# Lifshitz hydrodynamics at generic z from a moving black brane

Aruna Rajagopal  Larus Thorlacius  University of Iceland, Science Institute,
Dunhaga 3, IS-107, Reykjavik, Iceland

###### Abstract

A Lifshitz black brane at generic dynamical critical exponent $z>1$, with non-
zero linear momentum along the boundary, provides a holographic dual
description of a non-equilibrium steady state in a quantum critical fluid,
with Lifshitz scale invariance but without boost symmetry. We consider moving
Lifshitz branes in Einstein-Maxwell-Dilaton gravity and obtain the non-
relativistic stress tensor complex of the dual field theory via a suitable
holographic renormalisation procedure. The resulting black brane hydrodynamics
and thermodynamics are a concrete holographic realization of a Lifshitz
perfect fluid with a generic dynamical critical exponent.

###### Contents

  1. I Introduction
  2. II The gravitational theory
    1. II.1 Black branes with linear momentum
    2. II.2 Noether charges
    3. II.3 Holographic renormalisation
    4. II.4 The asymptotic solution
  3. III Thermodynamics
    1. III.1 The near-horizon solution
    2. III.2 Static Lifshitz black brane
    3. III.3 Moving Lifshitz black brane
  4. IV Conclusion and Future outlook

##  I Introduction

Fluid dynamics LLfluid  efficiently captures the long-wavelength, low
frequency behavior of a large class of interacting physical systems at finite
temperature. Based on general symmetry principles and the conservation of
local currents, it is a universal theory, as even systems with varying
microscopic dynamics, can have the same macroscopic, hydrodynamic description.
An appropriate hydrodynamic model, incorporating the dissipative effects of
the thermal medium that are essential for a fluid to equilibriate after being
perturbed away from equilibrium, consists of a gradient expansion of the
conserved currents. At a given order, the conservation equations determine the
expansion up to a finite number of undetermined coefficients. These
coefficients may then be obtained either from measurements or from microscopic
computations. For strongly coupled quantum systems, explicit computations are
difficult and a gauge/gravity formulation Policastro_2001 ; Policastro_2002 ;
Policastro_2002b  can offer advantages. Einstein’s equations with a negative
cosmological constant, supplemented with appropriate regularity restrictions
and boundary conditions, reduce to the nonlinear equations of fluid dynamics
in an appropriate parameter range, and a systematic framework to construct
this universal nonlinear fluid dynamics, order by order in a boundary
derivative expansion has been developed Bhattacharyya:2008jc ; Baier:2007ix .

In standard anti-de Sitter (AdS) holography Maldacena_1997  the dual field
theory is relativistic and much of the dual gravitational formulation of
hydrodynamics has focused on relativistic systems. However, many strongly
correlated systems in nature are inherently non-relativistic and this has
motivated the application of holography to a wider setting involving
gravitational bulk theories with asymptotic geometries that are not AdS
Son_2008 ; Balasubramanian_2008 ; Kachru_2008 . Generic non-relativistic
quantum critical fluids have an emergent scaling symmetry of the form

$t\rightarrow\Lambda^{z}\,t,\qquad\vec{x}\rightarrow\Lambda\vec{x},$ (1)

commonly referred to as Lifshitz scaling and are characterised by a dynamical
critical exponent $z\geq 1$. For $z>1$, the scaling is asymmetric between the
time and spatial directions and the system has scale invariance without
conformal invariance. By now, there is a large body of work on holographic
systems that realize Lifshitz scaling of the form (1) at the asymptotic
boundary of the gravitational bulk space-time, including a number of papers
with a specific focus on hydrodynamic aspects, see e.g. Taylor:2008tg ;
Ross:2009ar ; Ross_2011 ; Mann:2011hg ; Bertoldi:2009dt ; Hoyos:2013eza ;
Hoyos:2013qna ; Kiritsis:2015doa ; Hartong_2016 .

Quantum critical fluids with Lifshitz symmetry, or Lifshitz fluids for short,
have interesting properties that set them aside from more conventional fluid
systems. In particular, a Lifshitz fluid with generic dynamical critical
exponent $z\neq 1,2$ cannot have boost symmetry de_Boer_2018 ; Grinstein_2018
. A general formalism for perfect fluids without boost symmetry has been
developed in de_Boer_2018 ; de_Boer_2020 ; Novak:2019wqg  and a primary
motivation for the present paper is to provide a concrete holographic setup
for the study of such fluids.

In previous work Fern_ndez_2019 , we have studied out-of-equilibrium energy
transport in a quantum critical fluid with Lifshitz scaling symmetry following
a local quench between two semi-infinite fluid reservoirs. The late time
energy flow was found to be universal and was accommodated via a steady state
occupying an expanding central region between outgoing shock and rarefaction
waves, in accordance with earlier work on critical fluids with Lorentz boost
symmetry Bhaseen:2013ypa ; Lucas_2016b ; Spillane_2016 ; Amado:2015uza ;
Pourhasan_2016 . In the relativistic case, a holographic version of the quench
problem was considered in Bhaseen:2013ypa ; Amado:2015uza  and it would be
interesting to extend this to Lifshitz fluids without boost symmetry. As a
first step in that direction, we consider a Lifshitz black brane at generic
$z$, moving with uniform velocity $v$ in one of the transverse directions, as
a holographic dual description of a non-equilibrium steady state of the
corresponding Lifshitz fluid.

The holographic model we choose to work with is a relatively simple variant of
Einstein-Maxwell-Dilaton gravity with exact analytic solutions describing
static Lifshitz black branes. The model can easily be generalised to include
charged matter in order to study holographic duality at finite temperature and
chemical potential. The exact static black brane solution provides a
convenient benchmark to test our formalism against, even if the moving brane
solution is beyond exact analytic control. A drawback of this model, that has
detracted from its usefulness for holography in the past, is the fact that the
asymptotic Lifshitz symmetry is obscured by a running dilaton and non-standard
asymptotic behaviour of the background vector field. It turns out, this can
easily be remedied by taking advantage of a built-in symmetry of the model and
a more or less standard holographic renormalisation is achieved by suitably
adapting the formalism developed for a holographic model involving a massive
vector field in Ross:2009ar . We find the following results:

  * •

A physically distinct class of moving Lifshitz black brane solutions in four
spacetime dimensions, dual to a uniformly moving 2+1 dimensional Lifshitz
fluid. The fluid velocity plays the role of chemical potential, dual to the
momentum density, in line with the perfect fluid formalism developed in
de_Boer_2018 ; de_Boer_2020 .

  * •

A renormalised boundary stress tensor complex whose components express the
energy density $\mathcal{E}$, energy flux $\mathcal{E}^{i}$, momentum density
$\mathcal{P}_{i}$, and spatial stress tensor $\Pi_{ij}$ of the dual Lifshitz
fluid.

  * •

The Ward identity associated to Lifshitz scaling leads to the equation of
state $z\mathcal{E}-\rho v^{2}=d_{s}P$, where $d_{s}=2$ is the number of
transverse dimensions. This is precisely the equation of state hypothesised
for a perfect Lifshitz fluid with an arbitrary dynamic exponent $z$ in
(Hartong_2016, ; de_Boer_2018, ).

