# Interacting holographic dark energy model in non-flat universe

###### Abstract

We employ the holographic model of interacting dark energy to obtain the equation of state for the holographic energy density in non-flat (closed) universe enclosed by the event horizon measured from the sphere of horizon named .

## 1 Introduction

The accelerated expansion that based on recent astrophysical data [1], our universe is experiencing is today’s most important problem of cosmology. Missing energy density - with negative pressure - responsible for this expansion has been dubbed Dark Energy (DE). Wide range of scenarios have been proposed to explain this acceleration while most of them can not explain all the features of universe or they have so many parameters that makes them difficult to fit. The models which have been discussed widely in literature are those which consider vacuum energy (cosmological constant) [2] as DE, introduce fifth elements and dub it quintessence [3] or scenarios named phantom [4] with , where is parameter of state.

An approach to the problem of DE arises from holographic principle that states that the number of degrees of freedom related directly to entropy scales with the enclosing area of the system. It was shown by ’tHooft and Susskind [5] that effective local quantum field theories greatly overcount degrees of freedom because the entropy scales extensively for an effective quantum field theory in a box of size with UV cut-off . As pointed out by [6], attempting to solve this problem, Cohen et al. showed [7] that in quantum field theory, short distance cut-off is related to long distance cut-off due to the limit set by forming a black hole. In other words the total energy of the system with size should not exceed the mass of the same size black hole i.e. where is the quantum zero-point energy density caused by UV cutoff and denotes Planck mass ( . The largest is required to saturate this inequality. Then its holographic energy density is given by in which is free dimensionless parameter and coefficient 3 is for convenience.

As an application of holographic principle in cosmology, it was studied by [8] that consequence of excluding those degrees of freedom of the system which will never be observed by that effective field theory gives rise to IR cut-off at the future event horizon. Thus in a universe dominated by DE, the future event horizon will tend to constant of the order , i.e. the present Hubble radius. The consequences of such a cut-off could be visible at the largest observable scales and particulary in the low CMB multipoles where we deal with discrete wave numbers. Considering the power spectrum in finite universe as a consequence of holographic constraint, with different boundary conditions, and fitting it with LSS, CMB and supernova data, a cosmic duality between dark energy equation of state and power spectrum is obtained that can describe the low features extremely well.

Based on cosmological state of holographic principle, proposed by Fischler and Susskind [9], the Holographic model of Dark Energy (HDE) has been proposed and studied widely in the literature [10, 11]. In [12] using the type Ia supernova data, the model of HDE is constrained once when c is unity and another time when c is taken as free parameter. It is concluded that the HDE is consistent with recent observations, but future observations are needed to constrain this model more precisely. In another paper [13], the anthropic principle for HDE is discussed. It is found that, provided that the amplitude of fluctuation are variable the anthropic consideration favors the HDE over the cosmological constant.

In HDE, in order to determine the proper and well-behaved system’s IR cut-off, there are some difficulties that must be studied carefully to get results adapted with experiments that claim our universe has accelerated expansion. For instance, in the model proposed by [10], it is discussed that considering particle horizon, ,

(1) |

as the IR cut-off, the HDE density reads to be

(2) |

that implies which does not lead to accelerated
universe. Also it is shown in [14] that for the case of
closed
universe, it violates the holographic bound.

The problem of taking apparent horizon (Hubble horizon) - the outermost surface defined by the null rays which instantaneously are not expanding, - as the IR cut-off in the flat universe, was discussed by Hsu [15]. According to Hsu’s argument, employing Friedman equation where is the total energy density and taking we will find . Thus either and behave as . So the DE results pressureless, since scales as like as matter energy density with the scale factor as . Also, taking apparent horizon as the IR cut-off may result the constant parameter of state , which is in contradiction with recent observations implying variable [16]. In our consideration for non-flat universe, because of the small value of we can consider our model as a system which departs slightly from flat space. Consequently we respect the results of flat universe so that we treat apparent horizon only as an arbitrary distance and not as the system’s IR cut-off.

On the other hand taking the event horizon, , where

(3) |

to be the IR cut-off, gives the results compatible with observations for flat universe.

It is fair to claim that simplicity and reasonability of HDE provides
more reliable frame to investigate the problem of DE rather than other models
proposed in the literature[2, 3, 4]. For instance the
coincidence or ”why now” problem is easily solved in some models of
HDE based on this fundamental assumption that matter and holographic
dark energy do not conserve separately, but the matter energy
density decays into the holographic energy density [17].
In fact a suitable evolution of the Universe is obtained when, in
addition to the holographic dark energy, an interaction (decay of
dark energy to matter) is assumed.

Some experimental data has implied that our universe is not a
perfectly flat universe and recent papers have favored the universe
with spatial curvature [18, 19]. As a matter of fact, we
want to remark that although it is believed that our universe is
flat, a contribution to the Friedmann equation from spatial
curvature is still possible if the number of e-foldings is not very
large [20]. Defining the appropriate distance, for the case
of non-flat universe has another story. Some aspects of the problem
has been discussed in [20, 21]. In this case, the event
horizon can not be considered as the system’s IR cut-off, because
for instance, when the dark energy is dominated and , where
is a positive constant, , we find , while we know that in this situation we must be in de Sitter
space with constant EoS. To solve this problem, another distance is
considered- radial size of the event horizon measured on the sphere
of the horizon, denoted by - and the evolution of holographic
model of dark energy in non-flat universe is investigated.

