\section{#1}}

\def{}
\def0.99\textwidth{0.99\textwidth}

\begin{document}

\title{The Composition of Cosmic Rays at the Knee
\footnote{Invited talk, given at the Centenary Symposium 2012: Discovery of
Cosmic Rays, June 2012, Denver.}}

\classification{96.50.S-, 96.50.sb, 96.50.sd, 98.70.Sa}
\keywords      {cosmic rays, composition, energy spectrum, knee, air showers}

\author{J\"org R.\ H\"orandel}{
  address={Department of Astrophysics/IMAPP, Radboud University Nijmegen,\\
    P.O. Box 9010, 6500 GL Nijmegen, The Netherlands ~---~
    http://particle.astro.ru.nl}
}

\begin{abstract}
 Recent results on the composition of cosmic rays in the energy region from
 about $10^{14}$ to $10^{18}$~eV are reviewed.
\end{abstract}

\maketitle

\Section{Introduction}
The all-particle energy spectrum of cosmic rays (ionized atomic nuclei from
outer space) follows a broken power law $dN/dE\propto E^\gamma$. At an energy
around $4\cdot 10^{15}$~eV the spectral index changes from $\gamma\approx-2.7$
at low energies to $\gamma\approx-3.1$.  This structure in the energy spectrum
is referred to as "the knee". Understanding the physics behind this "kink" in
the energy spectrum is thought to be crucial to understand the origin of
Galactic cosmic rays \cite{behreview}.

The origin of cosmic rays is threefold: particles with energies below about
100~MeV are accelerated during solar bursts in magnetic reconnections at the
surface of the Sun. Nuclei with energies up to several $10^{17}$~eV are usually
considered to originate within our Milky Way, most likely being accelerated in
Supernova Remnants. At the highest energies ($>10^{18}$eV), particles are
usually considered to be of extragalactic origin.
In the following, we will focus on the composition and origin of Galactic
cosmic rays, i.e.\ cosmic rays with energies between $\approx1$ and
$10^{9}$~GeV.

At energies around 1~GeV individual elements have been identified in direct
measurements of cosmic rays above the atmosphere, all elements from the
periodic table of elements are present in cosmic rays, e.g.\ \cite{cospar06}.
Their abundance follows roughly the abundance in the local Galactic
environment/solar system.  Energy spectra for the major cosmic-ray elements
from hydrogen to iron have been directly measured up to energies approaching
$10^{14}$~eV (e.g.\ Fig. 26.1 in \rref{Beringer:1900zz}). They follow power
laws with a spectral index close to $\gamma\approx-2.7$. Below about 10~GeV the
spectra are influenced by solar modulation, i.e.\ deflection/attenuation in the
Heliosphere.

The cosmic-ray flux is strongly falling as a function of energy.  Thus, at
energies exceeding $10^{14}$~eV large areas are needed to collect a
statistically significant data sample in a reasonable time. The large areas
needed, exceeding the size of a soccer field, can only be realized on ground,
i.e.\ below the atmosphere of the Earth. Thus, secondary particles are measured
which are produced in extensive air showers, initiated by high-energy cosmic
rays impinging onto the atmosphere.

The intrinsic fluctuations during the development of a particle cascade limit
the mass resolution of experiments registering air showers. Typical
mass-sensitive observables in such experiments are the depth of the shower
maximum in the atmosphere $X_{max}$ and the ratio of electrons to muons at
ground level $N_e/N_\mu$.  Using a Heitler type model for hadronic showers
\cite{matthewsheitler} it can be shown that these observables depend on the
nuclear mass number of the primary particle $A$ as $X_{max}\propto -X_0 \ln A$
and $\lg(N_e/N_\mu)\propto -0.065\ln A$, with the radiation length in air $X_0$
\cite{ricap07,jrherice06}.  Thus, if one requires e.g.\ a resolution of
$\Delta\ln A\approx1$, the experimental accuracies have to be better than
$\Delta X_{max}\approx 36$~g/cm$^2$\xspace or $\Delta(N_e/N_\mu)\approx 16\%$. These
values are close to the experimental uncertainties of current air shower
experiments. In state-of-the-art air-shower experiments a resolution of the
order of $\Delta\ln A\approx0.8$ has been achieved, i.e.\ the major cosmic-ray
elements from hydrogen to iron can be divided in up to 5 elemental groups
(instead of 26 individual elements as registered in direct measurements above
the atmosphere).

\Section{Recent Results}
\begin{figure}[t]
 \includegraphics[width=0.5\textwidth]{argo.eps}
 \includegraphics[width=0.5\textwidth]{kg.eps}
  \caption{\textit{Left}\xspace: Energy spectrum of the "light" cosmic-ray component (p+He),
             as measured by ARGO-YBJ \cite{argoicrc32}.
	   \textit{Right}\xspace: Energy spectrum of cosmic rays as determined with
	     KASCADE-Grande: all-particle spectrum (black)
             and heavy component (red) \cite{Apel:2011mi}.}
  \label{argokg}
\end{figure}

\paragraph{ARGO-YBJ}
The ARGO-YBJ experiment is situated at the Yangbajing International Cosmic-Ray
Observatory, located at an altitude of about 4300 m a.s.l.\ in Tibet
\cite{argoicrc32}.  The detector consists of an air-shower array, made of a
full-coverage RPC carpet.  The energy spectrum of the light cosmic-ray
component (protons and helium) has been measured in the energy range from 1 to
300~TeV, as depicted in \fref{argokg} (\textit{left}\xspace).  Results of the CREAM balloon
experiment are in good agreement with the air-shower data from ARGO. The latter
are also compatible with a parameterization of the world data \cite{pg}, as can
be inferred from the figure. This agreement in the energy region of overlap
between direct and indirect cosmic-ray measurements is a valuable test of the
simulation codes used to interpret the air shower data.

\paragraph{KASCADE(-Grande)}
The KASCADE experiment, located in Germany measured simultaneously all three
air shower components \cite{kascadenim}.  The number of electrons and muons in
air showers has been used to unfold the energy spectra for five groups of
elements \cite{ulrichapp,Apel:2008cd}. The results indicate that the knee is
caused by the fall-off of the energy spectrum of protons, followed by
subsequent cutoffs of heavier elements, approximately proportional to the
rigidity ($E/Z$) of the individual elements.
This observation has been confirmed by independent measurements, using a muon
tracking detector to determine the average production depth of muons in the
atmosphere \cite{Apel:2011zz}.

The experiment has been extended to cover an area of about 0.5~km$^2$ ---
KASCADE-Grande \cite{Apel:2010zz}. The all-particle energy spectrum has been
measured in the energy range from $10^{16}$ to $10^{18}$~eV, derived from
measurements of the total number of charged particles and the total muon number
of muons \cite{Apel:2012rm}. The energy spectrum exhibits strong hints for a
hardening of the spectrum at approximately $2\cdot 10^{16}$~eV and a
significant steepening at $8\cdot10^{16}$~eV.
The data have also been used to divide the primary cosmic rays in a light and a
heavy group and to determine the respective energy spectra, as shown in
\fref{argokg} (\textit{right}\xspace) \cite{Apel:2011mi}.  One recognizes a fall-off of the
heavy component at an energy around $8\cdot10^{16}$~eV.  Such a behavior is
expected, assuming a rigidity dependent cut-off of the individual cosmic-ray
elements.

Based one these measurements, one expects that the average composition becomes
heavier as a function of energy up to about $10^{17}$~eV and again lighter
above.  Such a behavior is confirmed by the following two observations.

\begin{figure}[t]
 \includegraphics[width=0.5\textwidth]{tunka.eps}
 \includegraphics[width=0.5\textwidth]{icecube.eps}
  \caption{\textit{Left}\xspace: Average depth of the shower maximum as a function of energy,
             measured by Tunka \cite{tunkaicrc32}.
           \textit{Right}\xspace: Mean logarithmic mass as a function of energy obtained by
             IceCube/IceTop \cite{Tamburro:2012ay}.}
  \label{tunkaice}
\end{figure}

\paragraph{Tunka}
The Tunka-133 air shower array with about 1~km$^2$ geometric acceptance area is
installed in the Tunka Valley (50 km from Lake Baikal) \cite{tunkaicrc32}. It
measures the \v{C}erenkov\xspace component of air showers with an array of 133 open
\v{C}erenkov\xspace detectors.  The average depth of the shower maximum as measured by
Tunka is presented in \fref{tunkaice} (\textit{left}\xspace). The lines represent
predictions for primary protons and iron nuclei. It can be recognized that as
a function of energy, the average cosmic-ray composition becomes heavier
(approaching iron) up to about $10^{17}$~eV and again lighter at higher
energies.

\paragraph{IceCube/IceTop}
The IceCube neutrino telescope at the South Pole is also a large km$^2$-scale
detector for muons from extensive air showers, complemented by an array of
detectors on the surface (IceTop) to register the charged particles in air
showers \cite{Tamburro:2012ay}.
First results on the mass composition of cosmic rays have been obtained from
data taken already during the construction of the detector.  The mean
logarithmic mass derived from one month of data with about half the detector is
depicted in \fref{tunkaice} (\textit{right}\xspace).
The measurements clearly indicate a rising mean mass as a function of energy.
Results up to energies exceeding $10^{17}$~eV are expected soon with the full
detector being operational since 2010 and it will be interesting to see, if a
trend to a lighter composition, as discussed above, will be found as well by
IceCube at energies exceeding $10^{17}$~eV.

\Section{The Composition of Galactic Cosmic Rays}
\begin{figure}[t]
 \includegraphics[width=0.99\textwidth]{elementspek1-denver.eps}
  \caption{Energy spectrum of protons in Galactic cosmic rays as measured by
     the direct experiments
     AMS \cite{amsp},
     ATIC-2 \cite{Panov:2011ak},
     BESS \cite{bess00},
     CAPRICE~98 \cite{caprice98},
     CREAM2011 \cite{Yoon:2011zz},
     PAMELA2011 \cite{Adriani:2011cu}, and
     RICH-II \cite{rich2}
     and the air shower experiments
     GRAPES \cite{grapes05},
     EAS-TOP (electrons and muons) \cite{Ghia:2005gn}
             (unaccompanied hadrons) \cite{eastopsh}, and
     KASCADE (electrons and muons) \cite{ulrichapp}
             (unaccompanied hadrons) \cite{kascadesh}.
     Two hadronic interaction models have been used to interpret the data
     from the GRAPES and KASCADE experiments.
     The line represents a parameterization according to the Poly Gonato model
     \cite{pg}.}
  \label{protons}
\end{figure}

\begin{figure}[t]
 \includegraphics[width=0.99\textwidth]{elementspek2-denver.eps}
  \caption{Energy spectrum of helium nuclei in Galactic cosmic rays as measured
     by the direct experiments
     AMS \cite{amsp},
     ATIC-2 \cite{Panov:2011ak},
     BESS \cite{bess00},
     CAPRICE~98 \cite{caprice98},
     CREAM2011 \cite{Yoon:2011zz},
     PAMELA2011 \cite{Adriani:2011cu}, and
     RICH-II \cite{rich2}
     and the air shower experiments
     GRAPES \cite{grapes05}, and
     KASCADE \cite{ulrichapp}.
     Two hadronic interaction models have been used to interpret the data
     from the GRAPES and KASCADE experiments.
     The line represents a parameterization according to the Poly Gonato model
     \cite{pg}.}
  \label{helium}
\end{figure}