##  II The gravitational theory

We restrict our attention to a holographic theory with Lifshitz scaling
defined in 4 bulk space-time dimensions but our results can easily be
generalised to an arbitrary number of dimensions. The model we work with is a
simple variant of Einstein-Maxwell-Dilaton (EMD) theory, which consists of
Einstein gravity along with a massless $U(1)$ gauge field, $A_{\mu}$ and a
dilaton field, $\phi$. Using a normalisation where $16\pi G_{4}=1$, the action
is given by,

$S_{0}=\int
d^{4}x\sqrt{-g}\left[R-2\Lambda-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{4}e^{\lambda\phi}F^{2}\right]+2\int
d^{3}x\sqrt{-\gamma}K,$ (2)

where the last term is the usual Gibbons-Hawking-York boundary term, which is
needed in order to have a well-defined variational problem for the metric. The
equations of motion for (2) are,

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}$
$\displaystyle=T_{\mu\nu}^{\phi}+T_{\mu\nu}^{(1)}$ (3)
$\displaystyle\nabla^{2}\phi-\frac{\lambda}{4}e^{\lambda\phi}F_{\mu\nu}F^{\mu\nu}$
$\displaystyle=0$ (4) $\displaystyle\nabla_{\mu}e^{\lambda\phi}F^{\mu\nu}$
$\displaystyle=0,$ (5)

with,

$\displaystyle T_{\mu\nu}^{\phi}$
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{4}g_{\mu\nu}(\partial\phi)^{2},$
(6) $\displaystyle T_{\mu\nu}^{F}$
$\displaystyle=\frac{1}{2}e^{\lambda\phi}\left(F_{\mu\sigma}F^{\sigma}_{\nu}-\frac{1}{4}g_{\mu\nu}F_{\sigma\rho}F^{\sigma\rho}\right).$
(7)

Models of this kind were introduced in the context of non-relativistic
holography in Taylor:2008tg . This is by no means the only possible model for
Lifshitz holography but it has an important advantage in that there exists a
full analytic solution to the field equations that describes a static black
brane in asymptotically Lifshitz space-time. The model can easily be
generalized to include charged black brane solutions that are Lifshitz analogs
of AdS-Reissner-Nordström black branes Tarr_o_2011 , which are key to a
holographic dual description of non-relativistic quantum critical matter at
finite temperature and chemical potential, see e.g. Keranen:2012mx . The bulk
theory can also include matter fields of various types, but this is not needed
for the main purpose of this paper, which is to carry out holographic
renormalisation for moving Lifshitz branes and establish thermodynamic
relations that hold for the resulting Lifshitz hydrodynamics.

The model (2) admits the so-called Lifshitz space-time as a solution,

$ds^{2}=\ell^{2}\left(-r^{2z}dt^{2}+\frac{dr^{2}}{r^{2}}+r^{2}d\vec{x}^{2}\right),$
(8)

where $\ell$ is a characteristic length scale of the geometry which we set to
unity for notational simplicity. The metric exhibits the required Lifshitz
scaling,

$t\rightarrow\Lambda^{z}t\ ,\qquad x\rightarrow\Lambda x\ ,\qquad
r\rightarrow\Lambda^{-1}r,$ (9)

with $z\geq 1$. Space-time geometries that are asymptotic to this metric
provide a holographic dual description of a scale-invariant non-relativistic
field theory formulated on a $\mathbb{R}^{t}\times\mathbb{R}^{2}$ boundary.

The metric (8) is a solution to the equations of motion provided the
parameters of the model satisfy $\lambda=-\frac{2}{\sqrt{z-1}}$ and
$\Lambda=-\frac{1}{2}(z+1)(z+2)$, and it is accompanied by a vector field and
dilaton background of the form,

$A_{t}=\sqrt{\frac{2(z-1)}{z+2}}\left(\frac{r}{r_{0}}\right)^{2}r^{z},\qquad
e^{\phi}=\left(\frac{r}{r_{0}}\right)^{2\sqrt{z-1}},$ (10)

where $r_{0}$ is an arbitrary constant. While this model has the advantage of
analytic control, it has the disadvantage of a logarithmically running dilaton
and diverging vector field at the boundary. However, this is not a very
serious disadvantage because the vector field only serves to provide the
background to support a Lifshitz geometry at the boundary and does not couple
to any non-gravitational fields.

Charged matter fields can be added to the model, but they should then be
charged under some additional gauge field and not couple directly to the
$A_{\mu}$ field considered here. The need for a further gauge field to
accommodate charged matter can also be seen from the fact that the background
field profile (10) is not of the standard form that corresponds to having a
chemical potential for an unbroken $U(1)$ symmetry in the dual field theory.
In fact, $A_{\mu}$ should not be viewed as a gauge field but simply as a
massless vector field that only interacts gravitationally (with a coupling
that depends on the dilaton). With this in mind, we do not have to respect the
$U(1)$ gauge symmetry of the bulk action in (2) when we construct boundary
counterterms for holographic renormalisation of the model.

The counterterms instead respect another symmetry. The action (2) is invariant
under a constant shift of the dilaton field while simultaneously absorbing a
constant normalisation factor into the gauge field,

$\phi\rightarrow\phi-\frac{2}{\lambda}\log\alpha,\qquad
A_{\mu}\rightarrow\alpha A_{\mu},$ (11)

and this will be a symmetry of our boundary action as well. The shift symmetry
is also helpful in analysing the asymptotic behaviour of bulk fields.

###  II.1 Black branes with linear momentum

We would like to construct the gravitational dual of a perfect Lifshitz fluid
moving at non-vanishing velocity. We are particularly interested in fluids at
generic $z$, which do not have boost symmetry. In this case, it is not enough
to perform a boost of the black brane along one of the transverse directions
and study the associated thermodynamics as this does not give us a genuine
dual of a moving Lifshitz fluid, but instead corresponds to studying a fluid
at rest from a moving coordinate frame. The way around this is to construct
from scratch a bulk solution that describes a moving Lifshitz black brane at
generic $z\neq 1$, with metric, dilaton and vector fields that encode the
fluid momentum. For this we adopt an ansatz employed in Hartong_2016  for a
$z=2$ moving brane and adapt it to the more general case. With the linear
momentum taken to be in one of the transverse directions, say the $y$
direction, the metric becomes

$ds^{2}=-F_{1}(r)r^{2z}dt^{2}+\frac{dr^{2}}{r^{2}F_{2}(r)}+r^{2}F_{3}(r)dx^{2}+F_{4}(r)\big{(}rdy+N(r)r^{z}dt\big{)}^{2},$
(12)

and the vector field is

$A=r^{z}G_{1}(r)dt+G_{2}(r)\big{(}rdy+N(r)r^{z}dt\big{)}.$ (13)

Note that this ansatz includes as a special case an exact solution for a
static black brane by simply setting $G_{2}=N=0$, $F_{3}=F_{4}=1$, and letting
$F_{1}=F_{2}=1-(r_{h}/r)^{z+2}$. The dilaton field of the static black brane
solution is unchanged from (10) provided we use the shift symmetry (11) to set
$r_{0}$ equal to $r_{h}$, the radial location of the event horizon.

There is no known analytic solution for a brane with non-vanishing linear
momentum. The field equations can be solved numerically, by imposing
appropriate boundary conditions at the black brane horizon, but we will not
pursue a numerical solution here. The results we are after can instead be
obtained by employing a combination of asymptotic analysis and radially
conserved charges along the lines of Hartong_2016 . In the following
subsections we first derive a set of Noether charges that are constant along
the radial coordinate. We then introduce boundary counterterms and implement
them in the holographic renormalisation of the boundary stress tensor.
Finally, we solve the linearised field equations around the Lifshitz fixed
point in (8) and (10). The different elements are then brought together in
Section III where we analyse the thermodynamics of the moving black brane.

###  II.2 Noether charges

The fields that enter in the moving brane ansatz (12) - (13) are functions of
the radial coordinate and in this case the action (2) can be reduced to an
integral over a one-dimensional Lagrangian by factoring off the integrals over
the time and transverse coordinates. The Gibbons-Hawking-York term cancels
against boundary terms resulting from integration by parts of terms in the
radial bulk action, leading to the result,

$\begin{split}L_{1D}=&r^{z+1}\sqrt{F_{1}F_{2}F_{3}F_{4}}\Big{[}-2(z^{2}+2z+3)+\frac{1}{2}\frac{F_{4}N^{2}}{F_{1}}\Big{(}z-1+\frac{rN^{\prime}}{N}\Big{)}^{2}-z\frac{rF_{1}^{\prime}}{F_{1}}\\\
&-(z+2)\frac{rF_{2}^{\prime}}{F_{2}}-\frac{rF_{3}^{\prime}}{F_{3}}-\frac{rF_{4}^{\prime}}{F_{4}}+\frac{r^{2}}{2}\Big{(}\frac{F_{1}^{\prime}}{F_{1}}\frac{F_{3}^{\prime}}{F_{3}}+\frac{F_{1}^{\prime}}{F_{1}}\frac{F_{4}^{\prime}}{F_{4}}+\frac{F_{3}^{\prime}}{F_{3}}\frac{F_{4}^{\prime}}{F_{4}}\Big{)}-\frac{2\Lambda}{F_{2}}-\frac{1}{2}(r\phi^{\prime})^{2}\\\
&+\frac{1}{2}e^{\lambda\phi}\Big{(}\frac{1}{F_{1}}(zG_{1}+rG_{1}^{\prime}+G_{2}((z-1)N+rN^{\prime}))^{2}-\frac{1}{F_{4}}(G_{2}+rG_{2}^{\prime})^{2}\Big{)}\Big{]},\end{split}$
(14)

where ${}^{\prime}\equiv\frac{d}{dr}$. The terms in the 1D Lagrangian are
arranged so as to make apparent the following two scaling symmetries,

