# A note on interacting holographic dark energy with a Hubble-scale cutoff

Ricardo G. Landim<EMAIL_ADDRESS>Technische Universität München,
Physik-Department T70,
James-Franck-Stra$\beta$e 1, 85748 Garching, Germany

###### Abstract

Holographic dark energy with the Hubble radius as infrared cutoff has been
considered as a candidate to explain the late-time cosmic acceleration and it
can solve the coincidence problem. In this scenario, a non-zero equation of
state is only possible if there is an interaction between dark energy and cold
dark matter. In this paper, a set of phenomenological interactions is assumed
and a detailed analysis of the possible values of the coupling constants is
carried out, however the resulting matter power spectrum and cosmic microwave
background temperature and polarization power spectra have a shape very far
from the observed ones. These results rule out any value for the free
parameters and it seems to indicate that the assumed interacting holographic
dark energy with a Hubble-scale cutoff is not viable to explain the
accelerated expansion of the Universe, when cosmological data are taken into
account.

##  I Introduction

The observational evidence of dark energy (DE) in 1998 perlmutter1999 ;
reiss1998  opened a new phase in the understanding of our Universe. While
cosmological data are continuously confirming the existence of the late-time
cosmic acceleration (see copeland2006dynamics ; pdg2022  for reviews), the
nature of the accelerated expansion is still an open issue. The simplest
candidate for DE is a cosmological constant $\Lambda$, which encompasses the
standard $\Lambda$-cold-dark-matter (CDM) model. The six free parameters of
the $\Lambda$CDM model are well constrained and are in agreement with
cosmological observations Aghanim:2018eyx , despite some tensions, e.g. the
Hubble tension Aghanim:2018eyx ; Riess:2021jrx  (a recent review is
DiValentino:2021izs ). The observed value of the vacuum energy is many orders
of magnitude smaller than the theoretically calculated Weinberg:1988cp ,
leading to the so-called ‘cosmological constant problem’. Additionally, the
evolution of CDM and DE are very different from each other, but their energy
densities today have the same order of magnitude. This coincidence may
indicate new physics and it is usually referred to as ‘coincidence problem’.

The lack of understating about the nature of the cosmological constant and the
aforementioned issues encourage alternative models of DE (for reviews see
copeland2006dynamics ; Bamba:2012cp ). Among the many candidates there are
scalar and vector fields peebles1988 ; ratra1988 ; Frieman1992 ; Frieman1995 ;
Caldwell:1997ii ; Padmanabhan:2002cp ; Bagla:2002yn ; ArmendarizPicon:2000dh ;
Brax1999 ; Copeland2000 ; Vagnozzi:2018jhn ; Koivisto:2008xf ; Bamba:2008ja ;
Emelyanov:2011ze ; Emelyanov:2011wn ; Emelyanov:2011kn ; Kouwn:2015cdw ;
Landim:2015upa ; Landim:2016dxh ; Banerjee:2020xcn , metastable DE
Szydlowski:2017wlv ; Stachowski:2016zpq ; Stojkovic:2007dw ; Greenwood:2008qp
; Abdalla:2012ug ; Shafieloo:2016bpk ; Landim:2016isc ; Landim:2017kyz ;
Landim:2017lyq , models using extra dimensions dvali2000 , alternative fluids
Landim:2021www ; Landim:2021ial , etc. Another explanation for DE comes from
the holographic principle, the so-called holographic DE (HDE) Hsu:2004ri ;
Li:2004rb ; Pavon:2005yx ; Nojiri:2005pu ; Wang:2005jx ; Wang:2005pk ;
Wang:2005ph ; Wang:2007ak ; Landim:2015hqa ; Li:2009bn ; Li:2009zs ; Li:2011sd
; Saridakis:2017rdo ; Mamon:2017crm ; Mukherjee:2016lor ; Feng:2016djj ;
Herrera:2016uci ; Forte:2016ben ; DAgostino:2019wko ; Nojiri:2020wmh ;
Nojiri:2021iko ; Nojiri:2021jxf ; Colgain:2021beg  (see Wang:2016och  for a
review). The holographic principle suggested by ’t Hooft 'tHooft:1993gx  and
Susskind Susskind:1993aa ; Susskind:1994vu , in turn based on the previous
works of Thorn Thorn:1991fv  and Bekenstein Bekenstein:1993dz , is a property
of quantum gravity, where at Planckian scale the world is best described by a
2-D lattice evolving with time, rather than 3+1-D. In this scenario, DE should
obey this principle and the fine-tuning problem is eliminated Cohen:1998zx .

