Secondary Cosmic Microwave Background Anisotropies in A

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SECONDARY COSMIC MICROWAVE BACKGROUND ANISOTROPIES IN A UNIVERSE REIONIZED IN PATCHES ANDREIGRUZINOV AND WAYNE HU1 Institute for Advanced Study, School of Natural Sciences, Princeton, NJ 08540 Received 1998 March 17; accepted 1998 July 2

ABSTRACT In a universe reionized in patches, the Doppler e†ect from Thomson scattering o† free electrons generates secondary cosmic microwave background (CMB) anisotropies. For a simple model with small patches and late reionization, we analytically calculate the anisotropy power spectrum. Patchy reionization can, in principle, be the main source of anisotropies on arcminute scales. On larger angular scales, its contribution to the CMB power spectrum is a small fraction of the primary signal and is only barely detectable in the power spectrum with even an ideal, i.e., cosmic variance limited, experiment and an extreme model of reionization. Consequently, patchy reionization is unlikely to a†ect cosmological parameter estimation from the acoustic peaks in the CMB. Its detection on small angles would help determine the ionization history of the universe, in particular, the typical size of the ionized region and the duration of the reionization process. Subject headings: cosmic microwave background È cosmology: theory È intergalactic medium È large-scale structure of universe

1. INTRODUCTION tion below the degree scale. Such e†ects rely on modulating the Doppler e†ect with spatial variations in the optical It is widely believed that the cosmic microwave back- depth. Incarnations of this general mechanism include the ground (CMB) will become the premier laboratory for the Vishniac e†ect from linear density variations (Vishniac study of the early universe and classical cosmology. This 1987), the kinetic Sunyaev-Zeldovich e†ect from clusters belief relies on the high precision of the upcoming MAP2 (Sunyaev& Zeldovich 1970), and the e†ect considered here: and PlanckSurveyor3 satellite missions and the high accu- the spatial variation of the ionization fraction. racy of theoretical predictions of CMB anisotropies given a Reionization commences when the Ðrst baryonic objects deÐnite model for structure formation(Hu et al. 1995). To form stars or quasars that convert a part of the nuclear or realize the potential of the CMB, aspects of structure forma- gravitational energy into UV photons. Each such source tion a†ecting anisotropies at only the percent level in power then blows out an ionization sphere around it. Before these must be taken into account. regions overlap, there is a period when the universe is A great uncertainty in models for structure formation is ionized in patches. The extent of this period and the time the extent and nature of reionization. Fortunately, this evolution of the size and number density of these patches uncertainty is largely not reÑected in the CMB anisotropies depend on the nature of the ionizing engines in the Ðrst because of the low optical depth to Thomson scattering at baryonic objects. Theories of reionization do not give the low redshifts in question. Reionization is known to be robust constraints (seeTegmark, Silk, & Blanchard 1994; essentially complete by z D 5 from the absence of the Gunn- Rees 1996; Aghanimet al. 1996; Loeb 1997; Haiman & Peterson e†ect in quasar absorption spectra(Gunn & Loeb1997; Silk & Rees 1998; Haiman & Loeb 1998). Peterson1965). SigniÐcant reionization before z D 50 will We therefore take a phenomenological approach to be ruled out once the tentative detections of the CMB studying the e†ects of patchy reionization on the CMB. We acoustic peaks at present are conÐrmed(Scott, Silk, & introduce a simple but illustrative three-parameter model White1995). It should be possible to deduce the reioniza- for the reionization process based on the redshift of its onset tion redshiftz through CMB polarization measurements i zi, the duration before completion dz, and the typical com- (Zaldarriaga,Spergel, & Seljak 1997). oving size of the patches R. It is then straightforward to Nevertheless, the duration of time spent in a partially calculate the CMB anisotropies generated by the patchiness ionized state will remain uncertain. Moreover, as empha- of the ionization degree of the intergalactic