# Emergent Cosmology in Models of Nonlinear Electrodynamics

###### Abstract

Nonlinear electrodynamics, which acts as a source of gravity Einstein field
equations, leads to emergent cosmology, an alternative solution which can
avoid Big Bang singularity. In this paper, we explore the emerging universe in
models of non-linear electrodynamics (described by dimensional parameter
$\beta$) by using the equation of state parameter $\omega$ and see how the
parameter $\beta$ helps the universe to cause a transition from a quasi-static
Minkowski phase to the inflationary phase of expansion through the point of
emergence and subsequently to the phase of normal thermal expansion. We
predict the spectral index parameter $n_{s}=0.97467$ (scalar spectral index),
$r=0.10133$ (tensor to scalar ratio) and $n_{T}=-0.01267$ (tensor spectral
index) of the inflationary perturbation in emergent cosmology of nonlinear
electrodynamics corresponding to $\beta$ = 0.1 and $B_{0}=10^{-10}$G.

Payel<EMAIL_ADDRESS>Prasanta Kumar
<EMAIL_ADDRESS>
Birla Institute of Technology and Science-Pilani, K. K. Birla Goa campus,
NH-17B, Zuarinagar, Goa-403726, India

##  1 Introduction

The present universe which is isotropic and homogeneous on large scale must
have passed through a phase of accelerated exponential expansion called the
inflation-in the very beginning of its expansion. Inflation is believed to
describe the physics of very early universe quite well and it also explains
several conceptual issues in Big-Bang cosmology [1, 2, 3, 4]. Most of the
ongoing research activities in the area of cosmic microwave background
radiation(CMBR) observations is to converge on the correct model of inflation
[5].
Despite of the impressive success of inflation, there are still a couple of
pressing issues/questions - related to inflation, period before the
commencement of inflation and successful exit from the inflation - are
unanswered/unattended till date which might pose few below mentioned
alternative possibilities.

  * •

The universe at its very beginning is described by a quantum gravity theory.
It entered into an inflationary phase through the quantum mechanical
tunneling.

  * •

Prior to the inflation, the universe was dominated by radiation (or some other
form of matters e.g. quintom matter, nonlinear electromagnetic field).

  * •

The universe underwent a non-singular bounce prior to inflation and
subsequently enters into the normal thermal expansion. Prior to the non-
singular bounce, the universe was contracting [6, 7].

  * •

The universe existed ‘eternally’ in a quasi-static Minkowski phase of non
vanishing minimal radius before it emerges into an inflationary phase and
subsequently enters into the normal thermal expansion [8, 9, 10].

None of the above listed alternatives is well established till date. A search
for the correct alternative (of non-singular cosmology) to Big-Bang cosmology
is the need of the hour.

In this work, we will investigate the emergent universe picture in model of
nonlinear electrodynamics within the frame of 4-dimensional Friedmann-
Robertson-Walker(FRW) metric. In the early time of the universe evolution
electromagnetic fields were very strong and quantum corrections should be
taken into account [11] and, as a result, Maxwell’s electrodynamics becomes
nonlinear electrodynamics (NED) [12, 13, 14]. This non-linear electromagnetic
fields, coupled strongly to gravity, can induce a non-vanishing trace anomaly
term (proportional $T^{\mu}_{\mu}$ (trace of the energy-momentum tensor) which
is zero for a theory containing only massless fields, and nonzero for a theory
containing massive field or in a theory of gravity-NLED fields). The trace
anomaly term which can be viewed as a quantum correction to the Einstein-
Hilbert action and breaks the scale invariance in the NLED + gravity theory,
can generate the negative pressure [15, 16, 17, 18, 19, 20] and hence drive
the universe to accelerate. It pushes the universe from its quasi-static
Minkowskian phase($-\infty<t\leq 0$) to the inflationary phase of accelerated
expansion and subsequently to the normal thermal expansion($t>0$).

The paper is organised as follows. In Section 2, we describe the general
requirements of the emergent universe. In Section 3, we present the analytic
and numerical solution of the emergent universe in our model of nonlinear
electrodynamics(NLED) with a parameterized EoS parameter $\omega$ near the
point of emergence(corresponding to $\omega=-1$). In Section 4, we discuss the
slow roll parameters of inflationary expansion in the emergent universe and
made an estimate of the CMB spectral index parameters in this NLED theory.
Finally, in Section 5, we summarize our findings and conclude.

##  2 Emergent Universe-it’s requirements

Maxwell’s electrodynamics usually leads to singular cosmology in various
models. It is worthwhile to see whether the non-singular cosmology can be
realized in model of nonlinear electrodynamics.
We begin with the spatially flat isotropic and homogeneous universe described
by Friedmann-Robertson-Walker(FRW) line element for a flat universe ($k=0$)
given by,

$ds^{2}=dt^{2}-a^{2}(t)\left[dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})\right]$ (1)

where $a(t)$ is the scale factor. From the Einstein’s equations, for a perfect
fluid of density $\rho$ and pressure $p$, we find the energy conservation law,
Friedmann equation and Raychaudhuri equation,

$\dot{\rho}+3H(\rho+P)=0,$ (2)
$~{}~{}~{}H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\rho}{3}$ (3)
$~{}~{}~{}\frac{\ddot{a}}{a}=-\frac{1}{6}(\rho+3P)$ (4)

and taking the time-derivative of the Friedmann, we find

$\dot{H}=\frac{\ddot{a}}{a}-\left(\frac{\dot{a}}{a}\right)^{2}=-\frac{1}{2}(\rho+P)$
(5)

where, $8\pi G=1$. Here $H=\frac{\dot{a}}{a}$ is the Hubble parameter where
$\dot{a}=\frac{da}{dt}$.
The emergent universe is characterised by a quasi-static phase which is
followed by an inflationary phase. It corresponds to non-singular cosmology.
In this scenario, the universe expands forever starting with a finite scale
factor $a(\neq 0)$ at infinite past i.e. its time derivative $\dot{a}$ is
almost zero. The Hubble parameter $H$ , which stays at zero in this static
phase, can not be negative. Since in the emergent phase
$\rho+P<0~{}\to\dot{H}>0$, one finds the EoS parameter($\omega$) in the quasi-
static phase

$\omega=\frac{P}{\rho}=-1-\frac{2~{}\dot{H}}{3~{}H^{2}}\ll-1$ (6)

After exiting quasi-Minkowski(steady) state, the universe enters into thermal
expanding phase which suggests $\omega$ should be $-1,1/3,0$. This requires a
transition from $\omega<-1$ phase to $\omega>-1$ phase. It is natural to
conclude that for emergent universe scenario, $H^{-1}=\infty$ for $t=-\infty$
which indicates Einstein static(steady) state(ESS) of our universe at infinite
past. The evolution of our universe from ESS state requires $H$ should
increase and $\dot{H}>0$ at some point - means the equation of state parameter
$\omega<-1$ which violates the null energy condition(NEC) i.e. $\rho+p<0$. At
later time universe enters into an inflationary phase through the
time($t_{E}$) of emergence (at which $\omega=-1$) and subsequently to normal
thermal expansion (radiation and matter dominated phase corresponding to
$\omega=1/3,0$).

##  3 Emergent universe with NLED

We consider the lagrangian of non-linear electrodynamics that describe the
early universe as ,[20]

$\mathcal{L}=-\frac{\mathcal{F}e^{-\beta\mathcal{F}}}{(\beta\mathcal{F}+1)^{2}}$
(7)

where $\mathcal{F}=\frac{B^{2}-E^{2}}{2}$, $\beta$ is the non linear parameter
having dimension $[M]^{-4}$. Note that in the limit $\beta\rightarrow 0$ or
$\beta\mathcal{F}\to 0$) (in late time cosmology) the lagrangian reduces to
classical Maxwell’s electrodynamics which is
$\mathcal{L}=-\mathcal{F}+\mathcal{O}(\beta\mathcal{F}^{2})\to-\mathcal{F}(=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu})$.
So, the Maxwell’s theory can be considered as an approximation of weak fields,
while for strong fields in early epoch we should use spacetime with NLED
fields.
The energy-momentum tensor derived for the above lagrangian (Eq. (7)) can be
obtained as

$T_{\mu\nu}=\frac{\partial{\mathcal{L}}}{\partial{\mathcal{F}}}\mathcal{F}_{\mu\alpha}\mathcal{F}^{\alpha}_{\nu}-g_{\mu\nu}\mathcal{L}$
(8)

where
$\frac{\partial{\mathcal{L}}}{\partial{\mathcal{F}}}=\frac{e^{-\beta\mathcal{F}}}{(1+\beta\mathcal{F})^{3}}\left[-1+2\beta\mathcal{F}+\beta^{2}\mathcal{F}^{2}\right]$.
The trace of the energy-momentum tensor $T^{\mu}_{\mu}$ can be obtained as

$T^{\mu}_{\mu}=-4\mathcal{L}+\frac{4e^{-\beta\mathcal{F}}}{(1+\beta\mathcal{F})^{3}}\left[-1+2\beta\mathcal{F}+\beta^{2}\mathcal{F}^{2}\right]\mathcal{F}$

which is non-vanishing in the presence of the nonlinear parameter $\beta\neq
0$, however the trace is zero for $\beta=0$. So, the scale invariance in the
NLED model is broken and this leads us the negative pressure. In the theory
where the nonlinear electromagnetic field acts as the source as gravity, to
restore the isotropy in Friedman-Robertson-Walker(FRW) space-time, we need to
take the average of electromagnetic field.

According to standard cosmological models a symmetry in the direction holds
(i.e. Universe is isotropic). Now, the stochastic fluctuations in electron-
positron plasma can lead to a stochastic magnetic field [21, 22] which will
fill the Universe. Assuming that the cosmic background is filled up by the
stochastic magnetic field, the averaged magnetic fields which guaranty the
isotropy of the Friedman− Robertson− Walker (FRW) space-time, should obey the
equations

$<{\bf{E}}>=<{\bf{B}}>=0,~{}<E_{i}B_{j}>=0,~{}<E_{i}E_{j}>=\frac{1}{3}E^{2}\delta_{ij},~{}<B_{i}B_{j}>=\frac{1}{3}B^{2}\delta_{ij}$
(9)

where $<>$ denotes an average over a volume larger than the the radiation
wavelength and smaller as compared with the curvature of space time.[23]
We set the electric field $\bf{E}=0$, as the electric field is screened by the
primordial charged plasma, while the magnetic field is not screened and the
Universe is called the magnetic Universe. [22].