In present paper, using the holographic model of dark energy in
non-flat universe, we obtain equation of state for interacting
holographic dark energy density in a universe enveloped by as
the system’s IR cut-off.

## 2 Intracting holographic dark energy density

In this section we obtain the equation of state for the holographic energy density when there is an interaction between holographic energy density and a Cold Dark Matter(CDM) with . The continuity equations for dark energy and CDM are

(4) | |||

(5) |

The interaction is given by the quantity . This is a decaying of the holographic energy component into CDM with the decay rate . Taking a ratio of two energy densities as , the above equations lead to

(6) |

Following Ref.[22], if we define

(7) |

Then, the continuity equations can be written in their standard form

(8) |

(9) |

We consider the non-flat Friedmann-Robertson-Walker universe with line element

(10) |

where denotes the curvature of space k=0,1,-1 for flat, closed and open universe respectively. A closed universe with a small positive curvature () is compatible with observations [18, 19]. We use the Friedmann equation to relate the curvature of the universe to the energy density. The first Friedmann equation is given by

(11) |

Define as usual

(12) |

Now we can rewrite the first Friedmann equation as

(13) |

Using Eqs.(12,13) we obtain following relation for ratio of energy densities as

(14) |

In non-flat universe, our choice for holographic dark energy density is

(15) |

As it was mentioned, is a positive constant in holographic model of dark energy()and the coefficient 3 is for convenient. is defined as the following form:

(16) |

here, , is scale factor and can be obtained from the following equation

(17) |

where is event horizon. Therefore while is the radial
size of the event horizon measured in the direction, is the
radius of the event horizon measured on the sphere of the horizon.
^{2}^{2}2 As I have discussed in introduction, in non-flat case the
event horizon can not be considered as the system’s IR cut-off,
because if we use as IR cut-off, the holographic dark energy
density is given by

(20) |

where . Using definitions and , we get

(21) |

Now using Eqs.(16, 17, 20, 21), we
obtain ^{3}^{3}3Now we see that the above problem is solved when
is replaced with . According to eqs.(12,
15), the ratio of the energy density between curvature and
holographic dark energy is

(23) |

By considering the definition of holographic energy density , and using Eqs.( 21, 23) one can find:

(24) |

Substitute this relation into Eq.(4) and using definition , we obtain

(25) |

Here as in Ref.[23], we choose the following relation for decay rate

(26) |

with the coupling constant . Using Eq.(22), the above decay rate take following form

(27) |

Substitute this relation into Eq.(25), one finds the holographic energy equation of state

(28) |

If we take , then is bounded from below for a fixed by

(29) |

According to relation , when , in this case also, therefore in flat universe, the holographic energy equation of state take following form

(30) |

which is exactly the result of [22]. From Eqs.(7, 27, 28), we have the effective equation of state as

(31) |

If we take , and taking for the present time, the lower bound of is . Therefore it is impossible to have crossing . This implies that one can not generate phantom-like equation of state from an interacting holographic dark energy model in non-flat universe.

## 3 Conclusions

In order to solve cosmological problems and because the lack of our knowledge, for instance to determine what could be the best candidate for DE to explain the accelerated expansion of universe, the cosmologists try to approach to best results as precise as they can by considering all the possibilities they have. It is of interest to remark that in the literature, the different scenarios of DE has never been studied via considering special similar horizon, as in [24], in the standard cosmology framworke, the apparent horizon, , determines our universe while in [25], in the Brans-Dicke cosmology framworke, the universe is enclosed by event horizon, . As we discussed in introduction, for flat universe the convenient horizon looks to be while in non-flat universe we define because of the problems that arise if we consider or (these problems arise if we consider them as the system’s IR cut-off). In present paper, we studied , as the horizon measured from the sphere of the horizon as system’s IR cut-off. Then, by considering an interaction between holographic energy density and CDM, we have obtained the equation of state for the interacting holographic energy density in the non-flat universe.