medium. sized byKaiser (1984), secondary anisotropies generated by We Ðnd that only the most extreme models of reioniza- the Doppler e†ect in linear perturbation theory are sup- tion can produce degree-scale anisotropies that are observ- pressed on small scales for geometric reasons (gravitational able in the power spectrum given the cosmic variance instability generates potential Ñows, leading to cancel- limitations. A large signal on degree scales requires early lations between positive and negative Doppler shifts). ionization,zi Z 30, long duration,dz D zi, and ionization in Higher order e†ects that are not generally included in the very large patches,R Z 30 Mpc. Thus the patchiness of theoretical modeling of CMB anisotropies are likely to be reionization is unlikely to a†ect cosmological parameter the main source of secondary anisotropies from reioniza- estimation from the acoustic peaks in the CMB(Jungman et al.1996; Zaldarriaga et al. 1997; Bond, Efstathiou, & Tegmark 1997). 1 Alfred P. Sloan fellow. On the other hand, the patchy reionization signal on the 2 See the MAP homepage at http://map.gsfc.nasa.gov. 3 See the Planck Surveyor homepage at http://astro.estec.esa.nl/ subarcminute scale can, in principle, surpass both the SA-general/Projects/Planck. primary and the secondary Vishniac signals. These may be 435 436 GRUZINOV & HU Vol. 508 detectable by the Planck Surveyor and upcoming radio 2.2. Order of Magnitude Estimates interferometry measurements(Partridge et al. 1997), if point Consider the following patchy reionization scenario. The sources can be removed at the *T /T D 10~6 level. universe was reionized in randomly distributed patches An explicit expression for the CMB anisotropies power with a characteristic comoving size R. The patches spectrum generated in a universe reionized in patches is appeared at random in space and time. Once a reionized given in° 2.1. A simple order of magnitude estimate of the patch appears, it moves with matter. The average ionization anisotropy from patches is given in° 2.2. In° 2.3 we give a fraction, that is the Ðlling fraction of fully ionized patches, rigorous deÐnition of our three-parameter reionization grows monotonically fromXe \ 0 at high redshifts to Xe \ model and calculate the patchy part of the power spectrum. 1 at low redshifts. We consider late reionization (optical We discuss illustrative examples in ° 2.4. depth to Thomson scattering is small) and small patches 2. CMB POWER SPECTRUM (smaller than the characteristic length scale of the peculiar velocity Ðeld). We assume that reionization occurred at red- 2.1. Explicit Expression shiftzi, and the patchy phase duration is given by dz. Temperature perturbations * 4 dT /T are generated by The angular scale of the patchy signal is given by the Doppler shifts from Thomson scattering. For small optical ratio of the size of patches to the distance to them in com- depths, oving coordinates,h D R/(g [ gi). Assuming that the P patches are uncorrelated, the0 spectrum of Ñuctuations * \[ dl Æ¿p nxe . (1) should be white noise above this scale, which agrees with T the exact result, as we shall see (eq. [18]). All quantities here are in physical units. The integral is The rms CMB temperature Ñuctuation * on scales h due along the line of site,¿ is the peculiar velocity of matter, to the patchiness can be estimated as follows. Since by c \ 1,p is the Thomson cross section, n is the number assumption di†erent patches are independent, * D N1@2* . T p density of free and bound electrons, andxe is the local Here N is the number of patches on a line of site, and*p is ionization fraction. a temperature Ñuctuation from one patch, *p D qp v(zi). To evaluateequation (1) explicitly, one must specify the Here v(z) \ (1 ] z)~1@2v(0) is the rms peculiar velocity at cosmological model. For simplicity, we take a universe with redshift z, andqp is the optical depth for one patch, critical density in matter throughout; we describe the gener- q D (1 ] z )2p n R. The number of patches N D dg/ p i T 0 alization to an arbitrary FRW universe in° 2.3. We further- R D (1 ] zi)~3@2 dzg /R. Collecting all the factors, we get more use comoving coordinates x and conformal time g 4 the following estimate0 for the rms anisotropies from / (1 ] z)dt \ (1 ] z)~1@2g , whereg \ 2/H is the present patches: particle horizon. We observe0 at x \0 0 and conformal0 time * D q Sv2T1@2(R/g )1@2(1 ] z )3@4(dz)1@2 , (7) g along the direction of a unit vectorcü ; light propagation is 0 0 i given0 byx \ cü (g [ g). Equation (1) can be written as 0 which again agrees with the exact result(eq. [15], up to a Pdg dimensionless multiplier). *(cü ) \[q g3 cü Æ¿[g,cü(g [g)]x [g, cü (g [ g)] . 