# Systematics of the low-energy electric dipole strength in the Sn isotopic
chain

M. Markova<EMAIL_ADDRESS>P. von Neumann-Cosel<EMAIL_ADDRESS>darmstadt.de E. Litvinova<EMAIL_ADDRESS>
###### Abstract

We present a systematic study of the mass dependence of the low-energy
electric dipole strength (LEDS) in Sn isotopes in the range $A=111-124$ based
on data obtained with the Oslo method and with relativistic Coulomb excitation
in forward-angle ($p,p^{\prime}$) scattering. The combined data cover an
energy range of $2-20$ MeV which permits, with minimal assumptions, a
decomposition of the total strength into the contribution from the low-energy
tail of the isovector giant dipole resonance (IVGDR) and possible resonance-
like structures on top of it. In all cases, a resonance peaked at about 8.3
MeV is observed, exhausting an approximately constant fraction of the Thomas-
Reiche-Kuhn (TRK) sum rule with a local maximum at 120Sn which might be
related to shell structure effects. For heavier isotopes ($A\geq 118$) a
consistent description of the data requires the inclusion of a second
resonance centered at 6.5 MeV, representing the isovector response of the
pygmy dipole resonance (PDR). Its strength corresponds to a small fraction of
the total LEDS only and shows an approximately linear dependence on mass
number. The experimental results are also compared to ab initio-based
microscopic calculations to investigate the importance of an inclusion of
quasiparticle vibration coupling (qPVC) for a realistic description of the
LEDS.

###### keywords:

111-124Sn , Oslo method , ($p,p^{\prime}$) scattering , Low-energy electric
dipole strength

††journal: Physics Letters B

[a]organization=Department of Physics, University of Oslo, city=N-0316 Oslo,
country=Norway

[b]organization=Institut fuer Kernphysik, Technische Universitaet Darmstadt,
city=D-64289 Darmstadt, country=Germany

[c]organization=Department of Physics, Western Michigan University,
city=Kalamazoo, postcode=Michigan 49008, country=USA [d]organization=National
Superconducting Cyclotron Laboratory, Michigan State University, city=East
Lansing, postcode=Michigan 48824, country=USA [e]organization=GANIL, CEA/DRF-
CNRS/IN2P3, city=F-14076 Caen, country=France

##  1 Introduction

The observation of a resonance-like structure in the electric dipole response
of heavy nuclei at energies around or below the neutron threshold, commonly
termed pygmy dipole resonance (PDR), has been a topic leading to considerable
experimental and theoretical activities in recent years [1, 2, 3, 4]. The
interest has been triggered by attempts to interpret the underlying structure
and also investigate its impact on the cross sections of ($n,\gamma$)
reactions relevant to a deeper understanding of the nucleosynthesis of heavy
elements [5]. Qualitatively, all mean-field-based models predict the
appearance of such a mode, however, with a broad range of strengths and energy
centroids depending on a chosen interaction, a model space, and possible
extensions beyond the random phase approximation (RPA) level.

Many theoretical studies interpret the PDR to arise from neutron skin
oscillations, which implies a dependence of the PDR strength on neutron excess
[6, 7, 8, 9, 10, 11, 12]. Accordingly, some models predict a correlation of
the PDR strength with neutron skin thickness, which in turn suggests that
information on the symmetry energy parameters of neutron-rich matter can be
derived (see e.g. [7, 9]). These claims, however, have been questioned [13,
14, 15, 16, 17]. The neutron skin oscillation interpretation has been mainly
based on a specific form of transition densities in the energy region of the
PDR with an approximately isoscalar (IS) radial dependence in the interior and
a peak of the neutron density at the surface. The dominance of the neutron
character of the $E1$ transitions forming the PDR is experimentally confirmed
by studies of the IS response with $(\alpha,\alpha^{\prime}\gamma$) [18, 19]
and (${}^{17}{\rm O},^{17}{\rm O}^{\prime}\gamma)$ [20, 21] reactions.
However, Ref. [22] points out that all experimental signatures of the PDR,
viz., large ground-state branching ratios, large isovector (IV) strengths
(with respect to average B(E1) transition strengths at low excitation
energies), significant IS strengths, as well as the characteristic form of the
transition density in heavy nuclei, are consistent with an interpretation as a
low-energy IS toroidal mode. The recent first experimental demonstration [23]
of the toroidal nature of low-energy $E1$ transitions with all the
experimental signatures quoted above, albeit in a $N\approx Z$ nucleus, and
successful description of these data by models predicting a dominantly
toroidal nature of the PDR further challenge the neutron skin oscillation
picture.

The best suited case for a systematic investigation of a dependence of PDR
properties on neutron excess is the Sn isotope chain between $A=100$ and 132.
Despite covering a variation from neutron shell closure to mid-shell, the
proton shell closure stabilizes the g.s. features leading to similar low-
energy structure along the chain. Thus, theoretical work has been focused on
the Sn chain, attempting to predict the evolution of the PDR photoabsorption
cross sections with neutron excess in order to test a possible relation to the
neutron skin thickness [6, 7, 8, 9, 10, 11, 12, 13, 17]. However, many of
these studies use a summed strength below some cutoff energy to represent the
PDR photoabsorption cross sections, despite this approach not being supported
by the data. Experimental studies of the IS component of the PDR have
demonstrated an isospin splitting (for 124Sn see [19, 21, 24]), where good
correspondence between transitions excited in the IS and IV response is
observed in a confined energy region at lower excitation energies, while at
higher excitation energies the IV response dominates.

Furthermore, a comparison of photoabsorption cross sections in
112,116,120,124Sn [25, 26] deduced from Coulomb excitation in forward-angle
$(p,p^{\prime})$ scattering [27] and from the $(\gamma,\gamma^{\prime})$
reaction [28, 29] shows good agreement in the energy region where the IS
strength was also found, but large differences at higher excitation energies.
The strengths deduced from the $(\gamma,\gamma^{\prime})$ experiments depend
on the g.s. branching ratio [30]. The dramatic reduction compared to the
$(p,p^{\prime})$ results points to a complex structure of the excited states
with small g.s. branching ratios, suggesting that they rather belong to the
low-energy tail of the IVGDR. Thus, a significant part of the low-energy
photoabsorption strength is not related to the PDR. This interpretation has
recently been confirmed by a new 120Sn$(\gamma,\gamma^{\prime})$ experiment
[31], where the contributions from the statistical decay were additionally
extracted.