The energy density($\rho$) and pressure($P$) can be calculated from Eq. (7)
as,

$\rho=-\mathcal{L}-E^{2}\frac{\partial\mathcal{L}}{\partial\mathcal{F}},~{}~{}~{}P=\mathcal{L}-\frac{E^{2}-B^{2}}{2}\frac{\partial\mathcal{L}}{\partial\mathcal{F}}$
(10)

For the magnetic universe the energy density $\rho$(=$\rho_{B}$) and pressure
$P$ (=$P_{B}$) are found to be,

$\rho_{B}=\frac{2B_{0}^{2}}{a^{4}}e^{-\beta
B_{0}^{2}/2a^{4}}\left(2+\frac{\beta B_{0}^{2}}{a^{4}}\right)^{-2}$ (11)

and

$P_{B}=-\frac{4B_{0}^{2}}{3a^{4}}\frac{e^{-\beta
B_{0}^{2}/2a^{4}}}{\left(2+\frac{\beta
B_{0}^{2}}{a^{4}}\right)^{3}}\left(\frac{\beta^{2}B_{0}^{4}}{a^{8}}+\frac{11}{2}\frac{\beta
B_{0}^{2}}{a^{4}}-1\right)$ (12)

where we take the magnetic field (following magnetic flux conservation)
$B=\frac{B_{0}}{a^{2}}$, $B_{0}=10^{-10}$ G is the present day value of
magnetic field[26].
The Hubble parameter can be expressed from Eq. (3) as,

$H^{2}=\frac{2B_{0}^{2}}{3a^{4}}e^{-\beta B_{0}^{2}/2a^{4}}\left(2+\frac{\beta
B_{0}^{2}}{a^{4}}\right)^{-2}$ (13)

This gives rise to the conservation of energy with effective potential
$V_{eff}$ as,

$\dot{a}^{2}+V_{eff}=0$ (14)

where $V_{eff}=-\frac{B_{0}^{2}}{6a^{2}}e^{-\beta
B_{0}^{2}/2a^{4}}\left(1+\frac{\beta B_{0}^{2}}{2a^{4}}\right)^{-2}$. Solving
Eq. (14), we find the evolution of the scale factor $a(t)$ and we express $t$
as a function of $a(t)$ as,

$t=\sqrt{\frac{3}{2}}\frac{1}{B_{0}}\left(a^{2}e^{\beta
B_{0}^{2}/4a^{4}}-\sqrt{\pi\beta}B_{0}Erfi\left(\frac{\sqrt{\beta}B_{0}}{2a^{2}}\right)\right)$
(15)

where $Erfi(x)$ is imaginary Error function and is defined as
$Erfi(x)=-iErf(ix)$ where,
$Erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt$. From Eq. (11) and Eq.
(12) we can calculate the equation of state parameter as,

$\omega(t)=\frac{P_{B}}{\rho_{B}}=\frac{2a^{8}-11\beta
B_{0}^{2}a^{4}-2\beta^{2}B_{0}^{4}}{6a^{8}+3a^{4}\beta B_{0}^{2}}$ (16)

where we set $B_{0}=10^{-10}$ Gauss [26].

Figure 1: (Color online) Plot shows the variation of scale factor $a(t)$,
hubble parameter $H(t)$ and EOS parameter $\omega(t)$ with time $t$. In
numerical calculation we took $\beta=0.1$ and $B_{0}=10^{-10}$ Gauss. The red
vertical line in all figures describe the emergent time $t_{E}=0.145$ s.

In Fig. (1), we have plotted $a(t)$ as a function of $t$ (top-left), $H(t)$
against $t$(top-right) and $\omega(t)$ against $t$(lower). From Fig. (1), we
can see that $a$ approaches non-zero constant as $t\to 0$ and stays at that
value till the infinite past i.e. $t\to-\infty$. Also, $a$ starts expanding as
$t\to 0+$. In Fig. (1)(top-right), we see that $H=H_{max}$ at the emergent
point $t=t_{E}$. From the plot of $\omega$ vs $t$(lower plot of Fig. (1)), we
see that the EoS parameter $\omega(t=t_{E})=\omega_{E}=-1$, where $t=t_{E}$ is
the time of emergence. One finds $t_{E}$ by solving $\omega_{E}=-1$ as,

$t_{E}=\sqrt{\frac{3}{2}}\left(\sqrt{\frac{\beta(-1+\sqrt{2})}{2}}e^{1/2(1+\sqrt{2})}-\sqrt{\beta\pi}Erfi(\frac{1}{\sqrt{2(1+\sqrt{2})}})\right)$
(17)

Note that $t_{E}$ coincides with $t=0$ in the case of usual electrodynamics
i.e.$t_{E}=t=0$ for $\beta=0$. Also at $t=t_{E}$, the Hubble parameter reaches
its maximum value $H_{E}=H_{max}$ where

$H_{E}=\frac{1+\sqrt{2}}{2+\sqrt{2}}\sqrt{\frac{e^{-1/1+\sqrt{2}}}{3\beta(1+\sqrt{2})}}$
(18)

and the scale factor $a$ at time $t_{E}$ is found to be

$a(t=t_{E})=a_{E}=\left(\frac{\beta B_{0}^{2}}{2}+\frac{\beta
B_{0}^{2}}{\sqrt{2}}\right)^{1/4}$ (19)

Using Eq. (11) and Eq. (12) we calculate $\rho_{B}+P_{B}$ and
$\rho_{B}+3P_{B}$ as

$\rho_{B}+P_{B}=-2\dot{H}=\frac{4B_{0}^{2}e^{-\beta
B_{0}^{2}/2a^{4}}(4a^{8}-4\beta B_{0}^{2}a^{4}-\beta^{2}B_{0}^{4})}{3(\beta
B_{0}^{2}+2a^{4})^{3}}$ (20)

and,

$\rho_{B}+3P_{B}=\frac{4B_{0}^{2}e^{-\beta B_{0}^{2}/2a^{4}}(2a^{8}-5\beta
B_{0}^{2}a^{4}-\beta^{2}B_{0}^{4})}{(\beta B_{0}^{2}+2a^{4})^{3}}$ (21)

In Fig. (2), we have plotted $\rho_{B}$(top-left), $P_{B}$(top-right) and
$\rho_{B}+3P_{B}$(lower) as a function of $t$.

Figure 2: (Color online) Plot shows the variation of $\rho_{B}$, $P_{B}$ and
$\rho_{B}+3P_{B}$ with time $t$. In numerical calculation we took $\beta=0.1$
and $B_{0}=10^{-10}$ Gauss. The red vertical line in all figures describe the
emergent time $t_{E}=0.145$ s.

In the left (of Fig. (2)), we see that $\rho_{B}=0$ (constant) during the
period $-\infty\leq t<0$ and starts deviating from zero at $t=0$ and becomes
maximum $\rho^{max}_{B}$ at the point of emergence $t=t_{E}(>0)$. On the
right(top) plot, we see $P_{B}=-\rho^{max}_{B}$ at $t=t_{E}$. On the lower
plot, we see that $\rho_{B}+3p_{B}<0$ for $t\leq 0$ which corresponds to the
violation of SEC(strong energy condition). At $t=t_{E}$, we see
$\rho_{B}+3P_{B}(=2P_{B})\geq 0$ i.e. strong energy condition(SEC) is
satisfied.
We next study the variation of $\dot{H}$ and $\rho_{B}+P_{B}$ with time $t$.
Using Eq. (13) and Eq. (20), we find

$\dot{H}=-\frac{1}{2}(\rho_{B}+P_{B})=\frac{-2B_{0}^{2}e^{-\beta
B_{0}^{2}/2a^{4}}(4a^{8}-4\beta B_{0}^{2}a^{4}-\beta^{2}B_{0}^{4})}{3(\beta
B_{0}^{2}+2a^{4})^{3}}$ (22)

In Fig. (3), we have plotted $\rho_{B}+P_{B}$(left) and $\dot{H}$(right) as a
function of $t$.

Figure 3: (Color online) Plot shows the variation of $\rho_{B}+P_{B}$ and
$\dot{H}$ with time. In numerical calculation we took $\beta=0.1$ and
$B_{0}=10^{-10}$ Gauss. The red vertical line in all figures describe the
emergent time $t_{E}=0.145$ s.

On the left plot, we see that $\rho_{B}+p_{B}<0$ for $t\leq 0$ which suggests
violation of NEC(null energy condition). For $t\geq t_{E}$, we find
$\rho_{B}+P_{B}\geq 0$ i.e. NEC is satisfied. On the right plot, we see that
$\dot{H}=0$ at the emergent point $t=t_{E}$ and $\dot{H}$ moves from
$\dot{H}>0$(NEC condition is violated) region to $\dot{H}<0$ region (NEC is
obeyed) through the point of emergence (i.e. $\dot{H}=0$).

Finally, in Fig. (4), we have plotted $V_{eff}$ against $t$(on the left) and
$V_{eff}$ against $a$ (on the right) .

Figure 4: (Color online) Plot shows the variation of $V_{eff}$ with time and
scale factor $a$ respectively. In numerical calculation we took $\beta=0.1$
and $B_{0}=10^{-10}$ Gauss. The red vertical line in all figures describe
$a_{E}$ (the value of $a$ at the emergent time $t_{E}=0.145$ s).

We see $V_{eff}=0$ in the quasi-static Minkowski phase during the period
$-\infty<t\leq 0$ and it leave this phase at $t\simeq 0$. Note that $V_{eff}$
is negative for all $t\geq 0$ (left plot) and all $a$ (right plot). On the
right plot, $V_{eff}$ is minimum at $a=a_{c}$ (as determined below). On the
left of the minimum i.e. $a=a_{c}$, we find $\frac{dV_{eff}}{da}<0$, while on
the right $\frac{dV_{eff}}{da}>0$. Noting that

$\dot{a}^{2}=-V_{eff}\to
2\dot{a}\ddot{a}=-\frac{dV_{eff}}{dt}=-\frac{dV_{eff}}{da}\dot{a}\Longrightarrow\ddot{a}=-\frac{1}{2}\frac{dV_{eff}}{da}>0~{}~{}\rm{if}~{}~{}\frac{dV_{eff}}{da}<0$
(23)

Again,

$\ddot{a}=-\frac{1}{2}\frac{dV_{eff}}{da}<0~{}~{}\rm{if}~{}~{}\frac{dV_{eff}}{da}>0$
(24)

Now $V_{eff}$ is minimum at $a=a_{c}$ where

$a_{c}=\left(\frac{\beta B_{0}^{2}(5+\sqrt{33})}{4}\right)^{1/4}=7.2\times
10^{-6}$ Here we have taken $\beta=0.1,B_{0}=10^{-10}$ Gauss. On the left of
$a=a_{c}$(right plot), we see $\frac{dV_{eff}}{da}<0~{}\to\ddot{a}>0$ which
corresponds to the accelerating universe.