0 0 g4 0 e 0 2.3. Power Spectrum (2) We can factor the general expression for the temperature Hereq 4 p n g is the optical depth to Thomson scat- correlation inequation (6) as tering across0 T the0 present0 particle horizon. The scales contributing to the peculiar velocity are still in P dg P dg C(h) \ q2 g4 1 2 the linear regime, therefore 0 0 g3 g3 1 2 g ] Scü Æ¿(x)cü Æ¿(x)TSx (g , x )x (g , x )T . (8) ¿(g, x) \ ¿(x) , (3) 1 1 2 2 e 1 1 e 2 2 g 0 This assumes thatxe and¿ are independent random Ðelds. where¿(x) is the peculiar velocity today. The Ðnal explicit This is not strictly correct. The ionization fractionxe must expression for the CMB temperature perturbation gener- be determined by the density perturbation d, and the ated during reionization is density perturbation is not independent of the peculiar velocity (for example, in the linear regime d \[1g$ Æ¿). Pdg 2 *(cü ) \[q g2 cü Æ¿[cü(g [g)]x [g, cü (g [ g)] . (4) However, the ionizing radiation is presumably coming from 0 0 g3 0 e 0 strongly nonlinear objects, where Ðrst stars or quasars are The correlation function of the temperature perturbations is lightening up. At high z, the length scales where the density deÐned as is nonlinear are >10 Mpc comoving, which is much smaller than the length scales contributing to the peculiar velocity. C(h) \ S*(cü )*(cü )T o . (5) Under the assumption of scale separation, velocity and 1 2 À1 ÕÀ2/cos h With temperature perturbations given byequation (4), this density (and hencexe) are indeed independent. becomes The correlation function for the local ionization fraction Sxe xeT is not known. Our model parameterizes the corre- P dg P dg lation function through the patch size R and a mean (cosmic C(h) \ q2 g4 1 2 0 0 g3 g3 time-dependent) ionization fraction Xe(g), 1 2 ] Scü Æ¿(x)cü Æ¿(x)x(g ,x)x(g ,x)T, (6) Sxe(g , x )xe(g , x )T \ Xe(g )Xe(g ) 1 1 2 2 e 1 1 e 2 2 1 1 2 2 1 2 where we denotex 4 cü (g [ g ) and x 4 cü (g [ g ). ] [X (g ) [ X (g )X (g )]e~*(x1~x2)2@2R2+ . (9) 1 1 0 1 2 2 0 2 e min e 1 e 2 No. 2, 1998 SECONDARY CMB ANISOTROPIES 437

Hereg \ min (g , g ). The Gaussian function is chosen The power spectrum is given by the spherical harmonics for simplicity;min it could1 have2 been any function of the separa- decomposition tion x that equals 1 at x \ 0 and gradually turns to zero at p P x [ R. For the mean ionization fractionXe, we assume a Cl( )\2n d cos hPl(cos h)wp(h) . (16) change from 0 to 1 at a redshiftzi, with the transition occurring in a redshift interval dz. We also assume dz > zi For l ? 1, (this is true in all of the models of reionization of which we P = are aware). C(p) B 2n h dhJ (lh)w (h) \ 2nAh2 e~(h02l2@2) . (17) l 0 p 0 CMB anisotropies generated (and erased) because of the 0 spatially constant part of the correlation function (eq. [9]), The power per octave is which are obviously the same as in the model with a p uniform time-dependent reionization. The spatially varying l2Cl( ) \ Al2h2 e~h02l2@2 . (18) part is responsible for generating new anisotropies; its con- 2n 0 tribution to erasing the primary anisotropies is negligible. The anisotropy power reaches the maximal value The anisotropy suppression is mainly determined by the total optical depth to Thomson scattering and is insensitive Al2C(p)B J2n R l \ q2Sv2T dz(1 ] z )3@2 (19) to the small-scale structure of the ionization fraction 2n 18e 0 g i max 0 xe(g, x). The CMB correlation function due to the patchy part at only is J2 g l \ 0 [1 [ (1 ] z )~1@2] . (20) max R i P g0 dg P g0 dg C(p)(h) \ q2 g4 1 2 0 0 g3 g3 0 1 0 2 2.4. Discussion ] I Scü Æ¿(x)cü Æ¿(x)Te~*(x1~x2)2@2R2+ , (10) The signal from patchy reionization in our model 12 1 1 2 2 depends on four quantities: the rms peculiar velocity where we denoteI 4 X (g ) [ X (g )X (g ). The corre- e e e Sv2T1@2 today, the redshift of reionizationzi, its duration dz, lation function is nonnegligible12 min only for1o x [2 x o [ R. By 1 2 and the characteristic comoving size of the patches R. The assumption, R is much smaller then the characteristic scale structure formation model speciÐes the power spectrum of of the peculiar velocity Ðeld. Also,o x [ x o [ R requires Ñuctuations, which in turn tells us the rms peculiar velocity. that the lines of sightcü andcü be nearly1 parallel.2 Then 1 2 Let us now consider the patchy reionized signal in the context of a speciÐc model for structure formation. Scü Æ¿(x)cü Æ¿(x)TB1Sv2T , (11) 1 1 2 2 3 For illustrative purposes, let us consider a cold dark where Sv2T is the mean squared peculiar velocity today. For matter model with h \ 0.5,)b \ 0.1, and a scale-invariant zi ? 