Here we present a systematic study of the LEDS in Sn isotopes with mass
numbers from 111 to 124. It is based on a new set of Gamma Strength Functions
(GSFs) from $\gamma$ decay after light-ion-induced compound reactions (the so-
called Oslo method) [32, 33, 34] combined with a recent study of stable even-
mass isotopes with relativistic Coulomb excitation in the $(p,p^{\prime})$
reaction [25, 35]. The GSFs are directly related to the photoabsorption cross
sections, provided the Brink-Axel hypothesis [36, 37] also holds for the LEDS.
This seems to be the case for Sn isotopes [32, 33, 34]. The combined data sets
allow a decomposition of the LEDS with minimal assumptions, providing
information on the evolution of the PDR strength and other components with
neutron excess. The extraction of the GSFs from the Oslo data and a critical
investigation of the assumptions underlying the decomposition and possible
variants thereof are discussed in detail in Ref. [38]. The experimental GSFs
are also compared to ab initio-based microscopic calculations to investigate
the importance of an inclusion of quasiparticle vibration coupling (qPVC) for
a description of the LEDS.

##  2 Experimental data

GSFs extracted with the Oslo method are available for 111-113,116-122,124Sn.
They typically cover a $\gamma$ energy range from about 2 MeV to $1-2$ MeV
below the respective neutron thresholds ($S_{n}$). The results for 111-113Sn,
116Sn, and 118Sn were obtained with a custom-designed Si telescope ring (SiRi)
[39] combined with the CACTUS $\gamma$ ball [40] made of 28
5${}^{\prime\prime}\times 5^{\prime\prime}$ NaI detectors. New experiments
were performed for 117,119,120,124Sn using the OSCAR detector system [41, 42],
consisting of 30 large-volume LaBr $\gamma$ detectors providing superior
efficiency, timing, and energy resolution compared to the previous
experimental setup. Therefore, the new data for 117,119Sn supersede previous
results. Details of the consistent extraction of nuclear level densities and
GSFs are described in Ref. [38].

An independent set of GSFs for the even-mass isotopes 112-120,124Sn is
available from relativistic Coulomb excitation in the $(p,p^{\prime})$
reaction at extreme forward angles [25]. They agree in all cases with the Oslo
method results within the respective error bars. This implies that the Brink-
Axel hypothesis underlying the Oslo method and controversially discussed for
the LEDS (see e.g. Refs. [43, 44, 45, 46]) holds in case of the Sn isotopes.
Data from the latter experiment are available in an energy range $6-20$ MeV,
covering the major part of the IVGDR and providing sufficient overlap with the
Oslo method results. It should be noted that previous photoabsorption
experiments using the $(\gamma,xn)$ reaction show considerable scatter in the
systematics of IVGDR parameters. In contrast, the results of Ref. [25] provide
centroid energies in line with empirical systematics and an almost constant
width, as expected from the similar g.s. structure. In the energy region close
to $S_{n}$, where discrepancies with older data are particularly pronounced,
they are also in good agreement with new ($\gamma,n$) experiments [47, 48]
using monoenergetic photon beams from laser Compton back-scattering.

##  3 Decomposition of the low-energy electric dipole strength

Figure 1:  Decomposition of the experimental GSF of 124Sn. The Oslo data (blue
data points and error) are shown together with the $(p,p^{\prime})$ results
(orange data points and error band)[25]. The fit of the IVGDR with the GLO
model, Eq. (3), is presented as solid blue line. The low-lying $E1$ components
are shown as shaded red and violet areas. The $M1$ data from the (p,p′)
experiment [25] and the corresponding Lorentzian fit are given by the green
data points and the dashed line, respectively. The upbend at low $\gamma$
energies is described by the grey dashed line. The total fit is shown as solid
magenta line.

A decomposition of the LEDS is performed for the example of 124Sn illustrated
in Fig. 1. The following contributions to the total GSF are considered: a low-
energy tail of the IVGDR, a spin-flip M1 resonance, an upbend at very low
energies and one or two (when demanded by the data) E1 resonances. The
generalized Lorentzian model (solid blue line in Fig. 1) is chosen to describe
the IVGDR part [49]

$\displaystyle f_{\rm IVGDR}(E_{\gamma})=$
$\displaystyle\frac{1}{3\pi^{2}\hbar^{2}c^{2}}\sigma_{\rm IVGDR}\Gamma_{\rm
IVGDR}$ $\displaystyle\times\biggl{[}E_{\gamma}\frac{\Gamma_{\rm
KMF}(E_{\gamma},T_{f})}{(E_{\gamma}^{2}-E_{\rm
IVGDR}^{2})^{2}+E_{\gamma}^{2}\Gamma_{\rm KMF}^{2}(E_{\gamma},T_{f})}$
$\displaystyle+0.7\frac{\Gamma_{\rm KMF}(E_{\gamma}=0,T_{f})}{E_{\rm
IVGDR}^{3}}\biggr{]}$ (1)

with $E_{\rm IVGDR}$, $\Gamma_{\rm IVGDR}$, $\sigma_{\rm IVGDR}$ being the
IVGDR centroid energy, width, and maximum cross section, respectively.
$\Gamma_{KMF}$ corresponds to a temperature-dependent width in the Kadmenskii-
Markushev-Furman model [50], where $T_{f}$ refers to final states. It is the
only empirical or microscopic model capable of accounting simultaneously for
the low-energy flank of the IVGDR and a relatively flat strength distribution
at very low energies ($2-4$ MeV).

Data on the M1 spin-flip resonance are available (green points) in the energy
region from 6 to about 12 MeV from Ref. [25] and show broad distributions.
They are parameterized by a single Lorentzian (dashed green line). The upbend
at very low energies is described by exponential functions (dashed gray line).
The determination of parameters is described in Ref. [38], but its details
have no influence on the present results. Finally, additional resonances on
top of the tail of the IVGDR are assumed to be Gaussian (shaded red and violet
areas). All parameters are determined by a simultaneous fit (dark red line) to
the combined Oslo (blue points and error band) and $(p,p^{\prime})$ (orange
points and error band) data described above. An in-depth discussion of the
decomposition is provided in Ref. [38].

Figure 2:  Systematics of the total LEDS integrated over the energy region
$4-10$ MeV and its decompostion into the contributions from the tail of the
IVGDR and one or two (for masses $\geq 118$) resonances on top. (a) Strengths
in % of the TRK sum rule. (b) centroid energies.

