Large-scale structure formation in cosmology with classical and tachyonic scalar fields
aa r X i v : . [ a s t r o - ph ] O c t Large-s ale stru ture formation in osmology with lassi al and ta hyoni s alar (cid:28)eldsO. Sergijenko (1) , Yu. Kulini h (1) , B. Novosyadlyj (1) , V. Pelykh (2)
November 12, 2018 (1)
Astronomi al Observatory of Ivan Franko National University of Lviv (2)
Pidstryha h Institute of Applied Problems in Me hani s and Mathemati s of NASUThe evolution of s alar perturbations is studied for 2- omponent (non-relativisti matterand dark energy) osmologi al models at the linear and non-linear stages. The dark energyis assumed to be the s alar (cid:28)eld with either lassi al or ta hyoni Lagrangian and onstantequation-of-state parameter w . The (cid:28)elds and potentials were re onstru ted for the set of os-mologi al parameters derived from observations. The omparison of the al ulated within thesemodels and observational large-s ale stru ture hara teristi s is made. It is shown that for w = const su h analysis an't remove the existing degenera y of the dark energy models.Introdu tionThe observations of the last de ade surely on(cid:28)rm the a eleration of the osmologi al ex-pansion. The explanation of this fa t needs the assumption that the main part (cid:21) approxi-mately (cid:21) of the energy density of the Universe belongs to the mysterious repulsive om-ponent alled (cid:16)dark energy(cid:17). The simplest model des ribing satisfa tory almost the wholeset of the experimental data is Λ CDM-one. Here dark energy is identi(cid:28)ed with the Λ -termin the Einstein equations. However, in this ase there are several interpretational problems,whi h suggest that another solution should be found. The most popular alternative approa hesare quintessential s alar (cid:28)elds, i.e. s alar (cid:28)elds with the equation-of-state (EoS) parameter − < w de ≡ p de /ρ de < − / . The simplest physi ally-motivated Lagrangians are the lassi aland ta hyoni ones. The (cid:28)rst of them is the simple generalization of the non-relativisti par-ti le Lagrangian to the (cid:28)eld while the se ond ( alled also the Dira -Born-Infeld one) (cid:21) of therelativisti parti le one [3, 11, 27, 28, 31℄. The Lagrangian of lassi al (cid:28)eld has the anoni alkineti term, the Lagrangian of ta hyon (cid:28)eld has the non- anoni al one.As soon as the analysis of dynami s of expansion of the Universe [32℄ doesn't allow us to hoose the most preferable by the observational data model of s alar (cid:28)eld dark energy, here wefo us on study of the evolution of s alar perturbations and the large-s ale stru ture formationin the Universe (cid:28)lled only with the non-relativisti matter and either lassi al or ta hyoni (cid:28)eldminimally oupled to it. It should be noted that the behavior of perturbations has already beenstudied for di(cid:27)erent lassi al s alar (cid:28)elds more widely [5, 6, 34℄ than for ta hyoni ones [1, 10℄.The parametrizations of s alar (cid:28)elds, their impa t on formation of the large-s ale stru ture ofthe Universe as well as on osmi mi rowave ba kground anisotropy are widely dis ussed in theliterature (see, for example, [9, 13, 14, 29, 30℄ and iting therein). In this paper we analyse the1odels with re onstru ted for w de = const potentials of the lassi al and ta hyoni s alar (cid:28)eldsand ompare the obtained results to the Λ CDM-ones.1 Cosmologi al ba kgroundWe onsider the homogeneous and isotropi (cid:29)at Universe with metri of 4-spa e ds = g ij dx i dx j = a ( η ) (cid:0) dη − δ αβ dx α dx