Density perturbation in an interacting holographic dark energy model
DDensity perturbation in an interacting holographic dark energy model
Srijita Sinha * and Narayan Banerjee † IISER Kolkata, Mohanpur Campus, Mohanpur, Nadia 741246, India
Abstract
The present work deals with the evolution of the density contrasts for a cosmological model where along with thestandard cold dark matter (CDM), the present Universe also contains holographic dark energy (HDE). The HDE is allowedto interact with the CDM. The equations for the density contrasts are integrated numerically. It is found that irrespectiveof the presence of an interaction, the matter perturbation has growing modes. The HDE is also found to have growth ofperturbation, so very much like the CDM, HDE can also cluster. The interesting point to note is that the density contrastcorresponding to HDE has a peak at a recent past and is decaying at the present epoch.
Keywords: Density perturbation, Interaction, Holographic dark energy
Ever since the discovery through the luminosity versus redshift surveys[1, 2, 3], that the Universe at the present epoch isexpanding with an acceleration, there has been a proliferation of proposals of a “dark energy” that gravitates in the wrongway. A cosmological constant Λ appears to be a very competent candidate although not a clear winner because of theinsurmountable discrepancy between the observational requirement and the theoretically predicted value. A scalar fieldwith a suitable potential, called a quintessence, is arguably the close second. There are excellent reviews that summarisethe list of candidates and their strength and weakness [4, 5, 6, 7]. Among the quintessence models, some may evolvein a way such that the equation of state parameter of the dark energy attains a value less that − et al. [14].Another most talked about form of dark energy is the so-called holographic dark energy (HDE) based on the holo-graphic principle in quantum gravity theory[15]. The holographic principle, following the ’t Hooft conjecture[16], thatthe information contained in a volume can be ascertained with the knowledge about the degrees of freedom residing onits boundary actually stems from Bekenstein’s idea that the entropy of a black hole is related to its area[17]. Based onthe holographic principle, one of the characteristic features of the HDE is the long distance cut-off, called the infra-red(IR) cut-off[18]. In the context of cosmology, this cut-off is not uniquely specified but rather realised in various ways.One of the natural choices is the Hubble radius[19]. The drawback of this model, as shown by Hsu[20], is that it does notprovide the recent acceleration. In this context, Zimdahl and Pav´on in [21, 22] showed that allowing a non-gravitationalinteraction in the dark sector of the Universe not only solves this problem but also alleviate the nagging coincidenceproblem[23, 24]. Other possibilities are the particle horizon as suggested by Fischler and Susskind[25] and Bousso[26]and the future event horizon as suggested by Li[27] and Huang and Li[28]. A more recent choice for the cut-off scale isthe Ricci scalar curvature used by Gao et al. [29], Feng[30, 31], to name a few. Some of the notable work in modifiedRicci Holographic DE include [32, 33, 34, 35, 36, 37]. A generalized holographic inflation as well as holographic bouncewith no specific IR cut-off has also been proposed[38, 39].Hoˇrava and Minica showed that the expectation value of the cosmological constant is zero in the context of holographicprinciple [40]. This feature made holographic dark energy quite an attractive candidate as the dark energy. There hadbeen an attempt to find a unified model giving an early inflation and a late time acceleration by Nojiri and Odintsov[41].Holographic dark energy models have also been considered in some modified theories of gravity, for example, in Brans-Dicke theory by Setare [42], Banerjee and Pav´on [43], in Gauss-Bonnet gravity by Saridakis [44], Setare [45]. As thereis a host of observational data in cosmology, almost all kinds of holographic dark energy models are now tested againstobservations. Some of them are by Campo et al. [46], Zhang and Wu[47], Li et al. [48], Feng et al. [49], Mukherjee et * Email: [email protected] † Email: [email protected] a r X i v : . [ g r- q c ] O c t l. [50] and a recent one by D’Agostino[51]. The list is far from being exhaustive. The interacting and non-interactingHDE models have been studied in detail using the Planck data by Li et al. [52], Li et al. [53], Zhang et al. [54] and Feng et al. [55]. The stability criteria for holographic dark energy have also been discussed widely[56, 57, 58]. Although theHDE model is quite lucrative in many ways, it cannot avoid the phantom Universe[59, 60]. One way of preventing thefuture “big-rip” singularity is to allow some phenomenological interaction between the dark matter and dark energy asshown by Wang et al. [61, 62]. Thereafter the interacting HDE model has been studied extensively in[63, 64, 65, 66, 67]and references therein.The HDE, as it produces the recent acceleration of the Universe, will also affect the formation of large scale structures.Moreover, the evolution of the matter density perturbation can provide some knowledge about the components affectingit, in this case, the HDE. Since HDE is an evolving component of the Universe, it will have fluctuations like that of theDM and hence will not only affect the growth of matter perturbations[68] but also may cluster on its own[69, 70]. Thereare some work in the matter perturbation in HDE models, such as the one by Kim et al. [71] for a decaying HDE. Veryrecently, perturbation in a clustered HDE has been investigated by Mehrabi et al. [72] and Malekjani et al. [73].In this work, we will show how the evolution of DM, as well as DE, are affected in the presence of an interactionbetween them. In fact, this is the first full relativistic treatment of the perturbation of holographic dark energy modelsto start with. The closest work to this is the one by Mehrabi et al. [72], where a relativistic treatment is definitely given,but the evolution equation of the gravitational potential Φ is approximated to the standard Poisson’s equation. Anotherimportant new feature is certainly the inclusion of interaction between the HDE and DM in the perturbation. The inclusionof interaction leads to a brief transient oscillatory period for the density contrast for both HDE and DM. This existence ofgrowing mode with a transient oscillatory behaviour is an entirely new feature. A peak in the dark energy density contrastin the recent past (low values of z ) is another new feature which escaped the notice so far.The paper is organised as follows. Section 2 discusses the background, section 3 deals with a scenario where aninteraction between the cold dark matter and the holographic dark energy is allowed. Section 4 includes the relevantequations that describe the perturbation in the present case. The evolution of the density contrast is discussed in