Clustering of dark matter in interacting tachyon dark energy with Λ CDM background
aa r X i v : . [ a s t r o - ph . C O ] S e p Clustering of dark matter in interacting tachyon darkenergy with Λ CDM background
Ankan Mukherjee Centre for Theoretical Physics,Jamia Millia Islamia, Jamia Nagar, New Delhi-110025, India.
Abstract
One of the non-canonical descriptions of scalar field dark energy is the tachyon. Thepresent work is devoted to study the dynamics of dark matter overdensity in a conformallycoupled tachyon field dark energy model. The model is tuned to mimic the ΛCDM cosmologyat background level. The semi-analytic spherical collapse model of dark matter overdensity isadopted to study the nonlinear evolution. The effects of non-minimal coupling in the energybudget on the clustering of dark matter is investigated. It is observed that the growth rateof matter overdensity is higher in presence of the non-minimal coupling. The critical densityat collapse is suppressed in case of interaction. Further the number counts of dark matterhalos or galaxy clusters along redshift are studied using the Press-Schechter and Sheth-Tormen halo mass functions. Suppression in the number of dark matter halos is observedwhen the interaction is allowed. A comparison of cluster number count in the present modeland ΛCDM is carried out. The present model allowing the interaction produces much lowernumber of galaxy clusters compared to ΛCDM, but without interaction the cluster numbercount is slightly higher than ΛCDM cluster count.
Cosmological observations revels that the present universe is going through a phase of acceleratedexpansion [1, 2, 3, 4]. Within the regime of General Relativity, cosmic acceleration can beexplained by introducing an exotic component in the energy budget of the universe. This exoticcomponent is dubbed as dark energy . There are various theoretical explanations regarding thephysical entity of dark energy, starting from the vacuum energy density [5, 6, 7] to differentcanonical and non-canonical scalar fields [8, 9, 10, 11, 12, 13] or some exotic fluid with specialnature of the equation of state [14, 15]. Email: [email protected]
Though dark energy has started dominating the dynamics of the universe in the recent past(around redshift z < . σ ) from Planck-ΛCDMand the value of σ measured from the redshift space distortion. The evolution of the perturbationand clustering of dark matter bear the signature of dark energy properties. Thus the study of darkmatter clustering is useful to distinguish dark energy models which are degenerate at backgroundlevel.Tachyon is a non-canonical description of scalar field dark energy. In the present work, thebackground is tuned to mimic the ΛCDM by imposing the equality of Hubble parameter andits first derivative for the present model and the ΛCDM. Further the tachyon filed is allowed tointeract with the matter field through a non-minimal coupling. Fluctuation of matter density ina coupled quintessence with ΛCDM background is explored by Barros et al [41]. The sphericalcollapse of matter overdensity in coupled quintessence with ΛCDM background is studied byBarros, Barreiro and Nunes [37]. In the present analysis, the interaction between the tachyondark energy and the dark matter is constructed from phenomenological assumption about theinteraction function. The utility of fixing the background as the ΛCDM is that the scalar fileddark energy density and the equation of state parameter can be expressed independent of thescalar filed potential. Thus the analysis is independent of any specific choice of the scalar filedpotential. The effect of interaction on the evolution of matter overdensity and its collapse isinvestigated. The number of collapsed objects or the dark matter halos along the redshift isalso emphasized. The halo mass function formula is important to study the number count ofdark matter halos. Two different formalism of halo mass function, namely the Press-Schechtermass function and Sheth-Tormen mass function are utilized in the present context. Both themass functions formalisms are based on the assumption of Gaussian matter density filed. Thedifference between the number counts of dark matter halos in these two formalism is emphasized.Further the results are compared with the corresponding ΛCDM scenario.The manuscript is organized as the following. In section 2, the theoretical formulation ofinteracting tachyon dark energy in ΛCDM background is discussed. In section 3, evolution ofmatter overdensity and the spherical collapse are discussed in the context of the present model.In section 4, the number count of dark matter halos or galaxy clusters are studied. Finally theresults are summarized in the discussion section (section 5). Λ CDM background
