Signatures of Spatial Curvature on Growth of Structures
aa r X i v : . [ a s t r o - ph . C O ] J un Signatures of Spatial Curvature on Growth of Structures
Mohammad H. Abbassi ∗ and Amir H. Abbassi † Department of Physics, School of Sciences, Tarbiat Modares University. P.O. Box 14115-175, Tehran, Iran (Dated: June 12, 2020)We write down Boltzmann equation for massive particles in a spatially curved FRW universe andsolve the approximate line-of-sight solution for evolution of matter density, including the effectsof spatial curvature to the first order of approximation. It is shown that memory of early timegravitational potential is affected by presence of spatial curvature. Then we revisit Boltzmannequation for photons in the general FRW background. Using it, we show that how the frequency ofoscillations and damping factor (known as Silk damping) changed in presence of spatial curvature.At last, using this modified damping factor in hydrodynamic regime of cosmological perturbations,we find our analytic solution which shows the effects of spatial curvature on growing mode of matterdensity.
I. INTRODUCTION
It is believed that all of the current data are fitted pretty well with a flat FRW universe. The latest data (Planck2018 temperature and polarization of power spectra [1]) indicates a constraint that Ω K = − . +0 . − . . In spiteof the fact that flat FRW universe is strongly supported by different cosmological observations, but constraint onspatial curvature assumes a specific cosmological model. This means that most of this results are cosmological modeldependent. Many model independent methods to measure the spatial curvature of the universe are proposed [2–5].These results shows that non-zero Ω K can not be easiliy ruled out by current observations. In addition, there arevast majority of study to constraint topology of the universe using cosmological data [6–9]. Beside this practicalconcerns, from the theoretical perspective it is also interesting to know how the topology of the universe can affectour cosmological phenomena. Furthermore, data indicates a positive cosmological constant, which leads to a de Sitteruniverse for vacuum solutions. Knowing that from all the de Sitter solutions only the Lorentzian de Sitter spacetimewhich is spatially closed, is maximally symmetric, maximally extended and geodesiclally complete, increased ourtheoretical interests for studying cosmology around this background [10]. It’s well-known that spatial curvaturecauses shifts of angular scale of acoustic peaks, change primordial spectrum and evolution of long wavelength modesthrough Integrated Sachs-Wolf effect. In this work we want to show that how spatial curvature affects evolution ofmatter density by studying relativistic Boltzmann equation for matter. We drive an analytical formula that showshow frequency and damping of acoustic oscillations are affected by spatial curvature. Especially, we try to givean analytical solution to our equations so that we can have a clear interpretation and perception about behaviorof spatial curvature in our solutions. In section II we write relativistic Boltzmann equation for dark matter for aspatially curved background. We find an integro-differential solution for zeroth moment of it which gives evolution ofmatter density. In section III by writing relativistic Boltzmann equation for photons we find the effects of curvatureon damping parameter and oscillation frequency. In section VI we use the derived damping factor in hydrodynamiclimit of cosmological perturbation to show how spatial curvature affects the growth of matter density and especiallythe ripples of Baryon Acoustic Oscillations. In the last section we summarize our results and give some comments.Throughout this paper we use − + ++ signatures. Greek alphabets is set for 4 dimensional indices and Latin alphabetfor 3 dimensional ones. II. RELATIVISTIC BOLTZMANN EQUATION FOR MATTER
Through out this paper we assume a background of general curved FRW universe with the metric :¯ g = − , ¯ g i = 0 , ¯ g ij = a γ ij (1)where γ ij is the 3-dimensional spatial metric. In a semi-Cartesian coordinate it can be written as: γ ij = δ ij − Kx i x j (2) ∗ [email protected] † [email protected] where K is the constant of curvature of the spatial metric. There is a useful relation for spatial metric in semi-Cartesiancoordinates which helps us to simplify the following calculations: γ ii ′ γ jj ′ ∂ k γ i ′ j ′ = − ∂ k γ ij (3)Furthermore, the non vanishing components of the Christoffel’s connection of the metric are:¯Γ kij = 12 γ kl ( ∂ j γ li + ∂ i γ lj − ∂ l γ ij ) (4)¯Γ j i = Hδ ji (5)¯Γ ij = a Hγ ij (6)We define the four-momentum of dark matter particles as: p µ = m dx µ dτ (7)In this way, the particle’s four-momentum is related to particle’s velocity by dx i dt = p i p and zeroth component of fourmomentum would be p = p m + g ij p i p j . To write down the Boltzmann equation, we should define the numberdensity of the particles as function of Cartesian coordinates x i , momenta p i and time t which are the usual phasespace parameters of a system of particles. Note that here we use lower index p i only because the later calculationsare less complicated in terms of them . With the help of geodesic equation, we get: dp i dt = ∂ i g mn p m p n p (8)Because the interactions of the cold dark matter particles are only restricted to the gravitational interaction, numberdensity of cold dark matter satisfies collisionless Boltzmann equation, which is nothing but the fact that numberdensity is conserved in phase space. After applying chain rules and with the help of (8) we will have: ddt n ( x i , p i , t ) = ∂ t n ( x i , p i , t ) + p k p ∂ k n ( x i , p i , t ) + p l p m p ∂ k g lm ∂n ( x i , p i , t ) ∂p k = 0 (9)Then, we write the metric perturbations around this background as: g ij = a γ ij + δg ij , g ij = a − γ ij − a − δg ij (10)where the covariant and contravariant forms of the metric perturbations in the above definitions, are related in thisway: δg ij = γ ii ′ γ jj ′ δg i ′ j ′ (11)After linearizing the zeroth and i-th components of 4-momenta of the particles with respect to the metric perturbationwe get: p = p m + p /a (cid:18) − a − p i p j δg ij m + p /a ) (cid:19) (12) p i = a − γ ij p j − a − p j δg ij (13)in which p is defined as p ≡ p γ ij p i p j . The number density of the particles can be written as its background valueplus perturbed number density where the background number density is function of a p g ij p i p j : n = ¯ n (cid:18) a q g ij p i p j (cid:19) + δn ( x i , p i , t ) (14)The factor a ( t ) is included in its argument so that ¯ n ( p ) becomes time independent. After linearizing the backgroundnumber density in terms of metric perturbations, we get:¯ n = ¯ n (cid:18) a q g ij p i p j (cid:19) = ¯ n ( p ) − ¯ n ′ ( p ) p i p j δg ij a p (15)After straightforward but tedious calculations it can be shown that: p k p ∂ k ¯ n + p l p m p ∂ k g lm ∂ ¯ n∂p k = 0 (16)Putting the above number density in Boltzmann equation (9) and using (16), we would have our Boltzmann equationfor perturbed number density: ∂ t δn + a − γ ij p j p m + p /a ∂δn∂x i + a − K~x.~pp k p m + p /a ∂δn∂p k = n ′ ( p ) p i p j ∂ t (cid:0) a − δg ij (cid:1) p (17)The Fourier expansion of the scalar modes in a spatially closed universe can be written in the following form: δn ( x i , p i , t ) = Z d ˆ q X q e iq arccos(ˆ q.~x ) p − (ˆ q.~x ) δn ( q, ˆ q i , p i , t ) (18)where q = 3 , , ... which is the well-known fact that wave numbers are discrete in spatially closed universe. In thesame way, we can expand the metric perturbation, δg ij . Putting these modes expansion back in Boltzmann equation(17) and working in limit of x ≪
1, the equation (17) takes this form, ∂ t δn ( q i , p i , t ) + a − p m + p /a O δn ( q i , p i , t ) = ¯ n ′ ( p )2 p p i p j ∂ t (cid:0) a − δg ij ( q i , t ) (cid:1) (19)where q i = q ˆ q i and we define the differential operator O as: O ≡ i~q.~p + iKp k p l ∂ ∂p k ∂q l (20)From now till the end of this section, we omit the arguments of the perturbation parameters, so that are relationslooks much simpler. But we should note that these perturbations are in Fourier space. The integral solution of abovedifferential equation has the following form: δn = Z tt dt ′ exp " − Z tt ′ dt ′′ a ( t ′′ ) p m + p /a ( t ′′ ) O ¯ n ′ ( p )2 p p i p j ∂ t (cid:0) a − δg ij (cid:1) (21)By taking different moments of number density, we can get each components of energy-momentum tensor. Here wewe only use its zeroth moment which is : T = 1 √− g Z d p ( p n ) (22)At the end, the fluid parameters can be extracted from different components of energy-momentum tensor, δT = − δρ .In this section, we want to choose the metric perturbations in Newtonian gauge. Because the form of our solutionswill be simpler in this gauge. From Scalar-Vector-Tensor decomposition, we know that our perturbed equations donot mix different types of perturbations. Because our goal is to derive matter density, it is just enough to focus onscalar part of the metric perturbations: ds = − (1 + 2Φ) dt + a (1 − γ ij dx i dx j (23)From ij component of Einstein perturbed equation, it is easy to see that in the absence of scalar stress tensor, Π s , theNewtonian potentials Φ and Ψ are equal.Here, the metric perturbation, Φ, behaves as a source on the right hand side of equation (19). To first order in metricperturbation we would have, p − det ( g ) = a √ − Kx (1 − ≈ a (1 − p and ¯ n are perturbedas (12) and (15). Inserting the integral solution of number density in (22) we will find: δρ = 1 a Z d p ( − ¯ n ( p )Φ p m + p /a (2 m + 3 p /a ) − ¯ n ′ ( p )Φ p p m + p /a + p m + p /a Z tt dt ′ exp " − Z tt ′ dt ′′ a ( t ′′ ) p m + p /a ( t ′′ ) O ¯ n ′ ( p ) p∂ t Φ) (24)After using by part in the second integral, it will be simplified like this, δρ = 1 a Z d p p m + p /a − ¯ n ( p )Φ + Z tt dt ′ exp " − Z tt ′ dt ′′ a ( t ′′ ) p m + p /a ( t ′′ ) O ¯ n ′ ( p ) p∂ t Φ ! (25)Using (22) and after linearizing the (25) in terms of K, δρ can be written as: δρ = − ¯ ρ Φ + 1 a Z d p p m + p /a Z tt dt ′ exp " − Z tt ′ dt ′′ i~q.