Accretions of various types of dark energies onto Morris–Thorne wormhole

In this work, we have studied the accretion of dark energies onto a Morris–Thorne wormhole. Previously, in ref. (González-Díaz, arXiv:hep-th/0607137), it was shown that for quintessence like dark energy, the mass of the wormhole decreases, and for phantom like dark energy, the mass of the wormhole increases. We have assumed two types of dark energy: the variable modified Chaplygin gas and the generalized cosmic Chaplygin gas. We have found the expression of the wormhole mass in both cases. We have found the mass of the wormhole at late universe and this is finite. For our choices of the parameters and the function B(a)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(a)$$\end{document}, these models generate only quintessence dark energy (not phantom) and so the wormhole mass decreases during the evolution of the universe. Next we have assumed the five kinds of parametrizations of well-known dark-energy models. These models generate both quintessence and phantom scenarios i.e., phantom crossing models. So if these dark energies accrete onto the wormhole, then for the quintessence stage, the wormhole mass decreases up to a certain value (a finite value) and then again increases to an infinite value for the phantom stage during whole evolution of the universe. That means that if the five kinds of DE accrete onto a wormhole, the mass of the wormhole decreases up to a certain finite value and then increases in the late stage of the evolution of the universe. We have also shown these results graphically.


I. INTRODUCTION
In recent observations it is strongly believed that the universe is experiencing an accelerated expansion.The type Ia Supernovae and Cosmic Microwave Background (CMB) [1,2] observations have shown the evidences to support cosmic acceleration.This acceleration is caused by some unknown matter which has the property that positive energy density and negative pressure satisfying ρ + 3p < 0 is dubbed as "dark energy" (DE) [3][4][5][6].If ρ + p < 0, it is dubbed as "phantom energy".The combined astrophysical observations suggests that universe is spatially flat and the dark energy occupies about 70% of the total energy of the universe, the contribution of dark matter is ∼ 26%, the baryon is 4% and negligible radiation.A cosmological property in which there is an infinite expansion in scale factor in a finite time termed as 'Big Rip'.In the phantom cosmology, big rip is a kind of future singularity in which the energy density of phantom energy will become infinite in a finite time.To realize the Big Rip scenario the condition ρ + p < 0 alone is not sufficient [7].Distinct data on supernovas showed that the presence of phantom energy with −1.2 < w < −1 in the Universe is highly likely [8].In this case the cosmological phantom energy density grows at large times and disrupts finally all bounded objects up to subnuclear scale.
A wormhole is a feature of space that is essentially a "shortcut" from one point in the universe to another point in the universe, allowing travel between them that is faster than it would take light to make the journey through normal space.So the wormholes are tunnels in spacetime geometry that connect two or more regions of the same spacetime or two different spacetimes [9].Wormholes may be classified into two categories -Euclidean wormholes and Lorentzian wormholes.The Euclidean wormholes arise in Euclidean quantum gravity and the Lorentzian wormholes [10] which are static spherically symmetric solutions of Einstein's general relativistic field equations [11].In order to support such exotic wormhole geometries, the matter violating the energy conditions (null, weak and strong), but average null energy condition is satisfied in wormhole geometries [12][13][14].For small intervals of time, the weak energy condition (WEC) can be satisfied [15].Also the traversable wormhole solutions of the field equations are obtained [16].Recently evolving wormhole solutions and their implications are discussed by several authors [17][18][19][20][21][22][23][24].