β (cid:1) , where the fa tor a ( η ) is the s ale fa tor, normalized to 1 at the urrent epo h η , η is onformaltime ( cdt = a ( η ) dη ). Here and below we put c = 1 , so the time variable t ≡ x has thedimension of a length, and the latin indi es i, j, ... run from 0 to 3, the greek ones (cid:21) over thespatial part of the metri : ν, µ, ... =1, 2, 3.If the Universe is (cid:28)lled with non-relativisti matter ( old dark matter and baryons) andminimally oupled dark energy, the dynami s of its expansion is ompletely des ribed by theEinstein equations R ij − g ij R = 8 πG (cid:16) T ( m ) ij + T ( de ) ij (cid:17) , (1)where R ij is the Ri i tensor and T ( m ) ij , T ( de ) ij (cid:21) energy-momentum tensors of matter ( m ) anddark energy ( de ) . If these omponents intera t only gravitationally then ea h of them satisfydi(cid:27)erential energy-momentum onservation law separately: T i ( m,de ) j ; i = 0 (2)(here and below (cid:16);i(cid:17) denotes the ovariant derivative with respe t to the oordinate x i ). Forthe perfe t (cid:29)uid with density ρ ( m,de ) and pressure p ( m,de ) , related by the equation of state p ( m,de ) = w ( m,de ) ρ ( m,de ) , it gives ˙ ρ ( m,de ) = − aa ρ ( m,de ) (1 + w ( m,de ) ) (3)(here and below a dot over the variable denotes the derivative with respe t to the onformaltime: (cid:16) ˙ (cid:17) ≡ d/dη ). The matter is onsidered to be non-relativisti , so w m = 0 and ρ m = ρ (0) m a − (here and below (cid:16)0(cid:17) denotes the present values).We assume the dark energy to be a s alar (cid:28)eld with either lassi al Lagrangian L clas = 12 φ ; i φ ; i − U ( φ ) (4)or Dira -Born-Infeld (ta hyoni ) one L tach = −U ( ξ ) p − ξ ; i ξ ; i , (5)where φ , ξ are the lassi al and ta hyoni (cid:28)elds respe tively while U ( φ ) , U ( ξ ) are the (cid:28)eldpotentials de(cid:28)ning the models. We suppose also the ba kground s alar (cid:28)elds to be homogeneous,so their energy densities and pressures depend only on time: ρ clas = 12 a ˙ φ + U ( φ ) , p clas = 12 a ˙ φ − U ( φ ) , (6) ρ tach = U ( ξ ) q − ˙ ξ /a , p tach = −U ( ξ ) s − ˙ ξ a . (7)2hen the onservation law gives the s alar (cid:28)eld evolution equations ¨ φ + 2 aH ˙ φ + a d U dφ = 0 , (8) ¨ ξ − aH ˙ ξ − (cid:16) ˙ ξ/a (cid:17) + 3 aH ˙ ξ + a U d U dξ = 0 , (9)where H = ˙ a/a is the Hubble onstant for any moment of the onformal time η .We spe ify the model of ea h (cid:28)eld using the EoS parameter w de ≡ p de /ρ de . It is obviousthat the s alar (cid:28)eld evolution equations have the analyti al solutions for w = const (here andbelow we omit index de denoting both (cid:21) lassi al and ta hyoni (cid:21) s alar (cid:28)elds for w de ). In this ase another important thermodynami al parameter (cid:21) the adiabati sound speed c a ≡ ˙ p de / ˙ ρ de (cid:21) is equal to w .The analysis of the dynami s of the Universe expansion for the re onstru ted (cid:28)elds with w = const was presented in [32℄. It doesn't depend on the s alar (cid:28)eld Lagrangian and (cid:21) as aresult (cid:21) doesn't allow us to distinguish su h models of s alar (cid:28)elds. So, in order to hoose themost adequate to observations type of dark energy we should study at least the linear stage ofthe evolution of s alar perturbations.2 Evolution of s alar linear perturbationsWe derive the equations of evolution of s alar linear perturbations in dark energy (cid:21) matter dom-inant era by varying of the Lagrange-Euler and Einstein equations in the onformal-Newtonianframe with spa e-time metri ds = a ( η )[(1 + 2Ψ( x , η )) dη − (1 + 2Φ( x , η )) δ αβ dx α dx β ] , (10)where Ψ( x , η ) and Φ( x , η ) are metri perturbations, whi h in the ase of zero proper anisotropyof medium (as for dust matter and s alar (cid:28)elds) satisfy the ondition Ψ( x , η ) = − Φ( x , η ) exa tly[4, 15℄. In the theory of linear perturbations all spatially-dependent variables are usualy Fourier-transformed, so, all perturbations (cid:21) of metri , (cid:28)elds, matter density and velo ity (cid:21) in equationsare presented by their Fourier amplitudes: Ψ( k, η ) ), δφ ( k, η ) , δξ ( k, η ) , δ ( m ) ( k, η ) , V ( m ) ( k, η ) et ., where k is the wave number. They are gauge-invariant (cid:21) as it is parti ularly dis ussed inthe original papers [4, 15℄ and numerous reviews (see, for example, [8, 9, 25℄ and iting therein).The energy density and velo ity perturbations of dark energy, δ ( de ) and V ( de ) , are onne tedwith the perturbations of (cid:28)eld variables δφ , δξ in the following way: δ ( clas ) = (1 + w ) ˙ δφ ˙ φ − Ψ + a δφ ˙ φ d U dφ ! , (11) V ( clas ) = kδφ ˙ φ , (12) δ ( tach ) = − ww ˙ δξ ˙ ξ − Ψ ! + 1 U d U dξ δξ, (13) V ( tach ) = kδξ ˙ ξ . (14)3ther non-vanishing gauge-invariant perturbations of s alar (cid:28)eld are isotropi pressure per-turbations π ( clas ) L = 1 + ww ˙ δφ ˙ φ − Ψ − a δφ ˙ φ d U dφ ! , (15) π ( tach ) L = 1 + ww ˙ δξ ˙ ξ − Ψ ! + 1 U d U dξ δξ (16)and intrinsi entropy Γ ( de ) = π ( de ) L − c a w δ ( de ) . (17)The density perturbation of any omponent in the onformal-Newtonian gauge D s ≡ δ , whi his gauge-invariant variable, is related to the other gauge-invariant variables of density pertur-bations D and D g as: D = D g + 3(1 + w ) (cid:18) Ψ + ˙ aa Vk (cid:19) = D s + 3(1 + w ) ˙ aa Vk , (18)where D s , D , D g and V orrespond to either m - or de - omponent.2.1 Evolution equationsEvolution equations for s alar (cid:28)eld perturbations δφ ( k, η ) and δξ ( k, η ) an be obtained eitherfrom Lagrange-Euler equation or from di(cid:27)erential energy-momentum onservation law T i i ( de ) =0 : ¨ δφ + 2 aH ˙ δφ + (cid:20) k + a d U dφ (cid:21) δφ + 2 a d U dφ Ψ − φ = 0 , (19) ¨ δξ + aH − aH ˙ ξa ! − U d U dξ ˙ ξ ˙ δξ + " k + a U d U dξ − (cid:18) U d U dξ (cid:19) ! × − ˙ ξa ! δξ − ˙Ψ ˙ ξ − ξ − ˙ ξa ! + 2Ψ a U d U dξ + 6 aH Ψ ˙ ξ ˙ ξa ! = 0 . (20)The linearised Einstein equations for gauge-invariant perturbations of metri and energy-momentum tensor omponents are ˙Ψ + aH Ψ − πGa k (cid:2) ρ m V ( m ) + ρ de (1 + w ) V ( de ) (cid:3) = 0 , (21) ˙ V ( m ) + aHV ( m ) − k Ψ = 0 , (22) ˙ D g ( m ) + kV ( m ) = 0 , (23) ˙ V ( de ) + aH (1 − c a ) V ( de ) − k (1 + 3 c a )Ψ − c a k w D ( de ) g − wk w Γ ( de ) = 0 , (24) ˙ D g ( de ) + 3( c a − w ) aHD ( de ) g + k (1 + w ) V ( de ) + 3 aHw Γ ( de ) = 0 , (25)4here w Γ ( clas ) = (1 − c a ) (cid:20) D ( clas ) g + 3(1 + w )Ψ + 3 aH (1 + w ) V ( clas ) k (cid:21) = (1 − c a ) D ( clas ) , (26) w Γ ( tach ) = − ( w + c a ) (cid:20) D ( tach ) g + 3(1 + w )Ψ + 3 aH (1 + w ) V ( tach ) k (cid:21) = − ( w + c a ) D ( tach ) . (27)In w = const - ase − c a = 1 − w , w + c a = 2 w , hen e the di(cid:27)eren e between equations for lassi al and ta hyoni (cid:28)elds isn't big (for w lose to − (cid:21) as it follows from the observabledata [16℄) and suggests the similarity of their solutions.So, in ea h ase we have the system of 5 (cid:28)rst-order ordinary di(cid:27)erential equations for 5unknown fun tions Ψ( k, a ) , D ( m ) g ( k, a ) , V ( m ) ( k, a ) , D ( de ) g ( k, a ) and V ( de ) ( k, a ) satisfying also the onstraint equation: − k Ψ = 4 πGa (cid:0) ρ m D ( m ) + ρ de D ( de ) (cid:1) . (28)2.2 