{H}_{\text{f}}P_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)\,P_{l+1}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)\Bigg{]}$
$\displaystyle\qquad+g_{k}\Bigg{[}(l+1)\mathcal{H}_{\text{f}}Q_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)\,Q_{l+1}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)\Bigg{]}$
$\displaystyle=\left(\frac{\mathcal{H}_{\text{f}}}{2}-\text{i}\omega_{k}\right)a_{k}\text{e}^{-\text{i}\omega_{k}\eta_{\text{f}}}+\left(\frac{\mathcal{H}_{\text{f}}}{2}+\text{i}\omega_{k}\right)b_{k}\text{e}^{\text{i}\omega_{k}\eta_{\text{f}}}.$

Then, we find for the amplitudes

$\begin{split}a_{k}=&-\frac{c_{k}e^{\text{i}\omega_{k}(\eta_{\text{f}}-\eta_{\text{i}})}}{2\omega_{k}\sqrt{-s}}\frac{\Gamma(l-m+1)}{\Gamma(l+m+1)}\\\
&\times\Bigg{\\{}\left[\left(\left(l+\frac{1}{2}\right)\mathcal{H}_{\text{f}}-\text{i}\omega_{k}\right)P_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)P_{l+1}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)\right]\\\
&\times\left[\left(\left(l+\frac{1}{2}\right)\mathcal{H}_{\text{i}}+\text{i}\omega_{k}\right)Q_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)Q_{l+1}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)\right]\\\
&-\left[\left(\left(l+\frac{1}{2}\right)\mathcal{H}_{\text{f}}-\text{i}\omega_{k}\right)Q_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)Q_{l+1}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)\right]\\\
&\times\left[\left(\left(l+\frac{1}{2}\right)\mathcal{H}_{\text{i}}+\text{i}\omega_{k}\right)P_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)P_{l+1}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)\right]\Bigg{\\}},\\\
b_{k}=&\frac{c_{k}e^{-\text{i}\omega_{k}(\eta_{\text{f}}+\eta_{\text{i}})}}{2\omega_{k}\sqrt{-s}}\frac{\Gamma(l-m+1)}{\Gamma(l+m+1)}\\\
&\times\Bigg{\\{}\left[\left(\left(l+\frac{1}{2}\right)\mathcal{H}_{\text{f}}+\text{i}\omega_{k}\right)P_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)P_{l+1}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)\right]\\\
&\times\left[\left(\left(l+\frac{1}{2}\right)\mathcal{H}_{\text{i}}+\text{i}\omega_{k}\right)Q_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)Q_{l+1}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)\right]\\\
&-\left[\left(\left(l+\frac{1}{2}\right)\mathcal{H}_{\text{f}}+\text{i}\omega_{k}\right)Q_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)Q_{l+1}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{f}}}{\sqrt{-s}}\right)\right]\\\
&\times\left[\left(\left(l+\frac{1}{2}\right)\mathcal{H}_{\text{i}}+\text{i}\omega_{k}\right)P_{l}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)+\text{i}\sqrt{-s}(l-m+1)P_{l+1}^{m}\left(\text{i}\frac{\mathcal{H}_{\text{i}}}{\sqrt{-s}}\right)\right]\Bigg{\\}}.\end{split}$
(202)