section 5.In the last section, section 6, we summarise and discuss the results. We also include an appendix, where the coefficientsof the second order differential equations are explicitly written, which are not done in the main text in order to improvethe readability. We consider a spatially flat, homogeneous and isotropic Universe, given by the Friedman-Robertson-Walker (FRW) met-ric, ds = a ( η ) (cid:0) − d η + γ i j dx i dx j (cid:1) , (1)where γ ij = δ ij . The Friedmann equations for this metric take the form3 H = − a κρ , (2) H + H (cid:48) = a κ p , (3)where a ( η ) is the scale factor, κ = π G N ( G N being the Newtonian Gravitational constant), ρ ( η ) and p ( η ) are the totaldensity and the total pressure of the matter distribution in the Universe respectively, H = a (cid:48) a is the Hubble parameter inconformal time η and prime ( (cid:48) ) denotes the derivative with respect to the conformal time. The conformal time η is relatedto the cosmic time t as a d η = dt . Hereafter, the Greek indices µ , ν . . . denote the space-time coordinates while theLatin indices i , j . . . denote the coordinates in the spatial hypersurface.It is assumed that the Universe is filled with a perfect fluid dominated by a pressureless (cold) dark matter (CDM) andholographic dark energy (HDE). The energy densities and pressure are such that ρ = ρ m + ρ de and p = p de respectively.Subscript ‘ m ’ denotes the contribution of the CDM while ‘ de ’ denotes that of the HDE. There is an interaction between thetwo components of the Universe, CDM and HDE, hence a transfer of energy between the two. The total energy balanceequation ρ (cid:48) + H ( ρ + p ) = , (4)is thus divided into two equations, ρ (cid:48) m + H ρ m = aQ , (5) ρ (cid:48) de + H ( + w de ) ρ de = − aQ , (6)where Q is the rate of energy density transfer, w de = p de ρ de is the equation of state (EoS) parameter for the HDE. It is clearthat equations (5) and (6) together give the conservation equation. The non-interacting scenario can be recovered simplyby setting Q =
0. If Q > et al [74]. Some very recent work indicate that the observational anomaly of the21-cm line excess at cosmic dawn can be the relevant observations in this connection[75, 76].The expression for the energy density of HDE is ρ de = C M P L , (7)where 3 C is a numerical constant introduced for convenience, M P = (cid:113) κ is the reduced Plank mass, L is the characteristiclength scale of the Universe which provides the IR cut-off of ρ de . Incidentally, in the present work this cut-off is chosenas the future event horizon as suggested by Li[27], L = a (cid:90) ∞ t d ˜ ta = a (cid:90) ∞ a d ˜ aH ˜ a , (8) H is the Hubble parameter in cosmic time t . It has already been mentioned in the introduction that this is by no means theonly choice or the best choice as the infra-red cut-off.The energy-momentum tensor of the fluid ‘A’ (which stands for either ‘ m ’ or ‘ de ’) is T µ ( A ) ν and is given by T µ ( A ) ν = ( ρ A + p A ) u µ ( A ) u ( A ) ν + p A δ µ ν , (9)where u ( A ) µ = − a δ µ is the comoving 4-velocity of the fluid ‘A’. The total energy-momentum tensor is T µ ν = ∑ A T µ ( A ) ν such that ( ρ + p ) u µ u ν + p δ µ ν = ∑ A ( ρ A + p A ) u µ ( A ) u ( A ) ν + ∑ A p A δ µ ν , (10)where u µ = − a δ µ is the total comoving 4-velocity . It is again clear from equation (10) that ρ = ∑ A ρ A , p = ∑ A p A .In presence of an interaction, the energy-momentum tensor of the individual components T µ ( A ) ν does not conserveindependently, and its divergence has the source term Q ( A ) ν . Thus the covariant form of the conservation equation forfluid ‘A’ is given as T µ ( A ) ν ; µ = Q ( A ) ν , where ∑ A Q ( A ) ν = . (11)The source term for the interaction is a 4-vector and has the form Q µ m = a (cid:16) Q m ,(cid:126) (cid:17) = a (cid:16) Q ,(cid:126) (cid:17) = a (cid:16) − Q de ,(cid:126) (cid:17) = − Q µ de . (12)It is assumed that there is no momentum transfer in the background universe. The energy balance equation for the fluid‘A’ takes the form ρ (cid:48) A + H ( + w A ) ρ A = aQ A , (13)where Q A = Q ( A ) , the time component of the four vector Q ( A ) µ . The evolution of the dimensionless HDE density parameter Ω de = ρ de ρ c where ρ c = H M P is the critical density of theUniverse, and the dimensionless Hubble parameter E , in the presence of an interaction are governed by the simultaneousdifferential equations [77] d Ω de dz = − Ω de ( − Ω de ) + z (cid:32) + (cid:114) Ω de C − Ω I ( − Ω de ) (cid:33) , (14)1 E dEdz = − Ω de + z (cid:32) + (cid:114) Ω de C + Ω I − Ω de (cid:33) , (15)where E = HH is the Hubble parameter scaled by its present value H . The evolution equations (14) and (15) are given interms of the cosmic redshift z , which is a dimensionless quantity and is related to the scale factor a as z = a a − a beingthe present value of the scale factor (taken to be unity). These two equations are obtained following the usual steps (alsoshown in [77]).The EoS parameter of DE, w de , is an intrinsic characteristic of DE. From the system of equations, given by Einstein’sequations and the conservation equations, w de imposes a constraint on Q (see [77]) as w de = − − (cid:114) Ω de C − Ω I Ω de , (16)3here Ω I = QH ρ c is the interaction term expressed in a dimensionless form.To study the effect of interaction, we need to take a specific form of the interaction term Q . Models with interactionterm Q proportional to either ρ m or ρ de or any combination of them have been studied extensively in literature[77, 78, 79,80, 81, 82, 74, 83, 84, 85, 86]. It should be noted that there is no theoretical or observational compulsion for any one ofthese choices. In the present work we consider Q ∝ ρ de . We have taken the covariant form of the source term Q µ m ( η ) as Q µ m = − Q µ de = β H ρ de u µ de a , (17)where β is the coupling constant whose magnitude determines the strength of the interaction rate. Here we consider theHubble parameter H to be a global variable without any perturbation. When β <
0, it is clear from equations (5) and(6) that DM redshifts faster than a − while DE redshifts slowly. This is physically problematic as more of the DM isexpected to be transferred to the DE budget in the late time, rather than in the beginning. For β >
0, this problem isavoided. As shown by Feng and Zhang in [79], for an HDE model, this form of interaction is favoured by geometricaldata. We consider β to be a free parameter. Using u µ de = a δ µ and equation (12) in equation (17) the interaction term Q isobtained as Q = β H ρ de a . (18)Since dark energy dominates at the present epoch, we assume w de < − . As the motivation of the present work is toinvestigate the perturbation for a model without a big rip singularity, we restrict w de (cid:62) −
1. For the non-interacting case ( β = ) , it is clear from (16) that for w de → − z = C → √ Ω de , Ω de being the value of the dark energy densityparameter at the present epoch. The present value of Ω de is taken from the Planck data[87] and is close to 0 . C = . (cid:39) .