The equations of motion for steady-state spherical symmetric flow of matter into or out of a condensed object (e.g.neutron stars, 'black holes', etc.) are discussed by Michel [25] and also obtained analytic relativistic accretion solution onto the static Schwarzschild black hole.The accretion of phantom dark energy onto a Schwarzschild black hole was first modelled by Babichev et al [26,27].They established that black hole mass will gradually decrease due to strong negative pressure of phantom energy and finally all the masses tend to zero near the big rip where it will disappear.Accretion of phantom like variable modified Chaplygin gas onto Schwarzschild black hole was studied by Jamil [28] who showed that mass of the black hole will decrease when accreting fluid violates the dominant energy condition and otherwise will increase.Also the accretion of dark energy onto the more general Kerr-Newman black hole was studied by Madrid et al [29] and Bhadra et al [30].Till now, several authors [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] have discussed the accretion of various components of dark energy onto black holes.Recently, there is a great interest of the study of dark energy accretion onto static wormhole [46][47][48].The phantom energy accretion onto wormhole is discussed by González-D íaz [49].Madrid et al [57] studied the dark energy accretion onto black holes and worm holes phenomena could lead to unexpected consequences, allowing even the avoidance of the considered singularities.Also Mart ín-Moruno [51] have considered a general formalism for the accretion of dark energy onto astronomical objects, black holes and wormholes.It has been shown that in models with four dimensions or more, any singularity with a divergence in the Hubble parameter may be avoided by a big trip, if it is assumed that there is no coupling between the bulk and this accreting object.If this is not the case in more than four dimensions, the evolution of the cosmological object depends on the particular model.The dark energy accretion onto wormhole in accelerating universe has also been discussed [52,53] recently.
In the following section, we assume the Morris-Thorne static wormhole in presence of dark energy filled universe.If dark energy accretes onto the wormhole, the rate of change of mass of the wormhole is expressed in terms of the density and pressure of dark energy and also find the expression of wormhole mass in terms of density.Our main motivation of the work is to examine the natures of the mass of the wormhole during expansion of the universe if several kinds of dark energies accrete around the wormhole.The candidates of dark energy are assumed to be variable modified Chaplygin gas (VMCG) and generalized cosmic Chaplygin gas (GCCG).Also we have assumed some kinds of parametrizations of dark energy candidates.The mass of the wormhole has been calculated for all types of dark energies and its natures have been analyzed during expansion of the universe.Finally, we give some concluding remarks of the whole work.

II. ACCRETION PHENOMENA OF DARK ENERGY ONTO WORMHOLE
Let us consider a spherically symmetrical accretion of the dark energy onto the wormhole.We consider a non-static spherically symmetric Morris-Thorne wormhole metric [9] given by where the functions K(r) and Φ(r) are the shape function and redshift function respectively of radial co-ordinate r.If K(r 0 ) = r 0 , the radius r 0 is called wormhole throat radius.So we want to consider the outward region such that r 0 ≤ r < ∞.
A proper dark-energy accretion model for wormholes should be obtained by generalizing the Michel theory [25] to the case of wormholes.Such a generalization has been already performed by Babichev et al [26,27] for the case of dark-energy accretion onto Schwarzschild black holes.We shall follow now the procedure used by Babichev et al [26,27], adapting it to the case of a Morris-Thorne wormhole [52].For this purpose, we consider the energy-momentum tensor for the dark energy (DE) in the form of perfect fluid having the EoS p = p(ρ), is where ρ and p are the energy density and pressure of the dark energy respectively and u µ = dx µ ds is the fluid 4-velocity satisfying u µ u µ = −1.We assume that the in-falling dark energy fluid does not disturb the spherical symmetry of the wormhole.Now we assume Φ(r) = 0.The relativistic Bernoulli's equation after the time component of the energy-momentum conservation law T µν ;ν = 0 provide the first integral of motion for static, spherically symmetric accretion onto wormhole which yields [52] where M is the exotic mass of the wormhole, u = dr ds (> 0) is the radial component of the velocity four vector and the integration constant C 1 has the dimension of the energy density.
Moreover, the second integration of motion is obtained from the projection of the conservation law for energymomentum tensor onto the fluid four-velocity, u µ T µν ;ν = 0, which gives [52] where C 2 (> 0) is dimensionless integration constant, ρ ∞ is the dark energy density at infinity.Further the value of the constant C 2 can be evaluated for different dark energy models.Now we denote c 2 s = dp/dρ, the square speed of sound of the accreted fluid.Critical point can be calculated by taking logarithmic differential of equations ( 3) and ( 4), and put the multipliers of dr/r and du/u to zero.After the calculation, we obtain c 2 s (r * ) = 0 and u * = 0 at the critical point r = r * .So for this accretion, the critical point cannot be found.If we include Φ(r) for accretion process, the critical point may be found [54].
