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Progress of Theoretical , Vol. 90, No.1, July 1993

A Simple Model of Cosmological

Shin SASAKI, Fumio TAKAHARA and Yasushi SUTO* Department of Physics, Tokyo Metropolitan University, Hachioji 192-03 * Uji Research Center, Yukawa Institute for

Kyoto University, Uji 611 Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021

(Received February 8, 1993)

A simple scenario of cosmological reionization is presented. In hierarchical clustering models of , numerous nonlinear objects on small scales would collapse much earlier than a z~ 10. Our scenario assumes that such objects dissipate their kinetic efficiently into and eventually provide copious ionizing which reionize the almost instantane­ ously. We derive a simple estimate of the energy conversion efficiency required for the full reioniza­ tion. Then we apply the formula to a specific hierarchical clustering model in which the power of fluctuation obeys a single power-law p(k)rxkn approximately on scales of interest (at standard recombination epoch). After comparing the resulting production rate of ioniz­ ing photons with cooling rates of the relevant processes, we find that models with the spectral index n 2 -1 can potentially reionize the universe at z 210. For flatter spectra, the universe remains effectively neutral. This index is somewhat larger than the conventional value around -2 on galactic scales in a dark universe, but is preferred phenomenologically in primordial isocurvature models, for instance. Therefore it is possible to achieve a reionization· of the universe naturally due to the energy release associated with the structure formation. We discuss briefly the implications of our scenario, in particular on the and distortion of the cosmic background, and on a characteristic of .

§ 1. Introduction Gravitational of cosmic structures has been extensively examined over last ten years, and remained one of the central issues in . In particular, linear theory of density fluctuations and numerical simulations of nonlinear structures have elucidated a of quantitative features of the underlying cosmological 1H scenarios. ) Thermal history of the universe, on the other hand, has been studied so far in a fairly crude manner and/or in less plausible situations. Nevertheless the details of the thermal evolution of the universe, in particular after the standard recombination at a redshift Zrec ~ 1000, would have important consequences on the observable present-day universe. A conventional picture assumes, to a first-order approximation, that start to evolve gravitationally after Zrec and their subsequent processes to form galaxies have negligible effects on the rest of the universe. However, there are several why one should critically examine the above simple picture and consider the possible reheating and reionization of the universe at Z< Zrec seriously. Most notably the absence of the observable Gunn-Peterson trough in spectra of high 7 redshift ) places a strong upper limit on the abundance of neutral , implying that the universe was reionized before Z~4.8) If the reionization of the universe occurred early enough, it would substantially affect the spatial correlation and the energy spectrum of the cosmic microwave background (CMB). Furthermore, such a reionization will possibly affect the mass function of collapsed objects and the 86 S. Sasaki, F. Takahara and Y. Suto form of collapsed objects including the characteristic mass of galaxies. Thus such a feedback effect might even change the large-scale distribution of galaxies.9 ),) There are a number of papers which discussed possible models of reionization including heating from gravitationally collapsed objects/I) ionizing from quasars and Pop. III starsIO ) and decaying exotic .!2),I3) CouchmanI4) and Stebbins and SilkI5) also discussed a possible thermal history of the universe before z = 10. While these papers have treated some aspects of the reheating and reionization of the universe, there is not yet a consensus on the mechanism and the epoch of Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021 reionization. Since COBE have shown that spectral distortion of CMB 6 is very small/ ) the of the intergalactic matter should have remained to be low, while the degree should be very close to unity. This fact suggests that the universe has been ionized radiatively. However, there are no convincing sources of the ionizing photons before redshift of a few. The purpose of this paper is to discuss a possibility that ionizing· photons have been supplied by the structure formation itself which we know has occurred between the standard recombination and the present. First, we consider general requirements for the production rate of ionizing photons for the cosmological reionization after standard recombination. Then we examine to what extept this amount of ionizing photons is expected in a hierarchical clustering model of gravitational instability picture. To be more specific, we assume a power spectrum of density fluctuations obeying a single power-law, and compute the dissipation rate of the kinetic energy associated with the growth, collapse and virialization of the nonlinear gravitating objects. It is true that there are a number of astrophysically important processes which are complicated and in fact interrelated, and therefore a simplified order-of­ magnitude argument might be quantitatively incorrect. Nevertheless such general consideration is of great value in understanding clearly the basic picture of the thermal evolution of the universe.

