# Perturbations and stability conditions of k-essence and kinetic gravity
braiding models in two-field measure theory

R. Cordero 1<EMAIL_ADDRESS>J. De-Santiago 2,3<EMAIL_ADDRESS>O. G. Miranda 3<EMAIL_ADDRESS>M. Serrano-Crivelli 1
<EMAIL_ADDRESS>1 Departamento de Física, Escuela Superior de
Física y Matemáticas del Instituto Politécnico Nacional
Unidad Adolfo López Mateos, Edificio 9, 07738 Ciudad de México, Mexico  2
Consejo Nacional de Ciencia y Tecnología. Av. Insurgentes Sur 1582, 03940,
Ciudad de México, México  3 Departamento de Física, Centro de Investigación y
de Estudios Avanzados del IPN
Apdo. Postal 14-740 07000 Ciudad de México, Mexico

###### Abstract

We study cosmological perturbations for k-essence and kinetic gravity braiding
models in the context of the two-field measure theory (TMT). Considering
scalar perturbations and the uniform field gauge, we obtain the sound speed of
the fields and present a stability analysis by means of the kinetic matrix and
the mass eigenvalues. For k-essence models, in the two-field measure theory,
the speed of propagation of the field is modified completely due to the new
measure field and it gives rise to crucial differences with respect to the
case without new measure. The stability analysis gives a physical viable model
for the Universe. For the kinetic gravity braiding models in the two-field
measure theory we get that, in general, the speed of perturbations is equal to
the speed of light which is a consequence of the properties of the new measure
field. In the later case, there is always a ghost field. Furthermore, we
calculate general expressions for the mass eigenvalues and find, for an
explicit example, the existence of tachyonic instabilities.

##  I Introduction

The discovery of the Universe’s accelerating expansion Riess et al. (1998);
Perlmutter et al. (1999) became one of the most interesting current problems
in cosmology Sahni (2004); Alam et al. (2004); Sahni and Starobinsky (2006);
Feng et al. (2005); Durrer and Maartens (2008); Bamba et al. (2012). To
explain this behaviour, scientists have proposed many different dark energy
models that can be classified into two kinds: the matter and gravity-modified
models (see, for example, Bianchi and Rovelli (2010); Ratra and Peebles
(1988); Copeland et al. (2006); Linder (2008); Deffayet et al. (2002); Shtanov
and Sahni (2003); Amendola and Tsujikawa (2015); Aviles et al. (2018)).

A recent alternative explanation to the observations of an accelerating
expanding Universe is the Two Field Measure Theory (TMT) Guendelman et al.
(2012); Ansoldi and Guendelman (2013). This model includes a new measure,
$\Phi$, in its action, besides the usual general relativity measure
($\sqrt{-g}$). Different actions have been analyzed in this context Guendelman
et al. (2013, 2015, 2016); Cordero et al. (2019); Bensity et al. (2021);
Benisty and Guendelman (2019). Recently, it has been argued that this kind of
formalism is useful in string and brane theories to generate a string tension
that appears dynamically Guendelman (2021a, b); Vulfs and Guendelman (2020).
One of the more interesting properties is that for general k-essence models,
this proposal unifies dark matter and dark energy Guendelman et al. (2012);
Ansoldi and Guendelman (2013). Moreover, the role of fermionic matter in TMT
has also been studied and it was found that its incorporation may lead to a
dynamically generated dark energy, for instance, through the neutrino sector
Guendelman and Kaganovich (2006a); Guendelman and Kaganovich (2013). However,
to test its viability as an acceptable cosmological model, it is necessary to
study the perturbations of these models to see if they evolve according to the
observations without developing ghosts or exponential growth. For example,
there exist ghost modes in the self-accelerating branch of the Dvali-
Gabadadze-Porrati cosmological model Luty et al. (2003), in a nonlinear
massive gravity Boulware and Deser (1972) and in the linear massive gravity
with a de Sitter background spacetime Higuchi (1987). Several aspects of
perturbations in TMT models have been studied in the k-essence context in
Guendelman and Kaganovich (2008); Guendelman et al. (2012) by a different
approach to the presented here.

Some of the most studied models with the possibility to drive the late-time
accelerated expansion of the Universe lie in the category of Horndeski’s
models Horndeski (1974), which are scalar-tensor theories with second-order
equations of motion. From the classification found in Deffayet et al. (2011),
we will focus on the first two types of Lagrangians, the k-essence Armendariz-
Picon et al. (1999, 2001, 2000); Melchiorri et al. (2003); Chiba et al.
(2000); Chiba (2002); Chimento and Feinstein (2004); Chimento (2004); De-
Santiago and Cervantes-Cota (2011); De-Santiago et al. (2013); Cordero et al.
(2017) and the kinetic braiding models Deffayet et al. (2010), which are not
excluded by recent observations of gravitational waves Abbott et al. (2017a,
b, c); Kase and Tsujikawa (2019); Kobayashi (2019). Other physical preferences
could be extracted from the study of the Hubble constant tension in
Hordenski’s models which can be found in Banerjee et al. (2021); Lee et al.
(2022). The k-essence models are characterized by the action111This kind of
models were proposed initially for inflation Armendariz-Picon et al. (1999);
Garriga and Mukhanov (1999).

$S=\int K(\varphi,X)\sqrt{-g}d^{4}x\,,$ (1)

where the Lagrangian $K$ is a general function of the k-essence field
$\varphi$ and its kinetic term
$X=-1/2g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi$, where
$g_{\mu\nu}$ is the metric of the spacetime with signature $(-+++)$. Later on,
it was realized that this model belongs to a wider class of models called
Galileons Nicolis et al. (2009); de Rham and Tolley (2010); Goon et al.
(2011a, b). In particular, some k-essence models can provide a mechanism to
produce a unified description of dark energy and dark matter Scherrer (2004);
De-Santiago and Cervantes-Cota (2011); De-Santiago et al. (2013). The second
very interesting subclass of Horndeski’s models is the so-called kinetic
gravity braiding models. These models have an action of the form

$S=-\int{G_{3}}(\varphi,X)\square\varphi\sqrt{-g}d^{4}x\,,$ (2)

that includes a D’Alambertian operator and a general function of the field and
its kinetic term Deffayet et al. (2010). 222The notation $G_{3}$ for the
Lagrangian comes from the classification of the generalized Galileon models
Deffayet et al. (2011). In this model, the pressure depends on second-order
derivatives of the scalar fields, and the expression for the energy density
contains the Hubble parameter. These characteristics produce exciting features
analyzed in Refs. Pujolas et al. (2011); Kimura and Yamamoto (2011); Kimura et
al. (2012); Maity (2013).

In this paper, we study the cosmological scalar perturbations of k-essence and
kinetic gravity braiding models in the context of two field measure theory. To
do this we perturb the action following the procedure in De Felice and
Tsujikawa (2012). In particular, we obtain propagation speed of the fields,
the kinetic matrix and the mass eigenvalues for the kinetic gravity braiding
model in TMT. We illustrate our calculations for a particular braiding field.

The paper is organized as follows. In Section II, we develop the main
ingredients of TMT. In Section III, we compute the scalar perturbations and
the speed of sound of the field for k-essence models and compare them with the
case when the TMT is included. In Section IV, we calculate the perturbations
of the kinetic gravity braiding model in TMT and the kinetic matrix. In
particular, we present the procedure to obtain the mass eigenvalues. Finally,
we give our conclusions. Additionally, as a guide to the reader, we include
table 1 with the symbols used in this work.

$\varphi$ | Horndeski’s field.
---|---
$\displaystyle X=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi$ | Kinetic term associated to the Horndeski’s field.
$\Phi$ | New measure, Eq. (4).
$\psi$ |
$\chi$ | Metric perturbations, Eq. (14).
$\phi$ |
$\Psi=a^{3}+\Phi_{0}$ | Auxiliary field used in subsection IV.2.
$\phi_{N}$, $\psi_{N}$ | Bardeen potentials.
Table 1: Reference to the variables used in this work.