The rate of change of mass Ṁ of the exotic wormhole is computed by integrating the flux of the dark energy over the entire two dimensional surface of the wormhole i.e., Ṁ = T r t dS, where T r t represents the radial component of the energy-momentum densities and dS = √ −gdθdφ = r 2 sin θdθdφ [26,55] is the element of the wormhole surface.Using the above equations we obtain the rate of change of mass as [52] where Q is a positive constant.For the relevant asymptotic regime r → ∞, the above equation reduces to We see that the rate for the wormhole exotic mass due to accretion of dark energy becomes exactly the negative to the similar rate in the case of a Schwarzschild black hole, asymptotically.Since the Morris-Thorne wormhole is static, so the mass of the wormhole depends on r only.When some fluid accretes outside wormhole, the mass function M of the wormhole is considered as a dynamical mass function and hence it should be a function of time also.So Ṁ of the equation ( 6) is time dependent and the increasing or decreasing of the wormhole mass M sensitively depends on the nature of the fluid which accretes upon the wormhole.If ρ + p < 0 i.e., for phantom dark energy accretion, the mass of the wormhole increases but if ρ + p > 0 i.e., for non-phantom dark energy accretion, the mass of the wormhole decreases.In the following, we shall assume different types of dark energy models such as variable modified Chaplygin gas, generalized cosmic Chaplygin gas and parametrizations of some kinds of well known dark energy models.The natures of mass function of wormhole will be analyzed for present and future stages of expansion of the universe when the above types of dark energies are accreting upon wormhole.

A. Variable Modified Chaplygin Gas as dark energy model
We consider the background spacetime is spatially flat represented by the homogeneous and isotropic FRW model of the universe which is given by where a(t) is the scale factor.We assume the universe is filled with Variable Modified Chaplygin Gas (VMCG) and the EoS is [56] given by Here B(a) is the function of scale factor a and for simplicity we choose B(a) = B 0 a −n , where B 0 > 0 and n > 0 are constants.
The Einstein's equations for FRW universe are (choosing 8πG = c = 1) Conservation equation satisfied by the dark energy model VMCG is where H = ȧ a is the Hubble parameter.Using equations ( 7) and ( 10), we have the solution of ρ as where C > 0 is an arbitrary integration constant and 3(1 + A)(1 + α) > n, for positivity of first term.
Using equations ( 6), ( 9) and ( 11), we have [51,57] which integrates to yield where, M 0 and ρ 0 are the present values of the wormhole mass and density of the dark energy respectively.Using eq. ( 9), we get the mass function in terms of the Hubble parameter H as in the form [51,57] where H 0 is the present value of the Hubble parameter.In the late stage of the universe i.e., a is very large (z → −1), the mass of the wormhole will be If we put the solution ρ from equation (12) in equation ( 14), the mass of wormhole M can be expressed in terms of scale factor a and then use the formula of redshift z = 1 a − 1, M will be in terms of redshift z.Now M vs z is drawn in figure 1.Our choice of the function B(a), the VMCG gives only the quintessence dark energy, not phantom dark energy.So mass of the wormhole always decreases for our case.From figure, we see that M decreases with z dereases.So mass of wormhole decreases if the VMCG accretes onto the wormhole.

B. Generalized Cosmic Chaplygin Gas as the dark energy Model
A new version of Chaplygin gas which is known as Generalized Cosmic Chaplygin Gas (GCCG) [58,59] obeys the equation of state (17) where C = A ′ (1+w) − 1, A ′ takes either positive or negative constant, −l < w < 0, 0 ≤ α ≤ 1 and l > 1.The EOS reduces to that of current Chaplygin unified models for dark matter and dark energy in the limit w → 0 and satisfies the conditions: (i) it becomes a de Sitter fluid at late time and when w = −1, (ii) it reduces to p = wρ in the limit that the Chaplygin parameter A ′ → 0, (iii) it also reduces to the EOS of current Chaplygin  unified dark matter models at high energy density and (iv) the evolution of density perturbations derived from the chosen EOS becomes free from the pathological behaviour of the matter power spectrum for physically reasonable values of the involved parameters at late time.This EOS shows dust era in the past and ΛCDM in the future.