~pa ( t ′′ ) p m + p /a ( t ′′ ) ¯ n ′ ( p ) p∂ t ′ Φ − iK ~p.~qq ( p ¯ n ′′ ( p ) + p ¯ n ′ ( p )) ∂ t ′ Φ ′ Z tt ′ dt ′′ a ( t ′′ ) p m + p /a ( t ′′ ) ! (26)Here the prime over Φ indicates derivative with respect to wave number q. The momentum space volume elementin the limit of small scales can be written as, d p = p dpdzdφ , where z = ~p.~qqp , is the cosine of angle between ~p and ~q , φ is azimuthal angle of ~p with respect to ~q . So we can re-parametrize the integral solution in terms of these newparameters: δρ = − ¯ ρ Φ + 2 πa Z ∞ p dp Z − dz p m + p /a Z tt dt ′ exp " − Z tt ′ dt ′′ iqpza ( t ′′ ) p p /a ( t ′′ ) + m ¯ n ′ ( p ) p∂ t ′ Φ − Z tt ′ dt ′′ iKzp a ( t ′′ ) p p /a ( t ′′ ) + m ( p ¯ n ′′ ( p ) + ¯ n ′ ( p )) ∂ t ′ Φ ′ ! (27)Doing this integral solution in general is very complicated, but it will have much more simpler form in non relativisticlimit where p /a ≪ m , which is a permissible assumption for cold dark matter particles: δρ = − ¯ ρ Φ + 2 πa Z dpp Z dz Z tt dt ′ exp (cid:20) − iqpzm Z tt ′ dt ′′ a ( t ′′ ) (cid:21) (cid:18) m ¯ n ′ ( p ) p∂ t ′ Φ − iKzp ( p ¯ n ′′ ( p ) + ¯ n ′ ( p )) ∂ t ′ Φ ′ Z tt ′ dt ′′ a ( t ′′ ) (cid:19) (28)Also we should note that in the non relativistic limit, the rate of oscillation in the above exponential is much smallerthan the rate of oscillation in metric perturbation, Φ [13]. So we can pull back the time derivative to the whole of theargument: exp (cid:20) − iqpzm Z tt ′ dt ′′ a ( t ′′ ) (cid:21) (cid:18) m ¯ n ′ ( p ) p∂ t ′ Φ − iKzp ( p ¯ n ′′ ( p ) + ¯ n ′ ( p )) ∂ t ′ Φ ′ Z tt ′ dt ′′ a ( t ′′ ) (cid:19) ≈ ∂ t ′ (cid:18) exp (cid:20) − iqpzm Z tt ′ dt ′′ a ( t ′′ ) (cid:21) (cid:18) m ¯ n ′ ( p ) p Φ − iKzp ( p ¯ n ′′ ( p ) + ¯ n ′ ( p ))Φ ′ Z tt ′ dt ′′ a ( t ′′ ) (cid:19)(cid:19) (29)Using this fact, we can simplify δρ in this manner: δρ = − ¯ ρ Φ+ 2 πa Z dpp dz (cid:18) m ¯ n ′ ( p ) p Φ( t ) − e − iqpzm R tt dt ′′ a t ′′ ) (cid:18) m ¯ n ′ ( p ) p Φ( t ) − iKzp ( n ′′ ( p ) p + n ′ ( p ))Φ ′ ( t ) Z tt dt ′′ a ( t ′′ ) (cid:19)(cid:19) (30)with the help of by part in the first term of the above integral and using the fact that m N a ≡ ma R πp dp ¯ n = ¯ ρ andafter integration over z, we get: δ ≡ δρ ¯ ρ = −
2Φ + S (31) S ≡ m Φ( t ) N ω q ( t ) Z p dp ¯ n ′ ( p ) Sin ( pω q ( t ) m ) − K Φ ′ ( t ) N q Z p dp (¯ n ′′ ( p ) p + ¯ n ′ ( p )) Cos ( pω q ( t ) m ) − Sin ( pω q ( t ) m ) pω q ( t ) m ! (32)where the parameter ω q ( t ) is defined as: ω q ( t ) ≡ Z tt qdt ′′ a ( t ′′ ) (33)In the absence of spatial curvature, the second term can be interpreted as memory of the gravitational field at earlytimes. An effect like this find in [13] for gravitational waves, where they show that the presence of dark matteraffects propagation of gravitational wave by its memory at the time of emission. Here it can be seen that the spatialcurvature modified this memory as the second term of (32). The function S ( t ) is a transient function that goes tozero exponentially at late time, when dark matter particles travel a distance larger than mode’s wavelength, or inother words when pω q ( t ) m ≫
1. To clarify this fact, let us assume that n ( p ) has familiar Maxwell-Boltzmann form of:¯ n ( p ) = Ae − p P (34)and we set the normalization constant to A = q π N πP so that R d p ¯ n = N , where P is an arbitrary constant.Then after performing by parts and doing integration over p we will end up with: S = (cid:18) Φ( t )( ¯ v ω q ( t ) −
3) + K Φ ′ ( t ) q ω q ( t ) ¯ v (25 − ω q ( t ) ¯ v + ω q ( t ) ¯ v ) (cid:19) e − ωq ( t )2 ¯ v (35)We define ¯ v = P m , which is the mean square coordinate velocity for distribution function of ¯ n . As we mentionedearlier, dark matter particles are non relativistic and so ¯ v /a ( t ) ≪
1. For the particles that travel less than wavelengthof the modes between t and t, we can assume ¯ v ω q ( t ) ≪
1. Then we have S = − t ). In addition, at very latetime, where ¯ v ω q ≫ S ( t ) is exponentially small and this memory effect of the early time will be erased. In thebetween time, when v ω q ∼
1, we can see that spatial curvature, shows itself in memory as second term of (35).