§ 2. Requirements on the production rate of ionizing radiation

First consider several requirements for the cosmological reheating and reioniza­ tion in a somewhat general context. For simplicity of the argument, we assume that the universe consists purely of , neglecting for the -being. The recombination rate per unit volume is given by

(1) where x is the ionization degree and n is the number density of hydrogen . The latter is given in terms of the baryon density parameter Q b and the Hubble constant Ho=100 hkms-IMpc-I as

(2)

Neglecting the recombination to the ground state, we have

(3) A Simple Model of Cosmological Reionization 87

The ionization rate per unit volume, on the other hand, is given by (4) where < > denotes the average over the energy spectrum, nr is the number density of ionizing photons, c is the velocity, and O" is the ionization cross section at the II of ionizing photons:

(5) Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021 with lit being the Rydberg frequency. When the mean free time of an ionizing photon is much shorter than the expan­ sion time scale of the universe, it is convenient to introduce the production rate of ionizing photons per unit volume nr . In this case, every ionizing photon ionizes a neutral hydrogen instantaneously and nr should be equal to the ionization rate n!on. When the mean recombination time of hydrogen atoms is also smaller than the expansion time of the universe, the ionization degree is computed assuming the ionization equilibrium nrec= nr. These approximations turn out to be valid for z > 10 for most cases in the following discussion. Under these approximations, we have

x a(T)1/2n . (6)

Thus a full ionization (x = 1) is achieved for (7) where T4= T/(104K). If all ionizing photons have the same energy tr, the correspond­ ing energy input rate is

34 2 Er=trnr.c=7.2 x 10- ( 13.~reV ) T4 -1/2Qb h4(1 + z)6ergocm-3s-1 . (8)

Let us define the energy production efficiency 7J per hydrogen atom per one cosmic expansion time: (9) where we adopt an expression for the cosmic expansion time scale:

t -I dt 1__ 1 1 (10) e- dln(l+z) - Ho jJ20(l+z)3-Ko(l+z)2+Ao ' where Qo, Ko and ,,10 denote the density parameter, parameter and ­ less , respectively. Then one obtains a complete ionization if

> =13X10-8 T,-1/2( t r ) . Qbh(1+z)3 (11) 7J-7Jc. 4 13.6eV jQo(1+z)3-Ko(1+z)2+Ao'

It is to be noted that an energy release 13.6 e V per hydrogen atom and per expansion 8 time scale corresponds to 7J = 1.5 X 10- , which is the minimum value to ionize a hydrogen atom. The above estimate of the required efficiency (11) should be regard- 88 S. Sasaki, F. Takahara and Y. Suto ed as a lower limit to completely reionize the universe since we have neglected so far the amount of energy required to keep the temperature at about 104K and since we have assumed that all the produced energy is transferred to that of ionizing photons. On the other hand, instantaneous ionization approximation should break down once the recombination time exceeds the expansion time scale teo This epoch roughly corresponds to Z~( T4QO/(Qbh)2)1!3, which is lower than 10 for most cases. Thus, after this epoch the ionization can be maintained even when the production rate of ionizing photons is lower than the value given by Eq. (7). Conversely, if 7] is given in a specific model, Eq. (11) gives the redshift of reioniza­ Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021 tion Zreion. Since we consider early epoch when te can be approximated by Ho -1!20 -112(1 + Z)-3/2, the reionization of the universe is possible only at

(12)