## References

  * Mukhanov and Winitzki [2007] V. Mukhanov and S. Winitzki,  _Introduction to Quantum Effects in Gravity_ (Cambridge University Press, 2007).
  * Birrell and Davies [1982] N. D. Birrell and P. C. W. Davies,  _Quantum Fields in Curved Space_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1982).
  * Fulling [1989] S. A. Fulling,  _Aspects of Quantum Field Theory in Curved Space-Time_ (Cambridge University Press, 1989).
  * "Wald [1995] R. M. "Wald,  _Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics_ , "Chicago Lectures in Physics" ("University of Chicago Press", "Chicago, IL", "1995").
  * Parker [1969] L. Parker, Phys. Rev. 183, 1057 (1969).
  * Hawking [1975] S. W. Hawking, Comm. Math. Phys. 43, 199 (1975).
  * Landau and Lifshitz [2013] L. D. Landau and E. M. Lifshitz,  _Quantum mechanics: non-relativistic theory_ , Vol. 3 (Elsevier, 2013).
  * Sakurai and Napolitano [2020] J. J. Sakurai and J. Napolitano,  _Modern Quantum Mechanics_ , 3rd ed. (Cambridge University Press, 2020).
  * Griffiths and Schroeter [2018] D. J. Griffiths and D. F. Schroeter,  _Introduction to Quantum Mechanics_, 3rd ed. (Cambridge University Press, 2018).
  * Schwabl [2007] F. Schwabl,  _Quantenmechanik (QM I): Eine Einführung_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2007) pp. 47–97.
  * Flügge [1999] S. Flügge, One-body problems without spin, in  _Practical Quantum Mechanics_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 1999) pp. 25–331.
  * Boya [2008] L. Boya, La Rivista del Nuovo Cimento 31, 75–139 (2008).
  * Calzetta and Hu [2008] E. A. Calzetta and B.-L. B. Hu,  _Nonequilibrium Quantum Field Theory_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 2008).
  * Viermann  _et al._ [2022] C. Viermann, M. Sparn, N. Liebster, M. Hans, E. Kath, Á. Parra-López, M. Tolosa-Simeón, N. Sánchez-Kuntz, T. Haas, H. Strobel, S. Floerchinger, and M. K. Oberthaler, Nature 611, 260 (2022).
  * Tolosa-Simeón  _et al._ [2022] M. Tolosa-Simeón, A. Parra-López, N. Sánchez-Kuntz, T. Haas, C. Viermann, M. Sparn, N. Liebster, M. Hans, E. Kath, H. Strobel, M. K. Oberthaler, and S. Floerchinger, Phys. Rev. A 106, 033313 (2022).
  * Sánchez-Kuntz  _et al._ [2022] N. Sánchez-Kuntz, A. Parra-López, M. Tolosa-Simeón, T. Haas, and S. Floerchinger, Phys. Rev. D 105, 105020 (2022).
  * Unruh [1981] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981).
  * Unruh [1995] W. G. Unruh, Phys. Rev. D 51, 2827 (1995).
  * Unr [2007] _Quantum Analogues: From Phase Transitions to Black Holes and Cosmology_ (Springer Berlin Heidelberg, 2007).
  * Volovik [2009] G. E. Volovik,  _The Universe in a Helium Droplet_ (Oxford University Press, Oxford, 2009).
  * Feynman [1982] R. P. Feynman, International Journal of Theoretical Physics 21, 467–488 (1982).
  * Jaksch  _et al._ [1998] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Physical Review Letters 81, 3108–3111 (1998).
  * Bloch  _et al._ [2012] I. Bloch, J. Dalibard, and S. Nascimbène, Nature Physics 8, 267–276 (2012).
  * Cirac and Zoller [2012] J. I. Cirac and P. Zoller, Nature Physics 8, 264–266 (2012).
  * Georgescu  _et al._ [2014] I. Georgescu, S. Ashhab, and F. Nori, Reviews of Modern Physics 86, 153–185 (2014).
  * Aidelsburger  _et al._ [2021] M. Aidelsburger, L. Barbiero, A. Bermudez, T. Chanda, A. Dauphin, D. González-Cuadra, P. R. Grzybowski, S. Hands, F. Jendrzejewski, J. Jünemann, G. Juzeliūnas, V. Kasper, A. Piga, S.-J. Ran, M. Rizzi, G. Sierra, L. Tagliacozzo, E. Tirrito, T. V. Zache, J. Zakrzewski, E. Zohar, and M. Lewenstein, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380, 10.1098/rsta.2021.0064 (2021).
  * Mil  _et al._ [2020] A. Mil, T. V. Zache, A. Hegde, A. Xia, R. P. Bhatt, M. K. Oberthaler, P. Hauke, J. Berges, and F. Jendrzejewski, Science 367, 1128–1130 (2020).
  * Yang  _et al._ [2020] B. Yang, H. Sun, R. Ott, H.-Y. Wang, T. V. Zache, J. C. Halimeh, Z.-S. Yuan, P. Hauke, and J.-W. Pan, Nature 587, 392–396 (2020).
  * Schweizer  _et al._ [2019] C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero, E. Demler, N. Goldman, I. Bloch, and M. Aidelsburger, Nature Physics 15, 1168–1173 (2019).
  * Görg  _et al._ [2019] F. Görg, K. Sandholzer, J. Minguzzi, R. Desbuquois, M. Messer, and T. Esslinger, Nature Physics 15, 1161–1167 (2019).
  * Barceló  _et al._ [2011] C. Barceló, S. Liberati, and M. Visser, Living Rev. Relativ. 14, 3 (2011).
  * Visser  _et al._ [2002] M. Visser, C. Barceló, and S. Liberati, Gen. Relativ. Gravit. 34, 1719 (2002).
  * Novello  _et al._ [2002] M. Novello, M. Visser, and G. E. Volovik, eds.,  _Artificial Black Holes_ (World Scientific Publishing, Singapore, 2002).
