\section{Introduction}

It is generally accepted that the rotationally modulated photometric
variability of magnetic CP stars originates in consequence of
non-uniform horizontal structure of their atmospheres induced by
strong global magnetic field and uneven surface distribution of
chemical elements.

Recently we successfully simulated the observed light variability of
the He-strong CP star HD\,37776 in $uvby$ colours providing an
inhomogeneous elemental horizontal distribution. Despite this star
displays a relatively complex surface pattern with several
relatively bright and/or dark regions (see Krti\v{c}ka {\it et al.}
2007) and a quadrupole dominated magnetic field (Thompson,
Landstreet 1985), its light curve shows a not very complicated
single wave form. However, HD\,37776 is not a typical representative
for its considerably complex magnetic field geometry which can make
the stellar surface a bit disordered. A still better subject could
be another Si star, HD\,177410, (Krti\v{c}ka {\it et al.} 2008,
these proceedings) with only two bright spots on the surface, and we
do not exclude the existence of mCP stars with even simpler
photometric structure having only one dominant spot tied with
spectroscopic or magnetic structures on their surfaces.

In this brief recognition study we attempt to look up and study mCP
stars which could carry on such easily readable surface structures.
The comprehension of the simplest case can be a way to common, more
complicated ones.

\section{Photometrically simply behaving mCP stars}

We showed (Mikul\'a\v{s}ek {\it et al.} 2007a) that the variable
part of a light curve (LCs), $\Delta m_c$~in the colour $c$ of an
mCP star, can be expressed as a linear combination of two normalized
orthogonal phase functions $f_1(\varphi),f_2(\varphi)$, which can be
found easily by means of the advanced principal component analysis
(Mikul\'a\v{s}ek, 2007):
\begin{eqnarray}
\Delta m_c(\varphi)\cong A_{1c}\,f_1(\varphi)+A_{2c}
f_2(\varphi);\label{pca}
\end{eqnarray}
\begin{eqnarray}
\sum_{j=1}^N f_1^2(\varphi_j)=\sum_{j=1}^N
f_2^2(\varphi_j)\cong\frac{N}{2};\quad \sum_{j=1}^N
f_1(\varphi_j)f_2(\varphi_j)=0,
\end{eqnarray}
where $N$ is the number of measurements, $\varphi_j$ is phase of
the $j$-th measurement.

We quantify the similarity of two colour light curves by a ratio
\begin{eqnarray}
 r^2=\sum_{c=1}^5 A_{2c}^2/\sum_{c=1}^5 A_{1c}^2,
\end{eqnarray}
and we adopt that light curves of an mCP star are mutually similar
if $r<0.15$. In such a degree of similarity the light curves
can be described sufficiently by a unique phase function, which
mathematically means the second term in Eq.\,\ref{pca} can be neglected.

Out of the 85 stars with known ephemeris and the $uvby$ and $H_{\rm
p}$ photometries contained in the {\it On-line database of
photometric observations of mCP stars} (Mikul\'a\v{s}ek {\it et
al.}, 2007b) only 19 mCP stars satisfied the above-mentioned
criterion of the similarity: two He-weak, twelve Si, and five
SrCrEu-type stars. It is rather surprising that shapes of their LCs
are very diverse (see Fig.\,\ref{obr}), what can be also documented
by the following quantification. We assume the phase function
$f_1(\varphi)$ in the form:
\begin{eqnarray}
f_1(\varphi)=\sqrt{1-a_{1}^2-a_{2}^2}\
\cos(2\pi\varphi)+a_{1}\cos(4\pi\varphi)
+a_{2}\sin(4\pi\varphi).\label{a12}
\end{eqnarray}
If the timing of the linear ephemeris is set so that the term with
$\sin(2\pi\varphi)$ is zero, then the basic extrema of the LCs occur
near phases 0 and 0.5. The variable part of the LCs of a PSB star is
then characterized by a set of amplitudes $A_{1c}$ which can be
arranged into a vector
$\overrightarrow{A}=[A_{1u},A_{1v},A_{1b},A_{1H_{\rm p}},A_{1y}]$,
and by the two parameters $a_1,\,a_2$ from equation \ref{a12}. A
light curve characterized by $a_{1}>0$ has a sharper maximum at the
phase~0 and a flatter minimum (see e.\,g. HD\,197322) and vice versa
for $a_{1}<0$. The parameter $a_2$ expresses the measure of
asymmetry of an LC. The larger $|a_2|$ the larger asymmetry. The
relation between parameters $a_1,\,a_2$ is depicted on
Fig.\,\ref{a1a2}~left.

\begin{figure}[t]
\centerline{\includegraphics[width=0.492\textwidth]{a1a2.pdf}
\includegraphics[width=.500\textwidth,height=.47\textwidth]{B12.pdf}}
\caption{{\bf Left:} The relation of LCs' parameters $a_1$ and
$a_2$, shows that LCs of our PSB stars are only slightly asymmetric.
LCs can be divided into four groups. Three stars whose LC's can be
interpreted by the one-spot model are denoted with their HD names.
{\bf Right:} Two basic dependencies of LCs' amplitudes on the
effective wavelength.} \label{a1a2}
\end{figure}

The components of the amplitude vectors $\overrightarrow{A}$ of a
PSB star are not independent. They can be expressed by a linear
combination of two basic normalized vectors $\overrightarrow{B_1}$,
$\overrightarrow{B_2}$ (courses of $B_{1,2}(\lambda)$ -- see
Fig.\,\ref{a1a2}~right, where the wavelength $\lambda$ denotes the
maximum transmissivity of a photometric filter concerned) with
coefficients $A_1,\,A_2$, namely
$\overrightarrow{A}=A_1\,\overrightarrow{B_1}+A_2\,\overrightarrow{B_2}$.
For the relationship of parameters $A_1$ and $A_2$ see
Fig.\,\ref{acko12}. This indicates that there are at least two
different mechanisms responsible for the contrast of a photometric
spot in respect to the surrounding stellar surface. Consequently,
there exist at least two types of photometric spots on the surface
of an mCP star differing in their colour grades. These spots can of
course positionally coincide what can either strengthen or suppress
their resulting contrast in particular photometric colours.

The first mechanism which is obviously effective in all types of mCP
stars creates bright spots in $uvby$  due to redistribution of
energy from the UV region to the optical one originating at least
partly in bound-free transitions of overabundant ions. In hot CP
stars the most important elements seems to be silicon, iron and
helium. The symptomatic feature of this mechanism is the monotonous
decline of the amplitude of light variations $B_2$ with the
increasing effective wavelength as shown on Fig.\,\ref{a1a2}~right.
The effect is well reproduced by our models (Krti\v{c}ka {\it et
al.} 2008, these proceedings).

The second mechanism, manifesting itself through strong variations
of the $c_1$ index, characterizing the height of the Balmer jump, is
present only in the coolest mCP stars. The photometric spots caused
by this mechanism are markedly dark in the $v$ colour, in which the
largest light variations are observed. We speculate that the strong
opacity caused by various overabundant elements in a cool atmosphere
suppresses the Balmer jump. The purest example of the light
variations controlled by this mechanism is HD\,110956B with the
largest amplitude among all mCP stars (see Fig.\,\ref{obr}).

\begin{figure}[t]
\centerline{\includegraphics[width=0.8\textwidth]{A12.pdf}}
\caption{The relation between the effective amplitudes $A_1$ and
$A_2$, corresponding to $\protect\overrightarrow{B_1},
\protect\overrightarrow{B_2}$. The late mCP stars (gr. 4) strongly
differ from other PSBS.} \label{acko12}
\end{figure}

\section{Parametrization and classification of PSB stars}

The set of the $uvby$ and $H_{\rm p}$ light curves of each of the PSB
 stars can be satisfactorily well described by the
quaternion of parameters: $\{a_1,a_2,A_1,A_2\}$.

According to these parameters we sort out all the PSB stars into four
groups.
\begin{itemize}
    \item {\textbf{Group 1}: $A_2\cong0,\ a_1\geq0.24$, double wave
    LCs with two unequally prominent bright spot centered on the phases
    $\varphi=0$~and 0.5. Example: HD\,177410.}
    \item {\textbf{Group 2}: $A_2\cong0,\ |a_1|\leq0.24$, single wave
    LCs. Example: HD\,215441.}
    \item {\textbf{Group 3}: $A_2\cong0,\ a_1\leq-0.24$, double wave
    LCs with two unequally prominent bright spots centered on the
    vicinity of $\varphi=0$. Example: HD\,49333.}
    \item {\textbf{Group 4}: $A_2\neq0$, cool CP stars with the
    largest variations in $v$ caused by a dominating dark spot (see
    HD\,110956B) with a minimum near $\varphi=0.5$. Besides this,

    another dark, as in HD\,7676, or a bright spot, as in HD\,119213
    can be present on the surface.}
\end{itemize}

\section{Conclusions}

All the stars with asymmetric LCs (e.g. HD\,171247), as well as all
the stars with doublewave LCs (groups 1 and 3) must be left out from
the short list of the 19 candidates which LCs can be explained by
the simple one-spot model. Similarly, all the stars of the group 2
with $a_1<0$ must be deleted as they have two or more
undistinguished bright spots centered on the $\varphi=0$. Thus out
of the stars in the groups 1, 2 and 3 only the Si-star HD\,193732 is
the candidate to be explained by the simple one-spot model. Our
analysis of the light curves produced by the one-spot model shows
that for $a_1<0.15$ the spot covers almost the whole hemisphere of a
star.

HD\,188041 and HD\,110956 are the candidates in the group 4.
However, it should be noted, that the dominant photometric spot can,
due to geometry, suppress the visibility of another spot.

We conclude that the presence of two or more photometric spots on
the mCP stars is quite typical. It seems that the photometrically
simply behaving stars do not differ from the others and their
belonging to the privilege group is more likely only the consequence
of the distributions of spots on the surface with respect to the
rotational axis.