Of course, early reionization requires a larger 7]. Consider a case 7]=10-6 and T4 =1 for instance; a cold universe with Qo=l, Qb=0.03 and h=0.5 can be potentially reionized at z~300. For a dominated universe with !2o=O.l, ,10 =0.9, h=l and Qb=O.Ol (a theoretically ugly but observationally favored of 7 parameters), reionization would occur at z~ 180. Even for 7]=10- , the critical redshift becomes 65 and 39 for the above two cases, respectively. Thus, the central question here is if there are any physical processes which provide 7] at least as large 7 8 as 10- • If 7] is smaller than about 10- , it is impossible to reionize the universe at Z ~ 10 for a realistic set of cosmological parameters. Let us check the energy balance of intergalactic matter as well in order to show that an extra heating rate to keep the temperature at around 104 K is not s6large. In this paper, we shall treat photo-ionization of intergalactic matter, then it is expected that the temperature is not so high. Thus, we consider temperature range as 103~ 105 17 K. The recombination cooling rate to excited states ) is given by (13) which is an less than the required energy input rate given by 18 Eq. (8). The collisional excitation cooling rate ) takes (14) which thus makes a significant contribution only if the is partially ionized. This rate, however, is completely negligible as the ionization degree x approaches unity. 18 The collisional ionization cooling rate ) is given by (15) which is for most cases less than the excitation cooling rate. Bremsstrahlung cooling 17 rate ) is

18 which is comparable to the recombination cooling rate. Compton cooling ) A Simple Model of Cosmological Reionization 89

for Tro=2.735 K and T~ Tr (17)

2 is important at high redshift and exceeds Er for z > 1150 T4 -3/2Qb h • Thus, we can conclude that in most plausible situations the amount of energy input required to maintain T ~ 104 K is not so large as that of ionizing photons (Eq. (8». The exception is the Compton cooling at high redshift for a very low Qb; Thus, Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021 if Q b is as small as 0.01, then the Compton cooling makes the temperature lower than 104 K for z> 10. In this case correspondingly larger 7J is needed to ionize the inter· galactic matter. It is also noted that if the temperature is much higher than 104 K radiative cooling rate increases much and the energy requirement becomes more severe. A merit of photoionization lies in the fact that temperature can be kept low, maybe even lower than 104 K.

§ 3. Energy input due to collapsing objects

The value of 7J should be estimated on specific cosmological scenarios. If the sources of ionizing photons are of baryonic origin, there are three possible sources; massive , onto black holes and gravitational energy release of density 4 fluctuations. The first possibility has been proposed by many authors (e.g. Couchmad ); Songaila, Cowie and Lilly19). If one percent of baryonic matter is converted to massive stars and 0.1 percent of the rest mass energy is converted to ionizing photons, 5 7J may be as large as 10- • Although this is rather an optimistic estimation, it is not . inconsistent with observed of absorption . However, since the fraction of ionizing photons in emitted radiation seems to be orders of magnitudes lower than the total and less massive stars do not contribute to ionizing radiation, the actual value of 7J may be much smaller. The second possibility would be important at the formation epoch of quasars. Quasars do emit much ionizing radiation, but still it is argued that it is short of the required amount of ionizing flux at redshift of a few to pass the Gunn-Peterson test (e. 20 21 22 g. Shapiro and Giroux l; see an alternative result Madau ) and Terasawa ). The mass density of the black holes in quasars in the present universe is ·estimated as 5 3 35 3 23 about 10 M",Mpc- or 10- gcm- • ) For a typical efficiency of 10 percent, 7J at the 7 2 present epoch is 10- IQbh • Thus, cosmological reionization may be achieved by this mechanism provided that black hole formation occurs at high . ll The final one was discussed by Hogan ) from the viewpoint of the direct dis­ sipative heating of intergalactic matter due to hierarchical clustering. Let us briefly summarize his estimate of the available heat generation rate. First we assume that the power spectrum of density fluctuations at the cosmic recombination epoch is given by a power-law with a spectral index n:

(18)

where k denotes a comoving wavenumber. Linear theory implies that mass fluctua­ tions grow in proportion to 1/(1 +z) for Z;;:::Qo-l. If one assumes that structures are 90 S. Sasaki, F. Takahara and Y. Suto formed when their average mass density fluctuations 0 reach unity and applies linear theory up to that epoch, one obtains the mass of objects which collapse at a redshift Zc:

(19) where Ma is the mass of objects which collapse at the present epoch. For n> -3, smaller objects collapse earlier. Corresponding velocity perturbation or the velocity dispersion of the collapsed objects at Zc is given by Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021 V= va(l + zc)(n-1)/2(n+3) , (20) and this remains constant up to the present epoch unless significant dissipation would occur. In the above two equations, all the model dependence is represented in Ma and Va. Note that strictly speaking Ma and Va are not the same as the mass and velocity scales at the present epoch except the Einstein- (for low density· we should use Ma and Va somewhat different from those defined at the present.) The dissipation rate per unit volume is very roughly given by

2 31 1 2 3 3 1 T= nmpv /te= 1.4 X 10- Q bQa / h ( 500 ~~s 1 Y(1 + Zc)(lln+25)/(2n+6)ergocm- s- . (21)

Then the efficiency parameter 7J in this case (Eq. (9» reduces to

7J = v 2 /c 2 = 2.8 X 10-6( Va )2(1 + Z )(n-1)/(n+3) (22) 500 kms 1 c , if all the dissipated energy is converted to the production of ionizing photons. Hoganll) assumed that all the kinetic energy is available to the heating of the intergalactic matter and found that the gas can be partially ionized. Our model here takes account of the possibility that dissipative heating occurs inhomogeneously, and thereby collapsed objects are once heated to the virial temperature and cool by emitting ionizing radiation. A simple scenario of such processes will be described in the next section, where we will show that at the redshift

4 ( 0.01 )( Va )2](2n+6)/(n+ll) (23) 1 + Z< [ 2.1 X 10 &0 Qbh 500 kms 1 . '

7J can exceed the minimum efficiency given by Eq. (11) and the universe can be fully ionized. The redshift where this condition is satisfied is shown in Fig. 1 for several parameters. As is seen, a large Va and a steeper index n ~ - 1 are required. If Va is as small as 100 kms-I, the critical redshift of the reionization turns out to be less than ~10 for n~O in general, and even for the most favorable parameters it is ~40. Alternatively, if Va is as large as 600 kms-I, the universe can be reionized at redshift greater than 10 for Qb~O.l with n~ -1. For example, for a cosmological model h =1, Qa=O.l and Qb=O.Ol, the critical redshift Zreion is 149 and 39 for n=O and n = -1, respectively. For Qa=Qb=O.l, Zreion turns out to be 16 for n=-1. Because the amplitude of velocity perturbations decreases with redshift for n < 1 A Simple Model of Cosmological Reionization 91

according to Eq. (20), the efficiency of no nb h reionization due to dissipative heating is -- 1.0 0.01 0=0 0.2 0.1 relatively low at redshifts greater than -- --- 0.2 100. Note that the spectral index n is here defined at the recombination epoch, 100 and thus should be in the range of 1 ~ - 3 depending on the scales of interest for the primordial Harrison-Zel'dovich spec­ Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021 trum: the observed structure at the pres­ ent universe favors n= -1 around the 10 clustering scales of galaxies. The velocity dispersions (Eq. (20» at the present universe are fairly uncertain, but are likely in the range of 300 - 1000 1 +Zreion kms-1 again depending on the scales considered. The above estimate indi­ o 200 400 600 800 1000 cates that cosmological reionization ac­ Vo (km/sec) Fig. 1. The redshift of reionization in the hierar· companying structure formation is pos­ chical clustering scenario plotted against the sible at a redshift greater than 10 pro­ velocity dispersion at the present epoch Vo. vided that dissipated heat is efficiently , dashed and dot· dashed lines indicate converted to ionizing photons for an results for ($20, Qb h)=(1.0, 0.01), (0.2, 0.01) and appropriate initial spectrum of density (0.2, 0.1), respectively. In each parameter set, we plotted three curves corresponding to the fluctuations. This efficiency is esti- spectral index of density fluctuations at the mated in the next section. recombination epoch n=O, -1 and -2. It is also noted that the heat input sources are formed from fluctuations which have a larger amplitude than an average value and tend to have large peculiar velocities. A distinction of collapsing objects from general intergalactic matter is another topic which should be examined. To estimate these details we need to examine detailed statistical models and nonlinear evolution, which will be described elsewhere.

