∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail<EMAIL_ADDRESS>e4e-mail<EMAIL_ADDRESS>
11institutetext: Department of Physics, Cooch Behar Panchanan Barma
University, Cooch Behar 736101, West Bengal, India. 22institutetext:
Department of Physics, Malda College, Malda 732101, West Bengal, India.

# Anisotropic generalization of Buchdahl bound for specific stellar models

Ranjan Sharma e1,addr1  Arpita Ghosh e2,addr1  Soumik Bhattacharya e3,addr1
Shyam Das e4,addr2

(Received: date / Accepted: date)

###### Abstract

Anisotropy is one factor that appears to be significantly important in the
studies of relativistic compact stars. In this paper, we make a generalization
of the Buchdahl limit by incorporating an anisotropic effect for a selected
class of exact solutions describing anisotropic stellar objects. In the
isotropic case of a homogeneous distribution, we regain the Buchdahl limit
$2M/R\leq 8/9$. Our investigation shows a direct link between the maximum
allowed compactness and pressure anisotropy vi-a-vis geometry of the
associated $3$-space.

###### Keywords:

Compact star Buchdahl limit Exact solution Anisotropic pressure.

††journal: Eur. Phys. J. C

##  1  Introduction

What is the maximum permissible compactness of a self-gravitating compact
stellar object? This question remains intriguing in the context of the
stability of a stellar configuration, in particular. Buchdahl theorem buch59
addresses the question which states that no uniform density stars with radii
smaller than $9/8M$ can exist. Otherwise, the central pressure diverges. In
other words, for a stellar configuration in equilibrium, the Buchdahl bound
implies $2M/R<8/9$. The interior of the star in this derivation is assumed to
be a homogeneous distribution of perfect fluid whose exterior region is
described by the Schwarzschild solution. Buchdahl bound prescribes an absolute
constraint of the maximum allowed mass ($M$) to radius ($R$) ratio of a
uniform density star as well as stellar configurations with radially
decreasing energy density. Later on, by considering a variety of matter
distributions in different background spacetimes, numerous studies have been
made to determine the upper bound on the compactness of a star (for a ready
reference, see dadh20  and references therein). Of particular interest is the
recent work of Dadhich dadh20  where it has been shown that the Buchdahl
compactness can be determined without any reference to the interior matter
distribution. The upper limit in this calculation gets determined solely in
reference to the Schwarzschild exterior solution for a neutral star and
exterior Reissner-Nordström solution for a charged sphere. The probe also
prescribes an upper bound on the charge to mass ratio $Q^{2}/M^{2}\leq 9/8$.
In a recent article, by developing a model for charged star which was shown to
be a generalization of the uniform density Schwarzschild interior solution,
Sharma et alsharma2021  regained the charged analogue of the Buchdahl
compactness bound

$\frac{M}{R}=\frac{8/9}{(1+\sqrt{1-\frac{8\alpha^{2}}{9}})},$

where $\alpha^{2}=Q^{2}/M^{2}$. Obviously, the bound reduces to $2M/R<8/9$ in
the uncharged case.

While numerous charged analogue of the Buchdahl bound is available in the
literature (see, for example, giul08 ; mak01 ; boh07 ), the study of the
maximum mass-radius ratio for compact stars has been mostly confined to
isotropic fluid spheres in which the tangential pressure ($p_{t}$) equals the
radial pressure ($p_{r}$). However, theoretical investigations show that
anisotropic stresses might exist beyond a certain density range at the
interior of compact stars rude72 ; bow74 ; herr97 . In a recent article, it
has been argued that pressure anisotropy cannot be ignored in the studies of
relativistic compact stars as it is expected to develop quite naturally by the
physical processes inside such ultra-compact stars Herrera2020 . Earlier,
Heintzmann and Hillebrandt hein75  claimed that no limiting mass could exist
for a neutron star with an anisotropic matter distribution. However,
considering an anisotropic distribution of matter, Guven and Murchadha guve99
determined an upper limit on $M/R$. The investigation showed that for a
stellar configuration with monotonically decreasing density profile and having
$p_{r}\geq p_{t}$, the upper bound $\frac{M}{R}\leq\frac{8}{9}$ holds if the
gravitational mass is replaced by the quasi-local mass. However, the bound can
not be recovered for configurations with $p_{t}\geq p_{r}$. Mak et al mak02
developed an anisotropic stellar model by assuming the anisotropic stress to
be proportional to $r^{2}$ and analyzed the subsequent bound.

The objective of the current investigation is to obtain an anisotropic
generalization of the Buchdahl bound. We plan to obtain the bound for
particular class of solutions describing the interior of static spherically
symmetric anisotropic stars. In our calculation, we choose two realistic
stellar models developed by Maurya et al maur16  and Das et al das20 ) so as
to get an insight into the effects of anisotropy on the maximum bound of a
star. Our choice of the particular class of solutions is motivated by the fact
that both the solutions have their respective isotropic limits which enables
one to regain the Buchdahl bound for a constant density star.

The paper is organized as follows: In sec. 2, we outline the techniques
adopted to generate the stellar models used in our analysis. In sec. 3, we
make use of the models to find anisotropic generalizations of the Buchdahl
bound. In sec. 4, concludes by highlighting the main results of our
investigation. We also outline the scope of a further probe in this context.

##  2  Particular class of stellar solutions

To develop the model of a static spherically symmetric star with anisotropic
matter distribution, the line element describing the interior of the star is
assumed to be of the form

$ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}),$
(1)

where, $\nu(r)$ and $\lambda(r)$ are the unknown gravitational potentials. The
energy-momentum tensor for the anisotropic fluid distribution is assumed in
the form

$T_{ab}=diag(-\rho,~{}p_{r},~{}p_{t},~{}p_{t}),$ (2)

where $\rho$, $p_{r}$ and $p_{t}$ are the energy density, radial and
tangential pressure, respectively. The comoving fluid velocity is given by
$\displaystyle u^{i}=e^{-\nu/2}\delta^{i}_{0}$. The Einstein field equations
for the line element (1) and the energy-momentum tensor (2) are then obtained
as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}(1-e^{-\lambda})+\frac{\lambda^{\prime}}{r}e^{-\lambda},$
(3) $\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle-\frac{1}{r^{2}}(1-e^{-\lambda})+\frac{\nu^{\prime}}{r}e^{-\lambda},$
(4) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{e^{-\lambda}}{4}\Big{(}2\nu^{\prime\prime}+{\nu^{\prime}}^{2}-\nu^{\prime}\lambda^{\prime}+\frac{2\nu^{\prime}}{r}-\frac{2\lambda^{\prime}}{r}\Big{)},$
(5)

where a $(^{\prime})$ denotes differentiation with respect to $r$.
$\Delta=p_{t}-p_{r}$ denotes anisotropy in this construction. To close the
system of equations (3) - (5), following techniques are adopted:

###  2.1  Solution developed by Das et al das20

To solve the system, one introduces the Vaidya and Tikekar vaid82  metric
ansatz

$e^{\lambda}=\frac{1-K(\frac{r^{2}}{L^{2}})}{1-\frac{r^{2}}{L^{2}}},$ (6)

where $K$ and $L$ are constant curvature parameters. The Vaidya and Tikekar
metric ansatz is motivated by the observation that the geometry of the $t=$
constant hypersurface of the associated spacetime when embedded in a
$4$-dimensional Euclidean space is not spherical but spheroidal in nature. The
parameter $K$ in (6) denotes the departure from the sphericity of associated
$3$-space. The anstaz has found huge application for the development and
studies of relativistic stellar models like neutron stars. Earlier, making use
of the Vaidya and Tikekar metric ansatz, Sharma et al shar06  obtained the
maximum permissible mass and radius of a relativistic compact star for a given
surface density. An upper bound on mass to radius ratio for an anisotropic
stellar model was obtained by Maurya et al maur19  where the Buchdahl-Vaidya-
Tikekar metric ansatz was utilized.

Since anisotropy provides an additional degrees of freedom in this
construction, in addition to the Vaidya and Tikekar ansatz, Das et al das20
also utilized the Karmarkar’s embedding condition of class-I kar48  to
determine the unknown metric potential $e^{\nu(r)}$. The procedure yields

$e^{\nu}=\Big{[}C+D\sqrt{(K-1)(r^{2}-L^{2})}\Big{]}^{2}.$ (7)

where $C$ and $D$ are constants to be determined from the boundary conditions.
Subsequently, physical quantities are obtained as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{(K-1)(Kr^{2}-3L^{2})}{(L^{2}-Kr^{2})^{2}},$ (8)
$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{(K-1)\Big{[}D(K-3)\sqrt{r^{2}-L^{2}}+C\sqrt{K-1}\Big{]}}{(L^{2}-Kr^{2})\Big{[}D(K-1)\sqrt{r^{2}-L^{2}}+C\sqrt{K-1}\Big{]}},$
(9) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{(K-1)}{(L^{2}-Kr^{2})^{2}\Big{[}D(K-1)\sqrt{r^{2}-L^{2}}+C\sqrt{K-1}\Big{]}}$
(10) $\displaystyle+$
$\displaystyle\Big{[}D(K-3)L^{2}\sqrt{r^{2}-L^{2}}+DKr^{2}\sqrt{r^{2}-L^{2}}$
$\displaystyle+$ $\displaystyle CL^{2}\sqrt{K-1}\Big{]},$
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{(K-1)Kr^{2}\Big{[}D(K-2)\sqrt{r^{2}-L^{2}}+C\sqrt{K-1}\Big{]}}{(L^{2}-Kr^{2})^{2}\Big{[}D(K-1)\sqrt{r^{2}-L^{2}}+C\sqrt{K-1}\Big{]}}.$
(11)

The mass contained within a radius $r$ is obtained as

$m(r)=\frac{(K-1)r^{3}}{2(L^{2}-Kr^{2})}.$ (12)

Eq. (11) shows that that while $K$ denotes deviation from sphericity of the
associated $3$-space, the parameter also turns out to be a measure of
anisotropy in this formulation. Most importantly, the $K=0$ case suggests
spherical homogeneous distribution with zero anisotropy and in that case the
solution reduces to

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\Big{[}C-DL\sqrt{1-\frac{r^{2}}{L^{2}}}\Big{]}^{2}dt^{2}+\frac{1}{1-\frac{r^{2}}{L^{2}}}dr^{2}$
(13) $\displaystyle r^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}),$

which is the Schwarzschild interior solution for an incompressible fluid.

For a given $K$, the constants of the model, namely $C$, $D$ and $L$ do get
fixed by (i) matching the interior solution to the Schwarzschild exterior
metric across the boundary ($R$) and $(ii)$ by imposing the condition that the
radial pressure should vanish at a finite boundary i.e., $p_{r}(r=R)=0$. In
terms of total mass $M$ and radius $R$, the constants are obtained as

$\displaystyle L$ $\displaystyle=$ $\displaystyle R\sqrt{\frac{R+2KM-
KR}{2M}},$ (14) $\displaystyle C$ $\displaystyle=$
$\displaystyle(R-2M)(3-K)\sqrt{\frac{R(1-K)}{8M(L^{2}-R^{2})}},$ (15)
$\displaystyle D$ $\displaystyle=$ $\displaystyle-\sqrt{\frac{M}{2R^{3}}}.$
(16)

###  2.2  Solution developed by Maurya et al maur19

Maurya et al maur16  assumed a specific form of the metric potential $g_{rr}$

$e^{\lambda}=1+\frac{(a-b)r^{2}}{1+br^{2}},$ (17)

where $a$ and $b$ are constants with $a\neq b$. Using (17) and Karmarkar’s
embedding condition, it is possible to find a closed form solution for the
system (3) - (5) in the form

$e^{\nu}=\Big{[}\frac{Ab+B\sqrt{(a-b)}\sqrt{(1+br^{2})}}{b}\Big{]}^{2},$ (18)

where $A$ and $B$ are constants to be determined from the boundary conditions.
Subsequently, one obtains

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle(a-b)\Big{[}\frac{(3+ar^{2})}{(1+ar^{2})^{2}}\Big{]},$ (19)
$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle(a-b)\Big{[}\frac{-Ab+B(3b-a)\sqrt{\frac{(1+br^{2})}{(a-b)}}}{(1+ar^{2})[Ab+B\sqrt{(a-b)(1+br^{2})}]}\Big{]},$
(20) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(a-b)}}{(1+ar^{2})^{2}[Ab+B\sqrt{(a-b)(1+br^{2})}]}$
(21)
$\displaystyle\Big{[}-Ab\sqrt{(a-b)}+\frac{B}{\sqrt{(1+br^{2})}}(3b-a+ab^{2}r^{4})\Big{]}.$

The anisotropy factor is obtained in the form

$\Delta=ar^{2}\sqrt{(a-b)}\Big{[}\frac{Ab\sqrt{(a-b)}+\frac{B}{\sqrt{(1+br^{2})}}(a-2b)}{(1+ar^{2})^{2}[Ab+B\sqrt{(a-b)(1+br^{2})}]}\Big{]},$
(22)

which shows that the anisotropy vanishes for $a=0$.