$\displaystyle F_{1}\rightarrow\alpha^{2}F_{1},\quad
F_{3}\rightarrow\alpha^{-1}F_{3},\quad F_{4}\rightarrow\alpha^{-1}F_{4},$ (15)
$\displaystyle N\rightarrow\alpha^{\frac{3}{2}}N,\quad G_{1}\rightarrow\alpha
G_{1},\quad G_{2}\rightarrow\alpha^{-\frac{1}{2}}G_{2},$ (16)

and

$\displaystyle F_{3}\rightarrow\beta^{2}F_{3},\quad
F_{4}\rightarrow\beta^{-2}F_{4},$ (17) $\displaystyle N\rightarrow\beta
N,\quad G_{2}\rightarrow\beta^{-1}G_{2}.$ (18)

They represent diffeomorphisms which preserve the volume element $dtdxdy$, and
can thus be thought of as a Noether symmetry inherited by $L_{1D}$. They are
also symmetries of the boosted metric ansatz (12), as long as the coordinates
transform as
$t\rightarrow\alpha^{-1}t,\;x\rightarrow\alpha^{\frac{1}{2}}x,\;y\rightarrow\alpha^{\frac{1}{2}}y$
and $x\rightarrow\beta^{-1}x,\;y\rightarrow\beta y$ respectively.

The two Noether charges associated with these symmetries are found to be,

$\begin{split}Q_{\alpha}&=r^{z+2}\sqrt{F_{1}F_{2}F_{3}F_{4}}\Big{[}2(z-1)+\frac{r}{2}\Big{(}\frac{2F_{1}^{\prime}}{F_{1}}-\frac{F_{3}^{\prime}}{F_{3}}-\frac{F_{4}^{\prime}}{F_{4}}\Big{)}-\frac{3}{2}\frac{F_{4}N^{2}}{F_{1}}\Big{(}z-1+\frac{rN^{\prime}}{N}\Big{)}\\\
&-\frac{1}{2}e^{\lambda\phi}\frac{G_{2}}{F_{4}}(G_{2}+rG_{2}^{\prime})-e^{\lambda\phi}\Big{(}\frac{G_{1}}{F_{1}}+\frac{3}{2}\frac{G_{2}N}{F_{1}}\Big{)}\Big{(}zG_{1}+rG_{1}^{\prime}+G_{2}N\Big{(}z-1+\frac{rN^{\prime}}{N}\Big{)}\Big{)}\Big{]},\end{split}$
(19)

$\begin{split}Q_{\beta}&=r^{z+2}\sqrt{F_{1}F_{2}F_{3}F_{4}}\Big{[}r\Big{(}\frac{F_{3}^{\prime}}{F_{3}}-\frac{F_{4}^{\prime}}{F_{4}}\Big{)}-\frac{F_{4}N^{2}}{F_{1}}\Big{(}z-1+\frac{rN^{\prime}}{N}\Big{)}\\\
&-e^{\lambda\phi}\frac{G_{2}}{F_{1}}\Big{(}\frac{F_{1}}{F_{4}}(G_{2}+rG_{2}^{\prime})+(zG_{1}+rG_{1}^{\prime})N+G_{2}N^{2}\Big{(}z-1+\frac{rN^{\prime}}{N}\Big{)}\Big{]}\end{split}$
(20)

These charges are combinations of bulk fields that do not depend on the radial
bulk coordinate. This is particularly useful in relating horizon data to
boundary data, and plays an important role in determining the thermodynamic
equation of state when the exact bulk interpolating solution is not known, as
we will see later on.

There is another conserved charge in our system associated with the shift
symmetry of the dilaton (11). While it does not play a direct role in the
thermodynamics, it will be useful later on when we consider solutions of the
linearized field equations. It is given by,

$Q_{\phi}=r^{z+2}\sqrt{F_{1}F_{2}F_{3}F_{4}}\Big{[}e^{\lambda\phi}\frac{G_{2}}{F_{4}}(G_{2}+rG_{2}^{\prime})-e^{\lambda\phi}\frac{G_{1}}{F_{1}}\Big{(}zG_{1}+rG_{1}^{\prime}+G_{2}N\Big{(}z-1+\frac{rN^{\prime}}{N}\Big{)}-\frac{2r\phi^{\prime}}{\lambda}\Big{]}.$
(21)

In addition to the above symmetries, there is also a local gauge symmetry
involving $r$ diffeomorphisms, which we fix by setting $F_{3}(r)=1$ in (12).

###  II.3 Holographic renormalisation

In this section, we work out the renormalised stress-energy tensor at the
boundary for the EMD model. The conservation of the stress tensor gives us the
standard energy and momentum conservation equations, while its scaling
behaviour under a Lifshitz transformation will give us the equation of state
of a Lifshitz fluid. For holographic renormalisation of gravity models in
asymptotically Lifshitz spacetime see e.g. Ross:2009ar ; Ross_2011 ;
Zingg:2011cw ; Baggio:2011cp ; Mann:2011hg ; Tarrio:2012xx ;
langley2020quantum .

The variation of the action (2) around a solution to the field equations
reduces to a boundary term,

$\delta S_{0}=\int
d^{3}x\sqrt{-\gamma}\big{(}(K_{\alpha\beta}-K\gamma_{\alpha\beta})\delta\gamma^{\alpha\beta}-e^{\lambda\phi}n^{\alpha}F_{\alpha\beta}\,\delta
A^{\beta}-n^{\alpha}\nabla_{\alpha}\phi\,\delta\phi\big{)},$ (22)

where $n^{\alpha}$ refers to the outward directed unit normal at the boundary.
As usual, regularisation and renormalisation are needed in order to make sense
of this expression. For regularisation, we place the boundary at a large but
finite value of the radial coordinate $r$ and introduce boundary counterterms
to cancel divergences that would otherwise appear as the boundary regulator is
taken to infinity. A number of possible counterterms present themselves, but,
as we will see, there is a simple construction that removes all divergences
associated with the field variations in (22). Furthermore, the resulting
finite stress tensor complex has precisely the form expected for a general $z$
Lifshitz fluid. The remainder of this subsection draws heavily on reference
Ross:2009ar , where holographic renormalisation was carried out for gravity
coupled to a massive vector field in asymptotically Lifshitz spacetime.

The model at hand instead has a massless vector field that diverges at the
boundary and a logarithmically running dilaton field but the shift symmetry
(11) suggests a way around that. We can construct a shift-invariant scalar
combination of the fields, which takes a constant value, independent of $r$,
when evaluated on the pure Lifshitz solution,

$e^{\lambda\phi}A_{\mu}A^{\mu}\Big{|}_{\eqref{lifshitzbackground}}=-\,\frac{2(z-1)}{(z+2)}.$
(23)

In asymptotically Lifshitz spacetime any function of this scalar will
contribute at the same order in $r$ to the boundary action and we are led to
consider a counterterm action of the form,

$S_{c.t.}=\int d^{3}x\sqrt{-\gamma}\big{(}-4+f(e^{\lambda\phi}A^{2})\big{)}.$
(24)

where $f$ is some, as yet, undetermined function. The counterterm involving
the vector field explicitly breaks the gauge symmetry of the bulk theory. As
discussed in Section II, this is allowed because the only role of $A_{\mu}$ is
to provide the background source that supports the asymptotically Lifshitz
geometry and it does not couple directly to matter fields. It can in fact be
viewed as an advantage that the gauge symmetry is broken as there is then no
question of a conserved $U(1)$ charge in the dual field theory. Such a charge
can easily be introduced via an additional bulk gauge field as in Tarr_o_2011
.