\begin{figure}[t]
 \includegraphics[width=0.99\textwidth]{elementspek3-denver.eps}
  \caption{Energy spectrum of nuclei from the CNO group
     in Galactic cosmic rays as measured by the direct experiments
     ATIC-2 \cite{Panov:2011ak},
     CREAM \cite{Yoon:2011zz},
     CRN \cite{crn},
     HEAO-3 \cite{heao3},
     TRACER \cite{tracer03}
     and the air shower experiments
     GRAPES \cite{grapes05},
     EAS-TOP \cite{Ghia:2005gn},
     KASCADE \cite{ulrichapp}, and
     KASCADE-Grande (light) \cite{Apel:2011mi}.
     Two hadronic interaction models have been used to interpret the data
     from the GRAPES and KASCADE experiments.
     The direct measurements have single-element resolution, i.e.\ measure the
     flux of carbon and oxygen nuclei. Air shower experiments can only resolve
     elemental groups.
     The line represents a parameterization according to the Poly Gonato model
     \cite{pg}.}
  \label{CNO}
\end{figure}

\begin{figure}[t]
 \includegraphics[width=0.99\textwidth]{elementspek5-denver.eps}
  \caption{Energy spectrum of nuclei from the iron group
     in Galactic cosmic rays as measured by the direct experiments
     ATIC-2 \cite{Panov:2011ak},
     CREAM \cite{Yoon:2011zz},
     CRN \cite{crn},
     HEAO-3 \cite{heao3},
     TRACER 2003 \cite{tracer03} and 2006 \cite{Obermeier:2011wm}
     and the air shower experiments
     GRAPES \cite{grapes05},
     H.E.S.S. direct \v{C}erenkov\xspace light \cite{Aharonian:2007zja},
     EAS-TOP \cite{Ghia:2005gn},
     KASCADE \cite{ulrichapp}, and
     KASCADE-Grande (heavy) \cite{Apel:2011mi}.
     Two hadronic interaction models have been used to interpret the data
     from the GRAPES and KASCADE experiments.
     The direct measurements have single-element resolution, i.e.\ measure the
     flux of iron nuclei. Air shower experiments can only resolve
     elemental groups.
     The line represents a parameterization according to the Poly Gonato model
     \cite{pg}.}
  \label{iron}
\end{figure}

A compilation of world data from direct and indirect measurements of cosmic
rays for four elemental groups is given in \fref{protons} (protons),
\fref{helium} (helium nuclei), \fref{CNO} (CNO-group nuclei), and
\fref{iron} (iron-group nuclei).  Here we restricted ourself to "modern"
measurements. Older data are included in previous compilations
\cite{pg,cospar06}.
The energy is given as total energy per particle.
Direct measurements above the atmosphere (on balloons and space crafts) extend
to almost $10^6$~GeV and at higher energies air shower measurements set in.

To guide the eye the lines represent a parameterization according to the
Poly Gonato model with a rigidity dependent cut-off and a constant
$\Delta\gamma$ (see \rref{pg} for details) with the following parameter range
for the nuclear charge number $Z$:
\fref{protons} protons $Z=1$,
\fref{helium} helium $Z=2$,
\fref{CNO} CNO group $Z=5-12$,
\fref{iron} iron group $Z=26-92$.

These figures reflect the present status of our understanding of the elemental
composition of Galactic cosmic rays. Several common features can be recognized.
At low energies, the flux is influenced by magnetic fields in the heliosphere
(solar modulation).
At higher energies the spectra follow approximately a power law.
Finally, at energies exceeding $10^{15}$~eV the spectra exhibit a fall-off,
which is roughly proportional to the charge of the respective nuclei
$E_c\approx Z \cdot 4\cdot10^{15}$~eV.

A closer look reveals some more properties.
An often discussed issue is the spectral slope of protons and helium nuclei. As
can be inferred from \ffref{protons} and \ref{helium}, the spectrum of helium
is slightly flatter ($\gamma=-2.64\pm0.02$) as compared to protons
($\gamma=-2.71\pm0.02$).
Looking closely at the proton and helium spectra, a structure might be visible
around 200~GeV. Above this energy, the spectra follow power laws, which extend
into the air-shower energy region, where ultimately a cut-off is observed.
Below about 200~GeV, both proton and helium exhibit a "bump", before the solar
modulation yields to a depression at lowest energies.
This feature is sometimes referred to as "spectral hardening"
\cite{Yoon:2011zz}. However, from \ffref{protons} and \ref{helium} it appears
as there are two cosmic-ray components, one below 200~GeV and a second one at
higher energies.
It should also be noted that the effect is very subtle and one may ask if
systematic effects in the experiments are understood to such a precision, in
particular, since the energy corresponds roughly to the transition between two
experimental techniques: from magnet spectrometers (at low energies) to
calorimeters.

Looking at the CNO and iron groups, it may be noted that the recent
KASCADE-Grande data (\fref{argokg}, \textit{right}\xspace) extend previous measurements to
higher energies and a cut-off is now also clearly visible in the iron group.
Since protons and helium nuclei have already reached their cut-off energies,
the "light" component in \fref{argokg}, \textit{right}\xspace corresponds most likely to the
CNO group.
An interesting behavior can be observed for the iron group: two hadronic
interaction models (QGSJET and SIBYLL) have been used to interpret the air
shower data measured by GRAPES and KASCADE. For the interaction model QGSJET a
"dip" is visible for both experiments at energies around $10^6$~GeV.  To derive
the spectra the correlations between the number of electrons and muons in the
showers are investigated.  QGSJET is not compatible with the measured
distributions around energies of $10^6$ GeV. This yields the depression in the
iron spectrum, consistently observed by both experiments.  Such a behavior has
been observed earlier, for a detailed discussion see also \rref{ulrichapp} and
\cite{Apel:2008cd}.
It might also be worth to mention that the recent KASCADE-Grande data for the
heavy/iron component are right on top of the predictions of the Poly Gonato
model (published a decade before the measurements).

\Section{Hadronic Cross Sections and LHC Data}

Air showers measured with the Pierre Auger Observatory have been used to derive
the proton-air interaction cross section at a center-of-mass energy of 57~TeV
\cite{Collaboration:2012wt}.
For the proton-air
cross section at ($57\pm6$)~TeV a value of
$$ \sigma_{p-air}=\left[505\pm22 (\mbox{stat}) ^{+20}_{-15} (\mbox{syst})
\right]~\mbox{mb} $$
has been derived.  Using Glauber theory a value for the inelastic proton-proton
cross section has been derived:
$$\sigma_{pp}=\left[92\pm7 (\mbox{stat}) ^{+9}_{-11} (\mbox{syst}) \pm 7
(\mbox{Glauber}) \right]~\mbox{mb}. $$

\begin{figure}[t]
 \includegraphics[width=0.5\textwidth]{pp.eps}
  \caption{Inelastic proton-proton cross section as a function of the
   center-of-mass energy as measured by various accelerator experiments and
   recently derived from air shower data by the Pierre Auger collaboration
   \cite{Collaboration:2012wt}. The lines represent predictions from different
   hadronic interaction models.}
  \label{fig1}
\end{figure}

The Auger value is shown together with accelerator measurements, including
recent LHC data  in \fref{fig1}. Also shown are predictions from different
hadronic interaction models.  It should be noted, that the recent Auger results
and the recent LHC measurements are at the lower boundary of the range
predicted by the various interaction models.

The recent results indicate lower values for the proton-air and proton-proton
cross sections.  A lower cross section has already been predicted in 2003
\cite{wq}. In this work the mean logarithmic mass derived from air shower
observations has been systematically investigated. In particular, a discrepancy
has been found between experiments observing the depth of the shower maximum
and experiments registering the number of secondary particles at ground.  It
has been shown that a smaller proton-proton and consequently also proton-air
cross section reduces the discrepancy in the derived mean logarithmic mass from
the different classes of observables.  The recent results confirm this earlier
findings and this example illustrates how the new LHC measurements directly
influence the interpretation of air shower data.

In the past, various ideas have been discussed as possible origin of the knee
in the energy spectrum of cosmic rays, among them were ideas about new types of
(hadronic) interactions in the atmosphere, see e.g.\ \cite{origin}.  The recent
LHC results exhibit a fair agreement with predictions of hadronic interaction
models using standard physics, see e.g.\ \cite{pierogbeijing,jrhrap}. Hence, it
can be concluded that the knee in the energy spectrum is not caused by new
physics in the atmosphere, it is rather of astrophysical origin.

\Section{The Origin of Galactic Cosmic Rays}

\begin{figure}[t]
 \includegraphics[width=0.99\textwidth]{elementspek51-denver.eps}
  \caption{Energy spectrum of the cosmic-ray iron group.
     Experimental data as in \fref{iron}.
     The lines indicate spectra for models explaining the knee due to the
     maximum energy attained during the acceleration process according to
   Sveshnikova et al. \cite{sveshnikova} (---),
   Berezhko et al. \cite{berezhko} (-\,-\,-),
   Stanev et al. \cite{stanev} ($\cdot\cdot\cdot$), and
   Kobayakawa et al. \cite{kobayakawa} (-$\cdot$-$\cdot$).}
  \label{snr}
\end{figure}

The origin of the knee in the energy spectrum is often assumed to give hints
towards the origin of Galactic cosmic rays.  In the literature various
scenarios are discussed as possible origin for the knee. They can be grouped in
four classes \cite{origin}: (i) the finite energy reached during the
acceleration process (presumably in Supernova Remnants), (ii) leakage from the
Galaxy during the diffusive propagation process, (iii) interaction with
background particles during the propagation process, and (iv) new hadronic
interactions within the atmosphere, which transport a fraction of the energy
into unobserved channels.

The third class can most likely be excluded due to to measurements of the
composition of cosmic rays in the energy region around $10^{16}$ to
$10^{17}$~eV \cite{origin}. The interactions with background particles would
result in a  break-up of heavy nuclei, in which many light particles, such as
protons and $\alpha$ particles are produced. Thus, on expects a rather light
elemental composition in the energy range mentioned. However, recent data
indicate a heavy composition in this energy regime, see e.g.\ \fref{tunkaice}.

