Possible studies of gluon transversity in the spin-1 deuteron
at hadron-accelerator facilities

S. Kumano1,2$\star$ and Qin-Tao Song3

1 KEK Theory Center, Institute of Particle and Nuclear Studies, KEK,

Oho 1-1, Tsukuba, Ibaraki, 305-0801, Japan

2 J-PARC Branch, KEK Theory Center, Institute of Particle and Nuclear Studies,
KEK,

and Theory Group, Particle and Nuclear Physics Division, J-PARC Center,

Shirakata 203-1, Tokai, Ibaraki, 319-1106, Japan

3 School of Physics and Microelectronics, Zhengzhou University,

Zhengzhou, Henan 450001, China

*<EMAIL_ADDRESS><EMAIL_ADDRESS>
Proceedings for the XXVIII International Workshop
on Deep-Inelastic Scattering and Related Subjects,
Stony Brook University, New York, USA, 12-16 April 2021
10.21468/SciPostPhysProc.?

## Abstract

Chiral-odd gluon transversity distribution could shed light on a new aspect of
hadron physics. Although we had much progress recently on quark transversity
distributions, there is no experimental measurement on the gluon transversity.
The gluon trasversity does not exist in the spin-1/2 nucleons and it exists in
the spin-1 deuteron. Therefore, it could probe new hadron physics in the
deuteron beyond the basic bound system of a proton and a neutron because the
nucleons cannot contribute directly. Here, we explain that the gluon
transversity can be measured at hadron accelerator facilities, such as
Fermilab and NICA, in addition to charged-lepton scattering measurements at
lepton accelerator facilities by showing cross sections of the proton-deuteron
Drell-Yan process as an example.

##  1 Introduction

There are tensor-polarized structure functions for spin-1 hadrons, such as the
deuteron, in addition to the polarized ones for the spin-1/2 nucleons. The
twist-2 tensor-polarized structure function $b_{1}$ was measured by the HERMES
collaboration [1]. We have been investigating structure functions of spin-1
hadrons theoretically [2, 3]. On the other hand, there are future experimental
projects to measure these new structure functions at various accelerator
facilities. The structure function $b_{1}$ will be measured at the Jefferson
laboratory (JLab) [4], and there is a letter of intent to measure the gluon
transversity also at JLab [5]. These polarized-deuteron structure functions
will be also investigated at Fermilab [6] and Nuclotron-based Ion Collider
fAcility (NICA) [7] with the polarized-deuteron target and beam, respectively.
Furthermore, they should be also investigated at future electron-ion colliders
in US and China.

Recently, there was much progress in the quark transversity distribution on
both theoretical and experimental sides. Global analyses of experimental data
were done and optimum quark transversity distributions were proposed. However,
there is no experimental measurement on the gluon transversity. The gluon
transversity is a chiral-odd distribution, which needs a helicity-flip for a
gluon in the gluon-hadron helicity amplitude. Namely, the two-unit of spin
$\Delta s=2$ is necessary, so that the hadron spin should be equal to one or
larger, and it does not exist in the spin-1/2 nucleons. Because of this
property, the gluon transversity is a unique distribution for probing the non-
nucleonic component in the deuteron, it could provide a new hadron-physics
information beyond a simple bound system of nucleons.

There is an experimental plan to measure the gluon transversity at JLab by
measuring the azimuthal angle dependence of the deuteron spin polarization.
Since there are a number of hadron accelerators in operation and will be built
in the near future, it is nice that gluon transversity experiments will be
done as independent and complementary measurements to the JLab one. These
projects were not possible until recently because there was no theoretical
formalism with the gluon transversity for hadron-facility experiments.
Considering this situation, we provided such a formalism for the gluon
transversity studies at the hadron accelerator facilities by taking the
proton-deuteron Drell-Yan process as an example [3]. In the near future, we
expect that the gluon transversity experiments will be realized at Fermilab
[6] and NICA [7] by using proton and deuteron beams.

##  2 Linear polarization of spin-1 deuteron

The gluon transversity is defined by linear polarizations for the spin-1
deuteron and also the gluon. Although the linear polarization is often used
for photon, it is rarely used for hadron polarizations, so that it is briefly
explained in this section. The spin-1 deuteron polarizations are described by
the polarization vectors defined by $\vec{E}_{\pm}=\left(\,\mp
1,\,-i,\,0\,\right)/\sqrt{2}$, $\vec{E}_{0}=\left(\,0,\,0,\,1\,\right)$,
$\vec{E}_{x}=\left(\,1,\,0,\,0\,\right)$, and
$\vec{E}_{y}=\left(\,0,\,1,\,0\,\right)$. The polarizations $\vec{E}_{+}$,
$\vec{E}_{0}$, and $\vec{E}_{-}$ indicate the spin states with the $z$
component of spin $s_{z}=+1$, $0$, and $-1$; $\vec{E}_{x}$ and $\vec{E}_{y}$
are linear polarizations. The spin vector $\vec{S}$ and tensor $T_{ij}$ are
written by these polarization vectors, and then they are parametrized in the
deuteron rest frame as

$\displaystyle\vec{S}$
$\displaystyle=\text{Im}\,(\,\vec{E}^{\,*}\times\vec{E}\,)=(S_{T}^{x},\,S_{T}^{y},\,S_{L}),$
$\displaystyle T_{ij}$
$\displaystyle=\frac{1}{3}\delta_{ij}-\text{Re}\,(\,E_{i}^{\,*}E_{j}\,)=\frac{1}{2}\left(\begin{array}[]{ccc}-\frac{2}{3}S_{LL}+S_{TT}^{xx}&S_{TT}^{xy}&S_{LT}^{x}\\\\[5.69046pt]
S_{TT}^{xy}&-\frac{2}{3}S_{LL}-S_{TT}^{xx}&S_{LT}^{y}\\\\[5.69046pt]
S_{LT}^{x}&S_{LT}^{y}&\frac{4}{3}S_{LL}\end{array}\right).$ (4)

Polarizations | $\vec{E}$ | $S_{T}^{x}$ | $S_{T}^{y}$ | $S_{L}$ | $S_{LL}$ | $S_{TT}^{xx}$
---|---|---|---|---|---|---
Longitudinal $+z$ | $\frac{1}{\sqrt{2}}(-1,\,-i,\,0)$ | 0 | 0 |  $+$1 | $+\frac{1}{2}$ | 0
Longitudinal $-z$ | $\frac{1}{\sqrt{2}}(+1,\,-i,\,0)$ | 0 | 0 |  $-$1 | $+\frac{1}{2}$ | 0
Transverse $+x$ | $\frac{1}{\sqrt{2}}(0,\,-1,\,-i)$ |  $+$1 | 0 | 0 | $-\frac{1}{4}$ | $+\frac{1}{2}$
Transverse $-x$ | $\frac{1}{\sqrt{2}}(0,\,+1,\,-i)$ | $-1$ | 0 | 0 | $-\frac{1}{4}$ | $+\frac{1}{2}$
Transverse $+y$ | $\frac{1}{\sqrt{2}}(-i,\,0,\,-1)$ | 0 |  $+$1 | 0 | $-\frac{1}{4}$ | $-\frac{1}{2}$
Transverse $-y$ | $\frac{1}{\sqrt{2}}(-i,\,0,\,+1)$ | 0 | $-1$ | 0 | $-\frac{1}{4}$ | $-\frac{1}{2}$
Linear $x$ | $(1,\,0,\,0)$ | 0 | 0 | 0 | $+\frac{1}{2}$ | $-1$
Linear $y$ | $(0,\,1,\,0)$ | 0 | 0 | 0 | $+\frac{1}{2}$ | $+1$
Table 1: Longitudinal, transverse, and linear polarizations of the deuteron,
polarization vectors, and parameters of spin vector and tensor [3, 7]. The
other parameters vanish in the considered polarizations
($S_{TT}^{xy}=S_{LT}^{x}=S_{LT}^{y}=0$). The gluon transversity distribution
is associated with the polarization parameter $S_{TT}^{xx}$.

The parameters $S_{T}^{x}$, $S_{T}^{y}$, $S_{L}$, $S_{LL}$, $S_{TT}^{xx}$,
$S_{TT}^{xy}$, $S_{LT}^{x}$, and $S_{LT}^{y}$ are assigned to express the
vector and tensor polarizations of the spin-1 deuteron. The convariant forms
$S^{\mu}$ and $T^{\mu\nu}$ are given in Ref. [3].

The polarizations of the spin-1 deuteron are summarized in Table 1 by showing
the polarization $\vec{E}$ and the polarization parameters for the
longitudinal, transverse, and linear polarizations of the deuteron. For
example, the longitudinal polarization contains not only the longitudinal
polarization parameter $S_{L}$ but also the tensor polarization $S_{LL}$. The
transverse polarization has the transverse-polarization parameter
$S_{T}^{x,y}$, the tensor polarization $S_{LL}$, and the linear polarization
$S_{TT}^{xx}$. The linear polarization has both the tensor polarization
$S_{LL}$ and the linear polarization $S_{TT}^{xx}$. Therefore, for extracting
the gluon transversity $\Delta_{T}g$ associated with $S_{TT}^{xx}$, the linear
polarization asymmetry $E_{x}-E_{y}$ needs to be taken. Alternatively, the
transverse polarizations could be used if the $S_{T}^{x,y}$ and $S_{LL}$ terms
are removed by appropriate polarization combinations or calculated by using
corresponding PDFs [3].

##  3 Gluon transversity

The longitudinally-polarized and transversity distributions are defined for
quarks and gluons by matrix elements of nonlocal operators as

$\displaystyle\Delta q(x)$
$\displaystyle=\int\\!\frac{d\xi^{-}}{4\pi}\,e^{ixp^{+}\xi^{-}}\langle\,p\,s_{L}\left|\,\bar{\psi}(0)\gamma^{+}\gamma_{5}\psi(\xi)\,\right|p\,s_{L}\,\rangle_{\xi^{+}=\vec{\xi}_{\perp}=0},$
$\displaystyle\Delta_{T}q(x)$
$\displaystyle=\int\\!\frac{d\xi^{-}}{4\pi}\,e^{ixp^{+}\xi^{-}}\langle\,p\,s_{Tj}\left|\,\bar{\psi}(0)\,i\,\gamma_{5}\,\sigma^{j+}\psi(\xi)\,\right|p\,s_{Tj}\,\rangle_{\xi^{+}=\vec{\xi}_{\perp}=0},$
$\displaystyle\Delta_{T}g(x)$
$\displaystyle=\varepsilon_{TT,\alpha\beta}\int\\!\frac{d\xi^{-}}{2\pi}\,xp^{+}\,e^{ixp^{+}\xi^{-}}\,\langle\,p\,E_{x}\left|\,A^{\alpha}(0)\,A^{\beta}(\xi)\right|p\,E_{x}\,\rangle_{\xi^{+}=\vec{\xi}_{\perp}=0},$

by using the lightcone coordinates. Here, $\Delta q$ is the longitudinally-
polarized quark distribution function, and $\Delta_{T}q$ and $\Delta_{T}g$ are
quark and gluon transversity distributions. The $p$ is the hadron momentum,
the $s_{L}$ and $s_{Tj}$ ($j=1$ or $2$) indicate longitudinal and transverse
polarizations, $E_{x}$ is the linear polarization, $\psi$ and $A^{\mu}$ are
the quark and gluon fields, and $x$ is the momentum fraction carried by a
parton. The gauge links for the color gauge invariance are not explicitly
written. The transverse parameter $\varepsilon_{TT}^{\alpha\beta}$ is defined
by $\varepsilon_{TT}^{11}=+1$ and $\varepsilon_{TT}^{22}=-1$.

The longitudinally-polarized quark distribution functions are given by the
difference between the quark distributions with spin parallel to the hadron
spin and the ones with antiparallel spin: $\Delta q(x)=q_{+}(x)-q_{-}(x)$ for
the helicity operator $\Pi_{\parallel}=\Sigma_{\parallel}/2$ with
$\vec{\Sigma}=\gamma_{5}\gamma_{0}\vec{\gamma}$. The transversity distribution
is expressed as $\Delta_{T}q(x)=q_{\uparrow}(x)-q_{\downarrow}(x)$, where
$\uparrow$ and $\downarrow$ indicate parallel and anti-parallel quark
polarizations defined by the polarization operator
$\tilde{\Pi}_{\perp}=\gamma_{0}\Sigma_{\perp}/2$ [3] in the transversely
polarized hadron. Equation (3) indicates that the gluon transversity is the
distribution of linearly-polarized gluon distribution difference in the
linearly-polarized deuteron

$\displaystyle\Delta_{T}g(x)=g_{\hat{x}/\hat{x}}(x)-g_{\hat{y}/\hat{x}}(x),$

where $\hat{y}/\hat{x}$ indicate the gluon linear polarization
$\varepsilon_{y}$ in the deuteron with the polarization $E_{x}$.

Figure 1:  Parton-hadron forward scattering amplitudes.

Structure functions of hadrons are expressed by the imaginary part of parton-
hadron forward scattering amplitudes
$A_{\Lambda_{i}\lambda_{i},\,\Lambda_{f}\lambda_{f}}$ in Fig. 1, where the
initial and final hadron helicities are shown by $\Lambda_{i}$ and
$\Lambda_{f}$ and parton ones are by $\lambda_{i}$ and $\lambda_{f}$. These
helicities satisfy the conservation relation
$\Lambda_{i}-\lambda_{i}=\Lambda_{f}-\lambda_{f}$. The polarized quark and
gluon distribution functions defined in Eq. (3) are expressed by the helicity
amplitudes as

$\displaystyle\Delta q(x)\sim\text{Im}\,(A_{++,\,++}-A_{+-,\,+-})\,,\ \ \
\Delta_{T}q(x)\sim\text{Im}\,A_{++,\,-\hskip 0.85355pt-}\,,\ \ \
\Delta_{T}g(x)\sim\text{Im}\,A_{++,\,-\hskip 0.85355pt-}\ .$

Both quark and gluon transversity distributions are given by the amplitudes
with the helicity flips for both quark (gluon) and hadron. Since the nucleon
is spin 1/2, the quark-helicity flip is possible and it has the quark
transversity distribution. However, the hadron spin should be one or larger
for allowing the helicity flip of two units for the gluon, so that the gluon
transversity does not exist in the nucleon. As a spin-1 hadron or nucleus, the
deuteron is most appropriate for experimental investigations because it is a
stable and simple nucleus. These facts suggest a unique purpose for
investigating the gluon transversity of the deuteron. The deuteron is a weak
bound system of a proton and a neutron; however, these nucleons cannot
contribute to the gluon transversity directly. Therefore, if a finite
distribution is found experimentally, it sheds light on any new hadron physics
beyond the simple bound system of nucleons.