III. DAMPING OF PHOTONS DUE TO NON RELATIVISTIC MEDIUM
In the case of photons, we denote their distribution function by n ij and the upper indices are due to photon’spolarization. This function obeys the Boltzamann equation as: ∂ t n ij + p k p ∂ k n ij + p l p m p ∂ k g lm ∂n ij ∂p k + (Γ ikλ − p i p Γ kλ ) p λ p n kj + (Γ jkλ − p j p Γ kλ ) p λ p n ki = C ij (36)Here C ij is the collision term, which includes interactions of Thomson scattering of photons with electrons in theplasma. Then, we can write the perturbed distribution function of photon as: n ij = 12 ¯ n ( ap )( g ij − g ik g jl p k p l ( p ) ) + δn ij (37)We can follow the same procedure of the last section, now for photon distribution. Considering the metric perturbation(10), the perturbed distribution function becomes: ¯ n = ¯ n ( p ) − p i p j a p ¯ n ′ ( p ) δg ij . In addition, it will be easy to show that¯ n satisfies the same relation as (16). Using this and Christoffel values of (4) streaming part of Boltzmann equationwould be: ddt n ij = ∂ t δn ij + 1 a γ kl ˆ p l ∂ k δn ij + 2 Ka ˆ p k p l x l ∂δn ij ∂p k − p a ˆ p i ′ ˆ p j ′ ¯ n ′ ∂ t ( a − δg i ′ j ′ )( γ ij − γ ik γ jl ˆ p k ˆ p l ) + 2 ˙ aa δn ij (38)The ˆ p i is the unit vector of p i , which is defined as ˆ p i ≡ p i p . Furthermore, the collision part can be written as: C ij = − ω c δn ij + 3 ω c π Z d ˆ p [ δn ij ( x i , p ˆ p , t ) − γ ik ˆ p k ˆ p l δn lj ( x i , p ˆ p , t ) − γ jk ˆ p k ˆ p l δn il ( x i , p ˆ p , t )+ γ ik γ jl ˆ p k ˆ p l ˆ p m ˆ p n δn mn ( x i , p ˆ p , t )] − ω c a p k δu k ′ γ kk ′ ¯ n ′ [ γ ij − γ il γ jl ′ ˆ p l ˆ p l ′ ] (39)where the definition of p in above relation is : p = p (1 + (ˆ p − ˆ p ) k δu k ′ γ kk ′ a ) (40)Our goal is to derive damping of acoustic waves in a homogeneous and time-independent plasma in a spatially curvedbackground. So we can consider a static universe a = 1 with spatial curvature as: ds = − dt + γ ij dx i dx j (41)When the collision rate is much larger than sound frequency, we can also assume that ω c is time independent. Thisassumption of static universe is permissible, because the damping effect would only be noticeable for deep inside thehorizon modes. So we can put our Boltzmann equation in this form: ∂ t δn ij + γ kl ˆ p l ∂ k δn ij + 2 K ˆ p k p l x l ∂δn ij ∂p k = − ω c δn ij + 3 ω c π Z d ˆ p [ δn ij ( x i , p ˆ p , t ) − γ jk ˆ p k ˆ p k ˆ p l δn il ( x i , p ˆ p , t )+ γ ik γ jl ˆ p k ˆ p l ˆ p m ˆ p n δn mn ( x i , p ˆ p , t )] − ω c p k δu k ′ γ kk ′ ¯ n ′ ( γ ij − γ il γ jl ′ ˆ p l ˆ p l ′ ) (42)Then we can expand δn ij ( t, ~x, ~p ) and δu j ( t, ~x ) like the mode expansions of (18). Because we want to consider theacoustic modes deep inside the horizon, we can use the same approximation of the last section, x ≪
