| { | |
| "Abstract": [ | |
| "Abstract. We investigate cosmological signatures of uncorrelated isocurvature perturbations whose power spectrum is blue-tilted with spectral index 2 . p. 4. Such an isocurvature power spectrum can promote early formation of small-scale structure, notably dark matter halos and galaxies, and may thereby resolve the shortage of ionizing photons suggested by observations of galaxies at high redshifts (z≃7 −8) but that are required to reionize the universe at z∼10.We mainly focus on how the formation of dark matter halos can be modified, and explore the connection between the spectral shape of CMB anisotropies and the reionization optical depth as a powerful probe of a highly blue-tilted isocurvature primordial power spectrum. We also study the consequences for 21cm line fluctuations due to neutral hydrogens in minihalos. Combination of measurements of the reionization optical depth and 21cm line fluctuations will provide complementary probes of a highly blue-tilted isocurvature power spectrum. " | |
| ], | |
| "Contents ": [], | |
| "1 Introduction ": [ | |
| "The epoch of reionization is a milestone in cosmological structure formation history. Although recent cosmological observations are beginning to reveal the epoch of reionization, the process of cosmological reionization is still one of the most open questions in cosmology. In particular, observations of star-forming galaxies at high redshifts 6 . z. 8, suggest that the density of UV photons emitted from such galaxies may not be suÿcient to keep the universe ionized [(<>)1], if the number density of the galaxies are as predicted by the concordance cosmological model with a nearly scale-invariant curvature power spectrum consistent with current CMB observations [(<>)2, (<>)3], unless very optimistic assumptions are made about ionizing photon escape fractions and other parameters. Indeed this conclusion is further aggravated by the decline in star formation rate density recently found beyond z∼8 [(<>)4]. ", | |
| "To enhance the formation of collapsed objects in the high redshift universe, one possible solution is to allow matter fluctuations to have a large amplitude at small scales. However, in the inflationary cosmology, there are few degrees of freedom available to generate large curvature perturbations at small scales since they are found to be nearly-scale invariant on the scales of CMB and galaxy clustering (k. 0.1Mpc−1). On the other hand, isocurvature perturbations can have a highly blue-tilted power spectrum while satisfying CMB constraints on large scales1 (<>). ", | |
| "In this study, we consider how such blue-tilted isocurvature power spectra can a ect cosmological observables at high redshifts, and in particular the CMB via the reionization of the universe. Such an isocurvature power spectrum can promote the formation of dark matter halos and enhance the formation of stars and galaxies in the high-redshift universe. While there are several previous studies on various aspects of these issues [(<>)5, (<>)9–(<>)11]2 (<>), we here focus on e ects on reionization and 21cm fluctuations due to nonlinearities triggered by isocurvature power spectra and especially on the complementarity of these probes. Although several authors have discussed e ects of isocurvature power spectra on 21cm fluctuations [(<>)14, (<>)15], they have generally focused on the contribution of a homogeneous IGM well before the epoch of reionization (z& 20). Here we focus on collapsed halos which make dominant contributions at relatively low redshifts (z. 20). ", | |
| "Throughout this paper, we focus on matter isocurvature perturbations, where S(x) = δm −43 δ(δi is the initial density perturbation in a component i) that is uncorrelated with curvature fluctuations, ζ(x), i.e. S(x)ζ(x′)= 0. We in addition assume that there are no initial relative perturbations between baryon and CDM or so-called compensated isocur-vature perturbations [(<>)15]3 (<>), i.e. δc = δb. The power spectrum is defined as S(k)S(k′)= Piso(k)(2π)3 k). We assume the power spectrum Piso(k) is in power-law form, so thatδ(3)(k+′it can be parameterized as ", | |
| "(1.1) ", | |