The homogeneous and isotropic universe is described by Friedmann-Lemaitre-Robertson-Walker(FLRW) metric. In a spatially flat geometry, FLRW metric is written as, ds = − dt + a ( t ) δ ij dx i dx j , (1)where a ( t ) is the scale factor. The Hubble expansion rate or the Hubble parameter is definedas H = ˙ a/a , where the overhead dot denotes the differentiation with respect to time t . Thebackground evolution is governed by Friedmann equations. In terms of Hubble parameter andand its time derivative, the Friedmann equations are written as,3 H ( z ) = 8 πG ( ρ m + ρ r + ρ DE ) , (2)2 ˙ H + 3 H = − πG ( p r + p de ) , (3)where ρ m , ρ r , ρ de are respectively the matter, radiation and dark energy densities, and p r , p de arepressure contributions of radiation and dark energy. In the present context, the non-canonicalscalar field, namely the tachyon, is considered as the candidate of dark energy. The dark energydensity ρ de is further denoted as the scalar field energy density ρ φ . Tachyon field in the context ofdark energy was invoked by Padmanabhan [11]. Many more discussions on tachyon dark energyare there in literature [12, 42, 43, 44, 45, 46]. Aspects of spherical collapse in tachyon dark energyhave been discussed by Rajvanshi and Bagla [31] and by Setare, Felegary and Darabi [39]. Effectof inhomogeneous tachyon dark energy on dark matter clustering is studied by Singh, Jassal andSharma [40]. A tachyon field is described by the Lagrangian, L = − V ( φ ) q − ∂ µ φ∂ µ φ, (4)where V ( φ ) is scalar field potential. In case of dark energy, the scalar field is assumed to behomogeneous and evolving with time, i.e. φ = φ ( t ). The energy density ( ρ φ ) and pressure ( p φ ) ofthe tachyon field is given as, ρ φ = V ( φ ) q − ˙ φ , p φ = − V ( φ ) q − ˙ φ . (5)Consequently the equation of state parameter of the scalar field is given as, w φ = ˙ φ −
1. Theequation of state of tachyon field does not dependent on the choice of the scalar field potential. Asalready mentioned, in the present context a ΛCDM like background evolution is tuned. In ΛCDMcosmology, the dominant contributions to the energy density come from the vacuum energy density ρ Λ and the cold dark matter ρ cdm . The vacuum energy density ρ Λ remains constant throughoutthe evolution and ρ cdm ∝ a − . To ensure a ΛCDM like background in the present model, theHubble parameter and its first order time derivative for the present model are taken to be equalto that of ΛCDM. The equalities relate the energy densities as, ρ cdm + ρ Λ = ρ m + ρ φ , (6)and ρ cdm = ρ m + ρ φ + p φ . (7)Utilizing equation (5), (6) and (7), the scalar field potential is expressed as V ( φ ) = ρ Λ q − ˙ φ . (8)Similarly the matter density is expressed as, ρ m = ρ cdm − ρ Λ ˙ φ (1 − ˙ φ ) . (9)Thus physical quantities of a tachyon dark energy model are expressed in terms of φ , ˙ φ and ρ Λ and ρ cdm in the present context. Further, interaction between the scalar field and the dark matteris introduced. The total conservation of the energy momentum tensor ( T µν ) is obtained from thecontracted Bianchi identity T µν ; µ = 0. For homogeneous distribution of different components inFLRW geometry, the conservation equation of individual components are written as,˙ ρ m + 3 Hρ m = − Q, (10)˙ ρ de + 3 H (1 + w ) ρ de = Q, (11)˙ ρ r + 4 Hρ r = 0 . (12)The Q is the interaction function that determines characteristics of energy transfer between thedark matter and dark energy component. Independent conservation of radiation energy densityis assumed in the present analysis (equation 12). The conservation of ρ φ yields the evolutionequation of the scalar field as,¨ φ + (1 − ˙ φ ) " H ˙ φ + 1 V dVdφ = (1 − ˙ φ ) / V ˙ φ Q. (13)From equation (8) the differentiation of the scalar field potential V ( φ ) with respect to φ is ex-pressed as, 1 V dVdφ = ¨ φ (1 − ˙ φ ) . (14)Finally equation (13) can be written in the form which is independent of the scalar filed potential,2 ¨ φ + 3 H (1 − ˙ φ ) ˙ φ = (1 − ˙ φ ) ρ Λ ˙ φ Q. (15)The interaction function effects the evolution of the scalar field. There is no theoreticalcompulsion about the choice of the interaction function. Thermodynamical requirement demandsthat the energy transfer in the interaction should be from dark matter to dark energy [47]. Tosatisfy the condition of energy flow, the interaction function Q should be positive according to thechoice of signature in equations (10) and (11). The interaction dynamics of dark energy and darkmatter is usually studied using phenomenological assumption regarding the interaction function.In the present analysis, the form of the interaction functions is assumed as, Q = βHρ m . (16)The interaction function is proportional to the dark matter density ρ m and interaction rate islinear to the Hubble expansion rate. The dimensionless parameter β determines the strength ofcoupling between the scalar field and the matter field. Changing the argument of differentiationto the scale factor a , the scalar field equation (15) is obtained as, φ ′′ + a + H ′ H ! φ ′ + 3 φ ′ a (1 − a H φ ′ ) = (1 − a H φ ′ ) ρ Λ a H φ ′ Q. (17) β = × - × - × - × - a H ϕ β = a H ϕ Figure 1:
Plots of the redefined scalar field ˜ φ = H φ as a function of the scale factor a obtained from thenumerical solution of equation (20) with initial conditions, fixed at a i = 10 − , as ˜ φ i = 0 .