On the left plot, $V_{eff}$ is minimum at $t=t_{c}$. Now
$\frac{dV_{eff}}{dt}<0$ on the left of the minimum and hence

$\dot{a}^{2}=-V_{eff}\to
2\dot{a}\ddot{a}=-\frac{dV_{eff}}{dt}\Longrightarrow~{}~{}\ddot{a}=-\frac{1}{2\dot{a}}\frac{dV_{eff}}{dt}>0~{}~{}\rm{if}~{}~{}\frac{dV_{eff}}{dt}<0,~{}\dot{a}>0~{}(\rm{for}~{}t>0),$
(25)

i.e. the accelerating universe. The inflationary expansion corresponds to
$\dot{\rho_{B}}=0\to\omega=-1$ at the time of emergence $t=t_{E}$ (where
$0<t_{E}<t_{c}$). On the right of the minimum $\frac{dV_{eff}}{dt}>0$ and we
find

$\ddot{a}=-\frac{1}{2\dot{a}}\frac{dV_{eff}}{dt}<0~{}~{}\rm{if}~{}~{}\frac{dV_{eff}}{dt}>0,~{}\dot{a}>0~{}(\rm{for}~{}t>t_{c}),$
(26)

the decelerating universe. So, in a nutshell, after exiting the quasi-static
Minkowskian space, the universe enters into the inflationary phase of
acceleration through the time of emergence $t=t_{E}$ which corresponds to
$\omega=-1$ and $\rho=\rho_{B}^{max}=\rm{constant}$.

From Eq. (20) we see that $\rho_{B}+P_{B}$ is zero at the emergent time
$t=t_{E}$ with the scale factor $a=a_{E}=\left(\frac{\beta
B_{0}^{2}}{2}+\frac{\beta B_{0}^{2}}{\sqrt{2}}\right)^{1/4}=5.9\times 10^{-6}$
(with $\beta=0.1,B_{0}=10^{-10}$ ). We also see that $a_{E}<a_{c}$, where
$a_{c}=7.2\times 10^{-6}$ and $a_{E}=5.9\times 10^{-6}$ for
$\beta=0.1,B_{0}=10^{-10}$ Gauss. We also find $t_{E}=0.145~{}\rm{s}$ and
$t_{c}=0.453~{}\rm{s}$ for $\beta=0.1$, respectively.

###  3.1 Slow-roll Parameters and CMB constaints

The evolution of the scale factor $a$ with time $t$ is shown in Fig. (1) which
suggests the usual exponential expansion of $a$ associated with normal slow
rolling inflationary model at $t>t_{E}$(which is $>0,\rm{for}~{}\beta\neq 0$),
while at earlier time $t$ with $t<t_{E}$, $a$ approaches to constant non-zero
positive value which corresponds to non-singular cosmology(contrary to the
normal singular big-bang cosmology).

The duration of slow roll regime is described by two slow-roll parameters [4],

$\epsilon=-\frac{\dot{H}}{H^{2}},~{}\eta=\epsilon-\frac{1}{2\epsilon}\frac{d\epsilon}{dN}$
(27)

Where $N$ is the e-fold number that can be expressed as, $dN=Hdt$. Slow roll
approximation requires $\epsilon<<1$ and $\eta<<1$. For NLED case, Eq. (27)
can be expressed in terms of magnetic field as[20],

$\epsilon=2B\frac{H_{,B}}{H},~{}\eta=\epsilon-\frac{a}{2\epsilon}\frac{d\epsilon}{da}$
(28)

From Eq. (13), we can express the slow roll parameters $\epsilon$ and $\eta$
for emergent universe as,

$\epsilon=\frac{-\beta^{2}B_{0}^{4}-4\beta B_{0}^{2}a^{4}+4a^{8}}{a^{4}(\beta
B_{0}^{2}+2a^{4})}$ (29)

and,

$\eta=\frac{16a^{16}-56\beta
B_{0}^{2}a^{12}+6\beta^{3}B_{0}^{6}a^{4}+\beta^{4}B_{0}^{8}}{a^{4}(-\beta^{3}B_{0}^{6}-6\beta^{2}B_{0}^{4}a^{4}-4\beta
B_{0}^{2}a^{8}+8a^{12})}$ (30)

Next, we demostrate that emergent universe described by Eq. (7) is consistent
with recent measurement of Cosmic Microwave Background (CMB) by Planck 2018
[25]. As seen from Fig. (4), $V_{eff}$ vs $t$ plot, universe commences towards
Inflation after leaving quasi-Minkowski state. Quantum fluctuations generated
during Inflation can explain density perturbations observed in CMB. Those
primordial density fluctuations can be characterised by scalar spectral index
$n_{s}$, tensorial spectral index $n_{T}$ and tensor to scalar ratio $r$ and
in slow-roll approximation,they are related as

$r=4(1-n_{s})=-8n_{T}$ (31)

In slow roll approximation, $n_{s}$, $r$ and $n_{T}$ can be expressed as,

$n_{s}-1=2\eta-4\epsilon,~{}r=16\epsilon,~{}n_{T}=-2\epsilon$ (32)

From Eq. (32) and Eq. (29) we can derive $r$ as

$r=16\frac{-\beta^{2}B_{0}^{4}-4\beta B_{0}^{2}a^{4}+4a^{8}}{a^{4}(\beta
B_{0}^{2}+2a^{4})}$ (33)

and the scalar spectral index $n_{s}$ as

$n_{s}=1+\frac{4(\beta^{2}B_{0}^{4}+4\beta B_{0}^{2}a^{4}-4a^{8})}{a^{4}(\beta
B_{0}^{2}+2a^{4})}$ (34)

Below in Table 1, we have presented our predicted values of $r$, $n_{s}$ and
$n_{T}$ corresponding to different $\beta$ values. Also shown in the table are
the emergent time $t_{E}$, the scale factor $a_{E}$ (at time $t_{E}$) and the
non-linear parameter $\beta$ corresponding to the present day magnetic field
$B_{0}=10^{-10}$ G [26].

$\beta$ | $B_{0}$ | $t_{E}$ | $a_{E}$ | $n_{s}$ | $r$ | $n_{T}$ |
---|---|---|---|---|---|---|---
$0.1$ | $10^{-10}$ | 0.1450 | $5.90\times 10^{-6}$ | $0.97467$ | $0.10133$ | $-0.012670$ |
$0.01$ | $10^{-10}$ | $0.0458$ | $3.32\times 10^{-6}$ | $0.95727$ | $0.17093$ | $-0.021367$ |
$0.001$ | $10^{-10}$ | 0.0145 | $1.86\times 10^{-6}$ | $0.99942$ | $0.00231$ | $-0.000289$ |
Table 1:  Note that the scalar spectral index $n_{s}=0.97\pm 0.02$ (from SDSS-
III/BOSS DR9 data [24]) and $n_{s}=0.9649\pm 0.0042$ from PLANCK 2018 Data
[25].

From the Table 1, we see that the $n_{s}=0.97467(0.99942)$ corresponding to
$\beta=0.1(0.001)$ lies within within $1\sigma(2\sigma)$ of the SDSS-III/BOSS
DR9 data and within $3~{}\sigma$ of the Planck data corresponding to
$\beta=0.1$. We also find the tensor-to-scalar ratio $r\approx 0.10$ and
tensor spectral index $n_{T}(=-r/8)=-0.012670(-0.000289)$ corresponding to
$\beta=0.1(0.001)$ which is close to the Planck estimate.

##  4 Conclusion

We study the emergent universe (non-singular universe at time $t=0$) in a
class of models of nonlinear electrodynamics which is characterised by a
dimensionful parameter $\beta$. The scale factor $a(t)$ in this NLED scenerio
is found to remain constant during the phase of infinite past i.e.
$-\infty<t\leq 0$ (the phase is called the quasi-static Minkowski phase) and
remains so till time $t=0$. The non-zero NLED parameter $\beta$ allows the
universe from its quasi-static Minkowski phase to enter into the inflationary
phase of expansion at the time of emergence $t_{E}$ at which the EoS parameter
$\omega=\omega_{E}=-1$ and subsequently to the normal expanding phase
dominated by dust matter. We find that the Hubble parameter($H$) is maximum
$H=H_{max}$ and the Null Energy Condition(NEC) is violated at the time of
emergence $t_{E}$ which is found to be dependent on $\beta$. We found the time
of emergence $t=t_{E}=0.145$ s (which corresponds to scale factor
$a_{E}=5.9\times 10^{-6}$) for $\beta=0.1$. We estimate the scalar spectral
index $n_{s}=0.97467$ corresponding to $\beta=0.1(0.001)$ which lies within
within $1\sigma(2\sigma)$ of the SDSS-III/BOSS DR9 data and within
$3~{}\sigma$ of the Planck data corresponding to $\beta=0.1$. We also find the
tensor-to-scalar ratio $r\approx 0.10$ and tensor spectral index
$n_{T}(=-r/8)=-0.012670(-0.000289)$ corresponding to $\beta=0.1(0.001)$ which
can be compared with the estimate of the Planck 2018 collaboration.

##  5 Aknowledgement

PS would like to thank Department of Science and Technology, Government of
India for INSPIRE fellowship. The work of PKD is supported by CSIR Grant
No.25(0260)/17/EMR-II.

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# Testing the evolutionary pathways of galaxies and their supermassive black
holes and the impact of feedback from Active Galactic Nuclei via large
multiwavelength datasets

George Mountrichas 1 & Francesco Shankar2
1Instituto de Fisica de Cantabria (CSIC-Universidad de Cantabria), Avenida de
los Castros, 39005 Santander, Spain
2School of Physics and Astronomy, University of Southampton, Highfield, SO17
1BJ, UK
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

It is still a matter of intense debate how supermassive black holes (SMBH)
grow, and the role played by feedback from active galactic nuclei (AGN) in the
co-evolution of SMBHs and galaxies. To test the coevolution proposed by
theoretical models, we compile a large AGN sample of 5639 X-ray detected AGN,
over a wide redshift range, spanning nearly three orders of magnitude in X-ray
luminosity. The AGN have been detected in the COSMOS-Legacy, the
Bo$\rm\ddot{o}$tes, the XMM-XXL and the eFEDS fields. Using the specific star
formation rate estimates, we split the AGN host galaxies into star forming
(SF), starburst (SB) and quiescent (Q). Our results show that the AGN
accretion is increased in SB systems compared to SF and Q. Our analysis
reveals a mild increase of LX with M∗. The LX/SFR ratio has a weak dependence
on M∗, and at fixed M∗ it is highest in Q systems. The latter trend is mostly
driven by the significant drop in SFR in the Q state. The measured strong
variations in SFR from the SB/SF to Q mirror those predicted in merger models
with AGN feedback. However, the observed mild variations in LX are at variance
with the same models. We also study the evolution of SFR for a galaxy control
sample and found that it is very similar to that of X-ray AGN. This suggests
that either AGN play a minor role in the star formation quenching, or the
relative timescales of the two processes are different.