83. In the presence of interaction, C and β have a correlation. The big rip singularity can beavoided if the interaction rate, β lies between − (cid:114) Ω de C < β (cid:54) − (cid:114) Ω de C . (19)The numerical values of C and β can be further constrained from other physical quantities like the deceleration parameter, q . For the IHDE model, q depends on the parameters C and β . In the subsequent part of this section, we will see the effectof interaction on the different physical quantities and try to constrain the parameter space for C and β . Figures 1, 2 and 3show the evolution of the dark energy density parameter Ω de , the dimensionless Hubble parameter E and the dark energyEoS parameter w de respectively with redshift z for different values of C and β . For figures 1 and 2, equations (14) and(15) are solved numerically from z = z = Ω de ( z = ) = Ω de and E ( z = ) =
1. Forthe effect of interaction, two cases are considered, (i) C ranges from 0 . . C goes towards 0 . β taken as − . β ranges from − β to −
4) with C fixed at 0 .
75. In figure 1, Ω de rises from nearly zero in the past to 0 . ( ∼ . ) at present. For fixed non-zero interaction, the higher the value of C , themore gentle is the rise (figure 1a). Similarly, for a fixed value of C = .
75, higher the value of β the more gentle is theslope (figure 1b). In figure 2a, E does not vary much in the past for different values of C but in figure 2b, E varies quiteremarkably for different values of β — the smaller the value of β the higher the gradient.For figures 3 a and 3 b, the solution of equation (14) is used in equation (16). For any value of C , w de increases andasymptotically reaches − .
33 at higher redshifts when Ω de ∼
0. In both the figures w de varies from nearly − . . w de lies between − − .
33 are considered for further calculation. From figure3a, the allowed region for C is constrained as C ∈ [ . , . ] . From figure 3b and equation (19) the allowed regions of β are β ∈ ( − . , − . ) (for C = .
6) and β ∈ ( − . , − . ) (for C = . C and β properly.Figure 4 a shows the variation of the deceleration parameter q with ( + z ) in logarithmic scale for different sets of C and β . In all the cases, q increases with z and asymptotically approaches 0 . C (0 . . β ( − . − . β ( − . − . q isnearly equal to − . β ( < − . ) (larger magnitudes) will give decelerated Universeat present. Thus future event horizon as IR cut-off does not necessarily ensure accelerated Universe in the presence ofan interaction. For non-interacting case, acceleration comes naturally as said by Li[27]. This figure brings out some newfeatures, like it puts a limit on the strength of interaction. For achieving an accelerated model, β should be greater than − .
741 (for C = .
75) or greater than − .
292 (for C = . Ω I with with ( + z ) in logarithmic scale. For anypair of C and β ( (cid:54) = ) the interaction term is nearly zero at higher redshifts and then starts to increase in magnitude withdecrease in z . Since Ω I = β Ω de , the interaction increases in magnitude as the dark energy density parameter increases(figure 1). With β <
0, from the definition of Ω I , we can see Q <
0, which means energy is transferred from DM toDE. Thermodynamically energy transfer should be from DE to DM following Le Chˆatelier–Braun principle as shown by4 a) β =-0.50 2 4 6 8 10z0.00.10.20.30.40.50.60.7 Ω de β Figure 1: (a) shows plot of Ω de against z for C ranging from 0 . . β = − .
5. (b) shows plot of Ω de against z for β ranging from − . . C = .
75 . (a) β =-0.50 2 4 6 8 10z0510152025 E β Figure 2: (a) shows plot of E against z for C ranging from 0 . . β = − .
5. (b) shows plot of E against z for β ranging from − . . C = .
75 .Pav´on and Wang [88]. In case of an HDE model negative β (DM → DE) is slightly favoured by the data as shown byZhang et al. [77].
In what follows, a scalar perturbation of the metric (1) is considered. In longitudinal gauge, the perturbed metric takes theform ds = a (cid:2) − ( + Φ ) d η + ( − Ψ ) dx i dx j (cid:3) . (20)Here Φ ( η , x ) and Ψ ( η , x ) are the gauge-invariant variables, known as the Bardeen’s potential [89]. In absence of anyanisotropic stress, longitudinal gauge becomes identical to Newtonian gauge, as discussed in [90], making Φ = Ψ . Per-turbations in the form of δ ρ A ( η , x ) , δ p A ( η , x ) , δ u ( A ) µ ( η , x ) and δ Q ( A ) µ ( η , x ) are added to the expression for energy-momentum tensor (9). The components of the 4-velocity perturbation of the fluid ‘A’ are δ u ( A ) = − a Φ obtained fromthe normalisation condition and δ u ( A ) i = a ∂ i v A , v A being the peculiar velocity potential. Similarly, the total 4-velocityperturbation δ u µ ( η , x ) has the components δ u µ = − a ( Φ , ∂ i v ) such that ( ρ + p ) θ = ∑ A ( ρ A + p A ) θ A , (21)with θ = − k v being the divergence of the total fluid velocity [91] and k is the wave number in the corresponding Fouriermode. For an individual fluid ‘A’, the divergence of the corresponding velocity is θ A = − k v A . The perturbed energy-momentum transfer function δ Q ( A ) µ is split relative to the total 4-velocity u µ [92, 78] as δ Q ( A ) µ = δ Q A u µ + F ( A ) µ , u µ F ( A ) µ = , (22)5 a) β =-0.50 2 4 6 8 10z-1.4-1.2-1.0-0.8-0.6-0.4-0.2 w de β Figure 3: (a) shows plot of w de against z for C ranging from 0 . . β = − .