The boundary conditions mentioned above determine the unknown constants as

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{(3b-a)}{2b}\sqrt{\Big{[}\frac{1+bR^{2}}{1+aR^{2}}\Big{]}},$
(23) $\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{a-b}{1+aR^{2}}}.$ (24)

The total mass $M$ within a radius $R$ is obtained as

$M=\frac{R}{2}\Big{[}1-\frac{1+bR^{2}}{1+aR^{2}}\Big{]}.$ (25)

Note that the solution contains four constants $a,~{}b,~{}A,~{}B$. For given
values of $M$ and $R$, three of the constants do get fixed while one parameter
remains free. In our analysis, we assume the free parameter to be $a$. It is
noteworthy that $b\neq 0$ in this model as $A$ becomes infinite for $b=0$. For
$a=0$, equation (25) suggests that we must have $b<0$. Most importantly, by
setting $a=0$ in Eq. (17) and setting $b=-\frac{1}{L^{2}}$, we regain the
Schwarzschild incompressible fluid solution.

Thus, we have two sets of anisotropic stellar solutions, both having the
Schwarzschild interior solution as an isotropic limiting case. It can be shown
that both the solutions satisfy the energy conditions and other physical
requirements of a realistic star. In the following section, making use of the
anisotropic solutions, we intend to obtain an anisotropic generalization of
the Buchdahl bound.

##  3  Anisotropic generalization of Buchdahl bound

It is noteworthy that the solutions to be utilized to find a new bound should
fulfil the physical requirements that density and pressure should be non-
negative ($\rho,~{}p\geq 0$) and density should be a decreasing function of
radial distance ($\frac{d\rho}{dr}>0$). The generalized bound in the presence
of anisotropic stress is deduced by demanding that the pressure must not
diverge at the centre (note that at $r=0$, $p_{r}=p_{t}$). In the following
sub-sections, we derive two separate bounds on the mass to radius ratio.

###  3.1  Generalized Buchdahl bound utilizing the solution obtained in (2.1)

In eq. (9), the condition that central pressure must not diverge implies that
we must have

$\Big{[}C\sqrt{1-K}-DL(1-K)\Big{]}\geq 0.~{}~{}~{}~{}~{}~{}~{}(L\neq 0)$ (26)

To find the bound on $M/R$, we substitute the values of $C$, $D$ and $L$ in
eq. (26) and consider the extreme case (i.e., equate the left hand side to
zero) which yields

$\frac{(R-2M)(3+K^{2}-4K)}{(K-1)\sqrt{(R+2KM-KR)(R+2KM-KR-2M)}}=1.$ (27)

We rewrite (27) as

$(9-5K)y^{2}+(9K-17)y+(8-4K)=0,$ (28)

where $\displaystyle y=\frac{2M}{R}$. Solution of eq. (28) eventually leads to
the following inequality

$y=\frac{2M}{R}\leq\frac{4(K-2)}{(5K-9)}.$ (29)

Since the parameter $K$ can be linked to anisotropy in this model, eq. (29)
can be treated as a generalization of the Buchdahl bound for an anisotropic
stellar configuration. In Fig. (1), the bound on $M/R$ against $K$ is shown.
The plot indicates that as anisotropy increases, the upper bound decreases.
Since $K$ is also a measure of departure from homogeneous isotropic spherical
distribution, the result also shows that a departure from spherical geometry
reduces the upper bound on compactness. Setting $K=0$, we regain the bound
$2M/R\leq 8/9$.

###  3.2  Generalized Buchdahl bound utilizing the solution obtained in (2.2):

We utilize the solution obtained by Maurya et al and follow the same technique
to make an anisotropic generalization of the Buchdahl bound. In this case,
applying the condition that central pressure must not diverge in eq. (20), we
obtain the following constraint

$Ab+B\sqrt{(a-b)}\geq 0.$ (30)

We substitute the values of $A$ and $B$ in (30) and in the extreme case obtain
the following relation

$1-\frac{2M}{R}=\Big{(}\frac{a-b}{3b-a}\Big{)}^{2}\frac{1}{1+aR^{2}},$ (31)

which eventually leads to the following bound

$\frac{2M}{R}\leq\frac{4(\frac{a}{b})^{2}-12(\frac{a}{b})+8}{5(\frac{a}{b})^{2}-14(\frac{a}{b})+9},$
(32)

where we have substituted

$R^{2}=\frac{4a-8b}{(3b-a)^{2}}.$ (33)

Eq. (32) is an anisotropic generalization of the Buchdahl bound. Setting $a=0$
in eq. (32), we regain the bound $2M/R\leq 8/9$. In Fig. (2), the variation of
the maximum bound on compactness with respect to the dimensionless anisotropic
factor $a/b$ is shown. It is interesting to note that the maximum bound
decreases with the increase of anisotropy. In fact, both the models, exhibit
identical behaviour.

Figure 1: Upper bound on $M/R$ plotted against the anisotropic parameter $K$.
Figure 2: Upper bound on $M/R$ plotted against the anisotropic parameter
$a/b$.

##  4  Discussion:

The key results of our investigation are outlined below:

  * •

Making use of particular anisotropic stellar models, we have obtained
anisotropic generalizations of the Buchdahl bound. In the isotropic case, both
the solutions correspond to the Schwarzschild incompressible fluid solution,
and consequently, the Buchdahl bound $M/R\leq 4/9$ is regained once the
anisotropy is switched off. For any physically reasonable stellar
configuration, the generalized bound is obtained by demanding that the central
pressure must not be infinite.

  * •

Even though we have considered two different class of solutions, the impact of
anisotropy is similar in both cases. Anisotropy reduces the upper bound on
compactness, and it never exceeds the value of $2M/R=8/9$.

It remains to be seen whether the effects of anisotropy remain identical in
all kinds of anisotropic stellar models. It should be stressed here that we
have considered models where $p_{t}>p_{r}$. It remains to be seen whether a
similar outcome would follow for stellar configurations having $p_{r}>p_{t}$.
It is also a matter of further probe whether the technique adopted in this
paper provides the most stringent bound on compactness. All these issues are
beyond the scope of the current investigation and will be taken up elsewhere.

###### Acknowledgements.

RS and SD gratefully acknowledge support from the Inter-University Centre for
Astronomy and Astrophysics (IUCAA), Pune, India, under its Visiting Research
Associateship Programme.

##  5 Data repository

The data underlying this article is available in the public domain as cited in
the references.

