11institutetext: Institute of Theoretical Physics, University of Wroclaw, Max
Born Place 9, 50-204 Wroclaw, Poland 22institutetext: IT and Computing
Division, A. Alikhanyan National Laboratory, Alikhanian Brothers Str. 2,
Yerevan, 0036, Armenia 33institutetext: CONICET, Godoy Cruz 2290, Buenos
Aires, Argentina 44institutetext: Departamento de Física, Comisión Nacional de
Energía Atómica, Av. Libertador 8250, (1429) Buenos Aires, Argentina

# Special point "trains" in the M-R diagram of hybrid stars

David Blaschke 11<EMAIL_ADDRESS>Alexander Ayriyan 22
<EMAIL_ADDRESS>Mateusz Cierniak 11<EMAIL_ADDRESS>Ana Gabriela Grunfeld 3344<EMAIL_ADDRESS>Oleksii Ivanytskyi 11
<EMAIL_ADDRESS>Mahboubeh Shahrbaf 11
<EMAIL_ADDRESS>
###### Abstract

We present a systematic investigation of the possible locations for the
special point (SP), a unique feature of hybrid neutron stars in the mass-
radius diagram. The study is performed within the two-phase approach where the
high-density (quark matter) phase is described by the covariant nonlocal
Nambu–Jona-Lasinio (nlNJL) model equation of state (EOS) which is shown to be
equivalent to a constant-sound-speed (CSS) EOS. For the nuclear matter phase
around saturation density different relativistic density functional EOSs are
used: DD2p00, its excluded-volume modification DD2p40 and the hypernuclear EOS
DD2Y-T. In the present contribution we apply the Maxwell construction scheme
for the deconfinement transition and demonstrate that a simultaneous variation
of the vector and diquark coupling constants results in the occurrence of SP
"trains" which are invariant against changing the nuclear matter EOS. We
propose that the SP train corresponding to a variation of the diquark coupling
at constant vector coupling is special since it serves as a lower bound for
the line of maximum masses and accessible radii of massive hybrid stars.

##  1 Introduction

A key objective of modern NS observations is to infer constraints on the EOS
from precise mass and radius measurements. The basis for this inference is the
unique mapping between the EOS and the stellar structure which is provided by
the Tolman-Oppenheimer-Volkoff (TOV) equations. Recent highlights in this
endeavour are the combined mass-radius measurement on the high-mass pulsar PSR
J0740+6620 by NICER Miller:2021qha ; Riley:2021pdl , the multimessenger radius
constraint in the $1.4\leavevmode\nobreak\ M_{\odot}$ mass range
dietrich2020multimessenger  with the tidal deformability constraint from the
first binary NS merger GW170817 abbott2018gw170817 . While it is clear that
for the above mentioned complex of investigations one just needs a function
pressure vs. energy density $P(\varepsilon)$ for instance in a multi-polytrope
or multi-CSS form as input to the TOV equation in order to solve for the
characteristic mass-radius $M(R)$ curve. Doing this a million of times one
could straightforwardly extract a boundle of most probable EoS under given
observational constraints for $M(R)$ by using Bayesian analysis techniques.
This "agnostic" approach is not satisfactory because it does not allow to
infer, e.g., for the composition of matter and at present gives no evidence
for a QCD phase transition.

It is clear that between the nucleonic matter phase up to about saturation
density $n\lesssim n_{0}=0.15$ fm-3 and the range of validity of perurbative
QCD for $n\gtrsim 40\leavevmode\nobreak\ n_{0}$, there must be a dissociation
of hadrons into their quark and gluon constituents. Therefore, one should
strive to employ a physics educated approach of extracting the most probably
EOS, one that inherently includes the QCD phase transition. For such a type of
Bayesian analysis Ayriyan:2021prr ; Shahrbaf:2021cjz , where the EoS
parameters have a direct interpretation as parameters of an effective low-
energy QCD Lagrangian, it shall be of eminent importance that their variation
will determine the position and character of the deconfinement phase
transition. As it has been demonstrated by Yudin et al. Yudin:2014mla  for the
case of a simple CSS model of quark matter, varying the bag pressure parameter
$B$ determines the onset of deconfinement and leads to the occurrence of a so-
called "special point" (SP) in the $M(R)$ diagram. The SP is a small region,
like a focal point, through which the boundle of all hybrid star sequences
with different onset masses $M_{\rm onset}$ for quark deconfinement have to
pass. Two properties of the SP make it an inevitable ingredient to studies of
deconfinement in NS interiors: 1) its existence requires a phase
transformation so that it is absent in pure (hadronic or quark) phase stars
and can be regarded as a characteristics of hybrid star EOS; 2) its location
in the $M-R$ plane is independent of the choice of the hadronic EOS and thus
uniquely characterizes the high-density phase of matter. For a detailed
discussion of invariant properties and regions of the SP location in the
$M(R)$ diagram for the CSS family of quark matter EOS, see Cierniak:2020eyh ;
Blaschke:2020vuy ; Cierniak:2021knt ; Cierniak:2021vlf . A generic Lagrangian
model of cold dense quark matter in neutron stars has to have a vector meson
interaction channel that provides stiffness and a diquark interaction channel
to account for the diquark condensation signalling color superconductivity
Klahn:2006iw ; Klahn:2013kga ; Baym:2017whm . It came as a surprise when the
systematic simultaneous variation of the dimensionless vector meson coupling
$\eta_{V}=G_{V}/G_{S}$ and diquark coupling $\eta_{D}=G_{D}/G_{S}$ (normalized
to the scalar-pseudoscalar coupling strength $G_{S}$) within recent Bayesian
analyses studies Ayriyan:2021prr ; Shahrbaf:2021cjz  revealed the existence of
series of special points that were lined up in "trains". This phenomenon and
its possible consequences for NS phenomenology were discussed first in
Shahrbaf:2021cjz , but deserve further detailed study as will be outlined in
this contribution.

##  2 "Trains" of special points

The present study is a systematic investigation of the possible locations for
SPs, a unique feature of hybrid NS in the mass-radius diagram. It is performed
within the two-phase approach where the high-density (quark matter) phase is
described in the main part by the covariant nonlocal Nambu–Jona-Lasinio
(nlNJL) model EOS which was shown to be equivalent to a CSS EOS
Shahrbaf:2021cjz . The nuclear matter phase up to saturation density and
slightly beyond it is described by the relativistic density functional EOS
DD2p00 and its excluded-volume modification DD2p40. In section 5 we discuss
the existence of trains of SPs for the case of the confining relativistic
density functional approach to color superconducting quark matter and the
hypernuclear EoS DD2Y-T Ivanytskyi:2022bjc ; Ivanytskyi:2022oxv  in order to
check the robustness of the statements made in the preceeding sections.

Figure 1: Left panel: family of quark-hadron hybrid EoS obtained within the
two-phase approach by a Maxwell construction between the DD2p40 nuclear EOS
and the nlNJL quark matter one in its CSS representation. Right panel: mass-
radius relation for all hybrid star sequences for which the different cases of
$\eta_{V}$ are highlighted in colors and each value of $\eta_{D}$ corresponds
to a line style. Observational constraints on masses and radii are shown, for
details see text.

In the left panel of Fig. 1 we show the family of hybrid EOS that is obtained
by a Maxwell construction with the DD2p40 hadronic EoS and the nlNJL model
with the vector meson coupling $\eta_{V}$ and the diquark coupling $\eta_{D}$
as two free parameters which are varied in discrete steps. Details of the
model can be found in Ref. Shahrbaf:2021cjz . As it has been shown in
Shahrbaf:2021cjz , it is an excellent approximation to employ a CSS form of
EOS instead of the numerical solution of the nlNJL model. In that work also a
fit formula has been parametrized that maps the pair of free parameters
$(\eta_{V},\eta_{D})$ onto the those of the CSS EOS

$P(\mu)=A\left({\mu}/{\mu_{x}}\right)^{1+\beta}-B,$ (1)

where $\mu_{x}=1$ GeV defines the scale for chemical potential, $A$ is a slope
parameter in the units of the pressure, $B$ is the bag pressure and
$\beta=1/c_{s}^{2}$ is a parameter related to the squared speed of sound
$c_{s}^{2}=dP/d\varepsilon$.

On the right panel of Fig. 1, the solutions of the TOV equations for NS mass
versus radius are shown for the EOS depicted on the left panel. One can see
clearly that M-R curves corresponding to a fixed value of $\eta_{V}$ (shown in
the same color) but different values of $\eta_{D}$ (indicated by different
line styles) get collimated in a narrow region close to the maximum mass which
has been dubbed SP Yudin:2014mla .

Incrementing the value of $\eta_{V}$, a new SP is obtained, so that for our
set of EOSs a train of SPs in the $M-R$ diagram emerges. For better visibility
of the effect, we have selected a subset of 9 values for $\eta_{V}$ and show a
3x3 matrix of panels with the corresponding $M-R$ diagrams in Fig. 2.

Figure 2: Mass-radius relation for all hybrid stars obtained by a Maxwell
construction. The value of $\eta_{V}$ is taken to be fixed for each panel
while the value of $\eta_{D}$ is varied.

In Fig. 3 we provide a detailed inspection of the location of the special
points in the M-R diagram. We find the following systematics which would hold
for a simultaneous variation of the Lagrangian parameters of the quark matter
EOS:

  1. 1.

Varying $\eta_{V}$ and $\eta_{D}$ simultaneously while keeping the ratio of
variations fixed to $\xi=\delta\eta_{V}/\delta\eta_{D}$ defines a line
$M^{(\xi)}(R)=a_{\xi}R+b_{\xi}$ in the $M-R$ diagram along which SPs are
located. This line is called a SP train (SPT).

  2. 2.

All these SPTs meet in one point denoted by "X" with the coordinates
$(M_{X},R_{X})=(1.8663\leavevmode\nobreak\
\textmd{M}_{\odot},11.112\leavevmode\nobreak\ {\rm km})$.

  3. 3.

The slope of the SPTs follows a linear dependence on the parameter $\xi$
$a_{\xi}=\tan\phi(\xi)=\alpha-\beta\xi$ with $\alpha=0.47074$ and
$\beta=0.7252$, see appendix B of Ref. Shahrbaf:2021cjz .

  4. 4.

The SPT with largest slope corresponds to $\xi=0$, where each SP on that train
corresponds to a fixed $\eta_{V}$ so that $\delta\eta_{V}=0$. This SPT is a
lower bound to the line of maximum mass configurations for hybrid stars as a
function of $\eta_{V}$, i.e. the stiffness of quark matter EOS.

  5. 5.

We have repeated the analysis by replacing the stiff DD2p40 with the softer
DD2p00 EOS. The result is shown in the right panel of Fig. 3 and demonstrates
that the SPTs are independent of the hadronic EOS used in constructing the
hybrid EOS model.

Figure 3: Mass-radius relation for all hybrid stars obtained by a Maxwell
construction. Dashed lines connect special points with a fixed slope
$\xi=\delta\eta_{V}/(\delta\eta_{D})$ of simultaneous variation of the
Lagrangian parameters and meet in one point denoted as X. The lines remain
unchanged when the hadronic EOS is changed from the stiff DD2p40 EOS (left
panel) to the softer DD2p00 EOS (right panel). For a comparison, we show the
red trapezoidal region that is enclosed by the upper and lower limits of the
parameter $A$ (long edges) and the upper and lower limits of the constant
speed of sound (short edges) of the CSS fit to the nlNJL EOS Cierniak:2021vlf
. The bold blue line is obtained from the confining RDF model with conformal
limit Ivanytskyi:2022bjc  for $\xi=0$, see section 5.

Figure 4: The same as in Fig. 1 but for the confining RDF approach and the
hypernuclear EoS DD2Y-T replacing the nlNJL quark matter EoS and the
nonstrange hadronic EoS DD2p00 and DD2p40, resp. The pair of values given in
brackets in the legends stands for $(\eta_{V},\eta_{D})$. Figure taken from
Ivanytskyi & Blaschke in these Proceedings. Figure 5: Same as the right panel
of Fig. 1 but for DD2Y-T instead of DD2p40 and the confining RDF approach
replacing the nlNJL quark matter EOS. A smaller sample of $\eta_{V}$ and
$\eta_{D}$ values is used to demonstrate the existence and location of the SP
train for the slope $\xi=\delta\eta_{V}/(\delta\eta_{D})=0$ of simultaneous
variation of the Lagrangian parameters.

##  3 Invariance of SPTs under change of the hadronic EOS

An important property of the SPs found by varying the bag pressure parameter
in the CSS form of the quark matter EOS was its independence of the hadronic
EOS that was used to obtain the hybrid EOS by a Maxwell construction. This
property was proven analytically by Yudin et al. Yudin:2014mla  and was
demonstrated with several examples numerically in Cierniak:2020eyh .
Nevertheless, when in the systematic variation of the Lagrangian parameters
$\eta_{V}$ and $\eta_{D}$ in Shahrbaf:2021cjz  the appearance of SPs along
straight line trajectories came as a surprise, it was not obvious whether
these SPTs would remain invariant under changes of the hadronic EOS. To give
an example of SP non-invariance, in Ref. Sen:2022lig  a notable spread of SPs
in the mass-radius diagram was obtained for six different hadronic EOS. The
likely reason for this should be the usage of a density-dependent bag pressure
functional in the bag model description of the quark matter phase.