Figure 2 presents the evolution of the LEDS and its various components with
mass number when integrating within the $4-10$ MeV energy region. The total
LEDS is roughly constant (with slight indications of a local maximum at 120Sn)
with values ranging from 3.5% to 4% of the classical Thomas-Reiche-Kuhn (TRK)
sum rule. The contribution of the low-energy tail of the IVGDR is found to be
approximately constant as well as exhausting a TRK value of about 1.5%. The
strength on top of the tail of the IVGDR in the isotopes with $A\leq 118$ is
well parameterized by a single Gaussian function. For $A\geq 118$ the
$(p,p^{\prime})$ data demand the inclusion of a second peak at lower energies,
which is particularly pronounced in 124Sn (cf. Fig. 1). The dominant peak is
centered at about 8.3 MeV and its average strength is 2% of the TRK value.
Small variations are visible as a function of mass number with a possible
strength maximum at 120Sn. A similar local maximum of the LEDS at 120Sn has
been discussed e.g. in Refs. [1, 6, 51].

The centroid of the lower-energy peak lies at about 6.5 MeV with no energy
dependence within uncertainties. Its strength is small, ranging from 0.1% for
118Sn to a maximum of 0.5% of the TRK sum rule in 124Sn. Interestingly, this
contribution shows an approximately linear dependence on mass number (or
neutron excess). When extrapolated to lower masses, its predicted strength is
too small to be distinguished from the broad peak at higher energies. Studies
of the isoscalar E1 strength in 124Sn with the
$(\alpha,\alpha^{\prime}\gamma$) [19, 24] and (${}^{17}{\rm O},^{17}{\rm
O}^{\prime}\gamma)$ [21] reactions show a concentration between 5.5 and 7 MeV,
consistent with the properties of the lower-energy peak. Furthermore, studies
of the 112,116,120,124Sn$(\gamma,\gamma^{\prime})$ reaction [28, 29] typically
find the strongest transitions between 6 and 7 MeV, indicating large g.s.
branching ratios. These experimental signatures point towards an
interpretation of the lower-energy peak as the IV response of the PDR.

These numbers are significantly smaller than quoted in most theoretical
studies of the dependence of the PDR strength on neutron excess in the Sn
isotope chain (for a notable exception see Ref. [8]). They challenge an
interpretation of the PDR as neutron skin oscillation, which implies some
degree of collectivity. Of course, the distinction of strength belonging
either to the PDR or to the tail of the IVGDR is schematic. In reality, there
is some degree of mixing between transitions of different types. All QRPA
calculations agree on a gradual change of the radial transition densities from
an approximate IS behavior in the interior of the nucleus and a pronounced
peak of the neutron density at the surface (a signature of the PDR) to more
IV-type transitions at higher energies. However, for the confined energy
region where the PDR strength is located, effects on the IV response due to
variations of the transition densities are expected to be small.

The discrepancy probably arises in many cases from the fact that QRPA results
typically produce a single low-energy peak and its strength is assumed to
represent the PDR strength. The present work and experiments comparing the IS
and IV response on the same nucleus demonstrate that this is not correct. The
PDR strength exhausts a rather small fraction of the LEDS only (at most about
15% for the Sn isotopes studied here). Thus, quantitative predictions based on
QRPA have to be taken with some care, and one might have to go beyond it by
including complex configurations to achieve realistic low-energy strength
distributions. This is discussed in the next section.

##  4 Comparison with ab initio-based microscopic calculations

In the following we compare the experimental GSFs with ab initio-based
microscopic calculations using 120Sn as a representative example. An extensive
discussion of the models and a comparison with all experimental results can be
found in Ref. [38]. Nuclear response theory can be consistently derived in the
model-independent ab initio equation of motion (EOM) framework [52, 53, 54]
for the in-medium two-time two-fermion propagators. In superfluid media,
particle-hole ($ph$), particle-particle ($pp$), and hole-hole ($hh$)
propagators can be conveniently unified in one two-quasiparticle ($2q$)
propagator without loss of generality [55].

While the generic EOM for the correlated (four-time) two-fermion propagator is
the Bethe-Salpeter equation, the two-time character of the response leads to
the EOM of a Dyson form, which depends on a single time difference. The
interaction kernel of the Bethe-Salpeter-Dyson equation, before taking any
approximation, decomposes into the static and dynamical (time-dependent)
components. In the energy domain, they are represented by the energy-
independent and energy-dependent terms, respectively. The energy-dependent
contribution generates long-range correlations while making an impact on their
short-range static counterpart. Since the $2q$ EOM couples to a growing
hierarchy of higher-rank EOMs via the dynamical kernel, in practical
applications it is decoupled by making approximations with varying correlation
content.

The simplest decoupling scheme retains only the static kernel and is known as
the quasiparticle random phase approximation (QRPA). The heart of the
dynamical kernel is the four-fermion $4q$ fully correlated propagator
contracted with two interaction matrix elements. The approximation keeping the
leading effects of emergent collectivity in the $4q$ propagator reduces it to
$2q\otimes phonon$ configurations, where the $phonon$ represents correlated
$2q$ pairs (vibrations). This approach reproduces the phenomenological
(relativistic) quasiparticle time-blocking approximation ((R)QTBA) [56, 57] if
the bare interaction between nucleons is replaced by their effective
interaction. However, in the ab initio EOM method, the quasiparticle-vibration
coupling (qPVC) amplitudes are derived consistently from the underlying
interaction, while in (R)QTBA the qPVC self-energy is used as an input. The
EOM works with the two-time propagators from the beginning and hence does not
employ time blocking. Furthermore, the ab initio relativistic EOM (REOM)
framework links the nuclear response theory to the underlying scale of
particle physics and fundamental interactions, in the present case, to the
meson-exchange interaction and, more importantly, enables extensions of the
$4q$ dynamical kernel to more complex configurations [55].

Figure 3:  Comparison of experimental and calculated GSFs computed with an
artifical width of 200 keV for 120Sn in the energy regions (a) $4-22$ MeV and
(b) $3-10$ MeV. The blue and orange bands indicate the Oslo-method and
$(p,p^{\prime})$ data. The RQRPA and RQTBA calculations described in the text
are shown as magenta and violet lines, respectively.

The cluster decomposition of this kernel identifies the next-level complexity
non-perturbative approximation as the $2q\otimes 2phonon$. The implementation
of such configurations is becoming gradually possible with the increasing
computational capabilities [54, 58] and will be applied for systematic
calculations in the near future. While implementations with bare interactions
are not yet available, (R)EOM admits realistic implementations that employ
effective interactions adjusted in the framework of the density functional
theory. For such interactions, the qPVC in the dynamical kernel can be
combined with subtraction, restoring the self-consistency of the ab initio
framework [59], while reasonable phonons can be obtained already on the QRPA
level.