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††thanks: Wallenberg Academy Fellow

Present address: ]IFIC, CSIC-Universidad de Valencia, Valencia, Spain

XENON Collaboration

# Low-mass dark matter search using ionization signals in XENON100

E. Aprile  Physics Department, Columbia University, New York, New York, 10027,
USA  J. Aalbers  Nikhef and the University of Amsterdam, Science Park, 1098XG
Amsterdam, Netherlands  F. Agostini  INFN-Laboratori Nazionali del Gran Sasso
and Gran Sasso Science Institute, 67100 L’Aquila, Italy  Department of Physics
and Astrophysics, University of Bologna and INFN-Bologna, 40126 Bologna, Italy
M. Alfonsi  Institut für Physik & Exzellenzcluster PRISMA, Johannes Gutenberg-
Universität Mainz, 55099 Mainz, Germany  F. D. Amaro  Department of Physics,
University of Coimbra, 3004-516, Coimbra, Portugal  M. Anthony  Physics
Department, Columbia University, New York, New York, 10027, USA  F. Arneodo
New York University Abu Dhabi, Abu Dhabi, United Arab Emirates  P. Barrow
Physik-Institut, University of Zurich, 8057 Zurich, Switzerland  L. Baudis
Physik-Institut, University of Zurich, 8057 Zurich, Switzerland  B.
Bauermeister  Institut für Physik & Exzellenzcluster PRISMA, Johannes
Gutenberg-Universität Mainz, 55099 Mainz, Germany  Department of Physics,
Stockholm University, Stockholm, SE-10691 Sweden  M. L. Benabderrahmane  New
York University Abu Dhabi, Abu Dhabi, United Arab Emirates  T. Berger
Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic
Institute, Troy, New York 12180, USA  P. A. Breur  Nikhef and the University
of Amsterdam, Science Park, 1098XG Amsterdam, Netherlands  A. Brown  Nikhef
and the University of Amsterdam, Science Park, 1098XG Amsterdam, Netherlands
E. Brown  Department of Physics, Applied Physics and Astronomy, Rensselaer
Polytechnic Institute, Troy, New York 12180, USA  S. Bruenner  Max-Planck-
Institut für Kernphysik, 69117 Heidelberg, Germany  G. Bruno  Institut für
Kernphysik, Wilhelms-Universität Münster, 48149 Münster, Germany  INFN-
Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, 67100
L’Aquila, Italy  R. Budnik<EMAIL_ADDRESS>Department of Particle
Physics and Astrophysics, Weizmann Institute of Science, Rehovot, 7610001,
Israel  A. Buss  Institut für Kernphysik, Wilhelms-Universität Münster, 48149
Münster, Germany  L. Bütikofer  Albert Einstein Center for Fundamental
Physics, University of Bern, 3012 Bern, Switzerland  J. M. R. Cardoso
Department of Physics, University of Coimbra, 3004-516, Coimbra, Portugal  M.
Cervantes  Department of Physics and Astronomy, Purdue University, West
Lafayette, Indiana 47907, USA  D. Cichon  Max-Planck-Institut für Kernphysik,
69117 Heidelberg, Germany  D. Coderre  Albert Einstein Center for Fundamental
Physics, University of Bern, 3012 Bern, Switzerland  A. P. Colijn  Nikhef and
the University of Amsterdam, Science Park, 1098XG Amsterdam, Netherlands  J.
Conrad  Department of Physics, Stockholm University, Stockholm, SE-10691
Sweden  J. P. Cussonneau  SUBATECH, Ecole des Mines de Nantes, CNRS/In2p3,
Université de Nantes, Nantes 44307, France  M. P. Decowski  Nikhef and the
University of Amsterdam, Science Park, 1098XG Amsterdam, Netherlands  P. de
Perio  Physics Department, Columbia University, New York, New York, 10027, USA
P. Di Gangi  Department of Physics and Astrophysics, University of Bologna and
INFN-Bologna, 40126 Bologna, Italy  A. Di Giovanni  New York University Abu
Dhabi, Abu Dhabi, United Arab Emirates  E. Duchovni  Department of Particle
Physics and Astrophysics, Weizmann Institute of Science, Rehovot, 7610001,
Israel  A. D. Ferella  INFN-Laboratori Nazionali del Gran Sasso and Gran Sasso
Science Institute, 67100 L’Aquila, Italy  Department of Physics, Stockholm
University, Stockholm, SE-10691 Sweden  A. Fieguth  Institut für Kernphysik,
Wilhelms-Universität Münster, 48149 Münster, Germany  D. Franco  Physik-
Institut, University of Zurich, 8057 Zurich, Switzerland  W. Fulgione  INFN-
Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, 67100
L’Aquila, Italy  INFN-Torino and Osservatorio Astrofisico di Torino, 10125
Torino, Italy  M. Galloway  Physik-Institut, University of Zurich, 8057
Zurich, Switzerland  M. Garbini  Department of Physics and Astrophysics,
University of Bologna and INFN-Bologna, 40126 Bologna, Italy  C. Geis
Institut für Physik & Exzellenzcluster PRISMA, Johannes Gutenberg-Universität
Mainz, 55099 Mainz, Germany  L. W. Goetzke  Physics Department, Columbia
University, New York, New York, 10027, USA  Z. Greene  Physics Department,
Columbia University, New York, New York, 10027, USA  C. Grignon  Institut für
Physik & Exzellenzcluster PRISMA, Johannes Gutenberg-Universität Mainz, 55099
Mainz, Germany  E. Gross  Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot, 7610001, Israel  C. Hasterok  Max-
Planck-Institut für Kernphysik, 69117 Heidelberg, Germany  E. Hogenbirk
Nikhef and the University of Amsterdam, Science Park, 1098XG Amsterdam,
Netherlands  R. Itay  Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot, 7610001, Israel  B. Kaminsky  Albert
Einstein Center for Fundamental Physics, University of Bern, 3012 Bern,
Switzerland  G. Kessler  Physik-Institut, University of Zurich, 8057 Zurich,
Switzerland  A. Kish  Physik-Institut, University of Zurich, 8057 Zurich,
Switzerland  H. Landsman  Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot, 7610001, Israel  R. F. Lang
Department of Physics and Astronomy, Purdue University, West Lafayette,
Indiana 47907, USA  L. Levinson  Department of Particle Physics and
Astrophysics, Weizmann Institute of Science, Rehovot, 7610001, Israel  M. Le
Calloch  SUBATECH, Ecole des Mines de Nantes, CNRS/In2p3, Université de
Nantes, Nantes 44307, France  C. Levy  Department of Physics, Applied Physics
and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA  F.
Linde  Nikhef and the University of Amsterdam, Science Park, 1098XG Amsterdam,
Netherlands  S. Lindemann  Max-Planck-Institut für Kernphysik, 69117
Heidelberg, Germany  M. Lindner  Max-Planck-Institut für Kernphysik, 69117