HDE would then have an energy density given by $\rho_{\rm
de}=3c^{2}M_{Pl}^{2}L^{-2}$, where $c$ is a constant, $M_{Pl}$ is the reduced
Planck mass and $L$ is the infrared (IR) cutoff Hsu:2004ri ; Li:2004rb . The
first natural choice for the IR cutoff is the Hubble radius, however it led to
an equation of state that described pressureless matter Hsu:2004ri . This
problem was circumvented choosing the future event horizon as cutoff Li:2004rb
. Other choices for $L$ include the inverse of the Ricci scalar curvature
Gao:2007ep , the age of the Universe Cai:2007us , among others
Saridakis:2020zol ; Nojiri:2022aof ; Nojiri:2022dkr . Inspired by the
holographic principle and the AdS/CFT correspondence Maldacena:1997re , HDE
has been embedded in minimal supergravity Landim:2015hqa , while in
Nastase:2016sji  it was shown that HDE arises from generic quantum gravity
theory, assuming only the existence of a minimum length.

Another widely studied alternative to the $\Lambda$CDM paradigm is if DE
interacts with CDM Wetterich:1994bg ; Amendola:1999er ; Guo:2004vg ;
Cai:2004dk ; Guo:2004xx ; Bi:2004ns ; Gumjudpai:2005ry ; Sadjadi:2006qb ;
Yin:2007vq ; MohseniSadjadi:2008na ; Costa:2013sva ; Abdalla:2014cla ;
Costa:2014pba ; Landim:2015poa ; Landim:2015uda ; Costa:2016tpb ;
Marcondes:2016reb ; Landim:2016gpz ; Wang:2016lxa ; Farrar:2003uw ;
micheletti2009 ; Yang:2017yme ; Marttens:2016cba ; Yang:2017zjs ;
Costa:2018aoy ; Yang:2018euj ; Landim:2019lvl ; Vagnozzi:2019kvw  and it can
help alleviating the coincidence problem Olivares:2005tb  and the Hubble
tension DiValentino:2019ffd ; DiValentino:2019jae ; Lucca:2020zjb . Among the
many possible phenomenological interactions, one of the most famous and used
in the literature is at the background level, proportional to the sum of the
energy densities of CDM and DE (see Wang:2016lxa  for a review). Constraints
on the couplings constants and forecasts for several upcoming observational
programs are presented in Costa:2013sva ; Costa:2016tpb ; Costa:2019uvk ;
Bachega:2019fki ; Costa:2021jsk ; Xiao:2021nmk ; Landim:2022 .

Assuming an interaction between HDE (with a Hubble radius as IR cutoff) and
CDM not only gives the correct equation of state for DE, but also solves the
coincidence problem Pavon:2005yx . In this paper we investigate this HDE
model, using the aforementioned phenomenological interactions. We perform a
detailed analysis of the necessary values for the couplings that would give an
equation of state in agreement with the cosmic acceleration. It turns out that
the parameter space leads to a matter and cosmic microwave background (CMB)
power spectra in disagreement to what is observed. The resulting power spectra
are actually very similar to the $\Lambda$CDM model but without CDM, therefore
not being able to reproduce current cosmological observations.

This paper is organized as follows. Sec. II reviews some aspects of the HDE
model considered here, along with the phenomenological interactions, and
present the necessary equations. In Sec. III we show our results and Sec. IV
is reserved for conclusions. We use Natural units ($c=\hbar=1$) throughout the
text.