In this work, the effective meson-exchange interaction NL3* [60] was employed.
This interaction has a transparent link to particle physics, while the meson
masses and coupling constants are only slightly different from their vacuum
values (cf., for instance, Ref. [61]). The NL3* is an upgraded version of the
previously developed NL3 parametrization [62], which includes self-
interactions in the scalar $\sigma$ meson sector [63]. The self-interactions
represent three and four $\sigma$ meson vertices, i.e., concrete physical
processes that occur in correlated media. The NL3 and NL3* ansätze are
separable in the momentum representation, which speeds up the computation
considerably. With these interactions, the realization of the REOM approach on
the $2q\otimes phonon$ level technically corresponds to RQTBA, so this name is
retained in the present work.

Results of the RQRPA and RQTBA calculations for the electric dipole strength
in 120Sn are displayed in Fig. 3 together with the data. The ($p,p^{\prime}$)
results in the energy region of the IVGDR (orange band in Fig. 3(a)), are
reasonably well described by both approaches. The successful reproduction of
the gross features by the RQRPA calculations highlights the importance of
Landau damping for an understanding of IVGDR widths in heavy nuclei [64]. An
expanded view of the low-energy region is given in Fig. 3(b), where
significant differences between the RQRPA and RQTBA results are apparent. With
RQRPA, the LEDS is concentrated in a single transition at 8.6 MeV. The
strength at lower energies is an artefact of the 200 keV width in the
calculations only. Inclusion of qPVC provides a satisfactory description of
the energy dependence of the LEDS on a quantitative level down to energies of
about 3 MeV. The experimental structure centered at 8.5 MeV in 120Sn, cf. Fig.
2(b), is split in the RQTBA results into two major contributions at 7.8 and
8.8 MeV. A structure corresponding to the lower-energy peak at 6.5 MeV is also
visible. Some local deviation is seen around 5 MeV. However, as discussed
above, including $2q\otimes 2phonon$ configurations as the next level of
complexity promises to bring the theoretical results in even better agreement
with the data, in particular at lower energies.

The differences in the description of the LEDS underline the problems of an
interpretation of the PDR based on calculations on the QRPA level. The
predicted strengths correspond to the total LEDS rather than the PDR, and the
latter carries a small fraction only. Thus, an analysis of the structures
underlying the PDR and the dominant IV resonance at higher energies based on
mean-field models should include qPVC, at least in the leading approximation.

The agreement of RQTBA to the experimental LEDS data, although significantly
improved compared to RQRPA, is still imperfect. This is illustrated in Fig. 4,
where the experimental and theoretical strengths integrated between 4 and 10
MeV are compared as functions of mass number. Both model results show a
monotonous increase between mass numbers 112 and 124 from about 1.5% to 7.5%
of the TRK sum rule for RQRPA and 2.5% to 6.5% for RQTBA, respectively. In
contrast, the experimental strengths are approximately constant with values
$3.5-4$%. For the case of 120Sn, the models predict extra strength compared to
the data in the energy region $7-10$ MeV, leading to a total value of 6%. This
indicates that some mechanisms of the strength formation are still missing to
achieve spectroscopic accuracy. A complete response theory should take into
account the continuum, including the multiparticle escape, a more complete set
of phonons (in particular, those of unnatural parity and isospin-flip),
complex ground-state correlations, and higher-complexity configurations.

Figure 4:  Evolution of the experimental LEDS in the energy range $4-10$ MeV
for Sn isotopes with mass numbers $A=111-124$ compared with predictions of the
RQRPA (open squares) and RQTBA (open triangles) calculations described in the
text. The dashed lines are to guide the eye.

##  5 Summary and conclusions

We present a systematic study of the low-energy electric dipole strength in Sn
isotopes ranging from $A=111$ to 124 based on the data for
111-113,116,118-122,124Sn obtained with the Oslo method and the study of
112,114,116,118,120,124Sn with relativisitic Coulomb excitation in forward
angle ($p,p^{\prime}$) scattering. The tin chain is of particular interest
since the similarity of low-energy structure in the neighboring isotopes
allows to single out the impact of neutron excess on the evolution of the
LEDS. The combined data cover an energy range of $2-20$ MeV which permits a
decomposition into the contribution from the low-energy tail of the IVGDR and
possible resonance-like structures with minimal assumptions. One finds in all
cases a resonance at about 8.3 MeV, exhausting an approximately constant
fraction of the TRK sum rule with a local maximum at 120Sn which might be
attributed to shell structure effects.

For the isotopes with $A\geq 118$, a consistent description of the data
suggests introducing a second resonance centered at about 6.5 MeV. The
comparison with the results from the isoscalar probes and the
$(\gamma,\gamma^{\prime})$ reaction indicates that it might represent the IV
response of the PDR. Its strength corresponds to a relatively small fraction
of the total LEDS and demonstrates an approximately linear dependence on mass
number. Because of the schematic decomposition into contributions from the
IVGDR and resonances, the absolute values might have some model dependence.
Nevertheless, these results demonstrate that taking the full LEDS as
representative for the PDR strength (as done in many discussions of a possible
relation to the neutron skin thickness) is clearly unfounded. The small PDR
strength also challenges its interpretation of being due to neutron skin
oscillations, which requires some degree of collectivity. However, this
question cannot be settled based on the IV response alone.

Microscopic calculations rooted in ab initio EOM theory can reproduce the
experimental features in 120Sn remarkably well, provided qPVC is included.
Consideration of qPVC generally improves the description of the LEDS (cf. Ref.
[38]). However, even in the latter case, calculations are still failing to
reproduce the experimental mass dependence of the LEDS. While the experimental
strengths are roughly constant, the theoretical results predict a monotonous
increase with mass number.

Further work is required to elucidate the true nature of the PDR. One way are
systematic studies of the IV and IS response along isotopic or isotonic
chains, similar to the one presented here. For the Sn case, experiments at RIB
facilities in inverse kinematics, analogous to the pioneering study of E1
strength in 130,132Sn [65], but including the information on $\gamma$ decay
[66, 67] to extract the GSF below neutron thresholds, would be of particular
interest. New experimental observables like the particle-hole structure
investigated in transfer reactions [68, 69] or transition current densities
from transverse electron scattering [23] may help to clarify the underlying
structure. Theoretically, the role of the continuum, inclusion of more
complete sets of phonons, and in particular the impact of 6-quasiparticle
configurations need to be investigated.

## Acknowledgements

This work was supported in part by the National Science Foundation under Grant
No. OISE-1927130 (IReNA) and by the Norwegian Research Council Grants 325714
and 263030. P.v.N.-C. acknowledges support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Grant No. SFB
1245 (Project ID 279384907). The work of E.L. was supported by the GANIL
Visitor Program, US-NSF Grant PHY-2209376, and US-NSF Career Grant
PHY-1654379.