It was, however, a very rewarding experience when the invariance of the SPTs
could be demonstrated in the investigation of Shahrbaf et al. Shahrbaf:2021cjz
for the quark matter phase described by the CSS fit of the color
superconducting nlNJL model. With this invariance and the discovery of the
focal point $X$ for the SPTs, we have uncovered now a rich phenomenological
toolbox for the interpretation of mass-radius measurements in terms of the
unknown parameters of Lagrangian models of color superconducting quark matter
in the nonperturbative region of low-energy QCD. The special role of the
$\xi=0$ SPT among all invariant SPTs for the NS phenomenology is explained in
the next section.

##  4 Special role of the SPT with $\xi=0$

A closer inspection of the SPTs for $\xi=0$ in the $M-R$ diagrams of Figs. 2,
3 and 5 shows that the points where the maximum mass is attained by varying
$\eta_{D}$ at fixed $\eta_{V}$ lie on a parabola-shaped curve for which the
minimum (apex) coincides with the SP. When the maximum mass point (MMP) is to
the right of the SP, then the latter lies on the unstable branch and is not
interesting. When the MMP is to the left of the SP then the latter is below
the maximum mass. Therefore, the SPT for $\xi=0$ can be regarded as a lower
bound to the line of highest MMPs for hybrid stars!

Another interesting observation is that for those values of $\eta_{D}$ for
which the SP coincides with the MMP, the onset masses for quark deconfinement
are very close to each other. Therefore, we can identify a particular onset
mass $M_{\rm onset}^{*}$ for which this situation occurs and which is
characteristic for the class of quark matter EOS and the hadronic one that has
been used to construct the deconfinement phase transition. While the SP
position does not depend on the choice of the hadronic EOS (as we have seen
before), the onset of deconfinement does. We come to postulate a refined
dependence of the maximum mass on the onset mass for deconfinement as

$M_{\rm max}=M_{\rm SP}+\delta\left|M_{\rm onset}^{*}-M_{\rm
onset}\right|^{\kappa}\leavevmode\nobreak\ ,$ (2)

where $\delta$ is a positive constant and $\kappa=2$. For the nlNJL moodel
with a hadronic phase described by DD2p40 we find $M_{\rm
onset}^{*}=0.75\leavevmode\nobreak\ M_{\odot}$. A precise definition should be
investigated as the closeup view on the SP reveals that it is rather a special
region than a point.

##  5 SPTs for other quark matter EOS?

A systematic study with different classes of Lagrangian quark matter models is
under way. All models shall have in common that they have vector and diquark
coupling strengths that are not determined by vacuum observables so that they
are considered as free parameters like in the recent works in this direction,
e.g., Baym:2017whm ; Ayriyan:2021prr ; Shahrbaf:2021cjz ; Ivanytskyi:2022oxv .
Currently, the instantaneous nlNJL model with 3-momentum dependent nonlocality
formfactor Contrera:2022tqh  is under investigation.

In this contribution, we can report first results that are obtained within the
confining RDF approach to color superconducting quark matter
Ivanytskyi:2022bjc , where the existence of SPs was observed in
Ivanytskyi:2022bjc ; Ivanytskyi:2022oxv  for the regime when pairs of
couplings $(\eta_{V},\eta_{D})$ were chosen along a straight line in the
$\eta_{D}-\eta_{V}$ plane, see Fig. 4. The analogue of a $\xi=0$ line was
found (see Fig. 5) and it compares well with the one from the covariant nlNJL
model, see the bold blue line in both panels of Fig. 3. Also in this case the
relation (2) between $M_{\rm max}$ and $M_{\rm onset}$ holds, but with $M_{\rm
onset}^{*}=1.4\leavevmode\nobreak\ M_{\odot}$ being larger than for the nlNJL
case with DD2p40.

It is interesting to note that the confining RF approach with an early onset
of deconfinement leads to a dramatic compactification of the light hybrid star
for masses below $2\leavevmode\nobreak\ M_{\odot}$ in agreement with all
present radius constraints in three mass ranges that follow for the strangely
light central compact object in HESS J1731-347 doroshenko2022strangely , the
multimessenger radius constraint at $1.4\leavevmode\nobreak\ M_{\odot}$
dietrich2020multimessenger  and radius measurement on the high-mass pulsar
J0740+6620 by NICER Miller:2021qha ; Riley:2021pdl . Comparing with Fig. 1 one
observes that the early deconfinement hybrid star sequences for the nlNJL
model of color superconducting quark matter cannot describe the radius of
$R=10.4^{+0.86}_{-0.78}$ km at $M=0.77^{+0.20}_{-0.17}\leavevmode\nobreak\
M_{\odot}$ reported by Doroshenko et al. in doroshenko2022strangely .

What about the original CSS model for quark matter, which was used in the
original work by Yudin et al. Yudin:2014mla  who started the investigation of
the SP property? Their analytic investigation used a two-parameter bag model
EoS for which the speed of sound was fixed to $c_{s}^{2}=1/3$ and varying the
bag constant $B$ produced a varying onset of deconfinement and defined the SP.
It was shown analytically that the SP position depends only on the EOOS of the
high-density phase, not of the hadronic one. A variation of the speed of sound
parameter was not performed in Yudin:2014mla . The next systematic
investigation of the SP property of hybrid EOS was performed in
Cierniak:2020eyh  with a three-parameter CSS EoS (1) for quark matter. The
invariance of the SP against variation of the hadronic EOS was confirmed and
in follow-up works Cierniak:2021knt ; Cierniak:2021vlf  the location of the SP
(still defined by varying $B$) in dependence of the values for the remaining
two parameters $c_{s}^{2}$ and $A$ was explored. In a sense, also in these
studies trajectories in the mass-radius diagram were found upon variation of
the parameter pair $(c_{s}^{2},A)$. The accessible region is shown in Fig. 3
as a bold red trapezoid with borders defined by the limiting values for
$c_{s}^{2}$ and $A$ that follow from the CSS fit to the nlNJL quark matter
EOS. The discrepancy to the $\xi=0$ SPT obtained from the nlNJL model (black
dashed line) or the confining RDF model (bold blue line) is apparent.
Obviously it makes a difference how the SP position is defined. It would be
interesting to explore the trajectory of SPs for the CSS model when the bag
constant is kept fixed and only the parameter pair $(c_{s}^{2},A)$ is varied.

##  6 Conclusions

In the present contribution we have applied the Maxwell construction scheme
for the deconfinement phase transition and demonstrated that a simultaneous
variation of the vector and diquark coupling constants results in the
occurrence of SP "trains" which are invariant against changing the nuclear
matter EOS. We have shown that the phenomenon occurs also in more
sophisticated Lagrangian quark matter models such as the confining
relativistic density functional approach and should be generic to all classes
of multiparameter Lagrangian models. We have pointed out that Bayesian
approaches appropriate for testing the presence of a deconfinement phase
transition in NSs should generate SPTs upon parameter variation. We have
proposed that the SPT corresponding to a variation of the diquark coupling at
constant vector coupling is special since it serves as a lower bound to the
line of maximum masses and accessible radii of massive hybrid stars.

## Acknowledgements

We are grateful to K. Redlich for his interest in our work and stimulating
discussions. This work was supported by NCN, grant 2019/33/B/ST9/03059 (A.A.,
D.B., M.C., O.I., M.S.) and Consejo Nacional de Investigaciones Científicas y
Técnicas, grant PIP17-700 (A.G.G).

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Solid matter with zero shear modulus

in flat universe

Peter Mészáros***e-mail address<EMAIL_ADDRESS>
Department of Theoretical Physics, Comenius University, Bratislava, Slovakia

## Abstract

For a perfect fluid, the quantity defined through mixed components of the
stress-energy tensor
$\widetilde{w}=(T_{i}^{\phantom{i}i}/3)/(-T_{0}^{\phantom{0}0})$ is
independent on the choice of coordinates only for two values of the pressure
to energy density ratio $w=p/\rho$: for radiation with $w=1/3$, and for dark
energy with $w=-1$. With other choices of $w$, the quantity $\widetilde{w}$ is
coordinate dependent, and $\widetilde{w}=w$ only in the local rest frame of
the fluid. We show that the same is true also for solid matter with shear
stress Lamé coefficient set to zero in a flat
Friedmann–Lemaître–Robertson–Walker universe with perturbed metric as well as
stress-energy tensor. We call the two different solids with coordinate
independent $\widetilde{w}$ radiation-like solid and dark energy-like solid,
and we restrict ourselves to these two special cases. By analysing second
order perturbations we discover two one parametric sets of such solid matter
models containing special cases of radiation and dark energy as perfect
fluids. We also study equations for perturbations up to the second order for
both sets of models.

###### Contents

  1. 1 Introduction
  2. 2 Solid matter with zero shear modulus
  3. 3 Perturbations in flat universe
    1. 3.1 Perfect fluid as a special case of solid
    2. 3.2 Scalar perturbations without gauge fixing
    3. 3.3 Vector and tensor perturbations
  4. 4 Scalar perturbations with dark energy-like solid
    1. 4.1 Linear perturbations
    2. 4.2 Second order perturbations
  5. 5 Scalar perturbations with radiation-like solid
    1. 5.1 Linear perturbations
    2. 5.2 Second order perturbations
  6. 6 Summary and outlook
  7. G Mixed stress-energy tensor of perfect fluid

##  1 Introduction

Theory of cosmological perturbations [2] is the key tool for comparing
theoretical models with observations of the cosmic microwave background (CMB)
radiation [3, 4]. Thanks to this theory and modern observational data the
values of cosmological parameters are known with high accuracy [5]. Properties
of primordial perturbations serving as initial conditions for evolution of
cosmological perturbations within the $\Lambda$-CDM model can be deduced from
this comparison as well. This constrains the parameter space of models of
cosmic inflation [6, 7].

From the time after the end of inflation and reheating to the time when
nonlinear effects cause growth of the structure, the matter content of the
universe can be considered as a multicomponent fluid with viscosity due to
interaction between its components [8]. However, one can speculate on more
peculiar forms of matter filling the universe, or rather one has to do so in
order to study inflation or dark energy and dark matter sectors. The prominent
example of matter generalizing the perfect fluid is scalar field which can
drive the inflation [9, 10], describe dark energy [11], dark matter [12], or
even both of them at the same time [13, 14]. Among more exotic matter
components of the universe there may be also solid matter with elastic
properties - i.e. matter with nonzero shear modulus. Similarly as the scalar
field, the solid matter can be part of the dark matter sector [15], or it can
drive the inflation [16, 17, 18, 19]. A very recent and more systematic
approach generalizing the solid matter in cosmology is in [20].

Despite the framework of cosmological perturbations within the standard
$\Lambda$-CDM model and its inflationary extensions being successful in
fitting observational data, there are still several discrepancies [21]. The
most important problems include tension between the local value of the Hubble
constant and its value from CMB observations [22, 23], and anomalies of the
CMB anisotropies for low multipole moments [24, 25]. Especially the latter
problem must be studied with the use of the perturbation theory [26, 27]. In
this context a natural modification is based on considering anisotropy of the
background. Bianchi type anisotropy does not produce strong anomaly [28], but
a more recent works studying different kinds of the background anisotropy
[29], which may be caused by cosmic strings [30, 31] or domain walls [32, 33],
seem to be more promising. Another example of a recent attempt to explain this
CMB anomaly is based on cosmological model with bounce [34]. So far, these
problems remain open from both observational and theoretical points of view.

Here we focus on matter filling a universe with flat
Friedmann–Lemaître–Robertson–Walker (FLRW) metric which shares some properties
with both perfect fluid and solid matter. Up to the first order of the
perturbation theory it behaves in the same way as a perfect fluid, but at the
same time it obeys equation of state of a solid matter. This manifests in
deviation from the perfect fluid-like behaviour in second and higher orders of
the perturbation theory. Thus we have to use the theory of cosmological
perturbations with nonlinear effects [35, 36, 37, 38, 39, 40] taken into
account. We describe this matter in the framework of general relativistic
elasticity [41]. Our approach may be developed even further inspired by soft
matter in condensed matter physics, see for instance a very recent work
bringing this concept to cosmology [42], but in this paper we focus on the
special case of solid described above. This approach can be used for
describing also a perfect fluid, as long as its volume elements are not
displaced too far from their initial positions. Fortunately, an almost
homogeneous perfect fluid filling an almost homogeneous universe is such a
case. In other words, a slightly perturbed fluid filling the universe can be
considered as a special case of solid matter.

Our main goal is to study physically interesting models of solid matter with
zero shear modulus in a cosmological context. In order to do so, we restrict
ourselves to two special cases corresponding to the same pressure to energy
density ratio as for radiation, $w=1/3$, and for dark energy, $w=-1$. The
first case describes our early universe at the beginning of the standard
Friedmann expansion within the standard $\Lambda$-CDM model, while the second
case approximates either far future of the universe or the cosmic inflation
before the standard Friedmann expansion.