The first term in the counterterm action (involving $-4\sqrt{-\gamma}$) is a
standard counterterm in AdS gravity. With it in place, the variation in (22)
involving $\delta\gamma^{tt}$ vanishes for the Lifshitz background (8). This
restricts the function in the matter counterterm to be of the form
$f(-e^{\lambda\phi}A^{2})=\xi\sqrt{-e^{\lambda\phi}A^{2}}$, with some constant
$\xi$. The argument is the same as in Ross:2009ar  for the massive vector
theory, i.e. the matter counterterm should not contribute at all to the
$\delta\gamma^{tt}$ variation and this is ensured if the factor of
$\sqrt{-\gamma_{tt}}$ that comes from the $\sqrt{-\gamma}$ prefactor is
cancelled by a factor $\sqrt{-\gamma^{tt}}$ from
$\sqrt{-e^{\lambda\phi}A_{\mu}A_{\nu}\gamma^{\mu\nu}}$. By this simple
argument the counterterm action has been determined up to a single constant,

$S_{c.t.}=\int
d^{3}x\sqrt{-\gamma}\big{(}-4+\xi\sqrt{-e^{\lambda\phi}A^{2}}\big{)},$ (25)

and the variation of the full action $S=S_{0}+S_{c.t.}$ can be written,

$\delta S=\int
d^{3}x\Big{(}s_{\alpha\beta}\,\delta\gamma^{\alpha\beta}+s_{\alpha}\,\delta
A^{\alpha}+s\,\delta\phi\Big{)},$ (26)

with,

$\displaystyle s_{\alpha\beta}$
$\displaystyle=\sqrt{-\gamma}\Big{(}K_{\alpha\beta}-K\gamma_{\alpha\beta}+2\gamma_{\alpha\beta}-\frac{\xi
e^{\lambda\phi}}{2\sqrt{-e^{\lambda\phi}A^{2}}}(A_{\alpha}A_{\beta}-A^{2}\gamma_{\alpha\beta}\big{)}\Big{)},$
(27) $\displaystyle s_{\alpha}$
$\displaystyle=\sqrt{-\gamma}\Big{(}-e^{\lambda\phi}n^{\beta}F_{\beta\alpha}-\frac{2\xi
e^{\lambda\phi}A_{\alpha}}{2\sqrt{-e^{\lambda\phi}A^{2}}}\Big{)},$ (28)
$\displaystyle s$
$\displaystyle=\sqrt{-\gamma}\Big{(}-n^{\alpha}\nabla_{\alpha}\phi-\frac{\xi\,\lambda
e^{\lambda\phi}A^{2}}{2\sqrt{-e^{\lambda\phi}A^{2}}}\Big{)}.$ (29)

Remarkably, all the variations vanish for the Lifshitz background (8) for a
single value of the free parameter in the boundary counterterm,

$\xi=-\sqrt{2(z-1)(z+2)},$ (30)

and no further counterterms are needed for the problem at hand. For more
general backgrounds, including Lifshitz black branes that are asymptotic to
the Lifshitz solution, these variations will cancel at leading order and
interesting physics resides in subleading terms that do not cancel. For
instance, requiring the action to be invariant under the Lifshitz scaling (1)
leads to a Ward identity,

$z\,s^{t}_{t}+s^{x}_{x}+s^{y}_{y}+\frac{(z-2)}{2}s^{t}\,A_{t}-\sqrt{z-1}\,s=0,$
(31)

which encodes information about the fluid equation of state, as we will see in
Section III below.

The Ward identity brings out the different scaling behaviour of temporal and
spatial components and this difference should also be manifest in the
renormalised stress tensor complex. Indeed, in a non-relativistic system the
energy flux $\mathcal{E}^{i}$ and the momentum density $\mathcal{P}_{i}$ are
independent variables and this appears to be at odds with the symmetry of
$s_{\alpha\beta}$ under interchange of $\alpha$ and $\beta$. The resolution
involves a subtle issue in holographic renormalisation which arises when a
system has background vector fields Hollands:2005ya ; Ross:2009ar . In this
case, we are instructed to introduce orthonormal frame fields
$e_{\alpha}^{(A)}$, with tangent space index $A$, and re-express the metric
and vector variations in (26) in terms of variations involving frame
variables,

$\delta S=\int d^{3}x\Big{(}(-2s^{\alpha}_{\
\beta}+s^{\alpha}A_{\beta})\,e^{\beta}_{(B)}\delta
e^{(B)}_{\alpha}+s_{B}\,\delta A^{B}+s\,\delta\phi\Big{)}.$ (32)

The elements of the boundary stress tensor complex are obtained by considering
frame field variations while keeping fixed the dilaton and tangent space
components of the vector field Ross:2009ar ,

$\displaystyle\mathcal{E}$ $\displaystyle=2s^{t}_{\
t}-s^{t}A_{t},\qquad\quad\mathcal{E}^{i}=2s^{i}_{\ t}-s^{i}A_{t},$ (33)
$\displaystyle\mathcal{P}_{i}$ $\displaystyle=-2s^{t}_{\
i}+s^{t}A_{i},\qquad\Pi^{i}_{\ j}=-2s^{i}_{\ j}+s^{i}A_{j}.$ (34)

In a relativistic theory this prescription ensures a conserved stress tensor.
An alternative prescription involving metric variations, keeping fixed
spacetime components of a background vector field, would still produce a
finite stess tensor but one that does not satisfy the usual conservation law
Hollands:2005ya . In a non-relativistic theory, however, the tangent space
prescription is indispensable. Without the extra terms in (34) that come from
the vector field the elements of the stress tensor complex are not guaranteed
to be finite, let alone conserved Ross:2009ar . We will see this explicitly in
the following subsection. When we consider the asymptotic fields of a moving
Lifshitz black brane, the metric variation yields a divergence in the energy
flux which is exactly cancelled by the contribution from the vector sector.

It is worth noting that for generic $z$ the thermodynamic quantities in the
stress tensor complex have different scaling dimensions. Straightforward
dimensional analysis (see e.g. Ross_2011 ) gives the following scaling
behavior in $d_{s}$ spatial dimensions,

$\mathcal{E}^{\prime}=\Lambda^{-d_{s}-z}\,\mathcal{E}\,,\quad{\mathcal{E}^{\prime}}^{i}=\Lambda^{1-d_{s}-2z}\,\mathcal{E}^{i}\,,\quad\mathcal{P}_{i}^{\prime}=\Lambda^{-1-d_{s}}\,\mathcal{P}_{i}\,,\quad
P^{\prime}=\Lambda^{-d_{s}-z}\,P\,.$ (35)

###  II.4 The asymptotic solution

We now wish to consider general backgrounds that approach the Lifshitz
solution as $r\rightarrow\infty$. While the metric and gauge field ansatz (12)
and (13) is particularly well suited for deriving conserved Noether charges as
in Section II.2, it is less convenient for evaluating the stress tensor
complex. For the asymptotic analysis we instead adopt a set of orthonormal
frame fields,

$\displaystyle e^{(0)}=r^{z}\,H_{1}(r)\,dt+r\,v_{1}(r)\,dy,$ (36)
$\displaystyle e^{(R)}=\frac{dr}{r\,H_{2}(r)},$ (37) $\displaystyle
e^{(1)}=r\,H_{3}(r)\,dx,$ (38) $\displaystyle
e^{(2)}=r^{z}\,v_{2}(r)\,dt+r\,H_{4}(r)\,dy,$ (39)

and orient the frame field $e^{(0)}$ parallel to the vector field,
$A_{\mu}=A_{0}(r)\,e_{\mu}^{(0)}$, with

$A_{0}(r)=\sqrt{\frac{2(z-1)}{(z+2)}}\frac{r^{2}}{r_{h}^{2}}\>a_{0}(r).$ (40)

Fixing the radial gauge $F_{3}(r)=1$ amounts to setting $H_{3}(r)=1$.