\acknowledgements This work was supported by grants GA\,\v{C}R
205/06/0217, VEGA 2/6036/6, and MVTS \v{C}RSR 10/15 and 01506.


\section{Introduction}

The current burst of star formation (SF) dominates the spectral energy
distribution (SED) in H{\sc ii} galaxies even if previous stellar
populations are present, making difficult to know the star formation
history (SFH) of the galaxy. In order to understand how the SF takes
place we study the viability of a model \citep{man08} using
simultaneously the whole available information for the galaxy sample:
the ionized gas, which defines the present time state of the galaxy,
and spectrophotometric parameters, related to the galaxy SFH.

We assume 11 successive attenuated star-bursts along 13.2 Ga in a
region with a total gas mass of 10$^{8}M_{\odot}$. In each burst a
certain amount of gas is consumed to form stars with a given initial
efficiency. The SFH and the age-metallicity relation are given by the
chemical evolution code \citep[based on][]{fer94}. The SED of the
ionizing population is computed using the single stellar populations
(SSP's) from \cite{mol00}. Finally, the emission lines are calculated
by the photoionization code CLOUDY \citep{fer98}.

\section{Main Model Results}

The evolution of the oxygen abundance is shown in
Fig.~\ref{fig1}a. Models with initial efficiencies $\epsilon \sim $
10\% to 33\% reproduce the data range of H{\sc ii} galaxies,\citep[][dashed lines]{hoy06}.
Once the SED of the ionizing
continuum is computed, we obtain the emission lines, giving
information about the current burst of SF.  Fig.~\ref{fig1}b shows an
excitation diagram for the ionized nebula. The model reproduces the
observational data for H{\sc ii} galaxies. The effect of an
underlying population should be more easily seen in the observed
colours of these galaxies, since any star-burst previous to the
currently observed one will contribute substantially to the total
continuum luminosity at the different wavebands. The broad band
colours with and without the contribution by the stronger emission lines are
shown in Fig.~\ref{fig1}c (dashed and solid lines respectively).

\begin{figure*}[!ht]
\epsfclipon
\plottwo{fig1a.eps}{fig1b.eps}
\plottwo{fig1c.eps}{fig1d.eps}
\caption{a)Oxygen abundance with efficiencies $\epsilon= 0.3$ and 0.1. b)Diagnostic diagram,
c)Broad-band colors,d)Relation EW(H$\beta$) {\sl vs}
I(3729)/I(5010) ($\epsilon = 0.3$ model).}
\label{fig1}
\end{figure*}

Most of observed EW(H$\beta$) values for H{\sc ii} galaxies are lower
than 150\AA\, which implies the existence of an old non ionizing
population. We plot the evolution of EW(H$\beta$) {\sl vs} the
pseudo-color I(3729)/I(5010) for the successive bursts of our model
with $\epsilon=0.3$ in Fig~\ref{fig1}d.  The data trend is reproduced
by our model, not by any SSP. A metal-poor SSP (black solid line)
show bluer colours than observed. In order to decrease EW(H$\beta$)
and obtain redder colours, a more metal-rich ($Z\sim Z_{\odot}$) SSP
might be selected, but such high abundance is not observed. According
to our model, a stellar population of at least 1.3Ga must contribute
to the continuum, making it redder and, simultaneously, decreasing
EW($\beta$).  In summary, our model reproduces abundances, colours,
and emission lines of H{\sc ii} galaxies.


\section{\label{}}

\input{babar}
\input{mytex}
\newcommand{\multicolumn}{\multicolumn}
\newcommand{B{\small A}B{\small AR}}{B{\small A}B{\small AR}}
\newcommand{Belle}{Belle}

\section{Introduction}
Hadrons were once regarded as a source of
complication in the measurement of the dynamics of weak decays of
heavy quark states.  The realization that their
interactions can be used to help in understanding the phases
involved is quite a recent development.  There are several examples.
Interference between hadrons produced in decays of the $D^{0}$
and $\overline{D^{0}}$ mesons to the self-conjugate final state $K^{0}_s\pi^+\pi^-$
\footnote{Charge conjugate states are implied unless specified otherwise
throughout this talk}
has been used to measure $D^{0}$-$\overline{D^{0}}$ mixing parameters.
\cite{Abe:2007rd,Asner:2005sz}
It has also allowed the measurement of the CKM phase $\gamma$
from the decays $B^+\toD^{0}K^+$ and  $B^+\to\overline{D^{0}}K^+$.
\cite{Giri:2003ty,Aubert:2005iz,Poluektov:2006ia,Aubert:2007ii}
Interference between $S$- and $P$-waves in the $K^+\pi^-$ systems
produced in $B^{0}\to J/\psiK^+\pi^-$ decays has been used to resolve
the ambiguity in the sign of $\cos 2\beta$, where $\beta$ is the
CKM phase involved in these decays.

Central to all these analyses is the need to assign a partial
wave composition at each point in the hadron phase space.
Currently, models for the hadron interactions based on resonant
composition are mostly used to do this (the ``isobar model").
Sometimes a $K$-matrix description of the $S$-wave component
is included.  Uncertainties and ambiguities or questionable
assumptions, especially with respect to the $S$-wave behaviour
lead to significant systematic uncertainties in the results.
As the precision of the measurements improves, it becomes a more
pressing goal either to develop a good, model-independent
strategy for describing the hadron amplitudes or, at least, a
better understanding of how to describe the $S$-waves in the
decays of heavy quark mesons.

It is also of intrinsic interest to study the low mass
$S$-wave $K\pi$ and $\pi\pi$ systems to better understand the
scalar states, labelled here as $\kappa(800)$ or $\sigma(500)$,
that may exist
\cite{vanBeveren:2006ua}.
Measurements of these amplitudes in this region are sparse.
There is also merit, therefore, in pursuing the possibility of
using heavy quark meson decays to add experimental data in
these regions.  This talk focusses on recent progress and
future prospects on both fronts.

In the first section, information
presently available on both $K\pi$ and $\pi\pi$ $S$-waves is
briefly reviewed.  The expected relationship between scattering
amplitudes and those measured in decays of heavy quark mesons is
then examined.  The remainder of the talk discusses ways in
which experimental observations, mostly of the available $K\pi$
data, are being studied.

\section{\boldmath $S$-wave Scattering Data}
Excellent experimental information comes from model-independent
analyses of differential cross sections for reactions in which
production of $K\pi$ or $\pi\pi$ systems is dominated by a pion
exchange mechanism.  The amplitudes so determined show clear
evidence for the $K^*_0(1430)$ resonance in the iso-spin $I=1/2$
$K\pi$  $S$-wave and for the $f_0(980)$ in the $I=0$ $\pi\pi$
$S$-wave.  No exotic states are found in the $I=3/2$ $K\pi$ or
$I=2$ $\pi\pi$ waves.  Data at the very low mass regions, where
$\kappa(800)$ or $\sigma(500)$ poles may lurk far from the real
axis, are relatively poor.

\subsection{\boldmath The $K\pi$ System}

The SLAC E135 (LASS) experiment
\cite{Aston:1987ir,wmd:private}
provides the best information on the $K^-\pi^+$ system.  The data
are shown in Fig.~\ref{fig:LASS}.  Note that there are no data
below  825~MeV/c$^2$ from this experiment, though some is available
from an earlier era of experiments.  The LASS collaboration
determined that both $I=1/2$ and $I=3/2$ amplitudes $T(s)$ are
unitary up to $K\eta$' threshold ($\sqrt s=1454$~MeV/c$^2$)
\begin{eqnarray}
  T(s) = \sin\delta(s) e^{i\delta(s)}
  \label{eq:unitary}
\end{eqnarray}
where $s$ is the squared invariant mass and $\delta(s)$ is the
phase.

\begin{figure*}[hbt]
\centering
\epsfig{file=Figures/figure1ab.eps,
          width=0.66\textwidth,
          angle=0}
\epsfig{file=Figures/figure1cd.eps,
          width=0.33\textwidth,
          angle=0}
\caption{$S$-wave $K^-\pi^+$ scattering amplitude extracted by the LASS
 collaboration from their data on the reaction $K^- p\toK^-\pi^+ n$
 at 11~GeV/c.  The phase is shown in (a) and the magnitude in (b).
 The $I$-spin 1/2 component is represented by points on a solid curve
 and $I$-spin 3/2 is shown with no curve.
 $I=3/2$ phases measured in data from Ref.~\cite{Estabrooks:1977xe}
 are plotted as a function of $K^{\pm}\pi^{\pm}$ invariant mass ofr
 (c) $K^+ p\toK^+\pi^+ n$ and (d) $K^- p\toK^-\pi^-\Delta^{++}$.
\label{fig:LASS}
}
\end{figure*}

The $I=3/2$ phase, assumed to be unitary, was measured by
Estabrooks, {\sl etal}~\
\cite{Estabrooks:1977xe}
and is shown in Fig.~\ref{fig:LASS}(c) and (d).  The LASS
collaboration used this information to separate the $I=1/2$ and
$I=3/2$ components of the amplitude seen in
Fig.~\ref{fig:LASS}(a) and (b).  The fit to their data, shown
as the solid curves, are described by Eq.~(\ref{eq:unitary}) with
\begin{eqnarray}
  \begin{array}{rrcl}
    {\boldmath I=1/2:}
           & \delta^{1/2}(s) &=& \delta_R^{1/2}(s)+\delta_B^{1/2}(s) \\
           & \cot\delta_R^{1/2}(s) &=& (s_0-s)/(\sqrt s_0\Gamma_0) \\
           & q\cot\delta^{1/2}_B(s) &=& 1/a_{1/2} + b_{1/2}q^2 \\[8pt]
    {\boldmath I=3/2:}
           & q\cot\delta^{3/2}(s) &=& 1/a_{3/2} + b_{3/2}q^2 \\
  \end{array}
 \label{eq:lasswave}
\end{eqnarray}
where $q$ is the momentum of the $K^-$ in the $K\pi$ center of mass.
This parametrization is valid in the range $825<\sqrt s<1454$~MeV/c$^2$,
and includes one $I=1/2$ resonance of mass
$\sqrt s_0\sim 1435$~MeV/c$^2$ and width $\Gamma_0\sim 275$~MeV/c$^2$.
Non-resonant backgrounds in both waves are described by scattering
lengths $a$ and effective ranges $b$.

