The recent remarkable cosmological observations from high red shift Ia supernovae (SNIa) (Perlmutter et al.   , Riess et al.   , Astier et al.  , Spergel et al.  , Davis et al.  ) indicate that our universe is accelerating and confirmed later by cross-checks from cosmic microwave background radiation (Bennett et al.  , Spergel et al.  ) and large scale structure (Verde et al.  , Hawkins et al.  , Abazajian et al.    , Tegmark et al.  ) suggests that the universe is spatially flat and dominated by exotic component with large negative pressure. This component is usually referred to as dark energy (Weinberg et al.  , Carroll et al.  , Peebles et al.  , Padmanabhan et al.  ). Astronomical observations indicate that the universe consists of approximately 2/3 dark energy and 1/3 dark matter. The nature of dark energy and dark matter is unknown and many radically different models have been proposed. In order to explain anomalous cosmological observations in cosmic microwave background (CMB) at largest angles, Koivisto et al.  have suggested cosmological model with anisotropic and viscous dark energy.
Among all dark energy models, a holographic dark energy (HDE) models have received the remarkable attention (Cohen et al.  , Horava et al.  , Thomas et al.  , Li et al.  ). According to holographic principle, the number of degrees of freedom in a bounded system should be finite and related to the area of its boundary (Hooft et al.  ) and with the help of this principle, a field theoretical relation between a short distance (ultraviolet) cut off and a long distance (infrared) cut off was established (Cohen et al.  ) which ensures that the energy in a box of size L which has a cosmological length scale, does not exceed the energy of black hole of the same size. Different dark energy models are due to different types of these cut off.
Bianchi models have been studied by several authors to achieve a better understanding of the observed small amount of anisotropy in the universe. The simple Bianchi family containing flat FRW universe as a special case is the type-I space-times. The Bianchi type-V universe is a generalization of the open universe in FRW cosmology. Hence, its study as dark energy models with non-zero curvature (Coles et al.  ) in higher dimension is important. Holographic dark energy models have been tested and constrained by various astronomical observations (Zhang et al.  , Enqvist et al.  , Shen et al.  , Chang et al.  ). The special class models are the models in which holographic dark energy is allowed to interact with dark matter (Carvalho et al.  , Huang et al.  , Gong et al.   , Pavon et al.  , Wang et al.  , Perivolaropoulos et al.  , Nojiri et al.  , Guberina et al.   , Guo et al.    , Hu et al.  , Li et al.  , Setare et al.   , Sadjadi et al.  , Banerjee et al.  , Zimdahl et al.  ). Sarkar et al.    have studied non interacting holographic dark energy with linearly varying deceleration parameter in Bianchi type-I and Bianchi type-V universe and also interacting holographic dark energy in Bianchi type-II.
Spatially homogeneous and anisotropic cosmological models play a significant role in the description of large behavior of the universe, and many authors have been widely studied such models in the search of a relativistic picture of the early universe. Anisotropic Bianchi type-I, Bianchi type-II and Bianchi type-V dark energy models have been extensively studied by (Adhav K. S.   , Pradhan A. et al.  ). Kumar and Yadav  have constructed some Bianchi type-V cosmological models of accelerating universe with dark energy in general relativity by assuming constant deceleration parameter in order to solve Einstein’s field equations. The role of dark energy with variable equation of state parameter is studied in details within the evolution of Bianchi type-V universe and conjointly discovered that dark energy dominates the universe at the present epoch. Pradhan and Amirhashchi  have constructed an accelerating dark energy model and explored some new exact solutions of Einstein’s field equations in a spatially and anisotropic Bianchi type-V space-time with minimally interaction of perfect fluid and dark energy components. Adhav et al.  explored anisotropic and homogeneous Bianchi type-I universe field with interacting dark matter and holographic dark energy. Som and Sil  discussed the general approach of interacting holographic dark energy model.
Motivated by these investigations, we have constructed spatially homogeneous and anisotropic Bianchi type-V universe field with interacting dark matter and holographic dark energy. In this paper, we obtained the exact solutions of Einstein’s field equations by using variable deceleration parameter in the form .