Initial onditionsNow we are going to spe ify the adiabati initial onditions. The adiabati ity ondition intwo- omponent model gives D ( m ) g = D ( de ) g / (1 + w ) [7, 8, 15℄.Sin e the density of the w = const -(cid:28)elds is negligible at the early epo h ( a ≪ ), bothour models are initially matter-dominated. It is known that in su h ase the growing mode orresponds to Ψ = const . The (cid:28)eld equations of motion for the re onstru ted potentials arefollowing: δφ ′′ + (cid:18) − w de a − w − Ω de + Ω de a − w (cid:19) δφ ′ a + (cid:20) k H a (1 − Ω de + Ω de a − w ) +9(1 − w )4 a (cid:18) w + w Ω de a − w − Ω de + Ω de a − w (cid:19)(cid:21) δφ − a − w s πG Ω de (1 + w )1 − Ω de + Ω de a − w a Ψ ′ + 3(1 − w )Ψ a = 0 (29)for the lassi al (cid:28)eld and δξ ′′ − (cid:18)
12 + 3 w + 3 w de a − w − Ω de + Ω de a − w (cid:19) δξ ′ a − w (cid:20) k H a (1 − Ω de + Ω de a − w ) +92 a (cid:18) w Ω de a − w − Ω de + Ω de a − w (cid:19)(cid:21) δξ −√ wH √ − Ω de + Ω de a − w (1 − w ) a Ψ ′ − w Ψ √ a = 0 (30)for the ta hyoni one. Here and below a prime denotes the derivative with respe t to the s alefa tor a and Ω de = ρ de /ρ c , where ρ c ≡ H / πG .The ondition Ψ = const for a ≪ gives: δφ = 1 √ πG s Ω de (1 + w )1 − Ω de Ψ a − w , (31) δξ = 23 √ wH √ − Ω de Ψ a . (32)5ere Γ ( de ) = 0 .Using these solutions and equations (11)-(14), (18), (28), one an (cid:28)nd the initial values of D ( m ) g ( k, a ) , V ( m ) ( k, a ) , D ( de ) g ( k, a ) , V ( de ) ( k, a ) : V ( de ) init = 23 kH Ψ init √ − Ω de √ a init , (33) D ( de ) g init = − w )Ψ init , (34) V ( m ) init = 23 kH Ψ init √ − Ω de √ a init , (35) D ( m ) g init = − init , (36)whi h spe ify the growing mode of the adiabati perturbations.2.3 Numeri al analysisWe have integrated numeri ally the systems of equations for dust matter and dark energywith w = const for the adiabati initial onditions using the publi ly available ode DVERK1.We used the set of osmologi al parameters from http://lambda.gsf .nasa.gov/produ t/map,assumed Ψ init = − , a init = 10 − and integrated up to a = 1 . The evolution of perturbationsis s ale dependent, so we performed al ulations for k = 0 . , . , . and . Mp − .The models with the lassi al s alar (cid:28)eld are denoted as QCDM, with ta hyon (cid:21) as TCDM.For omparison we also solve the evolution equations for the Λ CDM-model.As it an be seen in Fig.1, the simple on lusion, that the behavior of the s alar linearperturbations in the model with the ta hyoni (cid:28)eld with w = const should be similar to thatin the model with the orresponding lassi al (cid:28)eld, is valid. The urves for both (cid:28)elds almostoverlap, so in this ase it is not possible to hoose the Lagrangian prefered by observations (seealso [35℄).In both models studied here the matter lusters while dark energy is smoothed out on sub-horizon s ales. Generally, at present epo h the growth of the matter density perturbations issupressed and (cid:21) unlike Λ CDM- ase (cid:21) su h supression is s ale dependent, however this depen-den e is very weak. The dark energy perturbations grow approximately up to the moment ofthe entering of parti le horizon and start to de ay after that (the density perturbation D ( de ) (cid:21)slowly).Note that the perturbations in su h s alar (cid:28)eld models are insensitive to the initial ondi-tions. Really, if we assume the dark energy to be initially homogeneous ( δφ = δφ ′ = 0 and δξ = δξ ′ = 0 ), the results of numeri al integration will be the same as in adiabati ase (thesimilar on lusion was made in [6, 19℄).The simplest