\subsection{\boldmath The $\pi\pi$ System}
Phases for the $I=0$ component, shown in Fig.~\ref{fig:Grayer}(a), have
been extracted in several analyses of $\pi^- p\to\pi^-\pi^+ n$ data from
Grayer, {\sl etal}~
\cite{Grayer:1974cr}
It is clear that data below 600~MeV/c$^2$ in the region of the
$\sigma(500)$ from these data are poor.

\begin{figure}[hbt]
\centering
\epsfig{file=Figures/figure2a.eps,
          width=0.48\textwidth,
          angle=0}
\\
\epsfig{file=Figures/figure2b.eps,
          width=0.48\textwidth,
          height=0.25\textwidth,
          angle=0}
\caption{(a) $I=0$ phase of the $S$-wave $\pi\pi$ scattering amplitude
 extracted from data for the reactions $\pi^- p\to\pi^-\pi^+ n$ from
 Ref.~\cite{Grayer:1974cr}.  The phase in degrees is plotted as a
 function of
 $\pi\pi$ invariant mass $M$.  (b) The $I=2$ phase shifts plotted as
 a function of $\pi\pi$ invariant mass $M$.  The $I=2$ amplitude is
 assumed to be unitary up to 1500 MeV/c and is obtained from data for
 the reactions $\pi^+ p\to\pi^+\pip n$ (12.5 GeV/c)
 \cite{Hoogland:1974cv}
 and $\pi^+ d\to\pi^-\pim pp_{\hbox{spectator}}$ (9 GeV/c)
 \cite{Durusoy:1973aj}.
} \label{fig:Grayer}
\end{figure}

The $I=2$ amplitude, assumed to be unitary up to
$\rho\rho$ threshold ($\sim 1500$~MeV/c$^2$), was derived from data
on $\pi^+ p\to\pi^+\pip n$ interactions at 12.5~GeV/c
\cite{Hoogland:1974cv}
and $\pi^+ d\to\pi^-\pim pp_{\hbox{spectator}}$ at 9 GeV/c
\cite{Durusoy:1973aj}.
It was fit to the form shown in Fig.~\ref{fig:Grayer}(b)
\cite{Achasov:2003xn}.
For $I=0$, the amplitude is unitary up to $K\bar K$ threshold where its
elasticity drops suddenly.  Slightly below this, the phase rises rapidly
indicating the presence of the $f_0(980)$ resonance.

\section{\boldmath Role of Scattering in $D$ Decays}
In terms of our two goals, it would be useful if we could draw on
these measurements to help in reducing ambiguities in models used
in analyzing decays of heavy quark mesons.  At the same time, we
need to learn how to interpret such decays in learning more about
the scattering amplitudes at small $S$ values.

Consider the decay $D\to (AB)C$ in which a two hadron system $f=AB$
and another system $C$ are produced.  A simple assumption about the
decay amplitude $F(s)$ for such a process is that it can be
factorized
into short and
long-range effects.
\begin{equation}
  F_f(s) = T_{fk}(s)Q_k(s)
  \label{eq:watson}
\end{equation}
Here $Q_k(s)$ describes the short-range decay of $D$ to $C$ and an
intermediate hadron system $k$, and $T_{kf}(s)$ describes the
subsequent re-scattering within the system $k$ to produce the final
state $f$.  We take $Q_k(s)$ to encode not only the relatively real
ratio of decay modes for the weak decay of the $D$ that would imply
an $s$-dependence of its magnitude, but also any other short-range
effects that may occur in the $C k$ system which might also
impart an $s$-dependence of its phase.

If $s$ is below the threshold where $AB$ scattering becomes
inelastic ($K\eta'$ for $L=$ even waves when $AB=K\pi$ and
$K^+K^-$ when $AB=\pi\pi$ systems, for example), then $T_{kf}(s)$ is
simply equal to the $AB$ elastic scattering amplitude.  If the phase
of $Q_k(s)$ is independent of $s$, then the phase of $F(s)$
will have the same $s$-dependence as $T(s)$, {\sl i.e.~}\ that observed in
elastic scattering.  This is a statement of the (more rigorously
derived) Watson theorem
\cite{Watson:1952ji}.
If partial wave expansions of $F(s)$ and $T(s)$ are made, then
this condition must hold for each partial wave.  It must hold,
in particular, for the $S$-waves.

The Watson theorem could provide a very useful constraint, therefore,
in the analysis of heavy quark states, so its range of applicability
is of great importance.  The considerations above lead us to expect
the following:
\begin{itemize}
 \item
 If $C$ is a di-lepton ($\ell\nu$, for example) then $Ck$ scattering
 would be unlikely and the phase of $Q_k$ would have little, if any,
 $s$-dependence.
 \item
 The same might also be true if $C$ were a massive hadron (such as a
 $J/\psi$) with a small interaction radius.
 \item
 In cases where $C$ is a hadron with mass comparable to $A$ or $B$,
 significant scattering in the $C k$ system Would be likely.  This
 could lead to a dependence of both the magnitude and the phase
 of $Q_k(s)$ upon $s$.
 \item
 The likelihood of $Ck$ scattering probably increases at shorter
 range.
\end{itemize}

We should expect, therefore, to find any deviations from the Watson
theorem to be most pronounced when the mass of $C$ is comparable with
that of $A$ or $B$, and at small $s$ in the $AB$ system.  We also note
that, the Watson theorem applies to every partial wave, and it makes
little sense to impose it only on the $S$-wave alone.

\section{\boldmath $K\pi$ Production from Heavy Quark Meson Decays}
\subsection{Decays with Leptons or Massive Hadrons}
In a study of semi-leptonic decays of $D$ mesons
\begin{eqnarray}
  D^{0} &\to&K^-\pi^+\ell\nu.
  \label{eq:kpilnu}
\end{eqnarray}
the FOCUS collaboration
\cite{Link:2005ge}
observed asymmetry in the distribution of $\cos\theta$, where
$\theta$ is the angle between $K^-$ and the $\ell\nu$ system in
the $K^-\pi^+$ rest frame.  The asymmetry results from
interference between $S$- and $P$-waves, and is proportional to
the cosine of the relative phase $\gamma$ between them.  As seen
in Fig.~\ref{fig:Focuslnu} this follows the behaviour observed in
the LASS data quite closely as the $K^-\pi^+$ invariant mass moves
through the $K^*(890)$ region.  This conforms to the first of our
expectations.

\begin{figure}[hbt]
\centering
\epsfig{file=Figures/figure4.eps,
          width=0.40\textwidth,
          angle=0}
\caption{Asymmetry in cosine of $K^-\pi^+$ helicity angle in
 $D^{0}\toK^-\pi^+\ell\nu$ data from the FOCUS collaboration
 \cite{Link:2005ge}
 as a function of invariant mass for the $K^-\pi^+$ system.
 The behaviour observed in the LASS data is indicated by the
 solid curve.
} \label{fig:Focuslnu}
\end{figure}

The B{\small A}B{\small AR}\ collaboration observed
\cite{Aubert:2004cp}
similar behaviour in the asymmetry of the helicity distribution
for $K^+\pi^-$ produced in
\begin{eqnarray}
  B^{0} &\to& J/\psiK^+\pi^-
  \label{eq:Jpsikpi}
\end{eqnarray}
decays.  Fig.~\ref{fig:BabarJpsikpi}
shows the two solutions for the $S$-$P$ phase difference $\gamma$
in the $K^*(890)$ invariant mass region.  The relative phase is
dominated by the rapid variation in the $P$-wave due to the
$K^*(890)$ and only one solution is physically meaningful.  It is
seen in the figure that this matches the LASS phase variation
(though shifted by $+\pi$ radians) very well.

\begin{figure}[hbt]
\centering
\epsfig{file=Figures/figure5.eps,
          width=0.45\textwidth,
          height=0.35\textwidth,
          angle=0}
\caption{Two solutions (open and closed circles with error bars)
 for the relative phase $\gamma=\delta_S-\delta_P$ between $S$-
 and $P$-wave amplitudes of $K^+\pi^-$ systems produced in
 $B^{0}\to J/\psiK^+\pi^+$ decays from the B{\small A}B{\small AR}\ collaboration
 \cite{Aubert:2004cp}.
 Values for $\gamma$ are plotted as a function of $K^+\pi^-$
 invariant mass.
 The solid circles are for the only physical solution and follow
 the LASS data points, plotted as triangles, closely.
} \label{fig:BabarJpsikpi}
\end{figure}

This conforms to the second of our expectations, but it is not
obvious why there is an overall phase shift of $+\pi$ radians.

\subsection{\boldmath Decays to Light Hadrons}
\label{sec:qphase}
The most detailed experimental information comes from studies of
$D^+\toK^-\pi^+\pip$ decays.  These Cabibbo favoured decays are known
to contain a large $S$-wave component.