  * Jacquet  _et al._ [2020] M. J. Jacquet, S. Weinfurtner, and F. König, Philos. Trans. Royal Soc. A 378, 20190239 (2020).
  * Barceló  _et al._ [2003] C. Barceló, S. Liberati, and M. Visser, Phys. Rev. A 68, 053613 (2003).
  * Fedichev and Fischer [2003] P. O. Fedichev and U. R. Fischer, Phys. Rev. Lett. 91, 240407 (2003).
  * Fedichev and Fischer [2004] P. O. Fedichev and U. R. Fischer, Phys. Rev. A 69, 033602 (2004).
  * Fischer [2004] U. R. Fischer, Mod. Phys. Lett. A 19, 1789 (2004).
  * Fischer and Schützhold [2004] U. R. Fischer and R. Schützhold, Phys. Rev. A 70, 063615 (2004).
  * Uhlmann  _et al._ [2005] M. Uhlmann, Y. Xu, and R. Schützhold, New J. Phys. 7, 248 (2005).
  * Calzetta and Hu [2005] E. A. Calzetta and B. L. Hu, Int. J. Theor. Phys. 44, 1691 (2005).
  * Liberati  _et al._ [2006a] S. Liberati, M. Visser, and S. Weinfurtner, Class. Quantum Gravity 23, 3129 (2006a).
  * Liberati  _et al._ [2006b] S. Liberati, M. Visser, and S. Weinfurtner, Phys. Rev. Lett. 96, 151301 (2006b).
  * Jain  _et al._ [2007] P. Jain, S. Weinfurtner, M. Visser, and C. W. Gardiner, Phys. Rev. A 76, 033616 (2007).
  * Prain  _et al._ [2010] A. Prain, S. Fagnocchi, and S. Liberati, Phys. Rev. D 82, 105018 (2010).
  * Bilić and Tolić [2013] N. Bilić and D. Tolić, Phys. Rev. D 88, 105002 (2013).
  * Weinfurtner  _et al._ [2006] S. Weinfurtner, S. Liberati, and M. Visser, J. Phys. A 39, 6807 (2006).
  * Weinfurtner  _et al._ [2009] S. Weinfurtner, P. Jain, M. Wisser, and C. W. Gardiner, Class. Quantum Gravity 26, 065012 (2009).
  * Eckel  _et al._ [2018] S. Eckel, A. Kumar, T. Jacobson, I. B. Spielman, and G. K. Campbell, Phys. Rev. X 8, 021021 (2018).
  * Chen  _et al._ [2021] C.-A. Chen, S. Khlebnikov, and C.-L. Hung, Phys. Rev. Lett. 127, 060404 (2021).
  * Tajik  _et al._ [2023] M. Tajik, M. Gluza, N. Sebe, P. Schüttelkopf, F. Cataldini, J. Sabino, F. Møller, S.-C. Ji, S. Erne, G. Guarnieri, S. Sotiriadis, J. Eisert, and J. Schmiedmayer, Proceedings of the National Academy of Sciences 120, e2301287120 (2023).
  * Wittemer  _et al._ [2019] M. Wittemer, F. Hakelberg, P. Kiefer, J.-P. Schröder, C. Fey, R. Schützhold, U. Warring, and T. Schaetz, Phys. Rev. Lett. 123, 180502 (2019).
  * Steinhauer  _et al._ [2022] J. Steinhauer, M. Abuzarli, T. Aladjidi, T. Bienaimé, C. Piekarski, W. Liu, E. Giacobino, A. Bramati, and Q. Glorieux, Nature Communications 13, 10.1038/s41467-022-30603-1 (2022).
  * Horstmann  _et al._ [2010] B. Horstmann, B. Reznik, S. Fagnocchi, and J. I. Cirac, Phys. Rev. Lett. 104, 250403 (2010).
  * Weinfurtner  _et al._ [2011] S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh, and G. A. Lawrence, Phys. Rev. Lett. 106, 021302 (2011).
  * Steinhauer [2016] J. Steinhauer, Nat. Phys. 12, 959 (2016).
  * Muñoz de Nova  _et al._ [2019] J. R. Muñoz de Nova, K. Golubkov, V. I. Kolobov, and J. Steinhauer, Nature 569, 688 (2019).
  * Jaskula  _et al._ [2012] J.-C. Jaskula, G. B. Partridge, M. Bonneau, R. Lopes, J. Ruaudel, D. Boiron, and C. I. Westbrook, Phys. Rev. Lett. 109, 220401 (2012).
  * Rodriguez-Laguna  _et al._ [2017] J. Rodriguez-Laguna, L. Tarruell, M. Lewenstein, and A. Celi, Phys. Rev. A 95, 013627 (2017).
  * Hu  _et al._ [2019] J. Hu, L. Feng, Z. Zhang, and C. Chin, Nat. Phys. 15, 785 (2019).
  * Gooding  _et al._ [2020] C. Gooding, S. Biermann, S. Erne, J. Louko, W. G. Unruh, J. Schmiedmayer, and S. Weinfurtner, Phys. Rev. Lett. 125, 213603 (2020).
  * Zenesini  _et al._ [2024] A. Zenesini, A. Berti, R. Cominotti, C. Rogora, I. G. Moss, T. P. Billam, I. Carusotto, G. Lamporesi, A. Recati, and G. Ferrari, Nature Physics 20, 558–563 (2024).
  * Jenkins  _et al._ [2024] A. C. Jenkins, J. Braden, H. V. Peiris, A. Pontzen, M. C. Johnson, and S. Weinfurtner, Phys. Rev. D 109, 023506 (2024).
  * Tolosa-Simeón  _et al._ [2023] M. Tolosa-Simeón, M. M. Scherer, and S. Floerchinger, Analog of cosmological particle production in moiré dirac materials (2023), arXiv:2307.09299 [cond-mat.mes-hall] .
  * Haller  _et al._ [2023] A. Haller, S. Hegde, C. Xu, C. De Beule, T. L. Schmidt, and T. Meng, SciPost Physics 14, 119 (2023).
  * Falque  _et al._ [2023] K. Falque, Q. Glorieux, E. Giacobino, A. Bramati, and M. J. Jacquet, Spectroscopic measurement of the excitation spectrum on effectively curved spacetimes in a polaritonic fluid of light (2023), arXiv:2311.01392 [cond-mat.quant-gas] .
  * Jacquet  _et al._ [2022] M. Jacquet, M. Joly, F. Claude, L. Giacomelli, Q. Glorieux, A. Bramati, I. Carusotto, and E. Giacobino, The European Physical Journal D 76, 10.1140/epjd/s10053-022-00477-5 (2022).