5. (b) shows plot of w de against z for β ranging from − . . C = .
75. For both of these plots Ω de = .
68 at z =
0. Here w de ranges from nearly − . . − . < w de (cid:54) − . -0.6-0.4-0.20.00.20.40.6 1 2 4 6 8 10 q C = 0.83, β = 0C = 0.60, β =-0.8C = 0.60, β =-1.5C = 0.60, β =-2.29C = 0.75, β =-0.5C = 0.75, β =-1.5C = 0.75, β =-1.74 -1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.20.00.2 1 2 4 6 8 10 Ω I C = 0.83, β = 0C = 0.60, β =-0.8C = 0.60, β =-1.5C = 0.75, β =-0.5C = 0.75, β =-1.5 Figure 4: (a) shows plot of q against ( + z ) and (b) shows the variation of Ω I against ( + z ) in logarithmic scale fordifferent values of C and β . The lines with solid circles corresponds to C = .
83 and β =
0. The lines with trianglescorrespond to the interacting models with C = . β = − . β = − . C = . β = − . β = − . C = .
6) and stars ( C = .
75) corresponds to the values of β for which there is no acceleration at present.where δ Q A is the perturbation in the energy transfer rate, F ( A ) µ = a ( , ∂ i f A ) is the perturbation in the momentum densitytransfer rate and f A is the momentum transfer potential. It is clear from equation (11) that ∑ A δ Q A = ∑ A f A = . (23)Writing the perturbation in Fourier modes, the ( ) , ( i ) ≡ ( i ) and ( i j ) components of the Einstein field equationsup to the first order in perturbation will read as[93],2 (cid:2) − H ( H Φ + Φ (cid:48) ) − k Φ (cid:3) = H ρ δ ρ , (24)2 k ( H Φ + Φ (cid:48) ) = − H ρ ( p + ρ ) θ , (25) − (cid:2)(cid:0) H (cid:48) + H (cid:1) Φ + H Φ (cid:48) + Φ (cid:48)(cid:48) (cid:3) δ i j = − H ρ δ p δ i j . (26)The temporal and spatial parts of the first order in perturbation of the energy balance equation of the fluid ‘A’ [78] are δ ρ (cid:48) A + ( ρ A + p A ) (cid:0) θ A − Φ (cid:48) (cid:1) + H ( δ ρ A + δ p A )= aQ A Φ + a δ Q A , (27) [ θ A ( ρ A + p A )] (cid:48) + H θ A ( ρ A + p A ) − k δ p A − k Φ ( ρ A + p A ) = − k a f A + aQ A θ . (28)For an adiabatic perturbation in interacting fluids, the pressure perturbation δ p A depends on δ ρ A as well as on the6nteraction term Q A as[94, 78, 95] δ p A = c s , A δ ρ A + (cid:0) c s , A − c a , A (cid:1) [ H ( + w A ) ρ A − aQ A ] θ A k , (29)where c a , A = p (cid:48) A ρ (cid:48) A = w A + w (cid:48) A ρ (cid:48) A / p (cid:48) A (30)is the adiabatic speed of sound of ‘A’ fluid and c s , A is the effective speed of sound of ‘A’ fluid, defined as c s , A = δ p A δ ρ A (cid:12)(cid:12)(cid:12)(cid:12) rest , A , (31)is the ratio of pressure fluctuation to density fluctuation in the rest frame of fluid ‘A’.As shown in [69], c s , de plays a significant role in DE clustering and hence DM clustering. When c s , de (cid:39)
1, the pressureperturbation should suppress any growth in DE perturbation whereas when c s , de (cid:28)
1, DE perturbation should grow likethat of DM. It is shown in [69, 70, 96] that DE can cluster like DM when c s , de =
0. Here for the case of CDM andIHDE we consider that 0 (cid:54) c s , de (cid:54) c s , de =
0. Then wecompare some of the results for different non-zero values of c s , de . It deserves mention that the effective sound speed c s , de ,defined in equation (31) is different from the adiabatic sound speed c a , de , defined in equation (30). We shall now frame the differential equations for density contrasts for both DM and DE that determine their evolutionwith redshift. For that, we need to know the perturbation in the interaction term itself. From equations (22) and (23), itfollows that δ Q m = − δ Q de = β H δ ρ de a , (32) f m = − f de = β H ρ de ( θ − θ de ) ak . (33)The density contrasts of CDM and IHDE are δ m = δρ m ρ m and δ de = δρ de ρ de respectively. Using (32) and (33), the equations(27) and (28) for CDM and IHDE can be written respectively as δ (cid:48) m + θ m − Φ (cid:48) = β H ρ de ρ m ( Φ − δ m + δ de ) , (34) θ (cid:48) m + H θ m − k Φ = − β H ρ de ρ m ( θ m − θ de ) , (35) δ (cid:48) de + H (cid:0) c s , de − w de (cid:1) δ de + ( + w de ) (cid:0) θ de − Φ (cid:48) (cid:1) + H (cid:2) H ( + w de ) (cid:0) c s , de − w de (cid:1)(cid:3) θ de k + H w (cid:48) de θ de k = − β H (cid:20) Φ + H (cid:0) c s , de − w de (cid:1) θ de k (cid:21) , (36) θ (cid:48) de + H (cid:0) − c s , de (cid:1) θ de − k Φ − k δ de c s , de ( + w de ) = β H ( + w de ) (cid:0) − c s , de θ de (cid:1) . (37)Eliminating θ m from equations (34), (35) and θ de from equations (36), (37), the coupled differential equations forCDM and IHDE are obtained respectively in terms of redshift as C ( m ) ∂ δ m ∂ z + C ( m ) ∂ δ m ∂ z + C ( m ) δ m + C ( m ) ∂ δ de ∂ z + C ( m ) ∂ δ de ∂ z + C ( m ) δ de + C ( m ) ∂ Φ ∂ z + C ( m ) ∂ Φ ∂ z + C ( m ) Φ = , (38) C ( de ) ∂ δ de ∂ z + C ( de ) ∂ δ de ∂ z + C ( de ) δ de + C ( de ) ∂ δ m ∂ z + C ( de ) ∂ δ m ∂ z + C ( de ) δ m + C ( de ) ∂ Φ ∂ z + C ( de ) ∂ Φ ∂ z + C ( de ) Φ = . (39)The coefficients C , C , C , C , C , C , C , C and C are given in the Appendix A. The coefficients C ( de ) , C ( de ) and C ( de ) are zero in equation (39) indicating that the evolution of DE is not directly affected by DM fluctuation but theconverse is not true. The coefficients C , C and C are non zero in both the equations (38) and (39) which implies thatthe potential Φ will affect the evolution of both DM and DE density contrasts. The evolution of Φ is governed by theequation (24) and is not approximated by the Poisson equation. The equations (38) and (39) along with the equation (24)7re solved numerically to find the evolution of the density contrasts of the CDM and IHDE. In order to do that, in thematter-dominated era, i.e. from z in = Φ ( z in ) = constant = φ and Φ (cid:48) ( z in ) =