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# Thermodynamics and remnants of Kiselev black holes in Rainbow gravity

P. H. Morais1, G. V. Silva1, J. P. Morais Graça1,2 and V. B. Bezerra1
<EMAIL_ADDRESS><EMAIL_ADDRESS>1Departamento de Física,
Universidade Federal da Paraíba,
Caixa Postal 5008, 58059-900, João Pessoa, PB, Brazil
2Instituto de Física, Universidade Federal do Rio de Janeiro, 21.941-972 - Rio
de Janeiro-RJ - Brazil

###### Abstract

We do a detailed study of the thermodynamics of black holes in Rainbow gravity
in the framework of the Kiselev metric, taking into account different cosmic
fluids which can mimic several kinds of dark energy scenarios. A discussion
related to the appearance of black hole remnants is presented and, unlike
other studies on the same topic, we found no evidence of the appearance of
remnants in Rainbow gravity, except in peculiar cases.

##  I Introduction

The special theory of relativity is grounded in two basic pillars, that
physics should be the same for all inertial frames and that the speed of light
is constant. These pillars were necessary to reconcile kinematics with
electrodynamics and to explain the Michelson-Morley experiment. Its
development by Albert Einstein in 1905 has guided theoretical physics since
then, leading to a generalization to non-inertial frames ten years later,
named the general theory of relativity.

Nowadays, the idea that special relativity should be modified to incorporate
new theoretical and observational evidence is a current area of research. From
the theoretical point of view, the major problem is the absence of a well-
defined quantum theory for gravity, and from an observational perspective, the
question of ultra energetic cosmic rays has posed a challenge on the
dispersion relation established by special relativity AmelinoCamelia:2000zs ;
AmelinoCamelia:2000ev .

A common feature of some quantum theories of gravity, such as string theory
Polchinski:1996na ; Forste:2001ah  and loop quantum gravity Rovelli:1997yv ;
Carlip:2001wq , is the modification of the dispersion relation, i.e., the
relation between momentum and energy of a particle (in momentum space). Among
these theories with modified dispersion relation (MDR) one can cite the so-
called double special relativity (DSR) AmelinoCamelia:2000ge ;
AmelinoCamelia:2002wr ; AmelinoCamelia:2000mn ; Magueijo:2002am , which can be
considered an extension to special relativity with not just one, but two
constants of nature: the speed of light and the Planck energy scale. Since
energy is not an invariant for Lorentz transformations, it is expected that
DSR leads to new (and non-linear) coordinate transformations.

To circumvent the fact that we do not know, a priory, the form of such
transformations, Magueijo and Smolin proposed a generalization to include
spacetime dynamics in the theory with no reference to position space
Magueijo:2002xx . From a general formula of the new dispersion relation,

$E^{2}f^{2}\left(\frac{E}{E_{p}}\right)-p^{2}g^{2}\left(\frac{E}{E_{p}}\right)=m^{2},$
(1)

where $E_{p}$ is the Planck energy scale, $f$ and $g$ are phenomenological
functions of energy of the particle, dubbed as Rainbow functions, and $m$ is
the mass of the particle, the gravitational field equations must now be given
by

$G_{\mu\nu}\left(\frac{E}{E_{p}}\right)=8\pi
GT_{\mu\nu}\left(\frac{E}{E_{p}}\right).$ (2)

This means that the metric of the spacetime as felt by a test particle with
energy $E$ must depend on its energy. For consistency, it is required that in
the limit of low-energy the Rainbow functions should be
$\text{lim}_{E_{p}\rightarrow 0}f(E/E_{p}),g(E/E_{p})=1$. This is the origin
of the terminology Rainbow gravity, the metric is no longer the same for all
test particles but an one parameter family of metrics with the energy of the
test particle as the parameter.

As stated above, at the limit of a low-energy scenario, Rainbow gravity should
reduce to general relativity, but in a regime of strong gravity, like in the
vicinity of black holes, we should expect discrepancies between theories. In
this scenario, with the creation of virtual pair of particles and the
subsequent emission of Hawking radiation Hawking:1974sw , one can study the
evolution and fate of black holes. This study is important for several
reasons, among them we can mention the fact that, if Hawking radiation stops
at some point before total evaporation, the appearance of a remnant black hole
Giddings:1992hh  can help solve two major problems in physics today: the
information paradox Chen:2014jwq  and, perhaps, an origin for dark matter
Chen:2002tu .

The main goal of this paper is to study black hole thermodynamics in rainbow
gravity Ling:2005bp ; Mu:2015qna  in the presence of a cosmic fluid that can
mimic several kinds of dark energy. The reason for incorporating a cosmic
fluid is because it is as relevant to cosmology today as it was in the early
universe. To mimic such a scenario, we will adopt, in this paper, the
Kiselev’s black hole Kiselev:2002dx  as the background.

The appearance of black hole remnants in Rainbow gravity has been debated in
the literature in recent years, with conflicting opinions among the authors
Ali:2014xqa ; Gim:2014ira ; Kiselev:2002dx . With or without the incorporation
of a cosmic fluid, we will show that we have found no evidence that Rainbow
gravity, by itself, will lead to their appearance, except for a very peculiar
case.

This article is divided as follows. In section two, we will present the
formalism of black hole thermodynamics in Rainbow gravity, as well as we will
discuss black hole thermodynamics in special cases, such as in the absence of
the cosmic fluid in Rainbow gravity, and in the presence of the cosmic fluid
in general relativity. The goal is that, in the next section, when all the
ingredients are brought together, the reader is already more familiar with the
results we are looking for. In section three, we will study thermodynamics and
black hole remnants in the full picture, in Rainbow gravity with the presence
of a cosmic fluid. Finally, in section four, we will present our conclusions.

##  II Black hole thermodynamics in rainbow gravity

One of the main issues in MDR theories is how to move from momentum space to
coordinate space and then to the metric of spacetime. The most accepted
proposal so far is that of Magueijo and Smolin Magueijo:2002xx , where the
metric can be written in terms of the functions $f(E/E_{p})$ and $g(E/E_{p})$
as $ds^{2}=-(A/f)dt^{2}+\sum_{i=1}^{3}(B_{i}/g)(dx^{i})^{2}$, where $A$ and
$B$ are functions of the coordinates.

