\section{Introduction}

Recent observational and theoretical studies suggest that
gravitational microlensing can induce  variability not only in
optical light, but also in the X-ray emission of lensed QSOs
\citep{Chart02a,Chart04,Dai03,Dai04,pj06,Pop01,Pop03a,Pop03b,Pop06a,Pop06b}.
Variability studies of QSOs indicate  that the size of the X-ray
emitting region is significantly smaller ($\sim$ several light
hours), than the optical and UV emitting regions ($\sim$ several
light days).

Gravitational lensing is achromatic (the deflection angle of a light
ray does not depend on its wavelength), but it is clear that if the
geometries of the emitting regions at different wavelengths are
different then chromatic effects could occur. For example, if the
microlens is a binary star or if the microlensed source is extended
\citep{Griest_Hu92,Griest_Hu93,Bog_Cher95a,Bog_Cher95b,Zakh97,Zakh_Sazh98,Pc04}
different amplifications in different spectral bands can be present.
Studies aiming to determine the influence of microlensing on the
spectra of lensed QSOs need to take into account the complex
structure of the QSO central emitting region \citep{Pc04}. Since the
sizes of the emitting regions are wavelength dependent, microlensing
by stars in the lens galaxy may lead to a wavelength dependent
magnification. For example, Blackburne et al. (2006) reported such a
'flux anomaly' in quadruply imaged quasar 1RXS J1131-1231. In
particular, they found discrepancies between the X-ray and optical
flux ratio anomalies. Such anomalies in the different spectral band
flux ratios can be attributed to micro- or milli-lensing in the
massive lensing halo. In the case of milli-lensing one can infer the
nature of substructure in the lensing galaxy, which can be connected
to Cold Dark Matter (CDM) structures (see e.g. Dobler \& Keeton
2006). Besides microlensing, there are several mechanisms which can
produce flux anomalies, such as extinction and intrinsic
variability. These anomalies were discussed in \citet{Pc04} where
the authors gave a method that can aid in distinguishing between
variations produced by microlensing from ones resulting from other
effects. In this paper we discuss consequences of variations due to
gravitational microlensing, in which the different geometries and
dimensions of emitting regions of different spectral bands are
considered.

The influence of microlensing on QSO spectra emitted from their
accretion discs in the range from the X-ray to the optical
spectral band is analyzed. Moreover, assuming different sizes
of the emitting regions, we investigate the microlensing time
scales for those regions, as well as a time dependent response
in amplification of different spectral bands due to
microlensing. Also, we give the estimates of microlensing time
scales for a sample of lensed QSOs.

In Section 2, we describe our model of the quasar emitting regions
and a model of the micro-lens. In Section 3 we discuss the time
scales of microlensing and in Section 4 we present our results.
Finally, in Section 5, we draw our conclusions.

\section{A model of QSO emitting regions and microlens}

\subsection{A model of the QSO emitting regions}

In our models we adopt a disc geometry for the emitting regions of
Active Galactic Nuclei (AGN)  since the most widely accepted
paradigm for AGN includes a supermassive black hole fed by an
accretion disc. \citet{Fabian89} calculated spectral line profiles
for radiation emitted from the inner parts of accretion discs and
later on such features of  Fe $K\alpha$ lines were discovered by
\citet{Tanaka95} in Japanese ASCA satellite data for Seyfert galaxy
MGC-6-30-15. Moreover, the assumption of a disc geometry for the
distribution of the emitters in the central part is supported by the
spectral shape of the Fe K$\alpha$ line in AGN (e.g.
\citet{Nandra_1997}, see also results of simulations
\citep{Zak_Rep99,Zak_Rep02,ZKLR02,Zak_Rep03a,Zak_Rep03b,ZR_Nuovo_Cim03,Zak_Rep03c,ZR_5SCSLSA,ZR_NA_05,Zak_SPIG04,ZR_NANP_05}).
On the other hand, very often a bump in the UV band is present in
the spectra of AGN, that indicates that the UV and optical continuum
originates in an accretion disk.

We should note here that probably most of the X-ray emission in the
1--10 keV energy range originates from inverse Compton scattering of
photons from the disc by electrons in a tenuous hot corona. Proposed
geometries of the hot corona of AGN include  a spherical corona
sandwiching the disc and a patchy  corona made of a few compact
regions covering a small fraction of the disc (see e.g.
\citet{mz07}). On the other hand, it is known that part of the
accretion disc that emits in the 1--10~keV rest-frame band (e.g. the
region that emits the continuum Compton reflection component and the
fluorescent emission lines) is very compact and may contribute to
X-ray variability in this energy range. In order to study the
microlensing time scales, one should consider the dimensions of the
X-ray emitting region which are very important for the integral flux
variations due to microlensing. The geometry of the emitting regions
adopted in microlensing models will affect the simulated spectra of
the continuum and spectral line emission \citep{Pc04,pj06}. In this
paper we assume that the AGN emission from the optical to the X-ray
band originates from different parts of the accretion disc.

Also we assume that we have a stratification in the disc, where the
innermost part emits the X-ray radiation and outer parts the UV and
optical continuum emission. To study the effects of microlensing on
a compact accretion disc we use the ray-tracing method, considering
only those photon trajectories that reach the sky plane at a given
observer angle (see, e.g. Popovi\'c et al. 2003a,b and references
therein). The amplified brightness with amplification $A(X,Y)$ for
the continuum is given by

\begin{equation}
I_{C} (X,Y;E_{obs}) = { {I_{P}}}  (E_{obs},T(X,Y)) \cdot A(X,Y),
\end{equation}
where $T(X,Y)$ is the temperature, $X$ and $Y$ are the impact
parameters which describe the apparent position of each point of the
accretion disc image on the celestial sphere as seen by an observer
at infinity, $E_{\rm obs}$ is the observed energy, $I_P$ is an
emissivity function.

In the standard Shakura-Sunyaev disc model \citeyearpar{Shakura73},
accretion occurs via an optically thick and geometrically thin disc.
The effective optical depth in the disc is very high and photons are
close to the thermal equilibrium with electrons. The surface
temperature is a function of disc parameters and results in the
multicolor black body spectrum. This component is thought to explain
the 'blue bump' in AGN and the soft X-ray emission in galactic black
holes. Although the standard Shakura-Sunyaev disc model does not
predict the power-law X-ray emission observed in all sub-Eddington
accreting black holes, the power law for the X-ray emissivity in AGN
is usually adopted (see e.g. \cite{Nandra_1999}). But one can not
exclude other approximations for emissivity laws, such as black-body
or modified black-body emissivity laws. Moreover, we will assume
that the emissivity law is the same through the whole disc.
Therefore we used the black-body radiation law. The disc emissivity
is given as (e.g. \cite{Jarosz_1992}):

$$I_P(X,Y;E)=B[E,T_s(X,Y)],$$
where
\begin{equation}
 B\left( {E,T_s(X,Y)} \right) = {\frac{{2E
^{3}}}{{h^2c^{2}}}}{\frac{{1}}{{e^{{{E
}
\mathord{\left/ {\vphantom {{h\nu}  {kT}}} \right.
\kern-\nulldelimiterspace} {kT_s(X,Y)}}} - 1}}},
\end{equation}
where $c$ is the speed of light, $h$ is the Planck constant,
$k$ is the Boltzmann constant and $T_s(X, Y)$ is the  surface
temperature. Concerning the standard accretion disc
\citep{Shakura73}, here we assumed that (see \citet{Pop06a}):
\begin{equation}
T_s(X, Y) \sim r^{-3/2}(X,Y)(1-r^{-1/2}(X,Y))^{4/5} \,{\rm K},
\end{equation}
taking that an effective temperature in the innermost part (where X-ray is
originated) is in an interval from 10$^7$ K to
10$^8$ K.

\begin{figure*}
\centering
\includegraphics[width=0.49\textwidth]{fig1a.eps}\hfill
\includegraphics[width=0.49\textwidth]{fig1b.eps}
\caption{Magnification map of the QSO 2237+0305A image (left) and of
a "typical" lens system (right). The white solid lines represent
three analyzed paths of an accretion disc center: horizontal
($y=0$), diagonal ($y = -x$) and vertical ($x=0$).}
\end{figure*}

\subsection{A model for microlensing}

To explain the observed microlensing events in quasars, one can use
different microlensing models. The simplest approximation is a
point-like microlens, where microlensing is caused by some compact
isolated object (e.g. by a star). Such a microlens is characterized
by its Einstein Ring Radius in the lens plane:
$ERR=\sqrt{\dfrac{4Gm}{c^2}\dfrac{D_lD_{ls}}{D_s}}$ or by the
corresponding projection in the source plane:
$R_E=\dfrac{D_s}{D_l}ERR=\sqrt{\dfrac{4Gm}{c^2}\dfrac{D_sD_{ls}}{D_l}}$,
where $G$ is the gravitational constant, $c$ is the speed of light,
$m$ is the microlens mass and $D_l$, $D_s$ and $D_{ls}$ are the
cosmological angular distances between observer-lens,
observer-source and lens-source, respectively. In most cases we can
not simply consider that microlensing is caused by an isolated
compact object but we must take into account that the
micro-deflector is located in an extended object (typically, the
lens galaxy). Therefore, when the size of the Einstein ring radius
projection $R_E$ of the microlens is larger than the size of the
accretion disc and when a number of microlenses form a caustic net,
one can describe the microlensing in terms of the crossing of the
disc by a straight fold caustic (Schneider et al. 1992). The
amplification at a point of an extended source (accretion disc)
close to the caustic is given by \citet{Chang84} (a more general
expression for a magnification near a cusp type singularity is given
by \citet{Schneider92,Zakharov95}):
\begin{equation}
A(X,Y)=A_0+K\sqrt{\frac{r_{\rm caustic}}{\kappa(\xi-\xi_c)}}\cdot
H(\kappa(\xi-\xi_c)),
\label{eq04}
\end{equation}
where $A_0$ is the amplification outside  the caustic and
$K=A_0\beta$ is the caustic amplification factor, where $\beta$ is
constant of order of unity (e.g. Witt et al. 1993). The "caustic
size" $r_{\rm caustic}$ is the distance in a direction
perpendicular to the caustic for which the caustic amplification is
1, and therefore this parameter defines a typical linear scale for
the caustic in the direction perpendicular to the caustic. $\xi$ is
the distance perpendicular to the caustic in gravitational radii
units and $\xi_c$ is the minimum distance from the disc center to
the caustic. Thus,
\begin{equation}
\xi_c={\sqrt{X_c^2+Y_c^2}},
\end{equation}

\begin{equation}
{\rm tg}\alpha=\frac{Y_c}{X_c},
\end{equation}
 and
\begin{equation}
\xi=\xi_c+\frac{(X-X_c){\rm tg}\phi+Y_c-Y}{\sqrt{{\rm tg}^2\phi+1}},
\end{equation}
 where
$\phi=\alpha+{\pi/2}$. $ H(\kappa(\xi-\xi_c))$ is the Heaviside
function, $H(\kappa(\xi-\xi_c))=1$, for $\kappa(\xi-\xi_c)>0$,
otherwise it is 0. $\kappa$ is $\pm 1$, it depends on the direction
of caustic motion; if the direction of the caustic motion is from
approaching side of the disc $\kappa=-1$, otherwise it is +1. Also,
in the special case of caustic crossing perpendicular to the
rotating axis $\kappa=+1$ for direction of caustic motion from $-Y$
to $+Y$, otherwise it is $-1$. A microlensing event where a caustic
crosses over an emission region can be described in the following
way: before the caustic reaches the emission region, the
amplification is equal to $A_0$ because the Heavisied function of
equation (4) is zero. Just as the caustic begins to cross the
emitting region the amplification rises rapidly and then decays
gradually towards $A_0$ as the source moves away from the
caustic-fold.