A study of $\sim 15,000$ such decays by the E791 collaboration
\cite{Aitala:2002kr}
provides an illustration.  The E791 Dalitz plot is shown in
Fig.~\ref{fig:E791dp} where significant $S$-$P$ interference is
evident from the asymmetry of the $K^*(890)$ bands.  This is plotted
in Fig.~\ref{fig:E791dp} vs. the Breit-Wigner phase
$\phi_{BW}=\tan^{-1}M_0\Gamma(M_0)/M_0^2-s)$, where $M_0$ is the mass
and $\Gamma(\sqrt s)$ the invariant mass-dependent width of the
$K^*(890)$.  The asymmetry is zero when $\phi_{\scriptscriptstyle BW}=56^{\circ}$,
almost $80^{\circ}$ below the phase found in the LASS data
\cite{Aston:1987ir}.
This shows that the phase of $Q_{K\pi}(s)$ at this invariant mass
is $\sim -80^{\circ}$ and is entered in Table~\ref{tab:wmd}.

\begin{figure}[hbt]
\centering
\epsfig{file=Figures/figure6.eps,
          width=0.45\textwidth,
          angle=0}
\caption{Dalitz plot for $D^+\toK^-\pi^+\pip$ decays.  Data are from E791
 \cite{Aitala:2002kr} and show distributions of squared invariant mass for
 one $K^-\pi^+$ combination plotted against the other (symmetrized).  The
 inset shows the asymmetry in the cosine of the $K^-\pi^+$ helicity angle
 (between $K^-$ and the other $\pi^+$ in the $K^-\pi^+$ rest frame) as a
 function of $\phi_{\scriptscriptstyle BW}$, the $K^*(890)$ Breit-Wigner phase.
} \label{fig:E791dp}
\end{figure}

\subsubsection{Isobar Model Fits}
In the earliest analyses of these decays, a model with interfering
resonances like the $K^*(890)$, the $L=0$ $K^*_0(1430)$ and a constant
non-resonant ``$NR$" 3-body amplitude could account for the voids and
asymmtries observed in the Dalitz plot.  With their larger sample,
the E791 isobar model analysis showed that additional
structure in the $S$-wave was required to achieve an acceptable fit,
and the addition of a $\kappa(800)$ Breit-Wigner isobar, which
interfered destructively with the $NR$ term, worked well.

The ``isobar model" description of the $L=0, 1$ and $2$ wave
amplitudes $F_L$ in the $K^-\pi^+$ systems for this fit can be
summarized as:
\begin{eqnarray}
  F_0(s) &=& c_{00}
             + \alpha_{10}BW_{K^*_0(1430)}(s) \nonumber \\
         & & + \alpha_{20}BW_{\kappa(800)}(s)
  \label{eq:s_isobar}
  \\
  F_1(s) &=& \alpha_{11}BW_{K^*_1(890)}(s) \nonumber \\
         & & + \alpha_{21}BW_{K^*_0(1688)}(s)
  \label{eq:p_isobar}
  \\

  F_2(s) &=& \alpha_{21}BW_{K^*_2(140)}
  \label{eq:d_isobar}
\end{eqnarray}
where the $BW(s)$ are relativistic Breit-Wigner functions with
$s$-dependent widths, and the $\alpha_{iL}$ are complex coefficients
determined in the fit.  The overall phase was defined by setting
$\alpha_{11}=1.0$, and $\alpha_{00}$ was the $NR$ term.

Two further isobar model analyses of this decay mode were recently
made, one by FOCUS
\cite{Pennington:2007se}
and the other by CLEOc
\cite{Bonvicini:2007nn}.
Each used samples $\sim 3.5$ times larger.  The conclusions, and
estimates of the resonant fractions
\footnote{The fraction for each resonance or $NR$ sub process is
defined in ref.~\cite{Aitala:2002kr}.}, of both were
in good general agreement with E791.

Parameters for $\kappa$ and $K^*_0(1430)$ $S$-wave Breit Wigner isobars
are compared for the three experiments in Table~\ref{tab:kappapars}.
The $K^*_0(1430)$ parameters in this model disagree significantly with
those obtained by the LASS experiment or with the World average
\cite{pdg06}.
There is general agreement that this description of the $S$-wave,
described by Eq.~(\ref{eq:s_isobar}), with two broad, Breit Wigner
resonances, one of which is also near threshold, is theoretically
problematic and could account for this discrepancy.  It would be
virtually certain that this amplitude would have an $s$-dependent
phase that would differ from the Watson theorem expectation.
\begin{table}[hbt]
\begin{center}
\caption{Breit Wigner Parameters for the $K^-\pi^+$ $S$-wave Isobar
 States.  All quantities are in MeV/c$^2$.}
 \[
 \begin{array}{|l|l|r||r|r|r|}
 \hline

   \multicolumn{2}{|c|}{ }
   & \hss\hbox{\bf E791}\hss
   & \hss\hbox{\bf Focus}\hss
   & \hss\hbox{\bf CLEO c}\hss
   & \hss\hbox{\bf PDG}\hss \\[3pt]
 \hline &&&&&\\[-10pt]
   \kappa          & M_{\circ}
                   &  797\pm 19\pm 42
                   &  883\pm 13
                   &  805\pm 11
                   &  672\pm 40 \\
                   & \Gamma_{\circ}
                   &  410\pm  43\pm  85
                   &  355\pm  13
                   &  453\pm 21
                   &  550\pm 34 \\[6pt]
 \hline &&&&&\\[-8pt]
   K^*_{\circ}     & M_{\circ}
                   & 1459\pm  7 \pm 12
                   & 1461\pm  4
                   & 1461\pm  3
                   & 1414\pm ~6 \\
                   & \Gamma_{\circ}
                   &  175\pm 12\pm 12
                   &  177\pm ~8
                   &  169\pm  5
                   &  290\pm 21 \\
 \hline
 \end{array}
 \]
\end{center}
\label{tab:kappapars}
\end{table}

\subsubsection{\boldmath Model-Independent Measurement}
A test of the Watson theorem requires a measurement of the
phase of $F(s)$ at several values of $s$ in a
model-independent way.  The first attempt to do this for the
$S$-wave for $K^-\pi^+$ produced in this decay mode was made
by the E791 collaboration
\cite{Aitala:2005yh}.

They replaced the analytical function describing the $S$-wave
in Eq.~(\ref{eq:s_isobar}) by a set of 38 complex values at discrete
values for $s$, using a spline interpolation for other values.
The $P$- and $D$-waves were paramerized, as before, by the form
in Eqs.~(\ref{eq:p_isobar}) and (\ref{eq:d_isobar}) and the
coefficients $\alpha_{iL}$ were allowed to float.  A fit was
then made to determine the best values (magnitude and phase)
for each of the 38 $S$-wave points.  The result, shown in
Fig.~\ref{fig:E791mipwa}, is compared with the LASS model for the
$I=1/2$ $K^-\pi^+$ system in Eq.~(\ref{eq:lasswave}).  Agreement is
good in the $K^*_0(1430)$ region (above $\sim$1100~MeV/c$^2$)
after a shift in phase of $75^{\circ}$ and an arbitrary scale
factor are applied to the LASS amplitude.  A very significant
discrepancy is, however, seen for lower values of invariant
mass.

\begin{figure}[hbt]
\centering
\epsfig{file=Figures/figure7.eps,
          width=0.49\textwidth,
          angle=0}
\caption{(a) Phase and (b) magnitude of the decay amplitude $F_0(s)$
 at 38 discrete values of $s$ (squared invariant mass of the
 $K^-\pi^+$ systems from $D^{0}\toK^-\pi^+\pip$ decays taken from
 Ref.~
 \cite{Aitala:2005yh}.
 The LASS model in Eq.~(\ref{eq:lasswave}), shifted by $-75^{\circ}$
 and scaled to the region where $s>1.2$~(GeV/c$^2$)$^2$ is shown as
 the blue, continuous curve.
} \label{fig:E791mipwa}
\end{figure}

This might conform to our 4th expectation, but there are two
problems.  First, though $I=1/2$ $K^-\pi^+$ production probably
dominates, $I=3/2$ production in these decays cannot be excluded.
Second, the isobar model form in Eqs.~(\ref{eq:p_isobar}) and
(\ref{eq:d_isobar}) for the $P$- and $D$-waves, upon which this
result depends, is questionable. The $P$-wave contains more than
one Breit-Wigner, and both waves are assumed to be dominated by
resonant behaviour.  More importantly, neither wave is likely to
follow the Watson theorem, so a test of the $S$-wave alone cannot
be conclusive.

The CLEO collaboration
\cite{Bonvicini:2007nn}
has attempted to overcome the latter difficulty.  Using their
high purity sample of $\sim 60,000$ events, they proceeded in the
same way as E791, interpolating the $S$-wave between discrete
values of $s$, while parametrizing the $P$- and $D$-waves
as above.  Their results were very similar.  They then fixed the
$S$-wave and, using a similar procedure, fit the $P-$wave parametrized
in the same way.  Then they repeated this for the $D$-wave.  In
each step, only one wave was allowed to float with the others fixed.

This procedure can converge only by simultaneously floating all
waves at once, and this was not done.  Also, the phases have to be
defined in some part of the phase space (as little as possible) in
order for this to work.

It seems that a truly model-independent measurement of the $S$-wave
decay amplitudes in these hadron decays is desirable, but has yet
to be made, and will probably need to await the much larger data
samples of the $B$ factories, BES or the Panda experiment.

\subsubsection{\boldmath $I$-spin Analysis by the FOCUS Collaboration}
Using a sample of about $50,000$ events with 96\% purity,
the FOCUS collaboration used the $K$-matrix method to separate
$I=1/2$ and $I=3/2$ production for this decay
\cite{Pennington:2007se}.

The $P$- and $D$-waves were described as in
Eqs.~(\ref{eq:p_isobar}) and (\ref{eq:d_isobar}), however, the $S$-wave
amplitude $F_0(s)$ in Eq.~(\ref{eq:s_isobar}) was replaced by the sum
of two terms, one for each $I$-spin.  Each was described by
\begin{eqnarray}
  F_f(s)    &=& (I-i\rho K(s))^{-1}_{fk}P_k(s) \\
  \label{eq:FK}
  T_{kf}(s) &=& (I-i\rho K(s))^{-1}_{ki}K_{if}(s)   \\
  \label{eq:TK}
  Q_f(s)    &=& K^{-1}_{fk}(s)P_k(s).