| "where Aiso and niso are the amplitude and the spectral index of Piso(k) at the pivot scale k0 = 0.002Mpc−1 , which roughly corresponds to ℓ= 10 in the CMB angular power spectrum. We define the ratio of the isocurvature to adiabatic power spectrum, α= Aiso/Aadi, where the adiabatic power spectrum is given by Hereafter we use two parameters αand p= niso to characterize the model with an isocurvature power spectrum. ", | |
| "The CMB and matter power spectra are computed using the CAMB code [(<>)17, (<>)18]. We take the concordance flat CDM model with power-law adiabatic power spectrum as the baseline cosmological model and set parameters to the best fit values from the WMAP 9-year result [(<>)2]. ", | |
| "This paper is organized as follows. To begin with, we briefly discuss current constraints on the blue-tilted isocurvature power spectrum from the shape of the CMB power spectrum in the next section. In order to see how such an isocurvature power spectrum a ects the late-time universe, we first investigate its e ects on the mass function of dark matter halos in Section (<>)3. For observational signatures, we focus on the reionization optical depth and 21cm line fluctuations, which are discussed in Section (<>)4 and (<>)5, respectively. A final section is devoted to our conclusions. " | |
| ], | |
| "2 CMB power spectrum and constraints ": [ | |
| "First, we see to what extent an isocurvature power spectrum is constrained from measured shapes of the CMB power spectrum. In Fig. (<>)1, we plot the CMB power spectrum with several di erent values of the amplitude and spectral indices of the isocurvature power spectrum. Fig. (<>)2 shows 1 and 2σconstraints in the p-αplane from the observed CMB power spectrum. We adopt the likelihood functions from the WMAP 9-year results [(<>)2, (<>)19] and the ACT 2008 results [(<>)20, (<>)21]. From the figure, we can see that for p. 3, WMAP is more sensitive than ACT. On the other hand, as the spectral index becomes larger p& 4, the CMB spectrum at higher multipoles ℓ& 1000, which can be measured by ACT, becomes more sensitive. ", | |
| "We however should note that here we have fixed other cosmological parameters and in addition contributions of foregrounds components that include the thermal Sunyaev-Zel’dovich e ect and point-sources. Varying these may non-negligibly loosen the constraints, in particular for ACT, which observes small angular scales where these foregrounds dominate primary anisotropies. In e ect, our constraints are at least an order of magnitude stronger than the constraints presented by Planck [(<>)22], although it is diÿcult to compare these with ", | |
| "Figure 1. CMB power spectrum for di erent amplitudes and spectral indices of the isocurvature power spectrum. In order from top to bottom, the temperature, E-mode polarization and their cross-correlation power spectra are plotted. Cases of p= 2, 3, 4 are shown in order from left to right. In each panel, cases of α= 10−1 (red solid), 10−3 (green dashed), 10−5 (blue dotted) are plotted. Thick and thin lines show contributions from the isocurvature power spectrum alone and the sum of the adiabatic and isocurvature spectra, respectively. ", | |
| "our constraints in a straightforward way, as the parameterizations for the isocurvature power spectrum are di erent 4 (<>). Our constraints here should be regarded as very optimistic. " | |
| ], | |
| "3 Mass function ": [ | |
| "Since the CMB power spectrum can measure primordial perturbations only on large scales, isocurvature perturbations are allowed to have large amplitudes on small scales if the power spectrum is blue-tilted. Such isocurvature perturbations with a blue-tilted spectrum can significantly a ect structure formation on small scales and can increase the number of dark ", | |
| "Figure 2. Constraints on the amplitude αand the spectral index pof the isocurvature power spectrum from CMB datasets. Results from WMAP9 (red) ACT2008 (green) and WMAP9+ACT2008 (blue) are shown The contours and shaded regions respectively show parameter regions excluded at 1 and 2σlevels. ", | |
| "matter halos in the high-redshift universe. To investigate this e ect, we evaluate the mass function of dark matter halos for di erent isocurvature parameters in this section. ", | |