1. After usingthis expansion in equation (42), we get: Z d p ( ω c − iω + i~q. ˆ p ) δn ij + 2 iK ˆ p k p l ∂∂p k ∂∂q l δn ij = − ω c n ′ p.δu ( γ ij − ˆ p i ˆ p j ) + 3 ω c π Z d ˆ p ( δn ij ( p ˆ p ) − ˆ p i ˆ p l δn lj ( p ˆ p ) − p j ˆ p l δn il ( p ˆ p ) + ˆ p i ˆ p j ˆ p m ˆ p n δn mn ( p ˆ p )) (43)Then, we multiply p j with kk component of equation (43), so that we come to: Z d p ( ω c − iω + i~q. ˆ p ) p j δn kk + 2 iK Z d p ˆ p l p l ′ p j ∂∂p l ∂∂q l ′ δn kk = − ω c Z d pp j p k δu k ¯ n ′ (44)Using by part for derivative of p l , the second term of left hand side of equation (44) gives:2 iK Z d p ˆ p l p l ′ p j ∂∂p l ∂∂q l ′ δn kk = − iK Z d p (3ˆ p l ′ p j + ˆ p l ′ p j + ˆ p l ′ p j ) δn kk ∂q l ′ = − iK Z d pp j ˆ p l ′ δn kk ∂q l ′ (45)For simplifying the term on the right hand side of (44), we can write R d pp j p k ¯ n ′ ( p ) = H ij . Contracting it with γ ij gives : γ ij H ij = Z d pp ¯ n ′ ( p ) = − Z d pp ¯ n ( p ) = − ρ γ (46)In which ¯ ρ γ is photon energy density. So we have: H ij = − ¯ ρ γ γ ij . Using (45) and knowing H ij we can rewriteequation (44): Z d p ( ω c − iω + i~q. ˆ p ) p j δn kk = 10 iK Z d pp j ˆ p l ∂∂q l δn kk + 43 ω c ¯ ρ γ δu j (47)The stress tensor for photon distribution function is defined like this: δT ij γ = γ ii ′ Z d pn kk p ˆ p i ′ ˆ p j , δT j γ = Z d pn kk p ˆ p j (48)For Baryon we have: δT Bij = 0 and δT B j = ¯ ρ B δu j . Using momentum conservation, δT j + δT ij = 0, we can write: ω ¯ ρ B δu j = − Z d pn kk ( p )ˆ p j (ˆ p i q i − ω ) (49)With the help of (49), the equation (47) simplifies to:( ωρ B + i ω c ρ γ ) δu j = i Z d pp j ( ω c δn kk − i K ˆ p l ∂∂q l δn kk ) (50)Now we can define form factors in this way: Z d ppδn ij = ¯ ρ γ ( Xδ ij + Y ˆ q i ˆ q j ) (51) Z d pp j δn kk = ¯ ρ γ Z ˆ q j (52) Z d pδn kk p j p j = ¯ ρ γ ( V δ ij + W ˆ q i ˆ q j ) (53)These integrals can be written in this way, because after integrating over ~p , the only parameter that contains directionis q i . Using these form factors in equation (50) leads to:( ω ¯ ρ B + i ω c ¯ ρ γ ) δu j = iω c ¯ ρ γ ˆ q j Z + 10 K ¯ ρ γ q ∂∂ ˆ q l ( V δ lj + W ˆ q l ˆ q j ) = iω c ¯ ρ γ Z ˆ q j + 4 W ˆ q j q (54)(1 − iωRt c ) δu j = 34 ˆ q j ( Z − i Kt c q W ) (55)where R = ρ γ ρ B . From the relation (55), the perturbed velocity can be written: δu j = q j − iωt c R ) ( Z − i Kt c q W ).Putting this back to equation (43), gives:( ω c − iω + i~q. ˆ p ) δn ij + 2 iK ˆ p l ˆ p l ′ ∂∂p l ∂∂ ˆ q l ′ δn ij = 3 ω c π Z d ˆ p ( δn ij ( p ˆ p ) − ˆ p i ˆ p l δn lj − ˆ p j ˆ p l δn il + ˆ p i ˆ p j ˆ p m ˆ p n δn mn ) − ω c n ′ ( δ ij − ˆ p i ˆ p j ) 3ˆ q k ˆ p k − iωt c R ) (cid:18) Z − i Kt c q W (cid:19) (56)We can now integrate right hand side of equation (56) over dpp : Z dpp RHS = 3 ω c π (( δ ij − ˆ p i ˆ p j ) X + Z − i Kt c q − iωt c R ˆ q k ˆ p k ! + Y (ˆ q i − ˆ p i ˆ p. ˆ q )(ˆ q j − ˆ p j (ˆ p. ˆ q ))) (57)After using ∂∂p k = ˆ p k ∂∂p , on the left hand side of (56), our Boltzmann equation becomes:( ω c − iω + i~q. ˆ p − iK ˆ p l ∂∂q l ) Z dpp δn ij = 3 ω c π (( δ ij − ˆ p i ˆ p j )( X + 43 δu k ˆ p k ) + Y (ˆ q i − ˆ p i (ˆ p. ˆ q ))(ˆ q j − ˆ p j (ˆ p. ˆ q )) (58)We define the differential operator Q like this: Q ≡ ω − ~q. ˆ p + 8 K ˆ p l ∂∂q l (59)Using this definition, the Boltzmann equation (58) gives us:4 π Z dpp δn ij = 3 ¯ ρ − i Q t c ) (( δ ij − ˆ p i ˆ p j )( X + Z − i Kt c W/q − iωt c R ˆ p. ˆ q ) + Y (ˆ q i − ˆ p i (ˆ p. ˆ q ))(ˆ q j − ˆ p j (ˆ p. ˆ q ))) (60)Then, expanding this equation to second order in t c gives:4 π Z dpp δn ij = 32 ¯ ρ γ (cid:0) i Q t c − t c Q (cid:1) (cid:18) ( δ ij − ˆ p i ˆ p j ) (cid:16) X +(1+ iωt c R − t c ω R )( Z − i Kt c q W )ˆ p. ˆ q (cid:17) + Y (ˆ q i − ˆ p i ˆ p. ˆ q )(ˆ q j − ˆ p j ˆ p. ˆ q ) (cid:19) (61)We start by an ansatz that X and Z are of order O ( t c ) and Y and W are of order O ( t c ). At the end we can checkthat our ansatz was correct. So to second order of t c , we have:4 π Z dpp δn ij = 32 ¯ ρ γ (cid:18) ( δ ij − ˆ p i ˆ p j ) (cid:16) X + it c Q X − t c Q X + ((1 + it c ωR − t c ω R ) Z − iKq t c W )ˆ p. ˆ q + it c Z (1 + it c ωR ) Q ˆ p. ˆ q − t c Z Q ˆ p. ˆ q (cid:17) + (1 + it c Q ) Y (ˆ q i − ˆ q i ˆ p. ˆ q )(ˆ q j − ˆ p j ˆ p. ˆ q ) (cid:19) (62)Now we should integrate them over d ˆ p . The calculation for each term of integration is written in the appendix. Thefinal result will be:4 π Z d ppδn ij = 4 πρ γ ( Xδ ij + Y ˆ q i ˆ q j ) = 4 πρ γ (cid:18)(cid:16) X + it c Xω − t c ω − K + q X + 8 Kq ( it c − t c ω (1 + R )) Z + ( − it c q t c ω ( R + 2) q Z + 1 + iωt c Y (cid:17) δ ij + (cid:16) − t c X + it c q Z − ωRZqt c iωt c )10 Y (cid:17) ˆ q i ˆ q j (cid:19) (63)Equating the coefficients of γ ij and ˆ q i ˆ q j on each side correspondingly, will lead to: X ( it c ω − ω t c + 8 Kt c − q t c ) + 1 + iωt c Y + ( 2 q − it c + t c ωR + 2 ωt c ) + 8 Kq ( − t c ω + it c − t c ωR )) Z = 0 (64) t c q X + ( −
310 + 7 i ωt c ) Y + ( it c q − R ωqt c ) Z = 0 (65)In addition, using equation (52) and after expanding it to second order in power of t c we get:4 π Z d pδn ii p j = 3 ¯ ρ Z d ˆ p ˆ p j (cid:16) X + Y Z + it c ωRZ (1 + it c ωR ) − it c KWq )ˆ p. ˆ q + it c Q ( X + Y it c Q ( Z − Y it c ωRZ )ˆ p. ˆ q − t c Q X − t c Q Z ˆ p. ˆ q − Y p. ˆ q (cid:17) (66)After calculating the integrals of each term (which is written the appendix), we will get:( − it c q + 2 ωt c q ) X − iqt c q Y + ( it c ω (1 + R ) − t c ω (1 + R + R ) − q t c + 24 Kt c ) Z − i Kt c Wq = 0 (67)At last, we can put the the system of equations derived for form factors into a matrix: it c ω − ω t c + 8 Kt c − q t c iωt c
10 2 q ( − it c + t c ωR + 2 ωt c ) + Kq ( − t c ω + it c − t c ωR ) 0 t c q − − i ωt c it c q − R ωqt c − it c q + 2 ωt c q − it c qω it c ω (1 + R ) − t c ω (1 + R + R ) − q t c + 24 Kt c − i Kt c q XYZW = 0 . (68)Setting the determinant of this matrix to zero and expanding it to first order in t c , we get the dispersion relation as: − q + 15(1 + R ) ω + 120 K − it c ω (5 + 5 R − R ) + 7 iωt c q − iKωt c = 0 (69)For solving the above dispersion relation, we split ω to real and imaginary part, ω = Ω + i Γ. Insert it in equation(69) and set the real an imaginary part of the relations separately to zero, we get: − q (5 + 7 t c Γ)+ 400 K (3 + 10 t c Γ) − R )Γ + 5( − − R + 3 R ) t c Γ + 15(1 + R )Ω + 15 t c ΓΩ (1 + R − R ) = 0 (70) − Kt c + 7 q t c + 30(1 + R )Γ − − − R + 3 R ) t c Γ R − R ) t c R = 0 (71)Now we can solve (71) for getting Ω:Ω = ± s Kt c − q t c + 15Γ( − − t c Γ + 3 R t c Γ − R (2 + t c Γ))5 t c ( − − R + 3 R ) (72)Putting the solution for Ω back to equation (70) and solving it for Γ upto first power of t c we get:Γ = − t c q R ) ( 1615 + R R ) + 4 Kt c R (3 + R R ) (73)Inserting Γ back in equation (72) and expand it to first power of t c we will get:Ω = ± p R ) r q − K R + R − − R + 3 R (74)For checking our results, we can look at the limit of K →
0, where we get back the results for flat universe as it isoriginally derived in [14].