##  4 Proton-deuteron Drell-Yan process and gluon transversity

Figure 2:  $q\bar{q}\to\gamma^{*}g$ subprocess.

Figure 3:  $qg\to\gamma^{*}q$ subprocess.

The purpose of our research is to propose a possibility to investigate the
gluon transversity at hadron accelerator facilities. As an example, we made
the theoretical formalism for the proton-deuteron Drell-Yan process
$p+d\to\mu^{+}\mu^{-}+X$ with the linearly-polarized deuteron. There are many
subprocesses which contribute to the Drell-Yan cross section as typically
shown in Figs. 3 and 3 for the subprocesses $q\bar{q}\to\gamma^{*}g$ and
$qg\to\gamma^{*}q$. The upper proton blob and the lower deuteron blob indicate
the collinear correlation functions, which are expressed by various parton
distribution functions (PDFs). The proton is not polarized, so that only the
unpolarized PDFs of the proton are needed for estimating the cross section. On
the other hand, we consider the linear polarizations $E_{x}$ and $E_{y}$ for
the deuteron, and their asymmetry $d\sigma(E_{x})-d\sigma(E_{y})$ is
calculated. We take the $z$ axis as the proton momentum direction. The
differential cross section is calculated for the variables
$\tau=M_{\mu\mu}^{2}/s=Q^{2}/s$, the dimuon (virtual photon) transverse
momentum squared $\vec{q}_{T}^{2}$, its azimuthal angle $\phi$, and rapidity
$y$ in the center-of-mass frame. Here, $M_{\mu\mu}$ is the dimuon mass, and
$s$ is the center-of-mass energy squared. For the linear polarization
asymmetry $E_{x}-E_{y}$, we obtained the leading cross-section expression by
integrating over the momentum fraction $x_{a}$ for partons in the proton as

$\displaystyle\frac{d\sigma_{pd\to\mu^{+}\mu^{-}X}}{d\tau\,d\vec{q}_{T}^{2}\,d\phi\,dy}(E_{x}-E_{y})$
$\displaystyle=-\frac{\alpha^{2}\,\alpha_{s}\,C_{F}\,q_{T}^{2}}{6\pi
s^{3}}\cos(2\phi)\int_{\text{min}(x_{a})}^{1}dx_{a}\frac{1}{(x_{a}x_{b})^{2}\,(x_{a}-x_{1})(\tau-
x_{a}x_{2})^{2}}$ $\displaystyle\ \hskip
85.35826pt\times\sum_{q}e_{q}^{2}\,x_{a}\\!\left[\,q_{A}(x_{a})+\bar{q}_{A}(x_{a})\,\right]x_{b}\Delta_{T}g_{B}(x_{b}),$

where $\alpha$ is the fine structure constant, $\alpha_{s}$ is the QCD running
coupling constant, $C_{F}$ is the color factor $C_{F}=(N_{c}^{2}-1)/(2N_{c})$
with $N_{c}=3$, and $e_{q}$ is the quark charge. The momentum fraction $x_{b}$
is given by $x_{b}=(x_{a}x_{2}-\tau)/(x_{a}-x_{1})$, and the kinematical
minimum of $x_{a}$ is $\text{min}(x_{a})=(x_{1}-\tau)/(1-x_{2})$ with
$x_{1}=e^{y}\sqrt{(Q^{2}+\vec{q}_{T}^{2})/s}$ and
$x_{2}=e^{-y}\sqrt{(Q^{2}+\vec{q}_{T}^{2})/s}$. The $q_{A}(x_{a})$ and
$\bar{q}_{A}(x_{a})$ are quark and antiquark distribution functions in the
proton, and $\Delta_{T}g(x_{b})$ is the gluon transversity.

The absolute value of the cross section in Eq. (4) is shown in Fig. 5 for the
Fermilab kinematics with $p_{p}=120$ GeV by taking $\phi=0$, $y=0.5$, and
$q_{T}=0.5$ or 1.0 GeV as the function of the dimuon-mass squared
$M_{\mu\mu}^{2}$. Here, the CTEQ14 PDFs are used for the unpolarized PDFs of
the proton, and the NNPDF1.1 is used for the gluon transversity by boldly
assuming that the longitudinally-polarized gluon distribution is equal to the
gluon transversity. Because of this bold assumption, the cross section and the
following spin asymmetry could be overestimated. The cross section
$d\sigma(E_{x}+E_{y})$ can be calculated by using the unpolarized PDFs of the
proton and deuteron with the assumption of neglecting the small tensor-
polarized PDF part, and the calculated spin assymmetry
$d\sigma(E_{x}-E_{y})/d\sigma(E_{x}+E_{y})$ is shown in Fig. 5. The
asymmetries are typically a few percent. If a finite value is found
experimentally, it could create a new field of hadron and nuclear physics
beyond the simple bound system of nucleons. Therefore, we hope that such an
experiment is realized in future. In fact, this experiment will be proposed
within the Fermilab-E1039 experiment [6], and a similar $J/\psi$ experiment
should be possible in the NICA project.

Figure 4:  Proton-deuteron Drell-Yan cross section
for the linear-polarization asymmetry $E_{x}-E_{y}$.

Figure 5: Spin asymmetry
$d\sigma(E_{x}-E_{y})/d\sigma(E_{x}+E_{y})$.

##  5 Conclusion

The gluon transversity has not been measured yet experimentally. If a finite
distributions is found, it indicates the existence of new hadronic physics. In
this work, we showed the general formalism to investigate the gluon
transversity at hadron accelerator facilities by taking the Drell-Yan process
as an example. In the 2020’s, we expect that the gluon transversity will be
measured at JLab, Fermilab, NICA, and other accelerator facilities, so that it
could become an exciting field of hadron physics in the near future.

## Acknowledgements

S. Kumano was partially supported by Japan Society for the Promotion of
Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant Number
19K03830. Qin-Tao Song was supported by the National Natural Science
Foundation of China under Grant Number 12005191 and the Academic Improvement
Project of Zhengzhou University.

## References

  * [1] A. Airapetian et al. (HERMES Collaboration), Phys. Rev. Lett. 95 (2005) 242001.
  * [2] F. E. Close and S. Kumano, Phys. Rev. D 42 (1990) 2377; S. Hino and S. Kumano, Phys. Rev. D 59 (1999) 094026; D 60 (1999) 054018; T.-Y. Kimura and S. Kumano, Phys. Rev. D 78 (2008) 117505; S. Kumano, Phys. Rev. D 82 (2010) 017501; W. Cosyn, Yu-Bing Dong, S. Kumano, and M. Sargsian, Phys. Rev. D 95 (2017) 074036; S. Kumano and Qin-Tao Song, Phys. Rev. D 94 (2016) 054022; D 103 (2021) 014025; arXiv:2106.15849.
  * [3] S. Kumano and Qin-Tao Song, Phys. Rev. D 101 (2020) 054011 & 094013\.
  * [4] J.-P. Chen et al., Proposal to Jefferson Lab PAC-38, PR12-11-110 (2011).
  * [5] M. Jones et al., A Letter of Intent to Jefferson Lab PAC 44, LOI12-16-006 (2016).
  * [6] D. Geesaman et al., Proposal to Fermilab PAC, P-1039 (2013); D. Keller, D. Crabb, and D. Day, Nucl. Instrum. Meth. A 981 (2020) 164504.
  * [7] A. Arbuzov et al., Prog. Nucl. Part. Phys. 119, 103858 (2021) 1.


# Topological quantum thermometry

Anubhav Kumar Srivastava<EMAIL_ADDRESS>ICFO - Institut de
Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain  Utso Bhattacharya  ICFO - Institut de
Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain  Maciej Lewenstein  ICFO - Institut de
Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain  ICREA, Pg. Lluis Companys 23, 08010
Barcelona, Spain  Marcin Płodzień<EMAIL_ADDRESS>ICFO - Institut de
Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain

###### Abstract

An optimal local quantum thermometer is a quantum many-body system that
saturates the fundamental lower bound for the thermal state temperature
estimation accuracy [L. Correa, et. al., Phys. Rev. Lett. 114, 220405 (2015)].
Such a thermometer has a particular energy level structure with a single
ground state and highly degenerated excited states manifold, with an energy
gap proportional to the estimated temperature. In this work, we show that the
optimal local quantum thermometer can be realized in an experimentally
feasible system of spinless fermions confined in a one-dimensional optical
lattice described by the Rice-Mele model. We characterize the system’s
sensitivity to temperature changes in terms of quantum Fisher information and
the classical Fisher information obtained from experimentally available site
occupation measurements.

Introduction— Quantum thermodynamics provides the necessary framework allowing
building quantum thermal devices like quantum heat engines [1, 2, 3, 4, 5] or
quantum batteries [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Since most
of these devices work in ultra-low temperature regimes of the order of nano-
and pico-Kelvin [19, 20, 21], there is a constant requirement for higher
accuracy of the temperature estimation of such quantum systems [22, 23]. The
temperature of a given quantum system is not a quantum mechanical observable
but rather a parameter of its quantum state [24, 25], and as a result,
temperature estimation corresponds to the quantum estimation problem. The
accuracy of any estimation is limited by the quantum Cramér-Rao bound [26, 27]
relating maximal sensitivity to the parameter changes given by quantum Fisher
information, with the estimated parameter [28, 29, 30, 31, 32, 33, 34]. The
two main paradigms in quantum thermometry are based on the prior knowledge
about the system’s temperature. In the global quantum thermometry, there is no
prior knowledge about the estimated temperature [35, 36, 37, 38]. On the other
hand, in the local quantum thermometry, it is assumed that the estimator for
the temperature is given, whereas the aim is to minimize the uncertainty of
the temperature estimator [39, 40, 41, 42, 43, 44].

The optimal local quantum thermometer, saturating the fundamental bound of the
system’s sensitivity to temperature changes, is a two-level many-body system
with a single ground state and degenerated manifold of the excited state, with
the energy gap proportional to the estimated temperature [39]. Current
theoretical efforts in local quantum thermometry are focused on finding the
optimal quantum setups that maximize the sensitivity of the system’s thermal
state to any temperature changes. Over the recent years, different systems
have been proposed as optimal quantum thermometers, including two-component
fermions in a one-dimensional harmonic trap [45, 46], thermoelectric systems
[47, 48], quantum critical systems [49, 50, 51, 52, 53], quantum dots [54, 55,
56, 57], color centers in diamonds [58], single qubit dephasing [59],
impurities in Bose-Einstein condensates [60, 61, 62, 63, 64] or Fermi gases
[65, 66, 67, 68, 69], as well as exotic models utilizing Unruh-DeWitt
detectors [70], utilizing the Berry phase [71], and recently proposed machine
learning techniques in various setups such as spin systems [72, 73, 74],
impurities in Bose gases [75], and even critical systems [76]. However, to the
best of our knowledge, there is no experimentally feasible proposition for a
quantum system realizing the energy level structure required by the optimal
local quantum thermometer.

In this work, we show that an idealized optimal local quantum thermometer with
$N$-fold degeneracy of the excited state, operating in the sub-nK regime, can
be realized in the experimentally feasible system of $N$ spinless fermions in
a one-dimensional optical lattice with $L=2(N+1)$ sites described by the Rice-
Mele Hamiltonian [77], operating in the topological regime. The proposed
quantum thermometer can be tuned to the optimal temperature by varying the
lattice dimerization and the staggered onsite potential. We characterize the
sensitivity of the considered quantum thermometer to the temperature changes
in terms of the quantum Fisher information (QFI) and the classical Fisher
information (CFI) obtained from the experimentally feasible site occupation
measurements.

The Rice-Mele model in the limit of vanishing staggered potential reduces to
the Su-Schrieffer-Heeger (SSH) model [78, 79] being one of the simplest
condensed matter systems that exhibit topological characteristics (for an
extensive review, see [80, 81, 82, 83]). The Rice-Mele and SSH models have
been experimentally realized in many quantum simulator platforms [84], such as
quantum gases in optical lattices [85, 86, 87, 88, 89, 90, 91], acoustic
systems [92], graphene [93], photonic crystals [94], time crystals [95], and
in Rydberg arrays [96, 97, 98].

Figure 1: Schematic representation of a standard thermometry protocol. (a) The
quantum thermometer is initially prepared in the pure ground state, (b) and
exchanges energy with the system in the Gibbs thermal state at temperature
$T_{A}$, eventually reaching a thermal state, and lastly, (c) the measurement
protocol is prepared to estimate the system’s temperature. Panel (d) presents
schematically the energy level structure of an optimal quantum thermometer
with a single ground state and $(D-1)$-fold degeneracy of the excited state
[39].

Preliminaries— Let us consider a quantum system described by a Hamiltonian
$\hat{H}_{A}$ in a thermal Gibbs state $\hat{\varrho}^{A}_{T_{A}}$ at an
unknown temperature $T_{A}$, and a quantum thermometer described by a
Hamiltonian $\hat{H}_{B}$ in a Gibbs state $\hat{\varrho}^{B}_{T_{B}}$. The
quantum thermometer is coupled to system A to exchange energy. After a
sufficiently long time, in an ideal scenario, when the larger system acts as a
bath, the composite system reaches thermal equilibrium and is described by
stationary density matrix $\hat{\rho}^{AB}$. The reduced density matrix of the
thermometer $\hat{\rho}^{B}_{T}=\Tr_{A}[\hat{\rho}^{AB}]$ is a thermal state
at temperature $T=T_{A}$, $\hat{\rho}^{B}_{T}\simeq\hat{\varrho}^{B}_{T}$. The
measurement protocol is now prepared to estimate the temperature $T$, now
encoded in the thermal state of the thermometer, with high precision as shown
in Fig. 1.

The full measurement protocol consists of global thermometry, i.e., estimation
of $T$ without prior knowledge about temperature, and local thermometry, which
minimizes the standard deviation $\Delta T$ of the estimated temperature [39,
42], which is however lower bounded by the quantum Cramér-Rao bound [27, 29]
$\Delta T\geq 1/\sqrt{{\cal N}\mathcal{F}_{T}}$, where ${\cal N}$ is the
prepared number of measurements and $\mathcal{F}_{T}$ is the quantum Fisher
information [28, 26, 30].