##  II Holographic dark energy

When the Hubble scale is considered as IR cutoff $L^{-1}=H$, the energy
density for DE is $\rho_{\rm de}=3c^{2}M_{Pl}^{2}H^{2}$, while for CDM the
energy density becomes $\rho_{\rm dm}=3(1-c^{2})M_{Pl}^{2}H^{2}$, where the
first Friedmann equation for a spatially flat Universe was used, safely
ignoring radiation and visible matter. The ratio $r\equiv\rho_{\rm
dm}/\rho_{\rm de}$ is thus $r=(1-c^{2})/c^{2}$ Pavon:2005yx , therefore
constant if $c$ is constant. When there is an interaction between DE and CDM
the total energy momentum tensor is still conserved, however not anymore for
the individual components. The continuity equations are

$\displaystyle\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}$ $\displaystyle=Q\,,$ (1)
$\displaystyle\dot{\rho}_{\rm de}+3H(1+w)\rho_{\rm de}$ $\displaystyle=-Q\,,$
(2)

where $w$ is the constant DE equation of state and a dot represents a time
derivative. We take the phenomenological interaction $Q=H(\lambda_{1}\rho_{\rm
dm}+\lambda_{2}\rho_{\rm de})$ He:2008si , where $\lambda_{1}$ and
$\lambda_{2}$ are constants.

Here we will use the original scenario of constant $c$. Using the expression
for $\rho_{\rm de}$ into Eq. (2) the equation of state is determined

$w=-\frac{1}{3}\Big{(}\lambda_{1}+\frac{\lambda_{2}}{r}\Big{)}(1+r)\,.$ (3)

This means that the equation of state is no longer a free parameter, as it is
usual in other interacting DE models. When the coupling constants are zero, a
pressureless fluid is recovered, as originally found in Hsu:2004ri .

The equation of state is constant and depends on the coupling constants, given
that the ratio $r$ is well known. In order for $w$ not to be zero, the
coupling constants should not be very small. The constant $c$ is also
completely determined by the ratio $r=r_{0}$, through $c^{2}=(1+r_{0})^{-1}$.

We can solve the corresponding continuity equations, which give the energy
densities for CDM and DE, respectively,

$\displaystyle\rho_{\rm dm}$ $\displaystyle=\rho_{\rm
dm,0}a^{-3+\lambda_{1}+\frac{\lambda_{2}}{r_{0}}}\,,$ (4)
$\displaystyle\rho_{\rm de}$ $\displaystyle=\rho_{\rm
de,0}a^{-3+\lambda_{1}+\frac{\lambda_{2}}{r_{0}}}\,.$ (5)

Both CDM and DE present the same background evolution, with an effective
equation of state $w_{\rm de}^{\rm eff}=w_{\rm dm}^{\rm
eff}=-1/3(\lambda_{1}+\lambda_{2}/r_{0})$, thus leading to the constant ratio
$r$ at all times. This already poses a problem in the description, because
both fluids will have the same evolution, therefore they both describe either
CDM (with $\lambda_{1}=\lambda_{2}\simeq 0$) or DE ($w^{\rm eff}_{\rm
de}<-1/3$). The first scenario is what Hsu found Hsu:2004ri  and the second
one is the incentive to add an interaction in the first place. However, if
both fluids describe DE then we would have a Universe without CDM, which is
ruled out by observations. We will return to this point in a moment.

The accelerated expansion can only be achieved if both couplings are not very
small, as it is depicted in Fig. 1. One may wonder if such relatively large
couplings are in agreement with observations, since in other models the
couplings are small Costa:2016tpb . This issue is investigated as follows.

Figure 1: Effective equation of state for DE as a function of the coupling
constants. The accelerated expansion of the Universe may happen for relatively
large couplings. If $\lambda_{1}=0$, then $\lambda_{2}$ should be considerably
larger than 1.