Moreover, for the specific event one can model the caustic shape to
obtain different parameters (see e.g. Abajas et al. 2005, Kochanek
2004 for the case of Q2237+0305). In order to apply an appropriate
microlens model, additionally we will consider a standard
microlensing magnification pattern for the Q2237+0305A image (Fig. 1
left). For generating this map we used the ray-shooting method
\citep{Kay86,sch1,sch2,WP90,WSP90}. In this method the input
parameters are the average surface mass density $\sigma$, shear
$\gamma$ and width of the microlensing magnification map expressed
in units of the Einstein ring radius (defined for one solar mass in
the lens plane).

First, we generate a random star field in the lens plane with use of
the parameter $\sigma$. After that, we solve the Poisson equation
$\nabla^2\psi=2\sigma$ in the lens plane numerically, so we can
determine the lens potential $\psi$ in every point of the grid in
the lens plane. To solve the Poisson equation numerically one has to
write its finite difference form:
\begin{equation}
\psi_{i+1,j}+\psi_{i-1,j}+\psi_{i,j+1}+\psi_{i,j-1}-4\psi_{i,j}=2\sigma_{i,j}.
\label{psi}
\end{equation}
Here we used the standard 5-point formula for the two-dimensional
Laplacian.  Next step is inversion of the equation (\ref{psi}) using
Fourier transform. After some transformations we obtain:
\begin{equation}
\hat{\psi}=\frac{\hat{\sigma}_{mn}}{2(\cos{\frac{m\pi}{N_{1}}}+\cos{\frac{n\pi}{N_{2}}}-2)},
\label{hatpsi}
\end{equation}
where $N_1$ and $N_2$ are dimensions of the grid in the lens plane.
Now, using the finite difference technique, we can compute the
deflection angle $ \vec{\alpha}=\nabla\psi $ in each point of the
grid in the lens plane. After computing deflection angle, we can map
the regular grid of points in the lens plane, via lens equation,
onto the source plane. These light rays are then collected in pixels
in the source plane, and the number of rays in one pixel is
proportional to the magnification due to microlensing at this point
in the source plane.

Typically, for calculations of microlensing in
gravitationally lensed systems one can consider cases where
dimensionless surface mass density $\sigma$ is some fraction of
1 e.g. 0.2, 0.4, 0.6, 0.8 \citep{Treyer03}, cases without
external shear ($\gamma=0$) and cases with $\gamma=\sigma$ for
isothermal sphere model for the lensing galaxy. In this article
we assume the values of these parameters within the range
generally adopted by other authors, in particular by
\citet{Treyer03}.

Microlensing magnification pattern for the Q2237+0305A image (Fig. 1
left) with 16~$R_E$ on a side (where $R_E\approx 5867\ R_g$) is
calculated using the following parameters: $\sigma=0.36$ and
$\gamma=0.4$ (see \cite{Pop06a}, Fig. 2), the mass of microlens is
taken to be 0.3$M_\odot$ and we assume a flat cosmological model
with $\Omega =0.3$  and $H_{0}= 75\ \rm km\ s^{-1} Mpc^{-1}$.

We also calculated microlensing magnification pattern for a
"typical" lens system (Fig. 1 right), where the redshifts of
microlens and source are: $z_l=0.5$ and $z_s=2$. In this case, the
microlens parameters are taken arbitrary: $\sigma=0.45$ and
$\gamma=0.3$ and the size of obtained microlensing pattern is also
$16\ R_E \times 16\ R_E$, where $R_E\approx 3107\ R_g$.

\section{Typical time scales for microlensing}

Typical scales for microlensing are discussed not only in books on
gravitational lensing \citep{Schneider92a,Zakh97,Petters01}, but in
recent papers also (see, for example, \cite{Treyer03}). In this
paper we discuss microlenses located in gravitational macrolenses
(stars in lensing galaxies), since optical depth for microlensing is
then the highest \citep{Wyithe02,Wyithe02b,Zakharov04,Zakharov05} in
comparison with other possible locations of gravitational
microlenses, as for example stars situated in galactic clusters and
extragalactic dark halos \citep{Tadros98,Totani03,Inoue03}.

Assuming the concordance cosmological model with $\Omega_{\rm tot}=1$,
$\Omega_{\rm matter}=0.3$ and $\Omega_{\Lambda}=0.7$ we
recall that typical length scale for microlensing is \citep{Treyer03}:
\begin{equation}
 R_E = \sqrt{2 r_s \cdot  \frac{D_s D_{ls}}{D_{l}}}
 \approx 3.2 \cdot 10^{16} \sqrt{\frac{m}{M_\odot}} h_{75}^{-0.5} \mathrm{~cm},
 \label{eq_suppl1}
\end{equation}
where "typical" microlens and sources redshifts are assumed to be
$z_l=0.5, z_s=2.$ (similar to \cite{Treyer03}),
$r_s=\dfrac{2Gm}{c^2}$ is the Schwarzschild radius corresponding to
microlens mass $m$, $h_{75}=H_0/((75 {\rm~km/sec})/{\rm Mpc})$ is
dimensionless Hubble constant.

The corresponding angular scale is \citep{Treyer03}
\begin{equation}
\theta_0=\frac{R_E}{D_s}
 \approx 2.2 \cdot 10^{-6} \sqrt{\frac{m}{M_\odot}} h_{75}^{-0.5}{\rm ~arcsec},
 \label{eq_suppl2}
\end{equation}

Using the length scale (\ref{eq_suppl1}) and velocity scale
(say $v_\bot \sim$~600~km/sec as \cite{Treyer03} did), one
could calculate the standard time scale corresponding to the
scale to cross the projected Einstein radius
\begin{equation}
t_E=(1+z_l)\frac{R_E}{v_\bot}
 \approx 25 \sqrt{\frac{m}{M_\odot}}v_{600}^{-1} h_{75}^{-0.5}{\rm ~years},
 \label{eq_suppl3}
\end{equation}
where a relative transverse velocity $v_{600}=v_\bot/(600$~km/sec).
The time scale $t_E$, corresponding to a point-mass lens and to a
small source (compared to the projected Einstein radius of the
lens), could be used if microlenses are distributed freely at
cosmological distances and if each Einstein angle is located far
enough from another one. However, the estimation (\ref{eq_suppl3})
gives long and most likely overestimated time scales especially for
gravitationally lensed systems. Thus we must apply another microlens
model to estimate time scales.

For a simple caustic model, such as one that considers a straight
fold caustic\footnote{We use the following approximation for the
extra magnification near the caustic: $\mu=\sqrt{\dfrac{r_{\rm
caustic}}{\xi-\xi_c}}$, $\xi>\xi_c$ where $\xi$ is the perpendicular
direction to the caustic fold (it is obtained from Eq. (\ref{eq04})
assuming that factor $K$ is about unity).}, there are two time
scales depending either on the "caustic size" ($r_{\rm caustic}$) or
the source radius ($R_{\rm source}$). In the case when source radius
is larger or at least close to the "caustic size" ($R_{\rm source}
\gtrsim r_{\rm caustic}$), the relevant time scale is the "crossing
time" \citep{Treyer03}:
\begin{eqnarray}
\label{eq_suppl4}
t_{\rm cross} & = & (1+z_l)\frac{R_{\rm source}}{v_\bot (D_s/D_l)} \nonumber \\
& \approx & 0.69\ R_{15}\ v_{600}^{-1} \left(\frac{D_s}{D_l}\right)^{-1} h_{75}^{-0.5}{\rm ~years} \\
& \approx & 251\ R_{15}\ v_{600}^{-1}\ h_{75}^{-0.5}{\rm ~days},\nonumber
\end{eqnarray}
where $D_l$ and $D_s$ correspond to $z_l=0.5$ and $z_s=2$,
respectively and $R_{15}=R_{\rm source}/10^{15}$~cm. As a matter of
fact, the velocity perpendicular to the straight fold caustic
characterizes the time scale and it is equal to $v_\bot \sin \beta$
where $\beta$ is the angle between the caustic and the velocity
$v_\bot$ in the lens plane, but in our rough estimates we can omit
factor $\sin \beta$ which is about unity. However, if the source
radius $R_{\rm source}$ is much smaller than the "caustic size"
$r_{\rm caustic}$ ($R_{\rm source} \ll r_{\rm caustic}$), one could
use the "caustic time", i.e. the time when the source is located in
the area near the caustic:
\begin{eqnarray}
\label{eq_suppl5}
t_{\rm caustic} & = & (1+z_l)\frac{r_{\rm caustic}}{v_\bot (D_s/D_l)} \nonumber \\
& \approx & 0.69\ r_{15}\ v_{600}^{-1} \left(\frac{D_s}{D_l}\right)^{-1} h_{75}^{-0.5}{\rm ~years} \\
& \approx & 251\ r_{15}\ v_{600}^{-1}\ h_{75}^{-0.5}{\rm ~days}, \nonumber
\end{eqnarray}
where $r_{15}=r_{\rm caustic}/10^{15}$~cm.

Therefore, $t_{\rm cross}$ could be used as a lower limit for
typical time scales in the case of a simple caustic microlens model.
From equations (\ref{eq_suppl4}) and (\ref{eq_suppl5}) it is clear
that one cannot unambiguously infer the source size $R_{\rm source}$
from variability measurements alone, without making some further
assumptions. In general, however, we expect that $t_{\rm cross}$
corresponds to smaller amplitude variations than $t_{\rm caustic}$,
since in the first case only a fraction of a source is significantly
amplified by a caustic (due to assumption that $R_{\rm source}
\gtrsim r_{\rm caustic}$), while in the second case it is likely
that the entire source could be strongly affected by caustic
amplification (due to assumption that $R_{\rm source} \ll r_{\rm
caustic}$).

\begin{figure}
\centering
\includegraphics[width=8cm]{fig2.eps}
\caption{The variations of normalized total continuum flux in
optical (3500--7000 \AA), UV (1000--3500 \AA) and X (1.24--12.4 \AA
\ i.e. 1--10 KeV) band due to microlensing by a caustic crossing
along $y=-x$ direction in the case of Schwarzschild metric. Time
scale corresponds to "typical" redshifts of microlens and source:
$z_l=0.5$ and $z_s=2$. The parameters of the caustic are: $A_0$=1,
$\beta$=1, $\kappa=+1$ and its "size" is 9000 $R_g$. Negative
distances and times correspond to the approaching side, and positive
to the receding side of accretion disc. In this case, due to
$\kappa=+1$, caustic motion is from the receding towards the
approaching side (i.e. from the right to the left). The source mass
is $10^8\ M_\odot$. The radii of optical emitting region are:
$R_{in}= 100\ R_g$, $R_{out}=2000\ R_g$, for UV emitting region:
$R_{in}= 100\ R_g$, $R_{out}=1000\ R_g$ and for X emitting region:
$R_{in}= R_{ms}$, $R_{out}=80\ R_g$.}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=8cm]{fig3.eps}
\caption{The same as in Fig. 2. but for $r_{\rm caustic}=2000\ R_g$.}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=8cm]{fig4.eps}
\caption{The same as in Fig. 2. in the case when time scale
corresponds to $z_l=0.04$, $z_s=1.69$ (i.e. to Q2237+0305 lens
system).}
\end{figure}

In this paper, we estimated the microlensing time scales for the
X-ray, UV and optical emitting regions of the accretion disc using
the following three methods:
\begin{enumerate}
\item by converting the distance scales of microlensing events
    to the corresponding time scales according to the formula
    (13) in which $R_{\rm source}$ is replaced by the distance
    from the center of accretion disc. Caustic rise times
    ($t_{HME}$) are then derived from the simulated variations
    of the normalized total flux in the X-ray, UV and optical
    spectral bands (see Figs. 2-4) by measuring the time from
    the beginning to the peak of the magnification event
\item by calculating the caustic times ($t_{\rm caustic}$)
    according to equation (14)
\item using light curves (see Figs. 5 and 6) produced when the
    source crosses over a magnification pattern. Rise times of
    high magnification events ($t_{HME}$) are then measured as
    the time intervals between the beginning and the maximum of
    the corresponding microlensing events (for more details, see
    the next section).
\end{enumerate}

\section{Results and discussion}

In order to explore different cases of microlensing and evaluate
time scales for different spectral bands, first we numerically
simulate the crossing of a straight-fold caustic with parameters
$A_0=1,\ \beta=1$, $\kappa=+1$ and $r_{\rm caustic}=9000$ $R_g$ over
an accretion disc with an inclination angle 35$^\circ$, that is
stratified into three parts:

(i) The innermost part that emits X-ray continuum (1.24 \AA\ -- 12.4
\AA \ or 1--10 keV). The inner radius is taken to be
$R_{inn}=R_{ms}$ (where $R_{ms}$ is the radius of the marginally
stable orbit: $R_{ms}=6\ R_g$ in the Schwarzschild metric) and outer
radius is $R_{out}=80$ $R_g$ (where $R_g=GM/c^2$ is the
gravitational radius for a black hole with mass $M$).