## References

- [1] S. Perlmutter et al, Nature (London), 391, 51, (1998); Knop. R et al., Astroph. J., 598, 102 (2003); A. G. Riess et al., Astrophy. J., 607, 665(2004); H. Jassal, J. Bagla and T. Padmanabhan, Phys. Rev. D, 72, 103503 (2005).
- [2] For review on cosmological constant problem: P. J. E. Peebles, B. Ratra, Rev. Mod. Phys., 75, 559-606, (2003); J. Kratochvil, A. Linde, E. V. Linder, M. Shmakova, JCAP, 0407, 001, (2004).
- [3] R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett., 80, (1998) 1582; I. Zlater, L. Wang and P. J. Steinhardt, Phys. Rev. Lett., 82, (1999), 896; T. Chiba, gr-qc/9903094; M. S. Turner and M. White Phys. Rev. D, 56, (1997), 4439.
- [4] R. R. Caldwell, Phys. Lett. B 545, 23, (2002); R. R. Caldwell, M. Kamionkowsky and N. N. Weinberg, Phys, Rev, Lett, 91, 071301, (2003); S. Nojiri and S. D. Odintsov, Phys. Lett., B 562, (2003), 147; S. Nojiri and S. D. Odintsov, Phys. Lett., B 565, (2003), 1; S. Nojiri and S. D. Odintsov, Phys. Rev., D, 72, 023003, (2005); S. Nojiri, S. D. Odintsov, O. G. Gorbunova, J. Phys., A, 39, 6627, (2006); S. Capozziello, S. Nojiri, S. D. Odintsov, Phys. Lett. B 632, 597, (2006).
- [5] G. ’t Hooft, gr-qc/9310026 ; L. Susskind, J. Math. Phys, 36, (1995), 6377.
- [6] Y. S. Myung, Phys. Lett. B 610, (2005), 18-22.
- [7] A. Cohen, D. Kaplan and A. Nelson, Phys. Rev. Lett 82, (1999), 4971.
- [8] K. Enqvist, S. Hannestad and M. S. Sloth, JCAP, 0502, (2005) 004.
- [9] W. Fischler and L. Susskind, hep-th/9806039.
- [10] M. Li, Phy. Lett. B, 603, 1, (2004).
- [11] D. N. Vollic, hep-th/0306149; E. Elizalde, S. Nojiri, S. D. Odintsov, and P. Wang, Phys. Rev. D71, 103504, (2005); B. Guberina, R. Horvat, and H. Stefancic, JCAP, 0505, 001, (2005); B. Guberina, R. Horvat, and H. Nikolic, Phys. Lett. B636, 80, (2006) ;H. Li, Z. K. Guo and Y. Z. Zhang, astro-ph/0602521; J. P. B. Almeida and J. G. Pereira, gr-qc/0602103; D. Pavon and W. Zimdahl, hep-th/0511053; Y. Gong, Phys. Rev., D, 70, (2004), 064029; B. Wang, E. Abdalla, R. K. Su, Phys. Lett., B, 611, (2005), M. R. Setare, and S. Shafei, JCAP, 09, 011, (2006).
- [12] Q. G. Huang, Y. Gong, JCAP, 0408, (2004),006.
- [13] Q. G. Huang, M. Li, JCAP, 0503, (2005), 001.
- [14] R. Easther and D. A. Lowe hep-th/9902088.
- [15] S. D. H. Hsu, Phys. Lett. B, 594, 13, (2004).
- [16] U. Alam, V. Sahni, T. D. Saini, A. A. Starobinsky, Mon. Not. Roy. Astron. Soc., 354, 275 (2004); D. Huterer and A. Cooray, Phys. Rev., D, 71, 023506, (2005), Y. Wang and M. Tegmark, astro-ph/0501351.
- [17] L. Amendola, Phys. Rev. D 60, 043501, (1999); L. Amendola and D. Tocchini-Valentini, Phys. Rev. D 64, 043509 (2001); W. Zimdahl, D. J. Schwarz, A. B. Balakin and D. Pavon, Phys. Rev. D 64, 063501 (2001); A. B. Balakin, D. Pavon, D. J. Schwarz and W. Zimdahl, New J. Phys. 5, 085 (2003); R. Horvat, Phys. Rev. D70, 087301, (2004); P. Wang and X. H. Meng, Class. Quant. Grav. 22, 283 (2005); R. G. Cai and A. Wang, JCAP 0503, 002 (2005); D. Pavon and W. Zimdahl, Phys. Lett. B 628, 206 (2005); W. Zimdahl, Int. J. Mod. Phys. D 14, 2319 (2005); B. Wang, Y. g. Gong and E. Abdalla, Phys. Lett. B 624, 141 (2005); B. Wang, C. Y. Lin and E. Abdalla, arXiv:hep-th/0509107; D. Pavon and W. Zimdahl, arXiv:hep-th/0511053; M. S. Berger and H. Shojaei, Phys. Rev. D 73, 083528, (2006); B. Hu, Y. Ling, Phys. Rev. D73, 123510 (2006) ; M. R. Setare, hep-th/0609104.
- [18] C. L. Bennett et al., Astrophys. J. Suppl. 148, 1 (2003); D. N. Spergel, Astrophys. J. Suppl. 148, 175, (2003).
- [19] M. Tegmark et al., astro-ph/0310723.
- [20] Q. G. Huang and M. Li, JCAP, 0408, 013, (2004).
- [21] B. Guberina, R. Horvat and H. Nikolić, Phys. Rev. D 72, 125011, (2005).
- [22] H. Kim, H. W. Lee and Y. S. Myung, Phys. Lett. B 632, 605, (2006).
- [23] B. Wang, Y. Gong, and E. Abdalla, Phys. Lett. B 624, 141 (2005).
- [24] P. C. Davies, Class. Quantum. Grav. L225, 4, (1987).
- [25] Y. Gong, Phys. Rev. D, 70, 064029, (2004).