Next Contents Previous


9.1. Mapping Hydrogen Before Reionization

The small residual fraction of free electrons after cosmological recombination coupled the temperature of the cosmic gas to that of the cosmic microwave background (CMB) down to a redshift, z ~ 200 [284]. Subsequently, the gas temperature dropped adiabatically as Tgas propto (1 + z)2 below the CMB temperature Tgamma propto (1 + z). The gas heated up again after being exposed to the photo-ionizing ultraviolet light emitted by the first stars during the reionization epoch at z < 20. Prior to the formation of the first stars, the cosmic neutral hydrogen must have resonantly absorbed the CMB flux through its spin-flip 21cm transition [131, 323, 367, 404]. The linear density fluctuations at that time should have imprinted anisotropies on the CMB sky at an observed wavelength of lambda = 21.12[(1 + z) / 100] meters. We discuss these early 21cm fluctuations mainly for pedagogical purposes. Detection of the earliest 21cm signal will be particularly challenging because the foreground sky brightness rises as lambda2.5 at long wavelengths in addition to the standard lambda1/2 scaling of the detector noise temperature for a given integration time and fractional bandwidth. The discussion in this section follows Loeb & Zaldarriaga (2004) [226].

Figure 53

Figure 53. The 21cm transition of hydrogen. The higher energy level the spin of the electron (e-) is aligned with that of the proton (p+). A spin flip results in the emission of a photon with a wavelength of 21cm (or a frequency of 1420MHz).

We start by calculating the history of the spin temperature, Ts, defined through the ratio between the number densities of hydrogen atoms in the excited and ground state levels, n1 / n0 = (g1/ g0)exp{-T* / Ts},

Equation 154 (154)

where subscripts 1 and 0 correspond to the excited and ground state levels of the 21cm transition, (g1 / g0) = 3 is the ratio of the spin degeneracy factors of the levels, nH = (n0 + n1) propto (1 + z)3 is the total hydrogen density, and T* = 0.068K is the temperature corresponding to the energy difference between the levels. The time evolution of the density of atoms in the ground state is given by,

Equation 155 (155)

where a(t) = (1 + z)-1 is the cosmic scale factor, A's and B's are the Einstein rate coefficients, C's are the collisional rate coefficients, and Inu is the blackbody intensity in the Rayleigh-Jeans tail of the CMB, namely Inu = 2kTgamma / lambda2 with lambda = 21 cm [306]. Here a dot denotes a time-derivative. The 0 -> 1 transition rates can be related to the 1 -> 0 transition rates by the requirement that in thermal equilibrium with Ts = Tgamma = Tgas, the right-hand-side of Eq. (155) should vanish with the collisional terms balancing each other separately from the radiative terms. The Einstein coefficients are A10 = 2.85 × 10-15 s-1, B10 = (lambda3 / 2hc) A10 and B01 = (g1 / g0)B10 [131, 306]. The collisional de-excitation rates can be written as C10 = 4/3 kappa(1 - 0) nH, where kappa(1 - 0) is tabulated as a function of Tgas [11, 406].

Equation (155) can be simplified to the form,

Equation 156 (156)

where Upsilon ident n0 / nH, H approx H0 (Omegam)1/2(1 + z)3/2 is the Hubble parameter at high redshifts (with a present-day value of H0), and Omegam is the density parameter of matter. The upper panel of Fig. 54 shows the results of integrating Eq. (156). Both the spin temperature and the kinetic temperature of the gas track the CMB temperature down to z ~ 200. Collisions are efficient at coupling Ts and Tgas down to z ~ 70 and so the spin temperature follows the kinetic temperature around that redshift. At much lower redshifts, the Hubble expansion makes the collision rate subdominant relative the radiative coupling rate to the CMB, and so Ts tracks Tgamma again. Consequently, there is a redshift window between 30 < z < 200, during which the cosmic hydrogen absorbs the CMB flux at its resonant 21cm transition. Coincidentally, this redshift interval precedes the appearance of collapsed objects [23] and so its signatures are not contaminated by nonlinear density structures or by radiative or hydrodynamic feedback effects from stars and quasars, as is the case at lower redshifts [404].

Figure 54

Figure 54. Upper panel: Evolution of the gas, CMB and spin temperatures with redshift [4]. Lower panel: dTb / ddeltaH as function of redshift. The separate contributions from fluctuations in the density and the spin temperature are depicted. We also show dTb / ddeltaH a propto dTb / ddeltaH × deltaH, with an arbitrary normalization.

During the period when the spin temperature is smaller than the CMB temperature, neutral hydrogen atoms absorb CMB photons. The resonant 21cm absorption reduces the brightness temperature of the CMB by,

Equation 157 (157)

where the optical depth for resonant 21cm absorption is,

Equation 158 (158)

Small inhomogeneities in the hydrogen density deltaH ident (nH - bar{n}H) / bar{n}H result in fluctuations of the 21cm absorption through two separate effects. An excess of neutral hydrogen directly increases the optical depth and also alters the evolution of the spin temperature. For now, we ignore the additional effects of peculiar velocities (Bharadwaj & Ali 2004 [41]; Barkana & Loeb 2004 [27]) as well as fluctuations in the gas kinetic temperature due to the adiabatic compression (rarefaction) in overdense (underdense) regions [29]. Under these approximations, we can write an equation for the resulting evolution of Upsilon fluctuations,

Equation 159 (159)

leading to spin temperature fluctuations,

Equation 160 (160)

The resulting brightness temperature fluctuations can be related to the derivative,

Equation 161 (161)

The spin temperature fluctuations delta Ts / Ts are proportional to the density fluctuations and so we define,

Equation 162 (162)

through delta Tb = (d Tb / d deltaH) deltaH. We ignore fluctuations in Cij due to fluctuations in Tgas which are very small [11]. Figure 54 shows dTb / ddeltaH as a function of redshift, including the two contributions to dTb / ddeltaH, one originating directly from density fluctuations and the second from the associated changes in the spin temperature [323]. Both contributions have the same sign, because an increase in density raises the collision rate and lowers the spin temperature and so it allows Ts to better track Tgas. Since deltaH grows with time as deltaH propto a, the signal peaks at z ~ 50, a slightly lower redshift than the peak of dTb / ddeltaH.