###### keywords:

galaxies: active; Galaxies, galaxies: evolution ; Galaxies, galaxies: star
formation ; Galaxies, X-rays: galaxies ; Resolved and unresolved sources as a
function of wavelength

††pubyear: 2015††pagerange: Testing the evolutionary pathways of galaxies and
their supermassive black holes and the impact of feedback from Active Galactic
Nuclei via large multiwavelength datasets –LABEL:lastpage

##  1 Introduction

Correlations have been observed in the local (e.g., Magorrian et al., 1998;
Ferrarese & Merritt, 2000) and sometimes in the distant Universe (e.g., Suh et
al., 2020), between the mass of the central supermassive black hole (SMBH) and
several properties of the host galaxies, or even of the host dark matter
haloes (e.g. Shankar et al., 2020). These correlations are often not well
established, possibly affected by biases (e.g., Bernardi et al., 2007; Shankar
et al., 2016, 2017, 2019), varying with galaxy morphology (e.g., Suh et al.,
2019) and/or with the degree of activity of the central SMBH (e.g., Reines &
Volonteri, 2015) and often presenting significant dispersions (e.g., Graham,
2016). Although some degree of correlation undoubtedly exists between SMBH and
host galaxy, pinning down the physical causes driving their evolution remains
elusive just relying on current observational constraints of their scaling
relations. Indeed, many theoretical models show strong degeneracies when
comparing with present-day data (e.g., Habouzit et al., 2021, 2022).

A step further in our understanding of SMBH growth within galaxies can be
performed by directly analysing samples of Active Galactic Nuclei (AGN), when
the SMBHs become active at different accretion rates at different epochs, and
knowledge of their host galaxies are available. Cold gas in the galaxy can
indeed fuel star formation rate (SFR) as well as accrete onto the central SMBH
triggering the AGN, thus correlations between SFR and SMBH accretion rate, as
recorded in the AGN luminosity, are expected.

Semi-analytic models and hydrodynamical simulations have studied the star
formation and the growth of the black hole in galaxy mergers. According to
most of these studies, after the starburst (SB) phase that is triggered by the
merger, both the star formation and the black hole accretion rate (BHAR) are
quenched. The magnitude of the suppression depends on the mass of the galaxy,
with more massive systems to present a steeper decrease of both the accretion
rate and the star formation (e.g., Di Matteo et al., 2005; Hopkins, 2012). In
the case of black holes the decrease of their accretion rates is due to the
accretion energy that is released. The star formation is suppressed by
outflows generated by strong winds, produced by the AGN (e.g., DeBuhr et al.,
2012) that remove or heat the star forming gas. However, this AGN feedback can
work both ways (e.g., Zinn et al., 2013). For instance, in the late gas-poor
phase, AGN feedback may quench star formation. In the gas-rich phase, AGN
outflows can overcompress cold gas and thus enhance star formation (e.g.,
Zubovas et al., 2013).

From the observational point of view, previous works found that the average
SMBH growth rate follows similar trends with stellar mass, M∗, and redshift as
the star formation rates (SFRs) of their host galaxies (e.g., Mullaney et al.,
2012; Yang et al., 2018). However, a stronger correlation has been found
between the BHAR and the SFR rather than M∗, at least for AGN hosted by bulge
dominated (BD) systems (Yang et al., 2019). Study of the BHAR/SFR ratio
revealed that it scales weakly with M∗ (e.g., Rodighiero et al., 2015; Yang et
al., 2018; Carraro et al., 2020), at least for star forming (SF) and starburst
(SB) galaxies (but see Mullaney et al., 2012), whereas a flat relation is
found for Q systems (Carraro et al., 2020). A drop of the BHAR/SFR ratio is
found during the SB phase of galaxies (Rodighiero et al., 2015), at odds with
the theoretical predictions. Although, the AGN activity is enhanced in
compact, SF galaxies, the bulk of the black hole growth takes place in
extended, Q systems (Aird et al., 2022). Moreover, the mean LX-M∗ relation is
independent of the AGN duty cycle, but depends strongly on the shape,
normalization and scatter of the underlying MBH-M∗ scaling relation and on the
characteristic Eddington ratio (Carraro et al., 2022).

In this work, we use X-ray detected AGN in four fields, namely the the
Bo$\rm\ddot{o}$tes, COSMOS-Legacy, the XMM-XXL and the eFEDS fields. Our
sample consists of more than 5500 X-ray sources, spanning a redshift range of
$\rm 0.5<z<2.5$ and nearly three orders of magnitude in X-ray luminosity,
while their stellar mass ranges from $10.0<\rm log\,[M_{*}(M_{\odot})]<12.0$.
The samples used in our analysis are described in Sect. 2. In Sect. 3, we
present the method we apply (SED fitting) to measure the host galaxy
properties (SFR, M∗) and the various selection criteria we use to select only
AGN with robust host galaxy measurements that also satisfy the mass
completeness limits of each survey. We also describe how we classify sources
into SB, SF and Q systems. Our goal is to study the X-ray luminosity (Sect.
4.1) and the LX/SFR ratio (Sect. 4.2), as a function M∗ and compare our
observational results with the predictions of theoretical models (Sect. 5).
Our conclusions are presented in Sect. 6.

Throughout this work, we assume a flat $\Lambda$CDM cosmology with
$H_{0}=70.4$ Km s-1 Mpc-1 and $\Omega_{M}=0.272$ (Komatsu et al., 2011).

##  2 Data

In our analysis, we use X-ray AGN detected in four fields, namely the
Bo$\rm\ddot{o}$tes, the COSMOS-Legacy, the eFEDS and the XMM-XXL-N fields. The
datasets are described in detail in Mountrichas et al. (2021c); Mountrichas et
al. (2022b); Mountrichas et al. (2022c) and Masoura et al. (2018) papers,
respectively.

Very good photometric coverage is essential to obtain robust measurements of
host galaxy properties (SFR, M∗), via spectral energy distribution (SED)
fitting (see Sect. 3). Therefore, all sources used in our analysis meet strict
photometric selection requirements. Specifically, all sources have
measurements in the following photometric bands: $u$, $g$, $r$, $i$, $z$, J,
H, K, W1 or IRAC1, W2 or IRAC2 and W4 or MIPS/24, where IRAC1, IRAC2, MIPS/24
are the 3.6, 4.5 and 24 $\mu$m photometric bands of $\it{Spitzer}$ and W1, W2
and W4 are the WISE photometric bands at 3.4, 4.6 and 22 $\mu$m. Approximately
$35\%$ of the AGN also have available far-IR photometry ($\it{Herschel}$).
Mountrichas et al. (2021b); Mountrichas et al. (2021c); Mountrichas et al.
(2022b) used data from the XMM-${\it{XXL}}$, Bo$\rm\ddot{o}$tes and COSMOS
fields and utilizing the CIGALE (Boquien et al., 2019; Yang et al., 2020,
2022) SED fitting code, showed that lack of far-IR photometry does not affect
the SFR calculations derived by CIGALE.

We also exclude from our analysis sources with bad SED fits and unreliable
host galaxy measurements (see Sect. 3.1). Furthermore, only AGN that satisfy
the mass completeness limits of each survey are included in our measurements.
These limits have been calculated following the method described in Pozzetti
et al. (2010). The process is described in detail in Sections 3 of Mountrichas
et al. (2021c); Mountrichas et al. (2022b); Mountrichas et al. (2022c). Below
we give a brief description of each dataset.

Table 1: Number of X-ray AGN that meet the photometric, quality and mass completeness criteria (see text for more details) in each redshift interval. field | total | $\rm 0.5<z<1.0$ | $\rm 1.0<z<1.5$ | $\rm 1.5<z<2.5$
---|---|---|---|---
Bo$\rm\ddot{o}$tes | 1020 | 590 | 298 | 132
COSMOS | 1073 | 328 | 272 | 473
eFEDS | 2860 | 1145 | 722 | 993
XMM-XXL-N | 686 | 409 | 118 | 159
all fields | 5639 | 2472 | 1410 | 1757
Table 2: Number of X-ray AGN in each of the stellar mass and redshift bins used in our analysis. $\rm log\,[M_{*}(M_{\odot})]$ | total | $\rm 0.5<z<1.0$ | $\rm 1.0<z<1.5$ | $\rm 1.5<z<2.5$
---|---|---|---|---
$10.0-10.5$ | 230 | 184 | 9 | 37
$10.5-11.0$ | 1299 | 809 | 282 | 208
$11.0-11.5$ | 2797 | 1165 | 757 | 875
$11.5-12.0$ | 1105 | 272 | 310 | 523

###  2.1 COSMOS

The COSMOS-Legacy survey (Civano et al., 2016) is a 4.6 Ms Chandra program
that covers 2.2 deg2 of the COSMOS field (Scoville et al., 2007). The central
area has been observed with an exposure time of $\approx 160$ ks while the
remaining area has an exposure time of $\approx 80$ ks. The limiting depths
are $2.2\times 10^{-16}$, $1.5\times 10^{-15}$ , and $8.9\times 10^{-16}\,\rm
erg\,cm^{-2}\,s^{-1}$ in the soft (0.5-2 keV), hard (2-10 keV), and full
(0.5-10 keV) bands, respectively. The X-ray catalogue includes 4016 sources.
Marchesi et al. (2016) matched the X-ray sources with optical and infrared
counterparts using the likelihood ratio technique (Sutherland & Saunders,
1992). Of the sources, 97% have an optical and IR counterpart and a
photometric redshift (photoz) and $\approx 54\%$ have spectroscopic redshift
(specz). Hardness ratios ($\rm HR=\frac{H-S}{H+S}$, where H and S are the net
counts of the sources in the hard and soft band, respectively) were estimated
for all X-ray sources using the Bayesian estimation of hardness ratios method
(BEHR; Park et al., 2006). The intrinsic column density, NH , for each source
was then calculated using its redshift and assuming an X-ray spectral power
law with slope $\Gamma=1.8$. This information is available in the catalogue
presented in Marchesi et al. (2016).

We only use sources within both the COSMOS and UltraVISTA (McCracken et al.,
2012) regions. UltraVISTA covers 1.38 deg2 of the COSMOS field (Laigle et al.,
2016) and has deep near-infrared (NIR) observations ($J,H,K_{s}$ photometric
bands) that allow us to derive more accurate host galaxy properties through
SED fitting (see below). There are 1718 X-ray sources that lie within the
UltraVISTA area of COSMOS. Out of these, 1073 satisfy the photometric criteria
mentioned above, have reliable SED fits (see section 3.1) and meet the mass
completeness requirements (see Sect. 3.4 in Mountrichas et al., 2022b).

The X-ray catalogue was cross-matched with the COSMOS photometric dataset
produced by the HELP collaboration (Shirley et al., 2019, 2021) to construct
the SEDs of the X-ray sources and fit them to obtain measurements of the host
galaxy properties. HELP includes homogeneous and calibrated multiwavelength
data from 23 of the premier extragalactic survey fields imaged by the Herschel
Space Observatory which form the Herschel Extragalactic Legacy Project (HELP).