In order for the space-time to be asymptotically Lifshitz, we require that
$H_{1}(r),H_{2}(r),H_{4}(r)$, and $a_{0}(r)$ tend to unity as
$r\rightarrow\infty$, while $v_{1}(r)$ and $r^{1-z}\,v_{2}(r)$ tend to zero.
With these assumptions, the field equations for a Lifshitz black brane with
linear momentum reduce to a set of coupled linear ordinary differential
equations for small deviations from the Lifshitz background in the asymptotic
region.

The linearised field equations corresponding to the $ty$ component of
Einstein’s equations and the $y$ component of Maxwell’s equations decouple
from the rest of the linearised equations and can be solved separately for
$v_{1}$ and $v_{2}$. The general solution of the resulting eigenvalue problem
consists of four eigenmodes,

$\displaystyle
v_{1}(r)=c_{1y}\,r^{z-1}+\frac{c_{2y}}{r^{3}}+\frac{z+2}{3z}\frac{c_{3y}}{r^{2z+1}}$
(41) $\displaystyle
v_{2}(r)=\frac{c_{4y}}{r^{z-1}}+\frac{c_{2y}}{r^{3}}+\frac{c_{3y}}{r^{2z+1}},$
(42)

The $c_{1y}$ and $c_{4y}$ modes are non-renormalisable source modes while
$c_{2y}$ and $c_{3y}$ will appear in the momentum density and energy flux,
respectively, in the renormalised stress energy complex Ross_2011 .

The remaining linearised Einstein, Maxwell and dilaton equations can be solved
in a straightforward manner along the lines of Keranen:2012mx . The full
linearised solution has eigenmodes that go as $r^{-z-2}$,
$r^{\frac{1}{2}(-z-2\pm\sqrt{(z+2)(9z+10)})}$ as well as $r^{0}$. A general
holographic analysis of asymptotically Lifshitz space-times (Ross:2009ar ;
Baggio:2011cp ; Zingg:2011cw ; Mann:2011hg ) shows that a $r^{-z-2}$ mode
carries finite energy and is accordingly of primary interest for the analysis
that follows. The remaining eigenmodes include a growing mode that would
disrupt the asymptotic Lifshitz geometry and must therefore be absent in a
physical solution and a mode that falls off at a faster rate than $r^{-z-2}$
and can thus be ignored in our asymptotic analysis. Finally, we also leave out
the constant mode which can be interpreted as a non-normalisable source mode.

We find that the finite energy mode takes the following form,

$\displaystyle H_{1}=1+\frac{c_{1}}{2}\,r^{-z-2},$ (43) $\displaystyle
H_{2}=1+\frac{c_{2}}{2}\,r^{-z-2},$ (44) $\displaystyle H_{3}=1$ (45)
$\displaystyle H_{4}=1+\frac{c_{3}}{2}\,r^{-z-2},$ (46) $\displaystyle
a_{0}=1+c_{4}\,r^{-z-2},$ (47)
$\displaystyle\phi=2\sqrt{z-1}\Big{(}\log\big{(}\frac{r}{r_{h}}\big{)}+\frac{c_{5}}{2}\,r^{-z-2}\Big{)},$
(48)

with the following relations between the various constants,

$c_{2}=c_{1}+c_{3},\qquad c_{5}=\frac{c_{3}}{2}.$ (49)

Radial conservation of the charge (21), associated with the shift symmetry of
the dilaton field, gives an additional relation,

$c_{4}=\frac{c_{1}}{2}-\frac{(z-2)}{4}c_{3},$ (50)

leaving us with two independent constants, $c_{1}$ and $c_{3}$.

Plugging in the asymptotic solution (43)-(48) into the renormalised stress
tensor complex (34) and eliminating constants using (49) and (50) gives the
following expression

$\displaystyle\mathcal{E}=-2c_{1}+zc_{3},$ (51)
$\displaystyle\mathcal{E}^{x}=0,$ (52)
$\displaystyle\mathcal{E}^{y}=-2(z-1)c_{3y},$ (53) $\displaystyle\Pi^{x}_{\
x}=zc_{1}-\frac{1}{2}(z^{2}-z-2)c_{3},$ (54) $\displaystyle\Pi^{y}_{\
y}=zc_{1}-\frac{1}{2}(z^{2}+z+2)c_{3},$ (55)
$\displaystyle\mathcal{P}_{y}=-2(z-1)c_{2y}.$ (56)

The modes $c_{2y}$ and $c_{3y}$ appear only in the off-diagonal terms of the
stress tensor complex and can be interpreted as the momentum density $\rho v$
and energy flux $(\mathcal{E}{+}P)v$ respectively. While they cannot be
related with the other modes in the absence of the full bulk solution, it
appears safe to make this statement, as they show the correct scaling
dimensions as required by (35).

In this calculation it was essential to include the vector field contribution
to the stress tensor complex in (33)-(34). The energy flux, given by
$\mathcal{E}^{i}=2s^{i}_{\ t}-s^{i}A_{t}$, is divergent if only the
contribution from the $s^{i}_{\ t}$ term is included but when both terms are
present the divergence is cancelled, leaving behind a finite result for
$\mathcal{E}^{i}$. Moreover, the resulting finite energy flux (53) and the
momentum density (55) are independent in that they involve different
eigenmodes of the linearised solution.

##  III Thermodynamics

Having constructed a finite renormalised stress energy complex, we can proceed
to study the thermodynamics of the fluid described by the above holographic
dual. We expect to find a perfect Lifshitz fluid, whose description is
invariant under time and space translations as well as rotations, but without
boost symmetry.

The thermodynamic relations that we wish to establish require us to supplement
the linearised solution in the asymptotic region with near-horizon data. This
can be done via the Noether charges that we obtained earlier, even in the
absence of an exact (or numerical) solution that interpolates between the
near-horizon and asymptotic regions.

###  III.1 The near-horizon solution

Using similar arguments to those made in Hartong_2016 , we find that an
appropriate near-horizon Taylor series expansion of the various metric and
gauge fields in (12) and (13) is given by,

$\displaystyle F_{1}$ $\displaystyle=f_{1}\frac{(r-r_{h})}{r_{h}}+\ldots$ (57)
$\displaystyle F_{2}$ $\displaystyle=h_{1}\frac{(r-r_{h})}{r_{h}}+\ldots$ (58)
$\displaystyle F_{3}$ $\displaystyle=1$ (59) $\displaystyle F_{4}$
$\displaystyle=p_{0}+p_{1}\frac{(r-r_{h})}{r_{h}}+\ldots$ (60) $\displaystyle
N$ $\displaystyle=n_{0}+n_{1}\frac{(r-r_{h})}{r_{h}}+\ldots$ (61)
$\displaystyle G_{1}$ $\displaystyle=g_{1}\frac{(r-r_{h})}{r_{h}}+\ldots$ (62)
$\displaystyle G_{2}$
$\displaystyle=m_{0}+m_{1}\frac{(r-r_{h})}{r_{h}}+\ldots$ (63)
$\displaystyle\phi$ $\displaystyle=l_{0}+l_{1}\frac{(r-r_{h})}{r_{h}}+\ldots,$
(64)

where, as before, we are working in the $F_{3}=1$ gauge. The location of the
horizon $r_{h}$ is where $g^{rr}$ vanishes. This means that $F_{2}$ will have
a first order zero at $r_{h}$. Also, regularity of the metric in Eddington-
Finklestein coordinates requires $F_{1}$ to have a first order zero at $r_{h}$
as well. Finally, $G_{1}$ should also have a first order zero at $r_{h}$, but
the remaining fields are regular there.