| "According to the Press-Schechter formalism [(<>)23], the mass function of dark matter halos is given by ", | |
| "(3.1) ", | |
| "where ρ¯m is the mean energy density of matter, f(ν) is the occupation function and we define ν≡(δcr/σ(M))2 with δcr and σ(M) respectively being the critical overdensity and the variance of matter fluctuations in a sphere of radius Rsatisfying M= 43ˇ ρ¯mR3 . ", | |
| "Given a matter power spectrum P(k), σ(M) is obtained by ", | |
| "(3.2) ", | |
| "where W(kR) is a top-hat window function with radius R, ", | |
| "(3.3) ", | |
| "Regarding the occupation function f(ν), we adopt the one proposed by Sheth and Tor-men [(<>)24], ", | |
| "(3.4) ", | |
| "where a= 0.3, q= 0.75 and A(a) is the normalization so that occupation function satisfies dνf(ν) = 1 or, equivalently, dMd ln dnM = ρ¯ m. ", | |
| "In Figs. (<>)3-(<>)5, we plot the mass function for various isocurvature power spectra specified by parameter sets (α,p). In these figures, pis varied from 2 to 4. In each figure, mass functions at di erent redshifts z= 20,16,12,8,4 are plotted from the top left panel to the bottom right panel. In each panel, di erent lines correspond to di erent amplitudes of the isocurvature power spectrum α. ", | |
| "From Fig. (<>)3, we can see that, for moderately blue-tilted isocurvature power spectra with p= 2, the e ects of isocurvature perturbations can be seen only for large amplitude α& 10−1; for smaller αthe mass function can hardly be distinguished from that for the vanilla model (the purely adiabatic case). For α& 10−1 , the presence of the isocurvature power spectrum amplifies the mass function independently of mass in the range [1M⊙,1011M⊙]. On the other hand, from Figs. (<>)4 and (<>)5, which show the e ects of more blue-tilted isocurvature spectra with p= 3 and p= 4, we can see qualitative di erences from those for p= 2. At larger masses M→1011M⊙, we see that the mass function is enhanced monotonically as αincreases, which is also true for p= 2 as already seen above. However, at the lower mass end M→1M⊙, the e ects on the mass function become more complicated: as αincreases from zero, the mass function first increases and, at some point in alpha, it tends to decrease. At smaller redshift and/or with larger p, the mass function at M= 1M⊙can become even smaller than that of the vanilla model with α= 0. ", | |
| "This somewhat counterintuitive behavior of the mass function comes from the normalization dMd ln dnM = ρ¯ m. To see this, it is convenient to define a critical mass Mcr as ", | |
| "(3.5) ", | |
| "From Eq. ((<>)3.4) as well as Figs. (<>)3-(<>)5, one can see that halos with masses up to Mcr or equivalently ν= 1 can be abundantly produced while the number of halos with larger masses are suppressed exponentially. In particular, Figs. (<>)4 and (<>)5 show that the presence of large isocurvature perturbations with blue spectra makes Mcr much larger than in the pure adiabatic vanilla model. Given fixed total mass in the universe ρ¯m = dMd ln dnM , any increase in the number of halos with large masses should be compensated with a decrease in those with small masses. ", | |
| "However, the above explanation may not be satisfactory enough as it lacks a physical pictures, which we try to describe below. First, we divide space into cells with comoving volume V1. We also consider the division of space into larger cells with V2 (>V1). The variance of matter fluctuations for V1 and V2 is given by σ(M1) and σ(M2) with the averaged mass M1 = ρ¯mV1 and M2 = ρ¯mV2 for V1 and V2, respectively. ", | |
| "Let us consider the case for which, while σ(M1) >δcr, σ(M2) <δcr (low density fluctuation case). When we divide space into cells with V1, the actual density fluctuations in a cell are larger than δcr in some cells, and dark halos are formed in such cells. For the division into larger cells, the actual density fluctuations in a large cell V2 cannot exceed δc. Therefore, large cells cannot collapse to dark halos (as discussed above, the number destiny of halos for M2 are suppressed exponentially). ", | |