Both radiation and dark energy satisfy another condition which may be imposed
on the solid in order to restrict its parameter space even more. In
particular, a relation for the components of the mixed stress-energy tensor

$\displaystyle\widetilde{w}=\frac{T_{i}^{\phantom{i}i}/3}{-T_{0}^{\phantom{0}0}}=\textrm{const.},$
(1.1)

with $T_{i}^{\phantom{i}i}$ denoting space part trace, holds in arbitrary
coordinates. However, in the local rest frame of the fluid $\widetilde{w}=w$
for any value of pressure to energy density ratio, not only for $w=1/3$ or
$w=-1$. For radiation, or photon gas, the relation (1.1) is the consequence of
scale invariance of the Maxwell theory. Due to conformal invariance, the full
trace of the mixed stress-energy tensor, which is invariant under space-time
diffeomorphisms, is zero. This property of the stress-energy tensor,
$T_{\mu}^{\phantom{\mu}\mu}=T_{0}^{\phantom{0}0}+T_{i}^{\phantom{i}i}=0$ in
arbitrary coordinates, implies that by passing to another coordinates
$T_{0}^{\phantom{0}0}$ changes by a factor opposite to change in
$T_{i}^{\phantom{i}i}$, leaving $\widetilde{w}$ in (1.1) unchanged. As for
dark energy, it can be described through the cosmological constant, so that
its stress-energy tensor must be proportional to the space-time metric. The
condition (1.1) then gives
$T_{0}^{\phantom{0}0}/T_{i}^{\phantom{i}i}=g_{0}^{\phantom{0}0}/g_{i}^{\phantom{i}i}=1/3$,
or $\widetilde{w}=w=-1$ in arbitrary coordinates.

Radiation and dark energy are the only two kinds of matter satisfying (1.1) in
arbitrary coordinates, while for other kinds of matter it holds only in their
local rest frame. A more technical approach to this fact, based on analysis of
the stress-energy tensor for perfect fluids can be found in appendix G.

The bulk of this paper can be schematically divided into three main parts:

  * •

In section 2 we introduce matter which departs from the perfect fluid only
when second or higher order perturbations are taken into account. It is solid
matter with shear modulus set to zero.

  * •

In section 3 we treat this matter pertubatively with flat FLRW space-time
metric as the background focusing on radiation-like, $w-1/3$, and dark energy-
like, $w=-1$, cases. We investigate consequences of the constraint on the
mixed stress-energy tensor – validity of (1.1) in arbitrary coordinates –
which not only allows only two above mentioned values of $w$, but also leads
to additional restrictions.

  * •

In sections 4 and 5 we present equations for up to second order scalar
perturbations for the model introduced in section 2 in two special cases
corresponding to constraints studied in section 3.

Finally, in section 6 we summarize and discuss the main results. Throughout
the paper we use the signature of the space-time metric $(-,+,+,+)$, together
with units in which $c=1$.

##  2 Solid matter with zero shear modulus

In order to describe a solid matter in general relativity one has to use map
from body space of the solid to the space-time, $\phi^{I}\mapsto x^{i}$, where
body coordinates $\phi^{I}$ label elements of the solid. If the solid is
relaxed, we may choose the space coordinates to coincide with the body
coordinates, $x^{i}=\delta^{i}_{I}\overline{\phi}^{I}$. It is convenient to
describe deformations of the solid by the body metric which is defined as a
pull-back of the space-time metric,

$\displaystyle
B^{IJ}=g^{\mu\nu}\phi^{I}_{\phantom{I},\mu}\phi^{J}_{\phantom{J},\nu},$ (2.1)

and in general the dependence of the solid matter energy density on its
deformations can be encoded by the equation of state of the form
$\rho=\rho\left(B^{IJ}\right)$. Furthermore, after restricting to the case of
solid with homogeneous and isotropic properties the equation of state must be
invariant with respect to global translations and rotations in the body space,

$\displaystyle\phi^{I}\mapsto\phi^{I}+T^{I}+R^{I}_{\phantom{I}J}\phi^{J},\quad
T^{I}\in\mathbb{R},\quad R^{I}_{\phantom{I}J}\in SO(3).$ (2.2)

Translational invariance is satisfied due to partial derivatives in the
definition of the body metric (2.1) automatically, and the rotational
invariance can be obtained by parametrizing the equation of state by
quantities which are invariant with respect to global rotations.

There are at most three independent quantities invariant with respect to
(2.2). They can be chosen as

$\displaystyle\left[\delta B\right]=\overline{B}_{IJ}\delta
B^{IJ},\quad\left[\delta B^{2}\right]=\overline{B}_{IJ}\delta
B^{JK}\overline{B}_{KL}\delta B^{LI},$ (2.3) $\displaystyle\left[\delta
B^{3}\right]=\overline{B}_{IJ}\delta B^{JK}\overline{B}_{KL}\delta
B^{LM}\overline{B}_{MN}\delta B^{NI},$

where $\overline{B}_{IJ}$ are components of the body metric corresponding to
the relaxed state, and $\delta B^{IJ}=B^{IJ}-\overline{B}^{IJ}$ is given by
the difference between the actual body metric and its relaxed configuration.
The equation of state with desired properties then can be parametrized as

$\displaystyle\rho=\overline{\rho}\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}C_{abc}\left[\delta
B\right]^{a}\left[\delta B^{2}\right]^{b}\left[\delta B^{3}\right]^{c},$ (2.4)

and the stress-energy tensor corresponding to such solid matter is

$\displaystyle T_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\frac{\partial\rho}{\partial
B^{IJ}}\phi^{I}_{\phantom{I},\mu}\phi^{J}_{\phantom{J},\nu}-\rho g_{\mu\nu}=$
$\displaystyle=$
$\displaystyle\overline{\rho}\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}C_{abc}\left[\delta
B\right]^{a}\left[\delta B^{2}\right]^{b}\left[\delta B^{3}\right]^{c}$
$\displaystyle\left[2\left(a\frac{\Psi^{(0)}_{\mu\nu}}{\left[\delta
B\right]}+2b\frac{\Psi^{(1)}_{\mu\nu}}{\left[\delta
B^{2}\right]}+3c\frac{\Psi^{(2)}_{\mu\nu}}{\left[\delta
B^{3}\right]}\right)-g_{\mu\nu}\right],$

where

$\displaystyle\Psi^{(0)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\overline{B}_{IJ}\phi^{I}_{\phantom{I},\mu}\phi^{J}_{\phantom{J},\nu},$
(2.6) $\displaystyle\Psi^{(1)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\overline{B}_{IK}\delta
B^{KL}\overline{B}_{LJ}\phi^{I}_{\phantom{I},\mu}\phi^{J}_{\phantom{J},\nu},$
$\displaystyle\Psi^{(2)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\overline{B}_{IK}\delta B^{KL}\overline{B}_{LM}\delta
B^{MN}\overline{B}_{NJ}\phi^{I}_{\phantom{I},\mu}\phi^{J}_{\phantom{J},\nu}.$

This parametrization is applicable up to arbitrary order of the perturbation
theory in models with general relativistic solid matter. When deriving
equations of motion for the $n$-th order perturbations all coefficients
$C_{abc}$ such that $a+2b+3c\leq n+1$ must be specified.

By setting coefficients $C_{abc}$ corresponding to up to second order terms in
the equation of state (2.4) as

$\displaystyle C_{000}=1,\quad C_{100}=\frac{w+1}{2},$ (2.7) $\displaystyle
C_{200}=\frac{\lambda+w+1}{8},\quad C_{010}=\frac{\mu-w-1}{4},$

and taking the Newtonian limit with weak deformation of the solid we retrieve
the same equation of state as in theory of elasticity within Newtonian
mechanics of continuum [43]. Coefficients $\lambda$ and $\mu$ in the equation
of state (2.4) then correspond to Lamé parameters
$\lambda=\lambda_{\textrm{Lam\'{e}}}$ and $\mu=\mu_{\textrm{Lam\'{e}}}$, and
$w$ is the pressure to energy density ratio. In this way we follow [44, 45],
but there are also works using different conventions for coefficients in the
equation of state for the solid matter [46, 15]. In this paper we will study
also second order perturbations. Hence we have to introduce also coefficients
$\nu_{1}$, $\nu_{2}$ and $\nu_{3}$, which parametrize the additional
coefficients $C_{abc}$ as

$\displaystyle C_{110}=\frac{1}{4}\nu_{1},\quad
C_{001}=\frac{1}{6}\nu_{2},\quad C_{300}=\frac{1}{12}\nu_{3}.$ (2.8)

Other coefficients $C_{abc}$ would be relevant only in higher orders of the
perturbation theory, and we may disregard them.

From now on we restrict ourselves to a special case in which the shear stress
coefficient $\mu$ vanishes, so that

$\displaystyle C_{010}=-\frac{w+1}{4}.$ (2.9)

Such solid matter then does not differ from a perfect fluid when only the
background together with the first order perturbations are taken into account.
In the next section we will demand independence of ratio of the time-time
component of the mixed stress-energy tensor to its space part trace
$T_{0}^{\phantom{0}0}/T_{i}^{\phantom{i}i}$ on the deformation of the solid,
i.e. we will make use of validity of the condition (1.1) in arbitrary
coordinates. This will lead to constraints of coefficient $w$, $\lambda$,
$\nu_{1}$, $\nu_{2}$ and $\nu_{3}$. Of course, one can study more complicated
models by relaxing the condition (1.1), but it is not the aim of this work.

##  3 Perturbations in flat universe

We can set any values of coefficients (2.7)-(2.9) to specify the solid matter,
choose any space-time metric, even add other matter components, and study
perturbations of such system up to the second order. However, we will restrict
ourselves to a specific case corresponding to homogeneous, isotropic and
spatially flat universe, and solid matter satisfying the condition on the
mixed tress-energy tensor (1.1) in arbitrary coordinates. For the background
space-time metric we choose the flat FLRW metric parametrized by cosmic time
$t$ and comoving space coordinates $x^{i}$,

$\displaystyle ds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j},$ (3.1)

with function $a(t)$ denoting the scale factor. We define the relaxed state of
the solid matter by identifying its body coordinates with the comoving space
coordinates, $\overline{\phi}^{I}=\delta^{I}_{i}x^{i}$. In this way stretching
of the solid together with the expanding universe does not count for its
deformations in the perturbation theory. In this section we add perturbations
to both space-time metric and solid matter, and derive constraints on
coefficients (2.7)-(2.9) following from the condition (1.1).

As mentioned above we will restrict ourselves only up to the second order of
the perturbation theory. Due to the linearity in the first order of
perturbations, scalar, vector and tensor perturbations are decoupled from each
other. First, we focus on only scalar perturbations even in the second order
in which these three kinds of perturbations are coupled. If we were interested
in evolution of scalar perturbations in our universe only up to the second
order, such simplification would be reasonable because of two reasons. The
first one is that the first order vector perturbations are decaying in the
course of the expansion of the universe [3], and the second reason is based on
observations of the cosmic microwave background radiation which suggest small
tensor-to-scalar ratio [6]. Equations for the second order scalar
perturbations are sourced by the first order perturbations including vector
and tensor ones, but because of their negligibility this source is dominated
by scalar perturbations. Of course, equations for the second order vector and
tensor perturbations are sourced in the same manner, and therefore due to
presence of the first order scalar perturbations they must be present even if
their first order parts are set to zero. We will include these perturbation
later after finding restrictions on coefficients (2.7)-(2.9) based on analysis
of only scalar perturbations.

In general relativistic perturbation theory we are free to make use of small
coordinate transformations which do not change the form of the background.
Transformations of tensor $\mathcal{A}$ under such transformations are given
by Lie derivatives [47], in particular[36]

$\displaystyle\widetilde{x}^{\mu}=e^{\xi^{\nu}\partial_{\nu}}x^{\mu}\quad\longrightarrow\quad\widetilde{\mathcal{A}}=e^{-\mathcal{L}_{\xi}}\mathcal{A}.$
(3.2)

Setting coordinates in this way is called gauge fixing because of similarity
with models with gauge symmetry in other physical contexts.

For scalar sector we use longitudinal gauge in which the perturbed metric is
given by two functions $\phi$ and $\psi$,

$\displaystyle
ds^{2}=a^{2}\left[-(1+2\phi)d\tau^{2}+(1-2\psi)\delta_{ij}dx^{i}dx^{j}\right],$
(3.3)

with cosmic time $t$ replaced by the conformal time $\tau=\int a^{-1}dt$. With
this choice the gauge freedom is already exhausted, and the solid matter
perturbation has to be parametrized in the most general way. Therefore we have
to add perturbation to the body coordinates $\phi^{I}$ parametrized by scalar
perturbations in the general form,

$\displaystyle\phi^{I}=\delta^{I}_{i}(x^{i}+\pi^{i})=\delta^{I}_{i}(x^{i}+\sigma_{,i}),$
(3.4)

but as the last equality reveals, there is only one scalar part of this
perturbation, and it is parametrized by function $\sigma$.