The near horizon metric is then given by,

$ds^{2}=-f_{1}\rho
r_{h}^{2z-1}dt^{2}+\frac{d\rho^{2}}{r_{h}h_{1}\rho}+r_{h}^{2}dx^{2}+p_{0}r_{h}^{2}\big{(}dy+N(r_{h})r_{h}^{z-1}dt\big{)}^{2},$
(65)

where $\rho=r-r_{h}$. Making a change of coordinates,

$\tilde{\rho}^{2}=\frac{4\rho}{r_{h}h_{1}},\>\tilde{t}=(f_{1}h_{1})^{\frac{1}{2}}\frac{r_{h}^{z}t}{2},$
(66)

the metric takes the following form,

$ds^{2}=-\tilde{\rho}^{2}d\tilde{t}^{2}+d\tilde{\rho}^{2}+r_{h}^{2}dx^{2}+p_{0}r_{h}^{2}\big{(}dy+N(r_{h})r_{h}^{z-1}dt\big{)}^{2}.$
(67)

After going to Euclidean coordinates, the Hawking temperature is found to be
$T=\frac{r_{h}^{z}}{4\pi}(f_{1}h_{1})^{\frac{1}{2}}$. Meanwhile (using units
where $16\pi G_{N}=1$), the entropy is given by $s=\frac{A}{4G_{N}}=4\pi
p_{0}^{\frac{1}{2}}r_{h}^{2}$, and we have

$Ts=(p_{0}f_{1}h_{1})^{\frac{1}{2}}r_{h}^{z+2}.$ (68)

In the following subsections we consider in turn the static and the moving
black branes and compare this expression to the Noether charge(s) obtained in
the near-horizon region. Via the conservation of the Noether charges, this can
then be related to the asymptotic solution and the stress tensor complex
evaluated at the boundary.

The radial gauge conditions $F_{3}(r)=1$ and $H_{3}(r)=1$ ensure that we can
identify the radial variable $r$ in our near-horizon solution with that in our
asymptotic solutions and then there is a straightforward mapping between the
frame fields defined in (36)-(39) and coordinate fields (12), which we can use
to evaluate the Noether charges at the boundary.

###  III.2 Static Lifshitz black brane

As a warm-up exercise, let us first study thermodynamics of a static Lifshitz
brane with general $z$ given by the static limit of (12) by evaluating the
renormalised stress tensor at the boundary and using the method of Noether
charges.

For a static black brane we have $c_{3}=0$ and the stress tensor complex in
(51)-(56) reduces to

$\displaystyle\mathcal{E}=-2c_{1},$ (69) $\displaystyle\mathcal{E}^{x}=0,$
(70) $\displaystyle\mathcal{E}^{y}=0,$ (71) $\displaystyle\Pi^{x}_{x}=zc_{1},$
(72) $\displaystyle\Pi^{y}_{y}=zc_{1},$ (73) $\displaystyle\mathcal{P}_{y}=0.$
(74)

One can easily verify that the scaling Lifshitz Ward identity (31) is
satisfied. In fact, $s_{\alpha}$ and $s$ vanish trivially for the static
brane. Identifying $-2c_{1}$ and $-z\,c_{1}$ as the energy and pressure $P$,
respectively, we obtain

$z\mathcal{E}=2P,$ (75)

which is the well-known equation of state for a static perfect Lifshitz fluid.

Plugging in the near horizon metric anstaz defined in (III.1) into our Noether
charges obtained in (II.2), we find that $Q_{\beta}$ vanishes and $Q_{\alpha}$
reduces to

$Q_{\alpha}=Ts,$ (76)

with $T$ the Hawking temperature and $s$ the entropy density of the static
brane. Plugging in the asymptotic metric ansatz (43)-(48) into the non-
vanishing Noether charge gives

$Q_{\alpha}=-(z+2)c_{1}.$ (77)

Equating the right hand sides of (76) and (77) and identifying
$\mathcal{E}=-2c_{1}$ from (69), we get precisely the Gibbs-Duhem relation
expected for a Lifshitz fluid,

$\mathcal{E}=\frac{2}{z+2}Ts,$ (78)

or, equivalently,

$\mathcal{E}+P=Ts.$ (79)

###  III.3 Moving Lifshitz black brane

Let us now proceed to repeat the above exercise for the case of interest, that
is, a Lifshitz black brane with linear momentum along one of the transverse
directions (which we chose to be the $y$-direction without loss of
generality).

For a perfect fluid moving in the y-direction the pressure is identified via
$\Pi^{x}_{\ x}=-P$ while $\Pi^{y}_{\ y}=-P-\rho\,v^{2}$ is the combined stress
due to pressure and momentum flow along the $y$ direction. The stress tensor
complex for the moving Lifshitz black brane was obtained in (51)-(56) and we
can read off the energy density, pressure, and momentum flux,

$\mathcal{E}=-2c_{1}+z\,c_{3},\quad
P=-z\,c_{1}+\frac{1}{2}(z^{2}-z-2)c_{3},\quad\rho\,v^{2}=(z+2)c_{3}.$ (80)

Using the linearised solution we can easily verify that the Lifshitz Ward
identity (31) is indeed satisfied and gives us the equation of state for a
perfect Lifshitz fluid in motion,

$z\mathcal{E}-\rho v^{2}=2P,$ (81)

as hypothesised in de_Boer_2018 ; de_Boer_2020 . As discussed above, the modes
$c_{2y}$ and $c_{3y}$ appear only in the off-diagonal terms of the stress
tensor complex. Although they cannot be related to the other modes in the
absence of an interpolating solution, they do have the right scaling to be
interpreted as the momentum density $\rho v$ and energy flux
$(\mathcal{E}{+}P)v$, respectively.

Plugging in the near horizon metric anstaz defined in (III.1) into our Noether
charges $Q_{\alpha}$ and $Q_{\beta}$ obtained in (II.2), we find that a simple
linear combination of the two in fact gives us (68). Specifically,

$Q_{\alpha}-\frac{3}{2}Q_{\beta}=Ts.$ (82)

Plugging in the asymptotic metric ansatz (43)-(48) into the same Noether
charges gives

$\displaystyle Q_{\alpha}$
$\displaystyle=\frac{(z+2)}{2}\,(-2c_{1}+z\,c_{3}),$ (83) $\displaystyle
Q_{\beta}$ $\displaystyle=(z+2)c_{3}=\rho\,v^{2}.$ (84)

Now use (80) and the equation of state (81) to express the Noether charges in
terms of fluid variables,

$\displaystyle Q_{\alpha}$
$\displaystyle=\mathcal{E}+P+\frac{1}{2}\rho\,v^{2},$ (85) $\displaystyle
Q_{\beta}$ $\displaystyle=\rho\,v^{2},$ (86)

and it immediately follows that the combination of Noether charges in (82)
gives the Gibbs-Duhem relation for a moving Lifshitz perfect fluid,

$\mathcal{E}+P=Ts+\rho v^{2}.$ (87)

Interpreting the various boundary data as field theory sources in this manner,
we find that the velocity of the fluid $v$ appears as the chemical potential
conjugate to the momentum density $\rho v$ in accordance with Hartong_2016 ;
de_Boer_2018 .

Finally, it is interesting to consider the special value of $z=2$, for which
the Lifshitz scaling symmetry is consistent with Galilean boost symmetry. In
this case, the energy density, pressure, and momentum flux in (80) reduce to

$\mathcal{E}=-2c_{1}+2c_{3},\quad P=-2c_{1},\quad\rho\,v^{2}=4c_{3},$ (88)

The energy density can be written as the sum of an internal energy density and
a kinetic energy density,
$\mathcal{E}=\mathcal{E}_{0}+\frac{1}{2}\rho\,v^{2}$. In a Galilean boost
invariant system $\mathcal{E}_{0}$ is simply the energy density in the rest
frame of the fluid and the pressure, $P=\mathcal{E}_{0}$, is frame
independent. For general $z\neq 2$, these last two statements do not hold.

##  IV Conclusion and Future outlook

In this paper, we have investigated the holographic dual of a perfect Lifshitz
fluid with an arbitrary dynamical exponent, $z$, moving with a velocity, $v$.
The moving black brane we have considered is a solution of the Einstein-
Maxwell-Dilaton theory, and is obtained by constructing a new class of metrics
that correspond to a black brane having a linear momentum, rather than
boosting a static black brane.

By evaluating the Noether charges associated with various scaling symmetries
of our metric, and equating their values at the horizon and boundary (as they
are radially conserved quantities), we were able to obtain the expected
equation of state for the perfect Lifshitz fluid with an arbitrary dynamical
exponent, $z$ (de_Boer_2018 ). Moreover we note that the velocity indeed
appears as a chemical potential, conjugate to the dual momentum density
(de_Boer_2018 , Hartong_2016 ).