test for identi(cid:28) ation of the sour e ausing the a elerated expansion of theUniverse an be based on the a tion of the studied (cid:28)elds on osmi mi rowave ba kground.Here the main attention should be paid to the temporal variation of the gravitational potentialwhi h auses the late-time integrated Sa hs-Wolfe (ISW) e(cid:27)e t. In Fig.2 the evolution of Ψ isshown for both s alar (cid:28)eld models and for Λ CDM-one for the s ales of perturbations k = 0 . , . , . and . M pc − . It an be easily seen that the s ale dependen e is weak and thereis no substantial di(cid:27)eren e between lassi al and ta hyoni dark energy. Su h s alar (cid:28)elds arein many senses similar to the osmologi al onstant and their behavior is loser to that of the Λ -term for EoS parameter values loser to − . The given dependen e for Λ k = 0 . , . , . and . M pc − from top to bottom. In the right olumn they are also k = 0 . , . , . and . M pc − from top to bottom for the density perturbations while for thevelo ity ones the urves orrespond to k = 0 . , . , . and . M pc − from top tobottom at a ≈ . The osmologi al parameters are: Ω de = 0 . , w = − . , Ω m = 0 . , h = 0 . .allow us to ex lude this model using the observational data, be ause the di(cid:27)eren e between itand those in models with the s alar (cid:28)eld dark energy is not substantial.It should be noted that negle ting of the dark energy perturbations leads to the (cid:16)quasi- Λ CDM(cid:17)-models, i.e. models in whi h the (cid:28)elds a(cid:27)e t the growth of the matter perturbationsonly through the ba kground. In these models the de ay of the gravitational potential is s aleindependent and lose to the small-s ale one in the models with perturbed dark energy (inagreement with the results of [19℄).Another possible test is based on the study of a tion of dark energy on the lustering proper-ties of dust matter. However, here we need the analysis of the evolution of s alar perturbationsat the non-linear stage. 7igure 2: Evolution of the gravitational potential for the s ales k = 0 . , . , . and . M pc − (from top to bottom). The urves for QCDM- (solid line) and TCDM- (dotted) modelswith the dark energy perturbations overlap.3 Spheri al ollapse in the models with homogeneous darkenergyThe simplest and most popular approa h used in the study of the non-linear stage of the large-s ale stru ture formation is the spheri al ollapse model. Within this framework we analysethe formation of the virialised haloes in the Λ CDM- and in the w = const QCDM- and TCDM-models with re onstru ted potentials, dis ussed in the previous se tions.The magnitudes of density perturbations of the lassi al and ta hyoni s alar (cid:28)elds with s aleless than the parti le horizon are lower than orresponding magnitudes of the matter densityones by few orders and pra ti ally do not a(cid:27)e t their growth. The amplitudes of matter densityperturbations in the QCDM- and TCDM-models grow almost equally in osmologies with thesame parameters. They are also lose to ones in the Λ CDM-models. So, in order to simplifythe dis ussion of the non-linear evolution of s alar perturbations we assume the dark energy omponent to tend to homogeneity. Other reasons for homogeneous distribution of dark energyin the regions of matter inhomogeneties see, for example, in [22, 23℄. Hen e, the temporaldependen e of the dark energy density is de(cid:28)ned by the orresponding ba kground equation.(Stru ture formation in inhomogeneous dark energy models has been analysed in [26℄.)The relative perturbation of mass of the dust omponent in the omoving volume v =4 πR / and metri ds = dt − M ( R ) y ( t, R ) dR − x ( t, R ) R (cos θdϕ + dθ ) is following[18℄: δ m = (cid:18) a ( t ) x ( t, R ) (cid:19) − , (37)where x ( t, R ) is the lo al s ale fa tor derived from the Einstein equation G = κ T (here G ij isthe Einstein tensor and κ is the Einstein onstant) [18, 17℄: ¨ x = − p de ρ c x −