2. Metric and Field Equations
The Bianchi Type-V metric can be written as
where , and are cosmic scale factors and is an arbitrary constant.
The Einstein’s field equations in natural limit (8πG = 1 and c = 1) are,
are energy momentum tensor for dark matter (pressureless, i.e. ) and holographic dark energy respectively. Here the quantity is the energy density of dark matter and , are energy density and pressure of holographic dark energy respectively.
In co-moving system, the Einstein field Equations (2.2) for the metric (2.1), using Equations (2.3) can be written as
where an overhead dot (?) represents derivative with respect to time t.
On integrating the Equation (2.8), we obtain
where is an integration constant.
On taking , without loss of generality, the volume scale factor V and average scale factor a is given by
Subtracting Equation (2.5) from Equation (2.6), Equation (2.6) from Equation (2.7), Equation (2.5) from Equation (2.7) and using Equation (2.10), we get
On integrating Equations (2.11a)-(2.11c) and using Equations (2.9) and (2.10), the scale factors , and can be written as,
where X and D are constants of integration.
The holographic dark energy density is given by,
i.e. with (Granda et al.  ), where and are constants.
For the universe, where dark energy and dark matter are interacting with each other, the total energy density satisfies the equation of continuity as,
It is assumed that the dark matter component is interacting with the dark energy through an interacting term Q, the continuity equation of matter and dark energy can be obtained as,
where is the equation of state parameter for the holographic dark energy and measures the strength of interaction. A vanishing Q implies that the dark matter and dark energy are separately conserved. In view of continuity equations, the interaction between dark energy and dark matter must be a function of the energy density multiplied by a quantity, with units of inverse of time, which can be chosen as the Hubble parameter H. It’s a freedom to choose the form of energy density which can be any combination of dark energy and dark matter. Thus interaction between dark energy and dark matter could be expressed phenomenologically in the form as (Guo et al.   , Amendola et al.  ),
where is the coupling constant.
Cai and Wang  have taken the same relation for interacting dark matter and phantom dark energy in order to avoid the coincidence problem.
Using Equations (2.15) and (2.17), we get the energy density of dark matter as,
where is a real constant of integration.
Using Equations (2.17) and (2.18), we get the interacting term Q as,
3. Cosmological Solution for Variable Deceleration Parameter
We consider the deceleration parameter to be a variable
and following Pradhan et al.  and Chawla et al.  , we assume the law of variation of scale factor as increasing function of time
Using (3.2) in Equations (2.12a)-(2.12c), we obtain the expressions for scale factors as,
where X and D are constants of integration.
Using Equations (2.10) and (3.2) in Equations (2.17) and (2.18), we obtained,
Using Equations (3.2)-(3.5) and (3.7) in Equation (2.4) we obtained the density of holographic dark energy as,
Using Equation (3.3)-(3.5) in Equation (2.7), we obtained the pressure as,
The EOS parameter of holographic dark energy is given by,
The physical parameters such as spatial volume V, Hubble parameter H, expansion scalar and the time varying deceleration parameter q are obtained as,
The shear scalar and mean isotropy parameter are given by,
In this paper, we have presented spatially homogeneous and anisotropic Bianchi Type-V universe field with interacting dark matter and holographic dark energy. With the consideration of variable deceleration parameter, we obtained the solutions of Einstein’s field equations.
It is found that the universe approaches to isotropy for large cosmic time as shown by different observational data and dark energy is responsible for expansion of universe. The concluding remarks of the model are as follows.
1) The sign of q represents that the universe is decelerating or accelerating i.e. a positive sign of q represents accelerating universe and negative sign of q represents decelerating universe.
In our model for and for i.e. the model represents the decelerating to accelerating phase and the values of deceleration parameter lie in the phase .
2) From the Equation (3.11), we can say that the spatial volume V is finite at t = 0 and expands as t increases and becomes infinite for .