  * Jacquet  _et al._ [2023] M. Jacquet, L. Giacomelli, Q. Valnais, M. Joly, F. Claude, E. Giacobino, Q. Glorieux, I. Carusotto, and A. Bramati, Physical Review Letters 130, 10.1103/physrevlett.130.111501 (2023).
  * Švančara  _et al._ [2024] P. Švančara, P. Smaniotto, L. Solidoro, J. F. MacDonald, S. Patrick, R. Gregory, C. F. Barenghi, and S. Weinfurtner, Nature 628, 66–70 (2024).
  * Giacomelli and Carusotto [2021] L. Giacomelli and I. Carusotto, Physical Review A 103, 10.1103/physreva.103.043309 (2021).
  * Delhom  _et al._ [2024] A. Delhom, K. Guerrero, P. Calizaya, K. Falque, A. Bramati, A. J. Brady, M. J. Jacquet, and I. Agullo, Entanglement from superradiance and rotating quantum fluids of light (2024), arXiv:2310.16031 [gr-qc] .
  * Floerchinger and Wetterich [2008] S. Floerchinger and C. Wetterich, Phys. Rev. A 77, 053603 (2008).
  * Pitaevskii and Stringari [2016] L. Pitaevskii and S. Stringari,  _Bose-Einstein Condensation and Superfluidity_ (Oxford University PressOxford, 2016).
  * Feshbach [1958] H. Feshbach, Annals of Physics 5, 357–390 (1958).
  * Heyen and Floerchinger [2020] L. H. Heyen and S. Floerchinger, Phys. Rev. D 102, 036024 (2020).
  * Cohl [2011] H. S. Cohl, Symmetry, Integrability and Geometry: Methods and Applications 10.3842/sigma.2011.108 (2011).
  * Donnelly [2002] H. Donnelly, in  _Minimal Surfaces, Geometric Analysis and Symplectic Geometry_ , Vol. 34 (Mathematical Society of Japan, 2002) pp. 15–30.
  * Carroll [2019] S. M. Carroll,  _Spacetime and Geometry_ (Cambridge University Press, 2019).
  * Visser [1999] M. Visser, Physical Review A 59, 427–438 (1999).
  * Dodelson [2003] S. Dodelson,  _Modern Cosmology_ (Academic Press, Amsterdam, 2003).
  * Hobson  _et al._ [2006] M. P. Hobson, G. P. Efstathiou, and A. N. Lasenby,  _General Relativity: An Introduction for Physicists_ (Cambridge University Press, 2006).
  * Weinberg [2008] S. Weinberg,  _Cosmology_ (Oxford University Press, Oxford, 2008).
  * Chen  _et al._ [2014] S. Chen, G. W. Gibbons, Y. Li, and Y. Yang, Journal of Cosmology and Astroparticle Physics 2014 (12), 035.
  * Griffiths and Taussig [1992] D. J. Griffiths and N. F. Taussig, American Journal of Physics 60, 883–888 (1992).
  * Kronig  _et al._ [1931] R. D. L. Kronig, W. G. Penney, and R. H. Fowler, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 130, 499 (1931).
  * Markos and Soukoulis [2008] P. Markos and C. M. Soukoulis,  _Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials_ (Princeton University Press, 2008).
  * G.Grosso and G.P.Parravicini [2000] G.Grosso and G.P.Parravicini, Electrons in one-dimensional periodic potentials, in  _Solid State Physics_ (Elsevier, 2000) p. 1–36.
  * Kalotas and Lee [1991] T. M. Kalotas and A. R. Lee, European Journal of Physics 12, 275–282 (1991).
  * Senn [1988] P. Senn, American Journal of Physics 56, 916 (1988).
  * Wigner [1948] E. P. Wigner, Phys. Rev. 73, 1002 (1948).
  * Weinberg [2003] S. Weinberg, Phys. Rev. D 67, 123504 (2003).
  * Mukhanov  _et al._ [1992] V. Mukhanov, H. Feldman, and R. Brandenberger, Physics Reports 215, 203 (1992).
  * Lyth [1985] D. H. Lyth, Phys. Rev. D 31, 1792 (1985).
  * [94] J. Martin, Inflationary cosmological perturbations of quantum-mechanical origin, in  _Lecture Notes in Physics_ (Springer-Verlag) p. 199–244.
  * Wellner [1964] M. Wellner, American Journal of Physics 32, 787–789 (1964).
  * Boya and Casahorrán [2007] L. J. Boya and J. Casahorrán, Internat. J. Theoret. Phys. 46, 1998 (2007).
  * Glenz and Parker [2009] M. M. Glenz and L. Parker, Physical Review D 80, 10.1103/physrevd.80.063534 (2009).
  * Álvarez-Domínguez  _et al._ [2023] A. Álvarez-Domínguez, J. A. R. Cembranos, L. J. Garay, M. Martín-Benito, A. Parra-López, and J. M. S. Velázquez, Phys. Rev. D 108, 065008 (2023).
  * Gel’fand and Levitan [1955] I. M. Gel’fand and B. M. Levitan (1955).
  * Marchenko [1955] V. Marchenko, Dokl. Akad. Nauk SSSR 104, 695 (1955).
  * Chadan  _et al._ [2013] K. Chadan, R. Newton, and P. Sabatier,  _Inverse problems in quantum scattering theory_, Theoretical and mathematical physics (Springer Berlin Heidelberg, 2013).
  * Bogolyubov [1947] N. N. Bogolyubov, J. Phys. (USSR) 11, 23 (1947).
  * Wands [1999] D. Wands, Phys. Rev. D 60, 023507 (1999).
  * Finelli and Brandenberger [2002] F. Finelli and R. Brandenberger, Phys. Rev. D 65, 103522 (2002).
  * Shaikh  _et al._ [2022] A. Y. Shaikh, S. Tarai, S. K. Tripathy, and B. Mishra, International Journal of Geometric Methods in Modern Physics 19, 10.1142/s0219887822501936 (2022).