(ii) An UV emitting part of the disc (contribute to the emission
from 1000 \AA\ -- 3500\AA ), with $R_{inn}=100$ $R_g$ and
$R_{out}=1000$ $R_g$.

(iii) An outer optical part of the disc with $R_{inn}=100$ $R_g$ and
$R_{out}=2000$ $R_g$ that emits in the wavelength band from 3500
\AA\ until 7000 \AA.

\begin{table*}
\centering \caption{The estimated time scales (in years) for
microlensing of the X-ray, UV and optical emission region for lensed
QSOs observed by Chandra X-ray Observatory \citep{Dai04}. The
calculated caustic times $t_{\rm caustic}$ are obtained according to
formula (14) for the following values of the cosmological constants:
$H_0=75\rm \ km\ s^{-1}Mpc^{-1}$ and $\Omega_0=0.3$. The caustic
rise times $t_{HME}$ are derived from caustic crossing simulations
(see Figs 2 -- 4). The black hole mass is assumed to be 10$^8\rm
M_\odot$.}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
 Object & $z_s$ & $z_l$ & \multicolumn{2}{c|}{X-ray} & \multicolumn{2}{c|}{UV}    & \multicolumn{2}{c|}{optical} \\
\cline{4-9}
        &       &       &  $t_{\rm caustic}$  & $t_{HME}$  & $t_{\rm caustic}$  & $t_{HME}$ & $t_{\rm caustic}$ & $t_{HME}$  \\
\hline
\hline
    HS 0818+1227   &  3.115 & 0.39 & 0.572 & 0.660 & 7.147  &  7.070 & 14.293 & 15.160 \\
  RXJ 0911.4+0551  &  2.800 & 0.77 & 0.976 & 1.120 & 12.200 & 12.080 & 24.399 & 25.880 \\
   LBQS 1009-0252  &  2.740 & 0.88 & 1.077 & 1.240 & 13.468 & 13.330 & 26.935 & 28.570 \\
     HE 1104-1805  &  2.303 & 0.73 & 0.918 & 1.050 & 11.479 & 11.370 & 22.957 & 24.350 \\
      PG 1115+080  &  1.720 & 0.31 & 0.451 & 0.520 & 5.634  &  5.570 & 11.269 & 11.950 \\
     HE 2149-2745  &  2.033 & 0.50 & 0.675 & 0.780 & 8.436  &  8.350 & 16.871 & 17.890 \\
      Q 2237+0305  &  1.695 & 0.04 & 0.066 & 0.080 & 0.828  &  0.820 & 1.655  &  1.760 \\
\hline
\end{tabular}
\end{table*}

\begin{table*}
\centering \caption{Average rise times ($<t_{HME}>$) of high
magnification events, their average number ($<N_{\rm caustic}>_0$)
per unit length ($R_E$) and their average number ($<N_{\rm
caustic}>_y$) per year in the light curves of Q2237+0305A (Fig. 5)
and "typical" lens system (Fig. 6).}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
& & \multicolumn{3}{|c|}{Q2237+0305} & \multicolumn{3}{c|}{"Typical" lens} \\
\hline
Disc path & Sp. band & $<t_{HME}>$ & $<N_{\rm caustic}>_0$ & $<N_{\rm caustic}>_y$ & $<t_{HME}>$ & $<N_{\rm caustic}>_0$ & $<N_{\rm caustic}>_y$ \\
\hline
\hline
      & X        & 0.37 & 0.95 & 0.20 & 3.94 & 1.63 & 0.06 \\
$y=0$ & UV       & 1.29 & 0.59 & 0.12 &10.96 & 0.78 & 0.03 \\
      & Optical  & 2.96 & 0.51 & 0.10 &15.81 & 0.62 & 0.02 \\
\hline
      & X        & 0.57 & 0.52 & 0.11 & 2.77 & 1.65 & 0.06 \\
$y=-x$ & UV       & 1.92 & 0.36 & 0.07 & 6.62 & 0.49 & 0.02 \\
      & Optical  & 3.98 & 0.26 & 0.05 &17.04 & 0.38 & 0.01 \\
\hline
      & X        & 0.68 & 1.61 & 0.33 & 4.52 & 1.40 & 0.05 \\
$x=0$ & UV       & 1.38 & 0.88 & 0.18 &13.25 & 0.54 & 0.02 \\
      & Optical  & 2.96 & 0.44 & 0.09 &31.62 & 0.31 & 0.01 \\
\hline
\end{tabular}
\end{table*}

Having in mind that the aims of this investigation are to study the
microlensing time scales for different emitting regions and time
dependent response of amplification in different spectral bands, we
considered microlensing magnification patterns only for image A of
Q2237+0305 and for a "typical" lens system. Our intention was not to
create a complete microlensing model for a specific lens system, and
therefore we did not analyze the differences between images (as for
instance the time delay between them). The variations in the total
flux in the different spectral bands are given in Figs. 2--6. In
Figs. 2 and 3 the simulations for a typical lens system with
$z_l=0.5$ and $z_s=2$ and for two different "caustic sizes" are
given. As one can see from Figs. 2 and 3, the microlensing time
scales are different for different regions, and the durations of
variations in the X-ray are on order of several months to a few
years, but in the UV/optical emission regions they are on order of
several years. Also, as one can see in Figs. 2 and 3, the time
scales do not depend on "caustic size" which, on the other hand,
affects only the maximal amplifications in all three spectral bands.
The results corresponding to the lens system of QSO 2237+0305
($z_l=0.04$, $z_s=1.69$) are given in Figs. 4 and 5. We considered
the straight-fold caustic (Fig. 4) and a microlensing pattern for
the image A of QSO 2237+0305 (Fig. 5). As one can see from Figs. 4
and 5, a higher amplification in the X-ray continuum than in the
UV/optical is expected. In this case, the corresponding time scales
are much shorter and they are from a few days up to a few months for
X-ray and a few years for UV/optical spectral bands. The similar
conclusion arises when we compare the latter results with those for
a "typical" lens system in the case of microlensing pattern (Fig.
6).

\begin{figure*}
\centering
\includegraphics[width=0.45\textwidth]{fig5a.eps}
\includegraphics[width=0.45\textwidth]{fig5b.eps} \\
\includegraphics[width=0.45\textwidth]{fig5c.eps}
\caption{Variations in the X-ray (solid), UV (dashed) and optical
(dotted) spectral bands corresponding to the horizontal (top left),
diagonal (top right) and vertical (bottom) path in the magnification
map of the QSO 2237+0305A (Fig. 1 left).}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=0.45\textwidth]{fig6a.eps}
\includegraphics[width=0.45\textwidth]{fig6b.eps} \\
\includegraphics[width=0.45\textwidth]{fig6c.eps}
\caption{The same as in Fig. 5 but for the magnification map of a
"typical" lens system  (Fig. 1 right).}
\end{figure*}

We also estimated time scales for seven lensed QSOs which have been
observed in the X-ray band \citep{Dai04}. For each spectral band two
estimates are made: caustic time ($t_{\rm caustic}$) - obtained from
equation (14) and caustic rise time ($t_{HME}$) - obtained from
caustic crossing simulations (see Figs. 2 -- 4). In the second case,
we measured the time from the beginning of the microlensing event
until it reaches its maximum (i.e. in the direction from the right
to the left in Figs. 2 -- 4). The duration of the magnification
event beyond the maximum of the amplification could not be
accurately determined because of the asymptotic decrease of the
magnification curve beyond the peak. Estimated time scales for
different spectral bands are given in Table 1. As one can see from
Table 1, the microlensing time scales are significantly smaller for
the X-ray than for the UV/optical bands.

Unamplified and amplified brightness profiles of the X-ray emitting
region, corresponding to the highest peak in Fig. 5 (top right) are
presented in Fig. 7. As one can see in Fig. 7, the assumed
brightness profile of the source is very complex due to applied ray
tracing method, which allows us to obtain an image of the entire
disc, not only its one dimensional profile. Therefore, we could not
use a simple source profile for the calculation of microlensing time
scales (as it was done by \citet{Witt95}). Instead, we estimated the
frequency of high amplification events (HMEs), i.e. the number of
such events per unit length, directly from the light curves
presented in Figs 5 and 6. For models with non-zero shear, this
frequency depends on the direction of motion and for both calculated
maps (Q2237+0305A and "typical" lens) we counted the number of high
magnification events along the following paths (see Fig. 1): i)
horizontal ($y=0$) in the direction from $-x$ to $x$, ii) diagonal
($y = -x$) in direction from $-x$ to $x$ and iii) vertical ($x=0$)
in direction from $y$ to $-y$. For each map the lengths of the
horizontal and vertical paths in the source plane are the same
(13.636 $R_E$ for Q2237+0305A and 12.875 $R_E$ for "typical" lens),
as well as are the corresponding crossing times (66.22 years for
Q2237+0305A and 337.32 years for "typical" lens). For Q2237+0305A
the length of the diagonal path in source plane is 19.284 $R_E$ and
the crossing time is 93.62 years, while the corresponding length in
the "typical" case is 18.208 $R_E$ and the crossing time is 477.04
years.

\begin{figure*}
\centering
\includegraphics[width=0.495\textwidth]{fig7a.eps}
\includegraphics[width=0.495\textwidth]{fig7b.eps}
\caption{Unamplified (left) and amplified (right) brightness profile
of the X-ray emitting region, corresponding to the highest peak in
Fig. 5 (top right). The profiles are obtained using the ray tracing
method (see, e.g., Popovi\'c et al. 2003a,b and references
therein).}
\end{figure*}

HMEs are asymmetric peaks in the light curves which depend not only
on microlens parameters but also on the sizes of emitting regions in
the following sense: the larger emitting regions are expected to
produce smoother light curves and more symmetric peaks
\citep{Witt95}. Consequently, it can be expected that the majority
of HMEs should be detected in X-ray light curves, less of them in UV
and the smallest number in optical light curves. Therefore, we
isolated only clearly asymmetric peaks in all light curves and
measured their rise times $t_{HME}$ as the intervals between the
beginning and the maximum of the corresponding microlensing events.
In the case of Q2237+0305A we found the following number of HMEs: i)
horizontal path: 13 in X-ray, 8 in UV and 7 in optical band, ii)
diagonal path: 10 in X-ray, 7 in UV and 5 in optical band and iii)
vertical path: 22 in X-ray, 12 in UV and 6 in optical band. In case
of "typical" lens these numbers are: i) horizontal path: 21 in
X-ray, 10 in UV and 8 in optical band, ii) diagonal path: 30 in
X-ray, 9 in UV and 7 in optical band and iii) vertical path: 18 in
X-ray, 7 in UV and 4 in optical band.