##  II The two field measure theory

In this Section we describe the main features of TMT, that we will use
throughout this work, following the discussions presented in Guendelman and
Kaganovich (2006b); Cordero et al. (2019). In this theory, the action has two
measures: the term with the usual metric $g_{\mu\nu}$ and an extra term with a
new measure $\Phi$, in the same space-time,

$S=\int\mathcal{L}_{1}\sqrt{-g}d^{4}x+\int\mathcal{L}_{2}\Phi d^{4}x.$ (3)

It is possible to construct the new measure using four scalar fields,
$\xi^{a}$ ($a=1,2,3,4$),

$\Phi=\epsilon^{\mu\nu\alpha\beta}\epsilon_{ijkl}\partial_{\mu}\xi^{i}\partial_{\nu}\xi^{j}\partial_{\alpha}\xi^{k}\partial_{\beta}\xi^{l}.$
(4)

In the present work we consider the Lagrangians $\mathcal{L}_{1}$,
$\mathcal{L}_{2}$ to be independent from the fields $\xi^{a}$ and each of them
to be independent of each other. The relations between the fields come from
the equations of motion.

As shown in Guendelman and Kaganovich (1998), the variation of the action
produces the equation of motion

$A^{\mu}_{i}\partial_{\mu}\mathcal{L}_{2}=0,$ (5)

where

$A^{\mu}_{i}=\epsilon^{\mu\nu\alpha\beta}\epsilon_{ijkl}\partial_{\nu}\xi^{j}\partial_{\alpha}\xi^{k}\partial_{\beta}\xi^{l}.$
(6)

Considering the relation

$\det\left(A^{\mu}_{i}\right)=\frac{4^{-4}}{4!}\Phi^{3}\,,$ (7)

and if we assume $\Phi\neq 0$, the constancy of the Lagrangian
$\mathcal{L}_{2}$ arises.

In the following sections we will assume that the first Lagrangian contains
the usual Einstein-Hilbert action plus a matter Lagrangian which will be
repeated in the second term. In this work we will consider the matter
Lagrangian to be either the k-essence or braiding scalar field, but it can
include the other matter-energy components of the Universe:

$\displaystyle\mathcal{L}_{1}$ $\displaystyle=$
$\displaystyle\frac{M_{p}^{2}}{2}R+\mathcal{L}_{M}$ (8)
$\displaystyle\mathcal{L}_{2}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{M},$ (9)

where $M_{\rm{P}}=1/\sqrt{8\pi G}$ is the reduced Planck mass ($c=\hbar=1$).
The considerations of this section imply the constancy of $\mathcal{L}_{M}$.

##  III K-essence model perturbations

In this section, we will first derive the cosmological gravitational
perturbations for a Universe filled with a purely kinetic k-essence field and
calculate its sound speed in the absence of the second measure. For this we
will follow De Felice and Tsujikawa De Felice and Tsujikawa (2012) who compute
the perturbations for a Universe filled with a general Horndeski’s model. We
will repeat the procedure for the model including the new measure and compare
the results.

We start by considering that the action of a k-essence field
$\mathcal{L}_{1}=\frac{M_{\rm{P}}^{2}}{2}R+K(\varphi,X)$ with
$\mathcal{L}_{2}=0$:

$\displaystyle S=\int
d^{4}x\>\sqrt{-g}\>\left[\frac{M_{\rm{P}}^{2}}{2}R\;+\;K\left(\varphi,X\right)\right].$
(10)

The flat Friedmann equations associated with this field are given by
Armendariz-Picon et al. (1999):

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3M_{\rm{P}}^{2}}\epsilon\,,$ (11)
$\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=$
$\displaystyle-\frac{1}{6M_{\rm{P}}^{2}}\left(\epsilon+3P\right)\,,$ (12)

with

$\displaystyle\epsilon_{k}=2XK_{,X}-K\,,$ $\displaystyle P_{k}=K\,.$ (13)

We will work with a flat Friedmann-Robertson-Walker spacetime plus scalar
perturbations, that has a metric of the form

$ds^{2}=-\left(1+2\psi\right)dt^{2}+2\partial_{i}\>\chi\>dt\>dx^{i}+a^{2}E_{ij}dx^{i}dx^{j}+a^{2}\left(1+2\phi\right)d\mathbf{x}^{2}\,.$
(14)

The selection of a gauge for this metric usually involves choosing two of the
perturbations as zero Ma and Bertschinger (1995). Here we will use the uniform
field gauge where the k-essence field perturbation vanishes
($\delta\varphi=0$) Maldacena (2003) and, to fix the remaining gauge freedom,
also set $E_{ij}=0$.

To obtain the first order equations of motion for the perturbation fields we
expand the action (10) up to second order as $S\approx
S^{(0)}+S^{(1)}+S^{(2)}$. Using the metric (14) as well as the background
Friedmann equations we get the expression:

$\displaystyle S^{\>(2)}$ $\displaystyle=$ $\displaystyle\int
d^{4}xa^{3}\left[\left(2\omega_{1}\dot{\phi}-\omega_{2}\psi\right)\frac{\partial^{2}\chi}{a^{2}}+\frac{\omega_{3}}{3}\psi^{2}+\frac{\omega_{4}}{a^{2}}\left(\partial\phi\right)^{2}-3\omega_{1}\dot{\phi}^{2}\right.$
$\displaystyle\quad\quad\quad\quad\left.+\left(3\omega_{2}\dot{\phi}-2\omega_{1}\frac{\partial^{2}\phi}{a^{2}}\right)\psi\right],$

where the $\omega_{i}$ coefficients are taken from Ref. De Felice and
Tsujikawa (2012) and we will use them to compare with the TMT case. Here they
reduce to

$\displaystyle\omega_{1}$ $\displaystyle=$
$\displaystyle\omega_{4}=M_{\rm{P}}^{2},$ (16) $\displaystyle\omega_{2}$
$\displaystyle=$ $\displaystyle 2M_{\rm{P}}^{2}H,$ (17)
$\displaystyle\omega_{3}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\dot{\varphi}^{2}\left(K_{,X}+\dot{\varphi}^{2}K_{,XX}\right)-9M_{\rm{P}}^{2}H^{2}.$
(18)

One can obtain the linear equations of motion for each scalar perturbation by
varying the second order action (III) with respect of each of them. In general
the expressions can be quite involved, but we can use the procedure in Ref. De
Felice et al. (2017) to obtain the speed of the perturbations directly from
the terms in the action. In the following paragraphs we summarize the steps
behind this procedure.

Considering some of the perturbation fields as Lagrange multipliers we can use
their equations of motion and some integrations by parts, to arrive at the
second order action in the general form:

$\displaystyle S^{(2)}=\int
d^{4}x\left[A_{ij}\dot{Q}_{i}\dot{Q}_{j}-C_{ij}\left(\partial
Q_{i}\right)\left(\partial
Q_{j}\right)-B_{ij}Q_{i}\dot{Q}_{j}-D_{ij}Q_{i}Q_{j}\right],$ (19)

where $Q_{i}$ are the dynamical perturbation fields of the theory and we are
summing over repeated indices. In general, $A,B,C$ and $D$ will be some square
matrices dependent only on background quantities. The associated Euler-
Lagrange equations become

$2A_{ij}\ddot{Q}_{j}+(2\dot{A}_{ij}+B_{ij}-B_{ji})\dot{Q}_{j}-2C_{ij}\partial^{2}Q_{j}+(2D_{ij}-\dot{B}_{ij})Q_{j}=0\,.$
(20)

In order to find the sound speed, we are interested in perturbations inside
the horizon, with $k\gg aH$. After Fourier-transforming the fields $Q_{i}$ we
keep only the higher terms in $\omega$ and $k$, arriving at

$(\omega^{2}A_{ij}-k^{2}C_{ij})\widetilde{Q}_{j}(\omega,k)=0\,,$ (21)

that can be solved for $Q_{j}\neq 0$ only if the determinant of the
coefficient is zero. Using $\omega=c_{s}k/a$ we obtain the condition

$\det\left(c_{S}^{2}A-a^{2}C\right)=0,$ (22)

which can be used to obtain the speed of propagation of the perturbations in
the limit of high frequencies. If $c_{s}^{2}$ is possitive, then the
perturbations propagate without growing, if it is zero, the perturbations grow
at the rate needed for the structure formation, if it is negative the
perturbations grow exponentially.