§ 4. Ionizing photons from collapsing objects

In the previous section, we simply estimated the heating rate due to structure formation. In order to ionize the intergalactic medium, however, the heat input should be converted to ionizing photons very efficiently. We assume that virialized objects contain a high temperature corresponding to the depth of the gravita­ tional potential. We assume that ionization of baryonic matter in virialized objects is made instantaneously by shocks. Of course, detailed numerical calculations are needed to obtain the resultant thermal and ionization state of shocked matter. However, because most dissipated energy is radiated while the temperature is high just behind the shock, the above assumption seems to be useful as a first order approximation. Comparing several cooling rates summarized in § 2, the Compton cooling dominates over other processes, which means that most of the dissipated energy would to a significant spectral deformations of CMB. Taking into 92 S. Sasaki, F. Takahara and Y. Suto account the clumpiness effect, however, this would not be the case in realistic situa­ tions. For virialized objects, the density is roughly two orders of magnitudes larger than the average density. Thus bremsstrahlung cooling can dominate over the Compton cooling there. Bremsstrahlung at mildly high , such as T 5 7 =10 - K, is, in fact, an efficient source for ionizing radiation. Remaining part of intergalactic medium can be photoionized by receiving an intense flux of ionizing radiation from virialized sources. In order for this to be possible, formation efficiency is required to be very low, and structure formation should occur hierarchically in a gaseous form. If a substantial part is converted to Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021 stellar form, subsequent clustering would not supply a sufficient amount of ionizing photons. Of course, if only massive stars are formed to provide copious ionizing photons, reionization proceeds much faster as was mentioned in the previous section. The same is true if there occurs the formation of incipient black holes which eventu­ ally become active galactic nuclei. While these processes can assist or even domi­ nate the cosmic reionization in principle, we do not discuss further these model­ dependent possibilities. Using the spherical collapse model of a homogeneous , the number density 24 of hydrogens in a collapsed object at a redshift zc ) is (24) and the virial temperature of object is

T =3.1 X 107(.Qoh2)1/3( 10iYM0 /,3(1 + zc)(n-l)/(n+3)K , (25)

2 which is defined via 2x(3/2)(MkB T)/(0.5 mp)=(3/5)GM /Rvir for a fully ionized homogeneous sphere, where Rv1r is the virial radius of the object. This temperature is greater than 13.6 e V, the ionization potential of a hydrogen atom, if AI, )2/3 ](n+3)/(I-n) 1 + Zc < [ 200 ( 1015 M0 (Qoh2)1/3 . (26)

Thus, for n=O or n= -1, hydrogen atoms' in collapsed objects can be ionized once during the virialization process at sufficiently high redshifts. In contrast, for n= -2, this occurs only at recent epochs and they do not provide sufficient ionizing photons. Ionized matter in the collapsed objects cools by bremsstrahlung and Compton scattering off microwave background photons. Their cooling time scales are esti­ mated as 2 =8 3X1017 (IJoh )1I6 ( Mo )2/3(1+Z )-(5n+19)/(2n+6)s (27) t brems '. 15 c (Qb h2) 10 M0 and (28) These should be compared to the expansion time scale of the universe. At redshift greater than 10, both cooling are shorter than the expansion time scale and dissipated energy is radiated away. Compton cooling time scale is shorter than the A Simple Model of Cosmological Reionization 93 cosmic expansion time scale if (29)

Compton cooling dominates over the bremsstrahlung cooling at redshift greater than

(30)