\end{eqnarray}
Here, the amplitudes $F$, $T$, $P$ and $Q$ were introduced for
each $I$-spin and their indices $K$ and $f$ labelled intermediate
and final states ($1=K\pi$, 2=$\eta'K$).
The production vector $P_k(s)$ describing the couplings of the $D$
decay to the two intermediate states was described by parameters
obtained from the fit, and $\rho_{kf}$ was the phase space matrix
for the two channels.

Real values for all elements of the $K$ matrices ($2\times 2$ for
$I=1/2$ and $1\times 1$ for $I=3/2$) were determined by a fit to
the LASS measurements of $T$.

The fit required a very large contribution from the $I=3/2$
$S$-wave ($40.50$\%) interfering destructively with $I=1/2$
($207.25$\%) giving a total $S$-wave ($83.23$\%).

The phases of the resulting decay amplitudes, $F(s)$, are shown in
Fig.~\ref{fig:FOCUSkpipi}, with the $K$-matrix fit to the LASS
elastic scattering data for $I=1/2$ superimposed.  The phase of
the total amplitude $F(s)$, similar to that found by E791 and
by CLEOc, differs significantly from the LASS $I=1/2$ data.
The $I=1/2$ component is shown alone in
Fig.~\ref{fig:FOCUSkpipi}(b) and shows better agreement.

\begin{figure*}[hbt]
\centering
\epsfig{file=Figures/figure8.eps,
          width=0.75\textwidth,
          height=0.25\textwidth,
          angle=0}
\caption{Decay amplitudes for $K^-\pi^+$ systems from $D^+\toK^-\pi^+\pip$
 decays from
 Ref.~\cite{Pennington:2007se}
 for a(a) the total $S$-wave and (b) the $I=1/2$ contribution.
 The black bands correspond to one standard deviation limits
 about the central fit in each of the amplitudes.
 Scattering amplitudes from a $K$-matrix fit to the LASS data
 \cite{Aston:1987ir}
 are shown as red continuous lines.
} \label{fig:FOCUSkpipi}
\end{figure*}

The parametrization of the production vector was chosen
specifically to allow an $s$-dependence in its phase, thereby
allowing a deviation from the Watson theorem.  Some deviations
are, indeed, evident in Fig.~\ref{fig:FOCUSkpipi}(b), as the
invariant mass approaches the $K^*_0(1430)$ region.  It is
possible that this is due to the $K^*_0(1430)$ pole and the
onset of effects from the $K\eta'$ channel.  However, another
interpretation may be possible.  The phase shown in this figure
is shifted by an arbitrary amount to achieve good agreement at
the lower invariant masses.  Were a different shift in phase
applied, agreement would, in fact, be good at the high invariant
mass end and poor at low mass, consistent with our expectation
number 4.

This analysis attempted to make a valid comparison between
scattering and decay.  It appears premature, however, to
conclude that the Watson theorem works for these hadronic decays
especially because the same is not required for the $P$- nor
$D$-waves in this analysis.

\section{\boldmath $\pi\pi$ Production from Heavy Quark Meson Decays}
Decays of $D^+\to\pi^-\pi^+\pip$ result in $S$-wave enriched
$\pi^+\pi^-$ systems.  Unfortunately, these decays are Cabibbo
suppressed.  Another decay rich in $S$-wave content is
$D^{0}\toK^{0}\pi^+\pi^-$.  This is partly Cabibbo favoured and partly
doubly suppressed, so is somewhat complex with many resonances
contributing.  Consequently, model-independent analyses of these
decays have yet to be attempted.

\subsection{\boldmath $D^+\to\pi^-\pi^+\pip$ Decays}
The largest sample yet studied comes from the CLEO experiment
\cite{Bonvicini:2007tc}.
It consists of only about $4,000$ events with a purity of
$\sim 95$\%.  It has been used to test a variety of models.

An isobar model fit to the Dalitz plot confirms, as does a
similar analysis by FOCUS
\cite{Link:2003gb},
the E791 collaboration conclusion
\cite{Aitala:2000xu}
that structure in the low mass $\pi^+\pi^-$ system is well
described by a scalar $\sigma(500)$ Breit Wigner destructively
interfering with a constant $NR$ amplitude.  The fit obtained
was of marginal quality.

CLEO tried several variations on the isobar model.  They included
an $I=2$ $\pi^+\pip$ $S$-wave contribution, slightly improving the
fit quality.  They also tried variations of $\pi\pi$ $S$-wave isobar
model.  In one, as suggested in
Ref.~\cite{Oller:2004xm},
they replaced the $\sigma$ Breit-Wigner by a simple pole
\begin{eqnarray}
  1/(m_0^2-s-im_0\Gamma)~\to~1/(s_0-s)
\end{eqnarray}
where $s_0=(0.47-0.22 i)$.
Other $S$-wave models used were an amplitude based on the linear
sigma model
\cite{Schechter:2005we}
and an amplitude derived by N.Achamov described in the CLEO paper
Ref.~\cite{Bonvicini:2007tc}.
All models provided fits to the data slightly more acceptable than
the isobar model, but no clear distinction was obvious.

\subsection{\boldmath $D^{0}\toK^{0}\pi^+\pi^-$ Decays}
Isobar model fits to large samples have been made by B{\small A}B{\small AR}\
\cite{Aubert:2005iz}
and Belle\
\cite{Poluektov:2006ia}
collaborations.  They each require two $\sigma$ states in the $\pi^+\pi$
$S$-wave, one similar to that found in isobar fits to the previous
channel by E791, FOCUS and CLEOc, and the other at a mass of
$\sim 1.0$~GeV/c$^2$.  Both collaborations have also used
$K$-matrix parametrizations of the $S$-wave $\pi^+\pi^-$ amplitude,
obtaining slightly better fits.  These fits involve no assumptions
about $\sigma$ states, and they do enforce the Watson theorem in
this system.  However, the  other waves are those defined by the
isobar model, with no such constraint.  None of the fits so far
published have acceptable quality.

Further progress in this system may come from the larger data samples
form the $B$ factories, or from the next generation of charm factories.

\subsection{Other Channels}
A number of $B$ and $D$ decays in which $K\pi$ systems are
produced have now been published.  A comparison between the
general characteristics of the $S$-wave amplitudes observed and
those of the LASS scattering data are summarized in
Table~\ref{tab:wmd}
\cite{wmd:private}.
The two examples above are included and
represent the best examples of the validity of the Watson theorem.
Hopefully, an underlying pattern for these characteristics will
eventually emerge, provided that such decays are studied with this
goal in mind.  Any pattern is certainly not yet evident.

\begin{table}[hbt]
\begin{center}
\caption{Qualitative comparison between $K\pi$ $S$-wave decay
 amplitude $F_0(s)$ and the LASS Model in a variety of cases studied.
 Three characteristics are compared.  The $S$-$P$ relative phase at
 the $K^*(890)$ mass.  The observed value minus that observed in
 LASS data, $\Delta\phi_{SP}$, is tabulated to the nearest
 15$^{\circ}$).  The general shape of $|F_0(s)|$ for $s$ values
 both below and above $s_0=1.0$~GeV/c$^2$ are also tabulated.}
\begin{tabular}{|ll|c|c|c|}
\hline  \multicolumn{2}{|c|}{\textbf Decay Process} &
               \boldmath\bf $\phi_{SP}^{\circ}$
                &
               \boldmath $|F_0|$
                &
               \boldmath $|F_0|$
\\
                &
                &
               \textbf{approx}
                &
               \boldmath $s\!<\!s_0$
                &
               \boldmath $s\!>\!s_0$
\\
\hline $B^0\to J/\psiK^+\pi^-$
 &\cite{Aubert:2004cp}~
 &
 $+180$ &
 Poorly &
 Similar
\\
 &&& defined & to LASS \\
\hline $B^+\to K^+\pi^-\pi^+$
 &\cite{Aubert:2005ce}
 &
 $0$ &
 Unknown &
 Similar \\
 &&&         & to LASS \\
\hline $B^+\to K^+\pi^-\rho$
 &\cite{Aubert:2006fs}
 &
 $+180$ &
 Unknown &
 Unknown \\
 &&&         &         \\
\hline $D^{0}\to\KmK^+\pi^{0}$
 &\cite{Mishra:2007dc}
 &
 $-90$ &
 Similar &
 Similar \\
 &&& to LASS & to LASS \\
\hline $D^+\toK^-\pi^+\pip$
 &\cite{Aitala:2005yh}
 &
 $-75$ &
 Very
            &
 Similar \\
 &&& different & to LASS \\
\hline $D_s^+\to\KmK^+\pi^+$
 &
 & $-90$ &
 Similar &
 Similar \\
 &&& to LASS & to LASS \\
\hline $D^+\toK^-\pi^+\ell\nu$
 &\cite{Link:2005ge}
 &
 $0$ &
 Similar &
 Similar \\
 &&& to LASS & to LASS \\
\hline
\end{tabular}
\label{tab:wmd}
\end{center}
\end{table}

\section{Conclusions}
The most reliable data on $S$-wave amplitudes are still those
from LASS or CERN-Munich data on elastic scattering.  Data at
the lowest energies, however, are somewhat sparse.  It would
surely help in the understanding of the pole structure relevant
to the firm establishment of existence or otherwise of
light scalar $\kappa(800)$ or $\sigma(500)$ states to improve
on this situation.  It is difficult, at present, to see how
hadronic decays of $D$ or $B$ mesons will help since $I$-spin
considerations, and uncertainties in the $D$ form-factor, make
such measurements difficult.  It appears, therefore, that such
information is most likely to come from high statistics studies of
$D$ semi-leptonic decays, or decays of $B$ mesons to $J/\psiK^+\pi^-$
(or $J/\psi\pi^+\pi^-$) from B{\small A}B{\small AR}\ or Belle\ or the new facilities
in BES or Darmstadt.

