We point out that during the reionization epoch of the cosmic history, the
plasma collective effect among the ordinary matter would suppress the large
scale structure formation. The imperfect Debye shielding at finite temperature
would induce a residual long-range electrostatic potential which, working
together with the baryon thermal pressure, would counter the gravitational
collapse. As a result the effective Jean's length, $\tilde{\lambda}_J$, is
increased by a factor, $\tilde{\lambda}_J/\lambda_J=\sqrt{8/5}$, relative to
the conventional one. For scales smaller than the effective Jean's scale the
plasma would oscillate at the ion-acoustic frequency. The modes that would be
influenced by this effect depend on the starting time and the initial
temperature of reionization, but roughly lie in the range $0.5 h{\rm Mpc}^{-1}<
k$, which corresponds to the region of the Lyman-$\alpha$ forest from the
inter-galactic medium. We predict that in the linear regime of density-contrast
growth, the plasma suppression of the matter power spectrum would approach
$1-(\Omega_{dm}/\Omega_m)^2\sim 1-(5/6)^2\sim 30%$.Comment: 4 pages and 2 figure

Chen, Pisin

Lai, Kwang-Chang

English

arXiv

National Taiwan University Repository

PLASMA SUPPRESSION OF LARGE SCALE STRUCTURE FORMATION IN THE UNIVERSE.

Plasma Suppression of Large Scale Structure Formation in the Universe

We point out that during the reionization epoch of the cosmic history, the plasma collective effect among the ordinary matter would suppress the large scale structure formation. The imperfect Debye shielding at finite temperature would induce a residual long-range electrostatic potential which, working together with the baryon thermal pressure, would counter the gravitational collapse. As a result the effective Jean's length, {tilde {lambda}}{sub J}, is increased by a factor, {tilde {lambda}}{sub J}/{lambda}{sub J} = {radical}8/5, relative to the conventional one. For scales smaller than the effective Jean's scale the plasma would oscillate at the ion-acoustic frequency. The modes that would be influenced by this effect depend on the starting time and the initial temperature of reionization, but roughly lie in the range 0.5hMpc{sup -1} &lt; k, which corresponds to the region of the Lyman-{alpha} forest from the inter-galactic medium. We predict that in the linear regime of density-contrast growth, the plasma suppression of the matter power spectrum would approach 1 - ({Omega}{sub dm}/{Omega}{sub m}){sup 2} {approx} 1 -(5/6){sup 2} {approx} 30%