###  2.2 BO$\rm\ddot{O}$TES

We also use X-ray AGN observed by the Chandra X-ray Observatory within the 9.3
deg2 Bo$\rm\ddot{o}$tes field of the NOAO Deep Wide-Field Survey (NDWFS). The
catalogue is compiled and fully described in Masini et al. (2020). It consists
of 6891 X-ray point sources with an exposure time of about 10 ks per XMM
pointing and a limiting flux of $4.7\times 10^{-16}$, $1.5\times 10^{-16}$ and
$\rm 9\times 10^{-16}\,erg\,cm^{-2}\,s^{-1}$, in the $\rm 0.5-7\,keV$, $\rm
0.5-2\,keV$ and $\rm 2-7\,keV$ energy bands, respectively. 2346 ($\sim 33\%$)
of the X-ray sources in this catalogue, have available spectroscopic redshifts
(specz). For the remaining sources, we use hybrid photometric redshifts
(photoz; Duncan et al., 2018b, a, 2019) that are available in the Masini et
al. (2020) catalogue. The X-ray absorption of each X-ray AGN is available and
is parameterized with $\rm N_{H}$. $\rm N_{H}$ has been calculated from HR
estimations, by applying the BEHR method. A fixed Galactic absorption of $\rm
N_{H,Gal}=1.04\times 10^{20}\,cm^{-2}$ is assumed. Mountrichas et al. (2021c)
cross-matched the X-ray catalogue with the Bo$\rm\ddot{o}$tes photometric
catalogue produced by HELP to enrich the dataset with photometry from optical
to far-IR . The final X-ray Bo$\rm\ddot{o}$tes sample used in our analysis
consists of 1020 AGN.

###  2.3 eFEDS

In our analysis, we include X-ray AGN observed in the eFEDS field. The
catalogue is presented in Brunner et al. (2022). eROSITA (extended ROentgen
Survey with an Imaging Telescope Array; Predehl et al., 2021) is the primary
instrument on the Spektrum-Roentgen-Gamma (SRG) orbital observatory (Sunyaev
et al., 2021). SRG was built to provide a sensitive, wide-field-of-view X-ray
telescope with improved capabilities compared to those of XMM-Newton and
Chandra, the two most sensitive targeting X-ray telescopes in operation. The
dataset includes 27910 X-ray sources detected in the $0.2-2.3$ keV energy band
with a flux limit of $\approx 7\times 10^{-15}$ erg cm${}^{-2}\,\rm
s{{}^{-1}}$ in the $0.5-2.0$ keV energy range. Details of the source detection
are given in Brunner et al. (2022) (see their Sect 3.3, Appendix A and Fig.
3). In brief, their method is based on selecting source candidates according
to the statistics of fitting source images with a point spread function
(PSF-)convolved model. First, a preliminary catalogue is created that contains
all the potential source candidates. Then, a background map is generated using
the preliminary catalogue. Next, the preliminary catalogue is used as input to
the PSF-fitting procedure, that selects reliable sources from this catalogue.
By comparing the best-fit source model with a zero-flux (pure background)
model, the algorithm calculates a detection likelihood $L$ for each source,
defined as $L=$ -ln$P$, where $P$ is the probability of the source being
caused by random background fluctuation. The final catalogue consists of
sources detected above a detection likelihood of 6 in the most sensitive
$0.2-2.3$ keV band (see also Fig. 5 in Salvato et al., 2022). Various criteria
are applied to exclude problematic sources from our analysis (for a detail
description see Sect. 2.1 in Mountrichas et al., 2022c). Furthermore, we
restrict our sources to those within the KiDS+VIKING area (Kuijken et al.,
2019; Hildebrandt et al., 2020). Near-infrared (NIR) photometry outside of
this region is shallow which significantly affects the accuracy and
reliability of the photoz calculations (Sect. 6.1 in Salvato et al., 2022).
This area encompasses 10294 extragalactic X-ray sources. Liu et al. (2022)
performed a systematic X-ray spectral fitting analysis on all the X-ray
systems, providing fluxes and luminosities —among other X-ray properties— for
the eFEDS sources. We use their posterior median, intrinsic (absorption-
corrected) X-ray fluxes in the $2-10$ keV energy band.

Mountrichas et al. (2022c) restricted the X-ray catalogue to $\rm 0.5<z<1.5$.
This was due to the unavailability of a large reference (non-AGN) sample at
higher redshifts, with which they could compare the SFR of their AGN. In our
analysis, we extend the X-ray catalogue to also include AGN at $\rm
1.5<z<2.5$, applying the same photometric and quality criteria as in
Mountrichas et al. (2022c). We also select only X-ray sources that satisfy the
mass completeness limits of the field at this redshift range, following the
method of Pozzetti et al. (2010) (for more details see Sect. 3.4 in
Mountrichas et al., 2022c). Our X-ray AGN catalogue consists of 2,860 sources
within a redshift range of $\rm 0.5<z<2.5$.

###  2.4 XMM-XXL

The XMM-Newton XXL survey (XMM-XXL; Pierre et al., 2016) is a medium-depth
X-ray survey that covers a total area of 50 deg2 split into two fields equal
in size, the XMM-XXL North (XXL-N) and the XMM-XXL South (XXL-S). The XXL-N
sample consists of 8445 X-ray sources. Of these X-ray sources, 5294 have SDSS
counterparts and 2512 have reliable spectroscopy (Menzel et al., 2016; Liu et
al., 2016). Mid-IR and near-IR was obtained following the likelihood ratio
method (Sutherland & Saunders, 1992) as implemented in Georgakakis & Nandra
(2011). For more details on the reduction of the XMM observations and the IR
identifications of the X-ray sources, see Georgakakis et al. (2017). The X-ray
absorption of the AGN has been calculated in Masoura et al. (2021), by
calculating the HR of the sources using the BEHR method (for details see Sect.
3.2 in Masoura et al., 2021). For those sources with no specz measurement, we
use their photoz calculations presented in Masoura et al. (2018). For their
estimation, a machine-learning technique has been applied (TPZ; Carrasco Kind
& Brunner, 2013), as described in Mountrichas et al. (2017); Ruiz et al.
(2018).

In our analysis, we use the X-ray sample presented in Masoura et al. (2018).
However, for consistency with the other X-ray datasets used in our analysis,
we apply the same photometric criteria mentioned in Sect. 2 as well as the
same quality criteria (see Sect. 3.1). Furthermore, we calculate the mass
completeness limits of the field at the three redshift intervals used in our
study and use only AGN that have M∗ above these limits. For this purpose, we
apply the method described in Pozzetti et al. (2010), utilizing equation (1)
in Mountrichas et al. (2021c) and using the $K$ band with $m_{lim}=20.0$ mag
(Georgakakis et al., 2017). Our calculations show that the mass completeness
of the sample is defined at $\rm
log\,[M_{*,95\%lim}(M_{\odot})]=10.55,11.00,11.20$ at $\rm 0.5<z<1.0$, $\rm
1.0<z<1.5$ $\rm and\,1.5<z<2.5$, respectively. Application of these criteria
results in 686 X-ray AGN in XXL-N.

Table 3: Number of X-ray AGN that live in quiescent (Q), star-forming (SF) and starburst (SB) galaxies, in the three redshift intervals used in our analysis. Only AGN with $10.0<\rm log\,[M_{*}(M_{\odot})]<12.0$ are included. In the parentheses we quote the fraction of each AGN population in the corresponding redshift interval. | total | $\rm 0.5<z<1.0$ | $\rm 1.0<z<1.5$ | $\rm 1.5<z<2.5$
---|---|---|---|---
SF | 3575 | 1775 (73%) | 920 (68%) | 880 (53.5%)
SB | 939 | 181 (7.5%) | 261 (19%) | 497 (30.3%)
Q | 917 | 474 (19.5%) | 177 (13%) | 266 (16.2%)

###  2.5 Final sample

Our final AGN dataset consists of 5639 X-ray sources, covering a redshift
range of $\rm 0.5<z<2.5$. Table LABEL:table_data, presents the number of
sources in each field and redshift interval. Fig. 1 presents the intrinsic
(absorption corrected) LX distribution of the AGN used in this study. Our
sources span more than three orders of magnitude in LX. The eFEDS field
contributes the majority of the luminous ($\rm
log\,[L_{X,2-10keV}(ergs^{-1})]>44$) sources used in our work, while most of
the AGN in the COSMOS field are low-to-moderate luminosity. Fig. 2 presents
the X-ray luminosity as a function of redshift, for the three redshift bins
used in our analysis. In our analysis, we also split the X-ray sources into
four M∗ bins. Table LABEL:table_data2, presents the number of sources in each
M∗ and redshift bin.

Figure 1: The intrinsic LX distributions of the total sample used in our
analysis (red shaded area) and of AGN in the four individual fields. eFEDS has
the most luminous AGN while the COSMOS field consists, mainly, of low-to-
moderate luminosity sources.  Figure 2: The intrinsic LX as a function of
redshift. The datapoints are colour-coded following the three redshift
intervals used in our measurements.

##  3 Analysis

In this section, we describe the SED fitting analysis applied to calculate the
AGN host galaxy properties and how this information is used to classify the
systems that AGN reside into quiescent, star-forming and starburst.

###  3.1 SED fitting

The host galaxy properties of the AGN in the the Bo$\rm\ddot{o}$tes, COSMOS-
Legacy, and the eFEDS fields, have been measured in Mountrichas et al.
(2021c); Mountrichas et al. (2022b); Mountrichas et al. (2022c), respectively,
by applying SED fitting. For that purpose, the CIGALE algorithm (Boquien et
al., 2019; Yang et al., 2020, 2022) has been utilized. CIGALE allows inclusion
of the X-ray flux in the fitting process and has the ability to account for
the extinction of the ultraviolet (UV) and optical emission in the poles of
AGN (Yang et al., 2020; Mountrichas et al., 2021a; Buat et al., 2021).

The same templates and parameter space has been used to fit the AGN in the
different datasets (Sect. 2). This minimises any systematic effects that may
be introduced due to the different modules and parametric grid used in the
fitting process. As already noted, in the case of the XXL-N sources, we do not
use the SFR and M∗ measurements of Masoura et al. (2018). Instead, we have
made new runs of CIGALE for these AGN, using the same SED fitting analysis, as
for the other three fields.

A detailed description of the fitting process is provided in Mountrichas et
al. (2021c); Mountrichas et al. (2022b); Mountrichas et al. (2022c). In brief,
a delayed star formation history (SFH) model with a function form $\rm
SFR\propto t\times exp(-t/\tau$) is used to fit the galaxy component. A star
formation burst is included (Ciesla et al., 2017; Małek et al., 2018; Buat et
al., 2019) as a constant ongoing period of star formation of 50 Myr. The
Bruzual & Charlot (2003) single stellar population template is used to model
the stellar emission. Stellar emission is attenuated following Charlot & Fall
(2000). The dust heated by stars is modelled following Dale et al. (2014). The
SKIRTOR template (Stalevski et al., 2012, 2016) is used for the AGN emission.
The values for the various parameters of the SED fitting are similar for the
AGN in all the four fields used in this work and are presented in Tables 1 in
Mountrichas et al. (2021c); Mountrichas et al. (2022b); Mountrichas et al.
(2022c).