Also the fourth class can be disregarded due to recent measurements.
Air shower data from the KASCADE experiment have been used to
check the predictions of hadronic interaction models. A quantitative analysis
demonstrates, that the predictions of the models agree with measured
observables on the 10\% level \cite{jenspune}.
A big step forward in the understanding of hadronic interactions
are the results from the Large Hadron Collider (LHC). The design center-of-mass energy of 14~TeV
corresponds to a laboratory energy of $10^{17}$~eV, i.e.\ well above the knee
in the energy spectrum.
The recent LHC results exhibit a fair agreement with predictions of hadronic
interaction models using standard physics, see e.g.\
\cite{pierogbeijing,jrhrap}.
All this information together leaves very little room for new type of
interactions inside the atmosphere.  Thus, it can be concluded that the knee in
the energy spectrum is not caused by new physics in the atmosphere, it is
rather of astrophysical origin.

\begin{figure}[t]
 \includegraphics[width=0.99\textwidth]{elementspek52-denver.eps}
  \caption{Energy spectrum of the cosmic-ray iron group.
     Experimental data as in \fref{iron}.
   The lines indicate spectra for models explaining the knee as an
   effect of the propagation process according to
   H\"orandel \& Kalmykov et al. \cite{prop} (---),
   Ogio et al. \cite{ogio} (-\,-\,-),
   Roulet et al. \cite{roulet} ($\cdot\cdot\cdot$), as well as
   V\"olk et al. \cite{voelk} (-$\cdot$-$\cdot$).}
  \label{prop}
\end{figure}

Predictions of astrophysical models are illustrated in \ffref{snr} and
\ref{prop} for the cosmic-ray iron group.  A comparison of the model
predictions with measured data for the light groups is given in
\rref{cospar06}.
Models describing the acceleration of cosmic rays in Supernova Remnants predict
a maximum energy attained during the acceleration process, which is
proportional to the nuclear charge number of the respective atomic nuclei $Z$
and the strength of the magnetic field $B$ in the acceleration region
$E_{max}\propto Z\cdot B$. The predictions of such models are depicted in
\fref{snr}.
A second class of models describes the diffusive propagation in the Galaxy. In
this models one finds a maximum rigidity $E/Z$ above which the nuclei cannot be
any more effectively magnetically bound to the Galaxy and the particles leak
out of the Milky Way. The predictions of such models are shown in \fref{prop}.
For a more detailed discussion of the models the reader is referred to
\rref{origin}.

The different models exhibit some scatter, however, the magnitude of the
differences is of the same order as the accuracy of the measurements. A general
trend can be observed: all models shown describe well the fall-off of the iron
group as seen in the measurements.
I would like to put some emphasize on the predictions of \rref{prop}.
It has been shown that the propagation process alone results in a very soft
bend of the individual spectra, not compatible with the measured strong fall-off
(see \ffref{protons} to \ref{iron}). In addition, a rigidity dependent maximum
energy in the sources had to be included to describe the measured sharpness of
the cut-offs.
Thus, I would like to conclude that the origin of the knee in the energy
spectrum is a combination of the maximum energy attained during the
acceleration process and leakage from the Galaxy.
This fits very well with the "standard picture" of Galactic cosmic rays, being
accelerated in Supernova Remnants and diffusively propagating through the Milky
Way.

\begin{theacknowledgments}
 The author thanks his colleagues from TRACER, KASCADE-Grande, and the
 Pierre Auger Observatory for fruitful discussions.
\end{theacknowledgments}

\bibliographystyle{aipproc}


\section{Introduction}

Observations of the redshifted 21-cm line of neutral hydrogen, planned
for the next decade, are expected to usher in a new era of direct
probing of the epoch of first stars. Though currently the main
observational focus is on the reionization epoch, there are
instruments hoping to observe the 21-cm signal from $z\sim
10-30$, e.g., LEDA \citep{Burns:2011},
DARE\footnote{http://lunar.colorado.edu/dare/}, and the SKA
\citep{Carilli:2004}.

The formation of the first stars is a relatively clean theoretical
problem, as they are formed in a metal-free environment via H$_2$
cooling \citep{Tegmark:1997, Machacek:2001, Abel:2002}. The radiation
produced by these first radiant objects changed the cosmic landscape
dramatically \citep{Madau:1997}. Three wavelength regimes of this
radiation are most important to consider: the Lyman-$\alpha$ photons
couple to the 21-cm line at high redshifts ($z\sim 30$) through the
Wouthuysen-Field effect \citep{Wouthuysen:1952, Field:1959}; X-ray
photons, produced by stellar remnants, heat the gas; and Lyman-Werner
(LW) photons dissociate molecular hydrogen, thus producing negative
feedback on star formation \citep{Haiman:1997} and decreasing the
heating rate.

The radiation spreads out to $\sim 100$ Mpc around each star, where
this finite effective horizon arises from redshift, time delay and
optical depth effects
\citep{Ahn:2009,21cmfast,Holzbauer:2011,Visbal:2012}. As star
formation progresses, the radiative backgrounds build up. Fluctuations
in the radiative backgrounds \citep{Barkana:2005,Pritchard:2006},
caused by the strongly fluctuating distribution of stars
\citep{Barkana:2004}, couple to the hyperfine transition of neutral
hydrogen and imprint fluctuations in the redshifted 21-cm
signal. Whereas Lyman-$\alpha$ coupling saturates at high redshifts
\citep{Holzbauer:2011, Visbal:2012},
the heating fluctuations couple to the 21-cm brightness temperature at
lower redshifts and thus are more interesting in terms of the
observational prospects. In particular, \citet{Visbal:2012} predict
detectible heating fluctuations from the first stars at $z = 20$ with
a distinctive signature of Baryon Acoustic Oscillations (BAOs)
imprinted by the supersonic relative (streaming) velocity between the
baryons and the dark matter \citep{Tseliakhovich:2010, Dalal:2010}.

In this paper we study the signature of the heating fluctuations
including the affect of relative velocity as in \citet{Visbal:2012},
and add for the first time a detailed three-dimensional calculation of
the inhomogeneous negative feedback by LW photons. We use the same
semi-numerical hybrid methods as in \citet{Visbal:2012} (based in part
on \citet{Tseliakhovich:2010} and \citet{21cmfast}) to build a
simulation where the stellar population and the radiative backgrounds
evolve simultaneously in time. The basic idea is to linearly evolve a
realistic sample of the Universe on large scales while using the
results of numerical simulations and analytical models to add in the
stars. For a detailed discussion of our computational methods, we
refer the interested reader to \citet{Visbal:2012}, in particular to
section S1 of this paper's Supplementary Information, and references
within. We use the standard set of cosmological parameters
\citep{WMAP7} along with an assumed star formation efficiency
(fraction of gas in star-forming halos that turns to stars) of $f_* =
10\%$, an X-ray photon efficiency of $10^{57}M_{\odot}^{-1}$ based on
observed starbursts at low redshift as in \citet{21cmfast}, and LW
parameters as explained in the next section.

\section{Incorporating the Negative Feedback}

The formation of the first stars via cooling of molecular hydrogen is
a highly non-linear process that can be mimicked by numerical
simulations, e.g., \citet{Abel:2002, Bromm:2002}. However, numerical
simulations in which primordial stars are created usually do not
consider the potentially fatal effect of the LW background on this
process. The negative feedback of the LW background on star formation
has been tested in the limited case of a fixed intensity $J_{\LW}$
\citep{Machacek:2001,Wise:2007,O'Shea:2008}. The feedback boosts the
minimal cooling mass, $M_{\rm cool}$, i.e., the mass of the lightest
halo in which stars can form, with the results of these simulations
well-described by the relation
\be \label{Mcool0} M_{\rm cool}\left(
J_{21}, z\right) = M_{\cool,0}(z)\times\left[1+6.96\left(4\pi
J_{21}\right)^{0.47}\right], \ee
where $J_{21} = J_{\LW}/(10^{-21}$erg
s$^{-1}$cm$^{-2}$Hz$^{-1}$sr$^{-1})$ in terms of the LW intensity
$J_{\LW}$; another common notation is the LW flux $F_{\LW} = 4\pi
J_{\LW}$. Here $M_{\cool,0}(z)$ is the value of the minimum cooling
mass in the standard case with no LW background.

This result is incomplete for two reasons. One is that it does not
account for the relative velocity $\vbc$, which has a strong impact on
the primordial star formation by (among other things) boosting the
minimum cooling mass \citep{Greif:2011,Stacy:2011,Fialkov:2011}. To
account for the velocity, we change $M_{\cool,0}(z)$ in
Eq.~(\ref{Mcool0}) to $M_{\cool,0}(z,\vbc)$, using the fit we
developed in \citet{Fialkov:2011} to the streaming-velocity
simulations (here we use their fit designed for Adaptive Mesh
Refinement (AMR) simulations as equation~(\ref{Mcool0}) was the result
of a fit to an AMR simulation). Thus, we combine two separate physical
phenomena, i.e., the relative motion and the LW flux, assuming that
they each have a fixed multiplicative effect on the minimum cooling
mass.  This simple ansatz for the dependence of $M_{\cool}$ on the two
parameters, $\vbc$ and $J_{21}$, should be checked by detailed
numerical simulation, which we hope to stimulate with this work. We
choose this multiplicative ansatz since both effects have an
independent effect on the minimal cooling mass. Therefore they likely
reinforce each other when they are both present. We note that while we
include the $\vbc$ and LW effects separately in Eq.~(1), our results
do account for the strong correlation between the velocity and the LW
flux (due to the effect of the velocity on star formation).

The second incompleteness of Eq.~(\ref{Mcool0}) is its validity only
in the case of a fixed background intensity during the formation of
the halo, whereas in reality the LW intensity is expected to rise
exponentially with time at high redshifts (e.g., see the Supplementary
Information in \citet{Visbal:2012}). Treating the intensity as fixed
at its final value would greatly overestimate the strength of the
feedback, since the cooling and collapse involved in star formation
should respond with a delay to a drop in the amount of H$_2$. For
instance, if the halo core has already cooled and is collapsing to a
star, changing the LW flux may not stop or reverse the collapse at
all, and certainly not immediately. Another indication for the gradual
process involved is that the simulation results can be approximately
matched \citep{Machacek:2001} by comparing the cooling time in halo
cores to the Hubble time (which is a relatively long timescale).
Though the relation in Eq.~(\ref{Mcool0}) is the best currently
available, more elaborate numerical simulations, which we again hope
to stimulate, are needed in order to find a more realistic dependence.
We overcome this limitation by using the above relation not with the
final value of $J_{21}$ at formation, but with the value at a mean,
characteristic time within the halo formation process.

The idea of looking at the flux at times well before virialization is
based on an analogy with the filtering mass defined in the
well-studied case of pressure \citep{Gnedin:1998,Gnedin:2000}. In the
latter case, the actual gas fraction in non-linear, virialized halos,
is close to the filtering mass, not the Jeans mass, and the filtering
mass is affected by the value of the Jeans mass at much earlier times.
The reason the gas fraction is affected by the Jeans mass at early
times is that the relationship between the gas temperature and the gas
density is very indirect; the temperature affects the pressure
gradient, and thus the acceleration, which affects the actual position
after a delay. In particular, if the temperature drops suddenly, the
gas that was far away from the halo center does not instantly fall
inside. In the case of cooling, there are the additional steps from
dissociation of H$_2$, which changes its abundance, through the
process of cooling which then affects the temperature. The cooling
history thus affects the distribution of gas. The LW flux rises
exponentially fast with time and was very small at early times. Thus,
the cooling was fast initially, and the gas cooled and started to
collapse. A sudden late rise in the flux may not be able to stop this
collapse.