When fixing the gauge we can fix the gauge separately for each order of the
perturbation theory [48]. Such order by order gauge fixing simplifies the
calculations considerably, however, a dictionary between sets of second order
variables with respect to two different ways of gauge fixing contains also
terms with first order variables [49]. For convenience, we chose the same
gauge for both first and second order perturbations. Then we can decompose
scalar perturbations listed above order by order,

$\displaystyle\phi=\phi^{(1)}+\phi^{(2)},\quad\psi=\psi^{(1)}+\psi^{(2)},\quad\sigma=\sigma^{(1)}+\sigma^{(2)},$
(3.5)

with the upper indices in parentheses indicating the order of the perturbation
theory.

For the stress-energy tensor of the form (2) with coefficients in it
parametrized as (2.7)-(2.9), with invariants (2.3) defined through body metric
(2.1) corresponding to body coordinates (3.4), and for the space-time metric
(3.3), the needed quantities appearing in the condition (1.1) expressed up to
the second order are

$\displaystyle\frac{T_{0}^{\phantom{0}0}}{\overline{\rho}}$ $\displaystyle=$
$\displaystyle-1-3(w+1)\psi^{(1)}-(w+1)\triangle\sigma^{(1)}-$
$\displaystyle-\frac{3}{2}\left(5w+5+3\lambda\right)\psi^{(1)2}-3(w+1)\psi^{(2)}-$
$\displaystyle-3\left(w+1+\lambda\right)\psi^{(1)}\triangle\sigma^{(1)}+$
$\displaystyle+(w+1)\left(\frac{1}{2}\sigma^{(1)}_{,ij}\sigma^{(1)}_{,ij}-\frac{1}{2}\left(\triangle\sigma^{(1)}\right)^{2}-\frac{1}{2}\sigma^{(1)\prime}_{,i}\sigma^{(1)\prime}_{,i}-\triangle\sigma^{(2)}\right)-$
$\displaystyle-\frac{1}{2}\lambda\left(\triangle\sigma^{(1)}\right)^{2},$
$\displaystyle\frac{T_{i}^{\phantom{i}i}}{\overline{\rho}}$ $\displaystyle=$
$\displaystyle 3w+9\lambda\psi^{(1)}+3\lambda\triangle\sigma^{(1)}+$
$\displaystyle+\frac{3}{2}\left(15\lambda+w+1+36\nu_{1}+8\nu_{2}+36\nu_{3}\right)\psi^{(1)2}+9\lambda\psi^{(2)}+$
$\displaystyle+\left(w+1+9\lambda+36\nu_{1}+8\nu_{2}+36\nu_{3}\right)\psi^{(1)}\triangle\sigma^{(1)}+$
$\displaystyle+(w+1)\left(-\sigma^{(1)}_{,ij}\sigma^{(1)}_{,ij}+\frac{1}{2}\left(\triangle\sigma^{(1)}\right)^{2}+\sigma^{(1)\prime}_{,i}\sigma^{(1)\prime}_{,i}\right)+$
$\displaystyle+\lambda\left(\frac{3}{2}\sigma^{(1)}_{,ij}\sigma^{(1)}_{,ij}+\frac{1}{2}\left(\triangle\sigma^{(1)}\right)^{2}-\frac{3}{2}\sigma^{(1)\prime}_{,i}\sigma^{(1)\prime}_{,i}+3\triangle\sigma^{(2)}\right)+$
$\displaystyle+\nu_{1}\left(6\sigma^{(1)}_{,ij}\sigma^{(1)}_{,ij}+4\left(\triangle\sigma^{(1)}\right)^{2}\right)+4\nu_{2}\sigma^{(1)}_{,ij}\sigma^{(1)}_{,ij}+6\nu_{3}\left(\triangle\sigma^{(1)}\right)^{2},$

where the prime denotes differentiation with respect to the conformal time.

We can see that satisfying the condition (1.1) in arbitrary coordinates up to
the second order of the perturbation theory is not a trivial matter. We have
to keep track of coefficients at every independent combination of
perturbations. Ratio of a coefficient at a term composed of perturbations in
$T_{0}^{\phantom{0}0}$ to a coefficient at the same perturbation term in
$T_{i}^{\phantom{i}i}$ either must be $-3w$ or both of the coefficients have
to be zero. This must be satisfied for every term in $T_{0}^{\phantom{0}0}$
and $T_{i}^{\phantom{i}i}$, which implies a set of constraints on coefficient
$w$, $\lambda$, $\nu_{1}$, $\nu_{2}$ and $\nu_{3}$. It turns out that there is
only one solution for this set of constraints. In this way, by taking into
account only the first order parts we find

$\displaystyle\lambda=w(w+1),$ (3.8)

and by including the second order terms we obtain three additional independent
restrictions

$\displaystyle(3w-1)(w+1)=0,\quad 3\nu_{1}+2\nu_{2}=0,\quad
2\nu_{1}+3\nu_{3}=0.$ (3.9)

The first one follows from comparing coefficients at
$\sigma^{(1)\prime}_{,i}\sigma^{(1)\prime}_{,i}$, the only term which is zero
for static deformations.

In summary, $w=1/3$ and $w=-1$ are the only two values of the pressure to
energy density ratio allowing the condition (1.1) to be satisfied in arbitrary
coordinates up to the second order of the perturbation theory with the
deformation of the solid dependent on time. Of course, in the local rest frame
the condition (1.1) holds automatically for any constant value of pressure to
energy density ratio, and $\widetilde{w}=w$. Note also that we have not
restricted ourselves to the case with $w=1/3$ or $w=-1$ first, and then
studied consequences of the additional condition (1.1) assumed to be valid in
arbitrary coordinates, but we have used only (1.1). Therefore, we have shown
that validity of (1.1) implies only $w=1/3$ and $w=-1$ and no other values of
the pressure to energy density ratio for the solid matter with zero shear
modulus. The same is true also for a perfect fluid, as shown in appendix G.

###  3.1 Perfect fluid as a special case of solid

By analysing perturbations up to the second order we have found that only the
radiation-like solid with $w=1/3$ and the dark energy-like solid with $w=-1$
satisfy the condition on the mixed stress-energy tensor components (1.1) in
arbitrary coordinates. The solid with zero shear modulus shares this property
with the perfect fluid, as shown in appendix G. Let us now approach the fluid
in a less usual way, as a special case of the solid. The perturbative solid
matter description is applicable also to an almost homogeneous fluid in an
almost homogeneous universe, because elements of the fluid are not displaced
too far from their initial comoving positions.

The equation of state for a solid (2.4) with zero shear modulus and satisfying
(1.1) in arbitrary coordinates which fills a flat FLRW universe written up to
the third order in its deformations reduces to

$\displaystyle\frac{\rho}{\overline{\rho}}$ $\displaystyle=$ $\displaystyle
1+\frac{1}{2}(w+1)\left[\delta
B\right]+\frac{1}{8}\left(w+1\right)^{2}\left[\delta
B\right]^{2}-\frac{1}{4}(w+1)\left[\delta B^{2}\right]+$
$\displaystyle+\nu\left(\frac{1}{4}\left[\delta B\right]\left[\delta
B^{2}\right]-\frac{1}{4}\left[\delta B^{3}\right]-\frac{1}{18}\left[\delta
B\right]^{3}\right),$

where $\nu_{1}$ is denoted as $\nu$, $w=1/3$ or $w=-1$, and other coefficients
has been written in terms of $w$ and $\nu$. We can see that for the dark
energy-like solid first and second order parts of the equation of state
vanish. It is in principle possible that this special form of equation of
state allows no deviation from a perfect fluid. In such case the description
of the continuum through body coordinates and body metric would not be needed
at all, and we might end this paper by referring to works studying higher
order perturbations in a universe filled with ordinary matter components [50].
In order to make sure that this is not the case, and also find out which
values of parameters $\nu$ correspond to the perfect fluid, we have to keep
using the formalism used so far to compare the equation of state of the
perfect fluid with the one for the solid matter.

The energy density of the perfect fluid depends only on volumes of its
elements, therefore it is a function of the determinant of the body metric. It
can be written as

$\displaystyle\frac{\rho_{\textrm{id}}}{\overline{\rho}}=F\left(\textrm{det}B\right)=\sum\limits_{n=0}^{\infty}c_{n}\left(\textrm{det}\left(\overline{B}B^{-1}\right)-1\right)^{n},$
(3.11)

where $\overline{B}$ denotes the background body metric,
$\overline{B}_{IJ}=a^{2}\delta_{IJ}$, and $B^{-1}$ is the inverse body metric
containing perturbations. By expanding this equation of state up to the third
order in terms of traces of the body metric perturbations and using the
formula for determinant of three by three matrices,
$\textrm{det}A=\left(\textrm{Tr}A\right)^{3}/6-\left(\textrm{Tr}A^{2}\right)\textrm{Tr}A/2+\textrm{Tr}A^{3}/3,$
we obtain

$\displaystyle\frac{\rho_{\textrm{id}}}{\overline{\rho}}$ $\displaystyle=$
$\displaystyle c_{0}+c_{1}\left[\delta
B\right]+\left(\frac{1}{2}c_{1}+c_{2}\right)\left[\delta
B\right]^{2}-\frac{1}{2}c_{1}\left[\delta B^{2}\right]+$
$\displaystyle+\left(\frac{1}{6}c_{1}+c_{2}+c_{3}\right)\left[\delta
B\right]^{3}-\left(\frac{1}{2}c_{1}+c_{2}\right)\left[\delta
B\right]\left[\delta B^{2}\right]+\frac{1}{3}c_{1}\left[\delta B^{3}\right].$

If we compare this equation of state with the equation of state of the solid
matter (3.1) we can express coefficients $c_{0}$-$c_{3}$ through $w$ and $\nu$
as

$\displaystyle c_{0}=1,\quad c_{1}=\frac{1}{2}(w+1),\quad
c_{2}=\frac{1}{8}(w-1)(w+1),$ (3.13) $\displaystyle
c_{3}=-\frac{1}{18}\nu-\frac{1}{24}(3w-1)(w+1).$

This procedure imposes seven conditions on six parameters $c_{0}$-$c_{3}$, $w$
and $\nu$, therefore, we also find restrictions

$\displaystyle\nu=-\frac{1}{2}(w+1)^{2}=-\frac{2}{3}(w+1),$ (3.14)

where the last equality is equivalent to the first condition in (3.9),
$(3w-1)(w+1)=0$, which has already been taken into account in (3.1).

In summary, both the radiation-like solid and the dark energy-like solid can
be reduced to a perfect fluid. For the radiation-like case it happens for
$\nu=-8/9$, and for the dark energy-like case the reduction occurs when
$\nu=0$. For other values of coefficient $\nu$ there is difference between the
perfect fluid and the solid matter with zero shear modulus, and indeed, in
this paper we are studying a non-trivial generalization of the fluid.

Conclusions made so far in this section correspond to only scalar
perturbations in the longitudinal gauge. In order to obtain complete results
we also have to examine different choices of the gauge or case without fixing
the gauge, and also cases including vector and tensor perturbations. This is
the content of the following subsections.

###  3.2 Scalar perturbations without gauge fixing

Now we relax the longitudinal gauge condition for perturbations. The FLRW
metric with scalar perturbations without any gauge fixing then reads

$\displaystyle ds^{2}=a^{2}\left\\{-\left(1+2\phi\right)d\tau^{2}-2B_{,i}d\tau
dx^{i}+\left[\delta_{ij}\left(1-2\psi\right)-2E_{,ij}\right]dx^{i}dx^{j}\right\\},$
(3.15)

and considering only scalar perturbations also for the body coordinates we
have perturbation $\sigma$ defined in (3.4). Additional scalar perturbations
$B$ and $E$ can by decomposed with respect to order of the perturbation theory
in the same way as scalar perturbations used so far, $B=B^{(1)}+B^{(2)}$,
$E=E^{(1)}+E^{(2)}$. We can now repeat calculations leading to results (3) and
(3) for the more general form of space-time metric (3.15), but with
constraints on parameters (3.8) and (3.9) already taken into account.