To exclude sources that are badly fitted and have unreliable host galaxy
measurements, we follow the criteria applied in previous studies (e.g.,
Mountrichas et al., 2021b; Mountrichas et al., 2021c; Buat et al., 2021;
Mountrichas et al., 2022b; Mountrichas et al., 2022c). Specifically, we reject
AGN with a reduced $\chi^{2}>5$. We also exclude sources for which CIGALE
could not constrain their SFR and M∗. For that purpose, we use the two
estimates that CIGALE provides for each galaxy property. One value is
evaluated from the best-fit model and the other weighs all models allowed by
the parametric grid, with the best-fit model having the heaviest weight
(Boquien et al., 2019). A large difference between these two values indicates
that the probability density function (PDF) is asymmetric and a simple model
for the errors is not valid. Thus, in our analysis we only keep sources with
$\rm\frac{1}{5}\leq\frac{SFR_{best}}{SFR_{bayes}}\leq 5$ and
$\rm\frac{1}{5}\leq\frac{M_{*,best}}{M_{*,bayes}}\leq 5$, where SFRbest and
M∗,best are the best-fit values of SFR and M∗, respectively, and SFRbayes and
M∗,bayes are the Bayesian values estimated by CIGALE (e.g., Mountrichas et
al., 2021b; Mountrichas et al., 2021c; Buat et al., 2021; Mountrichas et al.,
2022b; Mountrichas et al., 2022c).

Figure 3: Hδ vs. Dn4000 for the 52 of the X-ray AGN from our analysis that are
also included in the LEGA-C catalogue. AGN hosted by SB galaxies have the
youngest stellar populations (smallest Dn4000 values) and have experienced
recent star formation bursts (high Hδ values). Q systems have the oldest stars
(highest Dn4000 values) and present no signs of recent bursts (smallest Hδ
values). AGN that live in SF galaxies have intermediate Hδ and Dn4000 values.

###  3.2 Classification of sources

To identify AGN host galaxies that are quiescent, we use the criterion
presented in Mountrichas et al. (2021c); Mountrichas et al. (2022b);
Mountrichas et al. (2022c). This criterion uses the distribution of the
specific SFR ($\rm sSFR=\frac{SFR}{M_{*}}$) of each dataset, at different
redshift intervals. These distributions present a lower, second peak at low
sSFR values. The galaxies that populate these lower peaks are classified as
quiescent. Systems that have sSFR that is 0.6 dex higher than the mean value
of the dataset are considered starburst (e.g., Rodighiero et al., 2011, 2015).
The remaining of the AGN population is then classified as star-forming. To
examine whether our results and conclusions, presented in the next Sections,
are sensitive to the classification method applied, we have also used
different criteria for the AGN host galaxy classification. For instance, we
identify as quiescent sources that have sSFR values 0.3 dex below the mean
value and as SB systems that have 0.3 dex above the mean sSFR value. We
confirm that our results and conclusion are not affected by the classification
method applied.

Furthermore, we examine whether our classification is consistent with the ages
of the stellar populations of the AGN hosts. For that purpose, we use the
information included in the Large Early Galaxy Astrophysics Census (LEGA-C)
catalogue. The LEGA-C catalogue includes data obtained from the LEGA-C survey
(van der Wel et al., 2021; Straatman et al., 2016). The third data release
contains galaxy spectra for 3741 unique sources that cover a redshift range
from 0.6 to 1.3. The sources lie within the UltraVISTA region of the COSMOS
field. The catalogue includes measurements for two stellar age sensitive
tracers, the equivalent width (EW) of H$\delta$ absorption and the Dn4000
index (Worthey & Ottaviani, 1997; Balogh et al., 1999). Dn4000 is small for
young stellar populations and large for old, metal rich galaxies. The EW of Hδ
rises rapidly in the first few hundred Myrs after a burst of star formation,
when O- and B-type stars dominate the spectrum and then decreases when A-type
stars fade (e.g., Kauffmann et al., 2003; Wu et al., 2018; Mountrichas et al.,
2022a). We cross-match our X-ray dataset with the LEGA-C catalogue, using a
radius of 1″and the optical coordinates provided in each catalogue. There are
52 AGN that are included in both catalogues, after applying several criteria
to exclude sources with unreliable measurements (for more details see Sect.
2.2.3 in Mountrichas et al., 2022a). Our goal is to examine how our classified
AGN host galaxies are distributed in the Hδ-Dn4000 space. This is shown in
Fig. 3. AGN hosted by SB galaxies have the youngest stellar populations
(smallest Dn4000 values) and have experienced (recent) star formation bursts
(high Hδ values). Q systems have the oldest stars (highest Dn4000 values) and
present no signs of recent bursts (smallest Hδ values). AGN that live in SF
galaxies have intermediate Hδ and Dn4000 values. Although this subset is
small, the results of this exercise indicate that the criteria we have applied
to classify AGN host galaxies into SB, SF and Q are robust.

Table LABEL:table_class presents the number of Q, SB and SF galaxies that host
X-ray AGN, in the three redshift intervals used in our analysis. Only sources
with $10.0<\rm log\,[M_{*}(M_{\odot})]<12.0$ are taken into consideration. As
expected, the fraction of AGN hosted by SB galaxies increases as we move to
higher redshifts while the fraction of quiescent systems increases at lower
redshifts (e.g. Shimizu et al., 2015; Koutoulidis et al., 2022). However, the
bulk of the accretion density of the Universe is associated with star-forming
systems, at all redshift ranges probed by our sample, in agreement with
previous studies (e.g., Georgakakis et al., 2014; Rodighiero et al., 2015). At
$\rm 0.5<z<1.5$ the fraction of AGN hosted by quiescent systems is similar to
that found by Mountrichas et al. (2021b) ($\sim 12\%$, see their Fig. 6). At
higher redshifts ($\rm 1.5<z<2.5$) the fraction of quiescent galaxies that
host AGN ($\sim 16\%$) is consistent with that found by Rodighiero et al.
(2015) ($\sim 11\%$). Similar fractions of quiescent galaxies ($15-20\%$) has
also been found in non-AGN systems (e.g., Fontana et al., 2009). Regarding,
AGN hosted by SB galaxies, in our dataset this fraction is significantly
higher compared to that quoted by Rodighiero et al. (2015) ($\sim 30\%$ vs.
$\sim 6\%$). Albeit, our sample spans significantly higher X-ray luminosities
compared to theirs, in this redshift regime ($\rm
43.5<log\,[L_{X,2-10keV}(ergs^{-1})]<45$ vs. $\rm
41.8<log\,[L_{X,2-10keV}(ergs^{-1})]<43.6$) and AGN that span such high X-ray
luminosities have been associated with increased SFR compared to less luminous
sources (e.g., Masoura et al., 2021; Mountrichas et al., 2021c, 2022c;
Pouliasis et al., 2022).

Figure 4: LX as a function of M∗. The results are colour-coded based on the
classification of the sources. Each panel corresponds to a different redshift
interval. Errors have been calculated using bootstrap resampling. SB systems
showcase an increased median LX compared to SF galaxies and Q systems show a
deficit of LX compared to SF galaxies, at similar M∗. A mild increase of LX
(by $\sim 0.5$ dex) with M∗ is found for all AGN host galaxy classifications,
at all redshifts spanned by our dataset.

Figure 5: LX/SFR as a function of M∗. The amplitude of LX/SFR is higher for Q
compared to SF and SB and presents only a weak dependence on stellar mass.

Figure 6: LX as a function of SFR for SB, SF and Q AGN host galaxies, at three
redshift intervals. At all redshift ranges, LX increases by a smaller factor
compared to the increase of SFR.

##  4 Results

In this Section, we study the AGN accretion (LX) and the LX/SFR ratio as a
function of stellar mass. All measurements presented are the median values of
the properties in each bin. Bins that include less than 20 sources are
excluded from our analysis. All errors presented have been calculated using a
bootstrap resampling method (e.g., Loh, 2008). We note that in our analysis
only X-ray detected sources are included. Previous similar studies have also
included undetected AGN in their measurements, by applying stacking analysis
(e.g., Mullaney et al., 2012; Rodighiero et al., 2015; Yang et al., 2018).
This allows to average the black hole growth rate over long time intervals.
However, using only detected sources gives the advantage that the measured
X-ray luminosities provide a tracer of black hole luminosity independent of
the duty cycle (Carraro et al., 2022). In other words, the quantities we
present below trace the average properties of the active population of
galaxies at any given epoch.

###  4.1 LX vs M∗

Fig. 4 presents the X-ray luminosity vs. the host galaxy stellar mass, for SB,
SF and Q systems (blue squares, green circles and red triangles,
respectively), at different redshift intervals, as labeled at the top of each
panel. We notice that SB systems showcase an increased median X-ray luminosity
(and thus accretion rate) compared to SF galaxies, and Q galaxies show a
deficit of LX compared to SF systems, at similar M∗. This is in agreement with
the results of previous similar studies that attribute this behaviour, in
conjunction with the SFR-M∗ (MS) relation, to a gradual exhaustion of the gas
that fuels both the SMBH and the SFR (e.g., Rodighiero et al., 2015), possibly
accompanied by a proportional variation in Eddington ratios (e.g., Carraro et
al., 2022).

Furthermore, LX presents a mild increase (up to 0.5 dex) with M∗, in
particular at high redshifts ($\rm 1.5<z<2.5$). We verified (Appendix A) that
this trend is consistent in all four fields considered in this work. In
COSMOS, our average LX-M∗ relations generally follow similar patterns, but
have lower normalizations (in line with Carraro et al., 2022, that used
detected X-ray sources in COSMOS). This is because the COSMOS field is
characterized by an LX distribution skewed towards lower values (Fig. 1).
Similar trends have also been found in previous studies (e.g., Mullaney et
al., 2012; Rodighiero et al., 2015; Yang et al., 2018; Carraro et al., 2020,
2022), but a more direct comparison can be made with the results of Carraro et
al. (2022) that used only the detected X-ray sources of the Carraro et al.
(2020) sample. They find a (mild) increase of LX with M∗, similar to our
results.

The increase of LX with M∗ we detect, is most probably driven by the combined
effect of the underlying slope of the MBH-M∗ relation and the variation of the
mean Eddington ratio $\lambda\propto$LX/M∗ (Carraro et al., 2022). On the
assumptions of a relatively constant MBH-M∗ relation for AGN up to $\rm z\sim
2$ (e.g., Suh et al., 2019), the gradual flattening and overall decrease of
our LX-M∗ with cosmic time, could be ascribed to a decrease of the underlying
mean Eddington ratio, especially evident for the most massive galaxies
allegedly hosting, on average, the most massive SMBHs. Indeed, as we show in
Appendix B (Fig. 10), the increase in LX with M∗ at high redshift is mostly
driven by the most luminous sources, a possible signature that SMBHs in the
more massive galaxies tend to accrete at larger rates at earlier epochs.