12 ˙ x x + 1 x Ω f (38)(in this se tion (cid:16) ˙ (cid:17) ≡ d/H dt ). 8or the Λ -term we should put p de /ρ c = − Ω Λ while for the quintessential dark energy therelation is p de /ρ c = w Ω de . The lo al urvature parameter Ω f gives the amplitude of the initialperturbation: δ m ( t ) ≃ (Ω K − Ω f )Ω − m a ( t ) at a ≪ . Sin e in the Λ - ase Ω f ≡ Ω f ( R ) forthe dark energy we should put Ω f ≡ Ω f ( t, R ) [36, 17℄. It means that here we need an addi-tional equation de(cid:28)ning the evolution of the lo al urvature. However, for the homogeneousperturbations ( ∂∂R Ω f = 0 , x ≡ x ( t ) = y ( t ) ) using the ombination of the Einstein equations G + G + 2 G = κ ( T + T + 2 T ) we obtain the motion equation without time-dependent urvature [36℄: ¨ xx = − ρ c (3 p de + ρ de + ρ m ) , (39)where the dust matter density is ρ m = ρ (0) m x − . Combining of the equation (39) and Friedmannequations for the homogeneous Universe allows us to (cid:28)nd the evolution of the mass perturbation.In the analysis of halo formation the moment of rea hing of the dynami al equilibrium isimportant. A ording to the virial theorem at this moment the kineti energy be omes T vir = − U m,vir + U Λ ,vir . In the Λ CDM-model the energy onservation law T + U m + U Λ = E is obtainedby integration of the equation of motion (38), multiplied by x ˙ x . From this follows: T = ˙ x , U m = − Ω m x − , U Λ = − Ω Λ x and E = Ω f . In models with the Λ -term the total energy is onstant in time and at the rea hing of the dynami al equilibrium is equal E = U m,vir +2 U Λ ,vir .Alternatively, at the turnaround moment, when ˙ x = T = 0 , the total energy is E = U m,ta + U Λ ,ta .From these equations we obtain (cid:28)nally: U m,vir + 2 U Λ ,vir = U m,ta + U Λ ,ta . This identity, validfor the Λ - ase, doesn't hold for the dark energy, sin e the temporal variation of the urvaturein the perturbed region gives E ( t vir ) = E ( t ta ) . In other words, expli it temporal dependen e ofthe dark energy density ρ de ( t ) leads to the expli it dependen e of the potential energy of this omponent U de ( t, x ) on time and (cid:21) hen e (cid:21) to the expli it temporal dependen e of the totalenergy: E ( t ) = T ( ˙ x )+ U m ( x )+ U de ( t, x ) . It means that at di(cid:27)erent times we have di(cid:27)erent valuesof the total energy. For estimation of the moment of rea hing of the dynami al equilibrium forthe dark energy ase we use the equations (3.13)-(3.17) from [17℄. These equations des ribethe evolution of the spheri ally-symmetri perturbation with the arbitrary pro(cid:28)le in the modelwith dark energy. Assuming there Ω f = Ω f ( t ) , x ≡ x ( t ) = y ( t ) and V ≡ V ( t ) , we obtain theequations for the homogeneous spheri al loud. The additional ondition of homogeneity ofthe dark energy (the equality of (3) and (3.17) from [17℄) gives the expression for V . Usingit together with (3.15) from [17℄ we obtain the equation des ribing the temporal variation of urvature: ˙Ω f = 3 (cid:18) ˙ aa − ˙ xx (cid:19) (1 + w )Ω de x . (40)Combining of this equation with (38) leads to the energy onservation equation E ( t ) = T ( ˙ x ) + U m ( x ) + U de ( t, x ) in the following form: ˙ x −
12 Ω m x −