Although HDE is an effective description for the cosmological constant, the
perturbation of the energy density is non-zero, in contrast to $\Lambda$CDM.
The perturbation is $\delta_{\rm de}=2\delta H/H$, where the perturbation in
the Hubble rate is given by $\delta H=kv_{T}/3+\dot{h}/6$ Gavela:2010tm . We
can use the full set of linear order perturbation equations for CDM and DE to
investigate the CDM behavior. In the synchronous gauge they are Ma:1995ey ;
Valiviita:2008iv ; Gavela:2010tm ; Costa:2013sva ; Landim:2022

$\displaystyle\dot{\delta}_{\rm dm}=$ $\displaystyle-\theta_{\rm
dm}-\frac{\dot{h}}{2}+\mathcal{H}\lambda_{2}\frac{\rho_{\rm de,0}}{\rho_{\rm
dm,0}}(\delta_{\rm de}-\delta_{\rm
dm})+\bigg{(}\lambda_{1}+\lambda_{2}\frac{\rho_{\rm de,0}}{\rho_{\rm
dm,0}}\bigg{)}\bigg{(}\frac{kv_{T}}{3}+\frac{\dot{h}}{6}\bigg{)}\,,$ (6)
$\displaystyle\dot{\theta}_{\rm dm}=$ $\displaystyle-\mathcal{H}\theta_{\rm
dm}-\bigg{(}\lambda_{1}+\lambda_{2}\frac{\rho_{\rm de,0}}{\rho_{\rm
dm,0}}\bigg{)}\mathcal{H}\theta_{\rm dm}\,,$ (7)
$\displaystyle\dot{\delta}_{\rm de}=$ $\displaystyle-(1+w)\bigg{(}\theta_{\rm
de}+\frac{\dot{h}}{2}\bigg{)}-3\mathcal{H}(1-w)\delta_{\rm
de}+\mathcal{H}\lambda_{1}\frac{\rho_{\rm dm,0}}{\rho_{\rm de,0}}(\delta_{\rm
de}-\delta_{\rm dm})$
$\displaystyle-3\mathcal{H}(1-w)\bigg{[}3(1+w)+\lambda_{1}\frac{\rho_{\rm
dm,0}}{\rho_{\rm de,0}}+\lambda_{2}\bigg{]}\frac{\mathcal{H}\theta_{\rm
de}}{k^{2}}-\bigg{(}\lambda_{1}\frac{\rho_{\rm dm,0}}{\rho_{\rm
de,0}}+\lambda_{2}\bigg{)}\bigg{(}\frac{kv_{T}}{3}+\frac{\dot{h}}{6}\bigg{)}\,,$
(8) $\displaystyle\dot{\theta}_{\rm de}=$ $\displaystyle
2\mathcal{H}\theta_{\rm
de}\bigg{[}1+\frac{1}{1+w}\bigg{(}\lambda_{1}\frac{\rho_{\rm dm,0}}{\rho_{\rm
de,0}}+\lambda_{2}\bigg{)}\bigg{]}+\frac{k^{2}}{1+w}\delta_{\rm de}\,,$ (9)

where the adiabatic sound speed is assumed to be $w$, the DE effective sound
speed is one and the center of mass velocity for the total fluid $v_{T}$ is
defined as Gavela:2010tm

$(1+w_{T})v_{T}=\sum_{a}(1+w_{a})\Omega_{a}v_{a}\,.$ (10)

The DE equation of state $w$ is given by Eq. (3) with constant $r$.

In the synchronous gauge, the adiabatic initial conditions for CDM and DE are
He:2008si ; costa2014observational

$\displaystyle\delta_{\rm de}^{(i)}$ $\displaystyle=\delta_{\rm
dm}^{(i)}=\frac{3}{4}\delta_{r}^{(i)}\bigg{(}1-\frac{\lambda_{1}}{3}-\frac{\lambda_{2}}{3}\frac{1}{r_{0}}\bigg{)}\,,$
(11) $\displaystyle v_{\rm de}^{(i)}$ $\displaystyle=v_{r}^{(i)}\,,$ (12)

where the index ‘r’ represents radiation. The equations for the other species
remain as they are in the $\Lambda$CDM model. Finally, a comoving frame where
the CDM velocity is zero is chosen in order to fix the residual freedom of the
synchronous gauge.

##  III  Results

We implemented the background and perturbation equations in a modified version
of CLASS blas2011cosmic ; Lucca:2020zjb .