The optimal local quantum thermometry aims to find a quantum system maximizing
the quantum Fisher information of a thermal state with respect to changes in
temperature. The quantum Fisher information for a thermal state
$\hat{\rho}^{B}_{T}$ reads [26, 99]

${\cal
F}_{T}=4\sum_{l,j}p_{l}\frac{\absolutevalue{\bra{\varepsilon_{l}}\partial_{T}\hat{\rho}^{B}_{T}\ket{\varepsilon_{j}}}^{2}}{(p_{l}+p_{j})^{2}}=\frac{\Delta\hat{H}_{B}^{2}}{T^{4}},$
(1)

where
$\Delta\hat{H}_{B}^{2}\equiv\Tr[\hat{\rho}^{B}_{T}\hat{H}_{B}^{2}]-\Tr[\hat{\rho}^{B}_{T}\hat{H}_{B}]^{2}$,
and $\\{\varepsilon_{l},|\varepsilon_{l}\rangle\\}_{l=1}^{D}$ are eigenvalues
and eigenvectors of $\hat{H}_{B}$, and
$\hat{\rho}^{B}_{T}=e^{-\beta\hat{H}_{B}}/{\cal Z}$, where ${\cal
Z}=\sum_{l}^{D}e^{-\beta\varepsilon_{l}}$ and $\beta=1/k_{B}T$ (from here on,
we introduce the natural constants $k_{B}=\hbar=1$), and
$p_{l}=\bra{\varepsilon_{l}}\hat{\rho}^{B}_{T}\ket{\varepsilon_{l}}={\cal
Z}^{-1}e^{-\beta\varepsilon_{l}}$.

As shown in Ref. [39], the variance $\Delta\hat{H}_{B}^{2}$ for a thermal
state is maximized for the $D$-dimensional Hamiltonian $\hat{H}_{B}$ with a
two-level energy spectrum having a single ground state, and $(D-1)$-fold
degenerated excited state, with an energy gap proportional to estimated
temperature, i.e. $\varepsilon_{1}-\varepsilon_{0}=xT$, where $x>0$ is the
solution of the equation $e^{x}=(D-1)(2+x)/(2-x)$, see Fig. 1(d). In such a
case, the QFI for an optimal local quantum thermometer reads 111In the limit
of large degeneracy of the excited state, i.e. $D\to\infty$, one can
approximate $x\simeq\ln{D}$, $2\sqrt{f(D)}\simeq\ln{D}$, maximal QFI reads
$\overline{\cal F}_{T}=(\ln{D}/2T)^{2}$, and the relative temperature
estimation accuracy is bounded from below by $\Delta T/T\geq 2/\sqrt{{\cal
N}}\ln{D}$.

$\overline{{\cal
F}}_{T}=e^{x}x^{2}\frac{D-1}{(D-1+e^{x})^{2}}\frac{1}{T^{2}}.$ (2)

Figure 2: The energy level structure of the quantum thermometer, Eq. (3) in
the SSH limit, i.e. $m=0$, as a function of the dimerization parameter
$\delta$ with $N=L/2-1$ particles (panels (a)-(c)) and with $N=L/2$ particles
(panels (b)-(d)), for system sizes $L=14$ (panels (a)-(b)) and $L=16$ (panels
(c)-(d)). The optimal local quantum thermometer energy level structure is
realized at dimerization limit $\delta=-1$ (topological) with $2N$-fold
degeneracy in the first excited state manifold (panels (a)-(c)), and at
$\delta=1$ (trivial) with $N^{2}$-fold degeneracy (panels (b)-(d)).

The model— As a local quantum thermometer, we consider a system of $N$
spinless fermions in a one-dimensional lattice with $L$ sites described by the
Rice-Mele Hamiltonian

$\hat{H}_{B}=\sum_{i}\left(1+(-1)^{i}\delta\right)\left(\hat{c}^{\dagger}_{i+1}\hat{c}_{i}+h.c.\right)+m\sum_{i}(-1)^{i}\hat{n}_{i},$
(3)

where $\hat{c}^{\dagger}_{i}(\hat{c}_{i}$) are the fermionic creation
(annihilation) operators acting on the $i$-th lattice site fulfilling the
anti-commutation relation $\\{\hat{c}_{i},\hat{c}^{\dagger}_{i}\\}=1$,
$\hat{n}_{i}=\hat{c}_{i}^{\dagger}\hat{c}_{i}$ is the site occupation
operator, $\delta\in[-1,1]$ is the dimerization parameter, and $m\geq 0$ is
the staggered on-site potential strength. The Hilbert space dimension is given
by the binomial coefficient $D=\binom{L}{N}$. In the periodic boundary
condition geometry, assuming the lattice constant of unit length, the single
particle Hamiltonian, Eq. (3), can be expressed in the momentum representation
as a two-band model
$\hat{H}_{B}(k)=h_{x}(k)\hat{\sigma}_{x}+h_{y}\hat{\sigma}_{y}+m\hat{\sigma}_{z}$
with $h_{x}(k)=1+\delta+(1-\delta)\cos k$, $d_{y}(k)=(1-\delta)\sin k$, where
$\hat{\sigma}_{x,y,z}$ are the Pauli operators. At $m=0$, the Hamiltonian
reduces to the SSH model and can be characterized by the topological invariant
given by the winding number $\nu$ [101, 102, 103, 83]. The trivial
dimerization limit $\delta=1$ corresponds to the $\nu=0$, while $\delta=-1$
corresponds to the topological phase, $\nu=1$, supporting the two zero-energy
edge states in open boundary condition geometry.

The experimental platform for the considered quantum system has been realized
in spinful fermions using 171Yb atoms [88], and more recently with potassium
40K atoms [91]. The characteristic temperature related to the recoil energy in
this case is $T_{0}\sim 200$ nK.

Results— The Rice-Mele Hamiltonian, Eq. (3), can serve as a quasi-optimal
local quantum thermometer with open boundary conditions for specific fillings
$f=N/L$, with $N=L/2-1$ or $N=L/2$ particles, where the optimal energy gap can
be tuned by the dimerization parameter $\delta$, and the staggered potential
amplitude $m$.

We start with an analysis of the energy level spectrum for the SSH model, with
vanishing staggered potential amplitude $m=0$. We consider a lattice with
$L=14,16$ sites and open boundary conditions with $N=L/2-1$, and $N=L/2$
particles. With the full many-body exact diagonalization, we calculate energy
level spectra of the Hamiltonian Eq. (3) for each tuple $\\{L,N,\delta\\}$,
$\delta\in[-1,1]$, presented in Fig. 2. Panels (a)-(c) correspond to $N=L/2-1$
particles. The optimal energy level structure is realized at the topological
dimerization limit, $\delta=-1$, where the system has a single ground state
and degenerated first excited state with $2N$-fold degeneracy. At $\delta=1$,
energy levels contain quasi-degeneracy of the ground state, resembling the
energy level structure known from critical thermometry Ref. [53]. Panels
(b)-(d) correspond to $N=L/2$ particles, where the optimal local quantum
thermometer energy level structure is realized at trivial dimerization limit
$\delta=1$ with $N^{2}$-fold degeneracy of the first excited state. For the
$N=L/2-1$ particles and $\delta<0$ the system realizes the quasi-optimal
thermometer with quasi-degeneracy in the first excited state. In contrast, for
$\delta\geq 0$, the system enters a single quasi-degenerated manifold regime,
which is also characterized by high sensitivity to temperature changes [45,
53]. In the following, we study the thermometer’s sensitivity to temperature
changes quantified by the quantum Fisher information.

Figure 3: Top row: (log-log scale) The sensitivity of the SSH quantum
thermometer to the temperature changes in terms of QFI for $L=16$ sites. Panel
(a) corresponds to $N=L/2-1$, while panel (b) to $N=L/2$ particles. Solid thin
lines represent ${\cal F}_{T}$ as a function of temperature $T$, for different
values of the dimerization parameter $\delta$. The thick, solid blue line
represents an envelope of the maximal QFI. The dashed line represents the QFI
bound for an optimal local quantum thermometer with degeneracy corresponding
to the total Hilbert space dimension $D$, while the dash-dotted line
represents the QFI bound for the optimal quantum thermometer corresponding to
the $2N$-fold quasi-degenerated first excited state. For $T\gtrsim 10^{-1}$ in
panel (a), the higher energy levels start to contribute, and hence the
envelope may surpass the fundamental bound (dash-dotted line) where only the
first manifold is considered. Bottom row: the maximal specific heat capacity
$C^{*}$ (see main text) as a function of $\delta$, with increasing system size
$L$ for (c) $N=L/2-1$ and (d) $N=L/2$ fermions.

Quantum Fisher information for the SSH model— We assume the thermometer is
already in the thermal state at temperature $T$, given by
$\hat{\rho}^{B}_{T}$. We focus on the quantum Fisher information, ${\cal
F}_{T}$, Eq. (1) to characterize the system’s sensitivity to temperature
changes.

Fig. 3(a)-(b) represents the quantum Fisher information ${\cal F}_{T}$, as a
function of temperature $T$ for fixed $\delta$ (thin solid lines) for
$N=L/2-1$ and $N=L/2$ particles respectively. The thick, solid blue line
represents the maximal ${\cal F}_{T}$ envelope for different values of
$\delta$. The thermometer’s QFI is close to the QFI of the optimal local
quantum thermometer $\overline{{\cal F}}_{T}$, Eq. (2). In panels (a)-(b), the
dash-dotted black line corresponds to QFI of the optimal local quantum
thermometer with $2N$-fold degeneracy, while the dashed black line to
$(D-1)$-fold degeneracy, corresponding to the situation where all energy
levels form the optimal energy level structure. The QFI has a peaked structure
with the peak at an optimal temperature $T^{*}=\arg\max_{T}{\cal F}_{T}$, at
which the measurement accuracy is maximized. The position of the peak of QFI
can be controlled by tuning the dimerization strength $\delta$.

Figure 4: Top row: Energy level spectrum as a function of the staggered
potential amplitude $m$ for $L=16$ with (a) $N=L/2-1$ and $\delta=-1$, and (b)
$N=L/2$ with $\delta=1$. Insets present the fit (solid red lines) to the
energy gap (blue dots) between the ground state and the first excited energy
manifold. Bottom row: (log-log scale) quantum Fisher information ${\cal
F}_{T}$ for various $m$ (thin solid lines) and the corresponding maximal
quantum Fisher information envelope (thick solid blue line). The dashed line
represents the QFI bound for an optimal local quantum thermometer [39] with
degeneracy in the excited state manifold corresponding to the Hilbert space
dimension $D$, and the dash-dotted line denotes the QFI bound for optimal
local quantum thermometer with the quasi-degeneracy corresponding to the first
excited state manifold in the energy spectrum (panels (a)-(b)).

The potential of a given thermometer for thermometry tasks is encoded into a
more physically relevant quantity which is its specific heat capacity [39,
72]. Here, we consider the maximal specific heat capacity $C^{*}$ of the
thermometer, which is defined as $C^{*}={\cal F}_{T^{*}}T^{*2}$, where $T^{*}$
is the optimal temperature for a given set of model parameters
$\\{L,N,\delta\\}$. Fig. 3 (c)-(d) present $C^{*}$ vs $\delta$ for fixed
system size $L$ with $N=L/2-1$ and $N=L/2$ particles respectively. Specific
heat capacity shows the advantage of the topological dimerization regime
($\delta<0$) over the trivial one ($\delta>0$) in scaling with the system
size.

Quantum Fisher information for the Rice-Mele model— The crucial aspect of the
practical utilization of this system as a local quantum thermometer is its
energy gap tunability. Here, we show that controlling the staggered potential
amplitude $m$ allows adjusting the system’s sensitivity over a few orders of
magnitude of temperature changes. In Fig. 4(a)-(b), we present an energy level
spectrum as a function of the staggered potential amplitude $m$ with $N=L/2-1$
and $N=L/2$ particles, respectively, for $L=16$ sites. For $N=L/2-1$ the first
excited state has $N$-fold degeneracy, and $N^{2}$-fold degeneracy for $N=L/2$
particles. The energy gap between the ground state and the first excited state
manifold decreases exponentially 222$y=Ae^{-Bm}+C$;
$\\{A,B,C\\}=\\{0.76418,0.64130,0.13517\\}$. for $N=L/2-1$, while for $N=L/2$,
the gap increases 333$y=Am^{B}+C$;
$\\{A,B,C\\}=\\{0.35598,1.44097,1.74417\\}$. with increasing staggered
potential $m$.

In Fig. 4(c)-(d), we present the quantum Fisher information ${\cal F}_{T}$ of
the Hamiltonian with fixed dimerization parameter $\delta=-1$ (panel (c),
$N=L/2-1$) and $\delta=1$ (panel (d), $N=L/2$) for different values of the
staggered potential amplitude $m$ (thin solid lines). The control over the
staggered potential $m$ allows tuning the thermometer to the optimal
temperature over three orders of magnitude by changing the position of the
peak of quantum Fisher information ${\cal F}_{T}$.

Measurement protocol— Bounding the sensitivity given by the QFI requires
preparing a measurement operator constructed from the eigenstates of the
thermometer Hamiltonian $\hat{H}_{B}$, which is challenging from the
experimental point of view. To infer the sensitivity of the thermal state
$\hat{\rho}^{B}_{T}$ of the thermometer to temperature changes from the
measurements easily accessible in the experiment, we focus on the classical
Fisher information
$F_{T}=\sum_{i}\frac{1}{p_{i}(T)}\left(\frac{dp_{i}(T)}{dT}\right)^{2}$, where
$p_{i}(T)$ are the probabilities of measurement outcomes. Here, as a
measurement protocol, we consider the normalized lattice site occupation
$p_{i}(T)=\Tr[\hat{n}_{i}\hat{\rho}^{B}_{T}]/N$.

Figure 5: Classical and quantum Fisher information for $N=L/2-1$, $L=16$
lattice sites, topological dimerization limit $\delta=-1$, and staggered
potential $m=10$. The dashed blue lines present classical Fisher information,
$F_{T}$, obtained from lattice site occupation measurements, while the red
solid line represents the quantum Fisher information, ${\cal F}_{T}$, of the
system. The dashed and dash-dotted lines denote the same as in Fig. 4.