Using this prescription, a non-rotating black hole in Kiselev spacetime can be
described by the following spherically symmetric metric

$ds^{2}=-\left(1-\frac{2GM}{r}-\frac{c}{r^{3\omega+1}}\right)\frac{1}{f^{2}}dt^{2}+\left(1-\frac{2GM}{r}-\frac{c}{r^{3\omega+1}}\right)^{-1}\frac{1}{g^{2}}dr^{2}+\frac{r^{2}}{g^{2}}d\Omega,$
(3)

where $d\Omega=d\theta^{2}+\text{sin}\theta\;d\phi^{2}$, $c$ is a constant and
$\omega$ is a parameter related to the equation of state of the surrounding
matter. Considering a black hole in a space with a cosmological constant, then
$\omega=-1$. Other interesting scenarios are $\omega=-2/3$ and $\omega=-4/3$,
which mimic, respectively, a kind of quintessence and phantom matter.

In this paper, we will follow the traditional recipe to calculate the Hawking
temperature of the black hole,

$T_{H}=\frac{\kappa}{2\pi},$ (4)

where $\kappa$ is the surface gravity. For a spherically symmetric black hole,
it is given by

$\kappa=\lim_{r\rightarrow
r_{+}}\sqrt{-\frac{1}{4}g^{tt}g^{rr}\left(\frac{\partial g_{tt}}{\partial
r}\right)^{2}},$ (5)

where $r_{+}$ is the horizon radius. The temperature will depend on the
parameters of the theory, and concerning the Rainbow functions it will be
given by

$T_{H}=\frac{g(E/E_{p})}{f(E/E_{p})}\tilde{T_{H}},$ (6)

where $\tilde{T_{H}}$ is the temperature of the equivalent spacetime in
general relativity. This means that, when $f(E/E_{p})=g(E/E_{p})$, the Hawking
temperature in both theories will be the same. As mentioned in the
introduction, the Rainbow functions must be chosen through some theoretical or
phenomenological basis, and we will rely on advances in loop quantum gravity
to choose

$f(E/E_{p})=1,\hskip
20.0ptg(E/E_{p})=\sqrt{1-\eta\left(\frac{E}{E_{p}}\right)^{n}},$ (7)

where $\eta$ is a dimensionless parameter.

The mass of the black hole is given by the parameter $M$ of the metric (3)
when the radius is the radius of the horizon,

$M=\frac{r_{+}}{2G}-\frac{c}{2Gr_{+}^{3\omega}}.$ (8)

Entropy can then be calculated using the first law of thermodynamics,
$dU=TdS$, considering the mass of the black hole equivalent to the internal
energy $U(S)$. In this paper we will disregard any kind of work, as our goal
is the study the behaviour of temperature, specific heat, entropy and decrease
in mass as a function of the mass/radius of the black hole.

We have (almost) all the necessary ingredients to study black hole
thermodynamics and remnants, but before we proceed let us make some
digressions.

###  II.1 Black holes in Rainbow gravity

(a) $n=1,|\eta|=0.9$

(b) $n=4,|\eta|=0.9$

Figure 1: Plot of energy by $r_{+}$ for all the solutions of the modified
dispersion relation, for the cases $n=1$ and $n=4$.

The temperature of the black hole, given by the equation (6), will depend not
only on the parameters of the metric but also on the energy present in the
Rainbow functions. This is the energy of the particles that ”feels” the
metric; in this case, the energy of the virtual particles emitted by the black
hole. For the sake of simplicity, we will consider that only photons are
emitted.

Since the Hawking temperature must contain only parameters of spacetime, we
have to relate the energy of the photons to the parameters of the black hole
that radiates them. We will do this considering that the photons form an
ensemble with a well-defined temperature. The energy of the photons is given
by the MDR, equation (1), and we will relate the momentum of a photon with the
horizon radius through the Heisenberg uncertainty relation, $p=1/r_{+}$, using
$\hbar=1$.

We would like to draw attention to the fact that the author in Ali:2014xqa
used $p=E$ in the Heisenberg uncertainty relation. Using this dispersion
relation is the reason why the authors found black holes remnants in their
studies. This issue was mentioned and corrected in Gim:2014ira , where the
correct MDR was used, but many authors still claim that Rainbow gravity allows
remnants based on the results found in Ali:2014xqa . As we will see, our study
found no evidence of remnants in Rainbow.

Note that the MDR, equation (1), is a polynomial in the energy. The usual
dispersion relation is a quadratic polynomial and gives us two values for the
energy, one positive and another negative. For the MDR, and especially for
$n>2$, there will be more than two solutions, and we need to verify which of
them are physically acceptable.

For $n=0$, there is only one positive solution, as long as $\eta<1$. For
$n=1$, one can find an analytical solution,

$E_{n=1}=\frac{-\eta+\sqrt{{\eta}^{2}+4\,{r_{+}}^{2}}}{2\,r_{+}^{2}}$ (9)

where $\eta$ can be positive or negative. For $n=2$, we have

$E_{n=2}=\pm\frac{1}{\sqrt{{r_{+}}^{2}+\eta}}$ (10)

so that $\eta>-r_{+}^{2}$, and for $n=4$, there are four solutions, and their
expressions are too wide for us to show here; instead, we will depict it. In
figure (1), we plot both $n=1$ and $n=4$ solutions.

For $n=1$ we have two positive solutions; the first one goes to infinity as
$r_{+}\rightarrow 0$, the same behavior of the usual Schwarzschild black hole.
The second one introduces a new behavior and, as $r_{+}\rightarrow 0$, the
energy approaches a finite value. The same also happens for $n=2$.

For $n=4$ we have three real positive solutions, but only one is valid for all
$r_{+}$. Also, there is an interchange of solutions for $\eta<0$ that should
be better analyzed when we study black hole thermodynamics. Note that this
solution appears to be continuous, so one can consider it as a valid real
solution described by a piecewise function, even though it does not cover all
values for the horizon radius.

###  II.2 Horizons in a Kiselev Black Hole

Before proceeding to study thermodynamics in Rainbow Gravity, we will analyze
the horizons in a Kiselev Black Hole. Such horizons will be the same both in
GR as in Rainbow, since it does not depend on the Rainbow functions. The
horizons are given by $g_{tt}=0$, that results in the equation (8). As one can
see, positive values for the mass will depend on a relation between the
parameters of the theory and the horizon radius, $r_{+}$.

Figure 2: Plot of mass by $r_{+}$ for $\omega=-1,-2/3$ and $-4/3$. In the
center of the graph, a closer depict for small values of the horizon radius.
We used $c=0.01,G=1/E_{p}^{2}=1$ for the rest of the discussion.

In figure (2), we depict this relation for some values of $\omega$, and we
will set $G=1/E_{p}^{2}=1$ and $c=0.01$ for the rest of the discussion. As one
can see, the mass has an upper bound that corresponds to the case where two
horizons coincide: the black hole horizon and the cosmic horizon. This kind of
black hole is usually called Nariai black hole when the cosmic fluid is a
cosmological constant.