Coming back to the k-essence field, we can use the equation of motion for
$\chi,\psi,\phi$,

$\displaystyle E_{\chi}$ $\displaystyle\to$
$\displaystyle\partial^{2}(2\omega_{1}\dot{\phi}-\omega_{2}\psi)=0\,,$ (23)
$\displaystyle E_{\psi}$ $\displaystyle\to$ $\displaystyle
2\frac{\omega_{3}a^{3}}{3}\psi+3\omega_{2}a^{3}\dot{\phi}-2\omega_{1}a\partial^{2}\phi-\omega_{2}a\partial^{2}\chi=0\,,$
(24) $\displaystyle E_{\phi}$ $\displaystyle\to$
$\displaystyle-\frac{d}{dt}\left(2\omega_{1}a\partial^{2}\chi-6\omega_{1}a^{3}\dot{\phi}+3\omega_{2}a^{3}\psi\right)-2\omega_{4}a\partial^{2}\phi-2\omega_{1}a\partial^{2}\psi=0\,,$
(25)

to write the action in the form (19) in terms of a single perturbation field
$Q_{i}=(\phi)$. The speed of sound $c_{s}^{2}$ in this case is given by:

$\displaystyle
c_{S}^{2}=\frac{3\left(2\omega_{1}^{2}\omega_{2}H-\omega_{2}^{2}\omega_{4}-2\omega_{1}^{2}\dot{\omega}_{2}\right)}{\omega_{1}\left(4\omega_{1}\omega_{3}+9\omega_{2}^{2}\right)}.$
(26)

Taking into account that $\partial\epsilon_{k}/{\partial X}=K_{,X}+2XK_{,XX}$
and substituting the values of $\omega_{1},\omega_{2}$ and $\omega_{3}$ we
obtained the well known result Armendariz-Picon et al. (1999)

$c_{S}^{2}=\frac{\frac{\partial P}{\partial
X}}{\frac{\partial\epsilon}{\partial X}}=\frac{\partial P}{\partial\epsilon}.$
(27)

###  III.1 k-essence and the TMT

In the remaining of this section we repeat the previous calculation for the
k-essence field including the new measure. First we introduce the new measure
field as $\Phi=\Phi_{0}(t)+\delta\Phi(t,\mathbf{x})$, where $\Phi_{0}$ is the
value of the new measure at the background level and $\delta\Phi$ is its
perturbation. We start again with the corresponding action for the TMT case

$\displaystyle S=\int d^{4}x\sqrt{-g}\>\frac{M_{p}^{2}}{2}R+\int
d^{4}x\>\left(\sqrt{-g}+\Phi\right)\>K\left(\varphi,X\right).$ (28)

By writing the field equations at the background level, we arrive at the
energy density and pressure of the k-essence field as

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\dot{\varphi}^{2}K_{,X}-K+\dot{\varphi}^{2}K_{,X}\frac{\Phi_{0}}{a^{3}},$
(29) $\displaystyle P$ $\displaystyle=$ $\displaystyle K.$ (30)

Then we expand the action (28) up to second order in the perturbations fields,
and use the Friedmann equations with the new energy density and pressure to
write

$\displaystyle S^{\>(2)}$ $\displaystyle=$ $\displaystyle\int
d^{4}xa^{3}\left[\left(2\omega_{1}\dot{\phi}-\omega_{2}\psi\right)\frac{\partial^{2}\chi}{a^{2}}+\frac{W_{3}}{3}\psi^{2}+\frac{\omega_{4}}{a^{2}}\left(\partial\phi\right)^{2}-3\omega_{1}\dot{\phi}^{2}\right.$
$\displaystyle\quad\quad\quad\quad\left.+\left(3\omega_{2}\dot{\phi}-2\omega_{1}\frac{\partial^{2}\phi}{a^{2}}+\frac{W_{2}}{a^{3}}\left(3\Phi_{0}\phi-\delta\Phi\right)\right)\psi\right],$

where we defined the $\omega_{i}$’s in Eqs. (16),(17),(18) and two new
coefficients appear

$\displaystyle W_{2}$ $\displaystyle=$ $\displaystyle\dot{\varphi}^{2}K_{,X},$
(32) $\displaystyle W_{3}$ $\displaystyle=$
$\displaystyle\omega_{3}+\frac{3}{2}\left(3\dot{\varphi}^{2}K_{,X}+\dot{\varphi}^{4}K_{,XX}\right)\frac{\Phi_{0}}{a^{3}}\,.$

We can vary the action (III.1) to obtain linear equations of motion for the
fields, which can be written as

$\displaystyle E_{\chi}$ $\displaystyle\to$
$\displaystyle\partial^{2}(2\omega_{1}\dot{\phi}-\omega_{2}\psi)=0,$ (33)
$\displaystyle E_{\psi}$ $\displaystyle\to$
$\displaystyle-\omega_{2}a\partial^{2}\chi+\frac{2}{3}W_{3}a^{3}\psi+3\omega_{2}a^{3}\dot{\phi}-2\omega_{1}a\partial^{2}\phi+3W_{2}\Phi_{0}\phi-
W_{2}\delta\Phi=0,$ (34) $\displaystyle E_{\phi}$ $\displaystyle\to$
$\displaystyle-2\omega_{1}\dot{a}\partial^{2}\chi-2\omega_{1}a\partial^{2}\dot{\chi}-2\omega_{4}a\partial^{2}\phi+6\omega_{1}a^{3}\ddot{\phi}+18\omega_{1}a^{3}H\dot{\phi}-9\omega_{2}a^{3}H\psi-3\dot{\omega}_{2}a^{3}\psi-3\omega_{2}a^{3}\dot{\psi}$
$\displaystyle-2\omega_{1}a\partial^{2}\psi+3W_{2}\Phi_{0}\psi=0,$
$\displaystyle E_{\delta\Phi}$ $\displaystyle\to$ $\displaystyle-
W_{2}\psi=0\,.$ (36)

First we note that the new measure produces a equation of motion
$E_{\delta\Phi}$ that constrains $\psi$ to be zero. In fact this relation can
be obtained from the perturbation (in the uniform gauge) of the constancy of
${\cal{L}}_{2}$, since
$\delta{\cal{L}}_{2}=-\frac{1}{2}K(\varphi,X)_{,X}\partial_{\alpha}\varphi\partial_{\beta}\varphi\delta
g^{\alpha\beta}=-{K}_{,X}\dot{\varphi}^{2}\psi=0$. This relation is correctly
reproduced considering the perturbation of $\Phi$. Instead, if we take the
perturbations of the fields $\xi^{a}$, we can use the relation
$\delta\Phi=4A^{\mu}_{i}\partial_{\mu}\delta\xi^{i}$, and find that the
analogous relation to Eq.(36) is given by
$\partial_{\mu}\left(K_{,X}\dot{\varphi}^{2}\psi\right)=0$. This last relation
implies that $K_{,X}\dot{\varphi}^{2}\psi$ is a constant, and if we consider
it equal to zero we recover the result of Eq. (36).