1,the integral (eq. [10]) is dominated by small con- n \ 1 spectrum of initial Ñuctuations. Normalizing the spec- formal times, and we have o x [ x o2 Bh2(g [gi)2] trum to the COBE detection via the Ðtting formulae of (g [ g )2. Then 1 2 0 Bunn& White (1997; their eqs. [17]È[20]) and employing 1 2 the analytic Ðt to the transfer function ofEisenstein & Hu 1 (1998; their eqs. [15]È[24]), we Ðnd an rms velocity of C(p)(h) B q2 g4Sv2Te~*(g0~gi)2h2@2R2+ 3 0 0 Sv2T1@2 \ 3.9 ] 10~3. With the present optical depth of q \ 0.122) h \ 0.0061, we have a maximal anisotropy of 0 b P = dg P = dg R ] 1 2 I e~*(g2~g1)2@2 2+ . (12) Al2C B R g3 g3 12 l \ 2.41 ] 10~15 dz(1 ] z )3@2 (21) 0 1 0 2 2n Mpc i We assume thatg dz/(1 z ) ? R [withg 4 g(z )] and that max i ] i i i at during the patchy phase,zi [ z [ zi [ dz, the ionization fractionXe grows linearly from 0 to 1 such that eventually 16958 l \ [1 [ (1 ] z )~1@2] . (22) both hydrogen and helium are fully ionized. Then max R/Mpc i C(p)(h) \ Ae~(h2@2h02) , (13) The power spectrum of the model in principle also tells us the remaining parameters of the ionization: its redshift zi, where the characteristic angular scale is duration dz, and typical patch size R. Unfortunately, these quantities depend on details of the cooling and fragmenting R R (1 ] z )1@2 h \ \ i , (14) of the Ðrst baryonic objects to form the ionizing engines. We 0 g [ g g (1 ] z )1@2 [ 1 0 i 0 i therefore consider5 [ zi [ 50, which spans the range of and the amplitude is estimates in the literature(Tegmark et al. 1994; Rees 1996; Aghanimet al. 1996; Loeb 1997; Haiman & Loeb 1997; J2n R Silk& Rees 1998; Haiman & Loeb 1998). Reionization, A \ q2Sv2T dz(1 ] z )3@2 . (15) 36 0 g i once it commences, is generally completed in a short time, 0 compared with the expansion time at that epoch dz/ Note that a critical matter-dominated universe is assumed (1 ] zi) \ 1, by the coalescence of patches that are small in this expression. To generalize this result, replace g [ gi compared with the horizon at the timeR/gi > 1. Again, the with the comoving angular diameter distance in equation0 exact relations depend on the efficiency with which the Ðrst (14)and a factor of (1 ] zi) in equation (15) with the appro- objects form and create ionizing radiation (see, e.g., priate velocity growth factor. Tegmarket al. 1994). 438 GRUZINOV & HU Vol. 508

Let us consider an extreme example of zi \ 10, dz \ 3, is unlikely that patchy reionization will signiÐcantly a†ect and R \ 20 Mpc. Then the maximal power is parameter estimation through the CMB. B5.3 ] 10~12 at l B 590, and the primary signal at these We have called the(zi \ 10, dz \ 3, R \ 20) model scales is B3 ] 10~10: the contribution of the patchy reioni- extreme because of the size of patches; the reionization red- zation is small in comparison (seeFig. 1). However, in light shift and duration would be considered reasonable by a of the high-precision measurements expected from the number of theories. For example, the early quasar model of MAP and Planck satellites, such a signal is not necessarily Haiman& Loeb (1998) does predictzi D 10 and dz D 3. negligible. The ultimate limit of detectability through power However, their ““ medium quasar ÏÏ emits only D1067 ion- spectrum measurements is provided by so-called cosmic izing photons during its lifetime. These photons cannot variance. This arises because we can only measure 2l ] 1 ionize a bubble larger than R D 1 Mpc comoving. realizations of any given multipole such that power spec- Perhaps more interesting is the case where reionization trum estimates will vary by takes place at a higher redshift with, for example, zi \ 30, dz \ 5, and R \ 3 Mpc. The reduction in the