Next we calculate the angular power spectrum of the brightness temperature on the sky, resulting from density perturbations with a power spectrum Pdelta(k),

Equation 163 (163)

where deltaH(k) is the Fourier tansform of the hydrogen density field, k is the comoving wavevector, and < … > denotes an ensemble average (following the formalism described in [404]). The 21cm brightness temperature observed at a frequency nu corresponding to a distance r along the line of sight, is given by

Equation 164 (164)

where n denotes the direction of observation, Wnu(r) is a narrow function of r that peaks at the distance corresponding to nu. The details of this function depend on the characteristics of the experiment. The brightness fluctuations in Eq. 164 can be expanded in spherical harmonics with expansion coefficients alm(nu). The angular power spectrum of map Cl(nu) = < |alm(nu)|2 > can be expressed in terms of the 3D power spectrum of fluctuations in the density Pdelta(k),

Equation 165 (165)

Our calculation ignores inhomogeneities in the hydrogen ionization fraction, since they freeze at the earlier recombination epoch (z ~ 103) and so their amplitude is more than an order of magnitude smaller than deltaH at z < 100. The gravitational potential perturbations induce a redshift distortion effect that is of order ~ (H / ck)2 smaller than deltaH for the high - l modes of interest here.

Figure 55 shows the angular power spectrum at various redshifts. The signal peaks around z ~ 50 but maintains a substantial amplitude over the full range of 30 < z < 100. The ability to probe the small scale power of density fluctuations is only limited by the Jeans scale, below which the dark matter inhomogeneities are washed out by the finite pressure of the gas. Interestingly, the cosmological Jeans mass reaches its minimum value, ~ 3 × 104 Modot, within the redshift interval of interest here which corresponds to modes of angular scale ~ arcsecond on the sky. During the epoch of reionization, photoionization heating raises the Jeans mass by several orders of magnitude and broadens spectral features, thus limiting the ability of other probes of the intergalactic medium, such as the Lyalpha forest, from accessing the same very low mass scales. The 21cm tomography has the additional advantage of probing the majority of the cosmic gas, instead of the trace amount (~ 10-5) of neutral hydrogen probed by the Lyalpha forest after reionization. Similarly to the primary CMB anisotropies, the 21cm signal is simply shaped by gravity, adiabatic cosmic expansion, and well-known atomic physics, and is not contaminated by complex astrophysical processes that affect the intergalactic medium at z < 30.

Figure 55

Figure 55. Angular power spectrum of 21cm anisotropies on the sky at various redshifts. From top to bottom, z = 55,40,80,30,120,25,170.

Characterizing the initial fluctuations is one of the primary goals of observational cosmology, as it offers a window into the physics of the very early Universe, namely the epoch of inflation during which the fluctuations are believed to have been produced. In most models of inflation, the evolution of the Hubble parameter during inflation leads to departures from a scale-invariant spectrum that are of order 1/Nefold with Nefold ~ 60 being the number of e - folds between the time when the scale of our horizon was of order the horizon during inflation and the end of inflation [218]. Hints that the standard LambdaCDM model may have too much power on galactic scales have inspired several proposals for suppressing the power on small scales. Examples include the possibility that the dark matter is warm and it decoupled while being relativistic so that its free streaming erased small-scale power [48], or direct modifications of inflation that produce a cut-off in the power on small scales [192]. An unavoidable collisionless component of the cosmic mass budget beyond CDM, is provided by massive neutrinos (see [198] for a review). Particle physics experiments established the mass splittings among different species which translate into a lower limit on the fraction of the dark matter accounted for by neutrinos of fnu > 0.3 %, while current constraints based on galaxies as tracers of the small scale power imply fnu < 12 % [360].

Figure 56 shows the 21cm power spectrum for various models that differ in their level of small scale power. It is clear that a precise measurement of the 21cm power spectrum will dramatically improve current constraints on alternatives to the standard LambdaCDM spectrum.

Figure 56

Figure 56. Upper panel: Power spectrum of 21cm anisotropies at z = 55 for a LambdaCDM scale-invariant power spectrum, a model with n = 0.98, a model with n = 0.98 and alphar ident 1/2 (d2lnP / dln k2) = -0.07, a model of warm dark matter particles with a mass of 1 keV, and a model in which fnu = 10% of the matter density is in three species of massive neutrinos with a mass of 0.4 eV each. Lower panel: Ratios between the different power spectra and the scale-invariant spectrum.

The 21cm signal contains a wealth of information about the initial fluctuations. A full sky map at a single photon frequency measured up to lmax, can probe the power spectrum up to kmax ~ (lmax / 104) Mpc-1. Such a map contains lmax2 independent samples. By shifting the photon frequency, one may obtain many independent measurements of the power. When measuring a mode l, which corresponds to a wavenumber k ~ l / r, two maps at different photon frequencies will be independent if they are separated in radial distance by 1 / k. Thus, an experiment that covers a spatial range Delta r can probe a total of kDelta r ~ l Delta r / r independent maps. An experiment that detects the 21cm signal over a range Deltanu centered on a frequency nu, is sensitive to Delta r / r ~ 0.5 (Delta nu / nu)(1 + z)-1/2, and so it measures a total of N21cm ~ 3 × 1016 (lmax / 106)3 (Delta nu / nu) (z / 100)-1/2 independent samples.

This detection capability cannot be reproduced even remotely by other techniques. For example, the primary CMB anisotropies are damped on small scales (through the so-called Silk damping), and probe only modes with l leq 3000 (k leq 0.2 Mpc-1). The total number of modes available in the full sky is Ncmb = 2 lmax2 ~ 2 × 107 (lmax / 3000)2, including both temperature and polarization information.

The sensitivity of an experiment depends strongly on its particular design, involving the number and distribution of the antennae for an interferometer. Crudely speaking, the uncertainty in the measurement of [l(l + 1)Cl / 2pi]1/2 is dominated by noise, Nnu, which is controlled by the sky brightness Inu at the observed frequency nu [404],

Equation 166 (166)

where lmin is the minimum observable l as determined by the field of view of the instruments, lmax is the maximum observable l as determined by the maximum separation of the antennae, fcover is the fraction of the array area thats is covered by telescopes, t0 is the observation time and Deltanu is the frequency range over which the signal can be detected. Note that the assumed sky temperature of 0.7 × 104 K at nu = 50 MHz (corresponding to z ~ 30) is more than six orders of magnitude larger than the signal. We have already included the fact that several independent maps can be produced by varying the observed frequency. The numbers adopted above are appropriate for the inner core of the LOFAR array (, planned for initial operation in 2006. The predicted signal is ~ 1 mK, and so a year of integration or an increase in the covering fraction are required to observe it with LOFAR. Other experiments whose goal is to detect 21cm fluctuations from the subsequent epoch of reionization at z ~ 6-12 (when ionized bubbles exist and the fluctuations are larger) include the Mileura Wide-Field Array (MWA;, the Primeval Structure Telescope (PAST;, and in the more distant future the Square Kilometer Array (SKA; The main challenge in detecting the predicted signal from higher redshifts involves its appearance at low frequencies where the sky noise is high. Proposed space-based instruments [194] avoid the terrestrial radio noise and the increasing atmospheric opacity at nu < 20 MHz (corresponding to z > 70).

Figure 57

Figure 57. Prototype of the tile design for the Mileura Wide-Field Array (MWA) in western Australia, aimed at detecting redshifted 21cm from the epoch of reionization. Each 4m×4m tile contains 16 dipole antennas operating in the frequency range of 80 - 300MHz. Altogether the initial phase of MWA (the so-called "Low-Frequency Demostrator") will include 500 antenna tiles with a total collecting area of 8000 m2 at 150MHz, scattered across a 1.5 km region and providing an angular resolution of a few arcminutes.

The 21cm absorption is replaced by 21cm emission from neutral hydrogen as soon as the intergalactic medium is heated above the CMB temperature by X-ray sources during the epoch of reionization [88]. This occurs long before reionization since the required heating requires only a modest amount of energy, ~ 10-2 eV[(1 + z) / 30], which is three orders of magnitude smaller than the amount necessary to ionize the Universe. As demonstrated by Chen & Miralda-Escude (2004) [88], heating due the recoil of atoms as they absorb Lyalpha photons [237] is not effective; the Lyalpha color temperature reaches equilibrium with the gas kinetic temperature and suppresses subsequent heating before the level of heating becomes substantial. Once most of the cosmic hydrogen is reionized at zreion, the 21cm signal is diminished. The optical depth for free-free absorption after reionization, ~ 0.1 [(1 + zreion) / 20]5/2, modifies only slightly the expected 21cm anisotropies. Gravitational lensing should modify the power spectrum [287] at high l, but can be separated as in standard CMB studies (see [326] and references therein). The 21cm signal should be simpler to clean as it includes the same lensing foreground in independent maps obtained at different frequencies.

Figure 58

Figure 58. Schematic sketch of the evolution of the kinetic temperature (Tk) and spin temperature (Ts) of cosmic hydrogen. Following cosmological recombination at z ~ 103, the gas temperature (orange curve) tracks the CMB temperature (blue line; Tgamma propto (1 + z)) down to z ~ 200 and then declines below it (Tk propto (1 + z)2) until the first X-ray sources (accreting black holes or exploding supernovae) heat it up well above the CMB temperature. The spin temperature of the 21cm transition (red curve) interpolates between the gas and CMB temperatures. Initially it tracks the gas temperature through collisional coupling; then it tracks the CMB through radiative coupling; and eventually it tracks the gas temperature once again after the production of a cosmic background of UV photons between the Lyalpha and the Lyman-limit frequencies that redshift or cascade into the Lyalpha resonance (through the Wouthuysen-Field effect [Wouthuysen 1952 [388]; Field 1959 [131]]). Parts of the curve are exaggerated for pedagogic purposes. The exact shape depends on astrophysical details about the first galaxies, such as their production of X-ray binaries, supernovae, nuclear accreting black holes, and their generation of relativistic electrons in collisionless shocks which produce UV and X-ray photons through inverse-Compton scattering of CMB photons.

The large number of independent modes probed by the 21cm signal would provide a measure of non-Gaussian deviations to a level of ~ N21 cm-1/2, constituting a test of the inflationary origin of the primordial inhomogeneities which are expected to possess deviations > 10-6 [245].

9.2. The Characteristic Observed Size of Ionized Bubbles at the End of Reionization

Figure 59

Figure 59. 21cm imaging of ionized bubbles during the epoch of reionization is analogous to slicing swiss cheese. The technique of slicing at intervals separated by the typical dimension of a bubble is optimal for revealing different pattens in each slice.

The first galaxies to appear in the Universe at redshifts z > 20 created ionized bubbles in the intergalactic medium (IGM) of neutral hydrogen (H I) left over from the Big-Bang. It is thought that the ionized bubbles grew with time, surrounded clusters of dwarf galaxies [67, 143] and eventually overlapped quickly throughout the Universe over a narrow redshift interval near z ~ 6. This event signaled the end of the reionization epoch when the Universe was a billion years old. Measuring the unknown size distribution of the bubbles at their final overlap phase is a focus of forthcoming observational programs aimed at highly redshifted 21cm emission from atomic hydrogen. In this sub-section we follow Wyithe & Loeb (2004) [399] and show that the combined constraints of cosmic variance and causality imply an observed bubble size at the end of the overlap epoch of ~ 10 physical Mpc, and a scatter in the observed redshift of overlap along different lines-of-sight of ~ 0.15. This scatter is consistent with observational constraints from recent spectroscopic data on the farthest known quasars. This result implies that future radio experiments should be tuned to a characteristic angular scale of ~ 0.5° and have a minimum frequency band-width of ~ 8 MHz for an optimal detection of 21cm flux fluctuations near the end of reionization.

During the reionization epoch, the characteristic bubble size (defined here as the spherically averaged mean radius of the H II regions that contain most of the ionized volume [143]) increased with time as smaller bubbles combined until their overlap completed and the diffuse IGM was reionized. However the largest size of isolated bubbles (fully surrounded by H I boundaries) that can be observed is finite, because of the combined phenomena of cosmic variance and causality. Figure 61 presents a schematic illustration of the geometry. There is a surface on the sky corresponding to the time along different lines-of-sight when the diffuse (uncollapsed) IGM was most recently neutral. We refer to it as the Surface of Bubble Overlap (SBO). There are two competing sources for fluctuations in the SBO, each of which is dependent on the characteristic size, RSBO, of the ionized regions just before the final overlap. First, the finite speed of light implies that 21cm photons observed from different points along the curved boundary of an H II region must have been emitted at different times during the history of the Universe. Second, bubbles on a comoving scale R achieve reionization over a spread of redshifts due to cosmic variance in the initial conditions of the density field smoothed on that scale. The characteristic scale of H II bubbles grows with time, leading to a decline in the spread of their formation redshifts [67] as the cosmic variance is averaged over an increasing spatial volume. However the 21cm light-travel time across a bubble rises concurrently. Suppose a signal 21cm photon which encodes the presence of neutral gas, is emitted from the far edge of the ionizing bubble. If the adjacent region along the line-of-sight has not become ionized by the time this photon reaches the near side of the bubble, then the photon will encounter diffuse neutral gas. Other photons emitted at this lower redshift will therefore also encode the presence of diffuse neutral gas, implying that the first photon was emitted prior to overlap, and not from the SBO. Hence the largest observable scale of H II regions when their overlap completes, corresponds to the first epoch at which the light crossing time becomes larger than the spread in formation times of ionized regions. Only then will the signal photon leaving the far side of the HII region have the lowest redshift of any signal photon along that line-of-sight.