0. It is also assumedthat Ω m ( z in ) >> Ω de ( z in ) so that the term with the ratio Ω de ( z in ) Ω m ( z in ) can be neglected for δ m ( z in ) only. As discussed in [68],the initial conditions for δ m , δ (cid:48) m , δ de and δ (cid:48) de are δ mi = δ m ( z in ) = − φ (cid:34) + ( + z in ) k in H in (cid:35) , (40) δ (cid:48) mi = d δ m dz (cid:12)(cid:12)(cid:12)(cid:12) z = z in = − φ ( + z in ) k in H in (cid:20) − (cid:18) + w dei Ω dei Ω mi (cid:19)(cid:21) , (41) δ dei = δ de ( z in ) = δ mi − β Ω dei Ω mi { ( + w dei ) + β } , (42) δ (cid:48) dei = d δ de dz (cid:12)(cid:12)(cid:12)(cid:12) z = z in = (cid:16) dw de dz (cid:17) (cid:12)(cid:12)(cid:12) z = z in − β Ω dei Ω mi + δ (cid:48) mi − β Ω dei Ω mi { ( + w dei ) + β } + δ mi { ( + w dei ) + β } β (cid:104) ddz (cid:16) Ω de Ω m (cid:17)(cid:105) z = z in (cid:16) − β Ω dei Ω mi (cid:17) , (43)where H in = H ( z in ) , w dei = w de ( z in ) , Ω mi = Ω m ( z in ) , Ω dei = Ω de ( z in ) and k in is the mode entering the Hubble horizonat z in . The value of k in is taken as ( + z ) − H in . The numerical values for H in , w dei , Ω mi and Ω dei are obtained from thesolutions of the equations (14), (15) and (16). The value of φ is given by hand.For the Fourier mode, k in equations (38), (39) and (24), the value is considered in the linear regime given by thegalaxy power spectrum[97] 0 . h Mpc − (cid:46) k (cid:46) . h Mpc − , (44)where h = H km s − Mpc − is the dimensionless Hubble parameter at the present epoch. The value of H = .
27 istaken from the Planck data[87]. For k > . h Mpc − (smaller scales), non-linear effects become prominent whereas for k < . h Mpc − observations are not very accurate. So we consider k = . h Mpc − following [69]. For our calculationwe have considered φ = − and c s , de = δ m has over density (positive solution) while that of DE, δ de has under density (negativesolution). All the figures 5-9 are shown in logarithmic scale from z = z =
100 with C = .
83 and β = C = .
75 and β = − .
5. In all the figures, the densitycontrast is scaled by its present value. To study the effect of interaction in the growth of the density contrasts we haveconsidered different sets C and β .Figure 5 shows the variation of δ m and δ de with ( + z ) in logarithmic scale for the non-interacting case for φ = − and φ = − . When scaled by their respective present value, the nature of the growth of δ m and δ de is hardly sensitiveto the value of φ . This is clear from figures 5 a and 5 b. Figure 6 shows the same for the interacting case with C = . β = − .
5. One can clearly see from figures 5 a and 6 a that the interaction ( β (cid:54) =
0) makes the slopes different. Forthe variation of δ de , we see that it first grows up to a maximum and then decreases to unity at z =
0. The position as wellas the height of this peak is different in the figures 5 b and 6 b. The presence of an interaction has decreased the height ofthe maximum and made the growth a little flat.Figures 7 a and 8 a show the variation of δ m with ( + z ) in logarithmic scale for the same value of C but differentvalues of β . For C = .
6, as β decreases from − . − . C = . C = .
75. For δ de (figures 7 b and 8 b), the change inslope for smaller β is more prominent in smaller C value. In figure 7 b, for C = . δ de for β = − .
5, changes faster thanthat for β = − .
8. Similarly in figure 8 b, for C = . δ de for β = − . β = − .
5. Thechange in the direction of the growth rate takes place after the Universe starts accelerating and has a correlation with thedeceleration parameter q changing its sign. For figure 7 b, the maximum of δ de for β = − . β = − .
8, and for figure 8 b, the maximum of δ de for β = − . β = − . β is decreased below − .
5, no such correlation is seen.Figure 9 shows the variation of the density contrasts δ m and δ de for the different values of the effective speed of soundof dark energy perturbation, c s , de . From the expression of δ p de equation (29), we can see that the pressure perturbationnot only depends on the product c s , de δ ρ de but also on the background quantities like w de , ρ de , Q as well as the velocityperturbation through θ de . With c s , de =
0, the effect of the velocity perturbation present in the second term is prominent.The presence of the interaction Q actually decreases this effect (figures 5 b and 6 b). In presence of the c s , de (i.e. anynon-zero value), the effect of δ ρ de comes into play. The growth is more steep if the proportion of δ ρ de increases. Whenzoomed into smaller redshift region ( z = . z = . δ m and δ de with k / h at z =
0. In figure 11 a, for a given value of C and β , as k / h increases δ m increases — the growth rate of DM over densities increase for smaller scales entering the horizon. Thoughthe increase is not linear for k / h less than ∼ . h , but when scaled by δ m , the growth rates of δ m for different k -modesare independent of the k -modes. For figure 11 b, negative values of δ de increases with larger scales. The change in slopein this case is also not linear for modes smaller than ∼ . h and the growth of δ de / δ de for different modes are identical.8 .0 0.2 0.4 0.6 0.8 1.0 1.210 (a) C = 0.83, β = 0 δ m φ = 10 -5 φ = 10 -6 (b) C = 0.83, β = 0 δ de φ = 10 -5 φ = 10 -6 Figure 5: (a) shows plot of δ m against ( + z ) and (b) shows the plot of δ de against ( + z ) in logarithmic scale for C = . β =
0. The line shows the variation of δ m and δ de for the initial condition, φ = − and the solid circles representthe same corresponding to the initial condition, φ = − . (a) C = 0.75, β = -0.5 δ m φ = 10 -5 φ = 10 -6 (b) C = 0.75, β = -0.5 δ de φ = 10 -5 φ = 10 -6 Figure 6: (a) shows plot of δ m against ( + z ) and (b) shows the plot of δ de against ( + z ) in logarithmic scale for C = . β = − .