UNT (University of North Texas) Digital Library

Work supported in part by US Department of Energy contract DE-AC02-76SF00515Plasma Suppression of Large Scale Structure Formation in the UniversePisin Chen1, ∗ and Kwang-Chang Lai21Kavli Institute for Particle Astrophysics and Cosmology,Stanford Linear Accelerator Center, Stanford University, Stanford, CA 943092INAF Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, Firenze, 50125, Italy.(Dated: July 10, 2007)We point out that during the reionization epoch of the cosmic history, the plasma collective effectamong the ordinary matter would suppress the large scale structure formation. The imperfect Debyeshielding at finite temperature would induce a residual long-range electrostatic potential which,working together with the baryon thermal pressure, would counter the gravitational collapse. Asa result the effective Jean’s length, λ˜J , is increased by a factor, λ˜J/λJ =p8/5, relative to theconventional one. For scales smaller than the effective Jean’s scale the plasma would oscillate at theion-acoustic frequency. The modes that would be influenced by this effect depend on the startingtime and the initial temperature of reionization, but roughly lie in the range 0.5hMpc−1 < k, whichcorresponds to the region of the Lyman-α forest from the inter-galactic medium. We predict that inthe linear regime of density-contrast growth, the plasma suppression of the matter power spectrumwould approach 1− (Ωdm/Ωm)2∼ 1− (5/6)2 ∼ 30%.PACS numbers: 98.65.-r, 98.80.Bp, 98.80.EsThe matter power spectrum of the large scale struc-ture, alongside with the cosmic microwave background(CMB) fluctuations, provides important information onthe composition and the evolution history of the uni-verse. Unlike the CMB fluctuations, which was frozensince the recombination time, the matter power spec-trum continued to evolve until now, during which theuniverse has been essentially in a reionization, or plasma,state. Plasmas are known to support numerous electronand ion collective oscillation modes. Although in thepre-recombination era the universe was also in a plasmastate, the extreme dominance of radiation pressure overthat of matter (by a factor 1010) before decoupling ren-ders the contribution of (ep) plasma oscillations totallynegligible. In the post-decoupling era, the CMB pressurebecame negligible. Thus during the reionization epoch[1]one might expect collective plasma effects to exhibit in-fluence on the structure formation.A perfectly uniform plasma is charge-neutral and hasno long-range electrostatic potential. The inevitablethermal fluctuations of its electron density at finite tem-perature would, however, induce a non-vanishing resid-ual electrostatic potential among the ions that defies theDebye shielding. Such a residual electric force that ex-erts an additional pressure on the ions is non-collisionalin nature. When combined with the collision-inducedthermal pressure among them, these pressures can sup-port ion-acoustic oscillations, a phenomenon well-knownin plasma physics.[2, 3]In this Letter we point out that the inclusion of suchplasma collective effects would result in an increase ofthe sound speed and therefore an increase of the Jean’sscale by a factor λ˜J/λJ =√8/5. The modes that would∗Electronic address: chen@slac.stanford.edube influenced by this effect depend on the starting timeof reioniation and its initial temperature, but shouldroughly lie in the range 0.5hMpc−1 < k, which corre-sponds to the region of the Lyman-α forest. This plasmasuppression mechanism is effective to both linear andnonlinear regimes of density perturbations. We predictthat, the maximum