Using the characteristic value of $J_{21}$ with a realistically large
uncertainty should suffice for our main goal of spanning the possible
range of the effect of $J_{\LW}$ and $\vbc$ on the 21-cm background
during the X-ray heating era.
Specifically, we consider two possible feedback strengths which we
refer to as ``weak'' and ``strong'' feedback. Namely, for halos
forming (i.e., virializing) at some time $\tvir$, we adopt the
effective LW flux $J_{21}$ in Eq.~(\ref{Mcool0}) as the LW flux in the
same pixel at an earlier time $\tmid$, i.e., at the midpoint of the
halo formation process. In order to obtain a realistically large range
of uncertainty, with the spherical collapse model in mind we either
assume that ``formation'' spans the beginning of expansion up to
virialization (i.e., $t=0$ to $t=\tvir$, giving
$\tmid=\frac{1}{2}\tvir$: weak feedback), or just the collapse stage
starting at turnaround (i.e., $t=\frac{1}{2}\tvir$ to $t=\tvir$,
giving $\tmid=\frac{3}{4}\tvir$: strong feedback). We also compare to
the limiting cases (shown in \citet{Visbal:2012}) of no feedback or
saturated feedback. The latter correponds to assuming that star
formation is only possible via atomic cooling; in this scenario, the
LW feedback is so efficient that H$_2$ is completely dissociated early
on and stars form in atomic cooling halos ($M_{\cool} \ga 3
\times 10^7 M_{\odot}$), as opposed to $M_{\cool}\la
10^6 M_{\odot}$. For reference, we also consider the no feedback case
without the streaming velocity, in order to assess the importance of
the velocity effect. For given parameters, at each redshift the cosmic
mean gas fraction in stars decreases in the different cases in the
order: no feedback no velocity, no feedback, weak feedback, strong
feedback, and saturated feedback (where all cases except the first
include the streaming velocity effect).

\begin{figure*}
\includegraphics[width=3.4in]{FStellar.eps}
\includegraphics[width=3.4in]{Ratio.eps}
\caption{Star-formation contribution and effect of velocities
versus halo mass (LW feedback not included). {\bf Left}: The
logarithmic contribution of each halo mass to the total fraction of
gas in stars (i.e., $d f_{\rm stellar}/d \log(M)$ averaged over the
distribution of $\vbc$), including the $\log(M)$ modulation in
Equation~(\ref{eq:f*}) (solid) or with the standard assumption of a
fixed efficiency with mass (dashed). We consider $z = 13.6$ (red),
$z = 19.6$ (green), and $z = 25.6$ (blue). {\bf Right}: The ratio of the
cosmic mean stellar fraction with $\vbc$ to the value without the
velocity effect, i.e., $<f_{\rm stellar}(M,\vbc)>_{\vbc} / f_{\rm
stellar}(M,0)$. We include Equation~(\ref{eq:f*}), and consider the
same redshifts as in the left panel.}
\label{fig:dfdm}
\end{figure*}

In our hybrid simulations we also incorporate two elements of the
astrophysics that make our calculations more complete and
accurate. One aspect is that we include a gradual low-mass cutoff for
star formation, rather than a sharp cutoff at $M_{\rm cool}$ as we
(and others) have previously assumed. Since the cooling rate declines
smoothly with virial temperature, a smooth cutoff is expected
physically, and indeed \citet{Machacek:2001} found that the fraction
of highly-cooled, dense gas in their simulated halos is well described
as being proportional to $\log(M/M_{\rm cool})$. Since this is the gas
that can participate in star formation, we incorporate this by
generalizing the star-formation efficiency to include a dependence on
halo mass, $f_*(M)$. We assume our standard efficiency of $f_* = 10\%$
for $M \ge M_{\rm atomic}$, where $M_{\rm atomic}$ is the minimum mass
for atomic cooling ($\sim 3 \times 10^7 M_\odot$ but $z$-dependent).
In order for $f_*(M)$ to be a continuous function, we thus set
\be \label{eq:f*}
f_*(M) = \left\{ \begin{array}{ll}
f_* & \mbox{if $M \ge M_{\rm atomic}$} \\
f_* \frac{\log(M/M_{\rm cool})} {\log(M_{\rm atomic}/M_{\rm cool})}
& \mbox{if $M_{\rm cool} < M < M_{\rm atomic}$} \\
0 & \mbox{otherwise.}
\end{array} \right.
\ee

As shown in Figure~\ref{fig:dfdm} (left panel), the standard
assumption of constant $f_*$ makes halos with masses near $M_{\rm
cool}$ dominate the cosmic star formation rate, particularly at the
highest redshifts. Our more realistic model significantly reduces the
overall star formation rate (by a factor of 2.0 in the example shown
at $z=19.6$) and shifts the peak of the contribution to star formation
to a higher mass ($8.7\times M_{\rm cool}$ at $z=19.6$). Also shown in the
Figure (right panel) is the overall effect of the relative velocity
$\vbc$ broken down by mass. Since the velocity effect on halos is made
up of three distinct effects, with two of them dominant
\citep{Fialkov:2011}, the dependence on halo mass shows two separate
regimes. Near the cooling mass (and up to a factor of $\sim 2$ above
it), the velocity effect is very strong and also strongly dependent on
$M$, mainly due to the boosting of the cooling mass in regions with a
high $\vbc$. At higher masses, however, the velocity effect is weaker
and only changes rather slowly with halo mass, mainly due to the
suppression of the halo abundance. A small but non-negligible effect
remains even well above $M_{\rm atomic}$. Since the velocity effect is
strongest at the low-mass end (right panel), the shifting of the star
formation towards higher masses (left panel) reduces somewhat the
overall influence of the supersonic streaming velocities. Since the LW
feedback also affects low masses first, the $\log(M)$ modulation
delays the LW feedback.

The second aspect of realistic astrophysics that we incorporate is
more directly related to the LW feedback. The LW photons emitted by
each source are absorbed by hydrogen atoms as soon as they redshift
into one of the Lyman lines of the hydrogen atom; along the way,
whenever they hit a LW line they may cause a dissociation of molecular
hydrogen. Some previous papers \citep{Ahn:2009,Holzbauer:2011} assumed
a flat stellar spectrum in the LW region and a flat absorption profile
over the LW frequency range. We incorporate the expected stellar
spectrum of Population III stars from \citet{Barkana:2005} (based on
\citet{Bromm:2001}), which varies in the LW region typically by a few
percent but up to $17\%$. More importantly, we explicitly include the
full list of 76 relevant LW lines from \citet{Haiman:1997}. We
summarize the results with $f_{\rm LW}$, the relative effectiveness of
causing H$_2$ dissociation via stellar radiation. Specifically it is
the ratio between the dissociation rate of molecular hydrogen and the
naive total stellar flux (i.e., calculated without any absorption and
integrated over all wavelengths), normalized to unity in the limit of
zero source-absorber distance. This quantity is simply a function of
source-absorber distance at each redshift under the simplifying
assumption of a universe at the mean density. This assumption follows
our approach for X-rays (as in 21CMFAST, \citet{21cmfast}), and should
be sufficiently accurate since the strong bias of star-forming halos
at these high redshifts implies that fluctuations in star formation
(which drive the 21-cm fluctuations) are much larger than the
fluctuations in the underlying density. Thus, give our assumed stellar
spectrum we can pre-calculate $f_{\rm LW}$ and include this as an
effective optical depth that is spherically symmetric around each
source. Any such symmetric effect is easily incorporated within the
numerical method of 21CMFAST which uses Fourier transforms to rapidly
perform averages over spherical shells.

Figure~\ref{fig:fmod} shows $f_{\rm LW}$ versus the absorber-source
distance; we parametrize this distance in terms of the absorber-source
scale-factor ratio $R$, since $f_{\rm LW}$ versus $R$ is independent
of redshift. Beyond the max shown $R=1.054$ (which corresponds to 104
comoving Mpc at $z=20$), {\bf $f_{\rm LW}$} immediately drops by five
orders of magnitude. The Figure shows that LW absorption is poorly
approximated as being uniform in frequency. In reality, emission from
distant sources is absorbed more weakly. For example, in one of our
main examples in the following section (i.e., our strong LW feedback
case including velocities), assuming a uniform spectrum and absorption
profile at $z=20$ would imply that typically, an atom receives $50\%$
of its LW flux from sources out to a distance of 18.9 Mpc, $80\%$ from
up to 42.2 Mpc, and $90\%$ from up to 55.8 Mpc. Our more accurate {\bf
  $f_{\rm LW}$} reduces these numbers to 14.4, 33.0, and 46.2 Mpc,
respectively.  The accurate {\bf $f_{\rm LW}$} reduces the overall LW
intensity by $\sim 20\%$ (thus delaying the LW feedback), and makes it
more short-range (i.e., local) and variable.

\begin{figure}
\includegraphics[width=3.4in]{fLW.eps}
\caption{The relative effectiveness of causing H$_2$ dissociation in
  an absorber at $z_a$ due to stellar radiation from a source at
  $z_s$, shown versus the ratio $R\equiv (1+z_s)/(1+z_a)$ (solid). For
  comparison we show $f_{mod}$, a commonly used approximation from
  \citet{Ahn:2009} (dashed). Both functions are normalized to unity at
  $R=1$. (There was also a $1.45\%$ normalization difference after we
  carefully normalized as in \citet{Machacek:2001}, since we use their
  results for the LW feedback.)}
\label{fig:fmod}
\end{figure}

\section{Results}

\subsection{Mean Evolution}

The relative velocity amplifies heating fluctuations in the 21-cm
power spectrum, making it possible to observe the BAO in the first
stars \citep{Visbal:2012,McQuinn:2012}. In this section we consider
the effect of the negative LW feedback on this exciting observational
prospect. Our simulation evolves a realistic sample of the universe
from $z = 60$, roughly when the first stars turn on \citep{Naoz:2006,
Fialkov:2011}. We follow a box that is 384 Mpc on a side (all
distances comoving), with a pixel size of 3 Mpc. The X-ray and LW
backgrounds in each pixel are made up of contributions by stars
located within the corresponding effective horizons. Since we focus on
the era after Lyman-$\alpha$ coupling but prior to significant
reionization, the 21-cm brightness temperature (relative to the cosmic
microwave background (CMB) temperature $T_{\rm CMB}$)\cite{Madau:1997}
\begin{equation}
T_{\rm b} = 40 (1+\delta) \left( 1-\frac{T_{\rm CMB}} {T_{\rm K}}
\right) \sqrt{\frac{1+z}{21}} ~{\rm mK} \label{eq:Tb}
\ ,\end{equation}
where $\delta$ is the gas overdensity and $T_{\rm K}$ its kinetic
temperature.