5. The line shows the variation of δ m and δ de for the initial condition, φ = − and the solid circles representthe same corresponding to the initial condition, φ = − . The primary motivation of the present work is to study the effect of interaction on density perturbation in the dark sector ofthe Universe in a Holographic dark energy model. Among various possibilities, we have chosen the future event horizonas the IR cut-off for the HDE model for which the Universe can accelerate even in the absence of an interaction. For theinteraction between the DM and DE, we have chosen the interaction term to be proportional to the dark energy density ρ de . The interaction term is of the form Q = β H ρ de a , in which the dependence on cosmic time comes through the globalexpansion rate, the Hubble parameter H . The coupling constant β determines the strength of the interaction as well asthe direction of the energy flow. No interaction between DE and DM is characterised by β =
0. We restricted the modelparameters C and β in such a way that at present the DE EoS parameter, w de is sufficiently negative to produce the latetime acceleration but can avoid the “phantom menace” ( w de < − δ m ) as well as the HDE( δ de ) were obtained in the Newtonian gauge. Adiabatic initial conditions were used with the assumption that the DEdensity parameter is small compared to the DM density parameter ( Ω de Ω m (cid:28)
1) at the onset of the matter dominated epoch( z = δ m at z = z = z = δ m is increasing in the positive direction, whereas δ de is increasing in the negativedirection. However, as δ m and δ de are always scaled by their respective present values, this difference in signature is notreflected in the plots.A small negative value of β is found to be indicated for the density contrast to grow. This suggests that the dark matterdecays into dark energy and the interaction in the dark sector, if any, has to be small.The effect of effective sound speed of DE, c s , de on density perturbation was also looked at. The absence of aninteraction, with w de = − c s , de = .0 0.2 0.4 0.6 0.8 1.0 1.210 (a) C = 0.6 δ m β =-0.8 β =-1.5 (b) C = 0.6 δ de β =-0.8 β =-1.5 Figure 7: (a) shows plot of δ m against ( + z ) and (b) shows the plot of δ de against ( + z ) in logarithmic scale for C = . β . The line with circles is for β = − . β = − . (a) C = 0.75 δ m β =-0.5 β =-1.5 (b) C = 0.75 δ de β =-0.5 β =-1.5 Figure 8: (a) shows plot of δ m against ( + z ) and (b) shows the plot of δ de against ( + z ) in logarithmic scale for C = . β . The line with circles is for β = − . β = − . c s , de , the pressure perturbation remains non-vanishing as w de is now a varying function of redshift, z . Thepressure perturbation is then governed by the velocity perturbation through θ de and the background quantities ρ de , w de and Q de .The DE density contrast, δ de also grows almost in a similar fashion like its DM counterpart, δ m , for most of theevolution after the radiation dominated era, but right at the present moment is actually decaying after hitting a maximumin the recent past. This is true even in the absence of interaction (characterised by β = β . On decreasing the strength (smaller magnitude) of the interaction, the position of maximum shifts to lowerredshifts and the height decreases. This feature is observed for the zero value of the effective sound speed, c s , de (figure 9b). When c s , de = δ p de (see equation (29)) is zero and from the second part, we cansay that δ de reaching a maximum is characterised by θ de whereas Q de actually suppresses this feature (figures 5 b and6 b). When c s , de (cid:54) =
0, the contribution from the first part ( c s , de δ ρ de ) results in the steep rise in δ de at lower values of z (figure 9 b). This apparently is engineered by θ de . Except for the peak in the growth rate, similar features are also seenfor δ m (figure 9 a). For c s , de =
1, a rapid short-lived oscillations in the DE density contrast is found between z = . z = . c s , de = β (cid:54) = Φ (cid:48) is neglected for the estimations, but in the present work, its contribution isalso respected. The appearance of a peak δ de for c s , de = β = Φ (cid:48) in the estimation. For c s , de =
1, no growth for δ de is normally observed. The recent work of Batista and Pace[70] shows an almost negligible growth. The present work10 .0 0.2 0.4 0.6 0.8 1.0 1.210 (a) C = 0.75, β = -0.5 δ m c = 0.0c = 0.01c = 1.0 (b) C = 0.75, β = -0.5 δ de c = 0.0c = 0.01c = 1.0 Figure 9: (a) shows plot of δ m against ( + z ) and (b) shows the plot of δ de against ( + z ) in logarithmic scale for differentvalues of c s , de with C = .
75 and β = − .
5. The line with solid circles corresponds to c s , de =
0, the line with solid trianglescorresponds to c s , de = .
01 and the line with solid squares corresponds to c s , de = . -2.0-1.5-1.0-0.50.00.51.01.52.02.53.03.50.0 0.2 0.4 0.6 (a) C = 0.75, β = -0.5 δ m z c = 1.0 -2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.03.50.0 0.2 0.4 0.6 (b) C = 0.75, β = -0.5 δ de z c = 1.0 Figure 10: (a) shows plot of δ m against z and (b) shows the plot of δ de against z from z = z = . c s , de = . C = .