Figure 60

Figure 60. Spectra of 19 quasars with redshifts 5.74 < z < 6.42 from the Sloan Digital Sky Survey [128]. For some of the highest-redshift quasars, the spectrum shows no transmitted flux shortward of the Lyalpha wavelength at the quasar redshift (the so-called "Gunn-Peterson trough"), indicating a non-negligible neutral fraction in the IGM (see the analysis of Fan et al. [128] for details).

The observed spectra of some quasars beyond z ~ 6.1 show a Gunn-Peterson trough [163, 127] (Fan et al. 2005 [128]), a blank spectral region at wavelengths shorter than Lyalpha at the quasar redshift, implying the presence of H I in the diffuse IGM. The detection of Gunn-Peterson troughs indicates a rapid change [126, 288, 381] in the neutral content of the IGM at z ~ 6, and hence a rapid change in the intensity of the background ionizing flux. This rapid change implies that overlap, and hence the reionization epoch, concluded near z ~ 6. The most promising observational probe [404, 259] of the reionization epoch is redshifted 21cm emission from intergalactic H I. Future observations using low frequency radio arrays (e.g. LOFAR, MWA, and PAST) will allow a direct determination of the topology and duration of the phase of bubble overlap. In this section we determine the expected angular scale and redshift width of the 21cm fluctuations at the SBO theoretically, and show that this determination is consistent with current observational constraints.

Figure 61

Figure 61. The distances to the observed Surface of Bubble Overlap (SBO) and Surface of Lyalpha Transmission (SLT) fluctuate on the sky. The SBO corresponds to the first region of diffuse neutral IGM observed along a random line-of-sight. It fluctuates across a shell with a minimum width dictated by the condition that the light crossing time across the characteristic radius RSBO of ionized bubbles equals the cosmic scatter in their formation times. Thus, causality and cosmic variance determine the characteristic scale of bubbles at the completion of bubble overlap. After some time delay the IGM becomes transparent to Lyalpha photons, resulting in a second surface, the SLT. The upper panel illustrates how the lines-of-sight towards two quasars (Q1 in red and Q2 in blue) intersect the SLT with a redshift difference delta z. The resulting variation in the observed spectrum of the two quasars is shown in the lower panel. Observationally, the ensemble of redshifts down to which the Gunn-Peterson troughs are seen in the spectra of z > 6.1 quasars is drawn from the probability distribution dP / dzSLT for the redshift at which the IGM started to allow Lyalpha transmission along random lines-of-sight. The observed values of zSLT show a small scatter [127] in the SLT redshift around an average value of < zSLT > approx 5.95. Some regions of the IGM may have also become transparent to Lyalpha photons prior to overlap, resulting in windows of transmission inside the Gunn-Peterson trough (one such region may have been seen [381] in SDSS J1148+5251). In the existing examples, the portions of the Universe probed by the lower end of the Gunn-Peterson trough are located several hundred comoving Mpc away from the background quasar, and are therefore not correlated with the quasar host galaxy. The distribution dP / dzSLT is also independent of the redshift distribution of the quasars. Moreover, lines-of-sight to these quasars are not causally connected at z ~ 6 and may be considered independent.

We start by quantifying the constraints of causality and cosmic variance. First suppose we have an H II region with a physical radius R / (1 + < z >). For a 21cm photon, the light crossing time of this radius is

Equation 167 (167)

where at the high-redshifts of interest (dz / dt) = -(H0 Omegam1/2)(1 + z)5/2. Here, c is the speed of light, H0 is the present-day Hubble constant, Omegam is the present day matter density parameter, and < z > is the mean redshift of the SBO. Note that when discussing this crossing time, we are referring to photons used to probe the ionized bubble (e.g. at 21cm), rather than photons involved in the dynamics of the bubble evolution.

Second, overlap would have occurred at different times in different regions of the IGM due to the cosmic scatter in the process of structure formation within finite spatial volumes [67]. Reionization should be completed within a region of comoving radius R when the fraction of mass incorporated into collapsed objects in this region attains a certain critical value, corresponding to a threshold number of ionizing photons emitted per baryon. The ionization state of a region is governed by the enclosed ionizing luminosity, by its over-density, and by dense pockets of neutral gas that are self shielding to ionizing radiation. There is an offset [67] deltaz between the redshift when a region of mean over-density bar{delta}R achieves this critical collapsed fraction, and the redshift bar{z} when the Universe achieves the same collapsed fraction on average. This offset may be computed [67] from the expression for the collapsed fraction [52] Fcol within a region of over-density bar{delta}R on a comoving scale R,

Equation 168 (168)

where deltac(bar{z}) propto (1 + bar{z}) is the collapse threshold for an over-density at a redshift bar{z}; sigmaR and sigmaRmin are the variances in the power-spectrum linearly extrapolated to z = 0 on comoving scales corresponding to the region of interest and to the minimum galaxy mass Mmin, respectively. The offset in the ionization redshift of a region depends on its linear over-density, bar{delta}R. As a result, the distribution of offsets, and therefore the scatter in the SBO may be obtained directly from the power spectrum of primordial inhomogeneities. As can be seen from equation (168), larger regions have a smaller scatter due to their smaller cosmic variance.

Note that equation (168) is independent of the critical value of the collapsed fraction required for reionization. Moreover, our numerical constraints are very weakly dependent on the minimum galaxy mass, which we choose to have a virial temperature of 104 K corresponding to the cooling threshold of primordial atomic gas. The growth of an H II bubble around a cluster of sources requires that the mean-free-path of ionizing photons be of order the bubble radius or larger. Since ionizing photons can be absorbed by dense pockets of neutral gas inside the H II region, the necessary increase in the mean-free-path with time implies that the critical collapsed fraction required to ionize a region of size R increases as well. This larger collapsed fraction affects the redshift at which the region becomes ionized, but not the scatter in redshifts from place to place which is the focus of this sub-section. Our results are therefore independent of assumptions about unknown quantities such as the star formation efficiency and the escape fraction of ionizing photons from galaxies, as well as unknown processes of feedback in galaxies and clumping of the IGM.

