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SMALL-SCALE COSMOLOGICAL PERTURBATIONS: AN ANALYTIC APPROACH WAYNE HU1 AND NAOSHI SUGIYAMA2 Received 1995 December 7; accepted 1996 May 16

ABSTRACT Through analytic techniques veriÐed by numerical calculations, we establish general relations between the matter and cosmic microwave background (CMB) power spectra and their dependence on parameters on small scales. Fluctuations in the CMB, baryons, cold dark matter (CDM), and neutrinos receive a boost at horizon crossing. Baryon drag on the photons causes alternating acoustic peak heights in the CMB and is uncovered in its bare form under the photon di†usion scale. Decoupling of the photons at last scattering and of the baryons at the end of the Compton drag epoch freezes the di†usion-damped acoustic oscillations into the CMB and matter power spectra at di†erent scales. We determine the dependence of the respective acoustic amplitudes and damping lengths on fundamental cosmological parameters. The baryonic oscillations, enhanced by the velocity overshoot e†ect, compete with CDM Ñuctuations in the present matter power spectrum. We present new exact analytic solutions for the cold dark matter Ñuctuations in the presence of a growth-inhibiting radiation and baryon background. Com- bined with the acoustic contributions and baryonic infall into CDM potential wells, this provides a highly accurate analytic form of the small-scale transfer function in the general case. Subject headings: cosmic microwave background È cosmology: theory È dark matter È elementary particles È large-scale structure of universe

1. INTRODUCTION manner. Despite the simplicity of the Ðnal results, the study of small-scale Ñuctuations requires a rather technical expo- Small-scale Ñuctuations in the cosmic microwave back- sition of perturbation theory. For this reason, we divide this ground (CMB) and the matter density provide a unique paper into two components. The main text discusses results opportunity to probe structure formation in the universe. drawn from a series of appendices and illustrates the corre- As CMB anisotropy experiments reach toward smaller and sponding principles in the familiar context of adiabatic cold smaller angles, the region of overlap with large-scale struc- dark matter (CDM), adiabatic baryonic dark matter ture measurements will increase dramatically. Since the (BDM), and isocurvature BDM scenarios for structure CMB and matter power spectra encode information at very formation. di†erent epochs in the formation of structure, comparison We begin in° 2 with a discussion of the central approx- of the two will provide a powerful consistency test for com- imations employed, i.e., the tight coupling limit for the peting scenarios. Unfortunately, the simple relation photons and baryons and the external potential representa- between the two power spectrum at large scales(Sachs & tion for metric Ñuctuations. Details of these methods and Wolfe1967) does not hold at small scales. To establish the the justiÐcation of their use can be found inAppendix A. As general relation, we need to employ a more complete established inAppendix B, all Ñuctuations are given a boost analysis of how gravitational instability a†ects the matter at horizon crossing due to the driving e†ects of gravita- and CMB together. tional infall and dilation. Since the gravitational potential In previous papers (Hu & Sugiyama1995a, 1995b, here- subsequently decays, the driving e†ects disappear well after afterHSa, HSb), we developed a conceptually simple analy- horizon crossing, leaving Ñuctuations to evolve in their tic description of CMB anisotropy formation. The central natural source-free modes. For the photon-baryon system, advance over previous analytic works (e.g., Doroshkevich, these are acoustic oscillations whose zero point is displaced Zeldovich, & Sunyaev1978) involved the inÑuence of gravi- by baryon drag. As described in°° 34, and these oscil- tational potential wells, established by the decoupled lations are frozen into the photon and baryon spectra at last matter, on the acoustic oscillations in the photon-baryon scattering and at the end of the Compton drag epoch, Ñuid. In this paper, we reÐne the analysis for scales much respectively. Since these epochs are not equal, photon di†u- smaller than the horizon at last scattering as is relevant for sion(Silk 1968) sets a di†erent damping scale in the CMB large-scale structure calculations. Furthermore, we extend it and the baryons. This process is described by the acoustic to encompass the photon-baryon back-reaction on the visibility functions introduced in Appendix C. evolution of the cold dark matter and gravitational poten- The baryonic oscillations may be hidden in CDM models tial as well as baryonic decoupling and evolution. This by larger cold dark matter Ñuctuations. These are discussed makes it possible to extract the relation between CMB and in° 5. The baryon and radiation background also sup- matter Ñuctuations established in the early universe. presses CDM growth before the end of the Compton drag Analytic solutions in terms of elementary functions can be epoch.Appendix D treats this source-free evolution of the constructed in the small-scale limit and serve to illuminate CDM component analytically. After the end of the the physical processes involved in a model-independent Compton drag epoch, the CDM Ñuctuations provide potential wells into which the baryons fall. From the com- 1 Institute for Advanced Study, School of Natural Sciences, Princeton, NJ 08540; whu=sns.ias.edu bined analysis of CMB, baryon, and CDM Ñuctuations, we 2 Department of Physics, Kyoto University, Kyoto 606-01, Japan; obtain accurate analytic expressions for matter and photon sugiyama=tap.scphys.kyoto-u.ac.jp transfer functions in the small-scale limit. This improves 542 SMALL-SCALE COSMOLOGICAL PERTURBATIONS 543 greatly upon the Ðtting functions for the matter power spec- settle into the pure logarithmically growing mode in the trum in the literature(Bardeen et al. 1986, hereafter BBKS; radiation-dominated universe. Peacock& Dodds 1994; Sugiyama 1995) in the case of a signiÐcant baryon fraction. Our form is constructed out of 3. PHOTON FLUCTUATIONS Ðts fromAppendix E for parameters that depend on the detailed physics of recombination, i.e., the last scattering 3.1. Driving E†ects epoch, Compton drag epoch, photon damping length, and Before recombination, the photon-baryon system Silk damping length. behaves as a damped, forced oscillator. As discussed in ° 2, the self-gravity of the acoustic oscillations contributes to the 2. ACOUSTIC APPROXIMATION gravitational force on the oscillator at horizon crossing if Previously we developed an analytic description of the photons or baryons contribute signiÐcantly to the acoustic oscillations in the photons and baryons (HSa; density of the universe at that epoch. Even though the HSb).As discussed in more detail in Appendix A, this detailed behavior at horizon crossing in the radiation- approach is based on two simplifying assumptions: that a dominated epoch is complicated by feedback e†ects, the end perturbative expansion in the Compton scattering time result is extremely simple: Ñuctuations experience a boost between photons and electrons can be extended up to proportional to the potential at horizon crossing. As the recombination and that the gravitational potential may be Ðrst compression of the Ñuid is halted by photon pressure, considered as an external Ðeld. Combined, these two simpli- the self-gravitational potential decays. This leaves the oscil- Ðcations imply that photon pressure resists compression by lator in a highly compressed state and enhances the ampli- infall into potential wells and sets up acoustic oscillations in tude of the subsequent oscillation. For adiabatic initial the photon-baryon Ñuid which then damp by photon di†u- conditions, the amplitude of the potential at horizon cross- sion. However, both these approximations would seem to ing is essentially the initial curvature Ñuctuation. For iso- be problematic for small-scale Ñuctuations. Here the optical curvature initial conditions, it is simply related to the initial depth through a wavelength of the Ñuctuation at last scat- entropy perturbations. tering is small, implying weak rather than tight coupling We quantify these statements inAppendix B and calcu- (see, e.g.,Kaiser 1984; Hu & White 1996). This issue is late the exact boost factor (see eqs.[B5]È[B6]). It represents addressed more fully inAppendix C (below eq. [C3]), where an enhancement in temperature Ñuctuations of a factor of 3 it is shown that only acoustic contributions from the tight over its initial value in the adiabatic case or more impor- coupling regime are visible through the last scattering tantly a factor of5/[1 ] (4/15)f ] over the gravitational red- surface, i.e., those from a time slightly earlier than last scat- shift induced large-scale Ñuctuationsl in a scale-invariant tering when the optical depth to Compton scattering was model(Sachs & Wolfe 1967). Here the fraction of the radi- still high. ation energy density contributed by the neutrinos is f \ l On the other hand, the second issue poses a computa- o /(o ] o ) ^ 0.405 for the standard thermal history with tional problem. Inside the horizon in the radiation- threel l masslessc species. Its presence accounts for neutrino dominated epoch, the acoustic oscillations in the anisotropic stress, which provides the relation between the photon-baryon density feed back into the evolution of the gravitational potential ( and the curvature Ñuctuation ' gravitational potential through the generalized Poisson (seeeq. [A5]). Note that neutrino temperature Ñuctuations equation.Hu & White (1996) take the approach of includ- are boosted by a similar factor(Hu et al. 1995). ing the self-gravity e†ects consistently by separating them Three features are worth emphasizing: from the truly external potentials. For many cases, a simpler 1. Potential decay enhances temperature perturbations. approach suffices. By considering the self-gravity of the Ñuid 2. Enhancement only occurs for scales that cross the as contributing to an external potential, we retain the con- horizon before equality. ceptual simplicity ofHSa and HSb. Although a consistent 3. It occurs at horizon crossing causing phase-coherent evolution of the potential would seem to require a full oscillations in the wavenumber k. numerical solution of the coupled equations, it is in practice not generally necessary. The key point is that the feedback Together the Ðrst two points imply that as a function of k, e†ect on the potentials is not arbitrary. Since photon pres- the acoustic amplitude will increase throughk , the sure prevents the growth of photon-baryon Ñuctuations and horizon crossing wavenumber at equality, forming aeq poten- collisionless damping eliminates neutrino contributions, the tial envelope. By delaying equality, one moves the transi- potential decreases to zero after horizon crossing in the tional regime to larger scales and enhances the lower k radiation-dominated epoch. We shall examine how this oscillations. This explains the increasing prominence of the a†ects the acoustic oscillations more carefully in ° 3. Ðrst few acoustic peaks as the matter content) h2 is Due to Compton drag from the coupling, the baryons lowered or the number of relativistic species raised in0 adia- density Ñuctuations follow the photons as d5 \ 3d5 \ 3#0 batic models. It is interesting to note that the e†ect of equal- b 4 c 0 ordb \ 3# ] S and exhibit acoustic oscillations as well. ity on the CMB and matter power spectrum are Here overdots0 represent derivatives with respect to confor- anticorrelated, providing a powerful consistency test for its mal time,# is the isotropic temperature perturbation in redshiftz . As we shall see in° 3.2, though, a complication Newtonian gauge,0 and S is the constant photon-baryon arises becauseeq high-k oscillations are damped by photon entropy. On the other hand, the cold dark matter, if present, di†usion. is decoupled from photons and su†ers only the gravita- The third point follows because the gravitational driving tional e†ects of the oscillating radiation. Again, since the term is e†ective at sound horizon crossingg D 1/kcs, thus potential created by the radiation merely oscillates and mimicking a driving frequency ofu8 D kcs. Here g \ / dt/a is damps away after horizon crossing, the CDM density per- the conformal time, and we take c \ 1 throughout. Since turbationsdc experience a kick at horizon crossing only to the natural frequency of the oscillation is related to the 544 HU & SUGIYAMA sound speedcs asu \ kcs, the two scale in the same generation of viscosity and thus damping in the radiation- manner. Thus, the horizon crossing e†ect is timed to dominated universe(Kaiser 1983). The end result of the produce a coherent oscillation in k. SpeciÐcally, adiabatic calculation is a wavenumberkD(g) by which acoustic Ñuc- and isocurvature initial conditions yield cosine and sine tuations are damped as exp[[(k/kD)2] (see eq. [A14]). temperature oscillations, respectively. Combined with the potential envelope from the horizon crossing boost of° 3.1, this completes the acoustic envelope 3.2. Damping E†ects shown inFigure 1. Remaining after di†usion damping is the The photons and baryons are not in fact perfectly acoustic o†set of# ] ( \[R(, where R\3o /4o \ 0 b c coupled, leading to di†usion damping. The coupling 31.5) h2#~4(z/103)~1. Here# \ T /2.7 K is the scaled b 2.7 2.7 0 strength is quantiÐed by the Compton optical depth q. Cor- present temperature of the CMB,)b is the fraction of criti- respondingly, the mean free path of the photons in the cal density contributed by the baryons, and the Hubble baryons is given byq5 . Explicitly,q5 x n p a, where x constant is H \ 100 h km s~1 Mpc~1. We add the Newto- ~1 \ e e e 0 is the electron ionization fraction,n is the electronT number nian potential ( to the temperature perturbation# to e 0 density, andp is the Thomson cross section. As the remove the e†ect of gravitational redshift on the photons photons randomT walk across a wavelength of the pertur- (see Appendix A). The R( term represents the baryon drag bation, temperature Ñuctuations damp collisionally. More e†ect on the photons and is analogous to the Silk mecha- speciÐcally, di†usion generates viscosity or anisotropic nism(Silk 1968), with the roles of the photons and baryons stress in the photon-baryon Ñuid and causes heat conduc- interchanged. Infalling baryons drag the photons into tion across the wavelength(Weinberg 1972, p. 568). Both potential wells, leading to a displacement of the zero point these processes damp Ñuctuations. In 1 order of magnitude, of the acoustic oscillation(HSa). Thus, it is responsible for the damping length is the random walk distance(g/q5 )1@2. It the alternating peak heights and amplitude enhancement of exceeds the wavelength of a Ñuctuation when the optical intermediate scale acoustic oscillations (see Appendix A, depth through a single wavelengthq5 /k \ kg. If the di†usion Figure 8a). The zero-point shift remains even after di†usion scale is well under the horizon, kg ? 1, so thatq5 /k is still damping has eliminated the oscillations themselves. high at crossover. The photons and baryons are thus still strongly coupled, and the damping may be calculated under 3.3. Decoupling the tight coupling approximation. As the total optical depth to the present drops below A quantitative treatment of di†usion damping is given in unity,q(z ) 1, last scattering of the CMB photons freezes * \ Appendix A. Subtle e†ects such as the angular and polariza- the acoustic oscillations into the spectrum. The optical tion dependence of Compton scattering enhance slightly the depth drops rapidly as neutral hydrogen forms so that last