75 and β = − . -0.50.00.51.01.52.02.53.03.5 0 0.04 0.08 0.12 0.16 0.20(a) C = 0.75 δ m k/h β =-0.5 -1.0 -0.8 -0.6 -0.4 -0.2 0.00.1 0 0.04 0.08 0.12 0.16 0.20(b) C = 0.75 δ de0 k/h β =-0.5 Figure 11: (a) shows plot of δ m against k and (b) shows the plot of δ de against k at z = C = .
75 and β = − . z . This is there both in the presence and absence of interaction. For an interactingscenario, there is also a short-lived oscillatory period in the growing mode of δ de (figures 10 a and 10 b).Although we presented the calculations with φ = − and k = . h , these results are insensitive to changes in φ and k mode entering the horizon. 11 ppendix A Coefficients of the coupled differential equations The coefficients of equations (38) and (39) are given below.(i) The coefficients of equation (38) are: C ( m ) = − H E , (45) C ( m ) = − E (( β − ) H Ω de + H )( z + )( Ω de − ) − H E (cid:48) , (46) C ( m ) = − (cid:0) H β (cid:0) ( z + )( Ω de − ) Ω de E (cid:48) + E (cid:0) Ω de + ( β − ) Ω de − ( z + ) Ω (cid:48) de (cid:1)(cid:1)(cid:1) ( z + ) ( Ω de − ) , (47) C ( m ) = , (48) C ( m ) = (cid:0) H β E Ω de (cid:0) − H w de E + w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( β + ) c s , de − ( z + ) w (cid:48) de (cid:1)(cid:1)(cid:1)(cid:14) ( z + Ω de − (cid:0) − k ( z + ) − H w de E + w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( β + ) c s , de − ( z + ) w (cid:48) de (cid:1)(cid:1)(cid:1) , (49) C ( m ) =( H β ( E Ω de − E ( Ω de − ) Ω de − E ( Ω de + β E ( Ω de + c s , de k ( z + ) E ( Ω de − ) Ω de (cid:14) (cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1) + c s , de k z ( z + ) E ( Ω de − ) Ω de (cid:14) (cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1) − (cid:0) k ( z + ) w de E ( Ω de − ) Ω de ) / (cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1) − (cid:0) k z ( z + ) w de E ( Ω de − ) Ω de ) / (cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1) + ( z + )( Ω de − ) Ω de E (cid:48) +( z + ) E ( Ω de − ) Ω (cid:48) de − ( z + ) E Ω de Ω (cid:48) de (cid:1)(cid:1)(cid:14) ( z + ) ( Ω de − ) , (50) C ( m ) = H E , (51) C ( m ) = (cid:0) H (cid:0) β k E Ω de + β k zE Ω de + β k z E Ω de − β k w de E Ω de − β k zw de E Ω de − β k z w de E Ω de − E (cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1) + E Ω de (cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1) − β E Ω de (cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1) + ( z + )( − Ω de ) (cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1) E (cid:48) (cid:1)(cid:1)(cid:14) (( z + )( − Ω de ) (cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1)(cid:1) , (52) C ( m ) = − k H E + β H Ω de E (cid:48) ( z + )( Ω de − ) + ( H β E ( Ω de − ( Ω de − ) Ω de − ( Ω de + β ( Ω de + β k ( z + )( Ω de − ) Ω de (cid:14)(cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1) + β k z ( z + )( Ω de − ) Ω de (cid:14) (cid:0) k ( z + ) + H w de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1) + ( z + )( Ω de − ) Ω (cid:48) de − ( z + ) Ω de Ω (cid:48) de (cid:1)(cid:1)(cid:14) ( z + ) ( Ω de − ) . (53)(ii) The coefficients of equation (39) are: C ( de ) = − H E , (54)12 ( de ) = − (cid:0)(cid:0) H (cid:0) − k ( z + ) E (cid:0) − − β c s , de + ( Ω de − ( Ω de + ( z + ) w (cid:48) de + w de (cid:0) ( z + ) w (cid:48) de + β c s , de + (cid:1)(cid:1) + k ( z + ) (cid:0) ( Ω de − (cid:1) E (cid:48) + H ( z + )( w de + ) E (cid:0) ( β + ) (cid:0) − c s , de (cid:1) + (cid:0) β − c s , de + (cid:1) w de +( z + ) w (cid:48) de + ( Ω de (cid:1) E (cid:48) + H E (cid:0) c s , de + β c s , de − β c s , de − β c s , de + (cid:0) − β + c s , de − (cid:1) ( Ω de − ( Ω de + ( z + ) (cid:0) β + ( β − ) c s , de + (cid:1) w (cid:48) de + ( Ω de ( z + ) w (cid:48) de − β + ( β + ) c s , de − + ( z + ) w (cid:48)(cid:48) de + w de (cid:0) − β − β c s , de + β c s , de + β c s , de + c s , de − ( z + ) − (cid:0) − β + c s , de − (cid:1) w (cid:48) de + ( z + ) w (cid:48)(cid:48) de (cid:1)(cid:1)(cid:1)(cid:1)(cid:14) (cid:0) z + w de + (cid:0) − H ( Ω de − k ( z + ) E + w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( β + ) c s , de − ( z + ) w (cid:48) de (cid:1)(cid:1)(cid:1)(cid:1) , (55) C ( de ) = (cid:0) c s , de k ( z + ) − H k ( z + ) E − (cid:0) c s , de − (cid:1) ( z + ) w (cid:48) de + ( β + ) c s , de + c s , de + c s , de H k ( z + ) EE (cid:48) + H ( z + )( Ω de E E (cid:48) + c s , de H ( z + ) E (cid:0) ( β + ) c s , de − ( z + ) w (cid:48) de (cid:1) H ( Ω de E + E (cid:48) (cid:0) H E − (cid:0) c