Figure 62 displays the above two fundamental constraints. The causality constraint (Eq. 167) is shown as the blue line, giving a longer crossing time for a larger bubble size. This contrasts with the constraint of cosmic variance (Eq. 168), indicated by the red line, which shows how the scatter in formation times decreases with increasing bubble size. The scatter in the SBO redshift and the corresponding fluctuation scale of the SBO are given by the intersection of these curves. We find that the thickness of the SBO is <Delta z2 >1/2 ~ 0.13, and that the bubbles which form the SBO have a characteristic comoving size of ~ 60 Mpc (equivalent to 8.6 physical Mpc). At z ~ 6 this size corresponds to angular scales of thetaSBO ~ 0.4 degrees on the sky.

Figure 62

Figure 62. Constraints on the scatter in the SBO redshift and the characteristic size of isolated bubbles at the final overlap stage, RSBO (see Fig. 1). The characteristic size of H II regions grows with time. The SBO is observed for the bubble scale at which the light crossing time (blue line) first becomes smaller than the cosmic scatter in bubble formation times (red line). At z ~ 6, the implied scale RSBO ~ 60 comoving Mpc (or ~ 8.6 physical Mpc), corresponds to a characteristic angular radius of thetaSLT ~ 0.4 degrees on the sky. After bubble overlap, the ionizing intensity grows to a level at which the IGM becomes transparent to Lyalpha photons. The collapsed fraction required for Lyalpha transmission within a region of a certain size will be larger than required for its ionization. However, the scatter in equation (168) is not sensitive to the collapsed fraction, and so may be used for both the SBO and SLT. The scatter in the SLT is smaller than the cosmic scatter in the structure formation time on the scale of the mean-free-path for ionizing photons. This mean-free-path must be longer than RSBO ~ 60 Mpc, an inference which is supported by analysis of the Lyalpha forest at z ~ 4 where the mean-free-path is estimated [257] to be ~ 120 comoving Mpc at the Lyman limit (and longer at higher frequencies). If it is dominated by cosmic variance, then the scatter in the SLT redshift provides a lower limit to the SBO scatter. The three known quasars at z > 6.1 have Lyalpha transmission redshifts of [381, 127] zSLT = 5.9, 5.95 and 5.98, implying that the scatter in the SBO must be > 0.05 (this scatter may become better known from follow-up spectroscopy of Gamma Ray Burst afterglows at z>6 that might be discovered by the SWIFT satellite[26, 61]). The observed scatter in the SLT redshift is somewhat smaller than the predicted SBO scatter, confirming the expectation that cosmic variance is smaller at the SLT. The scatter in the SBO redshift must also be < 0.25 because the lines-of-sight to the two highest redshift quasars have a redshift of Lyalpha transparency at z ~ 6, but a neutral fraction that is known from the proximity effect [396] to be substantial at z > 6.2-6.3. The excluded regions of scatter for the SBO are shown in gray.

A scatter of ~ 0.15 in the SBO is somewhat larger than the value extracted from existing numerical simulations [152, 402]. The difference is most likely due to the limited size of the simulated volumes; while the simulations appropriately describe the reionization process within limited regions of the Universe, they are not sufficiently large to describe the global properties of the overlap phase [67]. The scales over which cosmological radiative transfer has been simulated are smaller than the characteristic extent of the SBO, which we find to be RSBO ~ 70 comoving Mpc.

We can constrain the scatter in the SBO redshift observationally using the spectra of the highest redshift quasars. Since only a trace amount of neutral hydrogen is needed to absorb Lyalpha photons, the time where the IGM becomes Lyalpha transparent need not coincide with bubble overlap. Following overlap the IGM was exposed to ionizing sources in all directions and the ionizing intensity rose rapidly. After some time the ionizing background flux was sufficiently high that the H I fraction fell to a level at which the IGM allowed transmission of resonant Lyalpha photons. This is shown schematically in Figure 61. The lower wavelength limit of the Gunn-Peterson trough corresponds to the Lyalpha wavelength at the redshift when the IGM started to allow transmission of Lyalpha photons along that particular line-of-sight. In addition to the SBO we therefore also define the Surface of Lyalpha Transmission (hereafter SLT) as the redshift along different lines-of-sight when the diffuse IGM became transparent to Lyalpha photons.

The scatter in the SLT redshift is an observable which we would like to compare with the scatter in the SBO redshift. The variance of the density field on large scales results in the biased clustering of sources [67]. H II regions grow in size around these clusters of sources. In order for the ionizing photons produced by a cluster to advance the walls of the ionized bubble around it, the mean-free-path of these photons must be of order the bubble size or larger. After bubble overlap, the ionizing intensity at any point grows until the ionizing photons have time to travel across the scale of the new mean-free-path, which represents the horizon out to which ionizing sources are visible. Since the mean-free-path is larger than RSBO, the ionizing intensity at the SLT averages the cosmic scatter over a larger volume than at the SBO. This constraint implies that the cosmic variance in the SLT redshift must be smaller than the scatter in the SBO redshift. However, it is possible that opacity from small-scale structure contributes additional scatter to the SLT redshift.

If cosmic variance dominates the observed scatter in the SLT redshift, then based on the spectra of the three z > 6.1 quasars [127, 381] we would expect the scatter in the SBO redshift to satisfy <Delta z2 >obs1/2 > 0.05. In addition, analysis of the proximity effect for the size of the H II regions around the two highest redshift quasars [396, 251] implies a neutral fraction that is of order unity (i.e. pre-overlap) at z ~ 6.2-6.3, while the transmission of Lyalpha photons at z < 6 implies that overlap must have completed by that time. This restricts the scatter in the SBO to be <Delta z2 >obs1/2 < 0.25. The constraints on values for the scatter in the SBO redshift are shaded gray in Figure 62. It is reassuring that the theoretical prediction for the SBO scatter of <Delta z2 >obs1/2 ~ 0.15, with a characteristic scale of ~ 70 comoving Mpc, is bounded by these constraints.