# The role of fast magnetic reconnection in acceleration zones of microquasars
and AGNs

###### Abstract

Fast magnetic reconnection events, like those associated to solar flares, can
be a very powerful mechanism operating at the jet launching region in the core
of microquasars and AGNs. We have recently found that the magnetic power
released by reconnection between the magnetic field lines of the coronal inner
disk region and the lines anchored into the black hole is able to accelerate
relativistic particles through a first-order Fermi process and produce the
observed radio luminosity from both microquasars and low luminous AGNs. We
also found that the observed correlation between the radio luminosity and the
mass of these sources, spanning $10^{9}$ orders of magnitude in mass, is
naturally explained by this process. Furthermore, recent observations of
$\gamma$-ray emission with strong variability from AGNs and galactic compact
sources indicate extremely small emission regions. In this work, assuming that
this high energy emission is probably originated in the same acceleration
zones that produce the radio emission, we have applied the scenario above of
magnetic reconnection-driven acceleration in the magnetized corona around the
accretion disks to investigate the origin of the high energy outcomes from an
extensive number of sources including high and low luminous AGNs,
microquasars, GRBs and the Crab nebula. As in the earlier analysis, we find
that the radio emission of all microquasars and low luminous AGNs are well
explained by our model. On the other hand, we find correlation of our model
with the $\gamma$-ray emission only for microquasars and a few low luminous
AGNs (e.g., M$87$), while none of the high luminous AGNs or GRBs are fitted,
neither in radio (as previously found) nor in $\gamma$ emission. We attribute
the lack of correlation of the $\gamma$ emission for most of the low luminous
AGNs to the fact that this processed emission does not depend only on the
local magnetic field activity around the source accretion disk (as the radio
Synchrotron emission), but also on other environmental factors like the photon
and density fields. We also find that the emission from both the high luminous
AGNs and the GRBs actually anti-correlates with our model indicating that
their emission is processed further out in the jet (as we claimed in [6], and
in consistency with recent findings in [24]). We conclude from these results
that the emission we see from the low luminous AGNs and microquasars comes
from the nuclear region of their sources and therefore, can be driven by
nuclear magnetic activity connected to the source/accretion disk corona, as in
our model. However, in the case of the high luminous AGNs (e.g. BL Lacs) and
GRBs, the nuclear emission is blocked by the surrounding density and photon
fields and, therefore, we can only see the jet emission further out (which has
already lost correlation with the nuclear conditions). This also explains why
high luminous AGNs do not fit the fundamental plane of luminosity source
versus mass correlation.