s , de + (cid:1) k ( z + ) E ( z + ) w (cid:48) de + β c s , de + k ( z + ) E (cid:48) − H ( z + ) (cid:0) − β + c s , de − (cid:1) E E (cid:48) (cid:1) + H E (cid:16) β ( β + ) c s , de − c s , de ( z + ) (cid:0) β c s , de + (cid:1) w (cid:48) de + (cid:0) c s , de − (cid:1) ( z + ) (cid:0) w (cid:48) de (cid:1) − c s , de ( z + ) w (cid:48)(cid:48) de (cid:1) +( Ω de (cid:0) c s , de k ( z + ) + c s , de H k ( z + ) (cid:0) − β + c s , de + (cid:1) E − c s , de H k ( z + ) EE (cid:48) + H ( z + ) E (cid:0) ( z + ) w (cid:48) de + β + c s , de − ( β + ) c s , de + (cid:1) E (cid:48) − H E (cid:0) β c s , de (cid:0) − β + c s , de − (cid:1) + (cid:0) c s , de − (cid:1) ( z + ) w (cid:48) de − ( z + ) w (cid:48)(cid:48) de (cid:1)(cid:1) + w de (cid:0) − H k ( z + ) E − c s , de k ( z + ) (cid:0)(cid:0) c s , de + (cid:1) ( z + ) w (cid:48) de − β c s , de − ( β + ) c s , de − (cid:1) − H H ( z + ) E + k ( z + ) EE (cid:48) − (cid:0) c s , de − (cid:1) ( z + ) w (cid:48) de + ( β + ) c s , de − ( β + ) c s , de H E + E (cid:48) (cid:16) β c s , de − β ( β + ) c s , de + ( z + ) (cid:0) ( β − ) c s , de + (cid:1) w (cid:48) de − ( z + ) (cid:0) w (cid:48) de (cid:1) − (cid:0) c s , de − (cid:1) ( z + ) w (cid:48)(cid:48) de (cid:17)(cid:17)(cid:17)(cid:46) (cid:0) H ( z + ) ( w de + ) E (cid:0) k ( z + ) + H ( Ω de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1)(cid:1) , (56) C ( de ) = C ( de ) = C ( de ) = , (57) C ( de ) = H ( w de − ) E , (58) C ( de ) = (cid:0) H k ( z + ) ( w de − ) (cid:0) + ( β − ) c s , de + β + (cid:0) β − β c s , de (cid:1) w de + (cid:0) c s , de − (cid:1) ( Ω de (cid:1) E + H k ( z + ) ( w de − ) ( w de + ) E (cid:48) + H ( z + ) (cid:0) ( Ω de − (cid:1) E (cid:0) ( β + ) (cid:0) − c s , de (cid:1) + (cid:0) β − c s , de + (cid:1) w de + ( Ω de +( z + ) w (cid:48) de (cid:1) E (cid:48) − H E (cid:0) c s , de + c s , de − β c s , de − β c s , de + (cid:0) c s , de + (cid:1) ( Ω de − ( z + ) (cid:0) − β + c s , de − (cid:1) w (cid:48) de + ( z + ) (cid:0) w (cid:48) de (cid:1) − ( Ω de (cid:0) ( z + ) w (cid:48) de − β + c s , de − c s , de − (cid:1) + w de (cid:0) − − c s , de + c s , de − β c s , de + β + β c s , de + β c s , de − ( z + ) (cid:0) − β + ( β + ) c s , de − (cid:1) w (cid:48) de + ( z + ) (cid:0) w (cid:48) de (cid:1) (cid:17) + w (cid:48)(cid:48) de + zw (cid:48)(cid:48) de + z w (cid:48)(cid:48) de + ( Ω de (cid:0) − − c s , de − c s , de + β − β c s , de + β − β c s , de + (cid:0) c s , de + (cid:1) ( z + ) w (cid:48) de − ( z + ) w (cid:48)(cid:48) de (cid:1)(cid:1)(cid:1)(cid:14) (cid:0) z + w de + (cid:0) − k ( z + ) − H ( Ω de E + w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( β + ) c s , de − ( z + ) w (cid:48) de (cid:1)(cid:1)(cid:1) , (59)13 ( de ) = (cid:16) H ( Ω de E − ( Ω de + k ( z + ) H (cid:0) − β + c s , de − (cid:1) E + H k ( z + ) E + H k ( z + ) E ( β + )( z + ) w (cid:48) de − β − (cid:0) β + β + (cid:1) c s , de + β H k ( z + ) EE (cid:48) − β H ( z + ) E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1) E (cid:48) + ( Ω de (cid:0) k ( z + ) + H k ( z + ) (cid:0) − β + c s , de − (cid:1) E + H E ( z + ) w (cid:48) de + β + β + c s , de − ( β + ) c s , de + − β H ( z + ) E E (cid:48) (cid:1) − ( Ω de (cid:0) H k ( z + ) E + k ( z + ) ( z + ) w (cid:48) de − ( β + ) − ( β + ) c s , de − H E (cid:0) ( β + ) + ( β + ) c s , de − (cid:0) β + β + (cid:1) c s , de − (cid:0) c s , de − (cid:1) ( z + ) w (cid:48) de (cid:1) + β H k ( z + ) EE (cid:48) − β H ( z + ) (cid:0) − β + c s , de − (cid:1) E E (cid:48) (cid:1) + H E (cid:0)(cid:0) β + β + β + (cid:1) c s , de − ( z + ) (cid:0) β ( β + ) + (cid:0) β + (cid:1) c s , de (cid:1) w (cid:48) de + ( z + ) (cid:0) w (cid:48) de (cid:1) − β ( z + ) w (cid:48)(cid:48) de (cid:17) − w de (cid:0) k ( z + ) − H k ( z + ) E β ( z + ) w (cid:48) de + ( β + ) + (cid:0) β − (cid:1) c s , de + β H ( z + ) E E (cid:48) (cid:0) ( z + ) w (cid:48) de + β − ( β + ) c s , de + (cid:1) − H E (cid:0) c s , de (cid:0) − β − β − β + (cid:0) β + β + (cid:1) c s , de − (cid:1) − ( z + ) (cid:0) β + β + c s , de − (cid:1) w (cid:48) de + ( z + ) (cid:0) w (cid:48) de (cid:1) − β ( z + ) w (cid:48)(cid:48) de (cid:17)(cid:17)(cid:17)(cid:46)(cid:0) H ( z + ) ( w de + ) E (cid:0) k ( z + ) + H ( Ω de E − w de (cid:0) H (cid:0) − β + c s , de − (cid:1) E + k ( z + ) (cid:1) + H E (cid:0) ( z + ) w (cid:48) de − ( β + ) c s , de (cid:1)(cid:1)(cid:1) . 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