Heidelberg, Germany  J. A. M. Lopes  Department of Physics, University of
Coimbra, 3004-516, Coimbra, Portugal  A. Lyashenko  Physics & Astronomy
Department, University of California, Los Angeles, California, 90095, USA  A.
Manfredini  Department of Particle Physics and Astrophysics, Weizmann
Institute of Science, Rehovot, 7610001, Israel  T. Marrodán Undagoitia  Max-
Planck-Institut für Kernphysik, 69117 Heidelberg, Germany  J. Masbou
SUBATECH, Ecole des Mines de Nantes, CNRS/In2p3, Université de Nantes, Nantes
44307, France  F. V. Massoli  Department of Physics and Astrophysics,
University of Bologna and INFN-Bologna, 40126 Bologna, Italy  D. Masson
Department of Physics and Astronomy, Purdue University, West Lafayette,
Indiana 47907, USA  D. Mayani  Physik-Institut, University of Zurich, 8057
Zurich, Switzerland  A. J. Melgarejo Fernandez  Physics Department, Columbia
University, New York, New York, 10027, USA  Y. Meng  Physics & Astronomy
Department, University of California, Los Angeles, California, 90095, USA  M.
Messina  Physics Department, Columbia University, New York, New York, 10027,
USA  K. Micheneau  SUBATECH, Ecole des Mines de Nantes, CNRS/In2p3, Université
de Nantes, Nantes 44307, France  B. Miguez  INFN-Torino and Osservatorio
Astrofisico di Torino, 10125 Torino, Italy  A. Molinario  INFN-Laboratori
Nazionali del Gran Sasso and Gran Sasso Science Institute, 67100 L’Aquila,
Italy  M. Murra  Institut für Kernphysik, Wilhelms-Universität Münster, 48149
Münster, Germany  J. Naganoma  Department of Physics and Astronomy, Rice
University, Houston, Texas 77005, USA  U. Oberlack  Institut für Physik &
Exzellenzcluster PRISMA, Johannes Gutenberg-Universität Mainz, 55099 Mainz,
Germany  S. E. A. Orrigo  [  Department of Physics, University of Coimbra,
3004-516, Coimbra, Portugal  P. Pakarha  Physik-Institut, University of
Zurich, 8057 Zurich, Switzerland  B. Pelssers  Department of Physics,
Stockholm University, Stockholm, SE-10691 Sweden  R. Persiani  SUBATECH, Ecole
des Mines de Nantes, CNRS/In2p3, Université de Nantes, Nantes 44307, France
F. Piastra  Physik-Institut, University of Zurich, 8057 Zurich, Switzerland
J. Pienaar  Department of Physics and Astronomy, Purdue University, West
Lafayette, Indiana 47907, USA  G. Plante  Physics Department, Columbia
University, New York, New York, 10027, USA  N. Priel  Department of Particle
Physics and Astrophysics, Weizmann Institute of Science, Rehovot, 7610001,
Israel  L. Rauch  Max-Planck-Institut für Kernphysik, 69117 Heidelberg,
Germany  S. Reichard  Department of Physics and Astronomy, Purdue University,
West Lafayette, Indiana 47907, USA  C. Reuter  Department of Physics and
Astronomy, Purdue University, West Lafayette, Indiana 47907, USA  A. Rizzo
Physics Department, Columbia University, New York, New York, 10027, USA  S.
Rosendahl  Institut für Kernphysik, Wilhelms-Universität Münster, 48149
Münster, Germany  N. Rupp  Max-Planck-Institut für Kernphysik, 69117
Heidelberg, Germany  J. M. F. dos Santos  Department of Physics, University of
Coimbra, 3004-516, Coimbra, Portugal  G. Sartorelli  Department of Physics and
Astrophysics, University of Bologna and INFN-Bologna, 40126 Bologna, Italy  M.
Scheibelhut  Institut für Physik & Exzellenzcluster PRISMA, Johannes
Gutenberg-Universität Mainz, 55099 Mainz, Germany  S. Schindler  Institut für
Physik & Exzellenzcluster PRISMA, Johannes Gutenberg-Universität Mainz, 55099
Mainz, Germany  J. Schreiner  Max-Planck-Institut für Kernphysik, 69117
Heidelberg, Germany  M. Schumann  Albert Einstein Center for Fundamental
Physics, University of Bern, 3012 Bern, Switzerland  L. Scotto Lavina
SUBATECH, Ecole des Mines de Nantes, CNRS/In2p3, Université de Nantes, Nantes
44307, France  M. Selvi  Department of Physics and Astrophysics, University of
Bologna and INFN-Bologna, 40126 Bologna, Italy  P. Shagin  Department of
Physics and Astronomy, Rice University, Houston, Texas 77005, USA  H. Simgen
Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany  A. Stein
Physics & Astronomy Department, University of California, Los Angeles,
California, 90095, USA  D. Thers  SUBATECH, Ecole des Mines de Nantes,
CNRS/In2p3, Université de Nantes, Nantes 44307, France  A. Tiseni
<EMAIL_ADDRESS>Nikhef and the University of Amsterdam, Science Park, 1098XG
Amsterdam, Netherlands  G. Trinchero  INFN-Torino and Osservatorio Astrofisico
di Torino, 10125 Torino, Italy  C. D. Tunnell<EMAIL_ADDRESS>Nikhef and
the University of Amsterdam, Science Park, 1098XG Amsterdam, Netherlands  M.
von Sivers  Albert Einstein Center for Fundamental Physics, University of
Bern, 3012 Bern, Switzerland  R. Wall  Department of Physics and Astronomy,
Rice University, Houston, Texas 77005, USA  H. Wang  Physics & Astronomy
Department, University of California, Los Angeles, California, 90095, USA  M.
Weber  Physics Department, Columbia University, New York, New York, 10027, USA
Y. Wei  Physik-Institut, University of Zurich, 8057 Zurich, Switzerland  C.
Weinheimer  Institut für Kernphysik, Wilhelms-Universität Münster, 48149
Münster, Germany  J. Wulf  Physik-Institut, University of Zurich, 8057 Zurich,
Switzerland  Y. Zhang  Physics Department, Columbia University, New York, New
York, 10027, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We perform a low-mass dark matter search using an exposure of 30 kg$\times$yr
with the XENON100 detector. By dropping the requirement of a scintillation
signal and using only the ionization signal to determine the interaction
energy, we lowered the energy threshold for detection to 0.7 keV for nuclear
recoils. No dark matter detection can be claimed because a complete background
model cannot be constructed without a primary scintillation signal. Instead,
we compute an upper limit on the WIMP-nucleon scattering cross section under
the assumption that every event passing our selection criteria could be a
signal event. Using an energy interval from 0.7 keV to 9.1 keV, we derive a
limit on the spin-independent WIMP-nucleon cross section that excludes WIMPs
with a mass of 6 GeV/$c^{2}$ above $1.4\times 10^{-41}$ cm2 at 90% confidence
level.