3) From Equation (3.16), we can conclude that for the large cosmic time (i.e. ), the anisotropy parameter . Therefore, for the large cosmic time, the anisotropy of the universe damped out and the universe approaches to an isotropy universe.
4) From the Equation (3.12), it is observed that the directional Hubble parameter diverges for t = 0 and converges for .
5) We observe that (i.e. pressure of dark energy) tends to a negative value for large cosmic time which shows that the universe is accelerating (SNeIa).
6) From the Equation (3.10), the EOS parameter for large cosmic time. In this case, the holographic dark energy looks like phantom energy, (Abazajian et al.  , Ade et al.  , Riess et al.   , Zimdahl  ). Our result is consistent with SNeIa.
7) For β = 0 in Equation (2.1), the investigated model approaches to Mete, et al.  .
 Chawla, C., Mishra, R.K. and Pradhan, A. (2012) Anisotropic Bianchi-I Cosmological Models in String Cosmology with Variable Deceleration Parameter. Romanian Journal of Physics, 58, 1000. arXiv:1203.4014 [physics.gen-ph]
 Riess, A.G., et al. (1998) Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. The Astronomical Journal, 116, 1009-1038. https://doi.org/10.1086/300499
 Riess, A.G., et al. (2004) Type Ia Supernova Discoveries at z > 1 from the Hubble Space Telescope: Evi-dence for Past Deceleration and Constraints on Dark Energy Evolution. The Astrophysical Journal, 607, 665-687. https://doi.org/10.1086/383612
 Davis, T.M., et al. (2007) Scrutinizing Exotic Cos-mological Models Using Essence Supernova Data Combined With Other Cosmological Probes. The Astrophysical Journal, 666, 716-725. https://doi.org/10.1086/519988
 Bennett, C.L., Halpern, M., Hinshaw, G., Jarosik, N., Kogut, A., et al. (2003) First Year Wilkinson Microwave Anisotropy Probe (WMAP) Ob-servations: Preliminary Maps and Basic Results. The Astrophysical Journal, 148, 1-43.
 Spergel, D.N., Verde, L., Peiris, H.V., Komatsu, E., Nolta, M.R., et al. (2003) First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations Determination of Cosmological Parameters. The Astrophysical Journal, 148, 175-194.
 Verde, L., et al. (2002) The 2dF Galaxy Redshift Survey: The Bias of Galaxies and the Density of the Universe. Monthly Notices of the Royal Astronomical Society, 335, 432-440. https://doi.org/10.1046/j.1365-8711.2002.05620.x
 Hawkins, E., et al. (2003) The 2dF Galaxy Redshift Survey: Correlation Functions, Peculiar Velocities and the Matter Density of the Universe. Monthly Notices of the Royal Astronomical Society, 346, 78-96.
 Koivisto, T. and Mota, D.F. (2008) Anisotropic Dark Energy: Dynamics of the Background and Perturbations. Journal of Cosmology and Astroparticle Physics, 2008, 18. https://doi.org/10.1088/1475-7516/2008/06/018
 Cohen, A.G., Kaplan, D.B. and Nel-son, A.E. (1999) Effective Field Theory, Black Holes, and the Cosmological Constant. Physics Review Letters, 82, 4971-4974.
 Enqvist, K., Hannestad, S. and Sloth, M.S. (2005) Searching for a holographic connection between Dark Energy and the Low/CMB Multipoles. Journal of Cosmology and Astroparticle Physics, 2005, 4. https://doi.org/10.1088/1475-7516/2005/02/004
 Shen, J., Wang, B., Abdalla, E. and Su, R.-K. (2005) Constraints on the Dark Energy from the Holographic Connection to the Small l CMB Suppression. Physics Letters B, 609, 200-205. https://doi.org/10.1016/j.physletb.2005.01.051
 Chang, Z., Wu, F.-Q. and Zhang, X. (2006) Constraints on Holographic Dark Energy from X-Ray Gas Mass Fraction of Galaxy Clusters. Physics Letters B, 633, 14-18. https://doi.org/10.1016/j.physletb.2005.10.095
 Carvalho, F.C. and Saa, A. (2004) Nonminimal Coupling, Exponential Potentials and the ω1 Regime of Dark Energy. Physical Review D, 70, Article ID: 087302.