By solving the linearised perturbations of the Einstein, Maxwell and dilaton
equations, we found the asymptotic solutions of the metric and gauge fields,
and used these to construct the boundary stress tensor of the fluid. By
working in an orthonormal frame and using counterterms similar to those
proposed in Ross:2009ar , we were able to construct the renormalised stress
tensor complex and read off the various thermodynamic variables. The result
has the expected form for a non-relativistic fluid, and satisfies the Ward
identity associated with Lifshitz scaling.

There are several interesting questions to be investigated. A straightforward
extension of the present work would be to study the gravitational dual of
charged perfect Lifshitz fluids via the addition of other $U(1)$ fields to the
theory, for which the boundary theory that one might need to study would be
that of non-relativistic electrodynamics coupled to torsionless Newton-Cartan
theory, as explored in festuccia2016symmetries . Another extension would be to
carry out the analysis performed in this paper on theories with a non-trivial
coupling between the dilaton and cosmological constant, that is, moving
hyperscale-violating Lifshitz geometries, and study the resulting
thermodynamics.

Lifshitz holography is still a relatively unexplored area as compared to it’s
relativistic counterpart. Following the work done in de_Boer_2020  where the
complete first-order energy-momentum tensor in curved space-time for a fluid
without boost symmetry was computed, one can now consider trying to find the
hydrodynamic modes of Lifshitz fluids from quasinormal modes and compare the
results. In this study, we have concentrated on perfect fluids without
impurities or lattice effects which break translational invariance. Physical
systems that break boost invariance, however, often also exhibit other
spontaneous symmetry breaking patterns. While the main objective of the work
carried out in Armas:2020mpr  was to provide a general hydrodynamic framework
for fluids that break boost symmetry, it also offers a controlled framework
for studying other patterns of symmetry breaking. It would be of interest to
study Lifshitz gravity duals that could model this behaviour, in order to
better understand their microscopics. With the current holographic model for a
moving Lifshitz fluid established, one can also proceed to calculate two point
correlation functions related to transport and other quantities of interest
from a gauge-gravity perspective.

Acknowledgements: We would like to thank Valentina Giangreco M. Puletti, Jelle
Hartong, Niels Obers, Nick Poovuttikul, and Watse Sybesma for useful
discussions and input. This research was supported in part by the Icelandic
Research Fund under grant 195970-052 and by the University of Iceland Research
Fund.

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11institutetext:  Departamento de FÃsica TeÃrica M-8 and CIAFF, Universidad
AutÃnoma de Madrid, Cantoblanco 28049, Madrid, Spain 22institutetext:
Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh,
EH9 3HJ, UK 33institutetext: Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
44institutetext: Dipartimento di Fisica, Sapienza Universitá di Roma, Piazzale
Aldo Moro, 5-00185 Roma, Italy 55institutetext: EURANOVA, Mont-Saint-Guibert,
Belgium

# Mass Estimation of Planck Galaxy Clusters using Deep Learning

Daniel de Andres 11<EMAIL_ADDRESS>Weiguang Cui 22 Florian Ruppin 33
Marco De Petris 44 Gustavo Yepes 11 Ichraf Lahouli 55 Gianmarco Aversano 55
Romain Dupuis 55 Mahmoud Jarraya 55

###### Abstract

Galaxy cluster masses can be inferred indirectly using measurements from X-ray
band, Sunyaev-Zeldovich (SZ) effect signal or optical observations.
Unfortunately, all of them are affected by some bias. Alternatively, we
provide an independent estimation of the cluster masses from the Planck PSZ2
catalog of galaxy clusters using a machine-learning method. We train a
Convolutional Neural Network (CNN) model with the mock SZ observations from
The Three Hundred (the300) hydrodynamic simulations to infer the cluster
masses from the real maps of the Planck clusters. The advantage of the CNN is
that no assumption on a priory symmetry in the cluster’s gas distribution or
no additional hypothesis about the cluster physical state are made. We compare
the cluster masses from the CNN model with those derived by Planck and
conclude that the presence of a mass bias is compatible with the simulation
results.

##  1 Introduction

Galaxy clusters are the biggest gravitational bound objects in the Universe
and their number density as a function of their mass and redshift is very
sensitive to the matter content of the Universe and the expansion history.
Therefore, they constitute one of the best cosmological probes to constrain
different cosmological parameters planck2020 . However, the total mass of a
cluster is not a direct measurable quantity. It is usually inferred from
several observables under some physical assumptions. An accurate determination
of the total mass of galaxy clusters is in fact a key problem in physical
cosmology.

Among the different cluster mass definitions, we are considering $M_{500}$
defined as the total mass contained in a sphere of radius $R_{500}$, within
which the density is 500 times the critical density:

$M_{500}=\frac{4\pi}{3}R^{3}_{500}500\rho_{\text{c}}(z),$ (1)

here $z$ is the redshift and $\rho_{\text{c}}(z)\equiv 3H(z)^{2}/8\pi G$ is
the critical density.

By observing the millimeter wavelength sky one can measure the spectral
distortions of the cosmic microwave background (CMB) photon due to the
inverse-Compton scattering by the free electrons of the intra cluster medium
(ICM), i.e. the Sunyaev-Zeldovich (SZ) effect SZeffect . Through the SZ
effect, thousands of clusters have been observed, and discovered, by the South
Pole Telescope (SPT; SPT ), the Atacama cosmology telescope (ATC; ATC ), and
the Planck telescope planckszcatalog . The intensity of the thermal component
of the SZ effect (tSZ) over the line of sight is quantified by the Compton
parameter $y$, defined as

$y=\frac{\sigma_{\text{T}}k_{\text{B}}}{m_{\text{e}}c^{2}}\int
n_{\text{e}}T_{\text{e}}dl\text{ ,}$ (2)

where $\sigma_{\text{T}}$ is the Thomson cross section, $k_{\text{B}}$ is the
Boltzmann constant, c the speed of light, $m_{\text{e}}$ the electron rest-
mass, $n_{\text{e}}$ the electron number density, $T_{\text{e}}$ the gas
temperature.

One procedure to derive the total mass of the cluster from the $y$-Compton
parameter map is using a scaling relation $Y_{500}-M_{500}$, where $Y_{500}$
denotes the integrated value of the $y$ parameter within $R_{500}$.

Nevertheless, in the Planck catalog, the calibration of this scaling relation
is carried out assuming hydrostatic equilibrium. Therefore the estimated mass
, $M_{\text{est}}$, is expected to be biased. This bias parameter $b$, is
usually defined as $1-b=M_{\text{est}}/M_{\text{true}},$ where
$M_{\text{true}}$ is the total mass of the clusters. The determination of
cosmological parameters, such as the matter density $\Omega_{\text{m}}$ and
the present amplitude of density fluctuations $\sigma_{8}$ is sensitive to the
particular value of $b$. Particularly, the bias value $b$ needed to find no
tension with CMB observations is $(1-b)=0.58\pm 0.04$ planckbiascmb , or the
latest result $(1-b)=0.62\pm 0.05$ salvaticmb . However, the bias value is
about $1-b\simeq 0.8$ from numerical simulations, which is clearly in a
disagreement with CMB observations (e.g. makiyabias ; giulia ). In addition,
recent analyses of weak lensing observations led to a value of the bias
$1-b=0.84\pm 0.04$ herbonetwl , which is in agreement with simulations.

In recent development, machine learning techniques have been used in
simulations to provide an unbiased estimation of the cluster total mass by
training the algorithms using mock observation in X-ray frequencies ntampaka ,
simulated SZ observations guptasz , and in a multi-channel approach combining
X-ray, SZ and stellar mock maps bahamas .

In this work, we aim at predicting the total mass of real galaxy clusters
selected from the Planck Compton-$y$ parameter maps by training a
Convolutional Neural Network (CNN) on simulated maps extracted from a large
dataset of hydrodynamic simulations provided by The Three Hundred
collaboration (threehundred, ).