In Fig. 5, we show the classical Fisher information $F_{T}$ for $L=16$ sites,
with $N=L/2-1$ at topological dimerization limit $\delta=-1$ with staggered
potential amplitude $m=10$. At the optimal temperature $T^{*}$, the quantum
Fisher information ${\cal F}_{T^{*}}$ has the same value as the quantum Fisher
information $\overline{{\cal F}}_{T^{*}}$ for an idealized optimal local
quantum thermometer with $N$-fold degeneracy of the excited state. The
classical Fisher information at the optimal temperature is $F_{T^{*}}\simeq
10^{2}$, indicating the system’s usefulness as a quantum thermometer.

Discussion and conclusions— We showed that the spinless fermions in a one-
dimensional optical lattice described by the Rice-Mele model in open boundary
conditions geometry can serve as an optimal local quantum thermometer
operating in the sub-nK regime. Based on the parameter estimation theory, we
analyzed the thermal sensitivity of the considered system in terms of quantum
Fisher information. We showed that the staggered field amplitude can control
the quantum thermometer’s energy gap to match the estimated temperature,
allowing us to maximize the sensitivity of the thermometer for a range of
temperature values. Finally, we showed that the classical Fisher information
obtained from more realistic and experimentally feasible lattice occupation
measurements is close to the quantum Fisher information limit.

The primary open question for utilizing the proposed quantum thermometer is
about thermalization dynamics after coupling to another quantum system. Such a
coupling can occur via contact interactions with another system of quantum
particles confined in a lattice or via induced long-range Rydberg-dressed
interactions leading to unconstrained energy transfer between the quantum
thermometer [106, 107] and the quantum gas in a harmonic trap [108].

Acknowledgements— We thank Grzegorz Rajchel-Mieldzioć and Arkadiusz Kosior for
carefully reading the manuscript. A.K.S. acknowledges support from the
European Union’s Horizon 2020 Research and Innovation Programme under the
Marie Skłodowska-Curie Grant Agreement No. 847517. M.P. acknowledges the
support of the Polish National Agency for Academic Exchange, the Bekker
programme no: PPN/BEK/2020/1/00317. ICFO group acknowledges support from: ERC
AdG NOQIA; MCIN/AEI (PGC2018-0910.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, Plan National STAMEENA PID2022-139099NB-I00 project
funded by MCIN/AEI/10.13039/501100011033 and by the “European Union
NextGenerationEU/PRTR” (PRTR-C17.I1), FPI); QUANTERA MAQS PCI2019-111828-2);
QUANTERA DYNAMITE PCI2022-132919 (QuantERA II Programme co-funded by European
Union’s Horizon 2020 program under Grant Agreement No 101017733), Ministry of
Economic Affairs and Digital Transformation of the Spanish Government through
the QUANTUM ENIA project call – Quantum Spain project, and by the European
Union through the Recovery, Transformation, and Resilience Plan –
NextGenerationEU within the framework of the Digital Spain 2026 Agenda;
Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social
Fund FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2023-1-0013); EU Quantum
Flagship (PASQuanS2.1, 101113690); EU Horizon 2020 FET-OPEN OPTOlogic (Grant
No 899794); EU Horizon Europe Program (Grant Agreement 101080086 — NeQST),
ICFO Internal “QuantumGaudi” project; European Union’s Horizon 2020 program
under the Marie Sklodowska-Curie grant agreement No 847648; “La Caixa” Junior
Leaders fellowships, La Caixa” Foundation (ID 100010434): CF/BQ/PR23/11980043.
Views and opinions expressed are, however, those of the author(s) only and do
not necessarily reflect those of the European Union, European Commission,
European Climate, Infrastructure and Environment Executive Agency (CINEA), or
any other granting authority. Neither the European Union nor any granting
authority can be held responsible for them.

## References

  * Quan  _et al._ [2007] H. T. Quan, Y. Liu, C. P. Sun, and F. Nori, Physical Review E 76, 031105 (2007).
  * Kosloff and Levy [2014] R. Kosloff and A. Levy, Annual Review of Physical Chemistry 65, 365 (2014).
  * Campisi  _et al._ [2015] M. Campisi, J. Pekola, and R. Fazio, New Journal of Physics 17, 035012 (2015).
  * Peterson  _et al._ [2019] J. P. S. Peterson, T. B. Batalhão, M. Herrera, A. M. Souza, R. S. Sarthour, I. S. Oliveira, and R. M. Serra, Physical Review Letters 123, 240601 (2019).
  * Myers  _et al._ [2022] N. M. Myers, O. Abah, and S. Deffner, AVS Quantum Science 4, 027101 (2022).
  * Gemmer  _et al._ [2009] J. Gemmer, M. Michel, and G. Mahler,  _Quantum thermodynamics: Emergence of thermodynamic behavior within composite quantum systems_, Vol. 784 (Springer, 2009).
  * Vinjanampathy and Anders [2016] S. Vinjanampathy and J. Anders, Contemporary Physics 57, 545 (2016).
  * Goold  _et al._ [2016] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, Journal of Physics A: Mathematical and Theoretical 49, 143001 (2016).
  * Binder  _et al._ [2018] F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, Fundamental Theories of Physics 195, 1 (2018).
  * Deffner and Campbell [2019] S. Deffner and S. Campbell,  _Quantum Thermodynamics_, 2053-2571 (Morgan & Claypool Publishers, 2019).
  * Bhattacharjee and Dutta [2021] S. Bhattacharjee and A. Dutta, The European Physical Journal B 94, 1 (2021).
  * Julià-Farré  _et al._ [2020] S. Julià-Farré, T. Salamon, A. Riera, M. N. Bera, and M. Lewenstein, Physical Review Research 2, 023113 (2020).
  * Mondal and Bhattacharjee [2022] S. Mondal and S. Bhattacharjee, Physical Review E 105, 044125 (2022).
  * Ghosh  _et al._ [2020] S. Ghosh, T. Chanda, and A. Sen(De), Physical Review A 101, 032115 (2020).
  * Ghosh  _et al._ [2021] S. Ghosh, T. Chanda, S. Mal, and A. Sen(De), Physical Review A 104, 032207 (2021).
  * Hoang  _et al._ [2023] D. T. Hoang, F. Metz, A. Thomasen, T. D. Anh-Tai, T. Busch, and T. Fogarty, Variational quantum algorithm for ergotropy estimation in quantum many-body batteries (2023), arXiv:2308.03334 [quant-ph] .
  * Boubakour  _et al._ [2023] M. Boubakour, T. Fogarty, and T. Busch, Physical Review Research 5, 013088 (2023).
  * Koch  _et al._ [2023] J. Koch, K. Menon, E. Cuestas, S. Barbosa, E. Lutz, T. Fogarty, T. Busch, and A. Widera, Nature 621, 723 (2023).
  * Leanhardt  _et al._ [2003] A. E. Leanhardt, T. A. Pasquini, M. Saba, A. Schirotzek, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, Science 301, 1513 (2003).
  * Medley  _et al._ [2011] P. Medley, D. M. Weld, H. Miyake, D. E. Pritchard, and W. Ketterle, Physical Review Letters 106, 195301 (2011).
  * Horodecki and Oppenheim [2013] M. Horodecki and J. Oppenheim, Nature Communications 4, 2059 (2013).
  * Carlos and Palacio [2015] L. D. Carlos and F. Palacio,  _Thermometry at the nanoscale: techniques and selected applications_ (Royal Society of Chemistry, 2015).
  * Hovhannisyan and Correa [2018] K. V. Hovhannisyan and L. A. Correa, Physical Review B 98, 045101 (2018).
  * Lipka-Bartosik  _et al._ [2023] P. Lipka-Bartosik, M. Perarnau-Llobet, and N. Brunner, Physical Review Letters 130, 040401 (2023).
  * Bhattacharjee  _et al._ [2018] S. Bhattacharjee, U. Bhattacharya, and A. Dutta, Physical Review B 98, 104302 (2018).
  * Braunstein and Caves [1994] S. L. Braunstein and C. M. Caves, Physical Review Letters 72, 3439 (1994).
  * Cramér [1946] H. Cramér,  _Mathematical Methods of Statistics (PMS-9), Volume 9_ (Princeton University Press, Princeton, 1946).
  * Helstrom [1969] C. W. Helstrom, Journal of Statistical Physics 1, 231 (1969).
  * Giovannetti  _et al._ [2011] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5, 222 (2011).
  * Pezzè  _et al._ [2018] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Reviews of Modern Physics 90 (2018).
  * Pati  _et al._ [2020] A. K. Pati, C. Mukhopadhyay, S. Chakraborty, and S. Ghosh, Physical Review A 102, 012204 (2020).
  * Demkowicz-Dobrzański  _et al._ [2020] R. Demkowicz-Dobrzański, W. Górecki, and M. Guţă, Journal of Physics A: Mathematical and Theoretical 53, 363001 (2020).
  * Müller-Rigat  _et al._ [2023] G. Müller-Rigat, A. K. Srivastava, S. Kurdziałek, G. Rajchel-Mieldzioć, M. Lewenstein, and I. Frérot, Quantum 7, 1152 (2023).
  * Frérot  _et al._ [2023] I. Frérot, M. Fadel, and M. Lewenstein, Reports on Progress in Physics 86, 114001 (2023).
  * Campbell  _et al._ [2017] S. Campbell, M. Mehboudi, G. D. Chiara, and M. Paternostro, New Journal of Physics 19, 103003 (2017).
  * Mok  _et al._ [2021] W.-K. Mok, K. Bharti, L.-C. Kwek, and A. Bayat, Communications Physics 4, 62 (2021).
  * Rubio  _et al._ [2021] J. Rubio, J. Anders, and L. A. Correa, Physical Review Letters 127, 190402 (2021).
  * Glatthard  _et al._ [2022] J. Glatthard, J. Rubio, R. Sawant, T. Hewitt, G. Barontini, and L. A. Correa, PRX Quantum 3, 040330 (2022).
  * Correa  _et al._ [2015] L. A. Correa, M. Mehboudi, G. Adesso, and A. Sanpera, Physical Review Letters 114, 220405 (2015).
  * De Pasquale  _et al._ [2016] A. De Pasquale, D. Rossini, R. Fazio, and V. Giovannetti, Nature Communications 7, 12782 (2016).
  * De Pasquale and Stace [2018] A. De Pasquale and T. M. Stace, in  _Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions_ (Springer International Publishing, Cham, 2018) pp. 503–527.
  * Mehboudi  _et al._ [2019a] M. Mehboudi, A. Sanpera, and L. A. Correa, Journal of Physics A: Mathematical and Theoretical 52, 303001 (2019a).
  * Boeyens  _et al._ [2021] J. Boeyens, S. Seah, and S. Nimmrichter, Physical Review A 104, 052214 (2021).
  * Mehboudi  _et al._ [2022] M. Mehboudi, M. R. Jørgensen, S. Seah, J. B. Brask, J. Kołodyński, and M. Perarnau-Llobet, Physical Review Letters 128, 130502 (2022).
  * Płodzień  _et al._ [2018] M. Płodzień, R. Demkowicz-Dobrzański, and T. Sowiński, Physical Review A 97, 063619 (2018).
  * Płodzień  _et al._ [2018] M. Płodzień, D. Wiater, A. Chrostowski, and T. Sowiński, Numerically exact approach to few-body problems far from a perturbative regime (2018), arXiv:1803.08387 [cond-mat.quant-gas] .
  * Zgirski  _et al._ [2018] M. Zgirski, M. Foltyn, A. Savin, K. Norowski, M. Meschke, and J. Pekola, Physical Review Applied 10, 044068 (2018).
  * Brange  _et al._ [2018] F. Brange, P. Samuelsson, B. Karimi, and J. P. Pekola, Physical Review B 98, 205414 (2018).
  * Brody and Rivier [1995] D. Brody and N. Rivier, Physical Review E 51, 1006 (1995).
  * Zanardi  _et al._ [2007] P. Zanardi, P. Giorda, and M. Cozzini, Physical Review Letters 99, 100603 (2007).
  * Invernizzi  _et al._ [2008] C. Invernizzi, M. Korbman, L. Campos Venuti, and M. G. A. Paris, Physical Review A 78, 042106 (2008).
  * Mehboudi  _et al._ [2016] M. Mehboudi, L. A. Correa, and A. Sanpera, Physical Review A 94, 042121 (2016).
  * Aybar  _et al._ [2022] E. Aybar, A. Niezgoda, S. S. Mirkhalaf, M. W. Mitchell, D. Benedicto Orenes, and E. Witkowska, Quantum 6, 808 (2022).
  * Walker  _et al._ [2003] G. W. Walker, V. C. Sundar, C. M. Rudzinski, A. W. Wun, M. G. Bawendi, and D. G. Nocera, Applied Physics Letters 83, 3555 (2003).
  * Haupt  _et al._ [2014] F. Haupt, A. Imamoglu, and M. Kroner, Physical Review Applied 2, 024001 (2014).
  * Seilmeier  _et al._ [2014] F. Seilmeier, M. Hauck, E. Schubert, G. J. Schinner, S. E. Beavan, and A. Högele, Physical Review Applied 2, 024002 (2014).
  * Chekhovich  _et al._ [2017] E. Chekhovich, A. Ulhaq, E. Zallo, F. Ding, O. Schmidt, and M. Skolnick, Nature Materials 16, 982 (2017).
  * Fujiwara and Shikano [2021] M. Fujiwara and Y. Shikano, Nanotechnology 32, 482002 (2021).
  * Razavian  _et al._ [2019] S. Razavian, C. Benedetti, M. Bina, Y. Akbari-Kourbolagh, and M. G. A. Paris, The European Physical Journal Plus 134, 284 (2019).
  * Sabín  _et al._ [2014] C. Sabín, A. White, L. Hackermuller, and I. Fuentes, Scientific Reports 4, 1 (2014).
  * Correa  _et al._ [2017] L. A. Correa, M. Perarnau-Llobet, K. V. Hovhannisyan, S. Hernández-Santana, M. Mehboudi, and A. Sanpera, Physical Review A 96 (2017).
  * Mehboudi  _et al._ [2019b] M. Mehboudi, A. Lampo, C. Charalambous, L. A. Correa, M. Á. García-March, and M. Lewenstein, Physical Review Letters 122, 030403 (2019b).
  * Khan  _et al._ [2022] M. M. Khan, M. Mehboudi, H. Terças, M. Lewenstein, and M. A. Garcia-March, Physical Review Research 4, 023191 (2022).
  * Glatthard and Correa [2022] J. Glatthard and L. A. Correa, Quantum 6, 705 (2022).
  * Lous  _et al._ [2017] R. S. Lous, I. Fritsche, M. Jag, B. Huang, and R. Grimm, Physical Review A 95, 053627 (2017).
  * Mitchison  _et al._ [2020a] M. T. Mitchison, T. Fogarty, G. Guarnieri, S. Campbell, T. Busch, and J. Goold, Physical Review Letters 125, 080402 (2020a).
  * Mitchison  _et al._ [2020b] M. T. Mitchison, T. Fogarty, G. Guarnieri, S. Campbell, T. Busch, and J. Goold, Physical Review Letters 125, 080402 (2020b).
  * Oghittu and Negretti [2022] L. Oghittu and A. Negretti, Physical Review Research 4 (2022).
  * Rodríguez  _et al._ [2023] R. R. Rodríguez, M. Mehboudi, M. Horodecki, and M. Perarnau-Llobet, Strongly coupled fermionic probe for nonequilibrium thermometry (2023), arXiv:2310.14655 [quant-ph] .
  * Robles and Rodríguez-Laguna [2017] S. Robles and J. Rodríguez-Laguna, Journal of Statistical Mechanics: Theory and Experiment 2017, 033105 (2017).
  * Martín-Martínez  _et al._ [2013] E. Martín-Martínez, A. Dragan, R. B. Mann, and I. Fuentes, New Journal of Physics 15, 053036 (2013).
  * Abiuso  _et al._ [2023] P. Abiuso, P. A. Erdman, M. Ronen, F. Noé, G. Haack, and M. Perarnau-Llobet, Optimal thermometers with spin networks (2023), arXiv:2211.01934 [quant-ph] .
  * Brenes and Segal [2023] M. Brenes and D. Segal, Physical Review A 108, 032220 (2023).
  * Dawid  _et al._ [2022] A. Dawid, J. Arnold, B. Requena, A. Gresch, M. Płodzień, K. Donatella, K. A. Nicoli, P. Stornati, R. Koch, M. Büttner, R. Okuła, G. Muñoz-Gil, R. A. Vargas-Hernández, A. Cervera-Lierta, J. Carrasquilla, V. Dunjko, M. Gabrié, P. Huembeli, E. van Nieuwenburg, F. Vicentini, L. Wang, S. J. Wetzel, G. Carleo, E. Greplová, R. Krems, F. Marquardt, M. Tomza, M. Lewenstein, and A. Dauphin, Modern applications of machine learning in quantum sciences (2022), arXiv:2204.04198 [quant-ph] .
  * Luiz  _et al._ [2022] F. S. Luiz, A. d. O. Junior, F. F. Fanchini, and G. T. Landi, Physical Review A 105, 022413 (2022).
  * Salvia  _et al._ [2023] R. Salvia, M. Mehboudi, and M. Perarnau-Llobet, Physical Review Letters 130, 240803 (2023).
  * Rice and Mele [1982] M. J. Rice and E. J. Mele, Physical Review Letters 49, 1455 (1982).
  * Su  _et al._ [1979] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Physical Review Letters 42, 1698 (1979).
  * Heeger  _et al._ [1988] A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su, Reviews of Modern Physics 60, 781 (1988).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Reviews of Modern Physics 83, 1057 (2011).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Reviews of Modern Physics 82, 3045 (2010).
  * Ryu  _et al._ [2010] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, New Journal of Physics 12, 065010 (2010).
  * Asbóth  _et al._ [2016] J. K. Asbóth, L. Oroszlány, and A. Pályi,  _A Short Course on Topological Insulators_ (Springer International Publishing, 2016).
  * Argüello-Luengo  _et al._ [2023] J. Argüello-Luengo, U. Bhattacharya, A. Celi, R. W. Chhajlany, T. Grass, M. Płodzień, D. Rakshit, T. Salamon, P. Stornati, L. Tarruell, and M. Lewenstein, Synthetic dimensions for topological and quantum phases: Perspective (2023), arXiv:2310.19549 [quant-ph] .
  * Atala  _et al._ [2013] M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, Nature Physics 9, 795 (2013).
  * Hayward  _et al._ [2018] A. Hayward, C. Schweizer, M. Lohse, M. Aidelsburger, and F. Heidrich-Meisner, Physical Review B 98, 245148 (2018).
  * Lohse  _et al._ [2015] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, Nature Physics 12, 350 (2015).
  * Nakajima  _et al._ [2016] S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Nature Physics 12, 296 (2016).
  * Reid  _et al._ [2022] G. H. Reid, M. Lu, A. R. Fritsch, A. M. Piñeiro, and I. B. Spielman, Physical Review Letters 129, 123202 (2022).
  * Cooper  _et al._ [2019] N. R. Cooper, J. Dalibard, and I. B. Spielman, Reviews of Modern Physics 91, 015005 (2019).
  * Walter  _et al._ [2023] A.-S. Walter, Z. Zhu, M. Gächter, J. Minguzzi, S. Roschinski, K. Sandholzer, K. Viebahn, and T. Esslinger, Nature Physics 19, 1471 (2023).
  * Li  _et al._ [2018] X. Li, Y. Meng, X. Wu, S. Yan, Y. Huang, S. Wang, and W. Wen, Applied Physics Letters 113, 203501 (2018).
  * Rizzo  _et al._ [2018] D. J. Rizzo, G. Veber, T. Cao, C. Bronner, T. Chen, F. Zhao, H. Rodriguez, S. G. Louie, M. F. Crommie, and F. R. Fischer, Nature 560, 204 (2018).
  * Li  _et al._ [2014] L. Li, Z. Xu, and S. Chen, Physical Review B 89, 085111 (2014).
  * Giergiel  _et al._ [2019] K. Giergiel, A. Dauphin, M. Lewenstein, J. Zakrzewski, and K. Sacha, New Journal of Physics 21, 052003 (2019).
  * de Léséleuc  _et al._ [2019] S. de Léséleuc, V. Lienhard, P. Scholl, D. Barredo, S. Weber, N. Lang, H. P. Büchler, T. Lahaye, and A. Browaeys, Science 365, 775 (2019).
  * Weber  _et al._ [2018] S. Weber, S. de Léséleuc, V. Lienhard, D. Barredo, T. Lahaye, A. Browaeys, and H. P. Büchler, Quantum Science and Technology 3, 044001 (2018).
  * Lienhard  _et al._ [2019] V. Lienhard, S. de Léséleuc, P. Scholl, D. Barredo, T. Lahaye, and A. Browaeys, in  _Quantum Information and Measurement (QIM) V: Quantum Technologies_ (Optica Publishing Group, 2019) p. F4B.2.
  * Jing  _et al._ [2014] L. Jing, J. Xiao-Xing, Z. Wei, and W. Xiao-Guang, Communications in Theoretical Physics 61, 45 (2014).
  * Note [1] In the limit of large degeneracy of the excited state, i.e. $D\to\infty$, one can approximate $x\simeq\ln{D}$, $2\sqrt{f(D)}\simeq\ln{D}$, maximal QFI reads $\overline{\cal F}_{T}=(\ln{D}/2T)^{2}$, and the relative temperature estimation accuracy is bounded from below by $\Delta T/T\geq 2/\sqrt{{\cal N}}\ln{D}$.
  * Resta [1998] R. Resta, Physical Review Letters 80, 1800 (1998).
  * Resta [1994] R. Resta, Reviews of Modern Physics 66, 899 (1994).
  * Sen and Gupta [2015] S. Sen and K. S. Gupta,  _Many-Body Physics, Topology and Geometry_ (World Scientific, 2015).
  * Note [2] $y=Ae^{-Bm}+C$;
$\\{A,B,C\\}=\\{0.76418,0.64130,0.13517\\}$.