\begin{figure*}
\centering
\includegraphics[width=0.495\textwidth]{fig8a.eps}\hfill
\includegraphics[width=0.495\textwidth]{fig8b.eps} \\
\vspace*{0.3cm}
\includegraphics[width=0.495\textwidth]{fig8c.eps}
\caption{Rise times ($t_{HME}$) of high magnification
events for all three spectral bands in the light curves of
Q2237+0305A (Fig. 5). Top left panel corresponds to the
horizontal, top right to the diagonal and bottom to the
vertical path of accretion disc.}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=0.495\textwidth]{fig9a.eps}\hfill
\includegraphics[width=0.495\textwidth]{fig9b.eps} \\
\vspace*{0.3cm}
\includegraphics[width=0.495\textwidth]{fig9c.eps}
\caption{The same as in Fig 8. but for the light
curves of "typical" lens system (Fig. 6)}
\end{figure*}

The average number of caustic crossings per unit length ($<N_{\rm
caustic}>_0$) and per year ($<N_{\rm caustic}>_y$) are given in
Table 2. This table also contains the average rise times
($<t_{HME}>$) in all three spectral bands, derived from the rise
times ($t_{HME}$) of individual HMEs which are presented in Figs. 8
and 9 in the form of histograms. These results, as expected, show
that the rise times are the shortest and the frequency of caustic
crossings is the highest in the X-ray spectral band in comparison to
the other two spectral bands. One can also see from Tables 1 and 2
that in the case of Q2237+0305A, the average rise times of HMEs for
all three spectral bands (obtained from microlens magnification
pattern simulations) are longer than both, caustic rise times
(obtained from caustic simulations) and caustic times (calculated
from equation (14)).

Microlensing can result in flux anomalies in the sense that
different image flux ratios are observed in different spectral bands
\citep{Pc04,Pop06b}. As shown in Figs 2--6. the amplification in the
X-ray band is larger and lasts shorter than it does in the UV and
optical bands. Consequently, monitoring of lensed QSOs in the X-ray
and UV/optical bands can clarify whether the flux anomaly is
produced by CDM clouds, massive black holes or globular clusters
(millilensing) or stars in foreground galaxy (microlensing).

\section{Conclusion}

In this paper we calculated microlensing time scales of different
emitting regions. Using a model of an accretion disc (in the center
of lensed QSOs) that emits in the X-ray and UV/optical spectral
bands, we calculated the variations in the continuum flux caused by
a straight-fold caustic crossing an accretion disc. We also
simulated crossings of accretion discs over microlensing
magnification patterns for the case of image A of Q2237+0305 and for
a "typical" lens system. From these simulations we concluded the
following:

(i) one can expect that the X-ray radiation is more amplified than
UV/optical radiation due to microlensing which can induce the so
called 'flux anomaly' of lensed QSOs.

(ii) the typical microlensing time scales for the X-ray band
are on order of several months, while for the UV/optical they
are on order of several years (although the time scales
obtained from microlensing magnification pattern simulations
are longer in comparison to those obtained from caustic
simulations).

(iii) monitoring of the X-ray emission of lensed QSOs can
reveal the nature of 'flux anomaly' observed in some lensed
QSOs.

All results obtained in this work indicate that monitoring the X-ray
emission of lensed QSOs is useful not only to discuss the nature of
the 'flux anomaly', but also can be used for constraining the size
of the emitting region.

\section*{Acknowledgments}

This work is a part of the project (146002) "Astrophysical
Spectroscopy of Extragalactic Objects" supported by the Ministry of
Science of Serbia. The authors would like to thank the anonymous
referee for very useful comments.


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\begin{document}

\setlist[itemize]{label={},leftmargin=7mm,rightmargin=7mm}
\Large
\noindent Astro2020 APC White Paper: State of the Profession Consideration\\

\noindent The Early Career Perspective on the Coming Decade, Astrophysics Career Paths, and the Decadal Survey Process \\

\normalsize
\vspace{1mm}

\noindent \textbf{Thematic Areas:} State of the profession considerations: Early career concerns for the coming decade, graduate and postdoctoral training, career preparation, career transitions, and structure and dissemination of decadal survey. \\

\noindent\textbf{Principal Authors:}

\noindent Names: Emily Moravec | Ian Czekala | Kate Follette

\noindent Institutions: University of Florida, 2018 NASEM Christine Mirzayan Science and Technology Policy Fellow | UC Berkeley | Amherst College

\noindent Emails: emoravec@ufl.edu | iancze@gmail.com | kfollette@amherst.edu \\

\noindent \textbf{Co-authors:} Zeeshan Ahmed, Mehmet Alpaslan, Alexandra Amon, Will Armentrout, Giada Arney, Darcy Barron, Eric Bellm, Amy Bender, Joanna Bridge, Knicole Colon, Rahul Datta, Casey DeRoo, Wanda Feng, Michael Florian, Travis Gabriel, Kirsten Hall, Erika Hamden, Nimish Hathi, Keith Hawkins, Keri Hoadley, Rebecca Jensen-Clem, Melodie Kao, Erin Kara, Kirit Karkare, Alina Kiessling, Amy Kimball, Allison Kirkpatrick, Paul La Plante, Jarron Leisenring, Miao Li, Jamie Lomax, Michael B. Lund, Jacqueline McCleary, Elisabeth Mills, Edward Montiel, Nicholas Nelson, Rebecca Nevin, Ryan Norris, Michelle Ntampaka, Christine O'Donnell, Eliad Peretz, Andres Plazas Malagon, Chanda Prescod-Weinstein, Anthony Pullen, Jared Rice, Rachael Roettenbacher, Robyn Sanderson, Jospeh Simon, Krista Lynne Smith, Kevin Stevenson, Todd Veach, Andrew Wetzel, and Allison Youngblood

\begin{center}
\includegraphics[width=\textwidth]{NAS.JPG}
\end{center}

\pagebreak

\noindent \textbf{Executive Summary:} In response to the need for the Astro2020 Decadal Survey to explicitly engage early career astronomers, the National Academies of Sciences, Engineering, and Medicine hosted the Early Career Astronomer and Astrophysicist Focus Session (ECFS)\footnote{http://sites.nationalacademies.org/SSB/SSB\_185166} on October 8-9, 2018 under the auspices of Committee of Astronomy and Astrophysics. The meeting was attended by fifty six pre-tenure faculty, research scientists, postdoctoral scholars, and senior graduate students\footnote{List of participants http://sites.nationalacademies.org/cs/groups/ssbsite/documents/webpage/ssb\_189919.pdf}, as well as eight former decadal survey committee members, who acted as facilitators. The event was designed to educate early career astronomers about the decadal survey process, to provide them with guidance toward writing effective white papers, to solicit their feedback on the role that early career astronomers should play in Astro2020, and to provide a forum for the discussion of a wide range of topics regarding the astrophysics career path.

This white paper presents highlights and themes that emerged during two days of discussion. In Section 1, we discuss concerns that emerged regarding the coming decade and the astrophysics career path, as well as specific recommendations from participants regarding how to address them. We have organized these concerns and suggestions into five broad themes. These include (sequentially): (1) adequately training astronomers in the statistical and computational techniques necessary in an era of ``big data", (2) responses to the growth of collaborations and telescopes, (3) concerns about the adequacy of graduate and postdoctoral training, (4) the need for improvements in equity and inclusion in astronomy, and (5) smoothing and facilitating transitions between early career stages. Section 2 is focused on ideas regarding the decadal survey itself, including: incorporating early career voices, ensuring diverse input from a variety of stakeholders, and successfully and broadly disseminating the results of the survey.

Recommendations presented here do not necessarily represent a universal consensus among participants, nor do they reflect the entirety of the discussions. Rather, we have endeavored to highlight themes, patterns, and concrete suggestions.

\clearpage

\Large
\begin{centering}
\textbf{The Early Career Perspective on the Coming Decade, Astrophysics Career Paths, and the Decadal Survey Process}
\end{centering}
\\

\normalsize
\section{Discussion Themes: Topics of Importance for Early Career Professionals in the Coming Decade}
\subsection{Theme 1 - The big role of big data and data science in the 2020s}
Big data and data science have become integral parts of astronomical research. In an era of data proliferation (both observational and simulated), professional and funding agencies must consider the provision of fiscal, physical, and computational resources when planning projects and evaluating proposals. Particular needs in the area of data science include: the development of infrastructure to manage large datasets, the training of early career professionals in the techniques necessary to work with sophisticated data analysis tools, and the expansion and development of career paths focused on software development, data collection, and data analysis. We recommend that management, development, and analysis of big data be considered both within its own decadal panel and holistically across the survey panels and committees.

During the ECFS, ``big data" and modern data analysis techniques featured prominently in discussions about preparation for academic career paths, as well as transitions into professions that employ astronomy Ph.D.s (data science, engineering, defense). There was a general concern among participants that graduate curricula do not provide sufficient training in the sophisticated computational and statistical methodologies necessary to efficiently handle data in general, and large datasets in particular. Participants felt strongly that in order to incentivize the improvement of graduate and postdoctoral training in this area, the decadal survey committee should make explicit recommendations to funding agencies, professional societies, and departments regarding improved training.

\paragraph{Recommendations}
\begin{itemize}
    \item \textbf{1.1.1} The decadal survey should include explicit prioritization of theoretical and computational questions in astronomy.
    \item \textbf{1.1.2} The decadal survey should consider whether the evolution of the profession means that a redistribution or re-prioritization of permanent positions is necessary. In particular, ECFS participants suggested that the panels consider ways in which grant funding might be restructured to incentivize semi-permanent software development and data management positions rather than postdoctoral positions (see also Theme 5).
    \item \textbf{1.1.3} Professional societies and funding agencies should provide support for conference workshops on data science. Wherever possible, these should be embedded rather than pre-conference workshops to minimize the additional cost of attendance for early career participants.
    \item \textbf{1.1.4} Departments should restructure graduate curricula to explicitly address the computational and statistical techniques necessary to succeed in the big data era (see also Theme 3).
    \item \textbf{1.1.5} Observatories, national labs, and large collaborations (LSST, DES, etc.) should increase the number of data science workshops and data-science themed postdoctoral fellowships.
\end{itemize}

\subsection{Theme 2 - The evolving challenges of large missions and collaborations}
A further concern among ECFS participants was the trend toward larger facilities and projects. While acknowledging that flagship missions and thirty meter class telescopes are critical for advancing the field, early career astronomers voiced a need for the astronomical community to maintain a broad portfolio. There was widespread concern about the closing of current facilities to ``make room" in budgets for large telescopes and missions. Continued operation of an array of facilities operating in various wavelength regimes will be key for the professional training of future astronomers, as well as being needed more broadly for followup observations, time-domain monitoring, etc. We urge the decadal survey committee to consider the need for smaller and more easily accessible facilities that will allow early career astronomers to lead proposals, execute observations, and obtain their own data.

While many early career astronomers are enthusiastic about the science enabled by large facilities and big collaborations, discussions at the ECFS revealed significant anxiety around the preparedness of Ph.D. astronomers to work in an era where `big' is the norm. Particular concerns in this regard are: the availability of a sufficient number of desirable long-term (non-postdoctoral) positions within academia to support large projects, the marketability of PhD astronomers in fields outside of or peripheral to academic astronomy, and concerns regarding whether and how contributions to large collaborations will be recognized by potential employers. The group predicted that more support scientist positions will be needed to design and maintain hardware and software for large facilities, but have observed a lack of sufficient funding for facility operations and maintenance under the current funding paradigm.