IV. HYDRODYNAMIC SOLUTIONS IN A SPATIALLY CURVED UNIVERSE
The 00-component of Einstein equation and conservation equations of energy and momentum are three coupleddifferential equations which are sufficient for getting hydrodynamic solutions of cosmological perturbations. Theseequations in synchronous gauge *1 in a curved spacetime are correspondingly: − πG ( δρ + 3 δp + ∇ Π s ) = ∂ t ( a ψ ) , (75) δp + ∇ Π s + ∂ t ((¯ ρ + ¯ p ) δu ) + 3 ˙ aa (¯ ρ + ¯ p ) δu + 2 K Π s = 0 , (76) δ ˙ ρ + 3 ˙ aa ( δρ + δp ) + ∇ ( 1 a (¯ ρ + ¯ p ) δu + ˙ aa Π s ) + (¯ ρ + ¯ p ) ψ = 0 (77)where ψ is defined as ψ = (3 ˙ A + ∇ ˙ B ). It is important to note that modification due to spatial curvature onlyappears here in the form of 2 K Π s . Ignoring Π s , we come to the same equations as we had in flat background. So theknown hydrodynamic solutions that was derived in [15] is applicable here. The fast mode solutions are: δu γ = a √ R q q (1 + R ) / e − R Γ dt Sin (cid:18)Z Ω dta (cid:19) (78) δ D = 48 πG ¯ ρ γ (2 + R )(1 + R ) / ( aq ) R Oq e − R Γ dt Cos (cid:18)Z Ω dta (cid:19) (79)Here, we should put what we find in (74) and (73) for frequency of oscillation Ω and damping factor Γ. Ignoringphoton and neutrino energy density combining equations (75),(76) and (77) we come to a second order differentialequation: ddt ( a ddt δ M ) = 4 πGa ¯ ρ M δ M (80)We can factorize the dependence of t and q by writing: δ M = ∆( q ) F ( t ), where ∆ satisfies:∆( q ) = βδ γq ( t L ) + (1 − β ) δ Dq ( t L ) − t L ψ q ( t L ) + βt L q a δu γq ( t L ) (81)So the time evaluation of F ( t ) is: ddt ( a dFdt ) = 4 πGa ¯ ρ M F (82)with initial condition F → ( tt L ) / . From Friedman equation we have:( ˙ aa ) = 8 πG ρ Λ + ¯ ρ M ) − Ka (83)Defining X ≡ ρ Λ ¯ ρ M = Ω Λ Ω M ( aa ) , The Friedman equation can be written in terms of X :˙ aa = H p Ω Λ s K Ω / Ω / M X / + 1 X (84) *1 Scalar metric perturbation in synchronous gauge is defined like this ds = − dt + a ((1 + A ) γ ij + ∂ ij B ) X C(X) Closed C(x) Flat0.1 1.0113 0.98260.2 1.0013 0.96670.3 0.9901 0.95200.5 0.9675 0.92560.7 0.9463 0.90251.0 0.9176 0.87251.5 0.8769 0.83142.0 0.8432 0.79812.5 0.8146 0.77023.0 0.7899 0.74623.5 0.7683 0.7254TABLE I. The values of C(x) which is the effect of dark energy on growth of matter, as a function of X ≡ ΩΛΩ M for closed andflat spacetime, We use 2018 Planck data [1] for its calculation, which is Ω M = 0 .
315 and for closed case Ω K = − . With the help of (84), the differential equation (82) can be written in terms of X. Then the solutions as function ofX is: F ∝ s Ω K Ω / Ω / M X / + XX Z x duu / (1 + Ω K Ω / Ω / M u / + u ) / (85)We know that F → aa L at early times. In this way, we can set coefficient of proportionality and write: F = 35 a ( t ) a L C ( Ω Λ Ω M ( aa ) ) (86)where, C ( X ) ≡ X − / s K Ω / Ω / M X / + X Z x duu / (1 + Ω K Ω / Ω / M u / + u ) / (87)We can calculate this integral numerically. The result for different values of X is written in the Table I. As it canbe seen from the table, for the flat universe, C ( X ) just have a suppression effect on growth of matter due to darkenergy. In the closed case, for small values of X , C ( X ) gives enhancement and then it gives suppression.Furthermore ignoring baryon’s effects (e.g. neglecting the terms of order β ), from(81) we have ∆( q ) = δ Dq ( t L ) − t L ψ q ( t L ), then we would get the solution in the form of ∆( q ) = q n R On τ ( κ )3 H L a L , the same as flat universe (where τ is transferfunction, t L = H L and H L = √ Ω M H (1 + z L ) / ). This will give us well-known hill-top shape power spectrum.When we consider the effects of the baryons, then what is the dominant term in (81) for fast mode is βt L q n a δu γ ( t L ),so using the fast mode solution of (78),we can write the q dependent part of matter density as:∆( q ) ≈ βt L ( qa L ) δu γ = 2 βq R Oq √ a L H L (1 + R L ) / e − R Γ dt Sin Z qdta p R ) (cid:18) − Kq R + R − − R − R (cid:19)! (88)Here also we should use the modified form of Γ as (73). This is the well-known acoustic oscillation forced by baryon’seffect. V. CONCLUSION