suppression of the power spectrum inthe linear regime should approach ∼ 1 − (Ωdm/Ωm)2 ∼1− (5/6)2 ∼ 30%.When the electrostatic potential is included, the stan-dard Einstein-Boltzmann equation in the Newtonianlimit is now extended into a Maxwell-Einstein-Boltzmann(MEB) equation. Let us designate the baryon perturba-tion as ∆ ≡ δρb/ρb. In the electrostatic and the New-tonian limits and with the density perturbation ∆ ≪ 1,the MEB equation reads,∆¨ + 2H∆˙−∇2[δPρb+eφemmb]= 4piGρ∆ . (1)where the electrostatic potential satisfies the Poissonequation: ∇2φem = 4pieρe. The source of gravity con-tains both dark matter and baryons, i.e., ρ = ρdm + ρb.We assume that the reionization epoch began at a cer-tain time a∗. The adiabatic condition dictates thatthe initial perturbations are identical between the darkand the baryonic matters, i.e., δρdm(a∗)/ρdm(a∗) =δρb(a∗)/ρb(a∗) = δρ(a∗)/ρ(a∗) ≡ ∆∗. After that, darkmatter and baryonic matter would still evolve hand-in-hand and grow linearly for scales larger than the Jean’sscale, but would evolve separately for scales smaller thanthat. It is this latter situation which we are focusing forour plasma effect. Thermodynamics governs thatδPρb=δPδρbδρbρb=γbkBTbmb∆ . (2)On the other hand, the electrostatic potential at a givenarXiv:0707.1340v1[astro-ph]SLAC-PUB-13042Submitted to Physical Review Letters2temperature is given byeφem =4pie2δρek2 + λ−2D≈ γekBTe∆ . (3)The approximation results from the fact that the typi-cal cosmic scale of interest is much larger than the De-bye length, i.e., λ ∼ 1/k ≫ λD ≡√γekBTe/4pie2ρe.We have further assumed that δρe/ρe ≃ δρb/ρb = ∆.This is because globally the fast-moving electrons tendto rapidly readjust themselves to follow the baryon den-sity perturbations. At first sight, it may seem counter-intuitive that the charge neutrality condition would al-low for a non-vanishing electrostatic potential. In re-ality, due to the inevitable thermal fluctuations of themuch lighter electrons the plasma is never perfectly neu-tral. Balance of forces on electrons requires that theelectron number density satisfies the Boltzmann’s re-lation: ne = ρe/me = ne0 exp(−eφem/kBTe). Thusat finite temperature the Debye shielding is never per-fect, and there is always a residual electrostatic poten-tial at the level of eφem ∼ kBTeδρe/ρe in the lead-ing order of Taylor expansion. This, that one assumesδρe/ρe = δρb/ρb and ∇ · E = ∇2φem 6= 0 at the sametime, is the so-called quasineutrality, or plasma approxi-mation in plasma physics[2].Fourier transforming Eq.(1) leads to∆¨k + 2H∆˙k +(ka)2c˜2s∆k = 4piGρ∆k , (4)wherec˜s ≡(γekBTe + γbkBTbmb)1/2=ωk(5)is the ‘sound speed’ of the ion-acoustic waves. We seethat the main change from the ordinary acoustic wave tothe ion acoustic wave is that there is an extra contribu-tion to the sound speed from the electron temperature.The fast-moving electrons are isothermal. So γe = 1.But for protons, γb = 5/3. In the case where electronsand ions are in thermal equilibrium, i.e., Te = Tb, the netchange is an increase of the sound speed from the ordi-nary acoustic wave,√5/3, to that of the ion-acousticwave,√1 + 5/3. Both waves, however, are constant-velocity pressure waves that satisfy linear dispersion re-lations in the long wavelength limit.So far the temperature and the on-set of the reioniza-tion are still unresolved. In the literature (e.g. [4, 5, 6]),the reionization temperature ranges from 103K to 107K.As for the on-set of reionization, the observation of theGunn-Peterson trough in the spectra of high redshiftquasars indicates that the reionization process was com-pleted by z∗ ≈ 6 [7, 8], while WMAP determines thatz∗ = 17± 5 [9, 10]. For simplicity and without compro-mising qualitative features of our effect, we model thereionization as an abrupt transition at a given instanta∗ = 1/(1 + z∗) where the plasma temperature raisedinstantly to T ∗b homogeneously throughout the universe.To solve Eq.