\paragraph{Recommendations}
\begin{itemize}
\item \textbf{1.2.1} The decadal survey committee, funding agencies, and observatories should carefully consider the prioritization of and funding for construction of new facilities vs. operation and maintenance of existing facilities. A broad portfolio of facilities should be maintained, including those where early career astronomers can reasonably expect to lead proposals and obtain telescope time for smaller projects.
\item \textbf{1.2.2} The reasonableness of cost estimates for missions and facilities proposed to the decadal survey should be carefully evaluated during project prioritization so that smaller facilities and support staff positions are not ``squeezed out" in agency budgets if and when the cost for these missions balloons.
\item \textbf{1.2.3} More astronomical journals should move toward a model where contributions to published papers are detailed specifically. This will incentivize early career astronomers to make contributions to large projects even when it will not lead to first authorship, and will provide a means for potential employers to gauge the role of applicants in coauthored papers.
\item \textbf{1.2.4} Professional societies and publishers should provide more opportunities for software development work to be recognized, published, and cited.
\end{itemize}

\subsection{Theme 3 - Career Preparation and Opportunities}
Early career astronomers are ideally situated to reflect on the experience of their education as well as assess its adequacy in preparing them for the challenge of navigating early career transitions. During the ECFS, participants focused on evaluating existing training, discussing the challenges that mark the transitions to postdoctoral and long-term positions, and preparing recommendations to improve these areas.

ECFS participants felt that some substantive changes are necessary to make the process of obtaining a Ph.D. more inclusive and supportive, and to reduce imbalances that lead to systemic inequalities in the profession (see Theme 5). In this section, we focus on recommendations around maximizing flexibility and marketability of Ph.D. astronomers, and around restructuring graduate curricula to reflect the changing nature of the profession.

\paragraph{Recommendations}
\begin{itemize}
\item \textbf{1.3.1} Departments should incorporate ethics and professional development seminars into graduate curricula. In addition to focusing on proper scientific and professional conduct, these seminars are an excellent venue to hone research and presentation skills.
\item \textbf{1.3.2} Departments should consider the flexibility of their curricula, de-emphasizing a burdensome number of core classes and allowing for more electives, particularly in statistical and computational methods.
\item \textbf{1.3.3} Grant components such as ``Mentoring Plans,'' and ``Facilities and Resources'' documents should be given more weight and structure. In particular, they should be tied specifically to curricular methodologies and professional development opportunities for all grants requesting graduate student or postdoctoral funding, and should be standardized across agencies.
\item \textbf{1.3.4} AAS should incentivize (e.g. through reduced fees for registration, exhibit space, and workshop facilitation) the participation of members of industry in annual conferences and should consider organizing an alternative careers networking event.
\end{itemize}

\subsection{Theme 4 - The Postdoctoral Years}
While the postdoctoral scholar period can be a time of great opportunity and scholarly freedom, it is also fraught with uncertainty. The traditionally transient nature of the postdoctoral years, where frequent cross-country (or international) moves are common and even encouraged, has a variety of hidden societal costs.
``2-body problems'' and relocation costs add stress and uncertainty to the career transition process and disproportionately affect women and underrepresented minorities.\footnote{See \url{http://womeninastronomy.blogspot.com/2013/02/figure-1-two-body-problem.html} and references therein.}
Efforts are also needed to de-stigmatize the actions of those astronomers who express interest in non-academic career options, including during the graduate admission process and during graduate and postdoctoral mentoring.
\paragraph{Recommendations}
\begin{itemize}
\item  \textbf{1.4.1} Five year postdoctoral appointments should be encouraged by funding agencies and appointments with terms of less than three years should be strongly discouraged.
\item \textbf{1.4.2} To increase the quality of applications while also reducing the considerable stress on applicants during job application season, the postdoctoral fellowship application process should be standardized in terms of the nature and length of required materials. Specific and measurable criteria should be posted in job ads to help applicants in evaluating their suitability for positions. For example, vague advertising for ``outstanding candidates'' preselects for candidates with high self-efficacy.
\item \textbf{1.4.3} The AAS should contract with social scientists to conduct regular membership surveys that include astronomers who have left academia, and should include the results in state of the profession reports.
\item \textbf{1.4.4} The social and psychological pressures of the Astrophysics Jobs Rumor Mill outweigh the benefits, and its information is often inaccurate or incomplete. It should be replaced by a commitment on the part of departments and agencies offering employment to release timely information to \textbf{all} job applicants regarding the search status, even if it has not concluded (e.g. ``we have created a short list and invited five applicants to visit").
\item \textbf{1.4.5} Development of teaching and communication skills should be incentivized and given greater emphasis in graduate curricula, postdoctoral training, and analysis of candidates for academic jobs. This can be accomplished through targeted graduate coursework, workshops, structured feedback to graduate students giving presentations (i.e. journal clubs), a greater emphasis on teaching and diversity statements in job applications, etc.
\item \textbf{1.4.6} The community should rethink the distribution of temporary (e.g. postdoc) vs. potentially permanent (e.g. staff scientist) positions and the ways in which they are funded (e.g. grants vs. operations budgets). In particular, support and software design for large surveys and telescopes should be done by long-term staff members and not by postdocs.
\end{itemize}

\subsection{Theme 5 - A Diverse Workforce and Equitable Community}
There was a strong consensus and desire among attendees that the astronomy community of the future should reflect the breadth of identities of the nation as a whole. Progress has been made in this area in the past decade, however there remains a great deal of work to be done. In particular, more work is needed to eliminate inequities and barriers to entry into the profession, which will enable more people of color, white women, members of the LGBTQ+ community, and people with disabilities to enter and, more importantly, to remain in the field. These efforts will benefit \textbf{all} members of the profession in myriad ways. In order to enable meaningful change, we hope to see the 2020 decadal survey make specific, high-priority, and far-reaching recommendations in this regard.

\paragraph{Recommendations}
\begin{itemize}
\item \textbf{1.5.1} We urge the decadal panels to consider white papers regarding the state of the profession to be of equal importance to science and facilities white papers. Although these papers will be read by separate panels, we hope that members of the state of the profession panels will work together with members of the science panels and survey officials to ensure that the broader recommendations of Astro2020 integrate science and state of the profession recommendations powerfully and effectively.
\item \textbf{1.5.2} Minoritized individuals often bear a disproportionate service burden, including during graduate school and postdoc years when these efforts are not generally a recognized responsibility. Graduate and postdoctoral fellowships should explicitly ask for information about, value, and reward this important work.
\item \textbf{1.5.3} AAS should consider establishing an award that specifically recognizes service in the area of improving representation in our profession.
\item \textbf{1.5.4} Training concerning gender and racial harassment should be required for all members of academic departments, including: faculty, postdocs, graduate students (especially teaching assistants), and staff. More training is needed across the board to combat gender and racial harassment in our field, particularly training with a focus on intersectionality.
\item \textbf{1.5.5} To mitigate concerns about harassment and bullying, graduate departments should move away from the single mentor model towards advising strategies that encourage interaction with multiple faculty members on a regular basis (e.g., larger advising committees with semi-regular meetings). This would have the added benefit of proving more opportunities for all students to seek support and advice from a range of mentors in addition to providing a safety net for students with difficult or abusive advising relationships.
\item \textbf{1.5.6} The AAS should establish a list of best-practices for parental and child leave policies for graduate students and postdocs. Similar to recommendation 1.3.3, this information should be included in grant documents that request funding for graduate students and postdocs.
\end{itemize}

\section{Recommendations Regarding the Structure and Dissemination of the Decadal Survey}
\subsection{Early career participation in the decadal survey}
A key goal of the ECFS was to solicit feedback regarding how early career astronomers would like to be represented throughout the decadal survey process. Many participants argued for the most direct form of representation: full membership on survey panels. Several concerns regarding full panel membership were raised, however. These included: 1) reduced scientific productivity during a vulnerable career stage, 2) uncertainty regarding recognition by tenure committees and potential employers of the importance of this service to the profession, and 3) the potential to become enmeshed in controversial decisions that could negatively affect future career prospects. Several ideas were put forward regarding ways to mitigate these concerns for early career participants on the panels, namely: funding (e.g., a ``decadal fellowship''), recognition (e.g. a specific title or award that could be included in job and tenure applications), commitments from panel chairs to write letters of recommendation for jobs and letters of support for tenure, commitments from senior panel members to support early career colleagues with invitations to give seminars and colloquia, and childcare support.

The principal alternative to full participation put forward was the formation of ``early career consultation groups." These groups would travel to or virtually participate in some subset of decadal survey meetings. This would give early career astronomers a ``seat at the table" but would also mitigate the potentially deleterious effects of full survey participation on the careers of participants in these groups.

Other ideas for early career involvement ranged from early career town halls, workshops, and conferences hosted by the American Astronomical Society and National Academies to inviting early career astronomers to make presentations to the decadal panels. While the assembled astronomers were not uniformly in support of a single recommendation, increased participation in some format (and perhaps multiple formats) was a universal theme in the discussions.
\paragraph{Recommendations}
In order to increase the early career representation in the decadal survey process, we recommend that the decadal survey committee:
\begin{itemize}
    \item \textbf{2.1.1} Implement incentives for early career astronomers to participate as full panel members (see above).
    \item \textbf{2.1.2} Invite early career astronomers to serve as consultants to the decadal panels.
    \item \textbf{2.1.3} Invite early career astronomers to deliver presentations to panels.
\end{itemize}

\subsection{Soliciting inclusive input to the 2020 decadal and beyond} Another discussion theme revolved around how to solicit broad input from the astronomical community, ensure that diverse voices are heard, and successfully disseminate the conclusions of the decadal survey to all of its ``stakeholders." First, town halls could be standardized (so that more places could easily arrange them) and conducted both digitally and in a wide range of geographic locations. Focused virtual town halls could explicitly target the challenges facing certain groups in astronomy (e.g. postdocs, observatory staff).

Second, we suggest that panel members are either present at the town halls, or that minutes from these events are compiled and circulated so that panel members are aware of what was discussed. Lastly, the survey committee should also consider inviting additional stakeholders not traditionally involved in the decadal survey process (e.g. native Hawaiians, amateur astronomers, K12 astronomy educators, data science employers) and certain additional subsets of the community (e.g. disabled astronomers, LGBTQ+ astronomers, etc.) to participate in Astro2020, either by making invited presentations to the committee or through focused town halls that are broadly advertised.

\paragraph{Recommendations}
In order to ensure equitable participation and dissemination to the broadest possible audience of stakeholders (within and beyond the astronomy community), we recommend that the decadal survey committee:
\begin{itemize}
    \item \textbf{2.2.1} Standardize town halls.
    \item \textbf{2.2.2} Organize official gatherings (town hall, focus session, etc.) that target specific groups of people whose perspectives have been historically missing in the decadal survey process and/or are important for the current decadal at hand (i.e. underrepresented people groups, geographically underrepresented groups, graduate students, postdocs, etc.).
    \item \textbf{2.2.3} Encourage participation by all stakeholders in the decadal survey process through invited presentations to the committee and targeted virtual town halls.
    \item \textbf{2.2.4} Make survey highlights available in multiple easily-digestible formats that adhere to the principles of universal design. For example: an ``Executive Summary" (1-page) document, publicly available slide decks highlighting priorities, digital representations that are captioned and easy to share on social media platforms (e.g. 5-10 minute videos).
\end{itemize}

\section{Conclusion}
The Early Career Astronomer and Astrophysicist Focus Session was a unique opportunity for a large group of early career professionals to gather and discuss the state of the astronomy profession. Lively and at times contentious discussions about the state of the profession centered around the five main themes identified in this document, namely: the role of big data in the future of astronomy, the evolving challenges of large missions and collaborations, graduate training and career preparation, the postdoctoral years, and creating a diverse workforce and equitable community. Although ECFS participants had many concerns about and ideas for improvements in these areas, there was also an overwhelming energy and enthusiasm about the future of our field and an optimism about the potential for meaningful change. A central goal of the event was to collect ideas about changes that early career astronomers would like to see made to the decadal survey process, and ideas that were generated in this regard are outlined in section 2. We, the participants, felt strongly that early career voices should be heard at every stage of the decadal process, and not just at the focus session. We look forward to participating in more of the Astro2020 decadal survey.