FIG. 1.ÈAcoustic oscillations receive a boost at horizon crossingaH due to driving from gravitational potential decay. The perturbations then settle into a pure acoustic mode and are subsequently damped by photon di†usion. Together the potential driving and di†usion damping e†ects form the acoustic envelope. After di†usion damping has destroyed the acoustic oscillations, the underlying baryon drag e†ect becomes apparent. Since ( is constant here for a ? a , o R( o increases with time. This contribution itself is damped at last scatteringa by cancellation. Well after last scatteringa ? a , isotropic * * perturbationseq damp collisionlessly, creating angular Ñuctuations in the CMB. The model here is adiabatic CDM, ) \ 1, h \ 0.5, ) \ 0.05. 0 b FIG. 2.ÈVisibility functions for the photons and baryons. In the undamped k ] 0 limit, the photon acoustic visibility and the Compton visibility are equivalent,VŒ \ V , and the baryon acoustic visibility equals the drag visibility,VŒ b \ Vb. If)b h2 [ 0.03, Vb peaks at later times thanV , i.e.,gd [g . For small scales, thec acousticc visibilities, which weight the acoustic contributions from the tight coupling regime, are suppressed increasinglyc at later times* by damping, and the width and amplitude of the acoustic visibilities decrease as k increases. FIG. 3.ÈPhoton transfer function. Plotted is the present rms photon temperature Ñuctuation relative to the initial curvature Ñuctuation '(0, k) for adiabatic Ñuctuations and the initial entropy Ñuctuation S(0, k) for isocurvature Ñuctuations assuming standard recombination. The intrinsic amplitude is approximated by the analysis inAppendix B for scales well inside the horizon at equalityk ? k . The damping is well approximated by the tight coupling eq expansion for high)b h2 and is overestimated slightly for low values. Below the damping tail, the baryon drag o†set clearly appears in (b) and (c), where the gravitational potential is not dominated by acoustic density Ñuctuations. An analytic treatment of this e†ect is given in AppendicesAC. and SMALL-SCALE COSMOLOGICAL PERTURBATIONS 547

FIG. 3.ÈContinued

scattering and recombination coincide approximately in the function describes how Ñuctuations at recombination are absence of subsequent reionization. This epoch is nearly frozen into the CMB. Equally useful for our purposes is the independent of cosmological parameters such as the matter acoustic visibility functionVŒ \ V exp[[(k/k )2]. By c c D and baryon content) h2 and)b h2 (Peebles 1968; Jones & including the intrinsic damping behavior of the Ñuctuations, Wyse1985). However,0 variations at the 10% level do occur it describes how acoustic oscillations are frozen into the across the full range of parameters. In Appendix E (eq. CMB (seeFig. 2). Due to the growth of the di†usion length [E1]), we present an extremely accurate analytic Ðt to the through recombination, this function is weighted to slightly last scattering epoch. earlier times thanV . At small scales, only the small frac- The phase of the acoustic oscillation is frozen in at the tion of photons whichc last scattered before z , and hence in * valuekrs(g ), where (HSa) the tight coupling regime, retain acoustic Ñuctuations. This * leads to a sharp decrease in acoustic Ñuctuations with k. J3 Sa J1 ] R ] JR ] R More speciÐcally, the damping envelope is given by (HSa) rs \ 2 () H2)~1@2 eq ln eq , 3 0 0 R 1 ] JR eq eq P g0 P g0 D (k) \ dgVŒ (g, k) \ dg V (g)e~*k@kD(g)+2 , (3) c c c (1) 0 0 is the sound horizon withR \ R(a ) and a \ 2.35 i.e., a near exponential damping in k. Through the Ðrst ] 10~5() h2)~1(1 [ f )~1#4 .eqIts variationeq witheq k pro- decade of the drop, it is well approximated by the form 0 l 2.7 duces an oscillatory pattern in the CMB temperature with m D (k) ^ e~*k@kDc+ c . (4) extrema at scales (HSb) c Gjn adiabatic , The damping angle is given by the same projection conver- k r (g ) \ (2) sion as the acoustic peaksl \ k r (g ). Accurate Ðtting cj s * ( j [ 1)n isocurvature . D Dc h * 2 functions forkD andm are given inAppendix E, equations The physical scale of the peaksk is projected onto an (E4)È(E7). Notec thatk c is the wavenumber at which di†u- cj Dc angular scale on the sky and provides a sensitive angular sion suppresses the Ñuctuation by e~1. It is interesting to diameter distance test for curvature in the universe. note thatkD~1 is related intimately to the width of the To determine the amplitude of the Ñuctuations, this Compton visibilityc function. This is because the thickness of instantaneous decoupling approximation must be modiÐed the last scattering surface is by deÐnition the di†usion to account for di†usion damping through recombination. length at last scattering. Combining the damping envelope The di†erential visibility functionV \ q5 e~q expresses the with the intrinsic amplitude of acoustic oscillations dis- probability that a photon last scatteredc within dg of g. This cussed in° 2 and Appendix B, we obtain the transfer func- 548 HU & SUGIYAMA Vol. 471 tion at small scales. This function represents the growth to forzd are given inAppendix E, equation (E2). Since recom- the present from the initial curvature or entropy pertur- bination occurs around z \ 103 for all models, the end of bation. InFigure 3, we plot examples for the adiabatic the drag epoch precedes last scattering if )b h2 Z 0.03. BDM, isocurvature BDM, and adiabatic CDM models The acoustic Ñuctuations in the baryons are frozen in at with standard recombination. Note that in more typical the drag epoch rather than at last scattering. Furthermore, isocurvature BDM models (Peebles1987a, 1987b), reioniza- unlike for the CMB, it is not the acoustic density Ñuctuation tion may occur soon after recombination. In this limit, the that forms peaks in the observable spectrum today but damping length continues to grow until it reaches the rather the acoustic velocity. This is because the baryon Ñuc- horizon at the new last scattering surface and destroys all tuations continue to evolve. InAppendix C, we give the oscillations in the photons. exact matching conditions atzd onto the growing and The corresponding anisotropy can be obtained by choos- decaying modes of pressureless perturbation theory. This ing an initial curvature and entropy spectrum '(0, k) and yields an accurate description for the subsequent evolution S(0, k) and employing the free streaming solution (see HSa, of the baryonic Ñuctuations in the presence of a background eq. [12]) for the monopole and dipole contributions to the radiation energy density and cold dark matter. Qualit- rms given inAppendix B, equation (B7). Note that for low atively, the acoustic velocity at the drag epoch dominates )b h2 models, the damping length is somewhat overesti- over the acoustic density for the growing mode of Ñuctua- mated by the tight coupling approximation. This is not tions due to the velocity overshoot e†ect(Sunyaev & Zeldo- ] surprising because as)b h2 0, the photon mean free path vich1970; Press & Vishniac 1980). The former moves approaches the horizon. In this case, the di†usion length matter and produces clumping in the baryon density. Since passes the wavelength of the Ñuctuation when the optical expansion damps peculiar velocities, this lasts for approx- depth through a wavelength is near unity and the tight imately an expansion timegd. Thus, only scales smaller than coupling expansion ofAppendix A breaks down. The the horizonkgd ? 1 experience a boost due to the velocity damping length is overestimated because the photons essen- at release. We show an example inFigure 4, where the tially free stream across the wavelength and do not su†er k-mode is chosen to be near a zero point of the acoustic collisional damping. A phenomenological correction for density oscillation atgd. The rapid regeneration of density this e†ect is given in Appendix E. Ñuctuations via velocity overshoot is due to the fact that In summary, the general features of the CMB Ñuctuation zeros of the density oscillation are maxima of the velocity spectrum are as follows: oscillation. Therefore, the peaks in the matter power spectrum due to baryonic acoustic oscillations occur at 1. Acoustic peaks at harmonics of the angle the sound horizon subtends at last scattering, sensitive to (a) the cur- G( j [ 1)n adiabatic , k r (g ) \ 2 (5) vature and (b) the adiabatic or isocurvature nature of the bj s d jn isocurvature , initial Ñuctuations; 2. An acoustic envelope for the oscillation amplitude sen- and they are roughly n/2 out of phase with the correspond- sitive to (a) The matter-radiation ratio, e.g., the adiabatic ing CMB Ñuctuations. acoustic envelope varies from (1/3)( to (5/3)( as one passes To obtain the amplitude of the acoustic Ñuctuations, we through the equality scale, (b) The baryon-photon ratio due must consider damping e†ects also. Photon di†usion in the to the a modulation of R( from baryon drag, and (c) The tight coupling regime damps baryon Ñuctuations as well thermal history from the exponential di†usion-damping due to Compton drag, i.e., via the Silk mechanism (Silk tail. 1968). Analogous to the photon case, we can construct the drag visibility functionV out of the drag optical depth q . The exact nature of the features will depend on details of the b d The acoustic visibility function then becomesVŒ \ V exp model. b b [[(k/kD)2](see Fig. 2). Similarly, the net damping as a function of scale is described by 4. BARYON FLUCTUATIONS P g0 In the tight coupling limit, the baryons participate in the Db(k) \ dgVŒ b(g, k) acoustic oscillations of the photons. Near recombination, 0 they decouple from the photons, though not exactly at last P g0 k kD k k m scattering. Scattering represents an exchange of momentum \ dgVb(g)e~* @ (g)+2 ^ e~( @ S) S , (6) between the two Ñuids and seeks to equalize their bulk 0 velocitiesV andV . However, the two momentum densities where the approximation is valid through the Ðrst decade of b c (o ] p )V \ (4/3)o V and(ob ] pb)Vb ^ ob Vb are not damping. Analytic Ðtting formulae fork andm are given in equal.c Thus,c c momentumc c conservation requires that the rate Appendix E,equations (E9)È(E10). AsS is the caseS with the of change of the baryon velocity due to Compton drag is photons, the latter accounts for the width of the visibility scaled by a factor ofR~1 \ (4/3)o /o compared with the function and is almost independent of cosmological param- c b photon case; i.e.,Pq5 d \ q5 / R . The explicit expression for the eters. Note that the Silk damping length is not the same as baryon momentum conservation or Euler equation is given the photon damping length despite the underlying simi- inAppendix A and is used in Appendix C to make the larity in cause. The small di†erence between last scattering qualitative statements here rigorous. The form of the coup- and the drag epoch can alter it signiÐcantly due to the rapid ling suggests that we can deÐne a drag depth qd(gd) \ change inkD around recombination (see Fig. 2). Moreover, / gd0 q5 d dg. Below drag depthqd(gd) \ 1, the baryons decouple the two scale di†erently with the baryon and matter dynamicallyg from the photons. For the standard recombi- content. nation scenario, this occurs near recombination but at a Together with the horizon crossing boost from Appendix di†erent value than last scattering. Analytic Ðtting formulae B, this deÐnes the contribution to the matter transfer func- No. 2, 1996 SMALL-SCALE COSMOLOGICAL PERTURBATIONS 549