###  4.2 Lx/SFR vs M∗

In Fig. 5, we plot the ratio $\rm log(L_{X}/SFR)$ as a function of M∗, for
different redshift intervals, as labelled at the top of each panel. Our data
suggest that SB have, at all redshifts explored here, a median LX/SFR ratio a
factor of $\sim 2-3$ and $\sim 10$ lower than SF and Q galaxies, respectively,
with only a weak dependence on stellar mass. Previous studies have found
either a positive correlation (Rodighiero et al., 2015; Yang et al., 2018;
Carraro et al., 2020) or a flat relation (Mullaney et al., 2012) between $\rm
log(L_{X}/SFR)$ (or BHAR/SFR) and M∗, with the caveat that these studies
included stacked sources in their analysis. We also note that the amplitude of
the $\rm log(L_{X}/SFR)$-M∗ relation in our results is about an order of
magnitude higher than the amplitude presented in the aforementioned studies.
This is mainly due to the inclusion, in our analysis, of X-ray sources
detected in wide fields (eFEDS, XMM-XXL, Bo$\rm\ddot{o}$tes), i.e., our sample
includes a larger number of luminous AGN (Fig. 1). Furthermore, application of
stacking analysis in these previous studies have allowed them to include very
faint sources in their calculations. The different luminosities probed among
our study and the previous works mentioned above, affect the amplitude of the
$\rm log(L_{X}/SFR)$-M∗ relation, however, we do not expect to affect the
overall trends (see Appendix B).

To examine if the observed trends in the $\rm log(L_{X}/SFR)$-M∗ relation are
driven by LX or SFR, in Fig. 6 we plot LX vs. SFR, for SF, SB and Q AGN host
galaxies, in the same redshift intervals as in the previous Figures. The
median SFR increases by a large factor (up to 3.5 dex) across M∗, when moving
from Q to SB galaxies. The corresponding increase of the mean LX is
significantly smaller ($\sim 0.7$ dex), at all redshift intervals and for all
AGN host galaxy classifications. We thus conclude that the strong increase
observed in the LX/SFR ratio from SB to Q at fixed M∗ is mostly driven by a
significant drop in SFR.

In Appendix B, we explore the $\rm log(L_{X}/SFR)$ as a function of M∗, for
luminous and low to moderate LX AGN. Similar trends are found for both AGN
populations (Fig. 11). Luminous AGN have higher $\rm log(L_{X}/SFR)$ amplitude
compared to less luminous X-ray sources, at low redshifts. However, this
picture reverses, for AGN hosted by Q galaxies, at the highest redshift
interval spanned by our dataset. This behaviour is due to how the LX and SFR
vary as a function of M∗ for the different AGN host galaxy classifications and
for different LX and redshifts (for more details see Appendix B).

We conclude that our results show that Q systems present the highest amplitude
in the $\rm log(L_{X}/SFR)$-M∗ relation due to their lowest SFRs, while SB
galaxies have the smallest $\rm log(L_{X}/SFR)$ ratio due to their highest
SFRs.

Figure 7: Evolution of SFR (top panels), LX (middle panels) and their ratio
$\rm logL_{X}/SFR$ (bottom panels) with time. Each row corresponds to a
different M∗ interval, as indicated as the top of each plot. Filled symbols
present the results for galaxies that host X-ray AGN. Open symbols connected
with dashed lines show the results for non-AGN galaxies (top panels). The
black lines present the predictions of hydrodynamical models that follow the
black hole growth and the star formation during a galaxy merger (Di Matteo et
al., 2005; Hopkins, 2012).

##  5 Discussion

As anticipated in Sect. 1, studying AGN statistical samples averaged in bins
of stellar mass, SFR and LX, provides clues to the overall connection among
these properties, but it does not necessarily provide insight into the actual
evolutionary phases, if any, between the starburst and star forming/quiescent
phases. To extract possible evolutionary sequences characterising AGN galactic
hosts, we need to connect the Lx and SFR of SB, SF and Q into a time sequence,
and check whether such sequences are comparable to predictions from
theoretical models (e.g., Rodighiero et al., 2015). To this purpose, in this
Section, we study how the LX/SFR ratio varies along the different phases of
the galaxy, from the SF to the SB and Q phases, and compare our observations
with theoretical light curves from SMBH-galaxy evolutionary models.

###  5.1 Evolution of LX/SFR at different galaxy evolutionary sequences

We start by examining how the LX/SFR evolves as the galaxy "transits" from one
phase to the other. We assume that the (host) galaxy is initially in a star
forming phase. Then, the galaxy merges with another galaxy and the gas is
converted to stars in intense bursts of star formation (SB phase), $\sim 1.5$
Gyr later. Finally, when nearly all the gas has been depleted, the galaxy
enters the Q state (at $\sim 2$ Gyr) (e.g., Di Matteo et al., 2005). The
transition times associated to each phase have been chosen on the assumption
that these mirror those suggested by theoretical models, which are, in a
typical binary merger scenario at $\rm z\sim 1-2$, $\sim 1$ Gyr between the SB
and SF phases and 0.5 Gyr between the SF and Q phase (e.g., Di Matteo et al.,
2005; Springel et al., 2005; Hopkins, 2012).

In Fig. 7, the top, middle and bottom panels show, respectively, the putative
evolution of the SFR, LX and the LX/SFR ratio, when the galaxy "transits" from
the SF to the SB and then to the Q phase, assuming that these are correlated
to each other. The results for different M∗ selections are presented, as
indicated in the top of each panel. We do not show results for the lowest
stellar mass bin ($\rm 10<log\,[M_{*}(M_{\odot})]<10.5$), since there are not
enough Q AGN host galaxies in this M∗ bin in our sample to allow robust
calculations. The estimated median quantities reported are largely independent
of the exact cuts in redshift and/or stellar mass interval chosen, and still
hold true when assuming some evolution in the host galaxy stellar mass,
following semi-empirical models, as discussed in the next section. We start
discussing the bottom panels of Fig. 7, where we show that the LX/SFR tends to
gradually increase by up to $\sim 1-1.5$ dex when the host galaxy becomes Q.

To examine what drives the variation of the LX/SFR with time, in the top and
middle panels of Fig. 7, we normalize the SFR and the LX, respectively, to the
SFR and LX of the SF (main sequence) galaxies and study them as a function of
time. Both SFR and LX increase as the galaxy moves from the SF to the SB and
then decrease in the Q state. We find that the SFR increases by nearly an
order of magnitude in the transition from the SF to the SB stage and then
drops by almost two orders of magnitude from the SB to the Q phase. On the
other hand, LX increases by, only, up to a factor of two in the SB phase and
then, in the Q phase, drops to similar values with those observed for SF
galaxies. These results suggest that the LX/SFR fluctuations we observe during
the different galaxy evolutionary sequences are largely driven by variations
in the SFR. It is also interesting to note that the aforementioned
fluctuations of SFR and LX are nearly independent of both redshift and galaxy
stellar mass.

Our next step is to compare our inferred SFR evolutionary tracks with those
from galaxies not detected as AGN. Mountrichas et al. (2021c); Mountrichas et
al. (2022b); Mountrichas et al. (2022c) constructed galaxy control samples
applying the same photometric requirements and SED fitting techniques (same
templates and parameter values) as in the X-ray sources adopted in this work.
Non-X-ray AGN systems have also been excluded from the control sample (for
more details see Sect. 3.3 in e.g., Mountrichas et al., 2022c). We, thus,
combine the galaxies from the Mountrichas et al. datasets and compile a sample
of $\sim 130,000$ sources within the mass completeness limits, and similarly
classify these galaxies into SB, SF and Q, following the same method as for
the X-ray AGN (see Sect. 3.2). We then normalize the SFR of each class based
on the SFR of the SF galaxies. The results are shown in the top panels of Fig.
7 (open symbols connected with dashed lines), for different M∗ regimes. We
find that for non-AGN sources the SFR increases in the SB phase and drops in
the Q stage in a similar manner as for the X-ray AGN. The similarity in the
variation of SFR from SB to Q galaxies between AGN and non-AGN galaxies,
suggest that the AGN has a negligible impact on the quenching. Theoretical
studies have suggested that the apparent lack of a (inverse) correlation
between AGN activity and SFR found in observational works, could be induced by
the different timescales characterizing the two processes (e.g., Hickox et
al., 2014; Ward et al., 2022). Theoretical models indeed suggest that a delay
of a few millions of years is required for the impact of a single episode of
nuclear activity to have a measurable effect on star formation. The presence
of such delays may explain, at least in part, why all galaxies of similar
stellar mass in our sample share similar drops of the SFR from the SB/SF to
the Q state, irrespective of their AGN activity. In other words, the
combination of a delay and of relative short AGN lifetimes (e.g., Martini,
2004; Shankar et al., 2004), could both contribute to wash out causal links
between AGN luminosities and host galaxies SFR.

Another viable solution could be to start from a SB phase, on the assumption
that the galaxy is forming out of the cooling of pristine gas clouds, and then
gradually converts into a SF and finally into a Q phase (e.g., Granato et al.,
2004, 2006; Lapi et al., 2006, 2018). Although, without clear age and
structural discriminators, our current data cannot safely distinguish between
these two evolutionary routes, our measurements presented in Fig. 4, suggest
that SB could be the starting phase in a galaxy’s lifetime.

In Fig. 8, we plot the SFR (top panel), the LX (middle panel) and their ratio
Lx/SFR (bottom panel), as a function of time, for AGN host galaxies with $\rm
11.0<log\,[M_{*}(M_{\odot})]<11.5$, assuming that the galaxy starts its life
in a SB phase. The x-axis in the top and middle panels present the galactic
age (in log) to allow us to compare our observational measurements with the
theoretical model of Granato et al. (2004) (see next Section). Similarly to
the results presented above, the LX/SFR ratio increases by up to $\sim 1-1.5$
dex when the host galaxy becomes Q. Both the SFR and LX drop as the galaxy
evolves from the SB to the SF and then the Q phase. This decrease is higher
for SFR ($\sim 2$ dex) compared to LX ($\sim 1$ dex). Similar results are
found for the other stellar mass bins.

Figure 8: Evolution of SFR (top panel), LX (middle panel) and their ratio $\rm
logL_{X}/SFR$ (bottom panel) with time, for AGN host galaxies with $\rm
11.0<log\,[M_{*}(M_{\odot})]<11.5$. In this exercise, we assume that the
galaxy starts its life in a SB phase and then it enters the SF and finally
becomes Q. The black lines show the predictions of the model presented in
Granato et al. 2004. The x-axis in the top and middle panels, present the
galactic age (in log), as shown in Figures 2 and 3 in Granato et al. 2004.