(4) for a given k-mode, it is essential to rec-ognize its relationship with the initial and the final Jean’sscales at a∗ and a, respectively. Since the universe ex-pands adiabatically during the matter-dominant epoch,the large-scale baryon/plasma temperature satisfies theequation of state, TbVγb−1 = const., where γb = 5/3 andV ∝ 1/ρ ∝ a3. Therefore Tb ∝ c˜2s ∝ a−2. Thus thepost-reionization (a > a∗) Jean’s scale evolves ask˜J (a) =√4piGρ(a)c˜s(a)/a=(3pi2mbGρ∗T ∗b)1/2√a∗a ≥ k˜∗J . (6)There are three regimes of k-modes that concern us. Fork < k˜∗J , the thermal and the plasma pressures are negli-gible from the outset. So the baryon perturbation growslinearly in the same way as that of the dark matter:δb ∝ δdm ∝ a. The decay mode, δb(dm) ∝ a−3/2, fadesaway in time and can be ignored.For k > k˜∗J , care must be taken regarding k’s furtherrelationship with the Jean’s scale at the time of interest.If k > k˜J(a) > k˜∗J , then the combined pressure dominatesover the gravity throughout the period (a∗, a). Thus theright-hand-side of Eq.(4) is negligible. Changing variablefrom t to a, with a ∝ t2/3 in the matter-dominant era,we findd2∆kda2+32ad∆kda+(k2c˜2saH20)∆k = 0 . (7)Let us introduce a parameter ωr which satisfies the rela-tion, k2c˜2s/H20 ≡ ω2ra−2, such that ωr is independent ofa. Since k˜J ∝ a1/2, we can reexpress it asω2r =c˜2sk2a2H20≃ 32ΩmΩ0( kk˜∗J)2a∗ . (8)It can be shown, through one more change of variableto x = a−2, that the solution to Eq.(7) is oscillatory:∆k ∝ exp(i2ωr/√a). Physically ωr/√a represents thephase of the ripples of the density contrast in k-space dueto the ion-acoustic oscillation. Such an oscillation beginsat the time of reionization, a∗. Although in reality c˜skis the physical ion-acoustic oscillation frequency, ωr canbe viewed as an effective ion-acoustic frequency since itrelates to c˜sk straight-forwardly.Ion-acoustic oscillations are known to suffer Landaudamping[2], which was not included in Eq.(7) a pri-ori. To include this effect, we invoke a simple empiricalformula[2] for the Landau damping dispersion relationthat relates the real and the imaginary parts of the ion-acoustic oscillation frequency,ωiωr≃ 1.1(TeTb)7/4e−(Te/Tb)2, 1 ≤ Te/Tb ≤ 10 . (9)Since the electron and proton temperatures in the reion-ization epoch are roughly equal, we find that ωi/ωr ∼ 0.4in our case. This means that within ∼ 0.4 of an oscil-lation, the ion-acoustic wave would be Landau-damped3to one e-folding of its initial amplitude. Now we replacethe ωr in the exponent of the solution to Eq.(7) by thecomplex frequency which satisfies the above dispersionrelation. Imposing the initial condition at a∗, we arriveat the perturbation for the k-mode whose scale has beenbelow the Jean’s scale all the time from a∗ to a, i.e.,k > k˜J(a) > k˜∗J ,∆k ≃ ∆∗ke(−0.8+i2)ωrη , (10)where η = η(a∗, a) ≡√1/a∗ −√1/a.For the intermediate regime k˜∗J < k < k˜J , the k-modemust have exited the Jean’s scale at an intermediate timeac: k = k˜J(ac), where a∗ < ac < a. Such a mode wouldoscillate for a period ∆a = ac − a∗. Then it would stopthe oscillation at a = ac and resume its gravitationalcollapse during the subsequent interval a−ac. Neverthe-less, such a resumed growth of perturbation is delayed,because of the ion-acoustic oscillations during the earliertime interval, by the same amount∆a ≡ ac − a∗ =[( kk˜∗J)2− 1]a∗ . (11)Thus the evolution of such a mode k˜∗J < k < k˜J(a) shouldbecome∆k = ∆∗k(a−∆aa∗)= ∆∗k[ aa∗+ 1−( kk˜∗J)2]. (12)Here we have ignored the fact that upon crossing theJean’s scale the k-mode may still carry a non-vanishingoscillation amplitude. We expect that such remainingamplitude should be small due to Landau damping.Our aim is to identify the imprints of this plasma ef-fect in the large scale structure formation through obser-vations. So we look for its modification to the matterpower spectrum[11],P (k, a) = 2pi2δ2HknHn+30T 2(k)D2(a) . (13)Here δH = 1.9×10−5 is the amplitude of primordial per-turbations, H0 = 100hkms−1Mpc−1 the Hubble param-eter at present, T (k) the transfer function, and D(a) thegrowth function. We limit our investigation in the lin-ear regime of power spectrum growth. Due to the onsetof the plasma effect, the conventional growth function,D(a), in Eq.(13) has to be modified. Let us denote themodified growth function as D˜(k, a). Clearly, the plasmasuppression would only affect the regime where k > k˜∗J ,which is what we will concentrate below. To track theplasma effect on the baryons for a > a∗, we writeD˜(k, a) = Ddm(a)ΩdmΩm+Db(a)ΩbΩm= Ddm(a∗)aa∗ΩdmΩm+Db(a∗)∆˜k∆∗kΩbΩm. (14)Invoking the initial condition, Ddm(a∗) = Db(a∗) =D(a∗), and under the conventional notion of lineargrowth, D(a) = D(a∗)(a/a∗), we arrive at the relativechange of the growth function asD˜(k, a)D(a)= 1− ΩbΩm[1− a∗a∆˜k∆∗k]. (15)Thus for a given k-mode with k > k˜∗J , the plasma-suppressed matter power spectrum relative to that of theconventional approach reads, for k > k˜J (a) > k˜∗J ,∣∣∣D˜(k, a)D(a)∣∣∣2≃(ΩdmΩm)2 {1 + 2( ΩbΩdma∗a)e−0.8ωrη cos(2ωrη)+( ΩbΩdma∗a)2e−1.6ωrη}. (16)On the other hand we have, for k˜∗J < k < k˜J (a),∣∣∣D˜(k, a)D(a)∣∣∣2≃(ΩdmΩm)2{1 + 2( ΩbΩdma∗a)[1 +aa∗−( kk˜∗J)2]+( ΩbΩdma∗a)2[1 +aa∗−( kk˜∗J)2]2}. (17)0.2 0.5 1 2 5 10k@h Mpc-1D0.60.70.80.91ÈDHk,aLDHaLÈ20.1 0.2 0.5 1 2 50.60.70.80.9FIG. 1: Suppression of the matter power spectrum. a∗ = 0.1and T ∗b = 5×106K. The red, purple and blue curves representthe suppressions at a = 1/6, 1/3 and 1, respectively. Thethree regimes of k can be clearly recognized.To appreciate what this implies in cosmology, we lookfor the physical scales relevant to the plasma suppres-sion effect. Let us assume that the reionization occurredat 1 + z∗ = 10, or a∗ = 0.1, with the initial plasmatemperature T ∗e ≃ T ∗b ∼ 5 × 106K. This correspondsto an ion-acoustic wave velocity of c˜∗s ∼ 7 × 10−4c andk˜∗J ∼ 0.63hMpc−1, and therefore the Jean’s length, λ˜∗J =2pi/k˜∗J ∼ 1.0h−1Mpc. The Jean’s scale at present wouldbe k˜J(a = 1) ∼ 2.0hMpc−1, or λ˜J (a = 1) ∼ 3.1h−1Mpc.Figure 1 plots the suppression factor, |D˜(k, a)/D(a)|2, atdifferent times: a = 1/6, 1/3 and 1, based on the ΛCDMmodel with ΩΛ = 0.7,Ωm = 0.3,Ωdm = 0.25,Ωb =0.05, a∗ = 0.1, a = 1 and T ∗b = 5 × 106K. The kinks of4each curve are associated with the transitions at k˜∗J andk˜J (ac), respectively. These kinks are the artifact of ourapproximation, which should be smeared in reality. Thethree regimes of k, namely, k < k˜∗J , k˜J(a = 1) > k > k˜∗J ,and k > k˜J(a = 1) > k˜∗J , can be readily recognized inthis plot. For a mode at k = k˜J(a = 1) = 2.0hMpc−1,for example, it would continue to oscillate until now. Itsoscillation ‘frequency’ is ωr ∼ 0.76. Thus 2ωrη ∼ 3.3 =0.52(2pi), i.e., it would have oscillated for roughly half acycle. By then its amplitude would be Landau-dampedby more than one e-fold. In the last regime, the suppres-sion reaches an asymptotic value of ∼ 70%. Fig.2 com-pares the conventional and the plasma-suppressed matterpower spectra. We have replaced the shape parameterΓ = Ωmh in the BBKS formula[12] for the transfer func-tion by the