The negative feedback suppresses star formation on average, which
leads to a slower rise of the radiative backgrounds, and delays
various milestones of the star formation history. Figure~\ref{fig:JLW}
shows the rise of the mean LW flux with time. Comparing the two
no-feedback cases, we see that the streaming velocity has a large
overall suppression effect at high redshifts, which reaches about an
order of magnitude at $z>40$ but becomes quite small at $z < 15$.
Feedback can potentially be very strong at high redshifts (as
indicated by the saturated feedback case), but in practice the LW
feedback is expected to begin only when the effective flux reaches a
level of $J_{21} \sim 10^{-5}$; this happens at around redshift 30
(weak feedback) or 40 (strong). In both realistic feedback cases, the
LW feedback effectively saturates at $z \sim 10$.

\begin{figure}
\includegraphics[width=3.4in]{JLW_z.eps}
\caption{The actual LW intensity
(solid lines) and the effective LW intensity for feedback on star
formation (dashed; shown only for the two realistic feedback
cases). We show the cosmic mean intensity (i.e., averaged over our
box) versus $1+z$ in the following cases: no feedback no $\vbc$
(purple), and with $\vbc$: no feedback (red), weak feedback (blue),
strong feedback (green), and saturated feedback (black).}
\label{fig:JLW}
\end{figure}

We can easily understand why the two feedback cases converge with
time. Initially, the effective LW flux for star formation [i.e.,
$J_{21}$ in Eq.~(\ref{Mcool0})] is much higher at a given $z$ for the
strong feedback case (which assumes a less delayed value, closer to
the value of $J_{21}$ at $z$). The strong resulting feedback leads to
a slower rise of the actual $J_{21}$ and thus, eventually, also of the
effective $J_{21}$, compared to the weak feedback case. Therefore, the
effective $J_{21}$ in the weak feedback case gradually catches up with
the strong feedback case. Also important is that the rate of increase
of the flux naturally slows with time (i.e., the curves flatten),
since star-forming halos become less rare (i.e., they correspond to
less extreme fluctuations in the Gaussian tail of the initial
perturbations). The weak feedback case effectively looks back to
$J_{21}$ at an earlier time, when the rise was faster.

Figure~\ref{fig:JLW} tracks the rise of the LW flux through several
milestone values. A reasonable definition of the central redshift
$z_{\LW}$ of the LW transition is a mean effective intensity of
$J_{21}=0.1$, at which the minimum halo mass for cooling (in the
absence of streaming velocities) is raised to $\sim 2 \times 10^6
M_{\odot}$ due to the LW feedback. This is a useful fiducial mass
scale, roughly intermediate (logarithmically) between the cooling
masses obtained with no LW flux and with saturated LW flux. The
central range of the LW feedback transition can be defined by the
effective LW flux coming within an order of magnitude of its central
value, so that the minimum $M_{\cool}$ goes from $8 \times 10^5
M_{\odot}$ to $5 \times 10^6 M_{\odot}$ during this period.

Feedback also slows down the heating of the Universe
(Figure~\ref{fig:TK}). For example, the average heating rate at
redshift 20 for the weak, strong and saturated feedbacks are $55.9\%$,
$33.7\%$ and $19.1\%$ of the heating rate with no feedback (all
including the streaming velocity). As a result, the heating transition
is delayed. There are two possible natural definitions for this
transition, the standard more physical definition as the redshift
$z_{\rm h}$ when the mean gas temperature equals that of the CMB, and
the more observational (or 21-cm-centric) definition as the redshift
(which we denote $z_0$) at which the cosmic mean 21-cm brightness
temperature vanishes $\langle T_{\rm b} \rangle =0$. We consider both
definitions, but due to our focus on observational predictions, we
mostly use $z_0$.

\begin{figure}
\includegraphics[width=3.4in]{TK.eps}
\caption{The cosmic heating transition. We show the cosmic mean
gas kinetic temperature versus $1+z$ in the following cases: no
feedback no $\vbc$ (purple), and with $\vbc$: no feedback (red), weak
feedback (blue), strong feedback (green), and saturated feedback
(black). Also shown for comparison is the CMB temperature (dotted),
which crosses the gas temperature at $z_{\rm h}$.}
\label{fig:TK}
\end{figure}

In our simulation, $z_{\rm h} = 17.1$ and $z_0=16.6$ for the
no-feedback case (no feedback and no velocity gives $z_{\rm h} = 17.7$
and $z_0= 17.4$), while saturated feedback would delay these
milestones to $z_{\rm h}=14.6$ and $z_0 = 14.2$. The realistic
feedback cases are intermediate: $z_{\rm h}=15.7$ and $z_0=15.2$ for
the weak feedback case (with a LW transition centered at $z_\LW=19.2$,
and a central range of $z=22.0-15.2$), while $z_{\rm h}=15.0$ and
$z_0=14.6$ for strong feedback (with $z_\LW=23.6$, and a central range
of $z=28.3-18.1$). In every case, the LW transition starts very early,
and passes through its central redshift before the heating transition
(with a much bigger delay between the two transitions in the strong
feedback case). We note that if all the fluctuations were linear, then
we would find $z_{\rm h}=z_0$ identically. The difference of $\Delta z
= 0.4-0.5$ between them is an example of the effect of non-linear
fluctuations (plus, in this case, of the non-linear dependence of the
brightness temperature on the gas temperature). This shows that
analyses of this era based on linear theory can only give rough
estimates, and a hybrid simulation like ours is necessary in order to
properly incorporate the non-linear fluctuations in stellar density
and other derived quantities.

\subsection{Spatial Fluctuations}

The most interesting 21-cm signature of the first stars is the
enhanced large-scale fluctuation level due to the supersonic streaming
velocity. A typical two-dimensional slice of our simulated volume is
shown in Figure~\ref{fig:maps} together with the spatial distribution
of the fluctuations in the density and in the relative velocity. A
snapshot of the Universe at a fixed redshift would look very
distinctive in the various feedback cases mainly due to the overall
delay in the heating due to the change in the mean heating rates. It
is more instructive to compensate for this shift and instead compare
each case to the others at the same time relative to the individual
heating redshift (we use $z_0$).

\begin{figure*}
\includegraphics[width=3.0in]{vbc.eps}
\includegraphics[width=3.0in]{delta.eps}
\includegraphics[width=7.2in]{T21.eps}
\caption{A two-dimensional slice of our simulated volume. All panels
show the same slice, i.e., with the same initial conditions. {\bf
Top:} Left: The magnitude of the relative velocity between baryons and
dark matter shown in units of the velocity's root-mean-square
value. Right: The relative fluctuation in density at redshift 20. {\bf
Bottom:} The 21-cm brightness temperature $T_{\rm b}$ (relative to the
cosmic mean in each case) in mK, shown at $z_0+3$. The cases shown
(from left to right) are: no feedback, no $\vbc$; no feedback; weak
feedback; strong feedback; and saturated feedback (where the last four
include the streaming velocities). The cosmic mean values (which have
been subtracted from the maps) are: $\langle T_{\rm b} \rangle = -81$
mK, -89 mK, -73 mK, -88 mK and $-108$ mK, respectively.  }
\label{fig:maps}
\end{figure*}

As seen in Figure~\ref{fig:maps}, the feedback weakens the effect of
the relative motion, since it boosts the minimal cooling mass so that
the stars form in heavier halos which are less sensitive to the
relative velocities. Thus, in Figure~\ref{fig:maps} the 21-cm signal
with saturated feedback shows the same pattern as in the no $\vbc$
case (with no feedback); namely, both of them follow the density
fluctuations but with a strong enhancement, where the bias is stronger
for the saturated feedback (since more massive halos are more strongly
biased). The no feedback case (with $\vbc$) shows a strong imprint of
the velocity field along with the influence of density fluctuations;
e.g., the enhanced $T_{\rm b}$ at the bottom right is mainly due to a
density enhancement, while the void at the top (just right of center)
is mainly due to a large relative velocity (but note that the 21-cm
maps are the result of a three dimensional calculation of the
radiation fields, so they cannot be precisely matched with
two-dimensional slices of the density and velocity fields).

The two realistic feedback cases are intermediate, showing a clear
velocity effect, though not as strong as in the no-feedback case. To
understand the comparison between the weak and strong feedback, we
note that the velocities cause a very strong suppression of star
formation up to a halo mass $M\sim 10^6 M_{\odot}$, but above this
critical mass the suppression and its $M$-dependence weaken
considerably (Figure~\ref{fig:dfdm}, right panel). Thus, once the LW
feedback passes through its central redshift, the remaining $\vbc$
effect changes only slowly with $M$, so that around the time of the
heating transition, the weak and strong feedback cases show a similar
fluctuation pattern. However, the strong dependence of bias on $M$
remains, so that the strong feedback case leads to larger fluctuations
on all scales.

These and other features can be seen more clearly and quantitatively
in the power spectra (Figure~\ref{fig:ps}). The 21-cm fluctuations
initially rise with time as the heating becomes significant (first in
the regions with a high stellar density). Eventually, as the heating
spreads, the 21-cm fluctuations decline, since the 21-cm intensity
becomes independent of the gas temperature once the gas is much hotter
than the CMB (Equation~\ref{eq:Tb}). Thus, the power spectrum reaches
its maximum height somewhat earlier than $z_0$. The comparison among
the various feedback cases is complex, since the negative LW feedback
has several different effects: 1) The lowest-mass halos are cut out,
reducing the effect of the streaming velocity; 2) The higher-mass
halos that remain are more highly biased; 3) The overall suppression
of star formation delays the heating and LW transitions to lower
redshifts; and 4) Since the higher-mass halos that remain correspond
to rarer fluctuations in the Gaussian tail, their abundance changes
more rapidly with redshift, making the heating transition more rapid
(i.e., focused within a narrower redshift interval). Thus, at $z_0+3$
the large-scale ($k=0.05$ Mpc$^{-1}$) peak is lower for the realistic
feedback cases than it would be with no feedback (due to effect \#1),
and higher for strong feedback than for the weak case (due to effect
\#2). Further back in time ($z_0+9$), weak feedback gives a higher
large-scale peak than both strong feedback (due to effect \#4) and no
feedback (due to effect \#2); at that redshift, saturated feedback
shows no velocity effect (due to effect \#4).

\begin{figure*}
\includegraphics[width=3.4in]{ps2.eps}
\includegraphics[width=3.4in]{ps1.eps}
\caption{Power spectra of the 21-cm brightness temperature for
no feedback (red), weak feedback (blue), strong feedback (green) and
saturated feedback (black). {\bf Left}: $z = z_0 + 3$ with (solid) or
without (dashed) the relative velocity. {\bf Right}: Including $\vbc$,
at redshifts $z = z_0$ (dashed), $z = z_0 + 3$ (solid) and $z = z_0 +
9$ (dotted).}
\label{fig:ps}
\end{figure*}

Lower redshifts offer improved observational prospects, due to the
lower foreground noise, which makes negative feedback advantageous due
to effect \#3, above. We find that the most promising redshift is
$z\sim z_0+3$ (Table~1). Assuming a first-generation radio telescope
array with a noise power spectrum that scales as $(1+z)^{5.2}$
\citep{McQuinn:2006, Visbal:2012}, the maximal S/N of the large-scale
($k=0.05$ Mpc$^{-1}$) peak is 3.24 for weak feedback (at $z=18.3$) and
3.91 for strong feedback (at $z=17.7$). For comparison, the
no-feedback case considered in \citet{Visbal:2012} gave (at
$z=z_0=20$) a S/N of only 2.0~. It would be particularly exciting to
detect the evolution of the 21-cm power spectrum throughout the
heating transition, as we suggested in \citet{Visbal:2012}. The S/N
remains above unity at all $z < z_0+7.9 = 23.1$ in the case of the
weak feedback and $z < z_0+7.2 = 21.9$ for the strong feedback (down
to $z=10$ where our simulations end). The streaming velocity clearly
plays a key role in creating this extended observable redshift range,
by boosting the large-scale power (Figure~\ref{fig:ps}).