###  5.2 Comparison with theoretical predictions

To set the stage between our data and theoretical models, we follow Rodighiero
et al. (2015) and compare in Fig. 7 our empirical renditions of AGN light
curves with those extracted from hydrodynamic simulations inclusive of AGN
feedback (Di Matteo et al., 2005; Hopkins, 2012) We take the typical light
curve of a galaxy hosted in a dark matter halo of 160 km/s (black lines in
Fig. 3 of Rodighiero et al., 2015) 111Our results would be similar for other
choices of dark matter haloes. As shown by Rodighiero et al. (2015), in fact,
light curves in larger haloes are predicted to be very similar.. The empirical
light curves suggest that the AGN luminosity increases only by a factor of a
few from the SF to the SB phase, and a similar drop is observed in the Q
phase. This behaviour is not entirely aligned with what predicted by models
which suggest that during the growth phase the BH grows by a factor of
$>100-1000$ depending on the assumed initial BH seed. We also checked, via
Monte Carlo simulations, that typical random measurement errors of 0.5 Gyr in
galaxy ages and of 0.15 dex in X-ray luminosities, are not capable to wash out
the peak of the AGN light curve and to reconcile data with models, thus
suggesting that this discrepancy we observe between predicted and observed AGN
X-ray light curves is robust.

On the other hand, the SFR is observed to actually behave as predicted, with
an increase of a factor of a few up to an order of magnitude, followed by a
drop of two orders of magnitudes, similar to what expected in AGN feedback
models where an AGN wind is expelling mass from the host galaxy or at least
injecting energy/momentum thus decreasing the possibility for any further gas
fragmentation.

In Fig. 8, we assume instead that a galaxy starts its life in a SB phase and
compare our observational measurements with the predictions of the model
presented in Granato et al. (2004). We find even in this case that the SFR
evolves from the SB to the SF and then to the Q phase, as predicted (top panel
of Fig. 8). However, LX (middle panel of Fig. 8) drops by $\sim 0.5$ dex,
compared to 1.5 dex predicted by the theoretical model. However, including
observational uncertainties may improve the match to the models in this second
type of models, at least at intermediate redshifts.

Taking these results at face value, we would conclude that in order to achieve
a condition where the AGN luminosity drops only by a factor of a few from the
peak whilst the SFR decreases significantly, a high AGN feedback efficiency
and/or coupling with the interstellar medium (ISM) should be characterising
AGN feedback processes.

The relatively limited pre-peak increase in the AGN luminosity, and to a
lesser extent in our SFRs, observed in our data, may signal the fact that in
our samples the SF and SB phases are close to each other. In addition, if our
simulations inclusive of observational errors, are not able to resolve the
tension in the discrepancy between predicted and observed AGN light curves, at
least for some models (e.g., Springel et al., 2005; Hopkins et al., 2006),
then it would imply that the post-peak phase of the AGN light curve does not
decrease as sharply as predicted by some models, and this is in fact what seen
by e.g., Lapi et al. (2014) to reproduce their FIR galaxy light curves, and
also by some continuity equation models that require extended light curves
(e.g., Shankar et al., 2013; Aversa et al., 2015).

Interestingly, we find that for each of our galaxy classes, the increase in
X-ray AGN luminosity is always smaller than the one in SFR within the same
time lag, and in the SB the X-ray luminosity is nearly constant, despite the
SFR still increasing by nearly an order of magnitude. Again, this behaviour in
our data is not entirely paralleled especially in the merger models, which
could either be a consequence of selection effect and/or observational errors,
but also it could reflect the fact that the BH has reached a self-regulated
state which prevents further substantial growth.

We note that all our results discussed above remain valid under the assumption
that the stellar mass does not grow more than a factor of $\sim 2$ within the
timeframe considered, i.e., $\sim 2$ Gyr, or even by following back in time
the stellar progenitors. We checked in fact that by selecting the SF and SB
progenitors of the quiescent galaxies of a given chosen M∗ following, e.g.,
the average stellar mass growth histories of Moster et al. (2018), or even
Grylls et al. (2019); Behroozi et al. (2019) (see Fig. 3 in Shankar et al.,
2020), we find very similar results for the LX, SFR evolution with time as the
ones reported in Fig. 7.

##  6 Conclusions

We used $\sim 5500$ X-ray AGN detected in four fields, namely the COSMOS-
Legacy, the Bo$\rm\ddot{o}$tes, the XMM-XXL and the eFEDS fields. The X-ray
sources span a redshift range of $\rm 0.5<z<2.5$ and about three orders of
magnitude in X-ray luminosity ($42.0<\rm log\,[L_{X,2-10keV}(ergs^{-1})]<45$),
while their stellar mass ranges from $10.0<\rm log\,[M_{*}(M_{\odot})]<12.0$.
These sources meet strict photometric selection requirements and various
selection criteria have been applied to ensure that only sources with robust
host galaxy measurements are included in the analysis. The latter have been
calculated via SED fitting, using the CIGALE code. Furthermore, our final
sample consists only of AGN that satisfy the mass completeness limits of the
field they belong. The X-ray sources are then classified into three classes,
based on the star formation activity of the host galaxy, i.e., SF, SB and Q.
For the classification, the sSFR of each system was used. Our final X-ray
sample, consists of 3575 SF galaxies, 939 SB and 917 Q systems. The fraction
of each class depends on the redshift, but at all redshift ranges used in our
analysis, the bulk of the black hole accretion occurs in SF systems. Our main
results can be summarised as follows:

  * $\bullet$

We found that SB systems have increased AGN accretion (LX) compared to SF
systems, at similar M∗. Q galaxies present the lowest LX among the three
classifications. We also find a mild increase of LX with M∗, possibly related
to a decrease of Eddington ratio with increasing stellar mass and at fixed
MBH-M∗ slope (e.g., Carraro et al., 2022). Our results also show a gradual
flattening and overall decrease with cosmic time of the LX at fixed M∗, which
suggests a decrease of the underlying mean Eddington ratio, in particular for
the most massive galaxies.

  * $\bullet$

The amplitude of the LX/SFR ratio is higher for Q systems compared to SF and
SB galaxies. We also find that the LX/SFR-M∗ relation is nearly flat, in
particular for the most massive galaxies.

  * $\bullet$

The ratio of LX/SFR decreases by $\sim 0.5$ dex when the galaxy enters the SB
phase and then increases by almost an order of magnitude in the Q phase. This
variation is mostly driven by variations in the SFR, whilst the LX remains
roughly constant from the SF to the SB and the Q states.

  * $\bullet$

The fluctuation of SFR is consistent with what predicted by theoretical
models, whereas the behaviour of LX is not in line with merger models that
predict an increase of LX by a factor of $>100-1000$ in the SB phase.

  * $\bullet$

We also study the evolution of SFR for a galaxy control sample of non-AGN
systems and found that it is very similar to that of X-ray AGN. This
similarity in the variation of SFR between the two populations may suggest
that AGN have a negligible impact on the star formation quenching or that the
two processes proceed on different timescales.

## Acknowledgements

GM acknowledges support by the Agencia Estatal de Investigación, Unidad de
Excelencia María de Maeztu, ref. MDM-2017-0765. FS acknowledges partial
support from the European Union’s Horizon2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 860744.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A LX vs M∗ in different fields

In this Section, we examine the LX-M∗ relation separately in each one of the
fields, used in our analysis. Our goal is to check whether the measurements in
each field are compatible with each other. For this exercise, we have used
sources at $\rm 0.5<z<1.0$ since in this redshift range our datasets have the
largest number of available sources (Table LABEL:table_data).

In Fig. 9, we present the LX as a function of M∗ for the Bo$\rm\ddot{o}$tes,
COSMOS, eFEDS and XMM-XXL fields, for AGN hosted by SF, SB and Q galaxies.
Based on the results, the measurements are consistent among the various field.
COSMOS has lower LX values, for all three classifications, but this is
expected since the vast majority of X-ray AGN in this field have low to
moderate luminosity sources ($\rm log\,[L_{X,2-10keV}(ergs^{-1})]<44$, Fig.
1). Most importantly, the trends observed are similar in all four fields.
Therefore, we conclude that the four fields give consistent results with each
other.

Figure 9: LX vs M∗ at each field separately, for sources with $\rm 0.5<z<1.0$.
All fields give consistent results within the same redshift, LX and M∗ bins.
COSMOS LX measurements are by $\sim 0.5$ dex lower, as expected from the LX
distribution of AGN in this field. Most importantly, the observed trends are
similar between the four fields.

##  Appendix B Dependence on LX

In this Section, we examine if the observed trends of LX and the ratio LX/SFR
as a function of M∗, presented in Sections 4.1 and 4.2, are different between
low to moderate LX ($\rm log\,[L_{X,2-10keV}(ergs^{-1})]<44$) and high LX AGN
($\rm log\,[L_{X,2-10keV}(ergs^{-1})]>44$).

Fig. 10, presents the LX-M∗ relation for the two AGN populations. The (mild)
increase of LX with M∗ seems to be mainly driven by the most luminous X-ray
sources.

In Fig. 11, we examine the ratio $\rm log(L_{X}/SFR)$ as a function of M∗, for
luminous and low to moderate LX AGN. We find similar trends for both AGN
populations, with luminous sources to have higher $\rm log(L_{X}/SFR)$
amplitude compared to less luminous X-ray sources, at low redshifts. However,
at higher redshifts, luminous AGN hosted by Q galaxies present a lower $\rm
log(L_{X}/SFR)$ amplitude compared to their lower LX counterparts. To
investigate this behaviour further, in Fig. 12, we plot the LX vs. SFR for the
two AGN populations. Based on these results, at low redshifts (top panel of
Fig. 12), for all AGN host galaxy classifications, luminous and less luminous
AGN have similar SFR. At higher redshifts (bottom panel of Fig. 12) this is
true only for the SB galaxies, while Q systems that host less luminous AGN
have significant lower SFR (by $\sim 1$ dex) compared to their more luminous
counterparts.

Figure 10: LX as a function of M∗, for AGN with $\rm
log\,[L_{X,2-10keV}(ergs^{-1})]>44$ and $\rm
log\,[L_{X,2-10keV}(ergs^{-1})]<44$. The mild increase of LX with M∗ is mostly
driven by the most luminous sources.

Figure 11: The difference of the amplitude of LX/SFR between luminous AGN
($\rm log\,[L_{X,2-10keV}(ergs^{-1})]>44$) and low-to-moderate luminosity AGN
($\rm log\,[L_{X,2-10keV}(ergs^{-1})]<44$). At $\rm 0.5<z<1.0$ luminous AGN
have higher LX/SFR values compared to their lower LX counterparts, for all AGN
host galaxy classifications. At higher redshifts, luminous AGN hosted by Q
systems present a lower LX/SFR compared to lower LX sources.

Figure 12: LX as a function of SFR for SB, SF and Q AGN host galaxies, at
three redshift intervals, for luminous and low-to-moderate LX AGN. At low
redshifts (top panel), both AGN populations have similar SFR. However, at high
redshifts (bottom panel) this is true only for the SB systems.