Progress in understanding ways to parametrize the $S$-wave decay
amplitudes in Dalitz plot analyses is slow, but the body of
information is growing.  The prize could be less systematic uncertainty
resulting from model dependence of such analyses in measurements of
important phenomena such as $D^{0}$-$\overline{D^{0}}$ mixing or of the CKM
angle $\gamma$.

A positive trend is that more techniques beyond the isobar model,
with its known problems and strong model-dependence, are being
developed.

\begin{acknowledgments}
The author would like to thank the organizers for arranging this
workshop, and for the invitation to speak.  The support of the
National Science Foundation is also gratefully acknowledged.
\end{acknowledgments}

\bigskip


\section{Introduction}

Current models of reionization, when compared with
QSO absorption line measurements and CMB polarization experiments, seem
to indicate that reionization is a complex process extending
over $6 < z < 15$. However, the sources which were primarily responsible
for the process still remain uncertain. Even if one makes
the (not-so-drastic) assumption that reionization is primarily
driven by UV photons from stellar sources, the exact nature of the
stars and the mass range of the hosting galaxies are still open
questions.

For example, semi-analytical models of
\citeN{cf06b}, which are consistent with a wide variety of
observational data sets,
predict that reionization is mostly driven by haloes of
mass $< 10^9 M_{\odot}$ harboring metal-free
stars at $z \approx 10$ \cite{cf07}. Radiative transfer simulations
of \citeN{imsp07} conclude that the constraints on the electron
scattering optical depth $\tau_{\rm el}$ \cite{sbd++07}
are satisfied by simply including
haloes above $10^8 M_{\odot}$; no exotic sources or minihaloes are required.
Using a comprehensive model for galaxy formation, \citeN{mlgzd07}
conclude that  the
IGM can be completely reionized at $z \approx 6-7$ by massive stars
within protogalactic spheroids with halo
masses $\sim 10^{10}-10^{11} M_{\odot}$
without resorting to any special stellar IMF; such models are also
found to be consistent with the bounds on $\tau_{\rm el}$.
On the other hand, using the observational constraints
on the Ly$\alpha$ optical depth at $z=6$, \citeN{bh07} conclude
that the reionization process is ``photon-starved'' and
considerable photon contribution at $z > 6$ is required to complete
reionization by $z=6$.
Numerical simulations of \citeN{gnedin07} predict negligibly small
escape of photons from haloes with $M < 10^{11} M_{\odot}$, and
hence it is quite difficult to produce enough photons so as
to reionize the IGM by $z=6$.
On the observational front, using the
observed value of the assembled mass at $z \simeq 5$ and
currently available
(but highly uncertain) rate of decline in the star formation history over
$5<z<10$, it can be concluded that a considerable fraction of star-formation
is not yet observed at high redshifts \cite{sbeel07}. This could be either due to significant dust extinction at early times or because of an abundant population of low-luminosity sources just beyond the detection limits of current surveys,
thus implying a reionization scenario by small galaxies.

Given such wide variety of conclusions in the literature, it is important to
examine in detail the kind of halo masses required to match the available
observational data. In particular, it would be interesting to check whether
models with only large galaxies (say, haloes with masses $> 10^9 M_{\odot}$)
with standard stellar spectra and IMF are able to match the data, or is there
a desperate need for minihaloes ($M \sim 10^6 M_{\odot}$) and/or metal
free (PopIII) stars or any other exotic source. To address this question,
we use the semi-analytical formalism of \citeN{cf05} and \citeN{cf06b}
(hereafter CF05 and CF06 respectively)
and consider a series of physically-motivated scenarios which differ
in the minimum mass of star-forming haloes. The main idea of this work
is to confront each of these scenarios with the QSO absorption line
data at $z \approx 6$ and the constraints on $\tau_{\rm el}$ and
determine if some of the scenarios can be conclusively ruled out.
Throughout the paper, we use the best-fit cosmological parameters from the 3-year WMAP data \cite{sbd++07}, i.e.,
a flat universe with $\Omega_m = 0.24$, $\Omega_{\Lambda} = 0.76$, and $\Omega_b h^2 = 0.022$, and $h=0.73$.  The
parameters defining the linear dark matter power spectrum are $\sigma_8=0.74$, $n_s=0.95$, ${\rm d} n_s/{\rm d} \ln k =0$.

\section{Basic features of the model}
The main features of the semi-analytical model used in this work
could be summarized along the following (for a more detailed description
see CF05 and CF06): The model accounts for IGM inhomogeneities by adopting a lognormal distribution with the evolution of volume filling factor of
ionized hydrogen (HII) regions $Q_{\rm HII}(z)$ being calculated
according to the method outlined in \citeN{mhr00};
reionization is said to be complete once all the low-density regions (say, with overdensities $\Delta < \Delta_{\rm crit} \sim 60$) are ionized.
Hence, the distribution of high density regions determines the
mean free path of photons
\begin{equation}
\lambda_{\rm mfp}(z) = \frac{\lambda_0}{[1 - F_V(z)]^{2/3}}
\label{eq:lambda_0}
\end{equation}
where $F_V$ is the volume fraction of ionized regions and
$\lambda_0$ is a normalization constant fixed
by comparing with low redshift observations
of Lyman-limit absorption systems \cite{smih94}.

The number of ionizing photons depends on the assumptions made regarding
the sources.
In this work, we have assumed two types of reionization sources:

{\it (i) Stellar sources:} We assume that the photon production rate
from stars within haloes is proportional to the formation
rate of haloes, which in turn is calculated using the
Press-Schechter formalism. All haloes
above a threshold mass $M_{\rm min}$ are allowed to form stars.
The stellar sources are assumed to have
metallicities $Z=0.2 Z_{\odot}$ and form with a Salpeter IMF in the mass
range $1 - 100 M_{\odot}$; the stellar emission spectra are obtained from
the population synthesis models of \citeN{bc03}.

Under the above assumptions, the characterization of the
stellar sources require only two free parameters as
far as reionization studies are concerned, namely, (i) the efficiency
parameter of stars $\epsilon \equiv \epsilon_* f_{\rm esc}$ where
$\epsilon_*$ is the fraction of baryonic mass within haloes converted
into stars and $f_{\rm esc}$ is
the escape fraction of ionizing photons from the host halo
and (ii) the minimum mass of haloes $M_{\rm min}$ which are able to form stars.
In this work, we assume $\epsilon$ to be independent of redshift and
halo mass, while different physically-motivated models for $M_{\rm min}$
are chosen and studied, as will
be discussed in the next section.

Note that the quantity $M_{\rm min}$ introduced above corresponds
to star-forming haloes {\it only within neutral regions}.
Reionization by UV sources is accompanied by
photo-heating of the gas, which results in a suppression of
star formation in low-mass
haloes within ionized regions, a process known as
radiative feedback. Hence, the minimum mass
of star-forming haloes within ionized regions $M_{\rm min}^{\rm RFB}$
could be substantially larger
than $M_{\rm min}$ introduced above.
We compute the value of $M_{\rm min}^{\rm RFB}$ self-consistently from the
evolution of the gas temperature in the ionized regions and is typically
$\sim 2-3 \times 10^8 M_{\odot}$ at $6 < z < 10$.

Note that we do not include any metal-free (i.e. PopIII) stars,
which is the main difference of this work compared to our previous works
(CF05, CF06).

{\it (ii) QSOs:}  In this work, we
compute the emissivity of QSOs using likelihood
estimations of the observed luminosity
function at $z < 6$ \cite{meiksin05}. The main uncertainty in the QSO
contribution comes from the slope of the faint end of the luminosity
function which is poorly constrained observationally \cite{sw07}.
In this work, we
include the contribution of only those QSOs whose luminosities are
above the break or characteristic luminosity; hence
the QSO contribution should be considered as a lower limit while the
actual emissivity could be a few times higher. Our estimates are similar
to or lower than that of \citeN{meiksin05} and \citeN{bh07}.

\nocite{fsb++06,ksm++06,tkk++06,gffc07,gsfc07}
\begin{figure*}
\rotatebox{270}{\resizebox{0.5\textwidth}{!}{\includegraphics{basicplots.ps}}}
\caption{Comparison of model predictions with observations for
different models described in the text and summarized in Table 1.
The different panels indicate:
(a): The volume-averaged neutral hydrogen
fraction $x_{\rm HI}$, with observational
limits from QSO absorption lines (Fan et al. 2006; diamond),
Ly$\alpha$ emitter luminosity function (Kashikawa et al. 2006; triangle) and
GRB spectrum analysis (Totani et al 2006; square). Also shown
are the constraints using dark gap statistics on QSO spectra
(Gallerani et al 2007a; open circles) and GRB spectra (Gallerani et al. 2007b;
filled circle).
(b): Electron scattering optical depth, with  observational constraint from
WMAP 3-year data release.
(c): Photoionization
rates for hydrogen, with estimates from numerical simulations (shown
by points with error-bars; Bolton et al. 2005, Bolton \& Haehnelt 2007).
The dotted line shows the lower limit of the QSO contribution.
(d): Evolution of Lyman-limit systems, with observed data points from Storrie-Lombardi et al. (1994).
(e): Emission rate of ionizing photons per comoving volume.
(f): The minimum mass of haloes which are allowed to form stars within
neutral regions. The dotted line denotes the corresponding minimum
mass within ionized regions obtained using the radiative
feedback prescription.
(g): Ly$\alpha$ effective optical
depth, with observed data points from Songaila (2004) and Fan et al. (2006).
(h): Ly$\beta$ effective optical
depth, with observed data points from Songaila (2004)  and Fan et al. (2006).
(i): Evolution of the photon mean free path in physical units.
}
\label{fig:basicplots}
\end{figure*}

The main observational data sets used in this work are those
of the transmitted fluxes $F_{\alpha}$ and $F_{\beta}$
in Ly$\alpha$ and Ly$\beta$ regions respectively, as obtained from
the QSO absorption spectra. We have taken the points tabulated
in \citeN{songaila04} and \citeN{fsb++06}. For calculating
$F_{\alpha}$, we have binned the data points
within redshift intervals of $\Delta z =0.2$ and calculated
the mean. The errors are calculated using
the extreme values of $F_{\alpha}$
along different lines of sight.
Hence the errors shown in this paper are typically
larger than other methods which compute the uncertainties using the
interquartile range \cite{bh07} or standard dispersion.
For calculating $F_{\beta}$, we note that the data points at $z < 5.5$ are
quite sparse \cite{songaila04} and hence do not require further binning;
we simply use the values and errors tabulated in \citeN{songaila04}. For
points at $z > 5.5$, we follow the method identical to the Ly$\alpha$ case.
The constraints on $\tau_{\rm el}$ are obtained from \citeN{sbd++07}, constraints
on $\Gamma_{\rm PI}^{\rm HII}$ from \citeN{bhvs05} and \citeN{bh07} and the
redshift distribution of Lyman-limit absorption
systems from \citeN{smih94}.