If Qb h2 is larger than about 0.05, bremsstrahlung dominate over Compton cooling at Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021 relevant redshifts. If Q b h2 is (0.01 ~0.02) as is indicated by the primordial nu­ cleosynthesis argument (e.g., Olive et al. 25»), Compton cooling always dominates over bremsstrahlung cooling. For the latter case the emission of ionizing photons is less efficient. However, the ratio of bremsstrahlung cooling to Compton cooling changes rather mildly with redshift for n= -1, a fair fraction of the thermal energy can still be radiated by bremsstrahlung. Moreover, while the above estimate assumed a homogeneous sphere, possible inhomogeneities can enhance bremsstrahlung cooling at least in denser regions. Actually at modest redshifts, cooling time is less than the expansion time scale (which is also comparable to the dynamical time scale of the collapsed objects) so that the collapse should be dissipative making baryon distribu­ tion more centrally condensed and the number density in the central region becomes greater than the above estimate. So we may expect that a fair fraction of the dissipated energy can be radiated in a form of ionizing photons. Thus, the efficiency for the production of ionizing photons 7J should be close to that given by Eq. (22), although it is very difficult to make a more quantitative estimate analytically. Since the cooling time is shorter than the expansion time scale which is roughly the same as the dynamical time scale of the collapsing objects, collapse is necessarily dissipative and realistic emission process is highly complicated. Some of the ionizing photons are absorbed by remaining neutral hydrogen in the collapsing objects. However, the collapse is likely to be highly inhomogeneous in hierarchical clustering scenario and thus majority of ionizing photons are expected to leave the source unabsorbed. The production rate of ioniz­ ing photons per unit volume can be calculated if the collapse rate of astronomical objects is given. We assumed that the distribution of ionizing sources is homogeneous and neglect­ ed possible effects of the discreteness of the sources; each ionizing source makes a HII region around it and the intergalactic gas is completely ionized when such HII regions 26 overlap. Such effects in cosmological situations were investigated by Shapir0 ) for the case of reionization by quasars. His conclusion was that a full ionization is possible if the ratio of the total number of ionizing photons emitted per expansion time scale to the total number of hydrogen atoms exceeds a few tens. This condition is similar to our estimate given by Eq. (11), which suggests that the overall features can be reproduced under the assumption of homogeneous source distribution.

§ 5. Discussion

Let us explore further consequences of our scenario and their astrophysical 94 S. Sasaki, F. Takahara and Y. Suto implications. First we briefly touch upon the ionization at recent past probed by Gunn-Peterson test. For a rough estimate, assuming that the mean free time of ionizing photons is comparable to the expansion time at around z=5, we estimate the ionizing flux from Eqs. (8) and (10). The resultant flux is very similar to the common­ ly adopted flux to explain the absence of the Gunn-Peterson trough 10-21ergs-1 cm-2 sr-1 Hz-1 for Qbh 2 =0.01. This coincidence suggests that dissipation processes in hierarchical clustering have played some roles in cosmic reionization. Although we have neglected the effects of helium, bremsstrahlung emission at reasonable tempera­ ture produces copious ionizing photons for helium and we expect that ionization of Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021 helium is also possible. Recent numerical simulations of structure formation have explored thermal properties of intergalactic matter as well27) and have shown that a part of intergalactic matter is ionized collisionally and radiatively, but, that the obtained ionization degree is too low to explain observations concerning the Gunn­ Peterson test. However, most simulations deal with rather recent epoch and seem to lack enough resolution of subgalactic scale. Simulations evolved from a high ­ shift and with higher spacial resolution would provide very detailed tests for our scenario. Secondly, we discuss a characteristic mass of galaxies. In the hierarchical clustering scenario, the cooling time scale becomes longer than the expansion time 15 scale below the redshift about 10. If we adopt n= -1 and Mo=10 M0 , the mass of the 12 objects forming at z=10 is 10 M0 which corresponds to a characteristic mass scale of galaxies. This may not be a pure coincidence. After z= 10, baryons in the collapsed objects only slowly cool and it takes much time to form stars, while before z=10 condensations can be formed efficiently. This feature of the existence of the charac­ teristic mass is independent of the reionization of intergalactic matter and has been 28 noticed earlier. )-30) This argument does not necessarily mean that the majority of stars have been formed before z=10. Most baryonic have remained to be in a gaseous form while experiencing collapse and virialization on successive mass scales. Radiative cooling, however, can make baryon distribution more condensed than the dark matter and give such objects a unique mass scale. Then we may identify 1012 M0 with the maximum mass of galaxies. On a mass scale more than 12 10 M0 , a collapsed object should consist of less massive objects which do not merge into a single object because of long cooling time; these may be identified with clusters or groups of galaxies. For this scenario to be valid, most baryonic matter should remain to be in a gaseous form, which seems to be observationally supported; the mass of a hot gas in rich clusters of galaxies is much larger than the mass of galaxies and the observed mass density of stars amounts to only ~30 % of the baryon density 31 derived by the cosmological (e.g., Persic and Salucci ). Finally, we estimate the effects on the CMB anisotropies and spectral distortions. Since the redshift of full reionization is rather modest, the optical thickness to the rr remains to be low. For example, Q o=O.l, Qb=O.Ol and Zreion =20, rr is as low as 0.1 and small scale anisotropies of CMB are not much affected. A possible effect arises if an epoch of partial ionization at x=O.l has continued to redshift up to z=200. Such a degree of ionization is achieved even if 7J is 1 % of that required for the full ionization. Although a simple estimate suggests that it is A Simple Model of Cosmological Reionization 95 marginally possible, at higher redshifts radiative cooling rate will make the electron temperature much lower than 104K, which enhances recombination. Then, it seems to be difficult to realize such a partial ionization stage for high redshifts. The effect on the Sunyaev-Zel'dovich spectral distortion is also small because the temperature remains to be rather low. Even if all the dissipated energy is deposited to the spectral distortion, the ratio of added to the original energy density of CMB UCMB is given by