Dark Matter, Direct Detection, Xenon

###### pacs:

95.35.+d, 14.80.Ly

††preprint: APS/123-QED

##  I Introduction

Astrophysical observations indicate that dark matter (DM) is needed to explain
structures ranging from the scales of galaxies to the largest observed scales
darktheory . Nevertheless, little is known about its nature. One theoretically
favored candidate is a weakly interacting massive particle (WIMP). These
particles may be detectable with experiments sensitive to WIMP-induced nuclear
recoils wimp .

Most WIMP models predict particles with a mass at the electroweak scale of
$\sim$100 GeV/$c^{2}$ 100gevwimp . However, there is also interest in light-
mass DM, below 10 GeV/$c^{2}$, prompted by, e.g., asymmetric models
asymmetricmodels ; asymmetricmodels1  and claims of DM observations dama ;
cdmssi . Light-mass DM would yield low-energy events that are close to the
experimental energy threshold of liquid-xenon detectors. Therefore, exploiting
an approach that lowers the threshold XENON10 , we investigate the spin-
independent WIMP-nucleon cross section versus mass parameter space extending
the XENON100 results for masses below $\sim$7.4 GeV/$c^{2}$.

##  II  The XENON100 detector

The XENON100 detector detector  is a dual-phase (liquid-gas) xenon time
projection chamber (TPC) located in the Laboratori Nazionali del Gran Sasso
(LNGS). The TPC detection principle allows for measurements of nuclear recoils
(NR) and electronic recoils (ER) through two signals: a prompt scintillation
signal S1 and an ionization signal S2. The S1 signal is scintillation light
from the rapid deexcitation of excited liquid xenon molecular states after an
ionizing particle deposits energy. This deposition also liberates electrons,
which drift in an electric field of 530 V/cm toward the liquid-gas interface,
where a larger field of $\sim$12 kV/cm extracts them from the liquid. These
accelerated electrons generate proportional scintillation in the xenon gas
above the liquid.

Two arrays of 178 1”-square Hamamatsu R8520-AL PMTs are installed above and
below the 62-kg xenon target. They detect both signals from the target. The
distribution of the S2 signal among the top PMTs gives the projection of the
interaction position on the PMT plane, while the relative time between the S1
and S2 signals provides the depth of the interaction, or $z$ coordinate. We
distinguish ER and NR by the ratio of their respective S1 and S2 signals. A
trigger identifies S2 signals, and the waveform of each PMT is digitized in
the interval between 200 $\mu$s before and after the trigger. The time for an
electron to drift from the cathode to anode, or the maximum drift time, is 176
$\mu$s detector . The TPC is surrounded by an active veto region consisting of
99 kg of liquid xenon, instrumented with 64 PMTs optically isolated from the
TPC.

In previous XENON100 analyses 225livedays ; spindependent , the recoil energy
has been determined using the size of the S1 signal and the relative
scintillation efficiency for the nuclear recoils,
$\mathcal{L}_{\textrm{eff}}$, relative to the 122 keV calibration $\gamma$
line of 57Co 225livedays . WIMPs with a mass below 10 GeV/$c^{2}$ create NRs
only up to a few keV, resulting in an S2 signal lower than a few hundred
photoelectrons (PE) and an S1 signal that is often not detectable. Therefore
for this analysis we only use the S2 signal to infer the energy.

##  III Analysis

This analysis is performed using the data from XENON100’s Science Run II,
which collected a 225 live-day exposure between February 28, 2011 and March
31, 2012 225livedays . For the WIMP analysis, we drop the requirement of
observing an S1 signal. This allows us to lower the effective threshold at the
cost of losing $z$ coordinate reconstruction from the S2-S1 peak time
difference and particle identification based on the S2/S1 signal ratio. We
perform a background-limited analysis on this previously unblinded data set.

Both a NR and an ER within liquid xenon will produce an S2 signal. We use
calibration data of ERs and NRs taken with external 60Co/232Th and 241AmBe
calibration sources, respectively. In these calibrations and in the DM search
data, photo-ionization and delayed extraction of electrons produce signals
that have a mean size of 20 PE per electron singleelectron . We restrict
ourselves to charge signals above 80 PE, where the trigger efficiency is still
at 80%, to minimize the background from these electrons. For the same reason,
this value will be used as the lower threshold for the WIMP analysis.

Many processes besides WIMP interactions can create S2 or S2-like signals in
our detector, e.g., radioactive backgrounds or photo-ionization of impurities
or metallic surfaces in the TPC singleelectron . We use selection criteria to
suppress these backgrounds in the DM search data. To begin, WIMPs are expected
to interact uniformly in the liquid xenon target. In the DM search data, the
event rate increases towards the radial edges of the detector because of
radioactive backgrounds. Therefore, we require that the reconstructed radius
of the event is less than 13.4 cm, which is approximately 2 cm from the TPC
walls. This cut removes events from external backgrounds, which are stopped
predominantly in the outer layers of the liquid target. The remaining liquid
xenon target mass is 48.3 kg analysispaper . Within this target, the events
are uniformly distributed radially, which means that a smaller fiducial volume
does not reduce the background density.

Given an event with an S1 signal in the DM search data, we can use the
information from that signal to isolate nuclear recoils using two additional
cuts. First, the Monte Carlo nuclear-recoil model of MarcWeber  is used to
determine a cut on the S1 size relative to the S2 size for any WIMP mass less
than 20 GeV/$c^{2}$. We parametrize this Monte Carlo model by requiring that a
nuclear recoil has–if present–an S1 signal less than
$[4.7+0.012\times(\text{S2}-80)]$ PE. This cut has an acceptance of 99.9%
determined from the same Monte Carlo. Second, we can estimate the $z$ position
of the interaction from the drift time between the S1 and S2 signals. We
require the $z$ position to be more than 1.9 cm below the liquid-gas interface
and more than 0.5 cm above the cathode. This condition decreases the fiducial
volume by 8% and we conservatively assume an acceptance of 92% also for events
without a detected S1 signal.

Figure 1: Rate of events (S2 $>$ 80 PE) as a function of the time difference
from the previous recorded event. A cut is set at 10 ms to remove a population
of events of small S2 signals (e.g., photo-ionization) that appears within a
few ms from the previous trigger.

Secondary S2 signals can create events in which the main S2 signal is preceded
or followed by similar nearby signals in the same event. These can be caused
by multiple scatters in the active volume (i.e., not WIMPs) or misidentified
detector artifacts. Additionally, any interaction in XENON100 can cause small
S2 signals appearing up to milliseconds after the trigger, which are partly
caused by photo-ionization on metal surfaces or impurities and possibly by
delayed charge extraction as well singleelectron .