\begin{table*}
\label{Tab:SN}
\begin{center}
\begin{tabular}{ |l ||c |  c | c |c | c || c | c| c | c |r | }
\hline
\multirow{2}{*}{$z-z_0$}&\multicolumn{5}{|c||}{$\delta T_{\rm b}
(k=0.05~{\rm Mpc}^{-1})$, S/N} &\multicolumn{5}{|c|}{BAO, S/N} \\
 & no $\vbc$ & no fbk & weak & strong & sat & no $\vbc$ & no fbk &
 weak & strong & sat \\
\hline
$-3$& 1.07 & 1.24  & 1.60 & 1.73 & 1.84 & 0.45 & 0.58 &  0.72 & 0.76 & 0.79\\
$0$& 1.68 & 2.33  & 2.35 & 2.69 & 3.09& 0.70& 1.26 &  1.16 &1.30 & 1.31\\
$3$ & 2.26 & 3.59  & 3.24 & 3.91 & 4.74 &0.91 & 2.18&  1.79 & 2.14 &2.00 \\
$6$ & 1.02 & 1.75 & 2.08 & 1.89 & 1.34 & 0.37& 1.17& 1.30&  1.18 & 0.54 \\
$9$ & 0.086 & 0.33  & 0.56 & 0.34& 0.31 & 0.051&  0.23  & 0.41 &0.25 & 0.14 \\
$12$ & 0.18& 0.23  & 0.24 & 0.27 &0.34&  0.083 & 0.099 & 0.11 &0.12& 0.15 \\
\hline
\end{tabular}
\caption{The signal to noise ratio S/N (i.e., the square root of the
ratio between the power spectra of the signal and noise), for a
projected first-generation radio array. We show the S/N of the
large-scale peak at the wavenumber $k=0.05$ Mpc$^{-1}$ (Left), and of
the BAO component (Right), at various redshifts, for five cases: no
feedback no $\vbc$. no feedback (with $\vbc$), weak, strong and
saturated feedback. The BAO S/N is defined as the square root of the
difference between the peak at $k=0.05$ Mpc$^{-1}$ and the trough at
$k=0.07$ Mpc$^{-1}$, each measured with respect to the non-BAO power
spectrum (i.e., the power spectrum smoothed out using a quartic fit),
and each normalized by the noise power spectrum at the same $k$ at the
corresponding redshift.}
\end{center}
\end{table*}

We suggested in \citet{Visbal:2012} that beyond just detecting the
power spectrum, it would be particularly remarkable to detect the
strong BAO signature, since this would confirm the major influence of
the relative velocity and the existence of small ($10^6 M_\odot$)
halos. We find that the S/N for the large-scale BAO feature of the
power spectrum is typically $\sim 0.5 - 0.7$ times that of the
large-scale peak itself (Table~1). In particular, the BAO S/N also
peaks at $z_0+3$, exceeds unity at $z_0-0.7 <z < z_0+ 6.9$ (weak) and
$z_0-1.1 < z < z_0+ 6.4 $ (strong) and reaches a maximum value of 1.79
(weak) or 2.14 (strong feedback).

We have assumed here the projected sensitivity of a thousand-hour
integration time with an instrument like the Murchison Wide-field
Array \citep{MWAref} but designed to operate in the range of 50--100
MHz. An instrument similarly based on the Low Frequency Array
\citep{LOFARref} should improve the S/N by a factor of $\sim 1.5$,
while a second-generation instrument like the SKA or a 5000-antenna
MWA should improve it by at least a factor of 3 or 4
\citep{McQuinn:2006, Visbal:2012}. Thus, future instruments may be
able to probe even earlier times, including the central stages of the
LW feedback.

\section{Conclusions}

We have presented new predictions for the signature of the first stars
in the heating fluctuations of the 21-cm brightness temperature. We
ran hybrid simulations that allow us to predict the large-scale
observable 21-cm signature while accounting on small-scales for
various effects on star-formation investigated by previous analytical
models and numerical simulations. In particular, we incorporated for
the first time the Lyman-Werner feedback on star formation, calculated
self-consistently including the effect of the supersonic streaming
velocity. A three-dimensional calculation of the Lyman-Werner and
X-ray backgrounds allowed us to calculate the heating history of the
gas and the resulting 21-cm intensity maps.

We have focused on the negative LW feedback, which begins at $z \sim
30-40$ but strengthens very gradually, passing its central point at
$z_\LW \sim 19-24$ and saturating only at $z \sim 10$. The heating
transition is centered at $z_0 \sim 15$ (including a delay of $\Delta
z \sim 1.5-2$ due to the feedback), when the LW transition is well
advanced but still far from saturated. The large-scale 21-cm power
spectrum is potentially observable over a broad redshift range of $z
\sim 10-22$ or 23. The best prospects are at $z \sim 18$, when the
large-scale peak reaches a signal-to-noise ratio (for a projected
first-generation radio telescope array) of 3.2 (for our weak feedback
case) or 3.9 (for strong feedback). At this redshift, the BAO
signature (which marks the velocity effect and the presence of $10^6
M_\odot$ halos) should also be observable with a S/N$\sim 2$. The BAOs
should be observable over a broad redshift range of $\Delta z \sim
7.5$~.

These numbers are obtained with our standard set of expected
astrophysical parameters, but they may shift around a bit depending on
the precise properties of the early stars and their remnants. We hope
these findings will stimulate additional numerical simulations of the
complex radiative feedback at $z \sim 10-30$, as well as future
observational efforts in 21-cm cosmology directed at the epoch prior
to reionization.

\section{Acknowledgments}
We are most grateful to Zoltan Haiman for supplying us with the atomic
data on the Lyman-Werner absorption lines. This work was supported by
Israel Science Foundation grant 823/09. A.F.\ was also supported by
European Research Council grant 203247.


\section{Introduction}

In neutral, homogeneous and isotropic fluids, the turbulent fluctuations are unpredictable, but their statistics are predictable and universal \citep{frisch95}; the  turbulent spectra follow the power-law $\sim k^{-5/3}$ for any local conditions ($k$ being the wave number). This empirical result was explained by  Kolmogorov \citep{Kolmogorov1941} assuming self similarity of the turbulent  fluctuations between the energy injection scale and the dissipation one $\ell_d$.

In the magnetized solar wind, collisions are very rare (the mean free path is of the order of 1~AU); the dissipation process at work and the dissipation length are not known  precisely. Moreover, in a magnetized plasma, it is difficult to imagine self-similarity over all scales where turbulent fluctuations are observed, since there exist several spatial and temporal characteristic scales, such as the ion Larmor radius $\rho_{i}=\sqrt{2kT_{i\perp}/m_i}/(2\pi f_{ci})$,  the ion inertia length $\lambda_{i}=c/\omega_{pi}$,  the corresponding electron scales $\rho_{e}, \lambda_{e}$, and the ion and electron cyclotron frequencies $f_{ci}$, $f_{ce}$. At these scales,  the dominant physical processes change,  which affects the scaling of the energy transfer time and furthermore the energy transfer rate,  leading to spectral shape changes.

The first clear spectral change appears at  ion scales.  At 1~AU, the ion scales are nearly equal, $\lambda_i\simeq \rho_i \simeq V/2\pi f_{ci}$,   so it is difficult to determine which of these scales is responsible for the ion break. Independent measurements at different distances from the Sun,  between 0.3 AU to 0.9 AU \citep{bourouaine12}, and a statistical study at 1~AU  \citep{Leamon2000}  indicate that the spectral break is related to the ion inertia length $\lambda_i$.  Nearly incompressible magnetic fluctuations cascading from the inertial range may undergo kinetic  effects in the vicinity of the ion scales. At these scales, ion temperature anisotropy instabilities occur \citep{gary01}  and can remove or inject energy in the turbulent cascade.    However, for most of the solar wind observations, the plasma is stable  \citep{matteini07,matteini11,bale09}. The energy re-distribution among the fluid and kinetic degrees of freedom  in the vicinity of ion scales is still a matter of debate and is probably at the origin of the spectral variations observed between 0.3  and 3~Hz in the satellite frame: the spectral index here varies between $-4$ and $-2$  \citep{Leamon1998,smith06,sahraoui10}.  This spectral range is usually attributed to the ion dissipation range \citep{Leamon1998,Leamon1999,Leamon2000,smith12}  or to another fluid cascade, which may continue down to electron scales \citep{Biskamp96,Stawicki2001,Li2001,Galtier2003b,GaltierBuchlin2007ApJ}.

Between ion and electron scales,  the fluctuations of the electron fluid form a small scale inertial range \citep{alexandr07,alexandr08apj}, or, following the nomenclature of \citet{smith12}, an electron inertial range. Here, indeed, a reproducible spectrum $\propto k_{\perp}^{-2.8}$ is observed \citep{alexandr09,chen10,sahraoui10}. Approaching electron scales, one may expect to observe an electron dissipation range, as was suggested by  \citet{alexandr08npg}. At such small scales,  there are only a few observations \citep{alexandr09,sahraoui10} and  the descriptions are different. Larger statistical studies are needed to establish more firmly the properties of turbulent spectra at electron scales.

In this paper  we present a large statistical study of magnetic spectra starting at ion scales and going beyond electron spatial scales. We use data from the STAFF instrument~\citep{cornilleau97} on the Cluster mission~\citep{Escoubet1997}, which is able to measure such a range of scales.  In a previous study, \citet{alexandr09} described   the electron inertial and the electron dissipation ranges separately:   a power-law $\sim k^{-2.8}$ for the inertial range  and a curved spectrum $\propto\exp(-\sqrt{k/k_0})$ for the dissipation range. This model is rather complicated and has a large number of free parameters. In the present study, we propose  a single algebraic description for both ranges,  namely, an exponential with a power-law pre-factor: $E(k_{\perp})=Ak_{\perp}^{-\alpha}\exp(-k_{\perp}\ell_d)$. We find that this model describes well the totality of the observed spectra  at scales smaller than $\lambda_i$ and $\rho_i$  and that its cut-off scale $\ell_d$ correlates with~$\rho_e$. The power-law exponent $\alpha$ is found to be close to $-8/3$. This model  (henceforth called ``the  {\it exp}--model")  has only one free parameter, the amplitude of the spectrum.

Previous authors \citep{sahraoui10} have used a double power-law model with a break to fit the observations  in the electron inertial and dissipation ranges. We  have applied  this model as well to our data,  and we find that the first power-law exponent is consistent with the previous studies \citep{alexandr09,chen10} while the second exponent varies a lot. Despite the fact that the double power-law model has  more free parameters than the exponential model used here,  we find that it describes only 30\% of the observed spectra and that the associated break scale does not present any clear correlation with an electron characteristic scale.