  * Note [3] $y=Am^{B}+C$;
$\\{A,B,C\\}=\\{0.35598,1.44097,1.74417\\}$.

  * Macrì and Pohl [2014] T. Macrì and T. Pohl, Physical Review A 89, 011402 (2014).
  * Malz and Cirac [2023] D. Malz and J. I. Cirac, PRX Quantum 4, 020301 (2023).
  * Płodzień  _et al._ [2017] M. Płodzień, G. Lochead, J. de Hond, N. J. van Druten, and S. Kokkelmans, Physical Review A 95, 043606 (2017).


# Modified Gravitational Equations on Braneworld with Lorentz Invariant
Violation

Arianto(1,2,3<EMAIL_ADDRESS>Freddy P. Zen(1,2<EMAIL_ADDRESS>Bobby E.
Gunara(1,2<EMAIL_ADDRESS>(1)Theoretical Physics Lab., THEPI Devision,
and
(2)Indonesia Center for Theoretical and Mathematical Physics (ICTMP)
Faculty of Mathematics and Natural Sciences,
Institut Teknologi Bandung,
Jl. Ganesha 10 Bandung 40132, INDONESIA.
(3)Department of Physics, Udayana University
Jl. Kampus Bukit Jimbaran Kuta-Bali 80361, INDONESIA.

###### Abstract

The modified gravitational equations to describe a four-dimensional braneworld
in the case with the Lorentz invariant violation in a bulk spacetime is
presented. It contains a trace part of the brane energy-momentum tensor and
the coefficients of all terms describe the Lorentz violation effects from the
bulk spacetime. As an application, we apply this formalism to study cosmology.
In respect to standard effective Friedmann equations on the brane, Lorentz
invariance violation in the bulk causes a modification of this equations that
can lead to significant physical consequences. In particular, the effective
Friedmann equation on the brane explicitly depends on the equation of state of
the brane matter and the Lorentz violating parameters. We show that the
components of five-dimensional Weyl curvature are related to the matter on
brane even at low energies. We also find that the constraints on the theory
parameters are depend on the equation of state of the energy components of the
brane matter. Finally, the stability of the model depend on the specific
choices of initial conditions and the parameters $\beta_{i}$.

###### pacs:

98.80.Cq, 98.80.Hw

##  I Introduction

There has been a growing appreciation of the importance of the violations of
Lorentz invariance recently. The intriguing possibility of the Lorentz
violation is that an unknown physics at high-energy scales could lead to a
spontaneous breaking of Lorentz invariance by giving an expectation value to
certain non Standard Model fields that carry Lorentz indices, such as vectors,
tensors, and gradients of scalar fields Kostelecky:1988zi . A relativistic
theory of gravity where gravity is mediated by a tensor, a vector, and a
scalar field, thus called TeVeS gravitational theory Bekenstein:2004ne ,
provides modified Newtonian dynamics (MOND) and Newtonian limits in the weak
field nonrelativistic limit. TeVeS could also explain the large-scale
structure formation of the Universe without recurring to cold dark matter
Skordis:2005xk , which is composed of very massive slowly moving and weakly
interacting particles. On the other hand, the Einstein–Aether theory
Jacobson:2000xp  is a vector-tensor theory, and TeVeS can be written as a
vector-tensor theory which is the extension of the Einstein–Aether theory
Zlosnik:2006sb . In the case of generalized Einstein–Aether theory
Zlosnik:2006zu , the effect of a general class of such theories on the solar
system has been considered in Ref. Bonvin:2007ap . On small scales the
Einstein-Aether vector field will in general lead to a renormalization of the
local Newton Constant Carroll:2004ai . Moreover, as has been shown by authors
in Ref. Li:2007vz , the Einstein–Aether theory may lead to significant
modifications of the power spectrum of tensor perturbation. The existence of
vector fields in a scalar-vector-tensor theory of gravity also leads to its
applications in modern cosmology and it might explain inflationary scenarios
Lim:2004js ; Kanno:2006ty ; Watanabe:2009ct ; Avelino:2009wj  and accelerated
expansion of the universe Zlosnik:2006zu ; Tartaglia:2007mh . Based on a
dynamical vector field coupled to the gravitation and scalar fields, we have
studied to some extent the cosmological implications of a scalar-vector-tensor
theory of gravity :2007xt . The models also allow crossing of phantom divide
line Nozari:2008ff .

Motivated by string theory and its extension M-theory, the standard model
particles may be confined on a hypersurface, called brane, embedded in a
higher dimensional space, called bulk. Only gravity and other exotic matter
such as the dilaton can propagate in the bulk Horava:1995qa . The braneworld
models have been shown to be extremely rich in phenomena leading to
modifications of General Relativity (GR) at both low and high energies
Maartens:2003tw . In the context of gravity and cosmology, models proposed by
Randall and Sundrum (RS) Randall:1999ee ; Randall:1999vf  have attracted much
attention, where four-dimensional gravity can be recovered at low energy
despite the infinite size of the extra dimension. In RS II model
Randall:1999vf , a positive tension brane is embedded in five-dimensional
anti-de Sitter (AdS) spacetime. To study gravity on the brane, it is useful to
derive the effective four-dimensional Einstein equation on the brane firstly
developed by Shiromizu, Maeda, and Sasaki (SMS) Shiromizu:1999wj . There are
two very important results that arise from the effective four-dimensional
Einstein equations on the brane. The first one is quadratic energy-momentum
tensor, $\pi_{\mu\nu}$, which is relevant in high energy and the second one is
the projected Weyl tensor, $E_{\mu\nu}$, on the brane which is responsible for
carrying on the brane the contribution of the bulk gravitational field. In the
RS II models, this term supplies an additional matter-like effect to the
brane. Thus, its contribution to the four-dimensional effective theory is of
crucial importance as it is non-negligible already even in low energy limit.
Then, the Friedmann equations on the brane, governing the cosmological
evolution of the brane, are non conventional in that the Hubble parameter
depends quadratically on the energy density instead of linearly as in standard
cosmology, and one radiation like term, usually referred to as a dark
radiation term in the homogeneous and isotropic background spacetime. This
dark radiation modifies the expansion of the background universe in the same
way as an usual radiation Ida:1999ui ; Kraus:1999it ; Mukohyama:1999qx ;
Ichiki:2002eh .

Recently, a braneworld scenario with bulk broken Lorentz invariance has been
developed, where a family of static self-tuning braneworld solutions was found
Koroteev:2009xd . In a different approach braneworld model, a bulk vector
field with a non-vanishing vacuum expectation value, allowing for the
spontaneous breaking of the Lorentz symmetry. The breaking of Lorentz
invariance the loss of this symmetry is transmitted to the gravitational
sector of the model. By assuming that the vacuum expectation value of the
component of the vector field normal to the brane vanishes, it found that
Lorentz invariance on the brane can be made exact via the dynamics of the
graviton, vector field, and the geometry of the extrinsic curvature of the
surface of the brane. As a consequence of the exact reproduction of Lorentz
symmetry on the brane, a condition for the matching of the observed
cosmological constant in four dimensions is found Bertolami:2006bf . The
notion of Lorentz violation in four dimensions is extended to a five-
dimensional braneworld scenario resulting the time variation in the
gravitational coupling and cosmological constant. There exist also a relation
between the maximal velocity in the bulk and the speed of light on the brane
Ahmadi:2006cr . Various Lorentz violating effects within the context of the
braneworld scenario have also been studied in Refs. Csaki:2000dm ;
Stoica:2001qe ; Libanov:2005yf ; Nozari:2008rg ; Farakos:2009ui .