patch size S 2 causes the signature to move to smaller angles, where the dC \ C(primary) . (23) l 2l ] 1 l primary signal is negligible because of dissipational e†ects at recombination. The increase in the optical depth at this Detection of a broad feature such as that from patchy reion- higher redshift is counterbalanced by the reduction in the ization is assisted in that we may reduce the cosmic variance rms Ñuctuation due to the number of patches along the line by averaging over many lÏs. We show an example of this of site such that the amplitude of the signal increases only averaging inFigure 1 (lower left boxes). In this model, the moderately. Here the maximal power is B6.2 ] 10~12 at patchy reionization signal can be detected at the several p l B 4650 (seeFig. 2). Patchy reionization e†ects exceed the level if cosmic variance was the main source of uncertainty. Vishniac signal at these scales (B3 ] 10~12), which is Of course, a realistic experiment also has noise and system- believed to be the leading other source of secondary aniso- atic errors. We also show the noise error contributions tropies(Hu & White 1996). expected from the MAP experiment in Figure 1. Although the morphology and amplitude of the patchy An important additional source of uncertainty is provid- reionization and Vishniac signals are similar, the Vishniac ed by other unknown aspects of the model. Indeed, it is e†ect is fully speciÐed by the ionization redshift and the hoped that the CMB power spectrum can be used to spectrum of initial Ñuctuations and hence may be removed measure fundamental cosmological parameters to high pre- once these are determined from parameter estimation at cision. Excess variance from patchy reionization can, in larger angular scales. Likewise, since the rms peculiar veloc- principle, cause problems for cosmological parameter esti- ity SvT and the ionization redshiftzi will be speciÐed by the mation from the CMB if not included in the model. It would large-scale observations, the amplitude of the signal can be remain undetected and produce parameter misestimates if used to estimate the duration of reionization dz and its its signal can be accurately mimicked by variations in the angular location the typical comoving size of the bubbles R. other parameters. Fortunately, the angular signature we In summary, the patchiness of reionization leaves a Ðnd here, l2 white noise until some cuto† due to the patch potentially observable imprint on the CMB power spec- size, does not resemble the signature of other cosmological trum but one that is unlikely to a†ect cosmological param- parameters that alter the positions and amplitudes of the eter estimation from the acoustic peaks in the CMB. We acoustic peaks (seeBond et al. 1997; Zaldarriaga et al. show how the signature scales with the gross properties of 1997). Coupled with the small amplitude of the e†ect on the reionization: its redshift, duration, and typical patch size. 10 arcminute to degree scale for even this extreme model, it

FIG. 2.ÈCMB anisotropy power spectra in a cold dark matter model FIG. 1.ÈCMB anisotropy power spectra in a cold dark matter model with early reionization. Shown here are the primary anisotropy suppressed with extreme patchiness. Shown here are the primary anisotropy and the by rescattering and the patchy reionization anisotropy,eq. (18) with zi \ patchy reionization anisotropy,eq. (18) with zi \ 10, dz \ 3, and R \ 20 30, dz \ 5, and R \ 3 Mpc. These signals are compared with the cosmic Mpc. These signals are compared with the cosmic variance of the primary variance of the primary anisotropy achievable by an ideal experiment in anisotropy and the noise of the MAP satellite (in logarithmic bins). the absence of galactic and extragalactic foregrounds. No. 2, 1998 SECONDARY CMB ANISOTROPIES 439

Observational detection of this signature would provide We thank R. Juszkiewicz, A. Liddle, M. Tegmark, and useful constraints on the presently highly uncertain reioni- M. White for discussions. This work was supported by zation scenarios but will likely require experiments with NSF PHY-9513835. W. H. was also supported by the angular resolution of an arcminute or better and fore- W. M. Keck Foundation. ground subtraction at better than the dT /T D 10~6 level.

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