0, ˜ φ ′ i = 0 . β = 0 . β = 0 . For the interaction function, given is equation (16), equation (17) yields as, φ ′′ = − a + H ′ H ! φ ′ − φ ′ a (1 − a H φ ′ ) + β (1 − a H φ ′ ) a H φ ′ ρ m ρ Λ ! . (18)Finally using the expression of ρ m from equation (9), the equation is written as, φ ′′ = − a + H ′ H ! φ ′ − φ ′ a (1 − a H φ ′ ) + β (1 − a H φ ′ ) a H φ ′ ρ cdm ρ Λ a − − a H φ ′ − a H φ ′ ! , (19)where ρ cdm is the present density of cold dark matter in ΛCDM cosmology. For convenience,the scalar field is redefined in a dimensionless way as ˜ φ = φH , where H is the present Hubbleparameter or the Hubble constant. The Hubble parameter H ( a ) is rescaled by H as h ( a ) = H ( a ) /H . In terms of the redefined scalar field, equation (19) can be written as,˜ φ ′′ = − a + h ′ h ! ˜ φ ′ − φ ′ a (1 − a h ˜ φ ′ ) + β (1 − a h ˜ φ ′ ) a h ˜ φ ′ ρ cdm ρ Λ a − a h ˜ φ ′ − a h ˜ φ ′ ! . (20)Equation (20) is studied numerically to obtain the evolution of the scalar field. The initial condi-tions are fixed at the scale factor a = 10 − and the initial conditions are ˜ φ i = 0 and ˜ φ ′ i = 0 . φ ( a ) for different values of the coupling parameter β .When the interaction coupling parameter is fixed to β = 0, that means no interaction is allowed,the scalar field shows a slowly varying nature. On the other hand, non zero coupling parameterallows a rapid evolution of the scalar field. The value of the ratio ( ρ cdm /ρ Λ ) is fixed from thePlanck-ΛCDM estimation by Planck 2018 [4]. Linear0.0 0.2 0.4 0.6 0.8 1.00.0010.0100.1001 a δ Nonlinear0.0 0.2 0.4 0.6 0.8 1.00.0010.0100.1001101001000 a δ Figure 2:
Linear and nonlinear evolution of δ ( a ) obtained from the numerical solutions of equation (23)and (24). The boundary conditions are fixed at a = 10 − as δ i = 3 . × − and δ ′ i = 0 .
0. The solidcurves are obtained for β = 0 .
001 and the dashed curves are obtained for β = 0 . In this section, the evolution of matter overdensity in the present dark energy model is discussed.The evolution equation of matter overdensity is conveniently written in terms of the matter densitycontrast, defined as δ = ∆ ρ m ρ m where ∆ ρ m is the deviation from homogeneous matter density ρ m .The overdense region initially expands with Hubble expansion. At the same time it gathers massdue to gravitational attraction. After certain amount of mass accumulation, the overdense regionstops the expansion and starts to collapse. The collapse of the overdense region is the fundamentalprocess of large scale structure formation in the universe. It is essential to study the nonlinearevolution of the matter overdensities to understand the dynamics of the structure formation.Spherical collapse model [19, 20, 21, 22] is the simplest approach to probe the evolution of thematter density contrast at the nonlinear regime. It is a semi-analytic approach that assumes theoverdense regions are spherically symmetric and the density inside the sphere is homogeneous.The nonlinear differential equation of matter density contrast in case of interacting dark energyis discussed in [24, 38, 48]. For the present interaction function the equation is written as,¨ δ + ˙ δ (2 − β ) H − πGρ m δ (1 + δ )(1 + 2 β ) −
43 ˙ δ δ = 0 , (21)and the linear version of equation (21) is given as,¨ δ + ˙ δ (2 − β ) H − πGρ m δ (1 + 2 β ) = 0 . (22)As the background is fixed to evolve like ΛCDM, the matter density ρ m in equation (21) and(22) is expressed by the relation given in equation (9). Thus the scalar field evolution effects a f g z c δ c Figure 3:
Left panel shows the linear growth rate f g as a function of the scale factor. Right panelshows the critical density at collapse δ c as a function of the redshift at collapse z c . The solid curves areobtained for β = 0 .