The possible presence of a significantly neutral IGM just beyond the redshift of overlap [396, 251] is encouraging for upcoming 21cm studies of the reionization epoch as it results in emission near an observed frequency of 200 MHz where the signal is most readily detectable. Future observations of redshifted 21cm line emission at 6 < z < 6.5 with instruments such as LOFAR, MWA, and PAST, will be able to map the three-dimensional distribution of HI at the end of reionization. The intergalactic H II regions will imprint a 'knee' in the power-spectrum of the 21cm anisotropies on a characteristic angular scale corresponding to a typical isolated H II region [404]. Our results suggest that this characteristic angular scale is large at the end of reionization, thetaSBO ~ 0.5 degrees, motivating the construction of compact low frequency arrays. An SBO thickness of <Delta z2 >1/2 ~ 0.15 suggests a minimum frequency band-width of ~ 8 MHz for experiments aiming to detect anisotropies in 21cm emission just prior to overlap. These results will help guide the design of the next generation of low-frequency radio observatories in the search for 21cm emission at the end of the reionization epoch.

The full size distribution of ionized bubbles has to be calculated from a numerical cosmological simulation that includes gas dynamics and radiative transfer. The simulation box needs to be sufficiently large for it to sample an unbiased volume of the Universe with little cosmic variance, but at the same time one must resolve the scale of individual dwarf galaxies which provide (as well as consume) ionizing photons (see discussion at the last section of this review). Until a reliable simulation of this magnitude exists, one must adopt an approximate analytic approach to estimate the bubble size distribution. Below we describe an example for such a method, developed by Furlanetto, Zaldarriaga, & Hernquist (2004) [143].

The criterion for a region to be ionized is that galaxies inside of it produce a sufficient number of ionizing photons per baryon. This condition can be translated to the requirement that the collapsed fraction of mass in halos above some threshold mass Mmin will exceed some threshold, namely Fcol > zeta-1. The minimum halo mass most likely corresponds to a virial temperature of 104 K relating to the threshold for atomic cooling (assuming that molecular hydrogen cooling is suppressed by the UV background in the Lyman-Werner band). We would like to find the largest region around every point that satisfies the above condition on the collapse fraction and then calculate the abundance of ionized regions of this size. Different regions have different values of Fcol because their mean density is different. In the extended Press-Schechter model (Bond et al. 1991 [52]; Lacey & Cole 1993 [212]), the collapse fraction in a region of mean overdensity deltaM is

Equation 169 (169)

where sigma2(M, z) is the variance of density fluctuations on mass scale M, sigmamin2 ident sigma2(Mmin, z), and deltac is the collapse threshold. This equation can be used to derive the condition on the mean overdensity within a region of mass M in order for it to be ionized,

Equation 170 (170)

where K(zeta) = erfc-1(1 - zeta-1). Furlanetto et al. [143] showed how to construct the mass function of ionized regions from deltab in analogy with the halo mass function (Press & Schechter 1974 [291]; Bond et al. 1991 [52]). The barrier in equation (170) is well approximated by a linear dependence on sigma2,

Equation 171 (171)

in which case the mass function has an analytic solution (Sheth 1998 [332]),

Equation 172 (172)

where bar{rho} is the mean mass density. This solution provides the comoving number density of ionized bubbles with mass in the range of (M, M + dM). The main difference of this result from the Press-Schechter mass function is that the barrier in this case becomes more difficult to cross on smaller scales because deltaB is a decreasing function of mass M. This gives bubbles a characteristic size. The size evolves with redshift in a way that depends only on zeta and Mmin.

One limitation of the above analytic model is that it ignores the non-local influence of sources on distant regions (such as voids) as well as the possible shadowing effect of intervening gas. Radiative transfer effects in the real Universe are inherently three-dimensional and cannot be fully captured by spherical averages as done in this model. Moreover, the value of Mmin is expected to increase in regions that were already ionized, complicating the expectation of whether they will remain ionized later. The history of reionization could be complicated and non monotonic in individual regions, as described by Furlanetto & Loeb (2005) [144]. Finally, the above analytic formalism does not take the light propagation delay into account as we have done above in estimating the characteristic bubble size at the end of reionization. Hence this formalism describes the observed bubbles only as long as the characteristic bubble size is sufficiently small, so that the light propagation delay can be neglected compared to cosmic variance. The general effect of the light propagation delay on the power-spectrum of 21cm fluctuations was quantified by Barkana & Loeb (2005) [29].

9.3. Separating the "Physics" from the "Astrophysics" of the Reionization Epoch with 21cm Fluctuations

The 21cm signal can be seen from epochs during which the cosmic gas was largely neutral and deviated from thermal equilibrium with the cosmic microwave background (CMB). The signal vanished at redshifts z > 200, when the residual fraction of free electrons after cosmological recombination kept the gas kinetic temperature, Tk, close to the CMB temperature, Tgamma. But during 200 > z > 30 the gas cooled adiabatically and atomic collisions kept the spin temperature of the hyperfine level population below Tgamma, so that the gas appeared in absorption [323, 226]. As the Hubble expansion continued to rarefy the gas, radiative coupling of Ts to Tgamma began to dominate and the 21cm signal faded. When the first galaxies formed, the UV photons they produced between the Lyalpha and Lyman limit wavelengths propagated freely through the Universe, redshifted into the Lyalpha resonance, and coupled Ts and Tk once again through the Wouthuysen-Field [388, 131] effect by which the two hyperfine states are mixed through the absorption and re-emission of a Lyalpha photon [237, 96]. Emission above the Lyman limit by the same galaxies initiated the process of reionization by creating ionized bubbles in the neutral cosmic gas, while X-ray photons propagated farther and heated Tk above Tgamma throughout the Universe. Once Ts grew larger than Tgamma, the gas appeared in 21cm emission. The ionized bubbles imprinted a knee in the power spectrum of 21cm fluctuations [404], which traced the H I topology until the process of reionization was completed [143].

The various effects that determine the 21cm fluctuations can be separated into two classes. The density power spectrum probes basic cosmological parameters and inflationary initial conditions, and can be calculated exactly in linear theory. However, the radiation from galaxies, both Lyalpha radiation and ionizing photons, involves the complex, non-linear physics of galaxy formation and star formation. If only the sum of all fluctuations could be measured, then it would be difficult to extract the separate sources, and in particular, the extraction of the power spectrum would be subject to systematic errors involving the properties of galaxies. Barkana & Loeb (2005) [28] showed that the unique three-dimensional properties of 21cm measurements permit a separation of these distinct effects. Thus, 21cm fluctuations can probe astrophysical (radiative) sources associated with the first galaxies, while at the same time separately probing the physical (inflationary) initial conditions of the Universe. In order to affect this separation most easily, it is necessary to measure the three-dimensional power spectrum of 21cm fluctuations. The discussion in this section follows Barkana & Loeb (2005) [28].