The equation of state constrained by results of analysis of scalar
perturbations in the longitudinal gauge up to second order of the perturbation
theory (3.1) for the radiation-like case with $w=1/3$ is

$\displaystyle\frac{\rho}{\overline{\rho}}$ $\displaystyle=$ $\displaystyle
1+\frac{2}{3}\left[\delta B\right]+\frac{2}{9}\left[\delta
B\right]^{2}-\frac{1}{3}\left[\delta B^{2}\right]+$
$\displaystyle+\nu\left(\frac{1}{4}\left[\delta B\right]\left[\delta
B^{2}\right]-\frac{1}{4}\left[\delta B^{3}\right]-\frac{1}{18}\left[\delta
B\right]^{3}\right),$

and the stress-energy tensor corresponding to the space-time metric (3.15) and
the equation of state written above gives

$\displaystyle\frac{T_{0}^{\phantom{0}0}}{\overline{\rho}}=-\frac{T_{i}^{\phantom{i}i}}{\overline{\rho}}$
$\displaystyle=$ $\displaystyle-1-4\psi^{(1)}-\frac{4}{3}\triangle
E^{(1)}-\frac{4}{3}\triangle\sigma^{(1)}-$
$\displaystyle-4\psi^{(2)}-\frac{4}{3}\triangle
E^{(2)}-12\psi^{(1)2}-8\psi^{(1)}\triangle E^{(1)}+$
$\displaystyle+\frac{2}{3}B^{(1)}_{,i}B^{(1)}_{,i}-\frac{4}{3}E^{(1)}_{,ij}E^{(1)}_{,ij}-\frac{8}{9}\left(\triangle
E^{(1)}\right)^{2}-$
$\displaystyle-\frac{16}{3}\psi^{(1)}\triangle\sigma^{(1)}-\frac{16}{9}\triangle
E^{(1)}\triangle\sigma^{(1)}-\frac{4}{3}\triangle\sigma^{(2)}+$
$\displaystyle+\frac{2}{3}\sigma^{(1)}_{,ij}\sigma^{(1)}_{,ij}-\frac{8}{9}\left(\triangle\sigma^{(1)}\right)^{2}-\frac{2}{3}\sigma^{(1)\prime}_{,i}\sigma^{(1)\prime}_{,i}.$

This is in agreement with the key condition (1.1).

For the dark energy-like case with $w=-1$ we have

$\displaystyle\frac{\rho}{\overline{\rho}}=1+\nu\left(\frac{1}{4}\left[\delta
B\right]\left[\delta B^{2}\right]-\frac{1}{4}\left[\delta
B^{3}\right]-\frac{1}{18}\left[\delta B\right]^{3}\right),$ (3.18)

and

$\displaystyle\frac{T_{0}^{\phantom{0}0}}{\overline{\rho}}=-1,\quad\frac{T_{i}^{\phantom{i}i}}{\overline{\rho}}=-3,$
(3.19)

so that the condition (1.1) remains valid in this case as well.

We can conclude that by relaxing the gauge fixing condition which sets the
form of the perturbed flat FLRW metric (3.3) we obtain no additional
constraints on parameters $w$ and $\nu$ describing the solid obeying the
condition (1.1) assumed to be valid in arbitrary coordinates.

###  3.3 Vector and tensor perturbations

If we add vector and tensor perturbations $V_{i}$ and $\gamma_{ij}$ to the
space-time metric using the longitudinal gauge for scalar perturbations and an
appropriate gauge for the rest of them, we can write the perturbed metric as

$\displaystyle ds^{2}=a^{2}\left\\{-\left(1+2\phi\right)d\tau^{2}-2V_{i}d\tau
dx^{i}+\left[\delta_{ij}\left(1-2\psi\right)-\gamma_{ij}\right]dx^{i}dx^{j}\right\\},$
(3.20)

where vector and tensor perturbations satisfy conditions $V_{i,i}=0$,
$\gamma_{ii}=0$ and $\gamma_{ij,j}=0$. We have to add vector perturbation
$v^{\perp}_{i}$ also to the body coordinates,

$\displaystyle\phi^{I}=\delta^{I}_{i}\left(x^{i}+v_{i}\right),\quad
v_{i}=\sigma_{,i}+v^{\perp}_{i},\quad v^{\perp}_{i,i}=0.$ (3.21)

Using order by order gauge fixing we can set the same gauge for the second
order perturbations as for the first order ones, and decompose the newly added
perturbations in the usual way, $V_{i}=V^{(1)}_{i}+V^{(2)}_{i}$,
$\gamma_{ij}=\gamma^{(1)}_{ij}+\gamma^{(2)}_{ij}$,
$v^{\perp}_{i}=v^{\perp(1)}_{i}+v^{\perp(2)}_{i}$.

We can assume that $\psi^{(1)}=\phi^{(1)}$, which follows from the traceless
part of the Einstein field equations [3, 4]. Note that this would not be true
if we did not set the shear modulus to zero [44, 45]. From the direct
calculation of the stress-energy tensor with respect to the metric (3.20) for
the radiation-like case with $w=1/3$ and the equation of state (3.2) then
follows

$\displaystyle\frac{T_{0}^{\phantom{0}0}}{\overline{\rho}}=-\frac{T_{i}^{\phantom{i}i}}{\overline{\rho}}$
$\displaystyle=$
$\displaystyle-1-4\phi^{(1)}-\frac{4}{3}v^{(1)}_{i,i}-4\psi^{(2)}-\frac{4}{3}v^{(2)}_{i,i}-$
$\displaystyle-12\phi^{(1)2}-\frac{16}{3}\phi^{(1)}v^{(1)}_{i,i}+\frac{2}{3}V^{(1)}_{,i}V^{(1)}_{,i}+$
$\displaystyle+\frac{2}{3}v^{(1)}_{i,j}v^{(1)}_{j,i}-\frac{8}{9}\left(v^{(1)}_{i,i}\right)^{2}-\frac{2}{3}v^{(1)\prime}_{i}v^{(1)\prime}_{i}-\frac{1}{3}\gamma^{(1)}_{ij}\gamma^{(1)}_{ij}.$

Note that this relation contains also scalar perturbation $\sigma$ in the way
consistent with the last three terms in (3.2), because according to (3.21) it
parametrizes the scalar part of $v_{i}$. Similarly, in the dark energy-like
case with $w=-1$ and the equation of state (3.18) we obtain

$\displaystyle\frac{T_{0}^{\phantom{0}0}}{\overline{\rho}}=-1,\quad\frac{T_{i}^{\phantom{i}i}}{\overline{\rho}}=-3.$
(3.23)

In both cases the condition (1.1) implies the same result as with only the
scalar perturbations in the longitudinal gauge. Hence the restrictions (3.8)
and (3.9) remain valid, and there is no independent restriction which should
be added to them.

##  4 Scalar perturbations with dark energy-like solid

In this section we focus on the case with $w=-1$, and analyse equations for
scalar perturbations up to the second order. In the case of perfect fluid with
$\nu=0$ they should have no solution of the form of perturbations. We will
investigate whether this holds for any value of parameter $\nu$.

First, we only write down equations and their solution for the unperturbed
case, which can be found in standard textbooks. The nontrivial Einstein
equations in the zeroth order of the perturbation theory are

$\displaystyle G^{(0)}_{00}=3\mathcal{H}^{2}=8\pi\kappa
T^{(0)}_{00}=8\pi\kappa a^{2}\overline{\rho},$ (4.1) $\displaystyle
G^{(0)}_{ij}=-\left(\mathcal{H}^{2}+2\mathcal{H}^{\prime}\right)\delta_{ij}=8\pi\kappa
T^{(0)}_{ij}=-8\pi\kappa a^{2}\overline{\rho}\delta_{ij},$ (4.2)

where $\mathcal{H}=a^{\prime}/a$, and the only nontrivial part of the energy-
momentum conservation law is

$\displaystyle
T^{(0)0\mu}_{\phantom{(0)0\mu};\mu}=\frac{\overline{\rho}^{\prime}}{a^{2}}=0.$
(4.3)

These equations are solved for

$\displaystyle a\propto\frac{1}{\tau},\quad\overline{\rho}=\textrm{const.},$
(4.4)

hence $\mathcal{H}=-1/\tau$. We will use this result as the background for
higher orders of the perturbation theory.

###  4.1 Linear perturbations

Not only the background equations, but also equations for the linearised
cosmological perturbations can be found in the standard textbooks. Here we are
including the first order equations for two reasons. The first one is that
they are a step towards analysis of the second order perturbations which will
be studied in the next subsection. The second reason is that textbooks tend to
omit them in the case with dark energy-like pressure to energy density ratio
$w=-1$, which is justified by lack of any nontrivial solution, as we will see
bellow. The needed equations in the first order of the perturbation theory are

$\displaystyle
G^{(1)}_{00}=2\left(\triangle\phi^{(1)}-3\mathcal{H}\phi^{(1)\prime}\right)=8\pi\kappa
T^{(1)}_{00}=6\mathcal{H}^{2}\phi^{(1)},$ (4.5) $\displaystyle
G^{(1)}_{0i}=2\left(\mathcal{H}\phi^{(1)}_{,i}+\phi^{(1)\prime}_{,i}\right)=8\pi\kappa
T^{(1)}_{0i}=0,$ (4.6) $\displaystyle
G^{(1)}_{ij}=2\left(6\mathcal{H}^{2}\phi^{(1)}+3\mathcal{H}\phi^{(1)\prime}+\phi^{(1)\prime\prime}\right)\delta_{ij}=8\pi\kappa
T^{(1)}_{ij}=6\mathcal{H}^{2}\phi^{(1)}\delta_{ij},$ (4.7)

with the energy-momentum conservation law
$T^{(1)\mu\nu}_{\phantom{(1)\mu\nu};\nu}=0$ implying only $0=0$ type of
equations. These equations has been simplified with the use of the background
solution (4.4), $\mathcal{H}^{\prime}=\mathcal{H}^{2}$, and the fact that
$\psi^{(1)}=\phi^{(1)}$ which follows from the traceless spatial part of
Einstein field equations,
$(\phi^{(1)}-\psi^{(1)})_{,ij}\stackrel{{\scriptstyle i\neq j}}{{=}}0$
together with the assumption that $\phi^{(1)}$ and $\psi^{(1)}$ are of the
form of perturbations. The latter simplification has already been used in
subsection 3.3.

The Fourier mode of perturbation $\phi^{(1)}$ corresponding to the solution of
the $00$-part of the Einstein field equations (4.5) is

$\displaystyle\phi^{(1)}_{k}=\alpha_{1}\tau e^{\frac{1}{6}k^{2}\tau^{2}},$
(4.8)

for the $0i$-part (4.6) it is

$\displaystyle\phi^{(1)}_{k}=\alpha_{2}\tau+\tau\int\limits_{\tau_{0}}^{\tau}\frac{d\tau^{\prime}}{\tau^{\prime}}f\left(\tau^{\prime}\right),$
(4.9)

and from the $ij$-part (4.7) follows

$\displaystyle\phi^{(1)}_{k}=\alpha_{3}\tau+\alpha_{4}\tau^{3},$ (4.10)

where $\alpha_{1}$-$\alpha_{4}$ and $\tau_{0}$ are constants and $f$ is an
arbitrary function. Consistency of three forms of the solution for the scalar
perturbation $\phi^{(1)}$ (4.8)-(4.10) can be satisfied only when
$\alpha_{1}=\alpha_{2}=\alpha_{3}$, $\alpha_{4}=0$, $f=0$ and $\tau_{0}=0$ and
only for mode with zero wavenumber, so that $\phi^{(1)}=\alpha_{1}\tau$.

The found $\phi^{(1)}$ is not of a form of perturbation and can be absorbed
into the background. The metric corresponding to this ”new” background can be
written as

$\displaystyle
ds^{2(0)}_{\textrm{new}}=\frac{1}{\tau^{2}}\left[-\left(1-\varepsilon\tau\right)d\tau^{2}+\left(1+\varepsilon\tau\right)\delta_{ij}dx^{i}dx^{j}\right],$
(4.11)

where the scale factor is chosen to be $a=1$ at the conformal time $\tau=1$,
and parameter $\varepsilon=-2\alpha_{1}$ must be small. It turns out that this
”new” background metric does not differ from the original background metric,
because by mere redefinition of the conformal time
$\widetilde{\tau}=\tau-\varepsilon\tau^{2}/2+\mathcal{O}(\varepsilon^{2})$ we
find

$\displaystyle
ds^{2(0)}_{\textrm{new}}=\frac{1}{\widetilde{\tau}^{2}}\left(-d\widetilde{\tau}^{2}+\delta_{ij}dx^{i}dx^{j}\right)+\mathcal{O}(\varepsilon^{2}).$
(4.12)

Hence in the universe with dark energy the first order perturbations must be
omitted.

We can also see that perturbation $\sigma^{(1)}$ corresponding to body element
displacement does not appear in equations for the first order perturbations
due to its absence in the stress-energy tensor, and thus, $\sigma^{(1)}$
remains arbitrary. In other words, in the dark energy-like case the first
order part of perturbation $\sigma$ is a redundant quantity.

###  4.2 Second order perturbations

Do the second order perturbations vanish not only, as expected in the case
with the dark energy as a perfect fluid, for $\nu=0$, but also for dark
energy-like solid with zero shear modulus and with $\nu\neq 0$? A naive answer
would be yes, because the condition (1.1) assumed to be valid in arbitrary
coordinates holds for any value of parameter $\nu$, and in the dark energy-
like case it is related to stress-energy tensor proportional to the space-time
metric. However, $T_{\mu\nu}\propto g_{\mu\nu}$ implies both validity of
condition (1.1) in arbitrary coordinates and absence of matter perturbation
$\sigma$ in equations in any perturbative order, the opposite is not true,
because the condition (1.1) does not necessarily imply $T_{\mu\nu}\propto
g_{\mu\nu}$. With nonzero coefficient $\nu$ we depart from the perfect fluid
case, and there appears traceless space part of the stress energy tensor,
which does not violate the condition (1.1), but it may affect the evolution of
perturbations.