From the conservation equation ( 10), we have the expression for energy density of GCCG in the form [58,59] In the late stage of the universe i.e., a is very large (z → −1), the mass of the wormhole will be If we put the solution ρ from equation (18) in equation ( 14), the mass of wormhole M can be expressed in terms of scale factor a and hence M will be in terms of redshift z.Now M vs z is drawn in figure 2. The GCCG gives only the quintessence dark energy, not phantom dark energy.So mass of the wormhole always decreases for our case.From figure, we see that M decreases with z decreases.So mass of wormhole decreases if the GCCG accretes onto the wormhole.

C. Some Parameterizations of dark energy Models
In astrophysical sense, the dark energy is popular to have a redshift parametrization (i.e., taking the redshift z as the variable parameter of the EoS only) of the EoS as p(z) = w(z)ρ(z).The EoS parameter w is currently constrained by the distance measurements of the type Ia supernova and the current observational data constrain the range of EoS as −1.38 < w < −0.82 [60].Recently, the combination of WMAP3 and Supernova Legacy Survey data shows a significant constraint on the EOS w = −0.97+0.07 −0.09 for the DE, in a flat universe [61].Two mainstream families of red shift parametrizations are considered here, viz., (i) Family I: w(z) = w 0 + w 1 z 1+z n .In this case, the conservation equation (11) gives the solution ρ = ρ 0 (1 + z) 3(1+w0) e 3(−1) −n w1{Beta[1+z,−n,1+n]+π cosec nπ} (20) (ii) Family II: where, w 0 and w 1 are two unknown parameters, which can be constrained by the recent observations and n is a natural number.For different values of n, we will get following three models of well known parametrizations (Models I, II, III).We shall also assumed other two parametrizations (Models IV, V).Since the following models generate both quintessence (w(z) > −1) and phantom (w(z) < −1) dark energies for some suitable choices of the parameters.So phantom divide is possible at ΛCDM stage w(z) = −1.At the first stage, it occurs quintessence and late stage it occurs phantom.So for quintessence stage, the mass of the wormhole decreases and decreasing upto a certain limit (of mass) and then again at phantom stage, the mass of the wormhole increases.
• Model I (Linear): For n = 0, family II reduces to the parametrization form w(z) = w 0 + w 1 z [62].This is known as "Linear" parametrization.For Linear parametrization, the solution becomes The above model generates phantom energy if w(z) < −1 i.e, z < − 1+w0 w1 provided w 1 > 0 and w 1 − w 0 > 1.If we drop this restriction, this model gives quintessence type dark energy.Since this model is the phantom crossing model, so if this dark energy accretes onto wormhole, for quintessence era, wormhole mass decreases upto a certain limit and after that for phantom era, the mass of the wormhole increases.We have shown this scenario in figure 3. We see that wormhole mass M decreases for redshift z decreases upto certain stage of z (ΛCDM stage) and then M increases (phantom era) as universe expands.
• Model II (CPL): For n = 1, both the families I and II lead to the same parametrization w(z) = w 0 +w 1 z 1+z .This is known as "CP L" parametrization [63,64].The solution becomes The above model generates phantom energy if w(z) < −1 i.e, z < − 1+w0 1+w1 provided w 1 > −1 and w 1 − w 0 > 0. If we drop this restriction, this model gives quintessence type dark energy.This model is also the phantom crossing model.From figure 4, we see that wormhole mass M decreases for redshift z decreases upto certain stage of z (ΛCDM stage) and then M increases (phantom era) as universe expands.