As the mass decreases, the horizons start to move away from each other; the
left horizon is the black hole horizon while the right one is the cosmological
horizon. This means that the maximum value for the black hole horizon is the
point where the mass reaches its critical point; above this value either we
are after the cosmological horizon or we have a naked singularity.

We are calling attention to this point because it will be relevant for the
study of the thermodynamics of the system. As we will see, this is exactly the
point where the temperature will reach a critical value and cannot be extended
for larges values of $r_{+}$.

###  II.3 Temperature, Entropy and Specific Heat

The temperature of the black hole in Rainbow gravity is given by

$T_{H}=\frac{r_{+}^{-3\omega-2}\,|3\,c\,\omega+r_{+}^{3\omega+1}|}{4\pi}\left(1-\eta\left(\frac{E}{E_{p}}\right)^{n}\right)^{1/2}$
(11)

and we need to find how $E$ relates to $r_{+}$. For this, we will use the
uncertainty relation and the dispersion relation we already found.

As we have shown, for $n=0$, $n=1$ and $n=2$, we have only one positive
solution for the energy, for each value of the parameter $\eta$ (positive or
negative). The case $n=4$ is more subtle, since we have two solutions for
$\eta<0$, but that apparently form a piecewise continuous function.

Entropy can be integrated as

$S=\int\frac{1}{T}\frac{dM}{dT}dT=\int\frac{1}{T}\frac{dM}{dr_{+}}dr_{+}$ (12)

and the specific heat can be derived as

$C\coloneqq\frac{dM}{dT}=\frac{dM}{dr_{+}}\left(\frac{dT}{dr_{+}}\right)^{-1}.$
(13)

One can find an analytic solution for the specif heat, but the formula will be
lengthy and will depend on the choice of the parameter $n$. Instead of s
lengthy formula, in the next section we will depict several graphs for the
entropy and specific heats.

##  III Temperature, entropy, specific heat and remnants

Let us now proceed to study the thermodynamics of black holes, in Kiselev
spacetime, in the framework of Rainbow gravity.

###  III.1 Temperature

Let’s start by ignoring the cosmic fluid and doing a preliminary study of
black holes thermodynamics in Rainbow gravity. This is not a new topic, but we
hope to clarify some issues that are not clear in the literature. For that, we
will look for solutions for different values of the exponent $n$ and
explicitly depicted the temperature as a function of the horizon radius, as
show in figure (3). We also plot the temperature for the Schwarzschild black
hole to compare Rainbow gravity with General Relativity.

(a) $\eta=0.9$

(b) $\eta=-0.9$

Figure 3: Temperature as a function of the horizon for Rainbow’s gravity
without a cosmic fluid.

For $\eta>0$, the temperature blows up for General Relativity and also for
$n=0$, otherwise it approaches a finite value. The shape of the curve is
different for different exponents, but their behavior is similar.

For $\eta<0$, all studied cases until $n<4$ leads to an infinite temperature
as the radius approaches zero, which is consistent with the energy of the
photons (or any massless particle) near the black hole when one uses these
MDRs. For $n=4$ we have two solutions with a bound to the horizon radius. At
this bound, the temperature is not zero, so it should continue radiating in a
kind of bouncing behavior. This is an interesting new behavior, since the
radius can first decrease and then increase (or the reverse), but the
temperature will always decrease (or always increase).

Figure 4: Temperature as a function of the horizon radius for Kiselev black
holes, in general relativity.

We will now briefly study the temperature of the black hole in Kiselev’s
spacetime, disregarding the Rainbow functions. The idea is to identify this
behavior from the one due to Rainbow. The temperature is depicted in figure
(4), and one can see that there is a point where the temperature is zero. This
point is precisely that of critical mass, where the horizons of the black hole
and cosmological coincide.

This occurs for all values of $\omega$ studied, and what varies is only the
critical horizon radius. As mentioned before, for horizon radius values
greater than this critical value, we are either considering the cosmological
horizon or what we have is a naked singularity. In any case, it is not an
acceptable physical situation.

(a) $\eta=0.9$

(b) $\eta=-0.9$

Figure 5: Temperature as function of the horizon radius, for Kiselev black
holes, $\omega=-4/3$, in Rainbow gravity.

Let us now consider the complete case of Kiselev black holes in Rainbow
gravity. Figure (5) shows the temperature of the black hole, for
$\omega=-4/3$, for different values of exponent $n$. On the left, we have the
case $\eta<0$, and on the right, the case $\eta>0$. As we can see, what
happens is a union of both behaviors; due to the incorporation of a cosmic
fluid, the temperature will now go to zero in a critical horizon radius,
instead of going to infinity, and the black hole horizon must be smaller than
such a critical horizon. For $n<4$, as the radius of the horizon decreases,
the temperature can go to infinity or a limit value. Either way, it doesn’t go
to zero. This is important for the study of remnants.

The exotic case occurs when $n=4$, when the temperature presents a kind of
loop, which indicates that the temperature can only decrease from a finite
value to zero, thus leaving a remnant. However, the final value of the horizon
radius would be equal to the cosmological horizon, which represents an extreme
situation; whether such a scenario is physically feasible or not is a
difficult question to answer.

(a) $\eta>0$

(b) $\eta<0$

Figure 6: Temperature as a function of the horizon radius, for Kiselev black
holes, $\omega=-4/3$, in Rainbow gravity, for a fixed exponent $n=4$. In this
graph, we vary the parameter $\eta$.

In figure (6), we depict the behavior of the temperature for different values
of the parameter $\eta$, for a fixed exponent $n=4$. As one can see, the
behavior is similar, but the loop size for $\eta<0$ and the final temperature
value for $\eta>0$ are different. For other values of the exponent, we found a
similar result, always maintaining the same behavior, but varying the path in
the $T-r_{+}$ plane.

(a) $\eta=0.9$

(b) $\eta=-0.9$

Figure 7: Temperature as a function of the horizon radius, for Kiselev black
holes, $\omega=-2/3$, in Rainbow gravity.

Finally, in figure (7), we plot the temperature for $\omega=-2/3$. The result
is also similar to the case $\omega=-4/3$, but because the critical horizon
radius is larger, the plot is less clear than the case of phantom energy. This
was the reason we choose to use $\omega=-4/3$ in most cases. For a
cosmological constant the behavior is also similar and that is the reason we
will not show it here.