We have extensively investigated the parameter space, and illustrative matter
and CMB power spectra are shown in Figures 2 and 3, respectively, where we
chose two different set of values for $\lambda_{1}$ and $\lambda_{2}$ and plot
also the case for $\Lambda$CDM. Independent of the chosen values for the
couplings, the power spectra are very different from $\Lambda$CDM.

Figure 2: Linear matter power spectrum at $z=0$ for two sets of representative
values for $\lambda_{1}$ and $\lambda_{2}$, and $\Lambda$CDM. The usual
cosmological parameters were fixed to the Planck 2018 best-fit values.

Figure 3: Dimensionless CMB temperature (top left), polarization (top right)
and cross (bottom) power spectra. Two sets of representative values for
$\lambda_{1}$ and $\lambda_{2}$ were taken, while the other cosmological
parameters were fixed to the Planck 2018 best-fit values. A comparison with
$\Lambda$CDM is also shown.

Several experiments have measured the CMB power spectrum since 1992, e.g. COBE
COBE:1992syq , TOCO Miller:1999qz , DASI Halverson:2001yy , Boomerang
Boomerang:2001tye , MAXIMA Hanany:2000qf , WMAP WMAP:2012fli ; WMAP:2012nax ,
and more recently Planck Planck:2018nkj . All of these experiments constrained
very well the CMB power spectrum, which is in agreement with the $\Lambda$CDM
model. Therefore, a deviation from the observed power spectrum, like the ones
shown in Fig. 3, is very disfavored. The same conclusion can be drawn for the
matter power spectrum. The matter power spectrum is well constrained by latest
observations, e.g. Planck 2018 CMB data Planck:2018nkj , DES Year 1 cosmic
shear DES:2017qwj , and SDSS galaxy and Ly $\alpha$ clustering Reid:2009xm ;
SDSS:2017bih ; Chabanier:2018rga ; Chabanier:2019eai . Thus all choices of
couplings are completely excluded from current observations.

A situation where the couplings are large enough to produce the cosmic
acceleration leads to a Universe without CDM, as pointed out before. In order
to compare the scenarios, we show in Fig. 4 the power spectra for the case
$\lambda_{1}=\lambda_{2}=0.6$ along with $\Lambda$CDM without CDM. We see that
the power spectra are very similar to each other, although not identical,
because of the different DE equation of state and perturbation equations. In
this case, the CMB power spectrum has all peaks increased, when compared to
the one for $\Lambda$CDM, due to the absence of CDM, while the third peak is
smaller than the second one. On the other hand, the matter power spectrum is
reduced mainly on small scales due to the absence of CDM.

Figure 4: Linear matter power spectrum (left) and dimensionless CMB
temperature (right) power spectrum, for $\lambda_{1}=\lambda_{2}=0.6$,
$\Lambda$CDM, and $\Lambda$CDM without CDM. The other cosmological parameters
were fixed to the Planck 2018 best-fit values.

##  IV Conclusions

In this paper we investigated an interacting HDE model with the Hubble-scale
as the IR cutoff. We assumed that the interaction between CDM and DE is driven
by the sum of the energy densities of both species, with constant coupling
constants. The evolution of the energy density for both components of the dark
sector is the same, leading to an always constant ratio $\rho_{\rm
dm}/\rho_{\rm de}$ and solving the coincidence problem. However, an analysis
of possible values for the couplings that would lead to the cosmic
acceleration shows that the corresponding CMB and matter power spectra are
very different from the ones in the $\Lambda$CDM model. Hence, this is in
disagreement with cosmological observations, indicating that the assumed
interacting HDE is not viable to describe the current phase of accelerated
expansion of the Universe.

We point out that the results presented here are valid for a constant $c^{2}$.
A time varying $c$ changes the DE equation of state and may lead to different
conclusions, but it is beyond the scope of the present work.

###### Acknowledgements.

R.G.L. thanks Matteo Lucca for useful comments and the Group T70 at the
Physics Department for the hospitality, while this work was in progress.

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