\end{document}


\section{Introduction}

Recently, substantial progress has been achieved in the search of non-supersymmetric
Minkowski/dS vacua in the context of string/M-theory compactifications. This was mainly related
to the understanding of the superpotentials generated by background fluxes \cite{GVW} and by
non-perturbative effects like gaugino condensation \cite{GC}, which generate a potential for
the moduli fields coming from the compactification and have suggested new
interesting possibilities for model building, like in particular those proposed in refs.~\cite{GKP,KKLT}.
From a phenomenological point of view, this type
of models must however posses some characteristics in order to be viable: supersymmetry must be
broken, the cosmological constant should be tiny, and all the moduli fields should be stabilized. In
the low energy effective theory all these crucial
features are controlled by a single quantity, the four-dimensional scalar potential, which gives
information on the dynamics of the moduli fields, on how supersymmetry is broken and on
the value of the cosmological constant. The characterization of the conditions under which a
supersymmetry-breaking stationary point of the scalar potential satisfies simultaneously the flatness
condition (vanishing of the cosmological constant) and the stability condition (the stationary point is
indeed a minimum) is therefore very relevant in the search of phenomenologically viable string models.
In this note we review the techniques presented in \cite{yo1,yo2,yo3} to study the
possibility of getting this type of vacua in the context of general supergravity
theories in which both chiral and vector multiplets participate to supersymmetry breaking.

\section{Viable supersymmetry breaking vacua}

The goal of this section is to find conditions for the existence of
non-supersymmetric extrema of the scalar potential of general supergravity theories fulfilling two basic
properties: i) they are locally stable and ii) they lead to a negligible cosmological constant. We will
first study this issue for theories with only chiral multiplets and then when
also vector multiplets are present.

\subsection{Constraints for chiral theories}

The Lagrangian of the most general supergravity theory with $n$ chiral superfields is entirely defined
by a single arbitrary real function $G$ depending on the corresponding chiral superfields $\Phi_i$ and
their conjugates $\bar \Phi_i$, as well as on its derivatives \cite{sugra}. The function $G$ can be
written in terms of a real K\"ahler potential $K$ and a holomorphic superpotential $W$ in the following
way\footnote{We will use the standard notation in which subindices $i$, $\bar \jmath$ mean
derivatives with respect to $\Phi^i$, $\bar \Phi^{j}$ and Planck units, $M_P=1$.}:
\begin{equation}
G(\Phi_i,\bar \Phi_i) = K(\Phi_i,\bar\Phi_i) + \log W(\Phi_i) +\log \bar W(\bar\Phi_i) \,.
\end{equation}
The quantities $K$ and $W$ are however defined only up to K\"ahler transformations acting as
$K \to K + f + \bar f$ and $W \to e^{-f} W $, $f$ being an arbitrary holomorphic function of the superfields,
which leave the function $G$ invariant. The scalar components of the chiral multiplets span an
$n$-dimensional K\"ahler manifold whose metric is given by $G_{i\bar \jmath}$, which
can be used to lower and raise indices.

The 4D scalar potential of this theory takes the following simple form:
\begin{equation}
V = e^G \Big(G^k G_k - 3 \Big) \,.
\label{genpot}
\end{equation}
The auxiliary fields of the chiral multiplets are fixed by the Lagrangian through the equations
of motion, and are given by $F_i = - \, e^{G/2}\, G_{i}$ where $e^{G/2}=m_{3/2}$
is the mass of the gravitino. Whenever $F_i\neq0$ at the vacuum, supersymmetry is spontaneously
broken and the direction given by the $G_i$'s defines the direction of the Goldstino eaten
by the gravitino in the process of supersymmetry breaking.

In order to find local non-supersymmetric minima of the potential (\ref{genpot}) with small non-negative
cosmological constant, one should proceed as follows: First impose the condition that the
cosmological constant is negligible and fix $V =0$. This flatness condition implies that:
\begin{equation}
\label{fc}
G^k G_k = 3 \,.
\end{equation}
Then look for stationary points of the potential where the flatness condition is satisfied.
This implies:
\begin{equation}\label{sc}
G_i + G^k \nabla_i G_k = 0 \,,
\end{equation}
where by $\nabla_i G_k=G_{ik}-\Gamma^n_{ik}G_n$ we denote the covariant derivative with
respect to the K\"ahler metric.

Finally, make sure that the matrix of second derivatives of the potential,
\begin{equation}
m^{2} = \left(
\begin{matrix}
m^2_{i \bar\jmath} &  m^2_{ij} \smallskip \cr
m^2_{\bar i \bar \jmath} & m^2_{\bar \imath j}
\end{matrix}
\right) \,,
\label{VIJ}
\end{equation}
is positive definite. This matrix has two different $n$-dimensional blocks, $m^2_{i \bar \jmath} =
\nabla_i \nabla_{\bar \jmath} V$ and $m^2_{i j} = \nabla_i \nabla_j V$, and after a straightforward computation these are found to be given by the following expressions:
\begin{eqnarray}\label{vij}
\begin{array}{lll}
m^2_{i \bar \jmath} \!\!\!&=&\!\!\! e^G\Big(G_{i \bar \jmath} + \nabla_i G_k \nabla_{\bar
\jmath} G^k - R_{i \bar \jmath p \bar q} G^p G^{\bar q}\Big) \,, \smallskip\ \\
m^2_{i j} \!\!\!&=&\!\!\! e^G\Big(\displaystyle{\nabla_i G_j + \nabla_j G_i + \frac 12 G^k
\big\{\nabla_i,\nabla_j \big\}G_k}\Big) \,, \\
\end{array}
\end{eqnarray}
where $R_{i \bar \jmath p \bar q}$ denotes the Riemann tensor with respect to the K\"ahler metric.
The conditions under which this $2n$-dimensional matrix (\ref{VIJ}) is positive definite are
complicated to work
out in full generality, the only way being the study of the behaviour of the $2n$ eigenvalues.
Nevertheless a necessary condition for this matrix to be positive definite can be encoded
in the condition that the quadratic form $m^2_{i \bar \jmath} z^i {\bar z}^{\bar \jmath}$ is positive
for any choice of non-null complex vector $z^i$. Our strategy will be then to look for a special
vector $z^i$ which leads to a simple constraint.

In this case there is only one special direction in field space, that is the direction given by $z^i = G^i$.
Indeed projecting in that direction we find the following simple expression:
\begin{equation}\label{eq}
m^2_{i \bar \jmath} G^i G^{\bar \jmath} = 6 - R_{i \bar \jmath p \bar q}\, G^i G^{\bar \jmath} G^p
G^{\bar q}
\,.
\end{equation}
This quantity must be positive if we want the matrix (\ref{VIJ}) to be positive definite
\footnote{Actually, as emphasized in \cite{yo4}, the Goldstino multiplet cannot receive any supersymmetric
mass contribution from $W$, since in the limit of rigid supersymmetry its fermionic
component must be massless. This means that, in order to study metastability, it is enough to study
the projection of the diagonal block $m^2_{i \bar \jmath} $ of the mass matrix along the
Goldstino direction $G^i$, as the rest of the projections can be given a mass with the help of
the superpotential.}. Using the
rescaled variables $f^i =  - \frac{1}{\sqrt{3}} G^i$ the conditions for the existence of
non-supersymmetric flat minima can then be written as:
\begin{eqnarray}\label{fs-1}
\left\{\!\!
\begin{array}{l}
G_{i\,\bar \jmath}f^if^{\bar \jmath} = 1\,,\\
R_{i \,\bar \jmath \,p \,\bar q}\, f^i f^{\bar \jmath} f^p f^{\bar q}< \displaystyle{\frac23}\,.
\end{array}\right.
\end{eqnarray}
The first condition, the flatness condition, fixes the amount of supersymmetry breaking
whereas the second condition, the stability condition, requires the existence of directions with
K\"ahler curvature less than {$2/3$} and constraints the direction of supersymmetry breaking
to be sufficiently aligned with it.

\subsection{Constraints for gauge invariant theories}

It can happen that the supergravity theory with $n$ chiral multiplets $\Phi^i$ we just described
has a group of some number $m$ of global symmetries, compatibly with supersymmetry. In this
subsection we consider the possibility of gauging such isometries with the introduction of
vector multiplets. The corresponding supergravity theory will then include in addition to the $n$ chiral
multiplets $\Phi^i$, $m$ vector multiplets $V^a$.

The two-derivative Lagrangian is specified in this case by a
real K\"ahler function $G(\Phi^k,\bar\Phi^{k},V^a)$, determining in particular the scalar geometry,
$m$ holomorphic Killing vectors $X_a^i(\Phi^k)$, generating the isometries that are gauged, and an
$m$ by $m$ matrix of holomorphic gauge kinetic functions $H_{ab}(\Phi^k)$, defining the
gauge couplings\footnote{The gauge kinetic function $H_{ab}$ must have an appropriate
behavior under gauge transformations, in such a way as to cancel possible gauge anomalies
$Q_{abc}$. Actually, the part
$h_{ab}={\rm Re}\,H_{ab}$ defines a metric for the gauge fields and must be gauge invariant.
On the other hand ${\rm Im}\,H_{ab}$ must have a variation that matches the coefficient of
$Q_{abc}$, namely $X_a^i h_{bc i} =  \frac i2 \, Q_{abc}$.}. In this case the minimal coupling between chiral and vector multiplets turn ordinary
derivatives into covariant derivatives, and induces a new contribution to the scalar potential
coming from the vector auxiliary fields $D^a$, in addition to the standard one coming from the
chiral auxiliary fields $F^i$. The 4D scalar potential takes the form:
\begin{equation}
\label{genpot2}
V =  e^G\Big( g^{i \bar \jmath}\, G_i G_{\bar \jmath}-3\Big) + \frac 12 h^{ab} D_a D_b\,.
\end{equation}
The auxiliary fields are fixed from the Lagrangian through the equations of motion  to be:
\begin{eqnarray}
&& F_i = - m_{3/2}\, G_i \,, \label{F} \\[1mm]
&& D_a = -G_a =  i \, X_a^i \, G_i = - i \, X_a^{\bar \imath} \, G_{\bar \imath} \,, \label{D}
\end{eqnarray}
where to get the relations in (\ref{D}) one should also use gauge invariance of the action.