\vspace{-0.5cm}
\section{Minimum mass of star-forming haloes}
\begin{figure*}
\rotatebox{270}{\resizebox{0.3\textwidth}{!}{\includegraphics{minihalo_halofraction.ps}}}
\caption{Photon contribution for the MH model.
{\it Left}: Number of ionizing photons per H-atom (accounting for the number of recombinations) contributed by haloes within a given halo mass range as a function of
redshift $z$.  The dotted line represents the evolution of the volume filling factor $Q_{\rm HII}$ of ionized regions.
{\it Right}: Cumulative fraction of the ionizing power $f_{\gamma}$ contributed by haloes of mass $>M$. The curves from
right to left correspond to $z = 6,7,8,9,10$ respectively.
}
\label{fig:minihalo_halofrac}
\end{figure*}

\begin{table}
\caption
{Parameter values for different models used in the paper.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Model & $\epsilon=\epsilon_* f_{\rm esc}$\\
\hline
MH & 0.008\\
SH & 0.009\\
LH & 0.013\\
\hline
\end{tabular}
\label{table:modpar}
\end{center}
\end{table}

In this Section, we consider three
physically motivated models which differ in the choice of
the value of $M_{\rm min}$ and check whether they are able to match
all the data sets. The models are described in the following:

(i) {\it Minihalo (MH)}: The minimum mass of star-forming haloes for this model
is set
by a virial temperature of $T_{\rm vir}=300$K,
which corresponds to a scenario where molecular cooling is fully efficient.
Note that $M_{\rm min}$ is redshift-dependent and is typically
$\sim 5 \times 10^5 M_{\odot}$ at $z = 6$. We should mention that
the star-forming efficiencies of such haloes are debatable \cite{hb06} as H$_2$ could
be easily dissociated by a Lyman-Werner background photons.
However, it has also been argued that such a background only delay the
star formation in minihaloes and do not necessarily suppress them \cite{on07}.

(ii) {\it Small Halo (SH)}: The minimum mass of star-forming haloes is set
by a virial temperature of $T_{\rm vir}=10^4$K; this is motivated
by the fact that all haloes having $T_{\rm vir} \geq 10^4$K are able
to cool via atomic transitions. This model has
$M_{\rm min} \sim 10^8 M_{\odot}$ at $z = 6$ and is usually considered
as standard in most semi-analytical works. We must mention again that
for both the SH and MH models,
the value of $M_{\rm min}$ corresponds to the neutral regions only;
the minimum mass of star-forming haloes is much larger in ionized regions
because of radiative feedback.

(iii) {\it Large Halo (LH)}: The minimum mass of star-forming haloes is set
by a virial temperature of $T_{\rm vir}=5 \times 10^4$K which corresponds
to $M_{\rm min}(z=6) \sim 10^9 M_{\odot}$. Such value is appropriate to
a scenario in which reionization is driven by large galaxies which are
largely unaffected by radiative feedback.

For each model, we find the maximum value of the efficiency $\epsilon$
such that it does not violate the upper bound on
$F_{\alpha}$ and $F_{\beta}$  at $z = 6$ and then check how it compares with
other observations, in particular whether it can produce
$\tau_{\rm el} > 0.06$, the 1-$\sigma$ lower limit from
WMAP3 \cite{sbd++07}.
The motivation for normalizing all the models by QSO absorption
line data is that the measurements of $F_{\alpha}$ and $F_{\beta}$ are
less affected by systematics and other uncertainties compared to
other data sets considered here. In contrast, the
constraints on $\tau_{\rm el}$ obtained from CMB polarization
measurements are still preliminary and the possibility of
major revision in future experiments cannot be ruled out.
Note that, the lower bounds on $F_{\alpha}$ and $F_{\beta}$  at $z = 6$
are practically zero and hence the minimum value
of $\epsilon$ cannot be obtained using QSO absorption line
data. However, the
upper bounds should be considered as robust; in fact we have been
quite conservative in this work and used the extreme maximum
value of $F_{\alpha}$ and $F_{\beta}$ allowed by the data.
The values of $\epsilon$ for different
models after normalizing to the upper limits
of $F_{\alpha}$ and $F_{\beta}$  at $z = 6$
are summarized in Table \ref{table:modpar}.
The results are shown in Figure \ref{fig:basicplots}.

It is clear from the figure that once the models
are normalized to the upper bounds on $F_{\alpha}$ and $F_{\beta}$ at
$z=6$, the MH and SH models are able to match the evolution
of $F_{\alpha}$ and $F_{\beta}$ up to lower redshifts [Panels (g) and (h)].
However, the
LH model, which does not include low mass ($< 10^9 M_{\odot}$) haloes,
gives a poor match with the low redshift observations.
Similar conclusions can be drawn from the constraints on
$\Gamma_{\rm PI}^{\rm HII}$ where the MH and SH models
can fit the data till $z \approx 3$ while the LH model fails
to do so.
More importantly, when compared with the
observed $\tau_{\rm el}$ [Panel (b)], we note that
only the MH model can match the data, while
the other two models fall short of the lower 1-$\sigma$ limit.
The LH model predicts $\tau_{\rm el} \approx 0.052$, which can
be considered as a poor match to the data.
The  SH model predicts
$\tau_{\rm el} = 0.058$, marginally lower than
the 1-$\sigma$ limit; given the uncertainties in the modeling
of the reionization, this could be considered as marginally acceptable.
Such low value of $\tau_{\rm el}$ is a severe problem for the LH model
because the only way to increase the value of $\tau_{\rm el}$ would be
to increase $\epsilon$ (the only free parameter) which would then
underpredict the Ly$\alpha$ and Ly$\beta$ optical depths at $z=6$.

Hence, models which do not include stars from haloes with masses
$< 10^9 M_{\odot}$ cannot match the GP and
the electron scattering optical depths simultaneously. In fact, one should
at least include haloes of masses $\sim 5 \times 10^7 M_{\odot}$ to get
a marginal match with the data (the SH model).
The analysis also brings out the
importance of including the QSO absorption line measurements explicitly
into any reionization model. For example, a model with only
$M >  10^9 M_{\odot}$ haloes with a efficiency $\sim 0.06$ would
reionize the universe around $z \approx 8$ and produce
$\tau_{\rm el} \approx 0.07$; however it would severely overpredict
the $F_{\alpha}$ and $F_{\beta}$ (and $\Gamma_{\rm PI}^{\rm HII}$ too)
at $z=6$ and hence
would not be acceptable.

Let us examine which haloes contribute most significantly to reionization; we
shall limit ourselves to the MH model as the other two models
are shown to be unable to match observations.
The number of ionizing photons per H-atom contributed by haloes in the mass range $[M_{\rm min}, M_{\rm max}]$
is given by\footnote{Note that, in our previous work,
we had defined $x_{\gamma}(z)$ as $\frac{n_{\gamma}(z)}{n_H} \frac{t_{\rm rec}(z)}{t_H(z)}$, which blows up when recombinations are negligible ($t_{\rm rec} \to \infty$). Under the present definition, $x_{\gamma}(z) \to \frac{n_{\gamma}(z)}{n_H}$when $t_{\rm rec} \to \infty$, which is the correct limit.}
\begin{equation}
x_{\gamma}(z) \equiv \frac{n_{\gamma}(z)}{n_H [1 + t_H(z)/t_{\rm rec}(z)]}
\end{equation}
where $n_H$ is the comoving number density of hydrogen
atoms while $n_{\gamma}$
is the time-integrated comoving photon density, calculated using the relation
\begin{equation}
n_{\gamma}(z) = \int_0^{t(z)} {\rm d} t ~ \dot{n}_{\gamma}(M_{\rm min} : M_{\rm max}, t)
\end{equation}
where $\dot{n}_{\gamma}(M_{\rm min} : M_{\rm max}, t)$ is the ionizing photon comoving emissivity
from haloes within $[M_{\rm min}, M_{\rm max}]$.
The term $[1 + t_H(z)/t_{\rm rec}(z)]$ accounts for the number of
recombinations in the IGM
where $t_H(z)$ is the Hubble time  and
$t_{\rm rec}(z)$ is the recombination time.
By construction, the IGM is reionized when $x_{\gamma} \gtrsim 1$.
A second quantity of interest is  the fractional instantaneous contribution of
haloes above a certain mass,
\begin{equation}
f_{\gamma}(>M, z) \equiv \frac{\dot{n}_{\gamma}(>M, z)}{\dot{n}_{\gamma}(z)}.
\end{equation}

The plots of $x_{\gamma}$ and $f_{\gamma}(>M, z)$ for the MH model
is shown in Figure \ref{fig:minihalo_halofrac}.
It is clear from the figure that
haloes of mass $< 10^8 M_{\odot}$ dominate the ionizing background
at high redshifts, their contribution decreasing gradually at $z < 8$ because
of radiative feedback.
However, these haloes are still the dominant contributors of ionizing
photons when integrated till $z=6$ (though the instantaneous
photon production rate at $z=6$ is dominated by $>10^9 M_{\odot}$
haloes). Hence models which do not include $M < 10^8 M_{\odot}$ haloes
would miss out a large fraction photons at high redshifts (before
radiative feedback is effective) and hence would underpredict
$\tau_{\rm el}$.
For the SH model, we find that
$< 10^8 M_{\odot}$ haloes produce only about 10\% of ionizing photons
when integrated till $z=6$, while about 50\% of photons come
from high mass $>10^9 M_{\odot}$ haloes.