Assuming that $\phi$ can be written as

$\phi(x,t)=\int\phi(k,t)e^{ik\cdot x}d^{3}k,$ (37)

the equation of motion for $\chi$ (33) implies

$\dot{\phi}(k,t)=0\,,$ (38)

or $\phi(k,t)=\phi(k)$. This is a known solution for the perturbation of a
flat FRW Universe dominated by dust as it is the case of Cold Dark Matter
(E.g. see Ref. Mukhanov (2005)). Here we obtain a similar solution but for a
general k-essence model in the presence of a second measure field. It would be
interesting to relate this result with the usual Cold Dark Matter approach.
The solution for Cold Dark Matter, however, is obtained in a Newtonian or
longitudinal gauge. In order to transform our results to the longitudinal
gauge we use a gauge transformation relation (see Ref. Malik and Wands (2009))

$\displaystyle\phi_{N}$ $\displaystyle=$
$\displaystyle\psi+H\chi+a\frac{d}{dt}\left(\frac{\chi}{a}\right)\,,$ (39)
$\displaystyle\psi_{N}$ $\displaystyle=$ $\displaystyle-\phi-H\chi\,,$ (40)

where the fields in the right-hand side of the previous equations correspond
to the fields used in this work, with solutions $\psi=0$ and $\dot{\phi}=0$.
From eq. (III.1)

$\frac{d}{dt}(a\chi)+a\phi=0\,.$ (41)

This model can reproduce the background behaviour of a Universe filled with
matter plus cosmological constant Guendelman et al. (2012); Cordero et al.
(2019). In the epoch dominated by matter, the scale factor $a\propto t^{2/3}$
leads to a metric perturbation $\chi=-3\phi t/5$. Substituting in equations
(39) and (40) we arrive at

$\displaystyle\phi_{N}$ $\displaystyle=$ $\displaystyle-\frac{3}{5}\phi\,,$
(42) $\displaystyle\psi_{N}$ $\displaystyle=$
$\displaystyle-\frac{3}{5}\phi\,.$

The Newtonian gravitational potentials (also known as Bardeen potentials) turn
out to be constant and equal, as it is expected for a Universe filled with
Dark Matter. In the following subsection we explore an alternative route to
convince ourselves that the perturbations of this model resemble those of a
matter component.

###  III.2 Geodesic perturbations

As obtained in section II the equation of motion associated to the new measure
implies the constancy of the matter Lagrangian. In this case

$K(\varphi,X)=\rm{constant},$ (43)

which can be written as follows:

$\nabla_{\alpha}K=K_{\varphi}\varphi_{,\alpha}+K_{X}\nabla_{\alpha}X=0\,.$
(44)

We can rewrite this equation by defining the 4-velocity of the field as

$U_{\alpha}\equiv\frac{\varphi_{,\alpha}}{\sqrt{2X}},$ (45)

then the constancy of the Lagrangian can be written as

$\left(\sqrt{2X}K_{\varphi}-K_{X}U^{\beta}\nabla_{\beta}X\right)U_{\alpha}-2XK_{X}U^{\beta}\nabla_{\beta}U_{\alpha}=0\,.$
(46)

Since the two terms are linearly independent we arrive at the equations

$\displaystyle\sqrt{2X}K_{\varphi}$ $\displaystyle=$ $\displaystyle
K_{X}U^{\beta}\nabla_{\beta}X\,,$ (47) $\displaystyle
U^{\beta}\nabla_{\beta}U_{\alpha}$ $\displaystyle=$ $\displaystyle 0\,.$ (48)

The last equation corresponds to the geodesic equation that is satisfied
independently of the k-essence Lagrangian. This result was first obtained in
Ref. Guendelman et al. (2012) by means of a different approach and it explains
that in the procedure of the previous subsection. The fact that the field
obeys the geodesic equation allows it to be a good candidate for Dark Matter
as it adequately reproduces the formation of structure in the Universe.

##  IV Kinetic gravity braiding model perturbations

In this section, we compute the perturbation equations, the kinetic matrix,
the mass eigenvalues and the sound speed for the kinetic gravity braiding
models Deffayet et al. (2010) plus the new measure field. We first follow De
Felice and Tsujikawa (2012) calculations without the new measure in order to
illustrate the procedure.

The action is given by

$\displaystyle S=\int
d^{4}x\>\sqrt{-g}\>\left[\frac{R\>M_{\rm{P}}^{2}}{2}\;-\;G_{3}\left(X,\varphi\right)\square\varphi\right].$
(49)

At the background level, the field equations we arrive are the same Friedmann
equations than for the k-essence field (11) and (12), where now the energy
density and pressure are:

$\displaystyle\epsilon_{B}=3\dot{\varphi}^{3}HG_{3,X}-\dot{\varphi}^{2}G_{3,\varphi}\,,$
$\displaystyle
P_{B}=-\dot{\varphi}^{2}\left(G_{3,\varphi}+\ddot{\varphi}G_{3,X}\right)\,.$
(50)

After we expand the action up to second order in perturbations, and use the
Friedmann equations and some integrations by parts, we obtain the analogous
expression as Eq. (III):

$\displaystyle S^{\>(2)}$ $\displaystyle=$ $\displaystyle\int
d^{4}xa^{3}\left[\left(2\omega_{1}\dot{\phi}-\omega_{2}\psi\right)\frac{\partial^{2}\chi}{a^{2}}+\frac{\omega_{3}}{3}\psi^{2}+\frac{\omega_{4}}{a^{2}}\left(\partial\phi\right)^{2}-3\omega_{1}\dot{\phi}^{2}\right.$
$\displaystyle\quad\quad\quad\quad\left.+\left(3\omega_{2}\dot{\phi}-2\omega_{1}\frac{\partial^{2}\phi}{a^{2}}\right)\psi\right],$

where the $\omega_{i}$’s introduced in De Felice and Tsujikawa (2012) are now:

$\displaystyle\omega_{1}$ $\displaystyle=$
$\displaystyle\omega_{4}=M_{\rm{P}}^{2},$ (52) $\displaystyle\omega_{2}$
$\displaystyle=$ $\displaystyle-2X\dot{\varphi}G_{3,X}+2M_{\rm{P}}^{2}H,$ (53)
$\displaystyle\omega_{3}$ $\displaystyle=$ $\displaystyle
18X^{2}\dot{\varphi}HG_{3,XX}+36X\dot{\varphi}HG_{3,X}-6X^{2}G_{3,\varphi
X}-6XG_{3,\varphi}-9M_{\rm{P}}^{2}H^{2}.$ (54)

The equation of motion for $\chi$ (23), will be used again to put the action
in the form of (19) and we only have one independent perturbation field
$\phi$. We can compute the speed of sound using the expression (22):

$\displaystyle
c_{S}^{2}=3\frac{2\omega_{1}^{2}\omega_{2}H-\omega_{2}^{2}\omega_{4}-2\omega_{1}^{2}\dot{\omega}_{2}}{\omega_{1}\left(4\omega_{1}\omega_{3}+9\omega_{2}^{2}\right)}.$
(55)

If we now turn the attention to the case with TMT, we will arrive at a
different and general result. In this case, the action for
$G_{3}(X,\varphi)\square\varphi$ with the new measure included is given by

$\displaystyle S=\int d^{4}x\>\frac{R\>M_{\rm{P}}^{2}}{2}\;\sqrt{-g}\>-\int
d^{4}x\>\left(\sqrt{-g}+\Phi\right)G_{3}\left(X,\varphi\right)\square\varphi.$
(56)

The energy density and pressure for the Horndeski field $\varphi$ become:

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\epsilon_{B}\left(1+\frac{\Phi_{0}}{a^{3}}\right)+6G_{3}\dot{\varphi}H\frac{\Phi_{0}}{a^{3}}-\dot{\varphi}G_{3}\frac{\dot{\Phi}_{0}}{a^{3}}+G_{3}\ddot{\varphi}\frac{\Phi_{0}}{a^{3}},$
(57) $\displaystyle P$ $\displaystyle=$ $\displaystyle
P_{B}\left(1+\frac{\Phi_{0}}{a^{3}}\right)-\dot{\varphi}G_{3}\frac{\dot{\Phi}_{0}}{a^{3}}-G_{3}\ddot{\varphi}\frac{\Phi_{0}}{a^{3}},$
(58)

where $\epsilon_{B}$ and $P_{B}$ are the energy density and pressure from the
kinetic gravity braiding model (50). After we vary the action (56) up to
second order, and simplify it using Friedmann equations and some integration
by-parts, the action can be written as:

$\displaystyle S^{(2)}$ $\displaystyle=$ $\displaystyle\int
d^{4}xa^{3}\left[\left(2\omega_{1}\dot{\phi}-V_{2}\psi+3\frac{\Phi_{0}}{a^{3}}\dot{\varphi}G_{3}\phi\right)\frac{\partial^{2}\chi}{a^{2}}+\frac{V_{3}}{3}\psi^{2}+\frac{\omega_{4}}{a^{2}}\left(\partial\phi\right)^{2}-3\omega_{1}\dot{\phi}^{2}+F_{1}\phi^{2}\right.$
(59)
$\displaystyle\quad\quad\quad\quad+\left(3V_{2}\dot{\phi}-2\frac{\omega_{1}}{a^{2}}\partial^{2}\phi+F_{2}\phi\right)\psi$
$\displaystyle\quad\quad\quad\quad+\left.\dot{\varphi}G_{3}\left(F_{3}\psi+3\dot{\phi}-\dot{\psi}-\frac{\partial^{2}\chi}{a^{2}}\right)\frac{\delta\Phi}{a^{3}}\right]\,,$

where

$\displaystyle V_{2}$ $\displaystyle=$
$\displaystyle\omega_{2}-\left(\dot{\varphi}^{3}G_{3,X}+\dot{\varphi}G_{3}\right)\frac{\Phi_{0}}{a^{3}},$
$\displaystyle V_{3}$ $\displaystyle=$
$\displaystyle\omega_{3}+\left(9H\dot{\varphi}G_{3}+21H\dot{\varphi}^{3}G_{3,X}+\frac{3}{2}H\dot{\varphi}^{5}G_{3,XX}+\frac{3}{2}\ddot{\varphi}G_{3}+2\dot{\varphi}^{2}\ddot{\varphi}G_{3,X}+\right.$
$\displaystyle\left.-\frac{3}{2}\dot{\varphi}^{2}G_{3,\varphi}-\frac{3\dot{\varphi}^{2}\ddot{\varphi}}{2}G_{3,X}-\frac{\dot{\varphi}^{4}}{2}G_{3,X\varphi}\right)\frac{\Phi_{0}}{a^{3}}-\frac{\dot{\varphi}^{3}}{2}G_{3,X}\frac{\dot{\Phi}_{0}}{a^{3}},$
$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\frac{9}{2}\dot{\varphi}^{2}\left(G_{3,\varphi}+G_{3,X}\ddot{\varphi}\right)\frac{\Phi_{0}}{a^{3}}+\frac{9}{2}\ddot{\varphi}G_{3}\frac{\Phi_{0}}{a^{3}}+\frac{9}{2}\dot{\varphi}G_{3}\frac{\dot{\Phi}_{0}}{a^{3}},$
$\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle\frac{3}{a^{3}}\left(\epsilon_{B}\Phi_{0}+6G_{3}\dot{\varphi}H\Phi_{0}-\dot{\varphi}G_{3}\dot{\Phi}_{0}+G_{3}\ddot{\varphi}\Phi_{0}\right),$
$\displaystyle F_{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{\dot{\varphi}G_{3}}\left(6H\dot{\varphi}G_{3}+3H\dot{\varphi}^{3}G_{3,X}+\dot{\varphi}^{2}\ddot{\varphi}G_{3,X}+2\ddot{\varphi}G_{3}\right)\,,$
(60)

and $\omega_{i}$ were defined by Eqs. (52)–(54).

To obtain the speed of sound, we cancel the fields $\psi$ and $\chi$ by using
their equations of motion, choosing $\phi$ and $\delta\Phi$ as the dynamical
variables. The action can be written in the form (19) with
$Q_{i}=(\phi,\delta\Phi)$ and $A$, $B$, $C$, $D$ will be $2\times 2$ matrices.
To this purpose we use the equation of motion for $\chi$ that this time has
contributions coming from the new measure:

$\displaystyle
E_{\chi}\to-2\omega_{1}\dot{\phi}+V_{2}\psi-3\dot{\varphi}G_{3}\frac{\Phi_{0}}{a^{3}}\phi+\dot{\varphi}G_{3}\frac{\delta\Phi}{a^{3}}=0.$
(61)

The equation for $\delta\Phi$ becomes:

$\displaystyle
E_{\delta\Phi}\to\dot{\varphi}G_{3}F_{3}\psi+3\dot{\varphi}G_{3}\dot{\phi}-\dot{\varphi}G_{3}\dot{\psi}-\dot{\varphi}G_{3}\frac{\partial^{2}\chi}{a^{2}}=0.$
(62)

From the $E_{\chi}$ (61), we can obtain $\psi$ as a function of
$Q=(\phi,\delta\Phi)$:

$\psi=P_{1}\dot{\phi}+P_{2}\phi+P_{3}\delta\Phi,$ (63)

where

$\displaystyle P_{1}$ $\displaystyle=$
$\displaystyle\frac{2\omega_{1}}{V_{2}},$ $\displaystyle P_{2}$
$\displaystyle=$
$\displaystyle\frac{3\dot{\varphi}G_{3}}{V_{2}}\frac{\Phi_{0}}{a^{3}},$ (64)
$\displaystyle P_{3}$ $\displaystyle=$
$\displaystyle-\frac{\dot{\varphi}G_{3}}{a^{3}V_{2}}.$

Using $E_{\chi}$ (61) and integrating by parts we can rewrite the action (59)
in terms of the perturbation fields $\phi$ and $\delta\Phi$

$\displaystyle S^{(2)}=\int
d^{4}x\left(a^{3}\left[\frac{V_{3}}{3}P_{1}^{2}-3\omega_{1}+3V_{2}P_{1}\right]\dot{\phi}^{2}+\left[\dot{\varphi}G_{3}P_{1}\right]\dot{\phi}\dot{\delta\Phi}\right.$
(65)
$\displaystyle\left.-a^{3}\left[-\frac{V_{3}}{3}P_{2}^{2}-F_{1}-F_{2}P_{2}\right]\phi^{2}-\left[-2\frac{a^{3}V_{3}}{3}P_{2}P_{3}-a^{3}F_{2}P_{3}-\dot{\varphi}G_{3}F_{3}P_{2}-P_{2}\frac{d}{dt}\left(\dot{\varphi}G_{3}\right)\right]\phi\delta\Phi\right.$
$\displaystyle\left.-\left[-\frac{a^{3}V_{3}}{3}P_{3}^{2}-\dot{\varphi}G_{3}F_{3}P_{3}-P_{3}\frac{d}{dt}\left(\dot{\varphi}G_{3}\right)\right]\delta\Phi^{2}-\left[-\omega_{4}a+\frac{d}{dt}\left(a\omega_{1}P_{1}\right)-2\omega_{1}aP_{2}\right]\left(\partial\phi\right)^{2}\right.$
$\displaystyle\left.-\left[-2\omega_{1}aP_{3}\right]\partial\phi\partial\delta\Phi-a^{3}\left[-2\frac{V_{3}}{3}P_{1}P_{2}-3V_{2}P_{2}-F_{2}P_{1}\right]\phi\dot{\phi}\right.$
$\displaystyle\left.-\left[-2a^{3}\frac{V_{3}}{3}P_{1}P_{3}-3V_{2}P_{3}a^{3}-\dot{\varphi}G_{3}F_{3}P_{1}-3\dot{\varphi}G_{3}-P_{1}\frac{d}{dt}\left(\dot{\varphi}G_{3}\right)\right]\delta\Phi\dot{\phi}\right.$
$\displaystyle\left.-\left[-\dot{\varphi}G_{3}P_{2}\right]\phi\dot{\delta\Phi}-\left[-\dot{\varphi}G_{3}P_{3}\right]\delta\Phi\dot{\delta\Phi}\right)\,.$

If we define $Q_{i}=(\phi,\delta\Phi)$ the previous action takes the form of
(19) where the matrices $A_{ij}$, $B_{ij}$, $C_{ij}$, $D_{ij}$ can be easily
read (see appendix A). The equation (22) can be used to obtain the sound speed
as:

$\displaystyle\begin{vmatrix}c_{S}^{2}A_{11}-a^{2}C_{11}&\quad
c_{S}^{2}A_{12}-a^{2}C_{12}\\\ c_{S}^{2}A_{21}-a^{2}C_{21}&\quad
0\end{vmatrix}=0,$ (66)

and implies that

$c_{S}^{2}=a^{2}\frac{C_{12}}{A_{12}}=a^{2}\frac{-\omega_{1}aP_{3}}{\dot{\varphi}G_{3}P_{1}/2}=a^{2}\frac{\omega_{1}a\frac{1}{a^{3}}\dot{\varphi}G_{3}}{\dot{\varphi}G_{3}\frac{2\omega_{1}}{2}}=1,$
(67)

which is independent of the model 333The same result is obtained if we use
$Q_{i}=(\phi,\psi)$ instead of $Q_{i}=(\phi,\delta\Phi)$.. This result arises
because the new measure field imposes the constraint of a constant value for
the matter Lagrangian. This value for the sound speed makes the kinetic
gravity braiding model with new measure unsuitable to represent the Dark
Matter of the Universe, as the perturbations will not grow as expected from
the structure formations. On the other hand, the field can still be a
candidate to the Dark Energy. In order to explore further the existence of
problems with the theory, in the following subsection we will investigate its
possible ghost or tachyonic instabilities.

###  IV.1 Ghost and tachyonic instabilities

In this section we study the ghost and tachyon instabilities of the kinetic
gravity braiding model when the new measure field $\Phi$ is added. To do this
we must transform the kinetic term in the perturbed action (65) to the
canonical form. First we Fourier transform the action to

$S^{(2)}=\int
d^{3}kdta^{3}(A_{ij}\dot{Q}_{i}\dot{Q}_{j}-B_{ij}\dot{Q}_{i}Q_{j}-\tilde{D}_{ij}Q_{i}Q_{j})\,,$
(68)

were $Q=(\phi,\delta\Phi)$ and the expressions for the matrices $A$, $B$ and
$D$ are written in Appendix A. Comparing this action with the general form
(19) we see that the Fourier transformation allowed us to merge the two last
terms as

$C_{ij}(\partial Q_{i})(\partial
Q_{j})+D_{ij}Q_{i}Q_{j}\to\left(-k^{2}C_{ij}+D_{ij}\right)Q_{i}Q_{j}\,,$ (69)

where we call the term in parentheses $\tilde{D}$.

The kinetic matrix has the structure

$A=\begin{pmatrix}A_{11}&A_{12}\\\ A_{12}&0\end{pmatrix}\,,$ (70)

with eigenvalues

$\lambda_{1,2}=\frac{1}{2}(A_{11}\pm\sqrt{A_{11}^{2}+4A_{12}^{2}})\,.$ (71)

From this expression we see that $\lambda_{1}>0$ and $\lambda_{2}<0$ which
implies that one of the resulting fields will have a negative kinetic term and
therefore will be a ghost. To see this explicitly we perform the following
field redefinition

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{A_{11}}}(\Psi_{1}+\Psi_{2})\,,$
$\displaystyle\delta\Phi$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{A_{11}}}{A_{12}}\Psi_{2}\,,$ (72)

which after integration by parts takes the Lagrangian to the form

$\mathcal{L}=\frac{a^{3}}{2}[\dot{\Psi}_{1}^{2}-\dot{\Psi}_{2}^{2}+K(\dot{\Psi}_{1}\Psi_{2}-\dot{\Psi}_{2}\Psi_{1})-M_{11}\Psi_{1}^{2}-2M_{12}\Psi_{1}\Psi_{2}-M_{22}\Psi_{2}^{2}]\,.$
(73)

where the expressions for the functions $K$ and $M_{ij}$ are written in
Appendix A. As expected, one of the resulting fields, in this case $\Psi_{2}$
has a negative kinetic term and therefore is a ghost Sbisa (2014), with the
unwanted property that its energy is unbounded from bellow.

In order to check for tachyonic instabilities, we also need to diagonalize the
mass term. To keep the kinetic part invariant while transforming the mass
term, we propose an hyperbolic rotation instead of the standard rotation De
Felice et al. (2017)

$\displaystyle\Psi_{1}$ $\displaystyle=$
$\displaystyle\cosh{\alpha}\xi_{1}+\sinh{\alpha}\xi_{2}\,,$
$\displaystyle\Psi_{2}$ $\displaystyle=$
$\displaystyle\sinh{\alpha}\xi_{1}+\cosh{\alpha}\xi_{2}\,,$ (74)

with

$\tanh{2\alpha}=-\frac{2M_{12}}{M_{11}+M_{22}}\,.$ (75)

The Lagrangian transforms to

$\mathcal{L}=\frac{a^{3}}{2}[\dot{\xi}_{1}^{2}-\dot{\xi}_{2}^{2}+(K+2\dot{\alpha})(\dot{\xi}_{1}\xi_{2}-\dot{\xi}_{2}\xi_{1})-\mu_{1}^{2}\xi_{1}^{2}-\mu_{2}^{2}\xi_{2}^{2}]\,,$
(76)

where the masses are given by

$\displaystyle\mu_{1}^{2}$ $\displaystyle=$
$\displaystyle\dot{\alpha}^{2}+\dot{\alpha}K+\frac{1}{2}(M_{11}-M_{22})+\frac{1}{2}\sqrt{(M_{11}+M_{22})^{2}-4M_{12}^{2}}\frac{|M_{11}+M_{22}|}{M_{11}+M_{22}}\,,$
$\displaystyle\mu_{2}^{2}$ $\displaystyle=$
$\displaystyle-\dot{\alpha}^{2}-\dot{\alpha}K-\frac{1}{2}(M_{11}-M_{22})+\frac{1}{2}\sqrt{(M_{11}+M_{22})^{2}-4M_{12}^{2}}\frac{|M_{11}+M_{22}|}{M_{11}+M_{22}}\,.$
(77)

In order to have a theory free from tachyonic instabilities both squared
masses need to be positive 444The former Lagrangian produces a very simple
form of the Hamiltonian which will be unbounded from below if the squared
masses are negatives.. In fact, it is convenient to compare these masses with
the Hubble parameter in the case when $\mu_{i}^{2}<0$. For $k=0$, we will have
strong tachyonic instability if $-\mu_{i}^{2}\gg H^{2}$ and the model is
unviable. However the model is physically viable if $|\mu_{i}^{2}|\leq H^{2}$
and the whole stability of the system will be not destroyed by the evolution
of this instability in time intervals shorter than the Hubble time De Felice
et al. (2017). Although the expressions in terms of the original functions in
the action (56) are rather involved, we exemplify the application of these
conditions in the following subsection in order to shed some light on how they
act in a concrete case.

###  IV.2 Example

As the expressions of the last subsection are too large to give some insight
on the tachyonic stability of the Braiding model, here we will apply the
stability criteria to a particular example of cosmological interest. We
propose a Braiding field which only depends on the kinetic term
$G_{3}(\varphi,X)=G_{3}(X)$. The Lagrangian then becomes invariant under the
transformation $\varphi\to\varphi+\rm{const}$, which leads to a conserved
quantity. At the background level we can write this quantity $I$ as:

$I=G_{3}(X)\left[\frac{d\Psi}{dt}-3h\left(2X\frac{G_{3,X}}{G_{3}}+1\right)\Psi\right]\,,$
(78)

where we are using555See section V of Ref. Cordero et al. (2019) for details.
$\Psi=a^{3}+\Phi_{0}$. A second conserved quantity arises from the variation
of the action (56) with respect to the new measure $\Phi$ which produces a
constant value for the factor $G_{3}(X)\square\varphi$, at the background
level this becomes

$J=G_{3}(X)\left(\ddot{\varphi}+3H\dot{\varphi}\right)\,.$ (79)

The energy density and pressure from this field can be written in terms of
both conserved quantities, substituting (78) and (79) in the expressions (57)
and (58) we obtain