Let us examine equations for the second order perturbations. The corresponding
Einstein field equations and the energy-momentum conservation law read

$\displaystyle G^{(2)}_{00}$ $\displaystyle=$ $\displaystyle
2\triangle\psi^{(2)}-6\mathcal{H}\psi^{(2)\prime}+$
$\displaystyle+3\phi^{(1)}_{,i}\phi^{(1)}_{,i}+12\phi^{(1)}\triangle\phi^{(1)}-12\mathcal{H}\phi^{(1)}\phi^{(1)\prime}+\phi^{(1)\prime
2}=$ $\displaystyle=$ $\displaystyle 8\pi\kappa
T^{(2)}_{00}=6\mathcal{H}^{2}\phi^{(2)},$ $\displaystyle G^{(2)}_{0i}$
$\displaystyle=$ $\displaystyle
2\mathcal{H}\phi^{(2)}_{,i}+2\psi^{(2)\prime}_{,i}-$
$\displaystyle-4\mathcal{H}\phi^{(1)}\phi^{(1)}_{,i}+2\phi^{(1)}_{,i}\phi^{(1)\prime}+4\phi^{(1)}\phi^{(1)\prime}_{,i}=$
$\displaystyle=$ $\displaystyle 8\pi\kappa T^{(2)}_{0i}=0,$ $\displaystyle
G^{(2)}_{ii}$ $\displaystyle=$ $\displaystyle
6\psi^{(2)\prime\prime}+6\mathcal{H}\left(\phi^{(2)}+2\psi^{(2)}\right)^{\prime}+$
$\displaystyle+18\mathcal{H}^{2}\left(\phi^{(2)}+\psi^{(2)}\right)+2\triangle\left(\phi^{(2)}-\psi^{(2)}\right)-$
$\displaystyle-72\mathcal{H}^{2}\phi^{(1)2}-7\phi^{(1)}_{,i}\phi^{(1)}_{,i}-8\phi^{(1)}\triangle\phi^{(1)}-$
$\displaystyle-60\mathcal{H}\phi^{(1)}\phi^{(1)\prime}-3\phi^{(1)\prime
2}-12\phi^{(1)}\phi^{(1)\prime\prime}=$ $\displaystyle=$ $\displaystyle
8\pi\kappa T^{(2)}_{ii}=18\mathcal{H}^{2}\psi^{(2)},$ $\displaystyle
G^{(2)}_{ij}$ $\displaystyle\stackrel{{\scriptstyle i\neq j}}{{=}}$
$\displaystyle-\left(\phi^{(2)}-\psi^{(2)}\right)_{,ij}+2\phi^{(1)}_{,i}\phi^{(1)}_{,j}+4\phi^{(1)}\phi^{(1)}_{,ij}\stackrel{{\scriptstyle
i\neq j}}{{=}}$ $\displaystyle\stackrel{{\scriptstyle i\neq j}}{{=}}$
$\displaystyle 8\pi\kappa T^{(2)}_{ij}\stackrel{{\scriptstyle i\neq
j}}{{=}}3\mathcal{H}^{2}\nu\left(-6\sigma^{(1)}_{,ik}\sigma^{(1)}_{,jk}+4\sigma^{(1)}_{,ij}\triangle\sigma^{(1)}\right),$
$\displaystyle T^{(2)i\mu}_{\phantom{(2)i\mu};\mu}$ $\displaystyle=$
$\displaystyle\frac{\nu\overline{\rho}}{a^{2}}\left(\frac{4}{3}\triangle\sigma^{(1)}_{,i}\triangle\sigma^{(1)}-2\sigma^{(1)}_{,ij}\triangle\sigma^{(1)}_{,j}-2\sigma^{(1)}_{,ijk}\sigma^{(1)}_{,jk}\right)=0,$
(4.17)

where these relations are simplified with the use of the background solution
together with relation $\psi^{(1)}=\phi^{(1)}$ in the same way as the first
order equations, $ij$-part of the Einstein field equations is split into its
trace and traceless part, and equation $T^{(2)0\mu}_{\phantom{(2)0\mu};\mu}=0$
is automatically satisfied. Since $\phi^{(1)}=0$, the second order Einstein
field equations reduce to

$\displaystyle\triangle\psi^{(2)}-3\mathcal{H}\psi^{(2)\prime}=3\mathcal{H}^{2}\phi^{(2)},$
(4.18) $\displaystyle\mathcal{H}\phi^{(2)}+\psi^{(2)\prime}=f^{(2)},$ (4.19)
$\displaystyle\psi^{(2)\prime\prime}+\mathcal{H}\left(\phi^{(2)}+2\psi^{(2)}\right)^{\prime}+3\mathcal{H}^{2}\phi^{(2)}+\frac{1}{3}\triangle\left(\phi^{(2)}-\psi^{(2)}\right)=0,$
(4.20)
$\displaystyle\left(\phi^{(2)}-\psi^{(2)}\right)_{ij}\stackrel{{\scriptstyle
i\neq
j}}{{=}}3\nu\mathcal{H}^{2}\left(6\sigma^{(1)}_{,ik}\sigma^{(1)}_{,jk}-4\sigma^{(1)}_{,ij}\triangle\sigma^{(1)}\right),$
(4.21)

with $f^{(2)}$ denoting any function of only time. In other words, equations
for the second order scalar perturbations are not sourced by their first order
parts, because they vanish, and here we are neglecting first order vector and
tensor perturbations.

The second order momentum conservation law (4.17) imposes a restriction on the
first order perturbation $\sigma^{(1)}$ which, assuming that it is of the form
of a perturbation, is satisfied only if this perturbation vanishes. As a
consequence of vanishing $\sigma^{(1)}$ the system of equations for the second
order perturbations reduces to the same system of equations as for the first
order perturbations. This means that the metric perturbation $\phi$ vanishes
not only in the first order but also in the second order of the perturbation
theory.

The fourth order conservation law $T^{(4)i\mu}_{\phantom{(4)i\mu};\mu}=0$
contains the same combinations of $\sigma^{(2)}$ as combinations of
$\sigma^{(1)}$ appearing in the second order equation
$T^{(2)i\mu}_{\phantom{(4)i\mu};\mu}=0$, and in principle only such
combinations of other perturbations that follow the structure
$\phi^{(1)}\phi^{(3)}$, $\phi^{(2)2}$, $\sigma^{(1)}\phi^{(3)}$,
$\phi^{(1)}\sigma^{(3)}$ and $\sigma^{(1)}\sigma^{(3)}$, which are zero. This
implies the same restriction on $\sigma^{(2)}$ as on $\sigma^{(1)}$. In the
case of the perfect fluid no such restriction emerges, so that perturbation
$\sigma^{(1)}$ remains unspecified. This means that for the dark energy as a
perfect fluid the body element displacement $\sigma_{,i}$ does not affect the
evolution of the metric perturbation $\phi$, which turns out to be zero as
well, and entire solid matter formalism can be simply replaced by the standard
perfect fluid approach. On the other hand, in the case with dark energy as a
solid with zero shear stress and $\nu\neq 0$ this body element approach is
needed, and it helps to reveal that perturbation $\sigma$ vanishes.

Therefore, regardless of choice of the parameter $\nu$ there are no scalar
perturbations in the dark energy-like solid up to the second order of the
perturbation theory. Moreover, due to the absence of perturbation $\sigma$ in
the stress-energy tensor the dark energy perturbations can be omitted also in
more realistic multicomponent cosmological models, as long as there is no
direct interaction of the dark energy sector with other matter components of
the universe.

The analysis presented in this section shows that by considering only scalar
perturbations, presence of only dark energy in the expanding universe does not
allow for their formation up to the second perturbative order in both standard
dark energy model as well as the extended model studied here. The same is true
also for vector perturbations, however, tensor perturbations are formed
already in the linear order, where is no difference between the standard dark
energy and our extension of it. In both models the Fourier modes of tensor
perturbations in the first order of the perturbation theory are given by
Bessel functions of the order $3/2$, in particular $(-k\tau)J_{3/2}(-k\tau)$
or $(-k\tau)Y_{3/2}(-k\tau)$, which have nondecaying parts. Note that in the
context of slow-roll inflation, in which condition for pressure to energy
density ratio $w=-1$ must be violated in order to avoid eternal inflation and
allow formation of scalar perturbations, these modes are slightly modified by
slow-roll parameters.

In spite of lacking a difference in results for up to second order scalar
perturbations between our dark energy model and the standard dark energy
model, such difference can be pursued by investigating terms quadratic in
first order tensor perturbations in equations for second order perturbations,
where they play the role of source terms. In this paper we omitted this
possibility motivated by limit on tensor to scalar ratio suggested by
observations, so that in such context our calculations represent only the
leading order approximation. Another exciting possibility, already hinted
above, is to extend our dark energy model even more by considering nonminimal
coupling to other kinds of matter or to gravity. This would, of course,
complicate the analysis considerably.

##  5 Scalar perturbations with radiation-like solid

We now examine scalar perturbations in the case with $w=1/3$. Up to the first
order of the perturbation theory we only reproduce results known from
textbooks [3], but we also reveal the way in which parameter $\nu$ enters
formulas for second order perturbations.

The only nontrivial background equations are

$\displaystyle G^{(0)}_{00}=3\mathcal{H}^{2}=8\pi\kappa
T^{(0)}_{00}=8\pi\kappa a^{2}\overline{\rho},$ (5.1) $\displaystyle
G^{(0)}_{ij}=-\left(\mathcal{H}^{2}+2\mathcal{H}^{\prime}\right)\delta_{ij}=8\pi\kappa
T^{(0)}_{ij}=\frac{8}{3}\pi\kappa a^{2}\overline{\rho}\delta_{ij},$ (5.2)
$\displaystyle
T^{(0)0\mu}_{\phantom{(0)0\mu};\mu}=\frac{\overline{\rho}^{\prime}+4\mathcal{H}\overline{\rho}}{a^{2}},$
(5.3)

with

$\displaystyle a\propto\tau,\quad\overline{\rho}\propto\frac{1}{\tau^{4}},$
(5.4)

being solution of these equations, so that $\mathcal{H}=1/\tau$.

###  5.1 Linear perturbations

Using the background solution (5.4) and the fact that $\psi^{(1)}=\phi^{(1)}$
which follows from the fist order traceless spatial part of Einstein field
equations and has already been used twice in this paper, we can write all
nontrivial first order equations as

$\displaystyle
G^{(1)}_{00}=2\left(\triangle\phi^{(1)}-3\mathcal{H}\phi^{(1)\prime}\right)=8\pi\kappa
T^{(1)}_{00}=2\mathcal{H}^{2}\left(2\triangle\sigma^{(1)}+9\phi^{(1)}\right),$
(5.5) $\displaystyle
G^{(1)}_{0i}=2\left(\mathcal{H}\phi^{(1)}_{,i}+\phi^{(1)\prime}_{,i}\right)=8\pi\kappa
T^{(1)}_{0i}=4\mathcal{H}^{2}\sigma^{(1)\prime}_{,i},$ (5.6) $\displaystyle
G^{(1)}_{ij}=2\left(-2\mathcal{H}^{2}\phi^{(1)}+3\mathcal{H}\phi^{(1)\prime}+\phi^{(1)\prime\prime}\right)\delta_{ij}=$
(5.7) $\displaystyle\phantom{G^{(1)}_{ij}}=8\pi\kappa
T^{(1)}_{ij}=\mathcal{H}^{2}\left(\frac{4}{3}\triangle\sigma^{(1)}+2\phi^{(1)}\right)\delta_{ij}.$

By combining the $00$-part of the Einstein field equations with the $ij$-part
we can exclude the perturbation $\sigma^{(1)}$ and write down the equation for
the metric perturbation

$\displaystyle\phi^{(1)\prime\prime}+4\mathcal{H}\phi^{(1)\prime}-\frac{1}{3}\triangle\phi^{(1)}=0,$
(5.8)

and then supplement the $0i$-part with its solution to find the solution for
perturbation $\sigma^{(1)}$ as well. In this way we find Fourier modes

$\displaystyle\phi^{(1)}_{k}$ $\displaystyle=$
$\displaystyle\frac{c_{1}}{\tau^{2}}\left(\frac{\cos\zeta}{\zeta}+\sin\zeta\right)+\frac{c_{2}}{\tau^{2}}\left(\frac{\sin\zeta}{\zeta}-\cos\zeta\right),$
(5.9) $\displaystyle\sigma^{(1)}_{k}$ $\displaystyle=$ $\displaystyle
c_{1}\left(\frac{\cos\zeta}{\zeta}+\frac{\sin\zeta}{2}\right)+c_{2}\left(\frac{\sin\zeta}{\zeta}-\frac{\cos\zeta}{2}\right),\quad\zeta\equiv\frac{k\tau}{\sqrt{3}},$
(5.10)

where we see that the corresponding sound speed is $1/\sqrt{3}$ as expected
for radiation. Because of the Bianchi identity this solution also satisfies
the energy-momentum conservation law,