In this work, first we write, Boltzmann equation for dark matter particles in a spatially curved spacetime. Theintegral solution to this equation shows that memory of gravitational field at early time affects growth at later times.1Specifically it is shown that spatial curvature modified this memory by the factor that is come in the last term of(35). Then we write Boltzmann equation for photons and using it find a dispersion relation in the presence of spatialcurvature. This gives us damping factor and frequency of acoustic oscillations, as (73) and (74). In the last sectionwe discuss that spatial curvature doesn’t affect evolution of matter density in the hydrodynamic regime. It is becausespatial curvature appears only in terms proportional to Π s . As it is shown, in this hydrodynamic regime spatialcurvature does not affect q -dependent part of matter density, ∆( q ). But for the time dependent part, in spite of thefact that dark energy always gives a factor of suppression to the growth of matter in a flat universe, but here whenratio of dark energy to matter is small it gives an enhancement and only when this ratio becomes larger, it changes tosuppression. When considering baryons, effects of spatial curvature shows itself in terms of modification of frequencyand damping parameters (the results of section III) of acoustic oscillations. Appendix A: Further Computations
The useful identities for following calculation is: Z d ˆ p = 4 π (A1) Z d ˆ p ˆ p i ˆ p j = 4 π δ ij (A2) Z d ˆ p ˆ p i ˆ p j ˆ p k ˆ p l = 4 π
15 ( δ ij δ kl + δ ik δ jl + δ il δ kj ) (A3)Here we write down the non-zero results of integration of each term in (62): Z d ˆ p ( δ ij − ˆ p i ˆ pj ) X = 4 πX ( δ ij − δ ij ) = 8 π Xδ ij (A4) it c Z d ˆ p ( δ ij − ˆ p i ˆ pj ) O X = it c ωXδ ij Z d ˆ p − it c Xw Z d ˆ p ˆ p i ˆ p j = 8 πit c Xw δ ij (A5) − t c Z d ˆ p ( δ ij − ˆ p i ˆ p j ) O X = − t c Z d ˆ p ( δ ij − ˆ p i ˆ p j )( ω X − ω ˆ q i ˆ p j X + ˆ q i ˆ q j ˆ p i ˆ p j X − KX )= − t c π ω X − KX ) δ ij − π t c q Xδ ij + 8 π t c Xq ˆ q i ˆ q j (A6) it c Z (1 + it c ωR ) Z d ˆ p ( δ ij − ˆ p i ˆ p j ) O ˆ p. ˆ q = it c Z (1 + iωt c R ) Z d ˆ p ( δ ij − ˆ p i ˆ p j )( ω ˆ p.~q − ˆ p.~q ˆ p.~q + 8 Kq )= it c Z (1 + iωt c R )( 8 Kq π δ ij − π q δ ij + 8 π q ˆ q i ˆ q j ) (A7) − t c Z d ˆ p ( δ ij − ˆ p i ˆ p j ) Z O ˆ p. ˆ q = − t c Z Z d ˆ p ( δ ij − ˆ p i ˆ p j )( − ωq k ˆ q k ′ ˆ p k ˆ p k ′ + 8 Kωq )= − t c Z ( − ω π q δ ij + 2 ω π q ( δ ij + 2ˆ q i ˆ q j ) + 8 Kωq π δ ij )= − t c Z ( 32 π ωq δ ij + 64 πKω q δ ij + 16 π ω ˆ q i ˆ q j ) (A8) Y Z d ˆ p (ˆ q i − ˆ p i ˆ p k ˆ q k )(ˆ q j − ˆ p j ˆ p k ′ ˆ q k ′ ) = 4 π Y ( δ ij + 7ˆ q i ˆ q j ) (A9)2 it c Y Z d ˆ p O (ˆ q i − ˆ p i ˆ p k ˆ q k )(ˆ q j − ˆ p j ˆ p k ′ ˆ q k ′ ) = it c Y ( 28 π ω ˆ q i ˆ q j + 4 π ωδ ij ) (A10)The calculation of integration for each term in (66) is presented here:( Z + iωt c RZ (1 + iωt c R ) − it c KWq )ˆ q i Z d ˆ p ˆ p i ˆ p j = 4 π iωt c R − ω t c R ) Z − it c KWq )ˆ q j (A11) it c Z d ˆ p ˆ p j O ( X + Y − π it c ( X + Y q j (A12) − t c Z d ˆ p ˆ p j O ( X + Y − t c Z d ˆ p ˆ p j (cid:16) ( X + Y ω − ω ( X + Y ~q. ˆ p +( X + Y ~q. ˆ p~q. ˆ p − K ( X + Y (cid:17) = 8 π ωt c ( X + Y q ˆ q j (A13) it c Z d ˆ p ˆ p j O ( Z + it c ωRZ − Y p. ˆ q = it c Z d ˆ p ˆ p j (cid:16) ω ˆ q. ˆ p ( Z − Y it c ωRZ ) − ~q. ˆ p ˆ q. ˆ p ( Z − Y iωRt c Z )+ 8 Kq ( Z − Y iωRt c Z ) (cid:17) = it c ω π Z − Y it c ωRZ )ˆ q j (A14) − t c Z d ˆ p ˆ p j O Z ˆ p. ˆ q = − π (cid:16) t c ω q − K (1 + q )) + t c q (cid:17) Z (A15)3 [1] N. Aghanim et al. (Planck), (2018), arXiv:1807.06209 [astro-ph.CO].[2] G. Bernstein, Astrophys. J. , 598 (2006), arXiv:astro-ph/0503276 [astro-ph].[3] S. Rsnen, K. Bolejko, and A. Finoguenov, Phys. Rev. Lett. , 101301 (2015), arXiv:1412.4976 [astro-ph.CO].[4] D. Sapone, E. Majerotto, and S. Nesseris, Phys. Rev. D90 , 023012 (2014), arXiv:1402.2236 [astro-ph.CO].[5] Y.-L. Li, S.-Y. Li, T.-J. Zhang, and T.-P. Li, Astrophys. J. , L15 (2014), arXiv:1404.0773 [astro-ph.CO].[6] M. Lachieze-Rey and J.-P. Luminet, Phys. Rept. , 135 (1995), arXiv:gr-qc/9605010 [gr-qc].[7] N. J. Cornish, D. N. Spergel, G. D. Starkman, and E. Komatsu, Phys. Rev. Lett. , 201302 (2004),arXiv:astro-ph/0310233 [astro-ph].[8] P. Bielewicz and A. J. Banday, Mon. Not. Roy. Astron. Soc. , 2104 (2011), arXiv:1012.3549 [astro-ph.CO].[9] R. Aurich and S. Lustig, Mon. Not. Roy. Astron. Soc. , 2517 (2013), arXiv:1303.4226 [astro-ph.CO].[10] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time , Cambridge Monographs on MathematicalPhysics (Cambridge University Press, 2011).[11] A. H. Abbassi, J. Khodagholizadeh, and A. M. Abbassi, Eur. Phys. J.
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