• Model IV: type of parametrization is considered as w , where A 0 , A 1 and A 2 are constants [66,67].This ansatz is exactly the cosmological constant w = −1 for A 1 = A 2 = 0 and DE models with w = −2/3 for A 0 = A 2 = 0 and w = −1/3 for A 0 = A 1 = 0.In this case, we get the solution The above model generates phantom energy if w(z) < −1 i.e, z < −1 − A1 A2 provided A 0 < 0, A 1 > 0, A 2 < 0 and A 0 + A 1 + A 2 < 0. For this condition, ρ is still positive.If we drop this restriction, this model gives quintessence type dark energy.This model is also the phantom crossing model.From figure 6, we see that wormhole mass M decreases for redshift z decreases upto certain stage of z and then M increases (phantom era) as universe expands.
• Model V: Other type of parametrization is assumed to be w(z) = w 0 + w 1 log(1 + z) [68,69].The solution is obtained as  The above model generates phantom energy if w(z) < −1 i.e, z < −1 + e − w 0 w 1 provided w 1 > 0. If we drop this restriction, this model gives quintessence type dark energy.This model is also the phantom crossing model.From figure 7, we see that wormhole mass M decreases for redshift z decreases upto certain stage of z and then M again increases (phantom era) as universe expands.

III. DISCUSSIONS
In this work, we have studied accretion of the dark energies onto Morris-Thorne wormhole.A proper dark-energy accretion model for wormholes have been obtained by generalizing the Michel theory [25] to the case of wormholes.Such a generalization has been already performed by Babichev et al [26,27] for the case of dark-energy accretion onto Schwarzschild black holes.We have followed the procedure used by Babichev et al [26,27], adapting it to the case of a Morris-Thorne wormhole.Here we have assumed the redshift function Φ(r) = 0. We have shown that if Φ(r) = 0, the critical point cannot be found.Astrophysically, mass of the wormhole is a dynamical quantity, so the nature of the mass function is important in our wormhole model for different dark energy filled universe.The sign of time derivative of wormhole mass depends on the signs of ρ + p.For quintessence like dark energy, the mass of the wormhole decreases and phantom like dark energy, the mass of wormhole increases.We have assumed recently proposed two types of dark energy like variable modified Chaplygin gas (VMCG) and generalized cosmic Chaplygin gas (GCCG).We have found the expression of wormhole mass in both cases.We have found the mass of the wormhole at late universe and this is found to be finite.Our dark energy fluids violate the strong energy condition (ρ + 3p < 0 in late epoch), but do not violate the weak energy condition (ρ + p > 0).So the models drive only quintessence scenario in late epoch, but do not generate the phantom epoch (in our choice).So wormhole mass decreases during evolution of the universe for these two dark energy models.Previously Babichev et al [26] have shown that the mass of black hole decreases due to phantom energy accretion.But for wormhole accretion, the mass of wormhole increases due to phantom energy, which is the opposite behaviour of black hole mass.Since our considered dark energy candidates do not violate weak energy condition, so the dynamical mass of the wormhole are decaying by the accretion of our considered dark energies, though the pressures of the dark energies are outside the wormhole.From figures 1 and 2, we observe that the wormhole mass decreases as z increases for both VMCG and GCCG, which accrete onto the wormhole in our expanding universe.Next we have assumed 5 kinds of parametrizations (Models I-V) of well known dark energy models (some of them are Linear, CPL, JBP models).These models generate both quintessence and phantom scenarios for some restrictions of the parameters.So if these dark energies accrete onto the wormhole, then for quintessence stage, wormhole mass decreases upto a certain value (finite value) and then again increases to infinite value for phantom stage during whole evolution of the universe.We also shown these results graphically clearly.Figures 3-7 show the mass of wormhole first decreases to finite value and then increases to infinite value.In future work, it will be interesting to show the natures of mass for various types of wormhole models if different kinds of dark energies accrete upon wormhole in accelerating universe also. Fig.1Fig.2

Figs. 1
Figs.1 and 2show the variations of wormhole mass M against redshift z for VMCG and GCCG models. Fig.3Fig.4