Now in order to find local non-supersymmetric minima of the potential (\ref{genpot2})
with small non-negative cosmological constant, we will proceed as in the previous subsection.
First we will impose the condition that the cosmological constant is negligible and fix $V =0$.
This flatness condition implies that:
\begin{equation}
- 3 + G^i G_i + \frac 12 \, e^{-G} D^a D_a = 0\,.
\label{flatness}
\end{equation}
The stationarity conditions correspond now to the requirement that $\nabla_i V = 0$, and
they are given by:
\begin{equation}
G_i + G^k \nabla_i G_k + e^{-G} \Big[D^a\Big(\nabla_i - \frac 12\, G_i \Big) D_a
+ \frac 12\,  h_{abi} D^a D^b \Big] = 0\,.
\label{stationarity}
\end{equation}

The $2n$-dimensional mass matrix (\ref{VIJ}) for small fluctuations of the scalar fields around the
vacuum has as before two different $n$-dimensional blocks, which can be computed as
$m_{i \bar\jmath}^2 = \nabla_i \nabla_{\bar\jmath} V$ and $m_{i j}^2 = \nabla_i \nabla_j V$. Using
the flatness and stationarity conditions, one finds, after a straightforward computation
\cite{FKZ,dudasvempati}:
\begin{eqnarray}
&&\hspace{-1.5cm}m_{i \bar \jmath}^2 = e^G \Big[g_{i \bar \jmath} - R_{i \bar \jmath p \bar q} G^p G^{\bar q}
+  \nabla_i G_k \nabla_{\bar \jmath} G^k \Big] - \frac 12\, \Big(g_{i \bar \jmath} - G_i G_{\bar \jmath} \Big) D^a D_a
-\,2\, D^a G_{(i} \nabla_{\bar \jmath)} D_a \label{mijbar} \\
&&\hspace{-.5cm} + \Big(G_{(i} h_{ab \bar \jmath)} + h^{cd} h_{a c i} h_{b d \bar \jmath} \Big)\, D^a D^b  -
2\, D^a h^{bc} h_{ab(i} \nabla_{\bar \jmath)} D_c
+ h^{ab} \nabla_i D_a  \nabla_{\bar \jmath} D_b + D^a \nabla_i \nabla_{\bar \jmath} D_a\,, \nonumber \\
&& \hspace{-1.5cm}m_{i j}^2 = e^G \Big[2 \, \nabla_{(i} G_{j)} + G^k \nabla_{(i} \nabla_{j)} G_k \Big] - \frac 12 \Big(\nabla_{(i} G_{j)} - G_i G_j \Big) D^a D_a+ h^{ab}\, \nabla_i D_a \nabla_j D_b
  \label{mij} \\
&&  \hspace{-.5cm}-\,2\, D^a G_{(i} \nabla_{j)}  D_a - 2\,D^a h^{bc} h_{ab(i} \nabla_{j)} D_c
+ \Big(G_{(i} h_{abj)} + h^{cd} h_{a c i} h_{b d j} - \frac 12 h_{a b i j} \Big) D^a D^b \,.\nonumber
\end{eqnarray}

We want to analyze now the restrictions imposed by the requirement
that the physical squared mass of the scalar fields are all positive. In general the theory displays a spontaneous breakdown of both supersymmetry and gauge symmetries, so in the study of the stability
of the vacuum it is
necessary to take appropriately into account the spontaneous breaking of gauge symmetries.
In that process $m$ of the $2n$ scalars, the would-be Goldstone
bosons, are absorbed by the gauge fields and get a positive mass, so we do not need to take them
into account for the analysis of the stability. Nevertheless the would-be
Goldstone modes correspond to flat directions of the unphysical mass matrix,
and get their physical mass through their kinetic mixing with the gauge bosons.
This means that positivity of the physical mass matrix implies semi-positivity of the unphysical mass
matrix in (\ref{mijbar}), (\ref{mij}). We can use then the same strategy as before but changing the
strictly positive condition to a semi-positive one.

In this case there exist two types of special complex directions $z^i$ one could look at. The first is
the direction $G^i$, which is associated with the Goldstino direction in the subspace of chiral multiplet
fermions. Projecting into this direction one finds, after a long but straightforward computation:
\begin{eqnarray}
m^2_{i \bar \jmath} G^i G^{\bar \jmath} \!\!\!&=&\!\!\!
e^G \Big[6 -R_{i \bar \jmath p \bar q} \, G^i G^{\bar \jmath} G^p G^{\bar q} \Big]
+\, \Big[\!-\! 2\, D^a D_a + h^{cd} h_{a c i} h_{b d \bar \jmath} \, G^i G^{\bar \jmath} D^a D^b \Big] \label{mGG}\\
\!\!\!&\;&\!\!\! +\,e^{-G} \Big[M^2_{ab} D^a D^b \!+\! \frac 34\,Q_{abc} D^a D^b D^c
\!-\! \frac 12 \Big(D^a D_a\Big)^2 \! \!+\! \frac 14 h_{ab}^{\;\;\;i} h_{cdi} D^a D^b D^c D^d \Big]\,, \nonumber
\end{eqnarray}
where $Q_{abc}=-2iX^i_ah_{bci}$.
The condition $m^2\hspace{-4pt}{}_{i \bar \jmath} \, G^i G^{\bar \jmath} \ge 0$ is then the generalization of the
condition in (\ref{eq}) for theories involving only chiral multiplets. In terms of the rescaled variables:
\begin{equation}
f_i=\frac{1}{\sqrt{3}}\frac{F_i}{m_{3/2}}=-\frac{1}{\sqrt{3}}G_i\,,\hspace{1.8cm}
d_a=\frac{1}{\sqrt{6}}\frac{D_a}{m_{3/2}}\,,
\end{equation}
the flatness and stability conditions take then the following form:
\begin{eqnarray}
\hspace{-.8cm}\left\{\hspace{-4pt}
\begin{array}{l}
\displaystyle{ \,f^i f_{i} + d^a d_a} = 1\,,\\
\displaystyle{R_{i \,\bar \jmath\, p\, \bar q} \, f^i f^{\bar \jmath} f^p f^{\bar q} \le \frac23
+ \frac23 \Big({M_{ab}^2}/{m^2_{3/2}} -2 h_{a\,b} \Big) d^a d^b
+2 h^{c\,d}h_{a\,c\,i}h_{b\,d\,\bar \jmath}f^if^{\bar \jmath}d^ad^b}\\
\displaystyle{\hspace{2.8cm}- \Big(2\, h_{a\,b} h_{c\,d}- h_{a\,b}^ih_{c\,d\,i}\Big)\, d^a d^b d^c d^d
+\sqrt{\frac 32} \frac{Q_{abc}}{m_{3/2}} d^a d^b d^c\,.}
\end{array}
\right.
\end{eqnarray}

Again we have that the flatness condition fixes the amount of supersymmetry breaking whereas
the stability condition constrains its direction. One could also consider the
directions $X_a^i$, which are instead associated with the Goldstone directions in the space of chiral
multiplet scalars. Nevertheless the constraint $m^2_{i\bar \jmath}X_a^iX_a^{\bar \jmath}\ge0$ turns
out to be more complicated and no useful condition seems to emerge from it.

\section{Analysis of the constraints}

The analysis of the flatness and stability conditions in the case where both chiral and vector multiplets
participate to supersymmetry breaking presents an additional complication with respect to the case
where only chiral multiplets are present, due to the fact that the auxiliary
fields of the chiral and vector multiplets are not independent of each other.
The rescaled auxiliary fields $f_i$ and $d_a$ are actually related in several ways.
One first relation (consequence of gauge invariance) can be read from
eq. (\ref{D}) and is given by:
\begin{equation}
d^a =  \frac{i\,X^a_i}{\sqrt{2} m_{3/2}} \,  f^i \label{DFkin} \,.
\end{equation}
This relation is satisfied as a functional relation valid at any point of the scalar field space.
It shows that the $d_a$ are actually linear combinations of the $f_i$. Using
now the inequality $|a^i b_i| \le \sqrt{a^i a_i} \sqrt{b^j b_j}$ one can derive a simple bound on
the sizes that the $d_a$ can have relative to the $f_i$:
\begin{equation}
|d_a| \le \frac {1}{2} \frac {M_{aa}}{m_{3/2}} \sqrt{f^i f_i}\,.
\label{DF}
\end{equation}
There is also a second relation between $f_i$ and $d_a$, that is instead valid
only at the stationary points of the potential.
It arises by considering a suitable linear combination of the stationarity conditions along the
direction $X_a^i$, in other words, by imposing $X_a^i \nabla_i V= 0$. This relation reads
\cite{KawamuraDFF,ChoiDFF} (see also \cite{KawaKobDFF}):
\begin{equation}
i\,\nabla_i X_{a\bar \jmath} \, f^i f^{\bar \jmath}
-  \sqrt{\frac23}m_{3/2}\Big(3 f^i f_i - 1 \Big) \,d_a - \, \frac{M^2_{ab}}{\sqrt{6} \,m_{3/2}} \, d^b
+Q_{abc}\,d^b d^c= 0\label{DFF}\,.
\end{equation}
These relations show that whenever the $f_i$ auxiliary fields vanish also the $d_a$ auxiliary fields
should vanish. Therefore we can say that the {$f_i$}'s represent the basic qualitative seed
for supersymmetry breaking whereas the {$d_a$}'s provide additional quantitative effects.
Along this section we will address
the problem of working out more concretely the implications of these constraints. In order to do so
we will concentrate on the case in which the gauge kinetic function is constant and diagonal:
$h_{ab} = g_a^{-2} \delta_{ab}$. In this case we can rescale the vector fields in such a way as to
include a factor $g_a$ for each vector index $a$. In this way, no explicit dependence on $g_a$ is left
in the formulas and the metric becomes just $\delta_{ab}$. Using this the flatness and stability
conditions take the following simple form:
\begin{eqnarray}\label{fs-2}
\left\{\hspace{-4pt}
\begin{array}{l}
 f^i f_i +\sum_a d_a^2 = 1 \,,\\
R_{i\, \bar \jmath\, p\, \bar q} \, f^i f^{\bar \jmath} f^p f^{\bar q} \le \displaystyle{\frac 23}
+ \displaystyle{\frac 43} \, \mbox{$\sum_{a}$} \Big(2\, m_{a}^2 -1 \Big)\, d_a^2
- 2\, \mbox{$\sum_{a,b}$} d_a^2 d_b^2\,,
\end{array}
\right.
\end{eqnarray}
where we have defined the quantity $m_{a} =  {M_{a}}/{(2\, m_{3/2})}$
measuring the hierarchies between scales. Denoting $v_a^i =  {\sqrt{2} X_a^i}/{M_a}$ and
$T_{a\, i \bar \jmath} = {i\,\nabla_i X_{a\,\bar \jmath}}/{M_a}$ the relations between
{$f^i$} and {$d^a$} read:
\begin{eqnarray}
&& d_a = i\, m_a v_a^i f_i  \hspace{.4cm}\Longrightarrow\hspace{.4cm}
|d_a| \le m_{a} \, \sqrt{ f^i f_i}  \,,\\
&& d_a = \sqrt{\frac 32} \, \frac {m_a \,T_{a\, i \,\bar \jmath}  \, f^i f^{\bar \jmath}}
{m_{a}^2 - 1/2 + 3/2\, f^i f_i}  \label{KIn}\,.
\end{eqnarray}

\subsection{Interplay between F and D breaking effects}

In this subsection we will study the interplay between the {$F$} and {$D$} supersymmetry
breaking effects. In order to do so it is useful to introduce the variables ${\hat f}^i =  {f^i}/
{\sqrt{1 - \sum_a d_a^2}}$. Using these variables the conditions for flatness and stability can
be rewritten as:
\begin{eqnarray}
\left\{\hspace{-4pt}
\begin{array}{l}
\displaystyle{ {\hat f}^i {\hat f}_{i} = 1 } \,,\\
\displaystyle{R_{i\, \bar \jmath\, p\, \bar q} \, {\hat f}^i {\hat f}^{\bar \jmath} {\hat f}^p {\hat f}^{\bar q}
\le \frac 23 \, K(d_a^2,m_a^2) }\,,
\end{array}
\right.
\end{eqnarray}
where the function $K(d_a^2,m_a^2)$ is given by:
\begin{equation}
K(d_a^2,m_a^2) = 1 + 4\, \frac {\sum_a m_a^2 d_a^2 - \big(\sum_a d_a^2\big)^2 \raisebox{-5pt}{}}
{\big(1 - \sum_b d_b^2\big)^2} \,.
\end{equation}