###  III.2 Entropy

The entropy of a black hole, in the case of general relativity, is given by
Bekenstein’s formula, $S=\pi r_{+}^{2}$ Bekenstein:1973ur. The horizon radius
clearly will depend on the parameter of the spacetime, but as long we are
interested on the values of the entropy as a function of the horizon radius,
the formula will not receive any correction. This means that, for Kiselev
black holes, the only effect will be to limit the value for the horizon
radius, as one can see in figure (10).

Figure 8: Entropy as function of the horizon radius, for Kiselev black holes,
in general relativity.

The same will not hold in Rainbow, since we cannot consider that the
Bekenstein’s formula is still valid. To plot the graphs, we will use the first
law of thermodynamics that gives us the entropy as equation (12).

(a) $\eta=0.9$

(b) $\eta=-0.9$

Figure 9: Entropy as a function of the horizon radius in Rainbow gravity.

That Rainbow gravity leads to a correction in entropy had already been
proposed in Ling:2005bp , and such corrections are explicitly depicted in
figure (9), without considering the cosmic fluid. For some values of the
exponent $n$, we can found an analytic formula for the entropy, but not for
all values of the exponent. For $n=4$, for example, the entropy need to be
calculated numerically.

When we consider the complete case of Kiselev black hole in Rainbow gravity,
these behaviors are complementary; the shape of the graph does not change, but
the horizon radius is limited. Because of this, there is no need to plot the
graphs.

###  III.3 Specific Heat

We are now going to study the specific heat. For the Schwarzschild black hole,
in general relativity, the specific heat is negative for all values of the
horizon radius. The same is a reflection of the fact that the temperature
decreased with increasing radius.

Figure 10: Specific heat as function of the horizon radius, for Kiselev black
holes, in general relativity.

For Kiselev black hole the same will hold, but with a different behavior. In
figure (10), we depict it for the cases of the parameter $\omega=-1,-2/3$ and
$-4/3$. The specific heat is still negative, but now it has an inflection
point, since the horizon radius has a limit.

(a) $\eta=0.9$

(b) $\eta=-0.9$

Figure 11: Specific heat as function of the horizon radius, without any cosmic
fluid, in Rainbow gravity.

We are now going to study the specific heat in Rainbow, without the cosmic
fluid. Note that when the temperature tends to infinity, the specific heat
tends to zero. When the temperature tends towards a finite value, the specific
heat tends towards minus infinite. This is depicted in figure (11).

The possible explanation should follow the following lines. A very small
specific heat allows a large increase in temperature, even for a small
variation in the heat exchanged with the environment. Conversely, a very large
specific heat (in absolute value) will result in a small variation in
temperature. Thus, the temperature and specific heat graphs are consistent.

A new case occurs for $n=4$, when the model has positive specific heat, a
characteristic not very common for black holes. This is because, in a certain
region, the temperature of the black hole can decreases with the horizon
radius. After reaching a critical value, the behavior changes, and the
temperature starts to decrease as the horizon radius increases (or the other
way around).

(a) $\eta=0.9$

(b) $\eta=-0.9$

Figure 12: Specific heat as function of the horizon radius, for Kiselev holes,
in Rainbow gravity.

Now let’s consider the complete case of Kiselev black hole in Rainbow, applied
to the case of phantom dark energy. This is depicted in figure (12). What we
perceive is a complementary behavior, the explanation of which is based on
previous comments.

Once again, the most interesting case occurs for $n=4$, when we have an
asymptotic behavior for specific heat. Note that this asymptotic behavior
occurs far from the critical radius and it is due to the change that occurs in
the relationship between temperature and horizon radius, which will allow the
loop one can see in figure (5).

###  III.4 Black Hole Remnants

One of the most interesting subjects in black hole physics is the concept of
remnants. What is a remnant and when can we be sure of its existence, however,
does not seem to be a well-defined issue. The main problem is that often the
solutions we seek are not valid for all the values of the horizon radius, so
it is not clear whether what the study gives us is a remnant or just a problem
in the coordinate system.

Black hole remnants in Rainbow has been studied in Mu:2015qna ; Gim:2014ira ;
Ali:2014xqa , and it has been claimed the existence of remnants. However, the
remainder would occur at a non-zero finite temperature, which does not seem
physically acceptable. A non-zero finite temperature black hole should
continue to radiate. In this work, we will consider that remnants must be
defined as objects with finite mass/radius whose temperature is zero or, in a
weaker condition, approaches zero.

With this condition, we found no evidence that Rainbow gravity allows the
existence of remnants by itself, at least for the studied Rainbow functions.
The only exception occurs for $n=4$ with $\eta<0$, however the remainder could
only be found in the presence of a cosmic fluid, and its final horizon radius
is that where the event horizon and the cosmological horizon coincide.

##  IV Conclusions

In this paper, we study thermodynamics and the possibility of black hole
remnants, immersed in a Kiselev metric, in Rainbow Gravity. We did a thorough
and graphic analysis of several parameters of the theory, such as
mass/internal energy, temperature, entropy and specific heat, both
prioritizing Kiselev’s metric and the Rainbow functions.

We have explicitly shown that there is a limit value for the event horizon, in
the case of Kiselev, which depends on the mass of the black hole. For this
reason, there is a limit value for the temperature/entropy of this spacetime.
It is for this reason that some authors find a negative temperature in their
work, when they do not limit the values for the horizon radius. Such negative
temperature is nothing more than an indication that beyond this critical
horizon there is no mass that provides such event horizon, that is, it is
actually a naked singularity.

When we consider Rainbow gravity, we have been able to find a finite value for
temperature as the horizon radius decreases. This behavior does not occur for
the Schwarzschild black hole, where the temperature tends to infinity as the
horizon radius decreases. Although this case does not result in a remnant, at
least it avoids the catastrophic behavior present in general relativity.

For a particular case of the MDR, where the energy is elevated to the fourth
order, we find a new behavior in the thermodynamics of black holes, where a
kind of bouncing occurs when we plot the temperature by the horizon radius. In
this case, we also find a range in parameter space where the specific heat is
positive, a behavior that is also unusual in general relativity.

When we incorporate the Kiselev metric, in the latter case, a loop occurs in
the temperature graph, which indicates the possibility of a remnant, albeit an
unusual remnant, as it would occur for a horizon radius where the event
horizon meets the cosmological horizon. With the exception of the latter case,
we believe that Rainbow gravity with the proposed Rainbow functions is not a
more open environment for the emergence of remnants than, for example, general
relativity.

###### Acknowledgements.

J.P.M.G. (grant 151701/2020-2) and V.B.B. would like to thank CNPq (Conselho
Nacional de Desenvolvimento Científico e Tecnológico - Brazil) for financial
support. This study was financed in part by the Coordenação de Aperfeiçoamento
de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. PHM would
like to thanks CAPES for financial support.

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