001 and the dashed curves are obtained for β = 0 . the dynamics of δ in this context. Taking the scale factor a as the argument of differentiation,equation (21) is obtained as, δ ′′ + − βa + h ′ h ! δ ′ − Ω cdm a h − (1 − Ω cdm ) ˜ φ ′ (1 − a h ˜ φ ′ ) ! δ (1 + δ )(1 + 2 β ) − δ ′ δ = 0 , (23)and similarly the linear equation (eq. 22) is obtained as, δ ′′ + − βa + h ′ h ! δ ′ − Ω cdm a h − (1 − Ω cdm ) ˜ φ ′ (1 − a h ˜ φ ′ ) ! δ (1 + 2 β ) = 0 . (24)Equation (23) and (24) are studied numerically. The linear and nonlinear evolution of δ are shownin figure 2 for different values of β . In figure 2 the initial conditions are fixed at a = 10 − as δ i = 0 . δ ′ i = 0 .
0. The linear and nonlinear evolutions are similar at early time. But atlater stage, the nonlinear growth rate is higher than the linear growth rate and eventually thenonlinear evolution leads to the singularity of δ . The singularity of δ indicates the collapse ofthe matter overdensity. The linear evolution is less effected by the change of coupling parameter β . But the effect of interaction is more prominent in case of nonlinear evolution. With the sameboundary conditions, the nonliear δ grows at faster rate in case of nonzero β . Linear growthfunction is defined as g ( a ) = δ ( a ) /δ , where δ is the present value of linear density contrast.The linear growth rate is defined as f g = d ln δd ln a . The linear growth function is obtained from thesolution of equation (24). The right panel of figure 3 shows the f g ( a ) curves for β = 0 . β = 0 . β at latetime.Another important quantity in the study of dark matter clustering in a spherically collapsingscenario is the critical density contrast at collapse ( δ c ). It is defined as the value of the lineardensity contrast at the redshift where the nonlinear density contrast diverges. Changing the initialvalue of δ in the differential equation (equation 23), the redsift of nonlinear collapse is changed.The initial value of δ required to have a collapse at z = 0 is δ i = 3 . × − for β = 0 . δ i = 3 . × − for β = 0 .
0. The δ c is determined from the linear equation of δ (equation(24)) with the same initial condition. The curves of δ c ( z ) for the present model are shown in theleft panel of figure 3. The critical density at collapse δ c ( z ) is essential to study the number ofcollapsed objects or dark matter halos along the redshift. In this section, the number count of collapsed object along the redshift is studied for the presentmodel. The collapsed objects are called the dark matter halos. The distribution of ordinarybaryonic matter follows the distribution of dark matter due to gravitational attraction. Theclusters of galaxies are actually embedded in dark matter halos. Thus the distribution of darkmatter halos can be tracked by observing the distribution of galaxy clusters. There are twodifferent mathematical formulations of halo mass to evaluate the number count of collapsed objectsor halos along the redshift. The first one is the Press-Schechter mass function formalism [49].Later a generalization of Press-Shechter mass function was proposed by Sheth and Tormen [50].Both these formalism stand up with the assumption of a Gaussian distribution of the matterdensity field. In the present analysis, both of these mass functions are utilized. The comvingnumber density of collapsed objects or dark matter halos at a certain redshift z having massrange M to M + dM is expressed as, dn ( M, z ) dM = − ρ m M ddM ln σ ( M, z ) ! f ( σ ( M, z )) , (25)where f ( σ ) is the mass function . The mathematical formulation of the mass function was firstproposed by Press and Schechter [49], which is given as, f P S ( σ ) = s π δ c ( z ) σ ( M, z ) exp " − δ c ( z )2 σ ( M, z ) . (26)The σ ( M, z ) is the corresponding rms density fluctuation in a sphere of radius r enclosing a massM. This can be expressed in terms of the linearized growth factor g ( z ) = δz/δ (0), and the rms ofdensity fluctuation at a fixed length r = 8 h − Mpc as, σ ( z, M ) = σ (0 , M ) (cid:18) MM (cid:19) − γ/ g ( z ) , (27)where M = 6 × Ω m h − M ⊙ , the mass within a sphere of radius r and the M ⊙ is the solarmass and h is the Hubble constant H scaled by 100 km.s − M pc . The γ is defined as γ = (0 . m h + 0 . (cid:20) .