FIG. 4.ÈBaryon Ñuctuation time evolution. The baryon density Ñuctuationdb follows the photons before the drag epochad yielding a simple oscillatory form foraH > a > ad. The Silk damping length is given by the di†usion length at the drag epoch. The portion of the baryon Ñuctuations that enter the growing mode is dominated by the velocity perturbation at the drag epochad due to the velocity overshoot e†ect (see alsoFig. 11). Since a Ñat " model is chosen here, the growth rate is slowed by the rapid expansion for a Z a \ () /) )1@3. " 0 " tion of the acoustic oscillations (seeFig. 5). Explicit expres- baryon fraction is signiÐcant, the power law is modiÐed. To sions for the three scenarios shown are given in Appendix Ðrst order in )b/) , D,equations (D22) and (D23). In the isocurvature BDM 0 G 3 ) 3 A 2 ) BH models, the transfer function at small scales is dominated by p \ 1 [ b , [ 1 [ b . (7) the initial entropy Ñuctuation S(0, k). Furthermore, unlike 5 ) 2 5 ) 0 0 the photon oscillations, baryon oscillations may survive CDM growth is thus inhibited by the presence of baryons. early reionization if it occurs more than an expansion time We give the exact solution to the evolution equation from after the drag epoch. In this case, the baryonic oscillations horizon crossing tozd inAppendix D. Because of the com- are subsequently surrounded by a homogeneous and iso- plexity of these expressions, it is also useful and instructive tropic distribution of photons. They then represent entropy to obtain approximate scaling relations. Ifad ? a and perturbations and are not damped by further photon di†u- ) /) > 1, the main e†ect is an amplitude reductioneq of the sion. Even in the rather unphysical event of near- b 0 CDM density perturbation dc(gd, k) by approximately instantaneous reionization, baryonic oscillations may b b (ad/a )~0.6) @)0 ^ (24) h2)~0.6) @)0 (see alsoFig. 10). As survive if reionization is accompanied by the formation of a ) h2eqis lowered, the drag0 epoch recedes into the radiation signiÐcant fraction of compact baryonic objects (see, e.g., domination0 epoch and the regime in which the growth rate Gnedin& Ostriker 1990). is a†ected,a \ a \ ad, vanishes. If the model contains cold dark matter, baryons su†er an At the drageq epoch, the baryons are released from the additional e†ect. After the drag epoch, they fall into poten- photons and behave dynamically as if they were CDM. tial wells established by the CDM. If the CDM to baryon Since baryonic infall into CDM wells subsequently contrib- ratio is high, this e†ect will dominate over the velocity over- utes to the self-gravity of the matter, the growth rates again shoot of the acoustic oscillations. To quantify this e†ect, we become p \ M1, [(3/2)N regardless of the baryon fraction. need to consider the evolution of CDM Ñuctuations. Thus, we follow the total nonrelativistic matter pertur- bationdm after the drag epoch. The relative contribution of 5. COLD DARK MATTER FLUCTUATIONS the CDM Ñuctuations at the drag epoch to the total nonrelativistic matter Ñuctuationsd scales as1 [ ) /) . A Let us begin with the evolution of CDM Ñuctuations m b 0 before the drag epoch. As shown in° 2 and Appendix B2, good Ðt to the net suppression is given by the form CDM Ñuctuations are given a kick at horizon crossing that A ) B sends them into logarithmically growing mode. As the uni- d ^ a 1 [ b lim d , (8) m ) m verse becomes matter dominated, this stimulates power-law 0 )b?0 growing and decaying modesd P ap. If CDM dominates wherea ^ (46) h2)~0.67)b@)0 for low) /) [ 0.5 and high c 0 b 0 the nonrelativistic matter, p \ M1, [(3/2)N. However, if the ) h2 ? 0.03. The change in the coefficients from the naive 0 FIG. 5.ÈMatter transfer function. The analytic estimates of the intrinsic acoustic amplitude are good approximations fork ? k . The Silk damping scale is approximated adequately although its value is underestimated by D10%. For isocurvature BDM and adiabatic CDM, the acousticeq contributions do not dominate the small-scale Ñuctuations. We have added in the contributions from the initial entropy Ñuctuations and the cold dark matter potentials according to the analytic treatment of Appendix E. SMALL-SCALE COSMOLOGICAL PERTURBATIONS 551