We remove events which occur less than 10 ms after any other recorded event,
resulting in a 2% live-time reduction. Figure 1 shows the event rate as a
function of the time difference from the previous event. Signals caused by
photo-ionization or delayed extraction are observed within a few ms from the
previous event and are removed by this selection.

Figure 2: S2 asymmetry parameter for 241AmBe calibration data in the liquid
and a population of events produced in the xenon gas phase. We select
interactions in the gas by requiring an S1 signal, small drift time, and a
large S2 width using 60Co and DM search data. An S2 asymmetry cut set at 0.17
is used to reject the gas event population in the dark matter data.

In the DM search data, we reject events with more than one S2 signal in the
same event, e.g., multiple scatter events. If an S2 signal larger than 10 (30)
PE is seen 176 $\mu$s before (after) the main S2 signal, the event is removed.
The threshold after the main S2 signal is less strict since even a 250 PE S2
signal will itself create a secondary single-electron S2 signal ($\approx$ 20
PE singleelectron ) by photo-ionization in 10% of the cases. The acceptance
loss is 3% at S2 = 100 PE and slowly increasing for larger S2 signals, as
estimated by a model of induced S2 signals similar to singleelectron , but
extended to low energies using the 10 ms time difference cut.

For the following selection criteria, we estimate the acceptance on
calibration data. For calibration events, to ensure that we only select valid
low-energy events, we additionally require that the S1 signal observed in the
TPC (at any size) has a coincident S1 signal in the veto region. We also apply
the fiducial volume and single scatter selections as described above. In this
way, we create a low-energy sample of real interactions. We use the fraction
of events removed by the individual selection condition in the 241AmBe
calibration data analysispaper . 241AmBe calibration was acquired before and
after the DM search data. The acceptance for 241AmBe taken at the end of the
run is $\simeq$ 6% lower compared to 241AmBe acquired at the beginning of the
run. Conservatively, we choose 241AmBe calibration data acquired at the end of
the run to model the WIMP acceptance.

Events which contain too much electronic noise activity cannot be evaluated
properly and are removed by comparing the area of the main S2 peak to the
remaining baseline area. An S2 size-dependent threshold (the S2 pulse should
contain at least 45% of the total area at 100 PE) was derived using 241AmBe
calibration data and leads to a 97% WIMP acceptance.

Table 1: Acceptances of the different data selections and number of DM candidate events passing the selections. The cuts are applied sequentially. The number of events is in the S2 energy range [80, 1000] PE. Description of cut |  Acceptance at S2=100 PE | Events
---|---|---
Radial cut (starting events) | 100% | 254901
Depth and electronic recoil | 92% | 103914
Detector noise | 97% | 57516
Single S2 and 10 ms cut | 95% | 49041
Interaction in the gas | 61% | 13560

Finally, we apply a cut to remove events where the S2 signal is produced by an
interaction in the gas between the anode and the top PMT screening electrode
detector . These are most likely caused by radioactivity from the top PMT
array. In these so-called “gas events,” a larger than average fraction of the
S2 light is seen by the top PMT array since the S2 signal is produced close to
it. The S2 signal is also wider than an S2 produced in the liquid since the
luminescence region is typically twice as wide and–if an S1 signal is
detected–it occurs very shortly before the S2. Therefore, we define an
asymmetry parameter (S2${}_{\text{top}}$ $-$ S2${}_{\text{bottom}}$) /
(S2${}_{\text{top}}$ $+$ S2${}_{\text{bottom}}$), corresponding to the
fraction of observed light in the top PMTs compared to the bottom PMTs.

In Fig. 2, the asymmetry parameter is shown for 241AmBe events that occurred
in the liquid xenon and a sample of events from interactions in the gas phase.
The gas events are taken from 60Co and DM search data, requiring an S1 signal
and selecting events where the S2 width at 10% peak height is inconsistent
with diffusion broadening given the drift time of the event. Both
distributions are normalized to the rate expected in the DM search data. The
events in the liquid should be primarily due to ERs from background $\gamma$s,
so we estimate the rate by comparing the rate of 60Co events and DM search
data events at energies far beyond the region of interest, as done in
analysispaper . The gas event rate was estimated from DM search data events
with an S2 asymmetry larger than 0.45 (again, well beyond the region of
interest), as seen in Fig. 2.

Figure 3:  The analysis acceptance (red triangles) and the trigger efficiency
(blue circles). The purple dashed line is the analysis threshold (80 PE).

We remove events with an S2 asymmetry parameter larger than 0.17 and smaller
than an S2 size-dependent threshold derived from 241AmBe ($-$0.32 at 100 PE).
The 0.17 threshold is chosen by optimizing the ratio of the liquid events over
the square root of gas events. Only 61% of liquid events with an S2 signal of
100 PE will pass the asymmetry cut (as determined from the 241AmBe data). The
low acceptance is necessary because of the gas event background in this
analysis. We also apply a loose S2 10%-width selection of [0.8, 2.7] $\mu$s
with an acceptance of 99.8% at S2=100 PE.

Figure 3 shows the analysis acceptance and the trigger efficiency
analysispaper  as a function of the S2 signal size. The trigger efficiency in
our region of interest is more than 80%. The product of the trigger efficiency
and analysis acceptance is our final signal detection efficiency. Table 1
shows the acceptance of the analysis selections discussed above, as well as
the number of events remaining at each stage. After applying the data
selection cuts summarized in Table 1 to the the entire data set of 30 kg
$\times$ yr, 13560 valid candidate events remain in the S2 range [80, 1000] PE
(see Fig. 4).

Figure 4:  Energy distribution of the events remaining in the data set after
all data selection cuts. As an example, the expected spectrum for a WIMP of 6
GeV/$c^{2}$ and a spin-independent WIMP-nucleon scattering cross section of
$1.5\times 10^{-41}$ cm2 is also shown. The corresponding nuclear recoil
energy scale is indicated on the top axis. The charge yield model assumed here
has a cutoff at 0.7 keV, which truncates the WIMP spectrum. The optimum
interval (thick red line) is found in the S2 range [98, 119] PE and contains
1173 events.

##  IV Results

The interpretation of the outcome of the data selection requires the
reconstruction of a nuclear recoil equivalent energy scale from the measured
S2 signals. It is based on two quantities: the first one is the charge yield
$Q_{y}$, shown in Fig. 5, which gives the number of ionization electrons per
keV liberated by a NR event. The second one is the secondary scintillation
gain $Y$, which is detector-dependent and gives the number of proportional
scintillation photoelectrons per electron extracted into the gas phase. In
this science run of XENON100, $Y$ is described by a normal distribution with
$\mu=(19.7\pm 0.3)$ PE/e- and $\sigma=(6.9\pm 0.3)$ PE/e- singleelectron .
Charge extraction from the liquid is almost unity at the XENON100 extraction
field detector .