##  1 Introduction

Galactic and extragalactic objects such as microquasars and AGNs often exhibit
variability and quasi-periodic ejections of matter that may offer important
clues about the physical processes that occur in their inner regions.

A model to explain the origin of these ejections and the associated
Synchrotron flare radio emission, which has characteristics that resemble
those of solar flares, was developed by de Gouveia Dal Pino and Lazarian ([5];
see also [6]) where they invoked a process of fast magnetic reconnection
between the magnetic field lines that arise from the accretion disk and the
lines of the magnetosphere of the central source. In accretion episodes where
the ratio between the effective disk pressure and magnetic pressure decreases
to values smaller than the unity and the accretion rate approaches the
critical Eddington rate, the magnetic reconnection may become very fast and
release large amounts of magnetic energy power. Part of this energy heats the
coronal and the disk gas and part accelerates particles to relativistic
velocities through a first-order Fermi-like process (see also [12], [13], [8])
that results in a Synchrotron radio power-law spectrum compatible with the
observations. Using this model, [5] and [6] also found that the observed
correlation between the radio luminosity and the mass from microquasars to low
luminous AGNs, spanning $10^{9}$ orders of magnitude in mass, is naturally
explained by this process.

In recent years, the very high energy emission (VHE) from AGNs have also
revealed strong variability, with timescales of the order of days (e.g.,
M$87$), which points to extremely compact emission regions (corresponding to
only a few Schwarzschild radii; e.g. [2]). However, the localization of the
emitting zones for several sources is until now unclear. Magnetic reconnection
events occurring close to the black holes (BH) could offer appropriate
conditions for producing particle acceleration and the associated VHE
$\gamma$-ray emission in these sources. Similarly, microquasars are also
expected to emit high-energy $\gamma$-rays owing to their general similarities
to quasars (e.g., [11]). Until now, only one source of this type, namely Cyg
X-$3$, has been unambiguously detected in the GeV $\gamma$-rays, by the Agile
and Fermi observatories ([21], [1]). At TeV energies, only the flux upper
limits are available, in spite of intensive monitoring. There is also some
evidence of sporadic GeV-TeV $\gamma$-ray emission from another source of this
type, Cyg X-$1$.

In the framework of high luminous AGNs and GRBs, Nemen et al. ([24]) have
recently shown that the jets produced by both classes of sources exhibit
nearly the same correlation between the kinetic power carried by accelerated
particles and the $\gamma$-ray luminosity. This is an indication that the bulk
of the high-energy emission comes from the jet region in these sources. On the
other hand, the lack of correlation with low luminous AGNs may be an
indication that the acceleration zones responsible for the high (and also low)
energy emission in these cases are mostly in the core region, rather than
further out in the jet, just like in the model proposed above. Since the
$\gamma$-ray emission is probably originated in the same acceleration zones
that produce the radio emission, in the present work we apply the de Gouveia
Dal Pino et al. scenario above of reconnection-driven acceleration in the
magnetized corona of the accretion disks ([5] and [6]) and investigate if the
high energy outcomes of microquasars and low luminous AGNs can be also
interpreted in the light of this mechanism.

##  2 Rate of magnetic energy released by magnetic reconnection: application
to radio emission

To evaluate the amount of magnetic energy that can be extracted through
violent magnetic reconnection, it is adopted the standard model for the
radiation-dominated accretion disk by [22] and the model by [4] to quantify
the parameters of the corona. Also, it is assumed that the inner radius of the
accretion disk corresponds approximately to the last stable orbit around the
BH ($R_{X}=3R_{S}$, where $R_{S}$ is the Schwarzschild radius). To determine
the accretion rate immediately before an event of violent magnetic
reconnection, it is assumed the equilibrium between the disk gas ram pressure
and the magnetic pressure of the magnetosphere anchored at the BH horizon. It
is assumed further that the intensity of the BH horizon field is of the order
of that of the inner disk. Under these conditions one can show that the
magnetic energy power released during violent magnetic reconnection for
microquasars is approximately given by [6]:

$\dot{W}_{B}\simeq
10^{35}\alpha_{0.5}^{-19/16}\beta_{0.8}^{-9/16}M_{14}^{1/2}l_{100R_{X}}^{11/16}\leavevmode\nobreak\
\leavevmode\nobreak\ erg/s,$ (1)

and for AGNs by [6]:

$\dot{W}_{B}\simeq
10^{40}\alpha_{0.5}^{-19/16}\beta_{0.8}^{-9/16}M_{8}^{1/2}l_{10^{4}R_{X}}^{11/16}\leavevmode\nobreak\
\leavevmode\nobreak\ erg/s,$ (2)

where $\alpha_{0.5}=\alpha/0.5$ is the disk viscosity and
$\beta_{0.8}=\beta/0.8$ is defined as the ratio between the effective disk
pressure and the magnetic pressure. The BH mass $M_{14}=M/14M_{\odot}$ and
$M_{8}=M/10^{8}M_{\odot}$ are parameters suitable for the microquasars and
AGNs, respectively, and the parameters $l_{100R_{X}}=l/100R_{X}$ and
$l_{10^{4}R_{X}}=l/10^{4}R_{X}$ are the scale height of the magnetic
reconnection zone in the corona.

Figure 1 extracted from [6] depicts a synthesis of the magnetic reconnection
scenario for relativistic sources including both microquasars and AGNs. The
diagram shows the calculated magnetic power released in violent magnetic
reconnection events as a function of the central source mass for a suitable
choice of the parameter space. The symbols correspond to the observed radio
luminosities of superluminal components (stars for microquasars, circles and
triangles for the low luminous AGNs, i.e. LINERs and Seyfert galaxies,
respectively, and squares for luminous AGNs).

Figure 1: $\dot{W}_{B}$ versus the BH mass for both microquasars and AGNs. The
stars represent the observed radio luminosities for microquasars. The circles,
triangles and squares are observed radio luminosities of jets at parsec scales
from LINERS, Seyfert galaxies, and luminous AGNs, respectively. The thick bars
correspond to the calculated magnetic reconnection power and encompass the
parameter space that spans $5M_{\odot}\leq M\leq 10^{10}M_{\odot}$,
$0.05\leq\alpha\leq 0.5$, $0.1\leq\beta\leq 1$, and $1R_{S}\leq l\leq
1000R_{S}$ (or $0.3R_{X}\leq l\leq 333R_{X}$), with $1R_{S}\leq l\leq 10R_{S}$
in black, $10R_{S}<l\leq 30R_{S}$ in light gray, and $30R_{S}<l\leq 1000R_{S}$
in dark gray. Extracted from [6].

The diagram indicates that the magnetic power released during violent
reconnection events obeys a correlation that is maintained throughout this
interval, spanning $10^{9}$ orders of magnitude. This correlation implies an
almost linear dependence (in a log-log diagram), which is approximately
independent of the physical properties of the accretion disks of these
sources. Moreover, it is compatible with the so-called “fundamental plane”
obtained empirically, which correlates the radio and X-rays emission of
microquasars and AGNs with the masses of their BH (see [3]). Thus, the model
of [5] provides a simple physical interpretation for the existence of this
empirical correlation as due to coronal magnetic activity in these sources.
The diagram also reveals that the more luminous AGNs do not obey the same
correlation, possibly because the density around the coronal region in these
sources is so high that it “masks” the emission due to the magnetic activity.
The radio emission in these cases is possibly due to regions further out at
the supersonic jet, where it has already expanded enough to become optically
thin and visible and where the relativistic electrons are probably accelerated
in shocks (see [6]). This conclusion in de Gouveia Dal Pino et al. [6] is
compatible with the recent correlation found by Nemem et al. ([24]).

##  3 The same scenario for the $\gamma$-ray emission

Employing the model described above of [6], we selected a preliminary set of
compact sources including high and low luminous AGNs, GRBs and galactic
sources (e.g., the pulsar Crab and the microquasars Cyg-X$1$ and Cyg-X$3$)
with observed $\gamma$ and radio emission and compared their luminosities with
the calculated magnetic reconnection power as a function of their mass.

Figure 2: Magnetic power versus the central source mass. The open blue symbols
correspond to the $\gamma$-ray emission from blazars, low luminous AGNs and
galactic black holes (Cyg-X$1$ and Cyg-X$3$). The blue vertical line and blue
star correspond, respectively, to the $\gamma$ emission from GRBs and the Crab
pulsar. The filled red symbols correspond to the radio emission of different
types of sources (Seyferts, liners, radio galaxies and microquasars).

Figure 2 depicts the preliminary results of this study. As in Figure 1, it
compares the calculated magnetic power released by fast magnetic reconnection
at the corona/accretion disk region as a function of the source mass with the
observed $\gamma$ and radio luminosities. We have considered the $\gamma$
emission (open blue symbols in Figure 2) of an extensive number of high
luminous AGNs and GRBs (all obtained from [24]), low luminous AGNs (Seyferts,
liners and radio galaxies; see [18], [9], [10] and [15]) and microquasars
(Cyg-X$1$ [14] and Cyg-X$3$ [16]). In the case of GRBs (blue vertical line in
the diagram), we have assumed that all sources have a mass around a few tens
of solar masses. We have also included the radio emission counterparts of
several of these sources (filled red symbols; obtained from [3], and [23]) and
the recently detected $\gamma$-ray emission from the Crab pulsar (blue star)
which apparently comes from the surrounds of the central source (see [20]).