FIG. 5.ÈContinued scaling relation is due to detailed matching of growing and fer function will therefore scale ask gd. Acoustic contribu- decaying modes (seeAppendix D, eq. [D18]). A highly tions increase in prominence if theS Silk scale is small accurate Ðtting formula for a in the general case is given in compared with the horizon at the drag epoch, i.e., in the ] Appendix E, equation (E12). For the)b 0 limit, an exact high)b h2 case. By including the suppression factor from expression in terms of elementary functions is given in equation (8)and numerical factors from Appendices B and Appendix D that improves the 10% accuracy of the stan- D, the maximum ratio of the acoustic amplitude to the dardBBKS Ðtting formula to better than the 1% level at CDM contribution in the transfer function scales crudely as small scales. (see Appendix D, eq. [D31]) InFigure 6, we show the resulting evolution of a scale under the Silk damping length for which the baryon Ñuc- )b 0.4k () h2)~1(1 ] 24) h2)~1@2(1 ] 32)b h2)~3@4a~1 , tuations at the drag epoch are negligible. To extend the S )c 0 0 scaling ofequation (8) to larger scales for Figure 5, we have (9) employed a generalization of theBBKS Ðtting function given inAppendix D. Notice that both (1) the change in the wherek , here in units of Mpc~1, and a are given explicitly growth rate between equality and the drag epoch and (2) the inAppendixS E. This relation incorporates the following: fractional contribution ofd tod at the drag epoch play a c m 1. Baryon acoustic oscillations; signiÐcant role in suppressing the Ðnal amplitude of matter 2. Baryon decoupling; Ñuctuations. Combined with the acoustic contributions 3. Silk damping; from° 4, this completes the matter transfer function in 4. Velocity overshoot; CDM models. 5. Baryon gravitational instability; We can now address the question of when acoustic oscil- 6. Baryon infall into CDM wells whose depth depends lations are prominent in the matter transfer function. The on (a) Acoustic feedback into the potential and (b) CDM main e†ect is simply due to the density ratio o /o ) /) . b c \ b c self-gravity versus the expansion rate. However, the acoustic and cold dark matter contributions have a di†erent dependence on scale. Relative to the cold The relation combines these in a consistent manner. dark matter, acoustic contributions scale as(kgd)Db due to DISCUSSION the velocity overshoot and di†usion damping factors. Since 6. Db incorporates an exponential cuto† at the Silk scale k We have established a framework for treating small-scale and velocity overshoot weights the spectrum toward smallS Ñuctuations in the realistic case of a coupled multiÑuid scales, acoustic contributions will be most visible just above system. In the limit in which Ñuctuations crossed the the Silk scale. The relative contribution to the matter trans- horizon in the radiation-dominated epoch, closed-form FIG. 6.ÈCDM and matter Ñuctuation time evolution. The cold dark matter Ñuctuations are constant outside the horizon scale and experience a boost into a logarithmically growing mode at horizon crossing in the radiation-dominated epoch. Between equality and the drag epoch, the presence of baryons suppresses the growth rate of CDM Ñuctuations (seeeq. [7]). By lowering) h2, this region of suppression can be reduced for Ðxed) /) . After the drag 0 b 0 epoch, we plotdm \ (db ob ] dc oc)/(ob ] oc) rather thandc because the two components thereafter contribute similarly to the total growth. Baryons also lower the contribution ofdc todm at the drag epoch. SMALL-SCALE COSMOLOGICAL PERTURBATIONS 553 analytic solutions are available. The fundamental elements 3. Oscillations in matter power spectra alone; early uncovered by this approach are the boost at horizon cross- reionization with )b Z )c. ing due to infall and dilation e†ects from potential decay, Large-scale structure observations already suggest there are the source-free solution of the component evolution equa- no dramatic oscillations in the matter power spectrum as tions, the baryon drag on the photons, and the Compton would be the case for classes (1) and (3) if )b ? )c (Peacock drag on the baryons. Together they establish general consis- & Dodds1994). However, low-amplitude oscillations, as tency relations between the matter and the radiation power might be expected if)b D )c, remain possible. Of course, spectra as well as expose their sensitivity to changes in the the exact form that these oscillations will take in the obser- background model. vations depends on issues such as redshift space distortions Unlike the case of the matter Ñuctuations, decay in the and nonlinear corrections. potential results in an ampliÐcation of acoustic oscillations If oscillations are discovered in neither spectra, the most in the CMB. These opposing manifestations of the same natural conclusion is that our universe has)b > )c and physical e†ect provide both a measure and a powerful con- su†ered early reionization. However, other possibilities sistency check on the redshift of equality. Baryon drag dis- include the formation of perturbations after recombination, places the zero point of the acoustic oscillations, which reionization within an expansion time after the drag epoch, remains as a temperature shift even after the oscillations and equal or random stimulation of adiabatic and iso- have damped by photon di†usion. Last scattering marks curvature mode acoustic Ñuctuations.3 These scenarios can the end of the baryon drag epoch at which the acoustic be distinguished by measuring the location of the damping oscillations with their characteristic di†usion damping scale scale. All scenarios obeying standard recombination, are frozen into the photon spectrum. Correspondingly, at regardless of the presence of actual peaklike structures, the end of the Compton drag epoch, the Silk-damped bary- follow the scalings for the damping length discussed here. In onic acoustic oscillations are frozen into the matter spec- all reionized models, the horizon at last scattering and at trum. We have provided convenient analytic Ðtting the drag epoch marks the damping scale for the photons formulae for these quantities as a function of the matter and and baryons, respectively. baryon content. These may be useful for the extraction of If acoustic oscillations are discovered in both the CMB cosmological information from the CMB and matter power and large-scale structure power spectra, we will possess a spectrum once observations become available. strong consistency test for the dynamics of the expansion, The photon-baryon system also a†ects the growth of i.e., a combination of the matter content, curvature, and CDM Ñuctuations. First it stimulates growth through the cosmological constant, as well as the adiabatic or iso- horizon crossing boost. Subsequently, it a†ects the balance curvature nature of the initial Ñuctuations. Furthermore, of the growth inhibiting expansion to the self-gravity of the the two contain complementary information on several spe- CDM. We have obtained an analytic solution for adiabatic ciÐc fundamental cosmological parameters. The scale of the initial conditions and the exact general solution for the two peaks in the matter power spectrum are determined mainly e†ects, respectively. To simplify these expressions, we have by the matter content) h2, whereas the angular scale of also provided an accurate Ðtting form for the transfer func- the CMB peaks is mostly0 sensitive to the curvature 1 [ ) tion in terms of elementary functions. Growth suppression [ ) . The two damping lengths also probe di†erent com-0 " due to the presence of baryons has implications for the Ðrst binations of) h2 and)b h2. The ratio of the peak heights generation of structure. The expressions derived here to the underlying0 CDM contribution in the matter power remain valid for linear perturbations to an arbitrarily small spectrum probes)b/) . Furthermore, by comparing similar scale at which direct numerical calculations are impractical. scales, dependence on0 the initial power spectrum can be Baryonic acoustic oscillations are of course not promi- eliminated, providing a clean test of the whole gravitational nent in presently popular models in which)b/) > 1 due to instability paradigm. the extra growth of the CDM Ñuctuations between0 horizon crossing and the drag epoch. However, they serve as a useful We would like to acknowledge useful discussions with complement to their CMB counterpart if either ) h [ 2 J. R. Bond, U. Seljak, J. Silk, and P. Steinhardt. W. H. 0.05 or big bang nucleosynthesis constraints are too0 strin- would like to thank J. R. Bond for calling to his attention gent(Gnedin & Ostriker 1990). Indeed, there are tentative the enhancement of di†usion damping through polarization indications from cluster inventories that the baryon fraction and M. White for pointing out several typos in the draft. may be as high as 15%(White et al. 1993). The presence or W. H. was supported by grants from the NSF and the absence of acoustic oscillations in the observations of the W. M. Keck Foundation. CMB and large-scale structure will in the future provide a robust distinction between general classes of scenarios: 1. Oscillations in CMB and matter power spectra: standard recombination with ) Z ) ; b c 3 This may be accomplished by balancing the initial conditions or 2. Oscillations in CMB alone: standard recombination hypothesizing a gravitational forcing potential that is external to the linear with )b > )c ; photon-baryon-neutrino-CDM system, e.g., cosmological defects. 554 HU & SUGIYAMA Vol. 471

APPENDIX A

TIGHT COUPLING LIMIT

A1. EVOLUTION EQUATIONS The Fourier transform of the Newtonian temperature Ñuctuation can be broken up into Legendre moments, related to the l direction cosines of the photon momenta ci, #(g, k, c) \ &l([i) #l Pl(k Æ c). The evolution equation for these moments is given by the Boltzmann hierarchy(Bond & Efstathiou 1984; HSb): k #0 \[ # ['0 , 0 3 1 A 2 B #0 \k # ]([ # [q5(# [V ), 1 0 5 2 1 b A2 3 B A9 1 1 B #0 \k # [ # [q5 # [ #Q [ #Q , 2 3 1 7 3 10 2 10 2 2 0 A l l ] 1 B #0 \ k # [ # [ q5 # (l [ 2) , (A1) l 2l [ 1 l~1 2l ] 3 l`1 l if the ratio of the wavelength to the curvature scale is much smaller than the angle considered, i.e., k ? l([K)1@2, where Q K \[H2(1 [ ) [ ) ). Recall that overdots are derivatives with respect to conformal time g \ / dt/a. Here#l is the CMB temperature0 perturbation0 " in the Stokes parameter Q. It accounts for polarization generated by Compton scattering of anisotropic radiation. The metric Ñuctuations in the mode are given byg \[(1 ] 2(Y ) andg \ a2(1 ] 2'Y )c , where 00 ik x ij ij cij is the three metric on a surface of constant curvature and Y is a plane wave e Õ in Ñat space or more generally a k-eigenfunction of the Laplacian. The presence of the curvature perturbation ' in the monopole equation represents the dilation e†ect. The form of the metric shows that it has the same origin as the photon redshift with the expansion. The gravitational potential ( in the dipole or velocity equation accounts for gravitational infall or redshift. The tight coupling approximation assumes that the Compton scattering rateq5 is sufficiently rapid to equilibrate changes in the photon-baryon Ñuid. It is an expansion in the Compton scattering timeq5 ~1, or more speciÐcally the inverse of the optical depth through a wavelengthq5 /k and through a period of the oscillationq5 /u \ q5 /kcs [q5/k, where 1 cs\ , (A2) J3(1 ] R) is the photon-baryon sound speed withR \ 3o /o . To Ðrst order, only the l \ 0 monopole (with density Ñuctuation b c d \ 4# ) and l \ 1 dipole (with bulk velocityV4 \ # ) survive and one obtains the forced oscillator equation for acoustic c c waves in0 the photon-baryon Ñuid(HSa).4 To second order,1 the acoustic oscillations of the monopole and dipole are damped due to the imperfect coupling between the photons and baryons. Photon di†usion creates heat conduction through # [ Vb and shear viscosity through # (Weinberg 1972; Bond 1996). 1 To close these equations, we2 need the continuity and Euler equations for the baryons, 5 db \[kVb [ 3'0 , a5 q5(# [V ) V0 \[ V ]k(] 1 b , (A3) b a b R the polarization hierarchy equations for the CMB(Bond & Efstathiou 1984; Kosowsky 1996), k C1 1 D #0 Q \[ #Q[q5 #Q[ (# ] #Q) , 0 3 1 2 0 10 2 2 A 2 B #0 Q \ k #Q [ #Q [ q5 # , 1 0 5 2 1 A2 3 B A 9 1 1 B #0 Q \ k #Q [ #Q [ q5 #Q [ # [ #Q , 2 3 1 7 3 10 2 10 2 2 0 A l l ] 1 B #0 Q \ k #Q [ #Q [ q5 #Q (l [ 2) , (A4) l 2l [ 1 l~1 2l ] 3 l`1 l

4 InHSa and HSb, we employed a hybrid gauge or ““ gauge-invariant ÏÏ representation of density Ñuctuations for computational convenience. Since there are no beneÐts of this choice below the horizon, we work entirely in the Newtonian gauge in this paper. Only the deÐnition of density Ñuctuations is a†ected: the total matter gauge*X \ dX ] 3(a5 /a)(1 ] pX/oX)VT/k, where X represents any of the particle components. No. 2, 1996 SMALL-SCALE COSMOLOGICAL PERTURBATIONS 555

FIG. 7.ÈPhoton di†usion scale. The photon di†usion scale grows rapidly near last scattering due to the increasing mean free path of the photons but remains well under the horizon scalek~1 \ (a5 /a) o at last scattering. The small di†erence betweena anda is sufficient to cause a signiÐcant di†erence in the * a* * d e†ective damping if)b h2 di†ers substantially from the crossover point 0.03. The inclusion of the angular dependence of Compton scattering enhances damping by a small factorf \ 9/10, as does the further inclusion of polarization f \ 3/4. 2 2 and the Einstein-Poisson equations for the metric or potential perturbations, C a5 V D k2' \ 4nGa2o d ] 3 (1 ] w ) T , k2(' ] () \[8nGa2p % , (A5) T T a T k T T if k ? ([K)1@2. HerewT \ pT/oT, where the subscript T denotes the total matter including all particle species and the total anisotropic stress is related to the radiation quadrupoles as p % \ 12 (p # ] p N ) , (A6) T T 5 c 2 c 2 withN as the neutrino temperature quadrupole. Notice that the baryon continuity equation can be rewritten as d5 \ 2 b [k(Vb [ # ) ] 3#0 because dilation e†ects on the photon temperature and baryon density Ñuctuations are analogous. This represents adiabatic1 0 evolution if V \ # . b 1 A2. ACOUSTIC DISPERSION RELATION Let us derive the dispersion relation for acoustic oscillations in the tight coupling limit. Consider Ðrst the e†ects of polarization. Since only the polarization monopole#Q and quadrupole#Q feed back into the temperature Ñuctuations, we may immediately expand the polarization hierachy inq50~1 to obtain 2 #Q \ #Q \ 1# , (A7) 2 0 4 2 which simpliÐes the temperature quadrupole evolution ofequation (A1) to #0 \ k(2# [ 3# ) [ q5 f # , (A8) 2 3 1 7 3 2 2 wheref \ 3. Other approximations commonly used aref \ 9/10 for unpolarized radiation (Chibisov 1972) andf \ 1 for further2 neglecting4 the angular dependence of Compton scattering2 (Weinberg 1972; Peebles 1980, p. 92; HSa). We2 keep the factorf implicit so that the separation of e†ects can be read directly o† the Ðnal results. Expanding the quadrupole equation (A8) to2 Ðrst order inq5 ~1, we obtain # \ q5 ~1f ~1 2k# . (A9) 2 2 3 1 A second-order expansion for the quadrupole is not necessary because its e†ect on the Ñuid equations through#0 is already of Ðrst order in q5 ~1. 1 556 HU & SUGIYAMA