$\displaystyle
T^{(1)i\mu}_{\phantom{(1)i\mu};\mu}=\frac{4\overline{\rho}}{3a^{2}}\left(2\phi^{(1)}_{,i}+\frac{1}{3}\triangle\sigma^{(1)}_{,i}-\sigma^{(1)\prime\prime}_{,i}\right)=0.$
(5.11)

###  5.2 Second order perturbations

List of the needed equations for the second order perturbations is

$\displaystyle G^{(2)}_{00}$ $\displaystyle=$ $\displaystyle
2\left(\triangle\psi^{(2)}-3\mathcal{H}\psi^{(2)\prime}\right)+$
$\displaystyle+3\phi^{(1)}_{,i}\phi^{(1)}_{,i}+12\phi^{(1)}\triangle\phi^{(1)}-12\mathcal{H}\phi^{(1)}\phi^{(1)\prime}+3\phi^{(1)\prime
2}=$ $\displaystyle=$ $\displaystyle 8\pi\kappa
T^{(2)}_{00}=2\mathcal{H}^{2}\bigg{(}2\triangle\sigma^{(2)}+3\left(\phi^{(2)}+2\psi^{(2)}\right)+$
$\displaystyle+30\phi^{(1)2}+12\phi^{(1)}\triangle\sigma^{(1)}+\frac{4}{3}\left(\triangle\sigma^{(1)}\right)^{2}-\sigma^{(1)}_{,ij}\sigma^{(1)}_{,ij}+\sigma^{(1)\prime}_{,i}\sigma^{(1)\prime}_{,i}\bigg{)},$
$\displaystyle G^{(2)}_{0i}$ $\displaystyle=$ $\displaystyle
2\mathcal{H}\phi^{(2)}_{,i}+2\psi^{(2)\prime}_{,i}-$
$\displaystyle-4\mathcal{H}\phi^{(1)}\phi^{(1)}_{,i}+2\phi^{(1)}_{,i}\phi^{(1)\prime}+4\phi^{(1)}\phi^{(1)\prime}_{,i}=$
$\displaystyle=$ $\displaystyle 8\pi\kappa
T^{(2)}_{0i}=\mathcal{H}^{2}\bigg{(}4\sigma^{(2)\prime}_{,i}+$
$\displaystyle+8\phi^{(1)}\sigma^{(1)\prime}_{,i}+\frac{16}{3}\sigma^{(1)\prime}_{,i}\triangle\sigma^{(1)}-4\sigma^{(1)}_{,ij}\sigma^{(1)\prime}_{,ij}\bigg{)},$
$\displaystyle G^{(2)}_{ii}$ $\displaystyle=$ $\displaystyle
6\psi^{(2)\prime\prime}+6\mathcal{H}\left(\phi^{(2)}+2\psi^{(2)}\right)^{\prime}-$
$\displaystyle-6\mathcal{H}^{2}\left(\phi^{(2)}+\psi^{(2)}\right)+2\triangle\left(\phi^{(1)}-\psi^{(1)}\right)+$
$\displaystyle+24\mathcal{H}^{2}\phi^{(1)2}-7\phi^{(1)}_{,i}\phi^{(1)}_{,i}-8\phi^{(1)}\triangle\phi^{(1)}-$
$\displaystyle-60\mathcal{H}\phi^{(1)}\phi^{(1)\prime}-3\phi^{(1)\prime
2}-12\phi^{(1)}\phi^{(1)\prime\prime}=$ $\displaystyle=$ $\displaystyle
8\pi\kappa
T^{(2)}_{ii}=\mathcal{H}^{2}\bigg{(}4\triangle\sigma^{(2)}+6\psi^{(2)}+$
$\displaystyle+12\phi^{(1)2}+8\phi^{(1)}\triangle\sigma^{(1)}-2\sigma^{(1)}_{,ij}\sigma^{(1)}_{,ij}+\frac{8}{3}\left(\triangle\sigma^{(1)}\right)^{2}+2\sigma^{(1)\prime}_{,i}\sigma^{(1)\prime}_{,i}\bigg{)},$
$\displaystyle G^{(2)}_{ij}$ $\displaystyle\stackrel{{\scriptstyle i\neq
j}}{{=}}$
$\displaystyle-\left(\phi^{(2)}-\psi^{(2)}\right)_{,ij}+2\phi^{(1)}_{,i}\phi^{(1)}_{,j}+4\phi^{(1)}\phi^{(1)}_{,ij}\stackrel{{\scriptstyle
i\neq j}}{{=}}$ $\displaystyle\stackrel{{\scriptstyle i\neq j}}{{=}}$
$\displaystyle 8\pi\kappa T^{(2)}_{ij}\stackrel{{\scriptstyle i\neq
j}}{{=}}\mathcal{H}^{2}\bigg{[}-16\left(1+\frac{9}{8}\nu\right)\sigma^{(1)}_{,ik}\sigma^{(1)}_{,jk}+$
$\displaystyle+\frac{32}{3}\left(1+\frac{9}{8}\nu\right)\sigma^{(1)}_{,ij}\triangle\sigma^{(1)}+4\sigma^{(1)\prime}_{,i}\sigma^{(1)\prime}_{,j}\bigg{]},$
$\displaystyle T^{(2)0\mu}_{\phantom{(2)0\mu};\mu}$ $\displaystyle=$
$\displaystyle\frac{4\overline{\rho}}{3a^{2}}\left(-2\phi^{(1)}_{,i}\sigma^{(1)\prime}_{,i}-\frac{1}{3}\sigma^{(1)\prime}_{,i}\triangle\sigma^{(1)}_{,i}+\sigma^{(1)\prime}_{,i}\sigma^{(1)\prime\prime}_{,i}\right),$
(5.16) $\displaystyle T^{(2)i\mu}_{\phantom{(2)i\mu};\mu}$ $\displaystyle=$
$\displaystyle\frac{4\overline{\rho}}{3a^{2}}\bigg{[}\left(\phi^{(2)}+\psi^{(2)}\right)_{,i}+\frac{1}{3}\triangle\sigma^{(2)}_{,i}-\sigma^{(2)\prime\prime}_{,i}+12\phi^{(1)}\phi^{(1)}_{,i}+$
$\displaystyle+\frac{8}{3}\phi^{(1)}_{,i}\triangle\sigma^{(1)}+2\phi^{(1)}\triangle\sigma^{(1)}_{,i}+2\phi^{(1)\prime}\sigma^{(1)\prime}_{,i}-2\phi^{(1)}\sigma^{(1)\prime\prime}_{,i}+$
$\displaystyle+\frac{5}{3}\sigma^{(1)\prime}_{,ij}\sigma^{(1)\prime}_{,j}-\frac{1}{3}\sigma^{(1)\prime}_{,i}\triangle\sigma^{(1)\prime}+\sigma^{(1)}_{,ij}\sigma^{(1)\prime\prime}_{,j}-\frac{4}{3}\sigma^{(1)\prime\prime}_{,i}\triangle\sigma^{(1)}+$
$\displaystyle+\left(\frac{4}{3}+\nu\right)\triangle\sigma^{(1)}_{,i}\triangle\sigma^{(1)}-\left(\frac{5}{3}+\frac{3}{2}\nu\right)\sigma^{(1)}_{,ijk}\sigma^{(1)}_{,jk}-$
$\displaystyle-\left(\frac{4}{3}+\frac{3}{2}\nu\right)\sigma^{(1)}_{,ij}\triangle\sigma^{(1)}_{,j}\bigg{]}.$

In order to find the solution we can rewrite the system of equations as

$\displaystyle T^{(2)i\mu}_{\phantom{(2)i\mu};\mu,i}=0$
$\displaystyle\implies$
$\displaystyle\phi^{(2)}+\psi^{(2)}+\frac{1}{3}\triangle\sigma^{(2)}-\sigma^{(2)\prime\prime}=\mathcal{Q}_{1},$
(5.18) $\displaystyle G^{(2)}_{00}=8\pi\kappa T^{(2)}_{00}$
$\displaystyle\implies$
$\displaystyle\phi^{(2)}+2\psi^{(2)}+\frac{1}{\mathcal{H}}\psi^{(2)\prime}-$
$\displaystyle-\frac{1}{3\mathcal{H}^{2}}\triangle\psi^{(2)}+\frac{2}{3}\triangle\sigma^{(2)}=\mathcal{Q}_{2},$
$\displaystyle
G^{(2)\phantom{0}0}_{\phantom{(2)}0}+G^{(2)\phantom{i}i}_{\phantom{(2)}i}=0$
$\displaystyle\implies$
$\displaystyle\psi^{(2)\prime\prime}+\mathcal{H}\left(\phi^{(2)\prime}+3\psi^{(2)\prime}\right)+$
$\displaystyle+\frac{1}{3}\triangle\left(\phi^{(2)}-2\psi^{(2)}\right)=\mathcal{Q}_{3},$
$\displaystyle G^{(2)}_{0i,i}=8\pi\kappa T^{(2)}_{0i,i}$
$\displaystyle\implies$
$\displaystyle\frac{1}{\mathcal{H}^{2}}\psi^{(2)\prime}+\frac{1}{\mathcal{H}}\phi^{(2)}-2\sigma^{(2)\prime}=\mathcal{Q}_{4},$
(5.21)

with
$G^{(2)\phantom{0}0}_{\phantom{(2)}0}+G^{(2)\phantom{i}i}_{\phantom{(2)}i}=0$
being valid only when condition (1.1) is satisfied with radiation-like
pressure to energy density ratio $w=1/3$, and
$\mathcal{Q}_{1}$-$\mathcal{Q}_{4}$ denoting the source terms of these
equations,

$\displaystyle\mathcal{Q}_{1}$ $\displaystyle=$
$\displaystyle\triangle^{-1}\bigg{[}-12\phi^{(1)}_{,i}\phi^{(1)}_{,i}-12\phi^{(1)}\triangle\phi^{(1)}-$
$\displaystyle-\frac{8}{3}\triangle\phi^{(1)}\triangle\sigma^{(1)}-\frac{14}{3}\phi^{(1)}_{,i}\triangle\sigma^{(1)}_{,i}-2\phi^{(1)}\triangle^{2}\sigma^{(1)}-$
$\displaystyle-2\phi^{(1)\prime}_{,i}\sigma^{(1)\prime}_{,i}-2\phi^{(1)\prime}\triangle\sigma^{(1)\prime}+2\phi^{(1)}_{,i}\sigma^{(1)\prime\prime}_{,i}+2\phi^{(1)}\triangle\sigma^{(1)\prime\prime}-$
$\displaystyle-\frac{4}{3}\sigma^{(1)\prime}_{,i}\triangle\sigma^{(1)\prime}_{,i}-\frac{5}{3}\sigma^{(1)\prime}_{,ij}\sigma^{(1)\prime}_{,ij}+\frac{1}{3}\left(\triangle\sigma^{(1)\prime}\right)^{2}+$
$\displaystyle+\frac{1}{3}\sigma^{(1)\prime\prime}_{,i}\triangle\sigma^{(1)}_{,i}-\sigma^{(1)\prime\prime}_{,ij}\sigma^{(1)}_{,ij}+\frac{4}{3}\triangle\sigma^{(1)\prime\prime}\triangle\sigma^{(1)}-$
$\displaystyle-\left(\frac{4}{3}+\nu\right)\triangle\sigma^{(1)}\triangle^{2}\sigma^{(1)}+\frac{1}{2}\nu\triangle\sigma^{(1)}_{,i}\triangle\sigma^{(1)}_{,i}+$
$\displaystyle+\left(3+3\nu\right)\sigma^{(1)}_{,ij}\triangle\sigma^{(1)}_{,ij}+\left(\frac{5}{3}+\frac{3}{2}\nu\right)\sigma^{(1)}_{,ijk}\sigma^{(1)}_{,ijk}\bigg{]},$
$\displaystyle\mathcal{Q}_{2}$ $\displaystyle=$
$\displaystyle-10\phi^{(1)2}-\frac{2}{\mathcal{H}}\phi^{(1)}\phi^{(1)\prime}+\frac{1}{2\mathcal{H}^{2}}\phi^{(1)\prime
2}+\frac{1}{2\mathcal{H}^{2}}\phi^{(1)}_{,i}\phi^{(1)}_{,i}+\frac{2}{\mathcal{H}^{2}}\phi^{(1)}\triangle\phi^{(1)}-$
$\displaystyle-4\phi^{(1)}\triangle\sigma^{(1)}-\frac{4}{9}\left(\triangle\sigma^{(1)}\right)^{2}+\frac{1}{3}\sigma^{(1)}_{,ij}\sigma^{(1)}_{,ij}-\frac{1}{3}\sigma^{(1)\prime}_{,i}\sigma^{(1)\prime}_{,i},$
$\displaystyle\mathcal{Q}_{3}$ $\displaystyle=$ $\displaystyle\phi^{(1)\prime
2}+\frac{5}{3}\phi^{(1)}_{,i}\phi^{(1)}_{,i}+\frac{8}{3}\phi^{(1)}\triangle\phi^{(1)}+4\mathcal{H}\phi^{(1)}\phi^{(1)\prime},$
(5.24) $\displaystyle\mathcal{Q}_{4}$ $\displaystyle=$
$\displaystyle\triangle^{-1}\bigg{[}\frac{2}{\mathcal{H}}\phi^{(1)}_{,i}\phi^{(1)}_{,i}+\frac{2}{\mathcal{H}}\phi^{(1)}\triangle\phi^{(1)}-\frac{1}{\mathcal{H}^{2}}\triangle\phi^{(1)}\phi^{(1)\prime}-$
$\displaystyle-\frac{3}{\mathcal{H}^{2}}\phi^{(1)}_{,i}\phi^{(1)\prime}_{,i}-\frac{2}{\mathcal{H}^{2}}\phi^{(1)}\triangle\phi^{(1)\prime}+4\phi^{(1)}_{,i}\sigma^{(1)\prime}_{,i}+4\phi^{(1)}\triangle\sigma^{(1)\prime}+$
$\displaystyle+\frac{8}{3}\triangle\sigma^{(1)\prime}\triangle\sigma^{(1)}+\frac{2}{3}\sigma^{(1)\prime}_{,i}\triangle\sigma^{(1)}_{,i}-2\sigma^{(1)\prime}_{,ij}\sigma^{(1)}_{,ij}\bigg{]}.$