92 + 13 log (cid:18) MM (cid:19)(cid:21) . (28)0 z ( Press - Schechter ) × z ( Sheth - Tormen ) Figure 4:
Plots show the cluster number count along redshift for the present dark energy model. The leftpanel is obtained for the Press-Schechter mass function and the right panel is obtained for Sheth-Tormenmass function. The solid curves are for β = 0 .
001 and the dashed curves are for β = 0. Finally the number of collapsed objects within a mass range M i < M < M s per redshift persquare degree yield as, N ( z ) = Z deg d Ω cH ( z ) "Z z cH ( x ) dx Z M s M i dndM dM. (29)This N ( z ) is called the number count of dark matter halos or cluster number count. Press-Schechter mass function formula is successful to depict a general nature of cluster number count.But it predicts higher abundance of galaxy cluster at low redshift and lower abundance at highredshift compared to the result obtained in simulation of dark matter halo formation [51]. Toalleviate this issue, a modification is proposed by Sheth and Tormen [50], which is given as, f ST ( σ ) = A s π " σ ( M, z ) δ c ( z ) ! p δ c ( z ) σ ( M, z ) exp " − aδ c ( z )2 σ ( M, z ) . (30)The Sheth-Tormen mass function formula, given in equation (30), introduces three new param-eters ( a, p, A ). For the values (1 , , ) of the set of parameters ( a, p, A ) the Sheth-Torman massfuntion actually becomes the Press-Schechter mass function. In the present work, while studyingthe cluster number count using Sheth-Tormen mass function formula, the values of the parameters( a, p, A ) are fixed at (0 . , . , . N ( z ) using Press-Schechter and Sheth-Tormen mass function for the present dark energy scenario. Values of cosmological parameters1 z S T - P S -
20 000020 00040 00060 00080 000100 000 z S T - P S Figure 5:
Plots show the difference between the cluster count obtained for Press-Schechter mass functionand Sheth-Tormen mass function. The solid curves are for β = 0 .
001 and the dashed curves are for β = 0.The right panel shows the same in the redshift range 0 ≤ z ≤ Ω m , H and σ are fixed at the best fit of latest measurements from Planck-ΛCDM along withCMB lensing and BAO data [4]. The values are Ω m = 0 . H = 67 . km.s − M pc − and σ = 0 . h − M ⊙ < M < h − M ⊙ . In figure 4, plots areobtained for two different values of the coupling parameter β = 0 .
001 and β = 0. The clusternumber count is highly suppressed when the interaction is allowed.The difference between Press-Schechter and Sheth-Tormen cluster number count is shown infigure 5. The Sheth-Tormen mass function produces much higher number of dark matter halosat high redshift compared to the number of halos produced in case of Press-Schechter. But atlow redshift ( z < . N = N Λ CDM − N
IT ach . The N Λ CDM is obtained for same values of cosmological parameters Ω m , H and σ . In case of β = 0, the number count in the present model is slightly higher than the ΛCDM.On the other hand, in case of β = 0 . β = -
25 000 -
20 000 -
15 000 -
10 000 - z Δ β = z Δ Figure 6:
Plots show the difference in cluster number count for the present dark energy scenario andthe ΛCDM, ∆ N = ( N Λ CDM − N
IT ach ). The left panel is for β = 0 and the right panel is for β = 0 . In the present work, the nonlinear evolution of matter overdensity is studied in a non-minimallycoupled tachyon scalar field dark energy with ΛCDM background. The nonlinear clustering ofdark matter is explored with the assumption of spherical collapse model of matter overdensity.It is observed that the non-minimal coupling causes a faster growth of matter density contrastcompared to the noninteracting scenario in the nonlinear regime (figure 2). The linear growthrate is also higher at low redshift in case of interaction (left panel of figure 3). In noninteractingscenario, the density contrast evolve for longer time before collapse. This causes a higher valueof critical density at collapse ( δ c ) in case of noninteracting scenario than the interacting scenario(right panel of figure 3).Further the number count of collapsed objects or the galaxy clusters along redshift is studiedfor two different mass function formalisms. It is observed that the cluster number count issubstantially suppressed in case of interaction for both the mass functions (figure 4). The clusternumber is higher at high redshit in case of Sheth-Tormen mass function compared to the clustercount for Press-Schechter mass function. At low redshift ( z < . Acknowledgment
The author would like to acknowledge the financial support from the Science and EngineeringResearch Board (SERB), Department of Science and Technology, Government of India throughNational Post-Doctoral Fellowship (NPDF, File no. PDF/2018/001859). The author would liketo thank Prof. Anjan A. Sen for useful discussions and suggestions.
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