On the other hand, a second-order expansion of the baryon Euler equation is necessary. Let us try a solution of the form # P exp i / u dg and ignore variations on the expansion timescalea5 /a in comparison with those at the oscillation frequency u.1 The electron velocity, obtained by iteration of the Euler equation, is to second order V \ # [ q5 ~1R(iu# [ k() [ q5 ~2R2u2# . (A10) b 1 1 1 Substituting this into the photon dipole equation(A1) and eliminating the zeroth-order term yields iu(1 ] R)# \ k[# ] (1 ] R)(] [ q5 ~1R2u2# [ 4 q5 ~1f~1 k2# . (A11) 1 0 1 15 2 1 This suggests that we try a solution of the form # ] (1 ] R)( P expi / u dg. Employing the monopole equation of (A1) and assuming again that variations at the oscillation frequency0 are sufficiently rapid that changes in R, ', and ( can be neglected, we obtain k2 A 4 B (1 ] R)u2\ ]iq5~1u R2u2] k2f ~1 . (A12) 3 15 2 With the Ðrst-order relation u2\k2/3(1 ] R), the solution to the resultant quadratic equation is k 1 C R2 4 1 D u \^ ] i k2q5 ~1 ] f ~1 . (A13) J3(1 ] R) 6 (1 ] R)2 5 2 1 ] R

Thus, to second order acoustic oscillations are damped as exp[[(k/k )2] with the damping length (Weinberg 1972; Kaiser 1983; Bond 1996) D 1 P 1 R2]4f~1(1 ] R)/5 k~2 \ dg 2 . (A14) D 6 q5 (1 ] R)2 IfR [ 1, polarization and the angular dependence of Compton scattering enhances damping through the generation of viscosity (seeFig. 7). Viscosity is related to photon di†usion because the quadrupole and higher moments are generated as photons from regions of di†erent temperatures meet.

A3. BARYON DRAG AND THE ADIABATIC INVARIANT The temperature perturbation oscillates around# ] (1 ] R)( \ 0 due to the baryon drag e†ect. This can be understood more easily by examining the Ðrst-order equation from0 which it originates, k2 k2 (1 ] R)# ] # ^ [ (1 ] R)( , (A15) 0 3 0 3 ignoring slow changes in R, ', and ( from the expansion. Notice thatm \ (1 ] R) plays the role of the e†ective mass of the oscillator and the gravitational potential provides the e†ective accelerationeff through infall. The photon pressure acts as the restoring force and is independent of the baryon content R. Equation (A15) has a simple solution 1 # \ [# (0, k) ] (1 ] R)(] cos (ug) ] #0 (0, k) sin (ug) [ (1 ] R)( , (A16) 0 0 u 0 where the frequencyu \ k/(3m )1@2 \ k/[3(1 ] R)]1@2 in agreement with the Ðrst-order dispersion relation of equation (A13). This solution describes an oscillatoreff whose zero point has been displaced by[m ( \[(1 ] R)( due to the gravitational force (seeFig. 8). The photons, although massless, su†er infall e†ects due to gravitationaleff blueshift. Since this is exactly cancelled as the photons stream out of the potential wells, we can consider# ] ( as the e†ective temperature perturbation. Thus, this part of the zero point shift has no net e†ect. However, the baryons0 also contribute to the e†ective mass of the Ñuid. Since the photons and baryons are tightly coupled, baryonic infall drags the photons into potential wells and contributes [R( to the displacement. Notice that baryon drag also increases the amplitude of the cosine oscillation because the initial conditions #(0, k) represent a greater displacement from the zero point. Thus, baryon drag accounts for both the alternating peak heights of the acoustic oscillations and their enhancement with)b h2 (HSa). After di†usion damping has eliminated the oscillations themselves, the zero-point shift [R( remains. Of course, for adiabatic BDM models ( is also reduced to zero by the di†usion. In reality, the variation of oscillator parameters, such as the e†ective mass and gravitational force, on the expansion timescale cannot be ignored over many periods of the oscillation. From equations(A1) and (A3), the full Ðrst-order equation is d k2 k2 d (1 ] R)#0 ] # \[ (1 ] R)( [ (1 ] R)'0 , (A17) dg 0 3 0 3 dg where the addition of the space curvature term comes from the dilation e†ect#0 \['0 . Notice that the left-hand side is precisely the equation of an oscillator with a time-varying massm \ (1 ] R). The0 homogeneous equation can be solved by employing the fact that variations over a single period of the oscillationeff are small. The adiabatic invariant associated with an oscillator is given by the energyE \ 1m u2A2 over the frequency u. Therefore, the amplitude scales as 2 eff FIG. 8.ÈBaryon drag e†ect in adiabatic CDM models. (a) Baryons cause a drag e†ect on the photons, leading to a temperature enhancement of [R( \ o R( o inside potential wells, which shifts the zero point of the oscillation (short-dashed lines). (b) This contribution yields alternating peak heights in the rms and is also retained after di†usion damping. Here numerical results are displayed. 558 HU & SUGIYAMA Vol. 471

A P u1@2 P (1 ] R)~1@4. This yields fundamental solutions of the form (1 ] R)~1@4 exp (^i / u dg)(Peebles & Yu 1970). The phase integral can be performed analytically, P P u dg \ k cs dg \ krs , wherers is the sound horizon given in equation (1). A4. SUMMARY Several points are worth emphasizing:

1. The Ðrst-order dispersion relation for acoustic oscillations is u \ kcs \ k/[3(1 ] R)]1@2. 2. Slow changes in the baryonic contribution to the e†ective mass cause the temperature oscillation to decay as (1 ] R)~1@4. 3. The oscillation phase is related to the sound horizonrs \ / cs dg by / u dg \ krs. 4. Photon di†usion alters the dispersion relation and leads to exponential damping. 5. The damping length increases roughly asjD D kD~1 D (g/q5 )1@2 or the geometric mean of the conformal time and the Compton mean free path, as one expects of a random walk. 6. If it is well under the horizon, the damping length surpasses the wavelength whenq5 /k \ kg ? 1 or when the optical depth through a wavelength is still high. 7. The angular dependence of Compton scattering and polarization increases the damping length when R [ 1. 8. The zero point of oscillations in# ] ( is [R' due to baryon drag and remains as a temperature shift after di†usion damping. 0

APPENDIX B

HORIZON CROSSING

B1. PHOTON-BARYON SYSTEM The amplitude of the acoustic oscillation is determined by the growth of Ñuctuations before and in particular during the epoch when the scale crosses the horizon. Since the second-order tight coupling expansion yields just a multiplicative di†usion damping factor, let us obtain the Ðrst-order solution, which we denote by carets. The full solution can be constructed through the relations # ] ( \ [#Œ ] (1 ] R)(] exp [[ (k/k )2] [ R( , 0 0 D # \ #Œ exp [[ (k/k )2], 1 1 D d \3# ]S(0, k), b 0 kV \ k# \[#0 ['0 , (B1) b 1 0 where recall S(0, k) is the initial entropy perturbation in the photon-baryon system S(0, k) \ db(0, k) [ 3d (0, k). Near or above the horizon atg , where the sources from potential growth and decay drive the oscillator, the full formalism4 c ofHSa is needed to follow the Ñuctuations* accurately. However, we are interested only in small-scale Ñuctuations that enter the horizon well before last scattering. Since radiation pressure prevents the growth of Ñuctuations in the radiation-dominated epoch, the gravitational potentials decay away after horizon crossing. On small scales, the acoustic oscillations therefore experience a boost at horizon crossing and thereafter settle into pure modes# (g, k) \ (1 ] R)~1@4[C (k)( coskr ) ] C (k)( sin kr )]. 0 A s I s Equation (A17) yields exact solutions forCA andCI for the fundamental adiabatic and isocurvature modes of the Ñuctuation if anisotropic stress%T is ignored so that ( \['. The adiabatic modes arises from an initial curvature perturbation '(0, k), whereas the isocurvature mode arises from an initial entropy perturbation S(0, k). The solutions are(Kodama & Sasaki 1986; HSb) 3 J6 k lim C (k) \ '(0, k) adiabatic , lim C (k) \[ eq S(0, k) isocurvature , (B2) A 2 I 4 k %?0 %?0 wherek \ (2) H2/a )1@2 \ 9.67 ] 10~2) h2(1 [ f )1@2#~2 Mpc~1 is the wavenumber that passes the horizon at equality. l Recall thateq f \ 0o /(0o ]eq o ). The two modes0 stimulate pure2.7 cosine and sine modes because the gravitational forcing function l l c l yields near-resonant driving with the phase Ðxed by '(0, k) \ constant and '(0, k) \ 0, respectively(HSb; Hu & White 1996). The amplitude of the Ñuctuations is easy to understand qualitatively. For the adiabatic case, # (0, k) \ 1'(0, k). If photon streaming is ignored,#0 \['0 , the dilation e†ect would raise the amplitude to # (g, k)\1'0(0, k) [ '2(g, k) ] '(0, k) \ 3'(0, k). Note that a decaying potential boosts the acoustic amplitude due to the gravitational0 forcing2 e†ect. A similar analysis for2 the isocurvature mode accounting for potential growth outside the horizon explains the isocurvature amplitude (HSb). The e†ect of anisotropic stress can be considered as a perturbation(HSa). The dominant term comes from the neutrino quadrupole because photon anisotropies are damped exponentially with optical depth before last scattering. The order or No. 2, 1996 SMALL-SCALE COSMOLOGICAL PERTURBATIONS 559 magnitude can be simply read o† the initial conditions for the growing mode of the perturbation, # (0, k) \[1((0, k) \ 1(1 ] 2 f )~1'(0, k) adiabatic , (B3) 0 2 2 5 l or # (0, k) \ ((0, k) \ '(0, k) \ 0 and 0 J2 A 2 B #0 (0, k) \ k 1 ] f S(0, k), 0 16 eq 15 l J2 A 2 B '0(0, k) \ k 1 ] f S(0, k) isocurvature , 16 eq 15 l J2 A 6 B (0 (0, k) \[ k 1 [ f S(0, k) . (B4) 16 eq 15 l Indeed, we Ðnd that the adiabatic amplitude is well approximated by 3 A 2 B~1 C (k) \ 1 ] f '(0, k) adiabatic , (B5) A 2 5 l and likewise J6 k A 4 B C (k) \[ eq 1 [ f S(0, k) isocurvature , (B6) I 4 k 15 l for the isocurvature mode. Explicitly, the Ðrst-order acoustic solution is #Œ ] ( ^ #Œ \ (1 ] R)~1@4[C cos (kr ) ] C sin (kr )] , 0 0 A s I s #Œ \ VŒ \[J3(1 ] R)~3@4[C sin (kr ) ] C cos (kr )] . (B7) 1 b A s I s InFigure 1, we display an adiabatic example. In this case, it is also useful to compare the acoustic amplitude to the Sachs-Wolfe e†ect(Sachs & Wolfe 1967) o # ] ( o (g , k) \ 1((g ,k). With the relation (see HSa, eq. [A18]), rms 0 3 0 ((g , k) \[9(1 ] 4 f )~1(1 ] 2 f )~1'(0, k) , (B8) 0 10 15 l 5 l valid at large scalesk > k , the relative amplitude becomes eq o 3C (k)/((g , k) o \ 5(1 ] 4 f )~1 , (B9) A 0 15 l and it represents a signiÐcant boost. B2. CDM COMPONENT The evolution of the cold dark matter Ñuctuations in the presence of acoustic oscillations is also interesting and relevant for determining the small-scale behavior of the matter transfer function (seeAppendix D). The cold dark matter evolution equations are of the same form as the baryon continuity and Euler equations save for the absence of coupling to the photons, a5 d ] d5 \[k2([3' . (B10) c a c In the radiation-dominated epoch, the metric terms on the right-handside are dominated by perturbations in the radiation and may be considered as external driving forces. The homogeneous equation has two fundamental solutionsdc P Mln a, 1N. The particular solution is constructed via GreenÏs method, P g a@ d \ C ln a ] C ] [ln a@ [ ln a] (k2( ] 3')dg@ . (B11) c 1 2 a5 @ 0 Adiabatic initial conditions requireC \ 0 and C \ 3# (0, k). Thus,dc remains constant outside the horizon and then gets a kick from infall and dilation that generates1 a logarithmic2 0 growing mode. Since the behavior of the potentials is self-similar in k, i.e., they are constant outside the horizon and decay to zero as a~2 inside of it, their e†ect ondc is the same for all k. Once the potentials have decayed to zero,dc settles into the logarithmic growing mode as A aB dc ^ I '(0, k)ln I aH>a>a , (B12) 1 2 aH eq where horizon crossing occurs at a 1 ] J1 ] 8(k/k )2 H \ eq , a 4(k/k )2 eq eq J2 k ^ eq k ? k . (B13) 2 k eq 560 HU & SUGIYAMA