Spin temperature history

As long as the spin-temperature Ts is smaller than the CMB temperature Tgamma = 2.725 (1 + z) K, hydrogen atoms absorb the CMB, whereas if Ts > Tgamma they emit excess flux. In general, the resonant 21cm interaction changes the brightness temperature of the CMB by [323, 237] Tb = tau ( Ts - Tgamma) / (1 + z), where the optical depth at a wavelength lambda = 21cm is

Equation 173 (173)

where nH is the number density of hydrogen, A10 = 2.85 × 10-15 s-1 is the spontaneous emission coefficient, xHI is the neutral hydrogen fraction, and dvr / dr is the gradient of the radial velocity along the line of sight with vr being the physical radial velocity and r the comoving distance; on average dvr / dr = H(z) / (1 + z) where H is the Hubble parameter. The velocity gradient term arises because it dictates the path length over which a 21cm photon resonates with atoms before it is shifted out of resonance by the Doppler effect [341].

For the concordance set of cosmological parameters [348], the mean brightness temperature on the sky at redshift z is

Equation 174 (174)

where bar{x}HI is the mean neutral fraction of hydrogen. The spin temperature itself is coupled to Tk through the spin-flip transition, which can be excited by collisions or by the absorption of Lyalpha photons. As a result, the combination that appears in Tb becomes [131] (Ts - Tgamma) / Ts = [xtot / (1 + xtot)] (1 - Tgamma / Tk ), where xtot = xalpha + xc is the sum of the radiative and collisional threshold parameters. These parameters are xalpha = 4 Palpha T* / 27 A10 Tgamma and xc = 4 kappa1-0(Tk) nH T* / 3A10 Tgamma, where Palpha is the Lyalpha scattering rate which is proportional to the Lyalpha intensity, and kappa1-0 is tabulated as a function of Tk [11, 406]. The coupling of the spin temperature to the gas temperature becomes substantial when xtot > 1.

Brightness temperature fluctuations

Although the mean 21cm emission or absorption is difficult to measure due to bright foregrounds, the unique character of the fluctuations in Tb allows for a much easier extraction of the signal [154, 404, 259, 260, 314]. We adopt the notation deltaA for the fractional fluctuation in quantity A (with a lone delta denoting density perturbations). In general, the fluctuations in Tb can be sourced by fluctuations in gas density (delta), Lyalpha flux (through deltaxalpha) neutral fraction (deltaxHI), radial velocity gradient (deltadrvr), and temperature, so we find

Equation 175 (175)

where the adiabatic index is gammaa = 1 + (deltaTk / delta), and we define bar{x}tot ident (1 + xtot) xtot. Taking the Fourier transform, we obtain the power spectrum of each quantity; e.g., the total power spectrum PTb is defined by

Equation 176 (176)

where bar{delta}Tb (k) is the Fourier transform of deltaTb, k is the comoving wavevector, deltaD is the Dirac delta function, and < ... > denotes an ensemble average. In this analysis, we consider scales much bigger than the characteristic bubble size and the early phase of reionization (when bar{delta_x{HII}} << 1), so that the fluctuations deltaxHI are also much smaller than unity. For a more general treatment, see McQuinn et al. (2005) [250].

The separation of powers

The fluctuation deltaTb consists of a number of isotropic sources of fluctuations plus the peculiar velocity term -deltadrvr. Its Fourier transform is simply proportional to that of the density field [191, 41],

Equation 177 (177)

where µ = cos thetak in terms of the angle thetak of k with respect to the line of sight. The µ2 dependence in this equation results from taking the radial (i.e., line-of-sight) component (propto µ) of the peculiar velocity, and then the radial component (propto µ) of its gradient. Intuitively, a high-density region possesses a velocity infall towards the density peak, implying that a photon must travel further from the peak in order to reach a fixed relative redshift, compared with the case of pure Hubble expansion. Thus the optical depth is always increased by this effect in regions with delta > 0. This phenomenon is most properly termed velocity compression.

We therefore write the fluctuation in Fourier space as

Equation 178 (178)

where we have defined a coefficient beta by collecting all terms propto delta in Eq. (175), and have also combined the terms that depend on the radiation fields of Lyalpha photons and ionizing photons, respectively. We assume that these radiation fields produce isotropic power spectra, since the physical processes that determine them have no preferred direction in space. The total power spectrum is

Equation 179 (179)

where we have defined the power spectrum Pdelta . rad as the Fourier transform of the cross-correlation function,

Equation 180 (180)

We note that a similar anisotropy in the power spectrum has been previously derived in a different context, i.e., where the use of galaxy redshifts to estimate distances changes the apparent line-of-sight density of galaxies in redshift surveys [191, 219, 178, 133]. However, galaxies are intrinsically complex tracers of the underlying density field, and in that case there is no analog to the method that we demonstrate below for separating in 21cm fluctuations the effect of initial conditions from that of later astrophysical processes.

The velocity gradient term has also been examined for its global effect on the sky-averaged power and on radio visibilities [366, 41]. The other sources of 21cm perturbations are isotropic and would produce a power spectrum PTb(k) that could be measured by averaging the power over spherical shells in k space. In the simple case where beta = 1 and only the density and velocity terms contribute, the velocity term increases the total power by a factor o < (1 + µ2)2 > = 1.87 in the spherical average. However, instead of averaging the signal, we can use the angular structure of the power spectrum to greatly increase the discriminatory power of 21cm observations. We may break up each spherical shell in k space into rings of constant µ and construct the observed PTb(k,µ). Considering Eq. (179) as a polynomial in µ, i.e., µ4 Pµ4 + µ2 Pµ2 + Pµ0, we see that the power at just three values of µ is required in order to separate out the coefficients of 1, µ2, and µ4 for each k.