In the limit in which the rescaled vector auxiliary fields are small ($d_a \ll 1$) we have that
${\hat f}^i \simeq f^i$ and therefore these variables $\hat f^i$ are the right variables to study
the effect of vector multiplets with respect to the case where only chiral multiplets are present. Note that
in such a limit the relation (\ref{KIn}) between F and D auxiliary fields can be written at first order as
$d_a\,\simeq\,\sqrt{3/2}\,m_a/(1+m_a^2)\,T_{a\,i\,\bar \jmath}\,{\hat f}^i\,{\hat f}^{\bar \jmath}$.
Using this we get:
\begin{equation}
K\,\simeq\,1+6\, \mbox{$\sum_a$} \,\xi^2_a(m)\,T_{a\,i\,\bar \jmath} \,T_{a\,p\,\bar q} \,
{\hat f}^i {\hat f}^{\bar \jmath}{\hat f}^p {\hat f}^{\bar q}\,,\hspace{1cm}\xi_a(m)=\frac{m_a^2}{1+m_a^2}\,,
\end{equation}
and we can write the flatness and stability conditions as:
\begin{eqnarray}
\left\{\hspace{-4pt}
\begin{array}{l}
\displaystyle{ {\hat f}^i {\hat f}_{i} = 1 } \,,\\
\displaystyle{\hat{R}_{i\, \bar \jmath\, p\, \bar q} \, {\hat f}^i {\hat f}^{\bar \jmath} {\hat f}^p
{\hat f}^{\bar q} \le \frac 23 }\,,
\end{array}
\right.
\end{eqnarray}
where $\hat{R}_{i\, \bar \jmath\, p\, \bar q} = R_{i\, \bar \jmath\, p\, \bar q} -
4\,\sum_a \xi^2_a(m)\,T_{a\,i\,(\bar \jmath} \,T_{a\,p\,\bar q)}$. This means
that the net effect in this case is to change
the curvature felt by the chiral multiplets. Note as well that in the case in which the mass of the
vectors is large this is not necessarily a small effect and can compete with the
curvature effects due to the chiral multiplets. Actually for heavy vector fields one can check that
integrating out the vector fields modifies the K\"ahler potential of the chiral multiplets
in a way that accounts for this shift in the K\"ahler curvature.

For larger values of {$d_a$} one can instead find an upper
bound to {$K$} (see \cite{yo3} for details):
\begin{equation}
\hspace{-.5cm}K \leq 1+6 \, \mbox{$\sum_a$} \, \xi^2_a(m) T_{a\,i\,\bar \jmath} \,T_{a\,p\,\bar q} \,
{\hat f}^i {\hat f}^{\bar \jmath}{\hat f}^p {\hat f}^{\bar q}\,,\hspace{1cm}
\xi_a(m)=\frac{m_a^2\, (1+\sum_bm_b^2)}{1+m_a^2+(m_a^2-\frac12)
\sum_bm_b^2}\,.
\end{equation}
So in this general case we get as well that the effect of vector multiplets can be encoded into an
effective curvature $\hat{R}_{i\, \bar \jmath\, p\, \bar q} = R_{i\, \bar \jmath\, p\, \bar q} -
4\,\sum_a \xi^2_a(m)\,T_{a\,i\,(\bar \jmath} \,T_{a\,p\,\bar q)}$.

In this section we have derived the implications of the flatness and stability conditions taking into
account the fact that  $f^i$ and $d^a$ are not independent variables. The strategy that we have followed
is to use the the relation (\ref{DFkin}) to write $d_a$ in terms of $f^i$. A second possibility would be
to use instead the relation (\ref{DFF}) to write $d^a$ in term of $f^i$ and a third one would be to
impose only the bound (\ref{DF}) to restrict the values of the $d^a$ in terms of the values of $f^i$.
It is clear that switching from the relation (\ref{DFkin}) to the relation (\ref{DFF}) and finally to
the bound (\ref{DF}) represents a gradual simplification of the formulas, which is also
accompanied by a loss of information. As a consequence, these different types of strategies will
be tractable over an increasingly larger domain of parameters, but this will be
accompanied by a gradual weakening of the implied constraints. A detailed
derivation of the implications of the flatness and stability conditions when the relations (\ref{DFF})
and (\ref{DF}) are used can be found in \cite{yo3}.

\section{Some examples: moduli fields in string models}

In this section we will apply our results to the typical situations arising for the moduli sector of string models. The K\"ahler potential and superpotential governing the dynamics of
these moduli fields typically have the general structure:
\begin{equation}
\label{stringk}
K = - \,\mbox{$\sum_i$}\, n_i\, {\rm ln} (\Phi_i + \bar\Phi_i) + \dots\,, \\
\end{equation}
where by the dots we denote corrections that are subleading in the derivative and loop expansions
defining the effective theory. The K\"ahler metric computed from (\ref{stringk}) becomes diagonal
and the whole Kahler manifold factorizes into the product of $n$ one-dimensional Kahler
submanifolds. Also the only non-vanishing components of the Riemann
tensor are the $n$ totally diagonal components ${R_{i \,\bar \jmath\, p\, \bar q} =
R_{i} \,g_{i\,\bar \imath}^2 \, \delta_{i\, \bar \jmath p \bar q}}$ where $R_i={2}/{n_i}$.
Recall now that when only chiral fields participate to supersymmetry breaking the flatness and
stability conditions take the form (\ref{fs-1}), so in this particular case they just read:
\begin{equation}\label{32}
\,\mbox{$\sum_i$}\, |f^i|^2 = 1\,,\hspace{1.5cm}\,\mbox{$\sum_i$}\, R_{i}\, |f^i|^4<  \frac 23 \,.
\end{equation}
These relations represent a quadratic inequality in the variable $|f^i|^2$ subject to a linear constraint.
This system of equations can be easily solved to get the condition $\sum_{i} R^{-1}_{i}
> \frac32$, which translates into:
\begin{equation}\label{con}
\hspace{4cm} \,\mbox{$\sum_i$}\, n_{i} > 3\,.
\end{equation}
Also eqs.~(\ref{32}) constrain the values that the auxiliary fields {$|f_i|$} can take.

When a single modulus dominates the dynamics the condition (\ref{con}) implies $n>3$ (this result was
already found in \cite{Brustein:2004xn} in a less direct way). For the universal dilaton $S$ we have
$n_S = 1$ and therefore
it does not fulfill the necessary condition (\ref{con}). This shows in a very clear way that
just the dilaton modulus cannot lead to a viable situation \cite{nodildom} unless subleading corrections
to its K\"ahler potential become large \cite{nonpertdil,yetmoredil}.
We can therefore conclude that the scenario proposed in ref.~\cite{dilatondom}, in which
the dilaton dominates supersymmetry breaking, can never be realized in a controllable way.
On the other hand, the overall K\"ahler modulus $T$ has $n_T = 3$, and violates
only marginally the necessary condition. In this case, subleading corrections to the K\"ahler potential
are crucial. Recently some interesting cases where subleading corrections can help in achieving a
satisfactory scenario based only on the $T$ field have been identified for example in
\cite{nonpertvol,nonpertvolbis}.

In this case where the dynamics is dominated by
just one field the K\"ahler potential of (\ref{stringk}) corresponds to a constant curvature manifold with
 $R = 2/n$ and it has a
global symmetry associated to the Killing vector $X = i\, \xi$, which can be gauged as long as the
superpotential is also gauge invariant. By doing so the potential would get a $D$-term contribution
that should be taken into account in the analysis of stability, as was explained in the previous section.
In such a situation the flatness condition in (\ref{fs-2}) can be solved by introducing
an angle $\delta$ and parametrizing the rescaled auxiliary fields as $f = \cos \delta$ and
$d = \sin \delta$. In terms of this angle the stability condition implies:
\begin{equation}
n > \frac {3}{1 + 4\, \tan^6\delta} \,.
\label{c2}
\end{equation}
From this expression, it is clear that it is always possible to
satisfy the stability condition for a large enough value of $\tan\delta$.
Note in particular that eq.~(\ref{c2}) implies that when $n$ is substantially less than
$3$, which is the critical value for stability in the absence of gauging, the contribution to
supersymmetry breaking coming from the $D$ auxiliary field must be comparable to the
one coming from the $F$ auxiliary field.

A final comment is in order regarding the issue of implementing the idea of uplifting with an uplifting
sector that breaks supersymmetry in a soft way. It is clear that such a sector will have to contain
some light degrees of freedom, providing also some non-vanishing $F$ and/or $D$ auxiliary field.
Models realizing an $F$-term uplifting are easy to construct. A basic precursor of such models
was first constructed in \cite{lutysundrum}.
More recently, a variety of other examples have been constructed, where the extra chiral multiplets
have an O' Raifeartaigh like dynamics that is either genuinely postulated from the beginning
\cite{Fup} or effectively derived from the dual description of a strongly coupled theory \cite{Fupiss}
admitting a metastable supersymmetry breaking vacuum as in \cite{ISS}. Actually, a very simple
and general class of such models can be constructed by using as uplifting sector any kind of sector
breaking supersymmetry at a scale much lower than the Planck scale \cite{yo1}. Models realizing
a $D$-term uplifting, on the other hand, are difficult to achieve. The natural idea of relying on some
Fayet-Iliopoulos term \cite{BKQ} does not work, due to the already mentioned fact that such terms
must generically be field-dependent in supergravity, so that the induced $D$ is actually
proportional to the available charged $F$'s. It is then clear that there is an obstruction in getting
$D$ much bigger than the $F$'s (see also \cite{cr}).
Most importantly, if the only charged chiral multiplet in the model
is the one of the would-be supersymmetric sector (which is supposed to have vanishing $F$) then
also $D$ must vanish, implying that a vector multiplet cannot act alone as an uplifting sector
\cite{ChoiDup,DealwisDup}. This difference between $F$-term and $D$-term uplifting is, as
was emphasized in the previous section,
due to the basic fact that chiral multiplets can dominate supersymmetry breaking whereas vector
multiplets cannot.

Finally we would like to mention that the  flatness and stability conditions simplify not only for
factorizable K\"ahler manifolds but also for some other classes of scalar manifolds that present
a simple structure for the Riemann tensor. This is the case for example for K\"ahler potentials
generating a scalar manifold of the form $G/H$ which arise for example in orbifold string
 models \cite{yo2,yo3}, and also for no-scale supergravities and
Calabi-Yau string models \cite{yo4}.

\section{Conlcusions}

In this note we have reviewed the constraints that can be put on gauge invariant supergravity
models from the requirement of the existence of a flat and metastable vacuum, following the
results of \cite{yo1,yo2,yo3}. We have shown that in a general {${\cal N}=1$} supergravity theory
with chiral and vector multiplets there are {strong necessary conditions} for the existence of
phenomenologically viable vacua. Our results can be summarized as follows. These necessary conditions severely {constrain the geometry} of the scalar manifold as well as {the direction} of supersymmetry breaking and {the size} of the auxiliary fields. When supersymmetry breaking is
dominated by the chiral multiplets the conditions restrict the {K\"ahler curvature}, whereas when
also vector multiplets participate to supersymmetry breaking the net effect is to alleviate the
constraints through a {lower effective curvature}. This is mainly due to the fact that the $D$-type
auxiliary fields give a positive definite contribution to the scalar potential, on the contrary of the
$F$-type auxiliary fields, which give an indefinite sign contribution. Nevertheless one should also
take into account the fact that the local symmetries associated to the vector multiplets also
restrict the allowed superpotentials. These results should be useful in discriminating
more efficiently potentially viable models among those emerging, for instance, as low-energy
effective descriptions of string models.

\begin{acknowledgement}
M.G.-R. would like to thank the organizers of the RTN meeting "Constituents, Fundamental Forces and
Symmetries of the Universe" held in Valencia, 1-5 October 2007, for the opportunity to present this work.
The work of C.A.S. has been supported by the Swiss National Science Foundation.
\end{acknowledgement}