##  4 Discussion and Conclusions

As in the previous works ([5] and [6]), we clearly see in Figure 2 that the
observed radio luminosity of the low luminous AGNs and microquasars can be
explained by the magnetic power released by fast reconnection in the core
region of these sources. This emission is due to Synchrotron radiation from
relativistic electrons which can be accelerated by a first-order Fermi process
directly within the magnetic reconnection site in the coronal region around
the accretion disk ([5], [12], [13], [8]). The corresponding $\gamma$-ray
emission from these sources, which is produced from the interaction of the
accelerated relativistic electrons and protons with the surrounding photon and
density fields (through inverse Compton, and/or pp inelastic collisions and
photon-meson decay) can in principle be also associated with the same emission
zone in the surroundings of the core of these sources. However, direct
evidence for this association, though found for the microquasars and the Crab,
is not found for most of the low luminous AGNs. This is explained by the fact
that the $\gamma$ emission (contrary to the radio Synchrotron emission) does
not depend only on the local magnetic fields, but also on the photon and
density fields in the surroundings of the source/accretion disk, as stressed
above, and these factors can provoke the loss of correlation with our nuclear
emission model (induced by magnetic activity around the accretion disk).

The high luminous AGNs and the GRBs in Figure 2, on the other hand, clearly do
not have their radio or $\gamma$ emission correlated with the magnetic
reconnection power released at the core regions. This result confirms the
previous findings of [6] and [24] (see also [7]) which suggested that the
$\gamma$ and radio emission observed in such sources is originated further out
at the relativistic jet associated to these sources (as the nuclear emission
is being screened by the surrounding strong photon and density fields).

The results above connecting both the radio and $\gamma$ emission from low
luminous compact sources to magnetically dominated reconnection processes in
their nuclear regions, though preliminary are very promising as they suggest a
unifying single process of relativistic particle acceleration in the core
region of low luminous AGNs and compact galactic sources which naturally
interpret the Fundamental Plane [3]. In forthcoming work, we intend to extend
the analysis of the diagram above including more radio compact sources with
$\gamma$-ray emission counterparts in order to reinforce the present
conclusions.

Acknowledgment: This work has been partially supported by the Brazilian
funding agencies FAPESP, CNPq and CAPES.

## References

  * [1] A.A. Abdo et al., Science 326 (2009) 1512
  * [2] A. Abramowski et al., ApJ 746 (2012) 151
  * [3] A. Merloni, S. Heinz and T. di Matteo, MNRAS 345 (2003) 1057
  * [4] B.F. Liu, S. Mineshige and K. Shibata, ApJ 572 (2002) 173
  * [5] E.M. de Gouveia Dal Pino and A. Lazarian, A&A 441 (2005) 845
  * [6] E.M. de Gouveia Dal Pino, P.P. Piovezan and L.H.S. Kadowaki, A&A 518 (2010a) A5
  * [7] E.M. de Gouveia Dal Pino, G. Kowal, L.H.S. Kadowaki, P. Piovezan, & A. Lazarian, International Journal of Modern Physics D, 19 (2010b) 729
  * [8] E.M. de Gouveia Dal Pino & G. Kowal, arXiv:1302.4374 (2013)
  * [9] F. Aharonian et al., A&A 403 (2003) L1
  * [10] F. Aharonian et al., ApJ Letters 695 (2009) L40
  * [11] G.E. Romero et al., in Relativistic Astrophysics Legacy and Cosmology Einstein s ESO Astrophysics Symposia 480 (2007)
  * [12] G. Kowal, E.M. de Gouveia Dal Pino, & A. Lazarian, ApJ, 735 (2011) 102
  * [13] G. Kowal, E.M. de Gouveia Dal Pino, & A. Lazarian, Physical Review Letters, 108 (2012) 241102
  * [14] J. Albert et al., ApJ Letters 665 (2007) 51
  * [15] J. Aleksić et al., ApJ 723 (2010) L207
  * [16] J. Aleksić et al., ApJ 721 (2010) 843
  * [17] J. Aleksić et al., A&A Letters 539 (2012) 2
  * [18] M. Ackermann et al., ApJ 747 (2012) 104
  * [19] M. Kadler et al., A&A 538 (2012) L1
  * [20] M. Tavani et al., Science 331 (2011) 736
  * [21] M. Tavani et al., Nature 462 (2009) 620
  * [22] N.I. Shakura and R.A. Sunyaev, A&A 24 (1973) 337
  * [23] N.M. Nagar, H. Falcke and A.S. Wilson, A&A 435 (2005) 521
  * [24] R.S. Nemmen et al., Science 338 (2012) 1445