 Wang, B., Gong, Y.G. and Abdalla, E. (2005) Transition of the Dark Energy Equation of State in an Interacting Holographic Dark Energy Model. Physics Letters B, 624, 141-146. https://doi.org/10.1016/j.physletb.2005.08.008
 Perivolaropoulos, L. (2005) Crossing the Phantom Divide Barrier with Scalar Tensor Theories. Journal of Cosmology and As-troparticle Physics, 2005, 1.
 Nojiri, S. and Odintsov, S.D. (2006) Unifying Phantom Inflation with Late-Time Acceleration: Scalar Phan-tom-Non-Phantom Transition Model and Generalized Holographic Dark Energy. General Relativity and Gravitation, 38, 1285-1304.
 Guberina, B., Horvat, R. and Nikolic, H. (2005) Generalized Holographic Dark Energy and the IR Cutoff Problem. Physics Review D, 72, Article ID: 125011.
 Sadjadi, H.M. (2007) The Particle versus the Future Event Horizon in an Interacting Holographic Dark Energy Model. Journal of Cosmology and Astroparticle Physics, 2007, 26. https://doi.org/10.1088/1475-7516/2007/02/026
 Sarkar, S. (2014) Holographic Dark En-ergy Model with Linearly Varying Deceleration Parameter and Generalised Chaplygin Gas Dark Energy Model in Bianchi Type-I Universe. Astrophysics and Space Science, 349, 985-993.
 Sarkar, S. (2014) Interacting Holographic Dark Energy with Variable Deceleration Parameter and Accreting Black Holes in Bianchi Type-V Universe. Astrophysics and Space Science, 352, 245-253. https://doi.org/10.1007/s10509-014-1876-0
 Sarkar, S. (2014) Interacting Holographic Dark Energy with Variable Deceleration Parameter and Tachyon Scalar Field Dark Energy Model in LRS Bianchi Type-II Universe. Astrophysics and Space Science, 350, 821-829.
 Adhav, K.S. (2011) State-finder Diagnostic for Variable Modified Chaplygin Gas in Bianchi Type-V Universe. As-trophysics and Space Science, 335, 611-617.
 Pradhan, A., Amirhashchi, H. and Saha, B. (2011) Bianchi Type-I Anisotropic Dark Energy Model with Constant Deceleration Parameter. International Journal of Theoretical Physics, 50, 2923-2938. https://doi.org/10.1007/s10773-011-0793-z
 Adhav, K.S., Tayade, G.B. and Bansod, A.S. (2014) Interacting Dark Matter and Holographic Dark Energy in an Anisotropic Universe. Astrophysics and Space Science, 353, 249-257. https://doi.org/10.1007/s10509-014-2015-7
 Amendola, L., Campos, G.C. and Rosenfeld, R. (2007) Consequences of Dark Matter-Dark Energy Inter-action on Cosmological Parameters Derived from Type Ia Supernova Data. Physical Review D, 75, Article ID: 083506.
 Cai, R.-G. and Wang, A. (2005) Cosmology with Interaction between Phantom Dark Energy and Dark Matter and the Coincidence Problem. Journal of Cosmology and Astroparticle Physics, 2005, 2. https://doi.org/10.1088/1475-7516/2005/03/002
 Pradhan, A., Jaiswal, R., Jotania, K. and Khare, R.K. (2012) Dark Energy Models with Anisotropic Fluid in Bianchi Type-VI0 Space-Time with Time Dependent Deceleration Parameter. Astrophysics and Space Science, 337, 401-413.
 Riess, A.G., et al. (2007) New Hubble Space Telescope Discoveries of Type Ia Supernovae at z ≥ 1: Narrowing Constraints on the Early Behavior of Dark Energy. Astrophysical Journal, 659, 98-121. https://doi.org/10.1086/510378