85 Progress of Theoretical Physics, Vol. 90, No.1, July 1993 A Simple Model of Cosmological Reionization Shin SASAKI, Fumio TAKAHARA and Yasushi SUTO* Department of Physics, Tokyo Metropolitan University, Hachioji 192-03 * Uji Research Center, Yukawa Institute for Theoretical Physics 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 structure formation, numerous nonlinear objects on small scales would collapse much earlier than a redshift z~ 10. Our scenario assumes that such objects dissipate their kinetic energy efficiently into heat and eventually provide copious ionizing photons which reionize the universe 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 spectrum of density 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 cold dark matter universe, but is preferred phenomenologically in primordial isocurvature baryon 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 anisotropies and distortion of the cosmic microwave background, and on a characteristic mass of galaxies. § 1. Introduction Gravitational evolution of cosmic structures has been extensively examined over last ten years, and remained one of the central issues in cosmology. In particular, linear theory of density fluctuations and numerical simulations of nonlinear structures have elucidated a number 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 baryons 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 reasons 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 quasars ) places a strong upper limit on the abundance of neutral hydrogen, 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 ),IO) There are a number of papers which discussed possible models of reionization including heating from gravitationally collapsed objects/I) ionizing radiation from quasars and Pop. III starsIO ) and decaying exotic particles.!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 observations have shown that spectral distortion of CMB 6 is very small/ ) the temperature of the intergalactic matter should have remained to be low, while the ionization 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 hydrogens, neglecting helium for the time-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 atoms. 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 photon energy spectrum, nr is the number density of ionizing photons, c is the light velocity, and O"ion is the ionization cross section at the frequency 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 atom 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, curvature parameter and dimension­ less cosmological constant, 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 dark matter universe with Qo=l, Qb=0.03 and h=0.5 can be potentially reionized at z~300.