\vspace{-0.5cm}
\section{Discussion}
We have used a semi-analytical reionization model, empirically calibrated on
a variety of observational data sets, to estimate the minimum mass of ionizing photon
sources required to match the current data. We find that models which
do not include haloes with mass $M < 10^9 M_{\odot}$ are not able to reproduce
the GP and electron scattering optical depths {\it simultaneously}. Such models (i)
contribute too few photons at high redshift,  and (ii) {\it produce too many photons
too late}. To get a marginally acceptable match with the data, one requires
haloes with masses as small as $5 \times 10^7 M_{\odot}$ at $z \approx 10$,
which would correspond to a virial temperature of $\sim 10^4$K. In such cases,
though the bulk of photons ($\sim 90\%$) is produced by $M > 10^8 M_{\odot}$ haloes,
the low mass haloes are important to contribute to $\tau_{\rm el}$
at high redshifts without violating the QSO absorption line constraints
at $z=6$.

A much better match to the data is obtained if minihaloes ($M \sim 10^6 M_{\odot}$)
are included in the analysis.
These haloes produce enough photons at high redshifts to give a
high $\tau_{\rm el}$. They are also easily destroyed once radiative
feedback becomes substantial and hence give no contribution to the
photoionization rate at $z \approx 6$, thus agreeing with the
$F_{\alpha}$ and $F_{\beta}$ upper bounds.
In case the minihaloes are not allowed to form stars because
of some photodissociating Lyman-Werner background,
it becomes almost impossible to construct reionization models
with standard stellar sources that are not in tension with data.
Given this, it is crucial to critically examine the assumptions and idealizations
made in our formalism which could allow reionization scenarios with
only large galaxies to be consistent with the data, which is done in the
following:

(i) $z$-dependence of the photon production efficiency: in this work,
we have assumed the
efficiency parameter $\epsilon$, the stellar IMF and the stellar spectrum
to be independent of $z$. In case the value of $\epsilon$ was higher
at high redshifts, it could, in principle, produce high $\tau_{\rm el}$ at
high redshifts without violating the GP constraints at $z=6$. Such behavior
of $\epsilon$ would mean that either stars were forming more efficiently
at early times and/or the escape fraction of photons was higher. A similar
effect could also be achieved if the stellar IMF was top-heavy
at high-$z$ or the spectra of the stars were harder. In short, one would
require  a very efficient production of photons per baryons at high-$z$. An obvious
candidate for achieving such effects would be the inclusion of metal-free
(PopIII) stars with or without a top-heavy IMF. Such models with
PopIII stars are found to be an excellent match to a wide variety of data sets
in the SH scenario (CF06), while they can possibly be tuned to match the data
in the LH case too.

(ii) Mass-dependence of $\epsilon$: similarly neglected here, is the possibility that
the efficiency parameter depends on the halo mass.
Note that, in order to make the LH scenario work, one would require $\epsilon$ to be higher
for smaller mass haloes so that the photon contribution
increases at $z > 6$. However, the mass-dependence, if any, is found
to be opposite, e.g., \citeN{khw++03} found that $\epsilon$ increases
with halo mass for $M < 3 \times 10^{12} M_{\odot}$
in the local Universe.  Given this, it is unlikely that a
mass-dependent $\epsilon$ would improve the performance of large-galaxy-only models.

(iii) Radiative feedback: One of the main uncertainties in theoretical
models of reionization is the implementation of radiative (i.e. photoionization) feedback. However,
note that this effect mostly affects haloes of masses $<10^9 M_{\odot}$ [Panel (f) of Figure \ref{fig:basicplots}]
and hence a different feedback prescription would have {\it no} effect on the LH model at all. For the SH model,
a less severe feedback mechanism, which allows the $10^8 M_{\odot} < M < 10^9 M_{\odot}$ haloes to survive
longer than what is used here \cite{gnedin00}, could produce enough photons at high-$z$ to get a better match to the data.
On the other hand, if the feedback is more severe on the $\sim 10^8 M_{\odot}$ haloes
(e.g., because of the photoionization rate boost arising from the clustering of galaxies, and not taken
into account here), the SH model would be ruled out.

(iv) QSOs: The contribution of QSOs considered here should be thought
of as a lower limit; the actual contribution could be much higher. However,
this does not affect our conclusions because a higher contribution
from QSOs at $z \sim 6$ would imply a lower value of $\epsilon$, which would
then produce a much lower $\tau_{\rm el}$.

(v) IGM inhomogeneities: The density distribution of the IGM has been
assumed to be lognormal, which is found to be a good match
to the QSO transmitted flux distribution \cite{gcf06}. However, it has been
argued that the density distribution obtained from simulations
has a different form \cite{mhr00}.
The density distribution can affect the results in three ways,
namely: (a) The evolution of $Q_{\rm HII}(z)$ could be altered
if the density distribution is different; however note that
there is not much freedom observationally in the qualitative
behavior of $Q_{\rm HII}(z)$ as QSO absorption line data
requires reionization to be completed around $z \gtrsim 6$.
(b) The evolution of $\lambda_{\rm mfp}$ could be different thus
modifying the photoionization rate $\Gamma_{\rm PI}^{\rm HII}$
which is discussed in the next point.
(c) For a given $\Gamma_{\rm PI}^{\rm HII}$, a different
density distribution would give
a different the value of $F_{\alpha}$ (and $F_{\beta}$). However, note that
the analysis presented in the paper could also be done using the
constraints on $\Gamma_{\rm PI}^{\rm HII}$ without any reference to
$F_{\alpha}$ or $F_{\beta}$, and the results would still
be qualitatively similar.

(vi) Photon mean free path: A related problem is that regarding the
value of $\lambda_{\rm mfp}$ at $z=6$. There are no observational
constraints on $\lambda_{\rm mfp}$ at $z > 4$, and the theoretical
estimates would depend on the density distribution of the IGM. In case
$\lambda_{\rm mfp}$ is found to be lower than that obtained in our models
($\lambda_{\rm mfp}(z=6) = 3.73$ and 2.37 proper Mpc for SH and LH respectively),
it would give a lower $\Gamma_{\rm PI}^{\rm HII}$ for the same value
of $\epsilon$, and hence could allow the SH and LH models to match with
observations. However, the typical values of $\lambda_{\rm mfp}$ found
using the density distribution of \citeN{mhr00}
are $\sim 5$ physical Mpc \cite{bh07,wbh07}, which
would clearly rule out the SH and LH models. A trivial extrapolation
of the observed $\lambda_{\rm mfp}$ at lower redshifts to high-$z$
would too give similar values.

(vii) Revised observational constraints: A good chance of
the large galaxies scenario to survive (without including PopIII stars
or other sources) would be to revise the
constraints on $\tau_{\rm el}$. We have already seen that the value of
$\tau_{\rm el}$ was lower in the WMAP3 data release than in the WMAP1
because of systematics. In case the value of $\tau_{\rm el}$ is found
to be $\sim 0.05$, it would be enough to allow the LH scenario. On the
other hand, in case the upper bounds on $F_{\alpha}$ and $F_{\beta}$ are
tightened with increase in QSO sample size, it could rule out the
LH (and possibly SH) scenario with a higher degree of confidence.
For example, we have
been conservative in estimating the errors and allowed
a $F_{\alpha}$ as high as 0.0125 at $z=6$. One should compare this
with the constraints $F_{\alpha} < 0.004$ used by \citeN{bh07}; such
severe constraints would clearly disfavor the LH and SH scenarios.
Another possibility is that the constraints on
the cosmological parameters are revised, e.g., the value of
$\sigma_8$ is found to be higher than what is used. A rigorous
exploration of the cosmological parameter space is beyond the
scope of this work. However, a model with higher value of $\sigma_8=0.9$
\cite{vhl06},
when normalized to Ly$\alpha$ and Ly$\beta$ flux at $z=6$,
gives $\tau_{\rm el} \approx 0.059$; this value is still well below
the corresponding 1-$\sigma$ bound on $\tau_{\rm el} \approx 0.1\pm 0.03$.

In spite all the model uncertainties, it
seems certain that reionization with large galaxies scenario
($M > 10^9 M_{\odot}$) can be conclusively
ruled out with the present data; such scenarios can only be allowed if
metal-free stars or other exotic sources at high redshifts
are included.
The scenario where only those haloes which can cool
via atomic transitions contribute is marginally acceptable.
In any case, there seems to be a requirement
for a large number of sources at $z \approx 10$, which are most
likely faint (i.e., low-mass) haloes.
Observationally, it
is important to put constraints on star formation within
these faint galaxies at high redshift which, however, seems to be a
challenging task. Nonetheless one should be optimistic as
most of such issues would be addressed with future experiments like JWST.
On the theoretical front, it is important to realize that reionization
models could be incomplete unless they are compared with both the $\tau_{\rm el}$ and GP constraints simultaneously.

\vspace{-0.5cm}
\section*{Acknowledgement}
SG acknowledges the support by the Hungarian National Office for
Research and Technology (NKTH), through the Pol\'anyi Program.

\vspace{-0.5cm}