| "Suppose that both σ(M1) and σ(M2) are larger than δcr (large density fluctuation case). Dividing space into small cells with V1, we still find that actual density fluctuations in a cell are larger than δcr in some cells, and dark matter halos are formed in such cells. We consider a large cell with V2 which includes such small cells. Since σ(M2) >δc, this large cell can have density fluctuations larger than δcr and a large halo is formed in this cell. In the formation of a large halo, small halos in the cell are merged into a large halo. ", | |
| "Therefore, as the density fluctuations are amplified, the number density of large halos increases. On the other hand, the number density of small halos decreases because small halos merge to form a large halo (or due to the normalization as discussed above). As we will see in Section (<>)5, this specific response of the mass function to αat relatively small masses M. 107M⊙leads to nontrivial behavior in 21 cm fluctuations. ", | |
| "Figure 3. Mass function dn/dln Mfor the case of isocurvature power spectrum with p= 2. In order from top-left to bottom-right, the panels show mass functions at redshifts z=20, 15, 10, 5. Each line corresponds to a di erent value of the amplitude of isocurvature power spectrum α= 0.1 (red solid), 1 ×10−3 (green short-dashed), 1 ×10−5 (blue dotted), 1 ×10−7 (magenta long-dashed), 1 ×10−9 (light blue dot-dashed), 1 ×10−11 (orange two-dot-chain). As a reference, the pure adiabatic case is also plotted (black three-dot-chain). " | |
| ], | |
| "4 Optical depth ": [ | |
| "As we have seen in the previous section, the presence of a blue-tilted isocurvature power spectrum significantly alters the number density of halos at high redshifts. In particular, the formation of massive halos is enhanced, which results in an increase in the number density of ionizing sources, or galaxies, at this epoch. This a ects the reionization history of the universe, to be discussed in this section. ", | |
| "To see how the isocurvature power spectrum a ects the reionization history, we here follow the analysis of Ref. [(<>)25]. The ionized fraction of IGM, χ, should be given by ", | |
| "(4.1) ", | |
| "where fs is the fraction of matter in the collapsed objects which can host galaxies emitting ionizing UV photons, fuvpp is the number of UV photons emitted into the IGM per proton in the collapsed objects, and fion is the net eÿciency of ionization by a single UV photon. Noting that a fraction 0.0073 of the rest mass is released in stellar burning of hydrogen to ", | |
| "Figure 4. Same as in Fig. (<>)3 but for p= 3. ", | |
| "helium, fuvpp can be further decomposed into ", | |
| "(4.2) ", | |
| "where mp is the proton mass, 1 −Yp ≃0.75 is the mass fraction of hydrogen in IGM, fburn is the mass fraction of hydrogen burned, fuv is the fraction of energy released as UV photons, and fesc is the fraction of UV photons that escape from galaxies. To compile parameters whose values are not precisely known, we define a parameter ", | |
| "(4.3) ", | |
| "which has a fairly large uncertainty. Then Eq. ((<>)4.1) can be rewritten as ", | |
| "(4.4) ", | |
| "The fraction of matter in collapsed objects fs should be given by ", | |
| "(4.5) ", | |
| "where M∗(z) corresponds to the minimum mass of collapsed objects whose virial temperature should be larger than 104K, above which atomic cooling becomes e ective. According to ", | |
| "Figure 5. Same as in Fig. (<>)3 but for p= 4. ", | |
| "Ref. [(<>)26], M∗(z) can be approximately given as ", | |
| "(4.6) ", | |
| "We note that similar calculation of reionization history is done in Ref. [(<>)11] for a blue-tilted isocurvature power spectrum. In particular for fnet = 10−4 , the reionization model we adopted here is almost the same as the model for POP II star in the reference. In this case, we indeed confirm that evolution of the reionization fraction in our calculation show a good agreement with Ref. [(<>)11]. ", | |
| "The optical depth of reionization is given by ", | |
| "(4.7) ", | |
| "where n¯ p(t) and σT are the mean number density of protons and the cross-section for Thomson scattering. ", | |
| "Fig. (<>)6 shows the contour plot of the optical depth τreion in the p-αplain and constraints on these parameters from the WMAP9 estimate of the optical depth τreion = 0.09±0.04 (95% C.L.) [(<>)2]5 (<>). The lowest value of the reionization eÿciency fnet = 10−6 we take here should be regarded as a case for very ineÿcient reionization following the discussion of Ref. [(<>)25]. ", | |