By numerical calculation of the integrals inequation (B11), we obtain I \ 9.11(1 ] 0.128f ] 0.029f2), 1 l l I \0.594(1 [ 0.631f ] 0.284f2) , (B14) 2 l l valid at the 1% or better level for the full range 0 ¹ f ¹ 1. As we shall see in Appendix D, this solution can be joined onto the growing mode in the matter-dominated epoch to describe the full time evolution of the CDM Ñuctuations.

APPENDIX C

DECOUPLING

C1. PHOTON DECOUPLING The tight coupling approximation is strictly valid only well before decoupling. However, the acoustic modes may be simply joined onto the free-streaming solutions once di†usion damping near decoupling has been taken into account. The full Boltzmann hierarchy has the formal solution (seeHSa, eq. [11]) P g0 (# ] ()(g , k, k) \ dg[(# ] ( [ ikV )q5 [ '0 ] (0 ]e~qeikk(g~g0) , (C1) 0 0 b 0 where kk \ k Æ c and curvature has been neglected. The terms in parentheses contribute at last scattering due to weighting by the visibility functionV \ q5 e~q, and the integrated Sachs-Wolfe metric terms play a role between last scattering and the c present. This formal solution is made practical by replacing the sources# andVb with their acoustic solution at last scattering. 0 It may seem that employing the tight coupling solution through decoupling would lead to erroneous results. In particular, the damping approximation should break down if the optical depth through a wavelength drops below unity. However, let us examine the tight coupling solution more carefully.Equation (B1) implies P g0 Pg0 [# ] (](g , k, k) \ dg(#Œ ] ( [ ikVŒ )MV e~*k@kD(g)+2Neikk(g~g0)] dgR(Me~*k@kD(g)+2 [ 1NV eikk(g~g0) . (C2) 0 0 b c c 0 0 Although baryon drag e†ects such as acoustic displacement and enhancement are already encorporated in the acoustic solution#Œ (seeeq. [A16]), the residual R( term appears beneath the di†usion scale. Let us ignore this for the moment. The e†ective visibility0 for the acoustic oscillations is given by VŒ \ V e~*k@kD(g)+2 . (C3) c c This function is plotted for a given model inFigure 2. UnlikeV , VŒ is damped exponentially at late times by the growing di†usion length and thus peaks at earlier times. Note that a 10%c shiftc in redshift represents a factor of 3 in optical depth near last scattering. Thus, the region in which we expect the approximation to break down is given little weight in the integral. More speciÐcally, the exponential damping ensures that most contributions come from before the epoch at which the di†usion length surpasses the wavelength. As we have seen inAppendix A, the optical depth through a wavelength is high at this time and justiÐes the tight coupling expansion. The damping of acoustic modes through last scattering can occur in general due to two di†erent mechanisms working in equation (C2): di†usion and cancellation. However, the e†ects are of greatly unequal magnitude. Cancellation occurs because on small scales many wavelengths of the perturbation span the Compton visibility function. Photons that scattered last at the crests of the perturbations will have their e†ect canceled by those that scattered at the troughs. Mathematically this occurs in equation (C2) because the oscillating plane wave is integrated over the visibility function. Cancellation leads to a power-law damping of Ñuctuations as the scale decreases below the width of the visibility function. However, in the case of di†usion damped acoustic contributions, it is not the width of the Compton visibility functionV that is relevant but rather the acoustic visibility functionVŒ . As one goes to smaller and smaller scales (high k), the widthc of this function decreases as well. Thus, even at high k the cancellationc regime is never fully reached, and one may approximate the integral (C2) by replacing VŒ by a delta function, i.e., c (# ] ()(g , k, k) ^ (#Œ ] ( [ ikVŒ )(g , k)eikk(g*~g0)D (k) , (C4) 0 0 b * c where P g0 P g0 D (k) \ dgVŒ \ dgV e~*k@kD(g)+2 . (C5) c c c 0 0 l The observable anisotropy follows by decomposingequation (C4) into Legendre moments# \ &([i) #l Pl(k) and summing over k-modes Cl \ (2/n) / k3o #l(g ,k)/(2l ] 1) o2d ln k (HSa). Decoupling thus increases the e†ective di†usion damping length due to the corresponding increase0 in the mean free path of the photons. The result is a near-exponential damping with scale that overwhelms completely the small residual cancellation damping. FIG. 9.ÈResidual baryon drag e†ect after last scattering in an adiabatic CDM model. On scales under the width of the visibility function, cancellation between contributions that came from potential wells and hills at last scattering damps Ñuctuations from the baryon drag e†ect. Note that cancellation damping is weak and scales as(kg )~1@2, in contrast to the exponential di†usion damping. Projection relates the rms Ñuctuation in (a) to the anisotropy power spectrum in (b). * 562 HU & SUGIYAMA Vol. 471

Cancellation damping does occur for the baryon drag e†ect [R( inequation (C2) that remains after di†usion damping. The amplitude of the resultant Ñuctuations can be estimated by noting that Pg0 (# ] ()(g , k, k) \[ dgV R(eikk(g~g0) , (C6) 0 c 0 is approximately a Fourier transform. Employing ParcevalÏs theorem, we obtain

1 P 1 n P g0 n P g0 o # ] ( o2 4 o # ] ( o2dk ^ o R(V o2dg ^ o ((g , k) o2 (RV )2dg . (C7) 0 rms 2 k c k * c ~1 0 0 Thus, the contribution to the rms is suppressed by roughly(kg ) due to cancellation. Note that the residual baryon drag * ~1@2 e†ect only appears in models in which ( itself is not damped away by di†usion, e.g., adiabatic CDM and isocurvature BDM models. InFigure 9a, we display this e†ect. The amplitude of the e†ect is overestimated slightly because baryon drag weakens through last scattering. Its e†ect on the anisotropy is shown in Figure 9b and can be obtained analytically in a computa- tionally simple matter through the approximations ofHu & White (1996). Since it is unlikely to be observable due to e†ects in the foreground of last scattering, we omit a detailed calculation here.

C2. BARYON DECOUPLING The formal solution to the baryon Euler equation(A3) is P g aV \ dg@a(q5 # ] k()e~qd , (C8) b d 1 0 whereq5 d \ q5 /R and quantities in the integrand are evaluated at g@. The two source terms are the Compton drag e†ect and infall into potential wells. The former forces the baryon velocity to follow the photon dipole (velocity) at high drag optical depthqd. The presence of the scale factor a in the equation represents the fact that baryon velocities decay as a~1 in the absence of sources. Sinceq5 e~qd is very nearly a delta function with respect to variations on the expansion timescale, this d ] equation is conceptually identical to its photon analog(C1) with the replacementq5 q5 d. The plane-wave factor is absent for the baryons because their particle velocities are low and the streaming can be neglected in comparison to the wavelength. Drag depth unityqd(zd) \ 1 marks the transition between the drag and infall epochs. For)b h2 Z 0.03, the drag epoch precedes last scatteringz [ z assuming standard recombination. If the universe is reionized after recombination to some d * ionization levelx , z \ 263() h2)1@5x~2@5(1 [ Y /2)~2@5#~8@5 and the drag epoch signiÐcantly precedes last scattering in e d 0 e p 2.7 most scenarios.5 HereYp is the helium mass fraction Yp ^ 0.23. By analogy to the photon case, it is useful to deÐne the drag visibility function d aq5 d e~q Vb \ , (C9) /g0 dgaq5 e~qd 0 d suitably normalized to have unity area. Di†usion damping modiÐes the acoustic visibility function as k kD VŒ b \ Vb e~* @ (g)+2 . (C10) Thus, we expect that immediately after the drag epoch the baryon velocity and density perturbations are approximately V (g , k) \ #Œ (g , k)D (k), d(g,k)\[dü (g,k)[S(0, k)]D (k) ] S(0, k) , (C11) b d 1 d b b d b d b where

P g0 P g0 k kD Db(k) \ dgVŒ b \ dgVb e~* @ (g)+2 , (C12) 0 0 and recall that#Œ anddü were given in equation (B7). 1 b

APPENDIX D

MATTER EVOLUTION After horizon crossing but before the end of the drag epoch, baryons follow the acoustic solution ofequation (C11) and the CDM follows its own pressureless evolution. After the drag epoch, the baryon evolution equation(A3) is identical to the cold dark matter, and their joint evolution can be expressed in terms of Ñuctuations in the total nonrelativistic matter density dm.

5 This di†ers from the treatment ofHSb in whichzd was deÐned to be the epoch when the perturbation joined the growing mode of pressureless linear theory. The presence of a decaying mode lowers this redshift by a factor of3 (see Appendix D, eq. [D19]). 5 No. 2, 1996 SMALL-SCALE COSMOLOGICAL PERTURBATIONS 563

Thus, the solution for the time evolution of the matter Ñuctuations requires knowledge of both the baryon and CDM perturbations at the drag epoch. The baryonic contribution was obtained inAppendix C. Let us now evaluate the CDM contributions.