Here we have assumed that the source terms are dominated by scalar
perturbations. By combining equation (5.18) with the conformal time derivative
of (5.2) we find

$\displaystyle\frac{1}{3}\triangle\sigma^{(2)}=\mathcal{Q}_{1}-\frac{1}{2}\mathcal{Q}_{4}^{\prime}-\frac{1}{2}\phi^{(2)}-\psi^{(2)}+\frac{1}{\mathcal{H}}\left(\frac{1}{2}\phi^{(2)\prime}+\psi^{(2)\prime}\right)+\frac{1}{2\mathcal{H}^{2}}\psi^{(2)\prime\prime}.$
(5.26)

By combining this equation with (5.2) we obtain

$\displaystyle-\psi^{(2)\prime\prime}-\mathcal{H}\left(\phi^{(2)\prime}+3\psi^{(2)\prime}\right)+\frac{1}{3}\triangle\psi^{(1)}=\mathcal{H}^{2}\left(2\mathcal{Q}_{1}-\mathcal{Q}_{2}-\mathcal{Q}_{4}^{\prime}\right),$
(5.27)

and after adding this equation to (5.2) we find the relation

$\displaystyle\phi^{(2)}-\psi^{(2)}=3\triangle^{-1}\left[\mathcal{Q}_{3}+\mathcal{H}^{2}\left(2\mathcal{Q}_{1}-\mathcal{Q}_{2}-\mathcal{Q}_{4}^{\prime}\right)\right].$
(5.28)

By inserting $\phi^{(2)}$ from this equation back into (5.2) we have

$\displaystyle\psi^{(2)\prime\prime}+4\mathcal{H}\psi^{(2)\prime}-\frac{1}{3}\triangle\psi^{(2)}=\mathcal{Q}_{0},$
(5.29)

where

$\displaystyle\mathcal{Q}_{0}=-\mathcal{H}^{2}\left(2\mathcal{Q}_{1}-\mathcal{Q}_{2}-\mathcal{Q}_{4}^{\prime}\right)-3\mathcal{H}\triangle^{-1}\left[\mathcal{Q}_{3}+\mathcal{H}\left(2\mathcal{Q}_{1}-\mathcal{Q}_{2}-\mathcal{Q}_{4}^{\prime}\right)\right]^{\prime}.$
(5.30)

Finally, the solution is

$\displaystyle\psi^{(2)}_{k}$ $\displaystyle=$
$\displaystyle-\frac{1}{\tau^{2}}\left(\frac{\cos\zeta}{\zeta}+\sin\zeta\right)\int\limits\mathcal{Q}_{0k}\frac{\tau^{3}}{\zeta}\left(\frac{\sin\zeta}{\zeta}-\cos\zeta\right)d\tau+$
$\displaystyle+\frac{1}{\tau^{2}}\left(\frac{\sin\zeta}{\zeta}-\cos\zeta\right)\int\limits\mathcal{Q}_{0k}\frac{\tau^{3}}{\zeta}\left(\frac{\cos\zeta}{\zeta}+\sin\zeta\right)d\tau,$

with $\mathcal{Q}_{0k}$ denoting Fourier image of $\mathcal{Q}_{0}$. Functions
$\phi^{(2)}$ and $\sigma^{(2)}$ then can be found with the use of this
solution and relations (5.28) and (5.26).

##  6 Summary and outlook

We have studied solid matter filling a flat FLWR universe with shear stress
Lamé parameter, or shear modulus, set to zero. Behaviour of such matter
departs from the case with perfect fluid only in nonlinear perturbative
regime. In order to focus on the most physically interesting models we
restricted ourselves to two cases, radiation-like solid with pressure to
energy density ratio $w=1/3$, and dark energy-like solid with $w=-1$.
Additionally, we have imposed validity of condition for time-time component
and space part trace of the mixed stress-energy tensor,
$\widetilde{w}=-3T_{0}^{\phantom{0}0}/T_{i}^{\phantom{i}i}=$ const. (1.1), in
arbitrary coordinates, and restricted the parameter space of the solid matter
model even more.

From the technical point of view, when restricting the parameter space of the
solid we have used only the condition (1.1) assumed to be valid in arbitrary
coordinates, because, as we have shown, it implies the above mentioned values
of the pressure to energy density ratio. This condition is inspired by the
fact, that there are only two kinds of perfect fluid for which it is satisfied
not only in its local rest frame, where $\widetilde{w}=w$ automatically, but
also in arbitrary coordinates, namely radiation and dark energy. In the case
with radiation it is consequence of traceless stress-energy tensor in the
Maxwell theory, $T_{\mu}^{\phantom{\mu}\mu}=0$, and in the case of dark energy
it follows from proportionality of the stress-energy tensor to the space-time
metric, $T_{\mu\nu}\propto g_{\mu\nu}$.

The second order perturbative analysis of the condition (1.1) assumed to be
valid in arbitrary coordinates leads to only two allowed values of the
pressure to energy density ratio, $w=1/3$ and $w=-1$, the same as for
radiation and dark energy as perfect fluids. Therefore, by passing from the
perfect fluid to solid matter with zero shear modulus we do not obtain another
physically interesting value of $w$, but there remains one additional
parameter $\nu$ which differentiates the solid with zero shear modulus from
the perfect fluid, and it can be arbitrary. Coefficients $w$ and $\nu$ then
parametrize the equation of state of the solid as in (3.1). For
$\nu=-2(w+1)/3$ the studied matter reduces to the special case coinciding up
to the third order with the perfect fluid.

In the second part of the paper, in sections 4 and 5, we have checked that in
the dark energy-like case with $w=-1$ no pure scalar perturbations up to
second order can be formed, not only in the case of perfect fluid, but also
for arbitrary value of the coefficient $\nu$. With $\nu\neq 0$ the stress-
energy tensor for the dark energy-like solid is no longer proportional to the
space-time metric as in the perfect fluid case with $\nu=0$, although the
condition (1.1) remains valid in arbitrary coordinates. Absence of scalar
perturbations then had to be proven through analysis of equations for
perturbations. In the same way, we have analysed scalar perturbations also in
the radiation-like case with $w=1/3$. In order to simplify the analysis we
omitted the effect of other than scalar perturbations. Such simplification is
reasonable in our universe, as long as we are not interested in equations for
second order vector and scalar perturbations which are sourced by first order
perturbations. The reason is that the first order vector perturbations decay
and observations suggest small tensor to scalar ratio. Consequently, the main
contribution to observable effects comes from the scalar perturbations.

Concepts studied in this paper including both dark energy-like and radiation-
like solids can be applied to cosmic inflation and reheating which follows it.
To include the dark energy-like solid into the inflationary context, one has
to slightly depart from the value of pressure to energy density ratio $w=-1$
by the slow-roll parameter, $w=-1+2\epsilon/3$, where
$\epsilon=-\dot{H}/H^{2}$. This work has been already done with the model of
solid inflation [16, 17] and in works following it [18, 19]. After the end of
inflation, during reheating, energy of inflaton fields is transferred to the
radiation. It is in principle possible that matter with properties similar to
solid matter studied in our work can be either formed during the reheating, or
it can be a remnant of the solid inflation or its generalizations. The model
of radiation-like case of the solid then could be partially applicable towards
the end of reheating and beginning of the radiation dominated era.

The model analysed in this paper can be studied further. This may include:

  * •

Investigating effect of linear order tensor perturbations, which are formed
also in the dark energy-like case, on evolution of second order perturbations;

  * •

relaxing the validity of condition (1.1) in arbitrary coordinates, which have
led to only two values of pressure to energy density ratio ( i.e. studying
cases with arbitrary $w$);

  * •

studying even higher orders of the perturbation theory, where one has to
define more coefficients parametrizing the equation of state of the solid, and
investigate constraints on them;

  * •

considering more general space-time metrics including also space-times beyond
a cosmological context, which may require also a non-perturbative approach.

We leave this for future work.

## Acknowledgements

I would like to thank Vladimír Balek for discussions, and pointing out a
mistake in a formula in the previous version of the paper (relations (G.4) and
(G.6) in this version).

##  G Mixed stress-energy tensor of perfect fluid

By definition a perfect fluid is isotropic in its local rest frame. It is then
fully specified by the equation of state,

$\displaystyle p=w\rho,$ (G.1)

which relates its local rest frame pressure $p$ to its local rest frame energy
density $\rho$. Stress-energy tensor of a perfect fluid is given by relation

$\displaystyle T_{\mu\nu}=\left(\rho+p\right)u_{\mu}u_{\nu}+pg_{\mu\nu},$
(G.2)

where $u^{\mu}=dx^{\mu}/d\tau$ is 4-velocity of the fluid elements with $\tau$
denoting their proper time. Due to normalization, $u_{\mu}u^{\mu}=-1$, its
zero component $u^{0}$ depends on its space part $u^{i}$, which is given by
the space velocity $v^{i}$,

$\displaystyle u^{i}=\frac{dx^{i}}{dx^{0}}\frac{dx^{0}}{d\tau}=v^{i}u^{0}.$
(G.3)

Thus, we can write

$\displaystyle u^{0}=\frac{1}{\sqrt{-g_{00}-2g_{0i}v^{i}-g_{ij}v^{i}v^{j}}}.$
(G.4)

The time-time component and the space part trace of the mixed stress-energy
tensor in arbitrary coordinates can be written as

$\displaystyle T_{0}^{\phantom{0}0}$ $\displaystyle=$
$\displaystyle\rho\left[-1+\left(w+1\right)\xi\right],$ (G.5) $\displaystyle
T_{i}^{\phantom{i}i}$ $\displaystyle=$
$\displaystyle\rho\left[3w-\left(w+1\right)\xi\right],$

where $\xi=-u_{i}u^{i}=1+u_{0}u^{0}$, which can be expressed as

$\displaystyle\xi=\frac{g_{0i}v^{i}+g_{ij}v^{i}v^{j}}{g_{00}+2g_{0i}v^{i}+g_{ij}v^{i}v^{j}}.$
(G.6)

Now we can check facts about $T_{0}^{\phantom{0}0}$ to $T_{i}^{\phantom{i}i}$
ratio stated in section 1. For the radiation with $w=1/3$ relations in (G.5)
imply $T_{0}^{\phantom{0}0}=-T_{i}^{\phantom{i}i}=\rho(-1+4\xi/3)$, and for
the dark energy with $w=-1$ we have
$T_{0}^{\phantom{0}0}=T_{i}^{\phantom{i}i}/3=-\rho$. In both cases the
condition (1.1) is satisfied in arbitrary coordinates, while for other choices
of pressure to energy density ratio relations in (G.5) imply that it does not.
This is obvious from expression of the quantity $\widetilde{w}$ defined in
(1.1) through quantities introduced here,

$\displaystyle\widetilde{w}=\frac{3w-\left(w+1\right)\xi}{3-3\left(w+1\right)\xi}.$
(G.7)

If $w\neq 1/3$ and $w\neq-1$ the relation (1.1) holds only in the local rest
frame of the fluid, in which $\xi=0$, and $\widetilde{w}=w$ automatically. For
example in the case with dust, $w=0$, and

$\displaystyle\left.\frac{T_{0}^{\phantom{0}0}}{T_{i}^{\phantom{i}i}}\right|_{w=0}=\frac{1}{\xi}-1\neq\textrm{const}.$
(G.8)

In conclusion, the condition (1.1) is satisfied for any perfect fluid with
arbitrary pressure to energy density ratio, but only in its local rest frame,
while validity of this condition in any coordinates is related to only two
cases, $w=1/3$ and $w=-1$.

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