\section{Observations}

\begin{figure}
\includegraphics[width=8.0cm]{fig1_proofs_10x8cm.eps}
\caption{Frequency spectra with a signal to noise ratio  greater than 3 measured by Cluster--1/STAFF in the free solar wind (for 27 intervals randomly chosen among 100). The dashed line shows the instrument noise level. The vertical dotted line corresponds to $f=3$~Hz.  The legend indicates the variations of some solar wind parameters for the studied data set of 100 spectra:  speed $V$, magnetic field $B$,  temperature ratio and the ion and electron plasma $\beta$. }
\label{fig:spec102}
\end{figure}
\begin{figure}
\includegraphics[width=9.0cm]{fig2.eps}
\caption{ Fit of the most intense spectrum of Figure~1 with the {\it exp}--model. The spectrum was measured by Cluster-1/STAFF on 22/01/2004.   Green crosses represent the SC measurements, red stars show the raw SA measurements without correction of the 1st three underestimated points, visible here around $0.1$km$^{-1}$. Diamonds indicate the STAFF-SA noise level. The blue arrows indicate inverse ion and electron Larmor radii, the black ones  correspond to the inertial lengths. The solid line gives the exp-model $Ak^{-8/3}\exp(- k\rho_e)$.  }
\label{fig:spec66}
\end{figure}

For our statistical study we select homogeneous intervals of 10 minutes (long enough to study kinetic scales)  within the five years interval (2001--2005) of Cluster. We eliminate  time intervals during which Cluster is magnetically connected to the bow-shock by using electrostatic wave spectrograms, which show clearly waves typical of the electron foreshock  \citep{etcheto84,lacombe85}, and by using the shock model described by \citet{Filbert1979}.   For small angles $\Theta_{BV} $ between the interplanetary magnetic field ${\bf B}$ and the solar wind velocity ${\bf V}$, Cluster is connected to the shock. Thus, our data set only contains intervals for which the angle $\Theta_{BV} > 60^{\circ}$. If the turbulent  fluctuations have a phase speed $V_{\phi}\ll V$,  Cluster detects  by Doppler shift the fluctuations with ${\bf k\| V}$. As ${\bf B}$ and ${\bf V}$ are quasi-perpendicular,  Cluster measures fluctuations with ${\bf k\perp B}$. We apply the Taylor hypothesis to get the wave-number from the frequency, $k_{\perp}=2\pi f/V$. However, about $\sim 10\%$ of the pre-selected intervals show the presence of   right hand polarised whistlers in quasi-parallel propagation. For these waves  the Taylor hypothesis is not applicable, because $V_{\phi} > V$. We  discard these intervals in the present study. This data selection process gives us 100~intervals.  Within this statistical sample, the plasma conditions vary as usually in the solar wind in fast and slow streams at 1~AU (see the legend of Figure~\ref{fig:spec102}).

Figure~\ref{fig:spec102} shows the total Power Spectral Density (PSD) of magnetic fluctuations, for  27 intervals randomly chosen among 100, as a function of frequency in the spacecraft frame $P(f)$.
These spectra are measured by the STAFF Search Coil sensor and analyzed onboard by the magnetic waveform unit (hereafter called SC) at $f\in[0.5,9]$~Hz  and by the Spectrum Analyser (hereafter called SA) at $f\geq 8$~Hz.

The spectra are analyzed only for the frequencies where the Signal to Noise Ratio (SNR) is larger than~3. The spectral parts below this threshold are not shown to avoid any  erroneous interpretation. As one can see from Figure~\ref{fig:spec102}, this instrumental noise limit allows us to use data up to $30-400$~Hz, depending on the turbulence intensity (i.e., for the most intense spectrum, we have valid observations up to $400$~Hz).  The analyzed range of frequencies corresponds to $f\in ]f_{ci},f_{ce}]$.

A poor calibration of the first 3 frequencies of SA (at 8, 11 and 14~Hz) [Y. de Conchy and N. Cornilleau, private communication, 2011], was corrected by an interpolation of these points between the highest SC frequency and the 4th point of the SA spectra. The linear interpolation between $\log_{10} P(f)$ and $\log_{10} f$ is possible as far as the spectra follow a power-law at these frequencies.  An example of a raw spectrum without the correction can be found in Figure~\ref{fig:spec66}.

\section{Algebraic description of turbulent spectra at scales smaller than $\rho_i$ and $\lambda_i$.}
\subsection{Exponential model}

Here we propose a model to describe the whole turbulent spectrum  at scales smaller than $\rho_i$ and $\lambda_i$ and down to a fraction of the electron scales with the  smaller  possible number of  parameters, namely an exponential  with a characteristic scale $\ell_d$ and with a power-law pre-factor
\begin{equation}\label{eq:exp}
E(k_{\perp})=A k_{\perp}^{-\alpha}\exp(- k_{\perp}\ell_d).
\end{equation}
This {\it exp}--model has three free parameters: the amplitude $A$, the spectral index $\alpha$ and the cut-off or "dissipation" scale~$\ell_d$.

We start by fitting the model (\ref{eq:exp}) to the 100 observed spectra (with a signal to noise ratio $>3$, as explained in section~2) for $k_{\perp}$ corresponding to $f>3$~Hz (see vertical dotted line in Figure~\ref{fig:spec102}), assuming that the three parameters have independent variations.

Figure~\ref{fig:spec66} gives  the fit with the most intense spectrum of Figure~\ref{fig:spec102} as a function of the wave-number $P(k_{\perp})=P(f)V/2\pi$, which is determined using the Taylor hypothesis and the energy conservation law $\int P(k_{\perp})dk_{\perp} = \int P(f)df$. Green crosses show the Morlet wavelet spectrum \citep{Torrence1998} of STAFF-SC measurements. Red stars display the STAFF-SA data for the same time period. (In this plot we keep the 3 first poorly calibrated data points, one can see them around $k=0.1$~km$^{-1}$ and compare with the result of the interpolation in Figure~1).  The error bars are estimated from the variance  over 10 minutes of the PSD at each frequency \citep{alexandr10}. This spectrum is valid up to $\simeq 400 Hz$, which gives us the maximum wave-vector $k\sim 4$~km$^{-1}$ (while $1/\rho_e \simeq 1$~km$^{-1}$).  This is the smallest scale ever measured with a good sensitivity at 1~AU in  the solar wind. The exp-model (\ref{eq:exp}) fitting is shown by the black solid line.  The parameters of the fit in this case are $\alpha = 2.70\pm 0.15$ and $\ell_d = (0.90 \pm 0.25)$~km, while $\rho_e=(0.95\pm 0.05)$~km.

\begin{figure}[htb]
\includegraphics[width=8.4cm]{fig3proofs.eps
\caption{ Results of the fitting with the {\it exp}--model for the 100 observed spectra: (a) histograms on the spectral index $\alpha$; (b) cut-off scale $\ell_d$ as a function of $\alpha$; (c) $\ell_d$ as a function of the electron Larmor radius $\rho_e$ and (d) $\ell_d$ as a function of the electron inertial length $\lambda_e$.}
\label{fig:fitexp}
\end{figure}

Figure~\ref{fig:fitexp} summarizes the results of the fitting for the 100 spectra. Panel (a) shows the histogram of the spectral index, $\alpha=2.63 \pm 0.15$,   the error being the standard deviation of the mean. Note that $\langle\alpha\rangle \simeq 8/3$.
 It appears that the variations of $\alpha$ and $\ell_d$ are not independent since  the dispersion in $\alpha$ is due to the  variations of the cut-off scale $\ell_d$ as observed in Figure~\ref{fig:fitexp}b.  A linear fit gives $\ell_d {\text (km)}=12.9-4.4\alpha$, i.e.
\begin{equation}\label{alpha-Ld}
\alpha=2.9-\ell_d/4.4
\end{equation}
i.e.  if $\ell_d$  was small, $\alpha$ would be approximatively equal to $2.9$, a value close to the one found by \citet{alexandr09}.

On the other hand, the variations of $\ell_d$ are related to the variations of  the electron Larmor radius, $\ell_d\sim 1.35 \rho_e$, as shown in  Figure~\ref{fig:fitexp}c, with a relatively high  correlation coefficient of $0.70$. Figure~\ref{fig:fitexp}d shows a positive but much weaker correlation of $0.34$ between the dissipation scale $\ell_d$ and the electron inertia length   $\lambda_e$.

The  results presented in Figure~\ref{fig:fitexp} suggest that within  the framework of  the exponential model there is  only one free parameter, the amplitude of the turbulent spectra, $A$, and the observed spectra can be described approximately by
\begin{equation}
E(k_{\perp}) \simeq Ak_{\perp}^{-8/3}\exp(-k_{\perp}\rho_e).
\end{equation}
\begin{figure}[htb]
\includegraphics[width=8.6cm]{fig4_proofs_diam.eps}
\caption{Upper panel: 107 superposed solar wind spectra, the 100 spectra in the kinetic range analysed in this paper and the 7 spectra covering fluid and kinetic scales from  \citep{alexandr09}, the constant being $E_0\in[0.4,95]$. All spectra have the signal to noise ratio greater than 3, as in Figure~\ref{fig:spec102}.  The blue line indicates $\propto k_{\perp}^{-5/3}$ and the red line shows $\propto k_{\perp}^{-8/3}\exp(-k_{\perp}\rho_e)$, that represents well all the spectra. Bottom panel:  the diamonds show 100 compensated spectra $E(k_{\perp}\rho_e)\cdot F(k_{\perp}\rho_e)$, with $F(k_{\perp}\rho_e)=(k_{\perp}\rho_e)^{8/3}\exp(k_{\perp}\rho_e)$, for $k_\bot \rho_e \geq 0.03$. The solid line denotes the  most intense compensated  spectrum.}
\label{fig:5}
\end{figure}
We verify this point in Figure~\ref{fig:5}, where we superpose  the 100 spectra analyzed here with the 7 spectra  covering fluid and kinetic scales from  \citep{alexandr09}.  The spectra are shifted vertically by a parameter $E_0$ (a relative spectral level), in the same way as in Fig.~2 of \citep{alexandr09}. The superposition of the 107 spectra is nice, which indicates the    generality of the turbulent spectrum $E(k_{\perp})$ in the solar wind: it follows $\propto k_{\perp}^{-5/3}$ at MHD scales and $\propto k_{\perp}^{-8/3}\exp(-k_{\perp}\rho_e)$ at scales smaller than the ion kinetic scales $\lambda_i$ and $\rho_i$ (i.e. $k\rho_e \geq 0.03$).  The bottom panel  shows the 100  spectra $E(k_{\perp}\rho_e)$ compensated by a function $F= (k_{\perp}\rho_e)^{8/3}\exp(k_{\perp}\rho_e)$  for $k\rho_e\geq 0.03$:  the resulting spectra are flat, indicating that the  {\it exp}--model with one free parameter describes well all the turbulent spectra in the solar wind at these scales and is valid for nearly 2 decades in wave numbers.  Note that a damping length $\ell_d$ variation of more than  20\% with respect to the mean $\rho_e$ values results in a strong departure of the compensated spectra.

The amplitude $A$  of the turbulent spectra (related to the  parameter $E_0$) is  found to be correlated with the ion thermal pressures $nkT_i$, as within the MHD range of turbulence \citep{Grappin90}, and with the ion temperature anisotropy $T_{i\perp}/T_{i\|}$ (not shown).  Other plasma parameters seem to be less important, but still it is impossible to exclude completely the influence  of the magnetic and kinetic energies in the solar wind (paper in preparation).