##  2 Data set

The mock Compton-$y$ parameter maps were computed from the results of the The
Three Hundred hydrodynamic simulations. They consist on the resimulations of
spherical regions of radius $15h^{-1}Mpc$ centered around the most massive 324
clusters found at $z=0$ MULTIDARK dark-matter-only simulation (MDPL2; MDPL2 )
with cosmological parameters from the Planck collaboration planckparameters .
Particularly, in this work we have used the results from the Gadget-X
murantegadgetx ; rasiagadgetx  runs. The cluster size halos are identified
with the AHF algorithm AHF . Then, we have chosen halos with masses greater
than $10^{14}{{\,h^{-1}{\rm{M_{\odot}}}}}$ at redshift $z<1$. Moreover, the
tSZ signal corresponding to these clusters is therefore computed using
equation 2 with the PYMSZ public software PYMSZ . The images are generated to
have a resolution of $1920\times 1920$ pixels with a fixed angular resolution
of 5” so that all images will cover at least $R_{200}$ for all clusters.

A Gaussian smoothing is then applied to these maps in order to mimic the
impact of the Planck Beam (FWHM = 10 arcmin) and we further add a similar
statistical noise to mimic the real Planck observations. In summary, the
simulated data set consists on 7016 selected clusters with 27 random rotations
each, amounting to a total of $191,862$ maps. They have been particularly
selected in order to cover the same mass and redshift range of the selected
Planck cluster sample. The real Planck cluster catalog consists of $1,096$
observed tSZ maps selected from the Second Planck Catalogue of Sunyaev-
Zeldovich Sources (PSZ2; PLSZ2 ) with known redshifts. In figure 1, we show
the Probability Distribution Function (PDF) corresponding to the distribution
in redshift and mass of the simulated Planck clusters and real Planck clusters
used in this work. An example of the $y$-maps for the simulated clusters is
presented in figure 2.

Figure 1: PDF of the cluster samples ( The Three Hundred simulation in solid
black line and Planck PSZ2 clusters in blue dashed line) corresponding to the
distribution in redshift (left panel) and mass (right panel). Figure 2:
Example of simulated tSZ maps. The left panel corresponds to the clean map and
the right panel represents the same image with Planck instrumental beam size
and noise. The value of this cluster mass is $M_{500}=1.23\times
10^{15}\,\rm{M_{\odot}}$ and the corresponding redshift is $z$=0.04. In
addition, the color bar shows the logarithmic value of the Compton-$y$
parameter.

##  3 Algorithm

In this work, we train a deep learning model (e.g. deeplearningbook ) using
the mock images of our simulated cluster sample. Our particular model is based
on a CNN model, that can find a mapping from a two dimensional image $I_{ij}$
to a scalar quantity, i.e. $M_{500}$. Our CNN architecture is a simplified
version of the architecture proposed by Simonyan & Zizzerman commonly known as
VGGNet vggnet . This architecture has already been used to infer cluster
masses from simulated observations in ntampaka  and bahamas . Furthermore, we
have checked that more dense and complicated VGGNet architectures reproduce
similar results based on our data set. To train our algorithm, we consider the
following loss function

$\mathcal{L}=\frac{1}{N}\sum_{i=1}^{N}(\log M_{\text{true}}^{i}-\log
M_{\text{pred}}^{i})^{2}\text{ ,}$ (3)

where $M_{\text{pred}}^{i}$ is the predicted CNN mass for cluster $i$ and $N$
is the total number of cluster images in the training set. We then split our
data set in 80% training, 10% validation, and 10% test. The validation data
set is used to ensure that the model produces an unbiased estimation of the
mass. The test set is used exclusively to asses the final accuracy of the
model. We have observed that mixing all the redshifts together yields to a
very poor performance. Therefore, we have trained 4 CNNs by diving the data
set in 4 different redshift bins: $z\leq 0.1$ ; $0.1<z\leq 0.2$ ; $0.2<z\leq
0.4$ and $0.4>z$.

Only open source python libraries have been used in this project.
Particularly, Keras keras  with Tensorflow tensorflow  GPU acceleration.

##  4 Results

Figure 3: The value of the relative error
$(M_{\text{pred}}-M_{\text{true}})/M_{\text{pred}}$ as a function of the
predicted mass $M_{\text{pred}}$ for The Three Hundred mock data. The dashed
black line represents the median for each mass bin while the error bars
represents the 68% confident interval. Furthermore, red dashed lines represent
the 0 relative error line and $\pm 20\%$ error. Only 1000 points (in red)
drawn randomly from the original distribution are displayed. The error bars
are computed using the full test set ($\simeq 20000$ maps).

Figure 4: The value of the relative error
$(M_{\text{pred}}-M_{\text{Planck}})/M_{\text{pred}}$ as a function of the
predicted mass $M_{\text{pred}}$ for the real Planck data set. The error bars
and lines are the same as figure 3.

After training the CNN models with Planck-like simulated Compton-$y$ parameter
maps, we predict the masses of the PSZ2 cluster catalog. In order to study the
performance of these predictions, we compute the relative error defined as the
relative difference of the Planck $M_{\text{Planck}}$ mass and the predicted
CNN mass $M_{\text{pred}}$: $\text{relative
error}=\frac{M_{CNN}-M_{Planck}}{M_{CNN}}\text{.}$

In figure 3, we show the relative error
$(M_{\text{pred}}-M_{\text{true}})/M_{\text{pred}}$ as a function of the
predicted mass $M_{\text{pred}}$ using simulated data. According to our
results depicted in the figure 3, the relative error with respect to the 3D
dynamical simulated mass is always smaller than 5% for this considered mass
range and the scatter (standard deviation $\sigma$) is 17%. A similar result
by applying the same CNN to real Planck maps is shown in figure 4. However,
note that the CNN mass is an unbiased estimator of the cluster mass as shown
in figure 3, while the Planck cluster mass show systematic difference to the
CNN mass. The bias parameter $b$ can then be computed as the mean of relative
difference between $M_{\text{pred}}$ and $M_{\text{Planck}}$:
$b\simeq(M_{\text{pred}}-M_{\text{Planck}})/M_{\text{pred}}$. We would like to
highlight that the relative error is biased $1-b=0.86$ for massive clusters
$M_{\text{pred}}>4\times 10^{14}\,\rm{M_{\odot}}$. However, in the low mass
range the CNN predictions seem unbiased with respect to the masses estimated
by Planck.

A possible explanation for that is by taking into consideration that Planck
collaboration planck2014  calibrated the scaling relation from a mass-proxy
relation kravtsov  with a slope of 1.79, whereas the Gadget-X clusters of The
Three Hundred simulations is in agreement with a self-similar scaling relation
threehundred  with a slope of $5/3$.

##  5 Conclusion

In this work, we have estimated the mass of clusters by exploring Compton-$y$
parameter maps provided by Planck. In order to do so, we have created a
catalog of almost $200,000$ mock Compton-$y$ maps mimicking the noise levels
of Planck observations. Then, a CNN is trained with the mock images aiming at
predicting the mass of real Planck observations.

Our results show the presence of a mass bias $b$ dependent on cluster mass.
The masses estimated by Planck are unbiased with respect to our predicted
masses for low mass clusters, while the bias for massive clusters is
$1-b=0.86^{+0.12}_{-0.7}$.

Otherwise, in hydrodynamic simulations, the value of this bias does not depend
on the cluster mass giulia  and its value is of order 10-20%. This is in
agreement with the fact that the parameters of The Three Hundred
$Y_{500}-M_{500}$ scaling relation are compatible with the self-similar
relation.

## Acknowledgments

We would like to thank the Red Española de Supercomputación for granting
access to the Marenostrum supercomputer where most of The Three Hundred
simulations have been performed. GY and DA would like to thank MICIN/Feder
(Spain) for partial financial support under project grant PGC2018-094975-C21.
WC is supported by the STFC AGP Grant ST/V000594/1 and the science research
grants from the China Manned Space Project with NO. CMS-CSST-2021-A01 and CMS-
CSST-2021-B01. MDP acknowledges support from Sapienza UniversitÃ di Roma
thanks to Progetti di Ricerca Medi 2020, prot. RM120172B32D5BE2.

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