D1. EXACT CDM SOLUTIONS The evolution of CDM Ñuctuations is described byequation (B10). Since the curvature or " terms are negligible before the drag epoch, this equation can be rewritten in terms of the equality-normalized scale factory \ a/a as eq d2 (2 ] 3y) d 3 ) d ] d \ c d . (D1) dy2 c 2y(1 ] y) dy c 2y(1 ] y) ) c 0 Here we have assumed that the radiation contributions to the gravitational potential have decayed to zero well after horizon crossing. In typical adiabatic models, the CDM contribution usually dominates the nonrelativistic matter. Consider Ðrst the limit of negligible baryon fraction,) /) > 1. In this case, the matching condition at the drag epoch becomes trivial because b 0 the baryons have no e†ect on the CDM. If)c \ ) , equation (D1) has the same solution before and after the drag epoch (see Peebles 1980, eqs. [12.5] and [12.9]), 0 2 15 C(1 ] y)1@2 ] 1D 45 U \ ] y , U \ (2 ] 3y)ln [ (1 ] y)1@2 , (D2) 1 3 2 8 (1 ] y)1@2 [ 1 4 before curvature or " domination. Matching to the radiation-dominated solution(B12), we obtain C3 A a B 4 D dT(g, k) ^ dc(g, k) \ I '(0, k) ln 4I e~3 eq U (g) [ U (g) , (D3) 1 2 2 aH 1 15 2 for k ? k . Let useq now solveequation (D1) for the case in which the contribution of baryon is not negligible. The two independent solutions are given in exact form through GaussÏs hypergeometric function F by A 1 1 1 B U \ (1 ] y)~aiF a , a ] ,2a] ; , (D4) i i i 2 i 2 1]y where i \ 1, 2 and 1 ] J1 ] 24) /) a \ c 0 , (D5) i 4

i with minus and plus signs for i \ 1 and 2, respectively. Note thatlimy? Ui \ y~a . Thus, the main e†ect of the baryons is to slow the power-law growth of CDM after equality. = It is easy to show that these solutions are identical toequations (D2) for) \ ) . They also take on elementary forms for c 0 two other special cases: )c \ 0, 1 C(1 ] y)1@2 ] 1D U \ 1, U \ ln , (D6) 1 2 2 (1 ] y)1@2 [ 1 and ) \ 1) , c 3 0 3 C(1 ] y)1@2 ] 1D U \ (1 ] y)1@2 , U \ (1 ] y)1@2ln [ 3 . (D7) 1 2 2 (1 ] y)1@2 [ 1 In order to map the solution for the radiation-dominated limit(B12) onto amplitudes ofU andU , we have to take the limit as y ] 0 ofequation (D4). By using a linear transformation of the hypergeometric function1 (see,2 e.g., eq. 15.3.9 in Abramowitz& Stegun 1965), we Ðnd !(2a ] 1/2) C A 1BD lim U \ i [ln y ] 2t(1) [ t(a ) [ t a ] , (D8) i !(a )!(a ] 1/2) i i 2 y?0 i i where !(x) and t(x) \ !@(x)/!(x) are gamma and digamma functions, respectively. Matching to the radiation-dominated solution (B12) yields d (g, k) \ I '(0, k)[A U (g) ] A U (g)] , (D9) c 1 1 1 2 2 where !(a )!(a ]1/2) C A a B A 1BD A \[ 1 1 ln I eq ] 2t(1) [ t(a ) [ t a ] . (D10) 1 !(2a ] 1/2)[t(a ) ] t(a ] 1/2) [ t(a ) [ t(a ] 1/2)] 2 a 2 2 2 1 1 1 2 2 H 564 HU & SUGIYAMA Vol. 471

FIG. 10.ÈCDM evolution in the Compton drag epoch. If baryons contribute a signiÐcant fraction of the total matter density, CDM growth will be slowed between equality and the drag epoch. Held by Compton drag, the baryons do not contribute their self-gravity. For the numerical results, we choose a model that never recombined, so that a ? a . d eq

A is obtained by replacing the subscripts 1 % 2 ofA . Since)b/) ¹ 1, it is useful to approximate the coefficients with a series expansion,2 1 0 A a B A \ B ln I eq ] B , (D11) 1 1 2 aH 2 with

B \ 3M1 [ 0.568() /) ) ] 0.094() /) )2]O[() /) )3]N , 1 2 b 0 b 0 b 0 B \ 3(ln 4 [ 3)M1 [ 1.156() /) ) ] 0.149() /) )2[0.074() /) )3]O[() /) )4]N , (D12) 2 2 b 0 b 0 b 0 b 0 valid at the percent level for) /) \ 1, and b 0 2 !(a )!(a ]1/2) C !(2a ] 1/2) D A \[ 2 2 1 A ] 1 . (D13) 2 !(2a ] 1/2) !(a )!(a ] 1/2) 1 2 1 1 For the three special cases, we can described by elementary functions. If) \ ) , d is given byequation (D2), whereas c c 0 c C A a B D dc(g, k) \ I '(0, k) ln 4I eq U [ 2U )c \ 0, 1 2 aH 1 2 C A a B 2 D 1 dc(g,k)\I '(0, k) ln 4I e~2 eq U [ U )c \ ) . (D14) 1 2 aH 1 3 2 3 0 InFigure 10, we show the time evolution ofdc before the drag epoch for several di†erent values of)b/)c. Numerical results in this Ðgure are for fully reionized models so that the drag epoch ends well after equality, unlike other examples in this paper.

D2. MATTER TRANSFER FUNCTION With the baryon and CDM Ñuctuations at the drag epoch from equations(C11) and (D9), respectively, we can now solve for the evolution to the present. After the drag epoch, baryons behave dynamically as CDM, and the combined nonrelativistic No. 2, 1996 SMALL-SCALE COSMOLOGICAL PERTURBATIONS 565 matter Ñuctuations ) A ) B d \ b d ] 1 [ b d , m ) b ) c 0 0 ) A ) B V \ b V ] 1 [ b V . (D15) m ) b ) c 0 0 follow the growing and decaying solutions for dm (Peebles 1980), 2 D \ ] y , 1 3 15 C(1 ] y)1@2 ] 1D 45 D \ (2 ] 3y)ln [ (1 ] y)1@2 , (D16) 2 8 (1 ] y)1@2 [ 1 4 before curvature or " domination. To account for e†ects from curvature and " ata ? a or y ? 1, one needs simply to replace a ] D, where eq 5 P da@ D(a) \ ) g(a) , g2(a) \ a~3) ] a~2(1 [ ) [ ) ) ] ) , (D17) 2 0 [a@g(a@)]3 0 0 " " is the growing mode of radiationless linear theory normalized to equal a at early times. By matching the Ñuctuations at the drag epoch, we obtain for the growing mode D0 D d (g, k) \ [G (g )d (g , k) ] G (g )kV (g , k)]D (g), G(g)\ 2 , G(g)\ 2 , (D18) m 1 d m d 2 d m d 1 1 D D0 [D0 D 2 D D0 [D0 D 1 2 1 2 1 2 1 2 and similarly for the decaying mode. The combined time evolution is plotted inFigure 4 for a BDM model withdm \ db. If z > z , equation (D18) reduces to the familiar form d eq a C3 1 D dm(g, k) \ dm(gd, k) [ (kgd)Vm(gd, k) . (D19) ad 5 5 Notice that on scales much less than the horizon at the drag epoch, the velocity atgd dominates the growing mode if the two values are comparable atgd (seeFig. 11). This ““ velocity overshoot ÏÏ e†ect occurs because the peculiar velocity moves the matter and creates density Ñuctuations kinematically. Expansion drag on the velocity eliminates it in an expansion time gd, and thus causality prevents this e†ect from generating density Ñuctuation above the horizon at the drag epoch kgd > 1. It is conventional to recast the evolutionary e†ects in terms of a transfer function. As withequation (D15), we can break the present-day transfer function up into a baryonic and cold dark matter contribution at the drag epoch ) A ) B T (k) \ b T (k) ] 1 [ b T (k) . (D20) ) b ) c 0 0 It should be kept in mind thatTb andTc do not represent the respective transfer functions today. Let us consider the adiabatic transfer function. Here one expresses the evolution of small-scale Ñuctuations in terms of those at large scales, i.e., o dT(g ,k) o2 P T 2(k)P(k) with normalization limk? T (k) \ 1 and initial power spectrum P(k) P o k2'(0, k) o2. The large-scale solution0 is given by (seeHSa, eq. [A15]) 0 A 4 BA 2 B~1 6 A k B2 lim d (g, k) \ 1 ] f 1 ] f D (g)'(0, k) . (D21) T 15 l 15 l 5 k 1 k?0 eq The acoustic contribution of the baryon is therefore 15 A 4 B~1Ak B2 A D B K T (k) \ 1 ] f eq D (k)(1 ] R)~1@4 cos kr [ 2 kc sin kr G . (D22) b 4 15 l k b s D0 s s 1 2 g/gd This equation is compared with numerical results for the adiabatic transfer function in Figure 5a. For isocurvature BDM models,T \ Tb, and it is conventional to deÐne it such that o dT(g , k) o2 P T 2(k) oS(0, k) o2 with normalizationlim T (k) \ 1. From equation (B6), the small-scale tail of the transfer function becomes0 k?= 3J6 A 4 B k A D BK T (k) \ 1 [ 1 [ f eq D (k)(1 ] R)~1@4 sin kr [ 2 kc cos kr . (D23) 4 15 l k b s D0 s s 2 g/gd This function is plotted in Figure 5b. In the) /) ] 0 limit, the CDM contributions can be expressed in terms of elementary functions as b 0 A 2 BA 4 B~1 5 Ak B2 A a B ln 1.8q lim T (k) \ I 1 ] f 1 ] f eq ln 4I e~3 eq ^ f \ 0.405 , (D24) 1 5 l 15 l 4 k 2 a 14.2q2 l )b?0 H 566 HU & SUGIYAMA Vol. 471

FIG. 11.ÈVelocity overshoot e†ect. Below the horizon at the drag epochkgd ? 1, the acoustic velocity atzd dominates the growing mode and hence the Ðnal transfer function. Near the horizon, the acoustic density becomes comparable and shifts the zero points of the oscillation. whereq \ (k/Mpc~1)#2 () h2)~1. This should be compared with the high-k tail of the standard Ðtting function to the numerical results (BBKS),2.7 0 ln (1 ] 2.34q) T (q) \ [1 ] 3.89q ] (16.1q)2](5.46q)3](6.71q)4]~1@4 , (D25) BBKS 2.34q i.e.,limq? T (q) \ ln (2.34q)/15.7q2, which di†ers by D10% from the analytic prediction at small scales. In fact, since the Ðtting formula= was designed to Ðt intermediate scales,equation (D24) is more accurate at extremely small scales. If the baryon fraction is nonnegligible, the contribution is expressed in terms of hypergeometric functions through equation (D4), A 2 BA 4 B~1 5 Ak B2 T \ I 1 ] f 1 ] f eq [G (A U ] A U ) [ G (A U0 ] A U0 )] o . (D26) c 1 5 l 15 l 6 k 1 1 1 2 2 2 1 1 2 2 g/gd Though exact, this expression is rather complicated. It is useful and instructive to seek a simple scaling relation for this form. Notice that for all cases the scale dependence of the transfer function may be written as ln 1.8bq T ^ a , (D27) c 14.2q2 ] forf \ 0.405. Here a and b are functions of) h2 and)b/) . Note that as q O the modiÐcation due to b becomes insigniÐcant.l A very accurate Ðt to both a and b is0 given inAppendix0 E, equations (E12) and (E13). The numerical, analytic, and Ðtted analytic results are compared with the empirical scalings ofPeacock & Dodds (1994) and Sugiyama (1995) in Figure 12. The analytic calculation is essentially exact while the Ðtted analytic form works to 1% accuracy. Notice that in this extreme case we have improved signiÐcantly upon previous results. Equation (D27) breaks down for intermediate to large scales. The CDM contribution can be scaled approximately from the BBKS form as k T (k) ^ T (q8 ), q6(k)\ a~1@2() h2)~1#2 . (D28) c BBKS Mpc~1 0 2.7 This expression is employed inFigure 5 with the coefficient 6.71 in equation (D25) replaced by 6.71(14.2/15.7) \ 6.07 to match the analytic small-scale tail. Notice that at the largest scales, this expression underestimates the matter transfer function. This No. 2, 1996 SMALL-SCALE COSMOLOGICAL PERTURBATIONS 567