\subsection{Break model}

Is there another simple model which represents well the observations with a small number of free parameters?  Let us compare the turbulent spectra  within the electron inertial and dissipation ranges (i.e. scales smaller than $\rho_i$ and $\lambda_i$) with  the  double power-law or {\it break}-model
\begin{equation}\label{eq:break}
\tilde{E}(k_{\perp})=A_1k_{\perp}^{-\alpha_1}(1-H(k_{\perp}-k_b)) + A_2k_{\perp}^{-\alpha_2}H(k_{\perp}-k_b),
\end{equation}
$H(k_{\perp}-k_b)$ being the Heaviside function, $k_b$ the wave number of the break scale $\ell_b=1/k_b$, $A_{1,2}$ the amplitudes of the two power-law functions with spectral indices $\alpha_{1,2}$ on both sides of $k_b$. This model has five free parameters.  Note that equation (\ref{eq:break}) is not differentiable for $k=k_b$. Near $k_b$ the  turbulent level  has to be the same on both sides. Thus, $k_b$ can be determined by the four other parameters of the model
\begin{equation}\label{eq:break-1}
\log_{10}k_b=\frac{\log_{10}(A_1/A_2)}{\alpha_1-\alpha_2} .
\end{equation}
Iterations to find the model parameters with condition (\ref{eq:break-1})  converge to different results, depending on the initial $k_b$. Therefore, minimising the error of the fit over $k_b$ is needed, so that finally the model has still five free parameters.

We  apply this {\it break}--model to the 100 observed spectra within the same $k_{\perp}$--range as was done for the {\it exp}--model.  Despite the fact that the {\it break}--model has more free parameters than the exponential model, we find that it  can be  applied only to 30 spectra;  for the 70 other spectra there is no solution with condition (\ref{eq:break-1}) verified,  indicating the absence of a clear break point, or, for a part of these spectra, not enough data points to isolate the second power-law.

 \begin{figure}[htb]
\includegraphics[width=8.4cm]{fig5proofs.eps}
\caption{\label{fig:break-model-res} same format as Figure~\ref{fig:fitexp}, but for the parameters of the {\it break}--model  determined from the fitting with 30 spectra, i.e. (a) histograms on the spectral indices; (b) break scale $\ell_b$ as a function of $\alpha_1$ (open circles) and $\alpha_2$   (filled circles); (c) $\ell_b$ as a function of the electron Larmor radius $\rho_e$ and (d) $\ell_b$ as a function of the electron inertial length $\lambda_e$.}
\end{figure}
Figure~\ref{fig:break-model-res} summarizes the  parameters of the  {\it break}--model determined from the fitting of the 30 spectra. For these spectra, the condition (\ref{eq:break-1}) is verified. Panel (a) shows histograms of the spectral indices: the mean values are  $\alpha_1 = 2.86\pm 0.08$, $\alpha_2 = 3.91 \pm 0.29$. Note the narrow dispersion of the spectral index $\alpha_1$. It is close to $\alpha$, when $\ell_d$ is negligible (see eq.~(\ref{alpha-Ld}))  so confirming the spectrum between ion and electron scales found by \citet{alexandr09}. So we can fix one of the parameters of the model. The second exponent $\alpha_2$ has a large dispersion, not found to be controlled by any plasma parameter.  The values of the spectral indices are correlated to the position of the break scale $\ell_b$ (see panel b).  Figure~\ref{fig:break-model-res}c shows $\ell_b$ as a function of $\rho_e$. No correlation is observed. Figure~\ref{fig:break-model-res}d shows $\ell_b$ as a function of $\lambda_e$: the correlation is positive but weak ($\simeq0.43$).  From the comparison of the observed turbulence spectra at plasma kinetic scales with the {\it break}--model,  one may conclude that this model has one fixed parameter, another is fixed by the condition (\ref{eq:break-1}), the other 3 parameters are free, not found to be determined by plasma parameters.

\begin{figure}[htb]
\includegraphics[width=8.6cm]{fig6_proofs.eps
\caption{Example of an observed spectrum (lower black line)  compared with $E(k)\sim k^{-2.7}\exp(-k\ell_d)$ (red dashed line), with $\ell_d^{-1}$  marked by a red diamond; the error of the fit is $\triangle_{exp}^2=1.8\times 10^{-3}$~(nT$^4$km$^2$).  The same spectrum, shifted by a factor $10^2$ and compared with  $E(k)\sim k^{-2.8}$ below  the break shown by  a blue diamond, and with $\sim k^{-4.0}$ above it (blue dashed line); here $\triangle_{break}^2=1.9\times 10^{-3}$~(nT$^4$km$^2$). Ion and electron characteristic scales are marked by arrows; a vertical dotted line indicates the beginning of the $k$--domain where the fits are performed. Insert: histograms of the fitting errors, $\triangle_{exp}^2$ (red, for 100 spectra) and $\triangle_{break}^2$ (blue, for 30 spectra).
}
\label{fig:event67}
\end{figure}
It is important to note that the errors of the fit of the {\it break} and {\it exp}--models are   of the same order.  Figure~\ref{fig:event67} shows an example of a solar wind spectrum (black lines) fitted with both models,   with the least mean square distance  between the observed spectra and the fit, $\triangle^2$, given in the caption; the insert shows histograms of $\triangle^2_{exp}$  (red) for 100 spectra and $\triangle^2_{break}$ (blue) for 30 spectra: the same mean values of the histograms are observed. This leaves us free to choose the model, based on other criteria than goodness of the fit, namely, on the number of degrees of freedom of the model and on the number of described cases.

\section{Discussion and Conclusion}

The {\it exp}--model $E(k_{\perp}) = A k_{\perp}^{-8/3}\exp(-k_{\perp}\rho_e)$ proposed in this study provides a single algebraic description of the solar wind spectrum at scales smaller than the ion characteristic scales, $\lambda_i$ and $\rho_i$, and going beyond the electron scales (i.e. within the electron inertial and dissipation ranges).  This model describes well the totality of the observed spectra and has only one free parameter -- the amplitude $A$ of the spectrum. The amplitude seems to be a function of the ion thermal pressure and the ion temperature anisotropy in the solar wind. However, it is difficult to exclude the role of the magnetic and kinetic energies: more work is needed to determine the exact relationship between the amplitude of the turbulent spectrum and the energy budget in the solar wind.

 The spectral index close to $-8/3$ observed in the solar wind at scales smaller than ion characteristic scales is in agreement with quasi-bidimentional strong Electron MHD turbulence ($k_{\perp} \gg k_{\|}$) when parallel cascade is weak  \citep{galtier05}. Recently, the same spectral index was found as well in strong kinetic Alfv\'en turbulence  \citep{boldyrev12}.

In usual fluid turbulence, the far dissipation range is described by $E(k) \sim k^{3}\exp(-c k\ell_d)$ (with $c\simeq 7$) \citep{chen93}.
This is due to the resistive damping rate $\propto k^2$ valid in a collisional fluid, which gives an exponential spectral tail.
In  the collisionless plasma of the solar wind there is no resistive damping, and thus this  coincidence deserves an explanation.

\citet{howes11pop} consider a model (``weakened cascade model'') which includes the nonlinear transfer of energy from large to small scales in Fourier space
and the damping of kinetic Alfv\'en waves (KAW's).
For completeness, we discuss now this model in some detail. The model reads for the magnetic energy $b^2$ at scale $k$:
\begin{equation}\label{eq:transf}
\partial_t b^2_k = - k_\bot \partial_k \epsilon - 2 \gamma b^2_k + S
\label{mou}
\end{equation}
with $\epsilon$ being the magnetic energy transfer rate, $S$ being the source term.
 The damping term $2 \gamma b^2_k$ is obtained by linearizing the Vlasov-Maxwell equations in the gyrokinetic limit ($k_\| \ll k_\bot$, with frequencies $f \ll f_{ci}$). In the limit $k \rho_i \gg 1$, this gives
 \begin{equation} \label{eq:gamma}
 \gamma \simeq k_\parallel V_a (k_\bot \rho_i)^2,
 \end{equation}
 see eq.~(63) in \citep{howes06}.
To complete the model, we must write down the expression for the magnetic energy flux $\epsilon$. This is given by
\begin{equation}
\epsilon = b^2 / \tau = k_\bot u b^2
\label{epsilon}
\end{equation}
where $\tau = 1/k_\bot u$ is the nonlinear time  and $u$ the  velocity fluctuation.  At MHD scales $k_\bot \rho_i \ll 1$, we have for Alfv\'en waves $u \simeq b$, but at small scales $k_\bot \rho_i \ge 1$, we have for KAW $u /b \simeq k_\bot \rho_i$, see eq.~(3) in \citep{howes11pop}. Using these expressions in eq.~(\ref{mou}) and (\ref{epsilon}), one obtains scaling laws for the magnetic energy spectrum as stationary solutions of the transfer equation (\ref{eq:transf}), neglecting the damping and source terms.
The spectral laws are respectively $E_k \propto k_\bot^{-5/3}$ at large scales and $E_k \propto k_\bot^{-7/3}$
between ion and electron scales. When taking into account the damping term, \citet{howes11pop} obtain numerically the same spectral laws, with a final curved tail at scales smaller than electron scales. Superficially, this spectrum thus resembles the analytic form which we have found to be valid to describe  the solar wind turbulence.

We last remark that the damping term in the model  of \citet{howes11pop}  for $k_\bot \rho_i \gg 1$ is (see eq. (7)) of the form
$\gamma \propto k_\| k_\bot^2$.
Taking into account the  assumption of critical balance $\tau=\tau_A$ (i.e. $k_{\perp}u=k_\| V_A$) \citep{Goldreich1995}, with $\tau_A$ the Alfv\'en time and $V_A$  the Alfv\'en speed, and the spectral index $-7/3$ (i.e. $u\sim k_\bot^{-2/3}$),  one gets $k_\| \propto k_\bot^{1/3}$. Therefore, the damping term takes the form  $\gamma \propto  k_\bot^{2+1/3}$.  The exponent of the damping rate is thus very close to the $k^2$ scaling of the Laplacian viscous term, which is known to lead in hydrodynamical turbulence to an exponential tail in the dissipation range.

 This model does not take into account the cyclotron damping. So, while the proposed phenomenology may explain the exponential tail of the $k_{\perp}$--spectrum studied here, it cannot describe more isotropic  wave vectors, which might be present as well in the solar wind.  It is possible that the dissipation mechanism could also be due to electron-cyclotron absorption of oblique short-wavelength whistler waves, or even of lower-hybrid waves. More observations under different field-to-flow angles $\Theta_{BV}$ are needed within the electron inertial and dissipation ranges to address this point.

To build a realistic model of the dissipation in the solar wind we need to resolve still an open question on the nature of the turbulent fluctuations. Some authors argue that the electron inertial range is a whistler mode turbulence \citep{saito10,narita-gary10}, others suggest KAW turbulence \citep{bale05,Schekochihin2009ApJS,sahraoui10,salem12}, or a combination of both types of linear waves \citep{podesta-gary11}. The model of \citet{howes11pop}, described above is based on KAW turbulence as well.  However, it is still not clear whether we can describe turbulence in the solar wind as a mixture of linear waves (weak turbulence) which will dissipate homogeneously in space (or in the plane perpendicular to ${\bf B}$), or if it is a strong turbulence with dissipation restricted to intermittent coherent structures.  What is the topology of these structures -- current sheets, shocks or coherent vortices?

In the present study we have limited ourselves to observations of the spectral shape in the electron inertial and dissipation ranges.  Our results give observational constraints for future theoretical models.