In this paper we address the issue of cosmological evolution on a brane in a
theory of gravity whose action includes, in addition to the familiar Einstein
term, a Lorentz violating vector field contribution. We generalize the
gravitational effects of the vector fields in four dimensions Jacobson:2000xp
; Kostelecky:2003fs  to include five dimensional braneworld gravity. In
particular, we put a vector $n^{a}$ in the direction of the extra dimension
such that the existence of the brane defines a preferred direction in the
bulk.

This paper is organized as follows. In Section II, we derive the four-
dimensional effective Einstein equations on the brane in the case with the
Lorentz invariant violation in a bulk spacetime. With non-ignoring of the
Lorentz violation effects, this equation is modified by the trace of the brane
energy–momentum tensor. Thus the relation between the projected Weyl tensor
and the brane matter may be understood. In Section III, we study the
cosmological implications of the modified four-dimensional effective Einstein
equations on the brane. In general, the effective four-dimensional Einstein
equations on the brane cannot be solved without knowing $E_{\mu\nu}$, because
it could have a non-trivial component of an anisotropic stress Maartens:2000fg
. However, it is possible to know some features of this tensor by using
constraint equations on the brane obtained by the four-dimensional Bianchi
identity. In the background spacetime, the four-dimensional equations are
sufficient to show that $E_{\mu\nu}$ induces the radiation fluid on the brane.
We will take this strategy to determine the Friedmann equation on the brane.
Interestingly, the Friedmann equation is found to depend on the equation of
state of the matter explicitly, and the Lorentz violation parameters. In
Section IV, we discuss a low energy limit of the theory. Remarkable, the
parameters of the theory can be determined by equation of state of the brane
matter. Section V is devoted to the conclusions.

##  II Modified SMS Effective Equation on the brane

In this section, we derive the $4$-dimensional effective gravitational
equations in a $Z_{2}$-symmetric braneworld using the geometrical projection
approach. For this purpose, we first write the $5$-dimensional field equations
in the form of the evolution equations along the extra dimension and the
constraint equations.

The action we consider consists of the vector field $n^{a}$ minimally coupled
to gravity:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{-\tilde{g}}\left({\cal R}-2\Lambda\right)+\int
d^{5}x\sqrt{-\tilde{g}}{\cal L}_{n}+\int d^{4}x\sqrt{-g}(-\sigma+{\cal
L}_{m})\ .$ (1)

Here, $\mathcal{R}$, $\kappa$, $\Lambda$, and $\tilde{g}$ are the scalar
curvature, the gravitational constant in $5$-dimensions, the bulk cosmological
constant, and the determinant of $5$-dimensional metric, respectively. ${\cal
L}_{m}$ and ${\cal L}_{n}$ are the Lagrangian density for the matter fields on
the brane and the vector field Lagrangian, respectively. A metric $g$ is the
induced metric on the brane while $\sigma$ denotes the brane tension. Note
that we have assumed no coupling between the matter fields and the vector
field in the action (1). Therefore, the brane observer does not feel the
present of the preferred frame.

We write the coordinate system for the bulk spacetime in the form

$ds^{2}=g_{ab}dx^{a}dx^{b}=dy^{2}+g_{\mu\nu}(y,x)dx^{\mu}dx^{\nu}\ ,$ (2)

and we may assume that the position of the brane is $y=0$ in this coordinate
system so that the induced metric on the brane is
$g_{\mu\nu}(x)=\tilde{g}_{\mu\nu}(y=0,x)$. We also assume a $Z_{2}$-symmetry
across the brane and the extrinsic curvature is defined as
$K_{\mu\nu}=-g_{\mu\nu,y}/2$.

The vector field Lagrangian, ${\cal L}_{n}$, is given by

$\displaystyle{\cal L}_{n}$ $\displaystyle=$
$\displaystyle-\beta_{1}\nabla^{a}n^{b}\nabla_{a}n_{b}-\beta_{2}\left(\nabla_{a}n^{a}\right)^{2}-\beta_{3}\nabla^{a}n^{b}\nabla_{b}n_{a}+\lambda(n^{a}n_{a}-1)\
,$ (3)

where $\beta_{i}$ are constant parameters and $\lambda$ is a Lagrangian
multiplier. In this setup, we assume that $n^{a}$ is a vector field along the
extra dimension and the preferred frame is selected by the constrained vector
field $n^{a}$ which violates Lorentz symmetry. We take $n^{a}$ as the
dimensionless vector. Hence, each $\beta_{i}$ has dimension of $(mass)^{3}$.
In other words, $\beta_{i}^{1/3}$ gives the mass scale of symmetry breakdown
in the bulk. Following the usual braneworld scenarios our spacetime is
orthogonal to the extra dimension. Then one can introduce the normal unit
vector $n^{a}$ which is orthogonal to the hypersurfaces at $y=const$. as
$n^{a}=\delta_{y}^{a}$. In particular, there is a background solution that
5-vector takes on a vacuum expectation value with components $(0,0,0,0,1)$,
thus allowing for the spontaneous breaking of the Lorentz symmetry.

Varying the action (1) with respect to the metric, $\lambda$, and $n^{a}$,
respectively, we have the field equations

$\displaystyle{}^{(5)}G_{ab}$ $\displaystyle=$ $\displaystyle-\Lambda
g_{ab}+\kappa^{2}(T_{ab}+{\cal
T}_{ab})+\kappa^{2}\delta_{a}^{\mu}\delta_{b}^{\nu}S_{\mu\nu}\delta(y)\ ,$ (4)
$\displaystyle g_{ab}n^{a}n^{b}$ $\displaystyle=$ $\displaystyle 1\ ,$ (5)
$\displaystyle\nabla_{a}J^{ab}$ $\displaystyle=$ $\displaystyle\lambda n^{b}\
,$ (6)

where current tensor $J^{a}{}_{c}$ is given by

$J^{a}{}_{b}=-\beta_{1}\nabla^{a}n_{b}-\beta_{2}\delta^{a}_{b}\nabla_{c}n^{c}-\beta_{3}\nabla_{b}n^{a}\
,$ (7)

and $S_{\mu\nu}=-\sigma g_{\mu\nu}+\tau_{\mu\nu}$ is the energy momentum
tensor on the brane, where $\tau_{\mu\nu}$ is the energy momentum tensor of
the brane matter other than the tension. $T_{ab}$ is the energy–momentum
tensor of the vector field. To be as general as possible, we also have
included a bulk energy–momentum tensor in (4), denoted by ${\cal T}_{ab}$.

Using the extrinsic curvature, the components of the left hand side of
Einstein equations (4) are

$\displaystyle{}^{(5)}G^{y}{}_{y}$ $\displaystyle=$ $\displaystyle-{1\over
2}R+{1\over 2}K^{2}-{1\over
2}K^{\alpha\beta}K_{\alpha\beta}=-\Lambda+\kappa^{2}T^{y}{}_{y}+\kappa^{2}{\cal
T}^{y}{}_{y}\ ,$ (8) $\displaystyle{}^{(5)}G^{y}{}_{\mu}$ $\displaystyle=$
$\displaystyle-
D_{\alpha}K_{\mu}{}^{\alpha}+D_{\mu}K=\kappa^{2}(T^{y}{}_{\mu}+{\cal
T}^{y}{}_{\mu})\ ,$ (9) $\displaystyle{}^{(5)}G^{\mu}{}_{\nu}$
$\displaystyle=$ $\displaystyle
G^{\mu}{}_{\nu}+(K^{\mu}{}_{\nu}-\delta^{\mu}_{\nu}K)_{,y}+{1\over
2}\delta^{\mu}_{\nu}(K^{2}+K^{\alpha\beta}K_{\alpha\beta})$ (10)
$\displaystyle=$
$\displaystyle-\Lambda\delta^{\mu}_{\nu}+\kappa^{2}(T^{\mu}{}_{\nu}+{\cal
T}^{\mu}{}_{\nu})+\kappa^{2}S^{\mu}{}_{\nu}\delta(y)\ ,$

where $G^{\mu}{}_{\nu}$ is the $4$-dimensional Einstein tensor and the
covariant derivatives $D_{\mu}$ is calculated with respect to the four-
dimensional metric $g_{\mu\nu}$. The components of the energy momentum tensor
of the vector field are given by

$\displaystyle T^{y}{}_{y}$ $\displaystyle=$
$\displaystyle\beta_{2}K^{2}+(\beta_{1}+\beta_{3})K^{\alpha\beta}K_{\alpha\beta}\
,$ (11) $\displaystyle T^{y}{}_{\mu}$ $\displaystyle=$ $\displaystyle 0\ ,$
(12) $\displaystyle T^{\mu}{}_{\nu}$ $\displaystyle=$ $\displaystyle
2(\beta_{1}+\beta_{3})K^{\mu}{}_{\nu}K+\beta_{2}\delta^{\mu}{}_{\nu}K^{2}-\delta^{\mu}_{\nu}(\beta_{1}+\beta_{3})K^{\alpha\beta}K_{\alpha\beta}-2(\beta_{1}+\beta_{3})K^{\mu}{}_{\nu,y}-2\beta_{2}\delta^{\mu}_{\nu}K_{,y}\
.$ (13)

Combining Eqs. (8) with (10) and using (11) and (13), we have

$\displaystyle-{1\over 3}\left(R^{\mu}{}_{\nu}-{1\over
4}\delta^{\mu}_{\nu}R\right)$ $\displaystyle=$ $\displaystyle{1\over
6}\delta^{\mu}_{\nu}\Lambda+{(1-\alpha_{0})\over
12}\delta^{\mu}_{\nu}K^{2}-{(1+\alpha_{1})\over
3}\left(KK^{\mu}{}_{\nu}-{3\over
4}\delta^{\mu}_{\nu}K_{\alpha\beta}K^{\alpha\beta}\right)$ (14)
$\displaystyle+{(1+\alpha_{1})\over 3}K^{\mu}{}_{\nu,y}-{(1-\alpha_{0})\over
3}\delta^{\mu}_{\nu}K_{,y}-{\kappa^{2}\over 3}\left({\cal
T}^{\mu}{}_{\nu}-{1\over 2}\delta^{\mu}_{\nu}{\cal T}^{y}{}_{y}\right)\ ,$

where we have defined

$\displaystyle\alpha_{0}=2\kappa^{2}\beta_{2},\quad\alpha_{1}=2\kappa^{2}(\beta_{1}+\beta_{3})\
.$ (15)

The trace of equation (14) yields

$\displaystyle(3-4\alpha_{0}-\alpha_{1})K_{,y}=2\Lambda-(\alpha_{0}+\alpha_{1})K^{2}+3(1+\alpha_{1})K_{\alpha\beta}K^{\alpha\beta}-{\kappa^{2}\over
3}\left({\cal T}^{\mu}{}_{\mu}-2{\cal T}^{y}{}_{y}\right).$ (16)

Substituting Eqs. (14) and (16) into the following components of the Weyl
tensor

$\displaystyle C_{y\mu y\nu}$ $\displaystyle=$ $\displaystyle-{1\over
3}\left(R_{\mu\nu}-{1\over 4}g_{\mu\nu}R\right)+{1\over
3}\left(KK_{\mu\nu}-{1\over 4}g_{\mu\nu}K^{2}\right)+{1\over
3}\left(K_{\mu}{}^{\alpha}K_{\alpha\nu}+{3\over
4}g_{\mu\nu}K_{\alpha\beta}K^{\alpha\beta}\right)$ (17) $\displaystyle+{2\over
3}\left(K_{\mu\nu,y}-{1\over 4}g_{\mu\nu}K_{,y}\right)\ ,$

we have

$\displaystyle\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}C_{y\mu y\nu}$
$\displaystyle=$ $\displaystyle{1\over 2}\Lambda
g_{\mu\nu}-\frac{3\alpha_{0}+(2+\alpha_{0})\alpha_{1}}{4(3+\alpha_{1})}g_{\mu\nu}K^{2}-\frac{(1+\alpha_{1})\alpha_{1}}{(3+\alpha_{1})}KK_{\mu\nu}+\frac{(1+\alpha_{1})(3+2\alpha_{1})}{(3+\alpha_{1})}K_{\mu}{}^{\lambda}K_{\lambda\nu}$
(18)
$\displaystyle+\frac{3(1+\alpha_{1})(4+\alpha_{1})}{4(3+\alpha_{1})}g_{\mu\nu}K_{\alpha\beta}K^{\alpha\beta}+(1+\alpha_{1})K_{\mu\nu,y}-(1-\alpha_{0})g_{\mu\nu}K_{,y}$
$\displaystyle+{\kappa^{2}\over 2}g_{\mu\nu}{\cal
T}^{y}{}_{y}-{\kappa^{2}\over 3}\left({\cal T}_{\mu\nu}+{1\over
2}g_{\mu\nu}{\cal T}^{\alpha}{}_{\alpha}\right)\ .$

Here, we have defined that the term ${\cal T}^{\alpha}{}_{\alpha}$ is the
trace defined with respect to the four-dimensional metric $g$, and not the
full trace defined with respect to $\tilde{g}$. Equation (10) can be expressed
as

$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle-\Lambda
g_{\mu\nu}+(1+\alpha_{1})KK_{\mu\nu}-{(1-\alpha_{0})\over
2}g_{\mu\nu}K^{2}-2(1+\alpha_{1})K_{\mu}{}^{\alpha}K_{\alpha\nu}-{(1+\alpha_{1})\over
2}g_{\mu\nu}K_{\alpha\beta}K^{\alpha\beta}$ (19)
$\displaystyle-(1+\alpha_{1})K_{\mu\nu,y}+(1-\alpha_{0})g_{\mu\nu}K_{,y}+\kappa^{2}{\cal
T}_{\mu\nu}\ .$