$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle
J\left(\frac{\Psi}{a^{3}}-1\right)-\frac{I\dot{\varphi}}{a^{3}}\,,$
$\displaystyle P$ $\displaystyle=$
$\displaystyle-J\frac{\Psi}{a^{3}}\left(\frac{2XG_{3,X}(X)}{G(X)}+1\right)+J-\frac{I\dot{\varphi}}{a^{3}}\,.$
(80)

As we are interested in a particular case, we set $I=0$ and

$G_{3}=rX^{\gamma}\,\,,$ (81)

with $\gamma$ and $r$ constants. Equation (78) solves to

$\Psi=La^{3(2\gamma+1)}\,,$ (82)

with $L$ an integration constant. This solution leads to an energy density and
pressure given by

$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle JLa^{6\gamma}-J\,,$
$\displaystyle P$ $\displaystyle=$ $\displaystyle-
JL(2\gamma+1)a^{6\gamma}+J\,.$ (83)

An interesting case happens with $\gamma=-1/2$ in which the energy density
looks like a matter component plus a cosmological constant. Unfortunately this
field is not suited to behave as the dark matter at the perturbation level, as
its speed of sound (67) equals the speed of light and it is incompatible with
the structure formation.

In order to study the tachyonic stability of the model we need to compute the
expressions in Eq. (77), which depend on the expressions in Appendix A and
ultimately are functions of $\Phi_{0}$ and $G_{3}(X)$. For the first one we
have from Eq. (82)

$\Phi_{0}=-a^{3}+La^{3(2\gamma+1)}\,.$ (84)

For $G_{3}(X)$ we need to solve equation (79) with $X=1/2\dot{\varphi}^{2}$
and for the particular Lagrangian (81) becomes

$J=\frac{r\dot{\varphi}^{2\gamma}}{2^{\gamma}}\left(\ddot{\varphi}+3H\dot{\varphi}\right)\,.$
(85)

This equation can be solved to

$\dot{\varphi}=\frac{1}{a^{3}}\left(\frac{2^{\gamma}(2\gamma+1)J}{r}\int{\frac{a^{6\gamma+2}}{H}da}+\tilde{C}\right)^{1/(2\gamma+1)}\,,$
(86)

where $\tilde{C}$ is an integration constant which can be fixed by given a
value to $\dot{\varphi}$ at an specific time. If we assume that the energy
density from (83) is the only component of the Universe, the Friedmann
equation becomes

$\frac{H^{2}}{H_{0}^{2}}=\Omega_{\gamma}a^{6\gamma}+1-\Omega_{\gamma}\,.$ (87)

The former integration constants can be written in terms of the free
parameters $\Omega_{\gamma}$ and $\gamma$ as

$\displaystyle J$ $\displaystyle=$ $\displaystyle
3M_{\rm{P}}^{2}H_{0}^{2}(\Omega_{\gamma}-1)\,,$ $\displaystyle L$
$\displaystyle=$ $\displaystyle\frac{\Omega_{\gamma}}{\Omega_{\gamma}-1}\,.$
(88)

With the above equations we can study the behaviour of the masses (77). Figure
1 gives the evolution of the masses for different values of the density
parameter $\Omega_{\gamma}$ for the case $\gamma=-1/2$. We can see changes in
the sign of the squared masses, which correspond to changes in behavior from
canonical to tachyonic. For small $\Omega_{\gamma}$ the tachyonic behavior
occurs only at the early Universe. The change from stable to tachyonic
behaviour is very fast in various regions, in particular for
$\Omega_{\gamma}=0.3$ near $a=0.52$. Furthermore, there are points where
singularities are present, some of them due to a zero value for the $A_{11}$
term (see Appendix). We have such divergent behavior, for instance, for
$\Omega_{\gamma}=0.3$ and $a\simeq 0.49$ in the left panel of Figure 1.

One could try a different approach by eliminating the perturbed new measure
field $\delta\Phi$ of the equations of motion for the perturbations and
considering instead the independent perturbations as $\phi$ and $\psi$.
However, even in this case, the stability behaviour remains. The canonical
field has a tachyonic behaviour and the other is free from tachyonic
instabilities, but it has a ghost.

If the Braiding field were intended to behave as a Dark Matter + Dark Energy,
then $\Omega_{\gamma}\sim 0.3$, but this case present several tachyonic
instabilities at different stages in the evolution of the Universe. An
extension of our study will need to be performed to see if these instabilities
(together with the ghost instability) can be avoided when we consider a
Universe filled with baryonic matter and radiation.

Figure 1: Evolution of the masses $\mu_{1}^{2}/H^{2}$ and $\mu_{2}^{2}/H^{2}$
from (77) for the particular case studied in subsection IV.2. The first mass
corresponds to the canonical fields while the second mass corresponds to the
ghost field. We see that in the early Universe these masses become imaginary,
corresponding to a tachyonic field. This behaviour can be modified with the
inclusion of extra matter components to the Universe. Velocity
---
| Without new measure | With new measure
$K(X)$ | $c_{S}^{2}=\frac{3\left(2\omega_{1}^{2}\omega_{2}H-\omega_{2}^{2}\omega_{4}-2\omega_{1}^{2}\dot{\omega}_{2}\right)}{\omega_{1}\left(4\omega_{1}\omega_{3}+9\omega_{2}^{2}\right)}.$ | Field follows geodesics
$G_{3}(X)\square\phi$ |  | $c_{S}^{2}=3\frac{2\omega_{1}^{2}\omega_{2}H-\omega_{2}^{2}\omega_{4}+4\omega_{2}\omega_{1}\dot{\omega}_{1}-2\omega_{1}^{2}\dot{\omega}_{2}}{\omega_{1}\left(4\omega_{1}\omega_{3}+9\omega_{2}^{2}\right)}.$
---
$c_{S}^{2}=1$
Table 2: The distinct results of the speed of perturbations is shown.

##  V Conclusions

The nature of the late time accelerating expansion of the Universe is one of
the most challenging aspects of modern cosmology. One of the more appealing
characteristics of Two Field Measure Theory is the possibility to have a
unifying description of dark matter and dark energy. In this paper we have
studied the perturbations of k-essence and braiding fields in the context of a
Two Field Measure Theory. We have found the stability characteristics and the
speed of sound of the scalars fields.

In the original Horndeski theory, the speed of perturbations is a function of
the field Lagrangian $K(\varphi,X)$ or $G_{3}(\varphi,X)$ as well as their
derivatives, as it is summarized in table 2, presenting a rich variety in the
speeds of sound depending on the particular Lagrangians. When the new measure
$\Phi$ is added, however, both cases present a fixed behaviour. For the
k-essence models we obtained that the field follows geodesics, a result
already obtained in Guendelman et al. (2012). This allows the field to be a
good candidate for the Dark Matter as it can grow its perturbations and form
the structure of the Universe. In contrast, kinetic gravity braiding model
$G_{3}\square\phi$ has a unitary speed of perturbations. This characteristic
is a consequence of the new measure field which fixed the value of the matter
Lagrangian.

For the braiding models we studied further their stability conditions. We
obtained that, in the presence of the new measure, there will always appear a
ghost instability. The field perturbations of the braiding field plus new
measure, can be mapped to a two field theory with a ghost field and a
canonical field, both with complicated mass expressions written in Eq. (77).
When the masses squared are negative we are in the presence of a tachyon
instability.

To shed some light on the tachyonic stability of the Braiding model, we
studied a particular example in which the field density evolves as a power law
in the scale factor plus a cosmological constant. In that case, we computed
the squared masses of the resulting fields, obtaining that they change their
sign as the Universe evolves. When the density of the Braiding model is low,
the tachyonic instability can be limited to the early Universe.

Both, ghost and tachyonic instabilities disfavour the braiding + TMT model as
explanation for either dark energy or dark matter, since its energy is
unbounded from below. An equally important problem for the braiding field to
represent the dark matter is that its sound speed is equal to the speed of
light, this prevents the perturbations to grow and form the structure of the
Universe, despite its background density to evolve as dust with $1/a^{3}$ as
can be seen in Eq. (83) for $\gamma=-1/2$. The tachyonic and braiding
instabilities might be alleviated if we had considered in our calculations a
Universe with more fields besides the braiding field, however this study was
outside of the scope of this work.