If the velocity compression were not present, then only the µ-independent term (times Tb2) would have been observed, and its separation into the five components (Tb, beta, and three power spectra) would have been difficult and subject to degeneracies. Once the power has been separated into three parts, however, the µ4 coefficient can be used to measure the density power spectrum directly, with no interference from any other source of fluctuations. Since the overall amplitude of the power spectrum, and its scaling with redshift, are well determined from the combination of the CMB temperature fluctuations and galaxy surveys, the amplitude of Pµ4 directly determines the mean brightness temperature Tb on the sky, which measures a combination of Ts and bar{x}HI at the observed redshift. McQuinn et al. (2005) [250] analysed in detail the parameters that can be constrained by upcoming 21cm experiments in concert with future CMB experiments such as Planck ( Once Pdelta(k) has been determined, the coefficients of the µ2 term and the µ-independent term must be used to determine the remaining unknowns, beta, Pdelta . rad(k), and Prad(k). Since the coefficient beta is independent of k, determining it and thus breaking the last remaining degeneracy requires only a weak additional assumption on the behavior of the power spectra, such as their asymptotic behavior at large or small scales. If the measurements cover Nk values of wavenumber k, then one wishes to determine 2 Nk + 1 quantities based on 2 Nk measurements, which should not cause significant degeneracies when Nk >> 1. Even without knowing beta, one can probe whether some sources of Prad(k) are uncorrelated with delta; the quantity Pun-delta(k) ident Pµ0 - Pµ22 / (4 Pµ4) equals Prad - Pdelta . rad2 / Pdelta, which receives no contribution from any source that is a linear functional of the density distribution (see the next subsection for an example).

Specific epochs

At z ~ 35, collisions are effective due to the high gas density, so one can measure the density power spectrum [226] and the redshift evolution of nHI, Tgamma, and Tk. At z < 35, collisions become ineffective but the first stars produce a cosmic background of Lyalpha photons (i.e. photons that redshift into the Lyalpha resonance) that couples Ts to Tk. During the period of initial Lyalpha coupling, fluctuations in the Lyalpha flux translate into fluctuations in the 21cm brightness [30]. This signal can be observed from z ~ 25 until the Lyalpha coupling is completed (i.e., xtot >> 1) at z ~ 15. At a given redshift, each atom sees Lyalpha photons that were originally emitted at earlier times at rest-frame wavelengths between Lyalpha and the Lyman limit. Distant sources are time retarded, and since there are fewer galaxies in the distant, earlier Universe, each atom sees sources only out to an apparent source horizon of ~ 100 comoving Mpc at z ~ 20. A significant portion of the flux comes from nearby sources, because of the 1 / r2 decline of flux with distance, and since higher Lyman series photons, which are degraded to Lyalpha photons through scattering, can only be seen from a small redshift interval that corresponds to the wavelength interval between two consecutive atomic levels.

There are two separate sources of fluctuations in the Lyalpha flux [30]. The first is density inhomogeneities. Since gravitational instability proceeds faster in overdense regions, the biased distribution of rare galactic halos fluctuates much more than the global dark matter density. When the number of sources seen by each atom is relatively small, Poisson fluctuations provide a second source of fluctuations. Unlike typical Poisson noise, these fluctuations are correlated between gas elements at different places, since two nearby elements see many of the same sources. Assuming a scale-invariant spectrum of primordial density fluctuations, and that xalpha = 1 is produced at z = 20 by galaxies in dark matter halos where the gas cools efficiently via atomic cooling, Figure 63 shows the predicted observable power spectra. The figure suggests that beta can be measured from the ratio Pµ2 / Pµ4 at k > 1 Mpc-1, allowing the density-induced fluctuations in flux to be extracted from Pµ2, while only the Poisson fluctuations contribute to Pun-delta. Each of these components probes the number density of galaxies through its magnitude, and the distribution of source distances through its shape. Measurements at k > 100 Mpc-1 can independently probe Tk because of the smoothing effects of the gas pressure and the thermal width of the 21cm line.

Figure 63

Figure 63. Observable power spectra during the period of initial Lyalpha coupling. Upper panel: Assumes adiabatic cooling. Lower panel: Assumes pre-heating to 500 K by X-ray sources. Shown are Pµ4 = Pdelta (solid curves), Pµ2 (short-dashed curves), and Pun-delta (long-dashed curves), as well as for comparison 2 beta Pdelta (dotted curves).

After Lyalpha coupling and X-ray heating are both completed, reionization continues. Since beta = 1 and Tk >> Tgamma, the normalization of Pµ4 directly measures the mean neutral hydrogen fraction, and one can separately probe the density fluctuations, the neutral hydrogen fluctuations, and their cross-correlation.

Fluctuations on large angular scales

Full-sky observations must normally be analyzed with an angular and radial transform [143, 314, 41], rather than a Fourier transform which is simpler and yields more directly the underlying 3D power spectrum [259, 260]. The 21cm brightness fluctuations at a given redshift - corresponding to a comoving distance r0 from the observer - can be expanded in spherical harmonics with expansion coefficients alm(nu), where the angular power spectrum is

Equation 181 (181)

with Gl(x) ident Jl(x) + (beta - 1) jl(x) and Jl(x) being a linear combination of spherical Bessel functions [41].

In an angular transform on the sky, an angle of theta radians translates to a spherical multipole l ~ 3.5 / theta. For measurements on a screen at a comoving distance r0, a multipole l normally measures 3D power on a scale of k-1 ~ theta r0 ~ 35/l Gpc for l >> 1, since r0 ~ 10 Gpc at z > 10. This estimate fails at l < 100, however, when we consider the sources of 21cm fluctuations. The angular projection implied in Cl involves a weighted average (Eq. 181) that favors large scales when l is small, but density fluctuations possess little large-scale power, and the Cl are dominated by power around the peak of k Pdelta(k), at a few tens of comoving Mpc.

Figure 64

Figure 64. Effect of large-scale power on the angular power spectrum of 21cm anisotropies on the sky. This example shows the power from density fluctuations and velocity compression, assuming a warm IGM at z = 12 with Ts = Tk >> Tgamma. Shown is the % change in Cl if we were to cut off the power spectrum above 1 / k of 200, 180, 160, 140, 120, and 100 Mpc (top to bottom). Also shown for comparison is the cosmic variance for averaging in bands of Delta l ~ l (dashed lines).

Figure 64 shows that for density and velocity fluctuations, even the l = 1 multipole is affected by power at k-1 > 200 Mpc only at the 2% level. Due to the small number of large angular modes available on the sky, the expectation value of Cl cannot be measured precisely at small l. Figure 64 shows that this precludes new information from being obtained on scales k-1 > 130 Mpc using angular structure at any given redshift. Fluctuations on such scales may be measurable using a range of redshifts, but the required Delta z > 1 at z ~ 10 implies significant difficulties with foreground subtraction and with the need to account for time evolution.

Next Contents Previous