Figure 5:  Charge yield ($Q_{y}$) as a function of energy for nuclear recoils
(keV). This analysis employs the conservative nuclear recoil charge yield
model of Bezrukov _et al._ (electric field independent) bezrukov , given by
the green line. It agrees with the measurement of XENON100 (E = 0.53 kV/cm)
MarcWeber  (red triangles). The NEST model (E = 0.73 kV/cm) nest  (dashed
black) and the recent measurement of LUX (E = 0.18 kV/cm) luxlowenergy  (blue
points) predict slightly higher yields. To account for the mild discrepancies
below 3 keV, we use the model from Bezrukov _et al._ and conservatively assume
$Q_{y}$=0 below 0.7 keV.

As shown in Fig. 5, there is some remaining uncertainty in $Q_{y}$, especially
at very low recoil energies, even though the LUX data demonstrate clearly that
$Q_{y}$ is nonzero above 0.7 keV luxlowenergy . In order to not base our WIMP
result on optimistic assumptions, we use the analytical model of Bezrukov _et
al._ bezrukov , which agrees with the XENON100 measurement MarcWeber , and the
NEST model nest  above $\sim$6 keV and is more conservative at lower energies.
We additionally introduce a cutoff at 0.7 keV, below which $Q_{y}$ is set to
zero, to penalize the result for the limited knowledge on the charge yield at
the lowest energies. This energy also corresponds to the threshold at which
signals will be above our 80 PE threshold.

Figure 6:  S2 spectrum of 241AmBe calibration data compared to simulations
using the $Q_{y}$ from Bezrukov _et al._ bezrukov  with no energy cutoff.

However, we note that a Monte Carlo model based on the Bezrukov _et al._
function without any cutoff leads to a good description of the measured charge
spectrum from 241AmBe calibration data (see Fig. 6). The data were selected
based on the same criteria as used in the WIMP analysis, with the exception of
the S2 asymmetry cut, which is not required due to the significantly higher
rate of the 241AmBe source compared to the gas event rate. Besides the
statistical uncertainty, the spectrum also includes a systematic uncertainty
of 8%, which is mainly due to the uncertainties in the S2 amplification
singleelectron  and the cut acceptance. The simulation follows the strategy
described in MarcWeber  but ignores the S1 light information.

Figure 7:  WIMP exclusion limit on the spin-independent WIMP-nucleon
scattering cross section at 90% confidence level. Limits from the LUX lux ,
XENON100 225livedays , SuperCDMS supercdms , CDMSlite cdmslite , XENON10
XENON10 , CRESST-II CREEST2015  and PICO-2L PICO2016  experiments are shown.
The claims from DAMA/LIBRA experimental data damachanneling  and CDMS-II (Si
detectors) cdmssi  are also shown. The limit from this analysis is shown with
the thick blue line and it improves the XENON100 result 225livedays  (dashed
blue line) for WIMP masses below $\sim$7.4 GeV/$c^{2}$.

The same Monte Carlo method is used to model the expected WIMP energy spectra.
The number of electrons released after a nuclear recoil of energy $E$ is given
by a Poisson distribution with mean $N=EQ_{y}$. The charge loss due to the
electron lifetime ($\tau_{e}$) is modeled per event as an exponential
reduction in the number of electrons, though this effect is small due to the
average $\braket{\tau_{e}}=570$ $\mu$s. The evolution of $\tau_{e}$ throughout
the 225 days is modeled as in previous work analysispaper . The secondary
scintillation is modeled using the measured parameters given above and in
singleelectron . A Maxwell-Boltzmann distribution with the asymptotic velocity
of the local system $v_{0}=220$ km/s, the solar velocity $v_{sun}=232$ km/s
and the galactic escape velocity $v_{esc}=544$ km/s is used to model the DM
halo, assuming a local WIMP density of $\rho_{0}=0.3$ GeV/($c^{2}$$\times$cm3)
green . As an example, Fig. 4 shows the NR spectrum, as parametrized in
lewinsmith , induced by a 6 GeV/$c^{2}$ WIMP at a spin-independent cross
section of $\sigma=1.5\times 10^{-41}$ cm2. We observe an event rate of
$\sim$0.5 events/(keV$\times$kg$\times$day) between 0.7 and 1.7 keV that drops
to $\sim$0.07 events/(keV$\times$kg$\times$day) between 3.4 and 9.1 keV.

In the absence of a full background model, which cannot be constructed as the
origin of the small-S2 background in the detector cannot be reliably
quantified, we assume that every event passing the analysis cuts could be due
to a DM interaction. The analysis employs the optimum interval method
optinterval  and will therefore always lead to an exclusion limit. The optimum
S2 interval varies with WIMP mass, but in all cases in this analysis, it
contains a minimum of 1000 events passing all cuts. The low-mass WIMP result
for this 30 kg $\times$ yr XENON100 exposure is based on all events remaining
in the 80-1000 PE interval (0.7-9.1 keV), the NR acceptance of Fig. 3, and is
shown in Fig. 7. At a WIMP mass of 6 GeV/$c^{2}$, XENON100 excludes spin-
independent WIMP-nucleon interaction cross sections of $1.4\times 10^{-41}$
cm2 at 90% confidence level. The moderate improvement upon the XENON10 low-
mass result XENON10 , despite the much larger exposure, is due to the
significantly higher background from photo-ionization events, which is
enhanced by the presence of larger metal surfaces inside the TPC. The new
result challenges a standard WIMP interpretation of the DAMA/LIBRA modulation
signal, excludes large fractions of the CDMS-II (Si) preferred region and
improves the result of the previous XENON100 result 225livedays  below
$\sim$7.4 GeV/$c^{2}$. We improve the LUX lux  (SuperCDMS supercdms ) results
below $\sim$3.7 (5.3) GeV/$c^{2}$.

##  V  Acknowledgments

We gratefully acknowledge support from the National Science Foundation, Swiss
National Science Foundation, Deutsche Forschungsgemeinschaft, Max Planck
Gesellschaft, Foundation for Fundamental Research on Matter, Weizmann
Institute of Science, I-CORE, Initial Training Network Invisibles (Marie Curie
Actions, PITN- GA-2011-289442), Fundacao para a Ciencia e a Tecnologia, Region
des Pays de la Loire, Knut and Alice Wallenberg Foundation, and Istituto
Nazionale di Fisica Nucleare. We are grateful to Laboratori Nazionali del Gran
Sasso for hosting and supporting the XENON project.

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