Using Eq. (18), Eq. (19) is expressed as

$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\Lambda
g_{\mu\nu}-\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}E_{\mu\nu}-\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}(K_{\mu}{}^{\alpha}K_{\alpha\nu}-KK_{\mu\nu})-\frac{6+4\alpha_{1}-(3+\alpha_{1})\alpha_{0}}{4(3+\alpha_{1})}g_{\mu\nu}K^{2}$
(20)
$\displaystyle+\frac{(1+\alpha_{1})(6+\alpha_{1})}{4(3+\alpha_{1})}g_{\mu\nu}K_{\alpha\beta}K^{\alpha\beta}+{\kappa^{2}\over
2}g_{\mu\nu}{\cal T}^{y}{}_{y}+{2\kappa^{2}\over 3}\left({\cal
T}_{\mu\nu}-{1\over 4}g_{\mu\nu}{\cal T}^{\alpha}{}_{\alpha}\right)\ ,$

where the projected Weyl tensor is $E_{\mu\nu}=C_{y\mu y\nu}|_{y=0}$. Note
that the coefficient of the four-dimensional Einstein tensor (20) is modified
by factor $(3+\alpha_{1})$. Here, we take $\alpha_{1}\neq-3$. The case
$\alpha_{1}=-3$ provides a relation between the extrinsic curvature and the
projected Weyl tensor. To eliminate the extrinsic curvature, we use the
junction conditions. It can be obtained by collecting together the terms in
field equations which contain a $\delta$-function. From Eqs. (10) and (13), we
then obtain

$\displaystyle\left[K^{\mu}{}_{\nu}-\delta^{\mu}_{\nu}K\right]|_{y=0}$
$\displaystyle=$ $\displaystyle{\kappa^{2}\over
2(1+\alpha_{1})}\left(S^{\mu}{}_{\nu}+\alpha_{2}\delta^{\mu}_{\nu}S\right)\ ,$
(21)

where

$\displaystyle\alpha_{2}=\frac{\alpha_{0}+\alpha_{1}}{3-4\alpha_{0}-\alpha_{1}}\
.$ (22)

For convenient we will take $\alpha_{1}\neq-3$ and $\alpha_{1}\neq-1$ in order
to avoid unreal singularities in Eqs. (20) and (21). Substituting (21) into
(20), we finally obtain the modified effective SMS equation on the brane as

$\displaystyle G_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\Lambda_{b}g_{\mu\nu}+8\pi
G\left(\tau_{\mu\nu}+{\alpha_{1}\over
12}g_{\mu\nu}\tau\right)+\kappa^{4}\pi_{\mu\nu}-\widetilde{E}_{\mu\nu}+F_{\mu\nu}\
,$ (23)

where we have defined the quantities

$\displaystyle\Lambda_{b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda+\frac{\kappa^{4}}{4(3-4\alpha_{0}-\alpha_{1})}\sigma^{2},$
(24) $\displaystyle 8\pi G$ $\displaystyle=$
$\displaystyle\frac{3\kappa^{4}}{2(3+\alpha_{1})(3-4\alpha_{0}-\alpha_{1})}\sigma,$
(25) $\displaystyle\pi_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{3}{4(3+\alpha_{1})(1+\alpha_{1})}\left[{(1-2\alpha_{0}-\alpha_{1})\over(3-4\alpha_{0}-\alpha_{1})}\tau\tau_{\mu\nu}-\tau_{\mu}{}^{\alpha}\tau_{\alpha\nu}+{(6+\alpha_{1})\over
12}g_{\mu\nu}\tau_{\alpha\beta}\tau^{\alpha\beta}\right.$ (26)
$\displaystyle\left.-{2(3-\alpha_{1})-(9+\alpha_{1})\alpha_{0}\over
12(3-4\alpha_{0}-\alpha_{1})}g_{\mu\nu}\tau^{2}\right]\ ,$
$\displaystyle\widetilde{E}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}E_{\mu\nu}\ ,$ (27)

and the bulk energy-momentum tensor projected on the brane is given by

$\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[{\kappa^{2}\over 2}g_{\mu\nu}{\cal
T}^{y}{}_{y}+{2\kappa^{2}\over 3}\left({\cal T}_{\mu\nu}-{1\over
4}g_{\mu\nu}{\cal T}^{\alpha}{}_{\alpha}\right)\right]_{y=0}\ .$ (28)

There are four features in the effective Einstein equations (23). The first
one is the presence of the bulk energy-momentum tensor. This term allows
exotic matter such as the dilaton can propagate in the bulk. The second
departure from the standard four-dimensional Einstein equation arises from the
presence of the Weyl tensor which is undetermined on the brane. The third is a
quadratic in the brane energy-momentum tensor. The last one is a linear in
addition to the brane energy-momentum tensor. It is our main result. This
trace part of the brane energy-momentum tensor is measured by local observers
at the brane and vanishes when
$\alpha_{1}=2\kappa^{2}(\beta_{1}+\beta_{3})=0$.

Equation (9) and the junction conditions (21) imply

$\displaystyle
D_{\mu}\tau^{\mu}{}_{\nu}+\alpha_{2}D_{\nu}\tau-(1+4\alpha_{2})D_{\nu}\sigma=-2(1+\alpha_{1}){\cal
T}^{y}{}_{\nu}\ .$ (29)

This equation tell us that the energy momentum tensor $\tau_{\mu\nu}$ is not
conserved on the brane. Taking the divergence of the four–dimensional
effective equations and using four–dimensional Bianchi identity, we obtain the
constraint equations for $E_{\mu\nu}$ as

$\displaystyle D_{\mu}\widetilde{E}^{\mu}{}_{\nu}$ $\displaystyle=$
$\displaystyle-D_{\nu}\Lambda_{b}+8\pi
G\left(D_{\mu}\tau^{\mu}{}_{\nu}+{\alpha_{1}\over
12}D_{\nu}\tau\right)+\kappa^{4}D_{\mu}\pi^{\mu}{}_{\nu}$ (30)
$\displaystyle+{\kappa^{2}\over 2}D_{\nu}{\cal T}^{y}{}_{y}+{2\kappa^{2}\over
3}\left(D_{\mu}{\cal T}_{\mu\nu}-{1\over 4}D_{\nu}{\cal
T}^{\alpha}{}_{\alpha}\right).$

Equations (29) and (30) indicate a time variation of the brane tension, the
cosmological constant, and the gravitational constant in general.

In the following section, we study analytically the cosmological consequences
of Eqs. (23), (29) and (30). Here, for simplicity, we consider constant
$\sigma$, because there are no theoretical observational arguments for the
evolution of $\sigma$ in time. For cosmology on the brane, we suppose here
that we can ignore the bulk matter, $F_{\mu\nu}=0$. The bulk matter is
important to get a well–behaved geometry in the bulk. We also assume that the
bulk cosmological constant is truly constant. Then, Eqs. (29) and (30) become

$\displaystyle D_{\mu}\tau^{\mu}{}_{\nu}=-\alpha_{2}D_{\nu}\tau\ ,$ (31)
$\displaystyle D_{\mu}\widetilde{E}^{\mu}{}_{\nu}=-8\pi
G\left(\alpha_{2}-{\alpha_{1}\over
12}\right)D_{\nu}\tau+\kappa^{4}D_{\mu}\pi^{\mu}{}_{\nu}\ .$ (32)

Note that the projected Weyl tensor is affected by the energy–momentum tensor
on the brane even at low energies. Thus, the model is quite different from the
conventional braneworld even at low energies.

##  III Braneworld cosmology

The projected Weyl tensor in the modified Einstein equation (23) is a priori
undetermined. This comes from the five-dimensional nature of the theory and
the fact that the system of equations is not closed on the brane. This tensor
mediates some information from the bulk to the brane. In this section, we will
try to solve Einstein equation to study the cosmology braneworld from equation
(23), by assuming that there is no cosmological constant on the brane and the
constant vacuum energy. Although these assumptions are usual in braneworld
scenario, we will show, which is the main result of present paper, the
effective Friedmann equations is modified by the effect of Lorentz violation,
and the components of the projected Weyl tensor are related to the matter on
the brane. We then discuss the method to obtain the components of the
projected Weyl tensor from the brane data. For cosmological applications, we
consider a metric of the form

$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}\ ,$ (33)

where $x^{i}$ are the three ordinary spatial coordinates and $a$ is the scale
factor. The Hubble parameter $H$ on the brane, describing the cosmological
dynamics of the Universe, is defined as $H=\dot{a}/a$. For simplicity, we
ignore the bulk matter for the cosmology on the brane. Hereafter, we will
consider only the matter on the brane. For further discussions on the
gravitational field equations in the braneworld model with Lorentz violation
and their cosmological applications see Ahmadi:2006cr . We restrict the
energy-momentum tensor on the brane of the form

$\displaystyle\tau_{\mu\nu}=(\rho,Pa^{2}\delta_{ij})\ ,$ (34)

where $\rho$ is the energy density and $P$ the pressure. We will assume that
the equation of state relating $\rho$ and $P$ has the form $P=\omega\rho$,
where $\omega$ is constant. Similarly, the projected Weyl tensor is of the
form

$\displaystyle E_{\mu\nu}=(\rho_{d},P_{d}a^{2}\delta_{ij})\ .$ (35)

The traceless property of $E_{\mu\nu}$ implies: $-\rho_{d}+3P_{d}=0$. We will
be interested in the relation between the components of the projected Weyl
tensor and the brane energy-momentum tensor. The components of the quadratic
in the energy-momentum tensor (26) are given by

$\displaystyle\pi_{00}$ $\displaystyle=$
$\displaystyle\frac{1+3\alpha_{3}}{4(3+\alpha_{1})(1+\alpha_{1})^{2}}\rho^{2}\
,$ (36) $\displaystyle\pi_{ij}$ $\displaystyle=$
$\displaystyle\frac{1+2\omega-3\alpha_{4}}{4(3+\alpha_{1})(1+\alpha_{1})^{2}}\rho^{2}a^{2}\delta_{ij}\
,$ (37)

where

$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{12(3-4\alpha_{0}-\alpha_{1})}\\{[7-9(2+\omega)\omega-3(1+\omega)^{2}(2-\alpha_{1})\alpha_{1}]\alpha_{0}-[17-3\omega(8+3\omega)$
$\displaystyle+2(1-12\omega-3\omega^{2}+(1+3\omega^{2})\alpha_{1}^{2})]\alpha_{1}\\}\
,$ $\displaystyle\alpha_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{12(3-4\alpha_{0}-\alpha_{1})}\\{[-1-2\omega+15\omega^{2}+3(1+\omega)^{2}(6+\alpha_{1})\alpha_{1}]\alpha_{0}-[15+32\omega$
(38)
$\displaystyle-15\omega^{2}-(1+3\omega^{2})\alpha_{1}^{2}-2(1+9\omega^{2})\alpha_{1}]\alpha_{1}\\}\
.$

Substituting metric (33) and tensors (34), (35) and (36), (37) in the
effective Einstein equations (23), one finds

$\displaystyle 3H^{2}=8\pi
G\left[1+\frac{(1-3\omega)\alpha_{1}}{12}\right]\rho+\frac{\kappa^{4}(1+3\alpha_{3})}{4(1+\alpha_{1})^{2}(3+\alpha_{1})}\rho^{2}-\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}\rho_{d}\
,$ (39) $\displaystyle-2\dot{H}-3H^{2}=8\pi
G\left[\omega-\frac{(1-3\omega)\alpha_{1})}{12}\right]\rho+\frac{\kappa^{4}(1+2\omega-3\alpha_{4})}{4(1+\alpha_{1})^{2}(3+\alpha_{1})}\rho^{2}-\frac{(1+\alpha_{1})}{(3+\alpha_{1})}\rho_{d}\
.$ (40)

Obviously, these equations are quite different from the usual braneworld
equations due to the effect of bulk Lorentz violation. From Eq. (31) and the
constraint equation for the projected Weyl tensor (32), we have

$\displaystyle[1+(1-\omega)\alpha_{2}]\dot{\rho}+3H\rho(1+\omega)=0\ ,$ (41)

and

$\displaystyle\dot{\rho}_{d}+4H\rho_{d}$ $\displaystyle=$
$\displaystyle\frac{8\pi
G(1+\omega)(1-3\omega)(3+\alpha_{1})(3-4\alpha_{0}-\alpha_{1})}{3(1+\alpha_{1})^{3}}\left[\alpha_{2}-\frac{(1+\alpha_{1})^{2}\alpha_{1}}{12(1-\omega\alpha_{1}-(1+\omega)\alpha_{0})}\right]H\rho$
(42)
$\displaystyle-\frac{\kappa^{4}(1+\omega)\alpha_{5}}{4(1+\alpha_{1})^{3}(1+(1-3\omega)\alpha_{2})}H\rho^{2},$

where

$\displaystyle\alpha_{5}$ $\displaystyle=$
$\displaystyle\frac{(1+\alpha_{1})}{2(3-4\alpha_{0}-\alpha_{1})}\\{12(1+\omega)^{2}\alpha_{0}^{2}\alpha_{1}+[9(1+3\omega^{2})+(1+3\omega^{2})\alpha_{1}^{2}-2(1+12\omega-9\omega^{2})\alpha_{1}]\alpha_{1}$
(43)
$\displaystyle+[3(1-3\omega^{2})-2(7+30\omega-9\omega^{2})\alpha_{1}+(7+6\omega+15\omega^{2})\alpha_{1}^{2}]\alpha_{0}\\}\
.$

For $\omega\neq-1$, equation (41) is solved to yield

$\displaystyle\rho=a^{-\frac{3(1+\omega)}{1+(1-\omega)\alpha_{2}}}\ .$ (44)

Here, we have absorbed a constant factor into the scale factor by rescaling
it. Equation (42) can be integrated. We find

$\displaystyle{\rho}_{d}$ $\displaystyle=$ $\displaystyle-{3C\over
a^{4}}+\frac{8\pi
G(1+\omega)(3+\alpha_{1})(3-4\alpha_{0}-\alpha_{1})^{2}}{9(1+\alpha_{1})^{4}}\left\\{[1+(1-3\omega)\alpha_{2}]\alpha_{2}-\frac{\alpha_{1}(1+\alpha_{1})^{2}}{4(3-4\alpha_{0}-\alpha_{1})}\right\\}a^{-\frac{3(1+\omega)}{1+(1-\omega)\alpha_{2}}}$
(45)
$\displaystyle-\frac{\kappa^{4}(1+\omega)\alpha_{5}}{8(1+\alpha_{1})^{3}[2(1-3\omega)\alpha_{2}-(1+3\omega)]}a^{-\frac{6(1+\omega)}{1+(1-\omega)\alpha_{2}}}\
,$

where $C$ is a constant of integration. This effect of the bulk acts as
radiation fluid, hence it is called as dark radiation. Substituting Eq. (45)
into Eq. (39), we obtain the effective Friedmann equation

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle{8\pi G_{eff}\over
3}\rho+A\rho^{2}+{\bar{C}\over a^{4}}\ ,$ (46)

where

$\displaystyle G_{eff}$ $\displaystyle=$
$\displaystyle\left\\{1-\frac{[1-\omega\alpha_{1}-(1+\omega)\alpha_{0}][(2+3\omega-(2+\alpha_{1})\alpha_{1})\alpha_{1}+3(1+\omega)\alpha_{0}]}{3(1+\alpha_{1})^{3}}\right\\}G\
,$ (47) $\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{\kappa^{4}}{12(3+\alpha_{1})(1+\alpha_{1})^{2}}\left[1+3\alpha_{3}-\frac{3(1+\omega)\alpha_{5}}{2[(1+3\omega)-2(1-3\omega)\alpha_{2}]}\right],$
(48) $\displaystyle\bar{C}$ $\displaystyle=$
$\displaystyle\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}C\ .$ (49)

Note that the effective Newton constant depends on the Lorentz violating
parameters and the equations of state. It is different from the conventional
braneworld cosmology in five-dimensional case even at low energy. If the
effects of Lorentz violations are ignored, $\beta_{i}=0$, we have $G_{eff}=G$,
$A=\kappa^{4}/36$ and $\bar{C}=C$. In the alternative theory of gravity
including Brans–Dicke theory, the effective Newton constant need not be
constant in time. Observational bounds on $\dot{G}/G$ then constrain the
theory. In our case, we have the relation (47), hence the Newton constant is
always constant.