The dark energy nature is one of the most intriguing question in modern
cosmology. The TMT theory has exciting properties. For example, any k-essence
model in the context of TMT can produce a unification of dark matter and dark
energy and explain the growth of structure in the Universe.

## Acknowledgments

This work was partially supported by SNI-México and CONACyT research grant:
A1-S-23238. Additionally the work of RC was partially supported by COFAA-IPN,
EDI-IPN and SIP-IPN grants: 20201841, 20210704 and 20221329.

##  Appendix A Braided field transformations

Here we write the matrix components associated to the action (68). First for
the kinetic term $A_{ij}\dot{Q}_{i}\dot{Q}_{j}$, the matrix $A_{ij}$ has the
following components

$\displaystyle A_{11}$ $\displaystyle=$ $\displaystyle
3M_{\rm{P}}^{2}\left(\frac{4V_{3}M_{\rm{P}}^{2}}{9V_{2}^{2}}+1\right)\,,$
$\displaystyle A_{12}=A_{21}$ $\displaystyle=$
$\displaystyle\frac{\dot{\varphi}G_{3}M_{\rm{P}}^{2}}{a^{3}V_{2}}\,,$
$\displaystyle A_{22}$ $\displaystyle=$ $\displaystyle 0\,,$ (89)

where the factors $V_{i}$ are defined in equations (60). The second term in
the action given by $B_{ij}\dot{Q}_{i}Q_{j}$ has the following components for
the matrix $B_{ij}$:

$\displaystyle B_{11}$ $\displaystyle=$ $\displaystyle-
G_{3}\dot{\varphi}\frac{\Phi_{0}}{a^{3}}\left(\frac{4M_{\rm{P}}^{2}V_{3}}{V_{2}^{2}}+9\right)-\frac{2M_{\rm{P}}^{2}F_{2}}{V_{2}}\,,$
$\displaystyle B_{12}$ $\displaystyle=$
$\displaystyle-\frac{6XG_{3}^{2}\Phi_{0}}{V_{2}a^{6}}\,,$ $\displaystyle
B_{21}$ $\displaystyle=$
$\displaystyle\frac{2M_{\rm{P}}^{2}}{V_{2}a^{3}}\left(\dot{\varphi}G_{3}\left[\frac{2V_{3}}{3V_{2}}-F_{3}\right]-\frac{d}{dt}[\dot{\varphi}G_{3}]\right)\,,$
$\displaystyle B_{22}$ $\displaystyle=$
$\displaystyle\frac{2XG_{3}^{2}}{a^{6}V_{2}}\,.$ (90)

Finally the last term in the action $\tilde{D}_{ij}=-k^{2}C_{ij}+D_{ij}$ has
the following components for the matrix $C_{ij}$:

$\displaystyle C_{11}$ $\displaystyle=$
$\displaystyle-\frac{M_{\rm{P}}^{2}}{a^{2}}+\frac{1}{a^{3}}\frac{d}{dt}\left(\frac{2aM_{\rm{P}}^{4}}{V_{2}}\right)-\frac{6M_{\rm{P}}^{2}\dot{\varphi}G_{3}}{V_{2}a^{2}}\frac{\Phi_{0}}{a^{3}}\,,$
$\displaystyle C_{12}$ $\displaystyle=$ $\displaystyle
C_{21}=\frac{M_{\rm{P}}^{2}\dot{\varphi}G_{3}}{a^{5}V_{2}}\,,$ $\displaystyle
C_{22}$ $\displaystyle=$ $\displaystyle 0\,,$ (91)

and for the matrix $D_{ij}$:

$\displaystyle D_{11}$ $\displaystyle=$
$\displaystyle-\frac{3V_{3}\dot{\varphi}^{2}G_{3}^{2}}{V_{2}^{2}}\frac{\Phi_{0}^{2}}{a^{6}}-F_{1}-\frac{3F_{2}\dot{\varphi}G_{3}}{V_{2}}\frac{\Phi_{0}}{a^{3}}$
$\displaystyle D_{12}=D_{21}$ $\displaystyle=$
$\displaystyle\frac{2V_{3}XG_{3}^{2}\Phi_{0}}{V_{2}^{2}a^{6}}+\frac{F_{2}\dot{\varphi}G_{3}}{2V_{2}a^{3}}-\frac{3F_{3}XG_{3}^{2}\Phi_{0}}{V_{2}a^{6}}-\frac{3\dot{\varphi}G_{3}\Phi_{0}}{2V_{2}a^{6}}\frac{d}{dt}\left(\dot{\varphi}G_{3}\right)\,,$
$\displaystyle D_{22}$ $\displaystyle=$
$\displaystyle\frac{2XG_{3}^{2}}{V_{2}a^{6}}\left(F_{3}-\frac{2V_{3}}{3V_{2}}\right)+\frac{\dot{\varphi}G_{3}}{V_{2}a^{6}}\frac{d}{dt}\left(\dot{\varphi}G_{3}\right)\,.$
(92)

The function $K$ in (73) is written in terms of the matrix components of the
original action as:

$K=\frac{B_{12}-B_{21}}{2A_{12}}+\frac{\dot{A}_{11}}{A_{11}}-\frac{\dot{A}_{12}}{A_{12}}\,.$
(93)

The mass matrix which appears in Eq. (73) is given by

$\displaystyle M_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{A_{11}}\left(\frac{1}{4}\frac{\dot{A}_{11}^{2}}{A_{11}}-\frac{\ddot{A}_{11}}{2}-\frac{\dot{B}_{11}}{2}+\tilde{D}_{11}-\frac{3H}{2}(\dot{A}_{11}+B_{11})\right)\,,$
$\displaystyle 2M_{12}$ $\displaystyle=$
$\displaystyle\left(\frac{\dot{A}_{11}}{A_{11}}-\frac{\dot{A}_{12}}{A_{12}}\right)\left(\frac{B_{12}-B_{21}-2\dot{A}_{12}}{2A_{12}}+3H\right)+\frac{\ddot{A}_{11}}{A_{11}}-\frac{\ddot{A}_{12}}{A_{12}}$
$\displaystyle-\frac{\dot{B}_{11}}{A_{11}}+\frac{\dot{B}_{21}+\dot{B}_{12}}{2A_{12}}+\frac{2\tilde{D}_{11}}{A_{11}}-\frac{2\tilde{D}_{12}}{A_{12}}-\frac{3H}{2}\left(\frac{2B_{11}}{A_{11}}-\frac{B_{21}+B_{12}}{A_{12}}\right)\,,$
$\displaystyle M_{22}$ $\displaystyle=$
$\displaystyle\left(\frac{\dot{A}_{11}}{A_{11}}-\frac{\dot{A}_{12}}{A_{12}}\right)\left(\frac{B_{12}-B_{21}+2\dot{A}_{12}}{2A_{12}}-3H\right)-\frac{\ddot{A}_{11}}{2A_{11}}-\left(\frac{\dot{A}_{11}}{2A_{11}}\right)^{2}+\frac{\ddot{A}_{12}}{A_{12}}$
(94)
$\displaystyle-\frac{\dot{B}_{11}}{2A_{11}}+\frac{\dot{B}_{21}+\dot{B}_{12}}{2A_{12}}-\frac{\dot{B}_{22}A_{11}}{2A_{12}^{2}}+\frac{\tilde{D}_{11}}{A_{11}}-\frac{2\tilde{D}_{12}}{A_{12}}+\frac{\tilde{D}_{22}A_{11}}{A_{12}^{2}}$
$\displaystyle-\frac{3H}{2}\left(\frac{B_{11}}{A_{11}}-\frac{B_{12}+B_{21}}{A_{12}}+\frac{A_{11}B_{22}}{A_{12}^{2}}\right)\,.$

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