Eisenstein-Hu[13] rescaled factor. Thus thebaryon effects prior to the reionization epoch has beenincluded in the transfer function, which is common toboth cases.1 2 3 4k@h Mpc-1D220200PHk,aL@h-3Mpc3D0.5 1 2 3 4220200FIG. 2: Matter power spectrum with and without the plasmaeffect. The conventions are the same as that in Fig.1.By including the long-range residual electrostatic po-tential, we have shown that the growth of the baryondensity contrast during the reionization epoch should besuppressed by the plasma collective effect in the formof ion-acoustic oscillations. Though subject to Landaudamping, these oscillations happen to have both the pe-riod and the damping time comparable to the Hubbletime. Thus there should exist certain imprints of theseoscillations in the large scale structure formation history.In the linear regime of density contrast growth and theasymptotic limit where the oscillations damp away, theamount of suppression of the matter power spectrum dueto the plasma effect reaches ∼ 30%, which is sizable. Inprinciple the plasma suppression is also effective in thenonlinear regime. It would be interesting to expand ourtreatment to the nonlinear regime of growth at late times(a→ 1) and track its impact.One major effort in modern cosmology is to under-stand the cosmic composition through the determinationof a set of cosmic parameters. In addition to the CMBfluctuations, large scale structure formation can help fur-ther constrain these parameters. In this regard, our pre-dicted plasma suppression of the power spectrum in theLyman-α forest region may be relevant. We note that theobserved power spectrum in the inter-galactic medium(IGM) region indeed appears to be systematically lowerthan that predicted by the conventional theory[14]. Itwould be very interesting to confirm this. Aside fromthis issue, the evolution of the plasma-suppressed powerspectrum should provide a unique window to reveal thedetail history and dynamics of reionization.AcknowledgmentsWe thank T. Abel, R. Blandford, W. Hu, Ue-Li Peng,J. Silk and R. Wechsler for helpful discussions. This workis supported in part by US Department of Energy underContract No. DE-AC03-76SF00515.[1] For a recent review, see, for example, R. Barkana, Science313, 931 (2006).[2] Francis F. Chen, Introduction to Plasma Physics andControlled Fusion, Vol. 1: Plasma Physics, 2nd Ed.(Plenum Press, New York, 1984).[3] N. A. Krall and A. W. Trivelpiece, Principles of PlasmaPhysics (McGraw-Hill, New York, 1973).[4] A.J. Benson, N. Sugiyama, Adi Nusser and C.G. Lacey,MNRAS 369, 1055 (2006) [astro-ph/0512364].[5] M. Tegmark, J. Silk, M. J. Rees, A. Blanchrd, T. Abeland F. Palla, ApJ 474, 1 (1997) [astro-ph/9603007].[6] I.T. Iliev, P.R. Shapiro, E. Scannapieco, G. Mellema, M.Alvarez, A.C. Raga, and Ue-Li Pen, Proceedings IAUColloquium No. 199, 369 (2005) [astro-ph/0505135].[7] R.H. Becker et al., ApJ 122, 2850 (2001).[8] X. Fan et al., AJ 123, 1247 (2002).[9] A. Kogut et al., ApJ Supp. 148, 161 (2003).[10] D.N. Spergel et al., ApJ Supp. 148, 175 (2003).[11] S. Dodelson, Modern Cosmology (Academic Press, 2003).[12] J.M. Bardeen, J.R. Bond, N. Kaiser and A.S. Szalay, ApJ15, 304 (1986).[13] D. J. Eisenstein and W. Hu, ApJ 496, 605 (1998)[astro-ph/9709112].[14] M. Tegmark and M. Zaldarriaga, Phys. Rev. D 66,103508 (2003) [astro-ph/0207047].

https://digital.library.unt.edu/ark:/67531/metadc895107/m2/1/high_res_d/921006.pdf

Plasma Suppression of Large Scale Structure Formation in the Universe

Abstract

Similar works

Full text

Available Versions

National Taiwan University Repository

National Taiwan University Repository

UNT (University of North Texas) Digital Library