FIG. 12.ÈAdiabatic CDM transfer function in a high)b/) \ 2 case. The analytic solution is essentially exact in the small-scale limit. Simple Ðts based on the BBKS form can cause large errors at the small scale(Peacock0 3 & Dodds 1994; Sugiyama 1995). The Ðtting function developed here (see equation [D27]) works at the 1% level, even for this extreme case. is because baryon contributions must be included properly. Although the limiting formlim T \ 1 is simple, the behavior k?0 b near the horizon scale atzd is not. Since this region is not the main focus of this work, we do not attempt to describe this analytically. If the baryonic oscillations are small or smoothed over, an approximate patch is given by k A ) B~1@2 T (k) ^ T (qü ), qü(k)\ 1 [ b a~1@2() h2)~1#2 , (D29) BBKS Mpc~1 ) 0 2.7 0 which extends thePeacock & Dodds (1994) approach to high )b/) . Finally, we can express the ratio of the acoustic peak heights to the0 CDM tail with equations(D20), (D22), and (D24), ) T J3 A 2 B~1C A a BD~1 lim T b b ^ kG (g )[1 ] R(g )]~3@4D (k) 1 ] f ln 4I e~3 eq )b?0 c , (D30) ) T I 2 d d b 5 l 2 a T c c 1 H c if the velocity overshoot e†ect dominates the acoustic contributions. We can simplify this expression by noting that G (g ) ^ 2 d 2() H2/a )~1@2(1 ] ad/a )~1@2. Furthermore, the functionkDb peaks at roughly0.8k with an amplitude of0.4k . With the scaling5 0 0 ofequationeq (D27),eq the peak relative amplitude of the acoustic oscillation is approximatelyS S ) k a C A 1.8b k BD~1 70 b S eq [1 ] R(g )]~3@4() h2)~1@2a~1 ln S . (D31) ) Mpc~1 (a ] a )1@2 d 0 ) h2 Mpc~1 c eq d 0 The logarithmic term is roughly unity and may be dropped for estimation purposes (seeeq. [9]). Equation (D31) roughly quantiÐes the prominence of the acoustic oscillations in a CDM model. For best accuracy, however, the solutions(D22) and (D26) for the baryons and CDM respectively should be employed.

APPENDIX E

RECOMBINATION FITTING FORMULAE Rather than recalculate the atomic physics of recombination each time one needs to consider e†ects at the last scattering and drag epochs, it is convenient to have accurate Ðtting formulae that incorporate the ionization history. In general, all quantities associated with the ionization history must be functions of) h2 and)b h2 alone once the CMB temperature T \ 2.726K (Mather et al. 1994), neutrino fractionf \ 0.405, and helium0 fractionY ^ 0.23 are Ðxed. Fitting functions in 0 l p 568 HU & SUGIYAMA Vol. 471 this Appendix are designed to be valid at the percent level for an extended range of parameter space, 0.0025 [ )b h2 [ 0.25 and0.025 [ ) h2 [ 0.64, and consequently they appear rather complicated. We employ a recombination calculation based on the improvements0 discussed inHu et al. (1995). The last scattering epoch is a very weak function of parameters and is given by

z \ 1048[1 ] 0.00124() h2)~0.738][1 ] g () h2)g2], * b 1 0 g \0.0783() h2)~0.238[1 ] 39.5() h2)0.763]~1 , 1 b b g \ 0.560[1 ] 21.1() h2)1.81]~1 . (E1) 2 b The drag epoch ends at a related redshift that depends somewhat more strongly on the parameters () h2)0.251 z \ 1345 0 [1 ] b () h2)b2], d 1 ] 0.659() h2)0.828 1 b 0 b \0.313() h2)~0.419[1 ] 0.607() h2)0.674], 1 0 0 b \0.238() h2)0.223 . (E2) 2 0 The two are approximately equal if )b h2 ^ 0.03. The di†usion damping envelope can be approximated through the Ðrst decade of damping by the form m D (k) ^ e~(k@kDc) c , (E3) c where the e†ective damping scalek has simple asymptotic scaling, Dc k GF ) h2 ? 0.1 , Dc \ 1 b (E4) Mpc~1 () h2)p1F ) h2 > 0.1 , b 2 b with F \ 0.293() h2)0.545[1 ] (25.1) h2)~0.648], 1 0 0 F \0.524() h2)0.505[1 ] (10.5) h2)~0.564], 2 0 0 p \0.29 . (E5) 1 The basic scaling in the low)b h2 limit can be understood by the Saha approximation in which the ionization fraction scales approximately asx P () h2)~1@2. Thus, the di†usion lengthj D k~1 D (g /q5 )1@2 P g1@2() h2)~1@4. Sinceg P () h2)~1@2 in e b D Dc * * b * 0 the matter-dominated high) h2 limit, this is approximately of the same form asequation (E5). For high)b h2, the corrections from an accurate treatment of0 the atomic levels become more important due to the high Lya opacity. These two simple limits can be joined accurately by a rather artiÐcial looking but highly accurate form, p p p p kD G2 Cn AF B 2@ 1 DH 1@ 2 c \ arctan 2 () h2)p2 F , Mpc~1 n 2 F b 1 1 p \ 2.38() h2)0.184 . (E6) 2 0 ] FromFigure 2, we see that this overestimates the true damping as)b h2 0 due to a breakdown of tight coupling. It is interesting to note that the full numerical results suggest that the Saha estimation ofp ^ 0.25 in equations (E5) and (E6) is a somewhat better phenomenological Ðt. 1 The steepness indexm of the di†usion damping envelope is a very weak function of cosmological parameters. In the limit that last scattering occurredc instantaneously,m ] 2. The Ðnite width of the visibility function modiÐes this as c m \ 1.46() h2)0.0303(1 ] 0.128 arctan Mln [(32.8) h2)~0.643]N) , (E7) c 0 b which only varies by D10% across the full range of parameter space. Silk damping for the baryons can likewise be approximated by k k ms Db(k) ^ e~( @ S) , (E8) with k 1 ] (96.2) h2)~0.684 S \ 1.38() h2)0.398()b h2)0.487 0 , (E9) Mpc~1 0 1 ] (346)b h2)~0.842 and the steepness index by () h2)~0.0297() h2)0.0282 m \ 1.40 b 0 . (E10) S 1 ] (781)b h2)~0.926 As is the case with the photons, the latter accounts for the width of the visibility function and is almost independent of cosmological parameters. No. 2, 1996 SMALL-SCALE COSMOLOGICAL PERTURBATIONS 569

TABLE 1 COMMONLY USED SYMBOLSa

Symbol DeÐnition Equation

# ...... CMB temperature perturbation (A1) # ...... CMB monopole perturbation (A1) #0 ...... V -CMB dipole perturbation (A1) 1 c %T ...... Anisotropic stress perturbation (A5) ' ...... Curvature perturbation (A5) ( ...... Newtonian potential (A5) )i ...... Critical fraction in i (1) a ...... Baryon T -suppression (E11) a a ...... U U power law (D5) b1...... 2 Baryon1 2 T -log correction (E11) g (gt) ...... Conformal time (at t) (A1) di ...... Density perturbation in i (8) q ...... Compton optical depth (A1) qd ...... Drag optical depth (C8) u ...... Acoustic frequency (A13) D ...... Photon damping factor (C5) c Db ...... Baryon damping factor (C12) V ...... Compton visibility (C1) VŒ c ...... CMB acoustic visibility (C3) c Vb ...... Drag visibility (C9) VŒ b ...... Baryon acoustic visibility (C11) A A ...... Boost CDM matching condition (D10) 1 2 CA ...... Adiabatic acoustic amplitude (B5) CI ...... Isocurvature acoustic amplitude (B6) D ...... Radiationless growth (D17) D D ...... Matter-radiation growth (D16) G1 G2 ...... Drag matching condition (D18) I 1I 2...... CDM boost integrals (B14) U1 U2 ...... Drag CDM growth (D4) 1 2 R (Rt) ...... b/c momentum (at t) (A2) T ...... Transfer function (D20) Tb ...... Baryon T drag contribution (D22), (D23) Tc ...... CDM T drag contribution (D28) Vi ...... Velocity in i (A1) a (at) ...... Scale factor (at t) (A1) aH ...... Horizon crossing a (B13) cs ...... Photon-baryon sound speed (A2) f ...... Neutrino fraction (B2) kl ...... Wavenumber (Laplace eigenvalue) (A1) kD ...... Di†usion wavenumber (A14) kD ...... CMB damping wavenumber (A14), (E6) k c...... Silk damping wavenumber (6), (E9) kS ...... Equality horizon wavenumber (B2) keq ...... jth photon peak wavenumber (2) cj kbj ...... jth baryon peak wavenumber (5) m ...... CMB damping steepness (4), (E7) mc ...... Baryon damping steepness (6), (E10) q S...... Scaled wavenumber (D25) rs ...... Sound horizon (A2) y ...... a/a (D1) z ...... Lasteq scattering redshift (E1) * zd ...... Drag redshift (E2) z ...... Equality redshift (1) eq a Fluid elements are i \ b, c, c, l, m, r for the baryons, CDM, photons, neutrinos, total nonrelativistic matter, and total relativistic matter, respectively. Special epochs include t \ \, d, eq, H for last scattering, the drag epoch, matter-radiation equality, and horizon crossing, respectively. Overdots represent conformal time derivatives. Finally, by employingequation (E2) for the drag epoch, the cumbersome analytic result for the CDM drag contribution to the small-scale transfer functionT \ () /) )T ] () /) )T fromequation (D26) can be Ðtted as b 0 b c 0 c ln (1.8bq) T ^ a , (E11) c 14.2q2 with

a \ a~)b@)0a~()b@)0)3 , 1 2 a \ (46.9) h2)0.670[1 ] (32.1) h2)~0.532], 1 0 0 a \(12.0) h2)0.424[1 ] (45.0) h2)~0.582] , (E12) 2 0 0 570 HU & SUGIYAMA as the suppression factor and

b~1 \ 1 ] b [() /) )b2 [ 1] , 1 c 0 b \ 0.944[1 ] (458) h2)~0.708]~1 , 1 0 b \ (0.395) h2)~0.0266 , (E13) 2 0 as the correction to the logarithm. A table of these and other commonly used symbols is provided in Table 1.

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