2 Downloaded from https://academic.oup.com/ptp/article/90/1/85/1825083 by guest on 28 September 2021 PbC 7j =4.2 X 10-6( Qbh2 )(~)(~) , (31) UCMB 0.01 10 1 + z

16 which is orders of magnitude smaller than the upper limit obtained by COBE. ) A possible effect may occur as the Sunyaev-Zel'dovich distortion due to individual clusters of galaxies at recent epoch z< 10. Those clusters may induce additional 32 anisotropies on very small angular scales. ),33)

§ 6. Conclusions

We have examined several requirements for the reionization of the intergalactic matter by the photoionization mechanism during z=10-1000. Assuming that the ionization equilibrium is established, the minimum production rate of ionizing photons and corresponding energy input rate are estimated. In terms of the efficiency of energy input rate per hydrogen atom per cosmic expansion time scale 7j, a full 7 reionization before a redshift of 10 requires 7j greater than about 10- • Although there are several candidates for sources of such ionizing photons such as first genera­ tion massive stars and incipient massive black holes which grow into quasars at recent past, they seem to be highly model-dependent. Then, we considered whether or not the gravitational energy release naturally accompanying structure formation can contribute to such ionizing photon sources. Estimating the rate of energy release and radiation rate of the collapsing objects, we showed that a fair fraction of the dissipated energy can be radiated by bremsstrahlung under several reasonable conditions. First, the spectral index of the density fluctua­ tions should be n? -1 at the recombination epoch, which is compatible with other observational facts. Second, baryon density should be as low as Qb=0.01-0.1 which 25 is also consistent with the prediction of standard nucleosynthesis. ) Finally, the baryon distribution in collapsing structure is fairly inhomogeneous in order to avoid excessive Compton cooling over bremsstrahlung, which is naturally expected in the hierarchical clustering scenario. It should be noted that all the above conditions are satisfied (or imposed phenomenologically) in the primordial isocur­ 34 vature baryon scenario. ),35),6) In our scenario, the efficiency of should not be so high and the majority of baryonic matter have remained to be in a gaseous form. Further, before z=10 the collapse of bound objects is dissipative, while after z=10 the collapse is nearly adiabatic. This would lead to a characteristic mass scale of galaxies of 12 10 M0 • The y-distortion of the CMB in the present scenario is likely to be small although the cumulative Sunyaev-Zel'dovich effect might be observable as CMB 96 S. Sasaki, F. Takahara and Y. Suto

33 anisotropies on small angular scales. ) Detailed numerical studies are needed for giving quantitative discussion to the process of cosmic reionization described in this paper. This research was supported in part by the Grant-in-Aid from the Ministry of Education, and Culture of Japan (03302012, 04352006 and 04740130).

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