If the effective cosmological constant is included, the Friedmann equation
(46) becomes

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle{1\over
3}\Lambda_{b}+{8\pi G_{eff}\over 3}\rho+A\rho^{2}+{\bar{C}\over a^{4}}\ ,$
(50)

where the relation between the vacuum energy and the effective cosmological
constant on a brane is given by Eq. (24). It is different from the usual four-
dimensional theory. In the RS braneworld, the vacuum energy in the brane is
not directly related to the cosmological constant on the brane in the
effective Einstein equation as in Eq. (24). In the RS braneworld, there should
be a cancellation between the four-dimensional and five-dimensional
contribution of the vacuum energy in order to have a vanishing cosmological
constant on the brane. This requires a fine-tuning for the parameters in the
action. In the present model the RS type relation is given by

$\displaystyle\sigma=\frac{6}{\kappa^{2}l}\left(1-{4\over 3}\alpha_{0}-{1\over
3}\alpha_{1}\right)^{1/2}\ ,$ (51)

and

$\displaystyle 8\pi G=\frac{3\kappa^{2}}{l(3+\alpha_{1})(1+\alpha_{1})^{1/2}}\
.$ (52)

Here, the bulk cosmological constant is defined as
$\Lambda=-6/\kappa^{2}l^{2}$, where $l$ is the scale of the bulk curvature
radius.

##  IV Low Energy Constraint on $\beta_{i}$

For a well-defined theory, the constraints on the theory parameters
$\beta_{i}$ are given by Lim:2004js  (see also Carroll:2004ai ):

  1. 1.

Subluminal propagation of spin-0 field:
$(\beta_{1}+\beta_{2}+\beta_{3})/\beta_{1}\leq 1$,

  2. 2.

Positivity of Hamiltonian: $\beta_{1}>0$,

  3. 3.

Non-tachyonic propagation of spin-0 field:
$(\beta_{1}+\beta_{2}+\beta_{3})/\beta_{1}\geq 0$,

  4. 4.

Subluminal propagation of spin-2 field: $\beta_{1}+\beta_{3}\leq 0$.

All these conditions together imply $(\beta_{1}+\beta_{2}+\beta_{3})\geq 0$
and $\beta_{2}\geq 0$.

At low energies, we can neglect the quadratic term of the Friedmann equation
(46). Then we have

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle{8\pi G_{eff}\over
3}\rho+{\bar{C}\over a^{4}}\ .$ (53)

Here, we have assumed $3A/8\pi G_{eff}<<1$. Therefore, one can set $A\approx
0$ without loss of generality. Solving Eq. (48) one finds

$\displaystyle\alpha_{1}=\frac{1-\alpha_{0}(1+\omega)}{\omega},~{}~{}\text{or}~{}~{}\alpha_{1}=\frac{2(1+3\omega)-3\alpha_{0}(1+\omega)^{2}}{1+3\omega^{2}}\
,$ (54)

where $\alpha_{0}$ and $\alpha_{1}$ is given by Eq. (15). In other word, the
effect of Lorentz violation in the bulk is dependent on the equation of state
of the energy components of the Universe. Remarkable, the first solution (54)
yields $G_{eff}=G$. In this case, using the above constraints we find

  1. 1.

For $\omega<-1$,

$\displaystyle\alpha_{0}>\frac{1+3\omega}{1+\omega},\qquad\alpha_{1}<-3\ ,$
(55)

and

$\displaystyle 1<\alpha_{0}<\frac{1+3\omega}{1+\omega},\qquad-3<\alpha_{1}<-1\
.$ (56)

  2. 2.

For $-1<\omega<0$,

$\displaystyle 1<\alpha_{0}\leq\frac{1}{1+\omega},\qquad-1<\alpha_{1}\leq 0\
.$ (57)

  3. 3.

For $\omega>0$,

$\displaystyle\frac{1}{1+\omega}\leq\alpha_{0}<1,\qquad-1<\alpha_{1}\leq 0\ .$
(58)

The above constraints give the correction in the coefficient of the dark
radiation. The second solution (54) gives the constraints:

  1. 1.

For $\omega<-1$,

$\displaystyle\alpha_{0}>\frac{5+6\omega+9\omega^{2}}{3(1+\omega)^{2}},\qquad\alpha_{1}<-3\
,$ (59)

and

$\displaystyle
1<\alpha_{0}<\frac{5+6\omega+9\omega^{2}}{3(1+\omega)^{2}},\qquad-3<\alpha_{1}<-1\
.$ (60)

  2. 2.

For $-1<\omega\leq-1/3$,

$\displaystyle\alpha_{0}>\frac{5+6\omega+9\omega^{2}}{3(1+\omega)^{2}},\qquad\alpha_{1}<-3\
,$ (61)

and

$\displaystyle
1<\alpha_{0}<\frac{5+6\omega+9\omega^{2}}{3(1+\omega)^{2}},\qquad-3<\alpha_{1}<-1\
.$ (62)

  3. 3.

For $\omega\geq-1/3$,

$\displaystyle\frac{2(1+3\omega)}{3(1+\omega)^{2}}\leq\alpha_{0}<1,\qquad-1<\alpha_{1}\leq
0\ .$ (63)

These constraints give the corrections both in the effective Newton constant
and the dark radiation.

##  V Conclusions

In the present paper, we have considered a five-dimensional braneworld model
with bulk Lorentz invariance violation, and derived the effective four-
dimensional Einstein equations on the brane. The main result of this paper is
the existence of the trace part of the brane energy-momentum tensor in the
modified Einstein equations on the brane, which is a modification of the SMS
effective equation Shiromizu:1999wj . Thus, the divergence of the projected
Weyl tensor is modified. Therefore, due to Lorentz violating effect, we have
obtained an expression for the projected Weyl tensor as a function of the
source on the brane. It becomes clear that the bulk effect can be determined
by matter localized on the brane even at low energies. As an application, we
have used the modified SMS effective equation to determine the Friedmann
equation on the brane. We have showed the effective Newton constant that
relates geometry to the matter density in Friedmann equation is dependent on
the equation of state of the energy component of the Universe, and the Lorentz
violating parameters. Note that if the brane was isotropic and homogeneous,
the matter part would have the additional property, $D^{\mu}\pi_{\mu\nu}=0$.
However, due to effect of Lorentz violation in the bulk, the effect of matter
still appears in Eq. (32). Thus, the brane matter will deform the bulk
geometry. In other word, the back-reaction of this to the brane will modify
the effective Friedmann equation even at low energies. It is interesting to
understand the low energy description of this braneworld model. The low energy
perturbation scheme proposed in Kanno:2006ty  is a major achievement as it
allows for the derivation of the effective theory on the brane and for the
full comprehension of the Weyl tensor contribution to the effective theory. We
leave this issue for future studies.

Finally, we also find that the effect of Lorentz violation in the bulk is
dependent on the equation of state of the energy components of the brane
matter. This model also provides a convenient framework within which one may
study dark energy.

###### Acknowledgements.

Arianto wishes to acknowledge all members of the Theoretical Physics
Laboratory, the THEPI Divison of the Faculty of Mathematics and Natural
Sciences, ITB, for the warmest hospitality. This work was supported by Hibah
Kompetensi DIKTI, 2009\.

## References

  * (1) V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989).
  * (2) J. D. Bekenstein, Phys. Rev. D 70, 083509 (2004) [Erratum-ibid. D 71, 069901 (2005)] [arXiv:astro-ph/0403694].
  * (3) C. Skordis, D. F. Mota, P. G. Ferreira and C. Boehm, Phys. Rev. Lett. 96, 011301 (2006) [arXiv:astro-ph/0505519] ;C. Skordis, Phys. Rev. D 74, 103513 (2006) [arXiv:astro-ph/0511591].
  * (4) T. Jacobson and D. Mattingly, Phys. Rev. D 64, 024028 (2001) [arXiv:gr-qc/0007031].
  * (5) T. G. Zlosnik, P. G. Ferreira and G. D. Starkman, Phys. Rev. D 74, 044037 (2006) [arXiv:gr-qc/0606039].
  * (6) T. G. Zlosnik, P. G. Ferreira and G. D. Starkman, Phys. Rev. D 75, 044017 (2007) [arXiv:astro-ph/0607411].
  * (7) C. Bonvin, R. Durrer, P. G. Ferreira, G. Starkman and T. G. Zlosnik, Phys. Rev. D 77, 024037 (2008) [arXiv:0707.3519 [astro-ph]].
  * (8) S. M. Carroll and E. A. Lim, Phys. Rev. D 70, 123525 (2004) [arXiv:hep-th/0407149].
  * (9) B. Li, D. Fonseca Mota and J. D. Barrow, Phys. Rev. D 77, 024032 (2008) [arXiv:0709.4581 [astro-ph]].
  * (10) V. A. Kostelecky, Phys. Rev. D 69, 105009 (2004) [arXiv:hep-th/0312310].
  * (11) S. Kanno and J. Soda, Phys. Rev. D 74, 063505 (2006) [arXiv:hep-th/0604192].
  * (12) M. a. Watanabe, S. Kanno and J. Soda, arXiv:0902.2833 [hep-th].
  * (13) P. P. Avelino, D. Bazeia, L. Losano, R. Menezes and J. J. Rodrigues, arXiv:0903.5297 [astro-ph.CO].
  * (14) E. A. Lim, Phys. Rev. D 71, 063504 (2005) [arXiv:astro-ph/0407437].
  * (15) A. Tartaglia and N. Radicella, Phys. Rev. D 76, 083501 (2007) [arXiv:0708.0675 [gr-qc]].
  * (16) Arianto, F. P. Zen, B. E. Gunara, Triyanta and Supardi, JHEP 0709, 048 (2007) [arXiv:0709.3688 [hep-th]]; Arianto, F. P. Zen, Triyanta and B. E. Gunara, Phys. Rev. D 77, 123517 (2008) [arXiv:0801.0331 [hep-th]]; F. P. Zen, Arianto, B. E. Gunara, Triyanta and A. Purwanto, arXiv:0809.3847 [hep-th].
  * (17) K. Nozari and S. D. Sadatian, Eur. Phys. J. C 58, 499 (2008) [arXiv:0809.4744 [gr-qc]].
  * (18) P. Horava and E. Witten, Nucl. Phys. B 460, 506 (1996) [arXiv:hep-th/9510209].
  * (19) For a review see, e.g., R. Maartens, Living Rev. Rel. 7, 7 (2004) [arXiv:gr-qc/0312059].
  * (20) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999) [arXiv:hep-ph/9905221].
  * (21) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999) [arXiv:hep-th/9906064].
  * (22) T. Shiromizu, K. i. Maeda and M. Sasaki, Phys. Rev. D 62, 024012 (2000) [arXiv:gr-qc/9910076].
  * (23) D. Ida, JHEP 0009, 014 (2000) [arXiv:gr-qc/9912002].
  * (24) P. Kraus, JHEP 9912, 011 (1999) [arXiv:hep-th/9910149].
  * (25) S. Mukohyama, Phys. Lett. B 473, 241 (2000) [arXiv:hep-th/9911165].
  * (26) K. Ichiki, M. Yahiro, T. Kajino, M. Orito and G. J. Mathews, Phys. Rev. D 66, 043521 (2002) [arXiv:astro-ph/0203272].
  * (27) P. Koroteev and M. Libanov, Phys. Rev. D 79, 045023 (2009) [arXiv:0901.4347 [hep-th]]; P. Koroteev and M. Libanov, JHEP 0802, 104 (2008) [arXiv:0712.1136 [hep-th]].
  * (28) O. Bertolami and C. Carvalho, Phys. Rev. D 74, 084020 (2006) [arXiv:gr-qc/0607043].
  * (29) F. Ahmadi, S. Jalalzadeh and H. R. Sepangi, Class. Quant. Grav. 23, 4069 (2006) [arXiv:gr-qc/0605038]; F. Ahmadi, S. Jalalzadeh and H. R. Sepangi, Phys. Lett. B 647, 486 (2007) [arXiv:gr-qc/0702103].
  * (30) C. Csaki, J. Erlich and C. Grojean, Nucl. Phys. B 604, 312 (2001) [arXiv:hep-th/0012143].
  * (31) H. Stoica, JHEP 0207, 060 (2002) [arXiv:hep-th/0112020].
  * (32) M. V. Libanov and V. A. Rubakov, JCAP 0509, 005 (2005) [arXiv:astro-ph/0504249].
  * (33) K. Nozari and S. D. Sadatian, JCAP 0901, 005 (2009) [arXiv:0810.0765 [gr-qc]].
  * (34) K. Farakos, arXiv:0903.3356 [hep-th].
  * (35) R. Maartens, Phys. Rev. D 62, 084023 (2000) [arXiv:hep-th/0004166].