| "Figure 6. Constraints on the isocurvature power spectrum from considerations on reionization. Here we take the eÿciency parameter for reionization fnet to 10−6 (red), 10−5 (green) and 10−4 (blue), and show contours of the reionization optical depth τreion for each fnet. For each value of fnet, shaded region shows parameter regions which give τreion ≥0.13, while thick solid and thin dashed lines respectively show parameter regions which give τreion = 0.09 and 0.05. Note that for fnet = 10−4 , τreion exceeds 0.05 even if the isocurvature power spectrum vanishes (i.e. α= 0), so that the blue thin dashed line does not appear in the plot. As reference, 1 and 2σconstraints from the CMB dataset (WMAP9+ACT2008) shown in Fig. (<>)2 are also plotted (black solid line). ", | |
| "From the figure, we can see that optical depth of reionization is sensitive to highly blue-tilted isocurvature spectrum with p& 3. Even if we assume a very low value of fnet = 10−6 , αshould not be large for p& 3. We found that WMAP constraints on τreion <0.13 (95% C.L.) leads to a constraint ", | |
| "(4.8) ", | |
| "which is valid for 10−6 ≤fnet ≤10−4 . Compared with the constraints directly obtained from the CMB power spectrum from WMAP9+ACT2008, Fig. (<>)6 shows that considerations of reionization history can give strong constraints on blue-tilted isocurvature power spectrum for large spectral indices p& 3 despite the large uncertainties in reionization eÿciency fnet. " | |
| ], | |
| "5 21cm fluctuations ": [ | |
| "Here we discuss signatures of a blue-tilted isocurvature power spectrum on redshifted 21cm line fluctuations. As we discussed in the previous section, a blue-tilted isocurvature power spectrum induces small-mass halos. Among them, we mainly focus on “minihalos” whose mass is too small to host a galaxy. Although minihalos are not considered to be luminous objects, a minihalo can make observable 21 cm line signals [(<>)26, (<>)27]. Our analysis is basically the same as those in Refs. [(<>)26, (<>)28]. As assumed in these references, we also assume that minihalos are modeled as a “truncated isothermal sphere” with physical radius rt(M), gas ", | |
| "temperature TK(l,M), dark matter velocity dispersion σV(l,M) and density profile ρ(l,M), where lis the distance from the center of a minihalo. We denote the frequency of 21cm line emission in the rest frame as ν0 = 1.4 GHz. We also note that in this section, any parameter dependences of quantities are significantly abbreviated. ", | |
| "Let us consider the brightness temperature of photons with frequency ν′at redshift zwhich penetrate a minihalo of mass Mwith an impact parameter r. The brightness temperature is given by ", | |
| "(5.1) ", | |
| "Here τ21cm(R) is the 21cm optical depth along the photon path at R, which is obtained from6 (<>)", | |
| "(5.2) ", | |
| "√where l′= R′2 + r2 , A10 = 2.85×10−15 s−1 and kBT∗= hν0 = 5.9×10−6 eV are respectively the spontaneous decay rate and emitted energy of the 21cm hyperfine transition, and nHI is the number density of neutral hydrogen atoms. The function φ(ν′,l′) is the line profile, which can be modeled by a thermal Doppler-broadening with ", | |
| "ν= (ν0/c) 2kBTK(l′)/mH , where mH is the hydrogen mass. The optical depth τ21cm in Eq. ((<>)5.1) is given as τ21cm = τ21cm(R→∞). The di erential brightness temperature observed today at frequency ν= ν′/(1 + z) can be written as ", | |
| "(5.3) ", | |
| "The mean surface brightness temperature for a halo with mass Mat zis provided by ", | |
| "(5.4) ", | |
| "where A(M,z) is the geometric cross-section of a halo with mass Mat z, A= πrt2 . The di erential line-integrated flux from this halo is given by [(<>)26] ", | |
| "(5.5) ", | |
| "For an optically thin halo, we can replace the integration with the multiplication by νe (z,b) ≡1 at ν′= ν0.(1+z)°(0,b) ", | |
| "Now we consider an observation with a finite bandwidth νand beam width θ. The mean di erential flux per unit frequency is expressed as ", | |
| "(5.6) ", | |
| "where ( )beam = π(θ/2)2 , and ν/z= ν0/(1 + z)2 , and Mmax(z) and Mmin(z) are respectively the maximum and minimum masses contributing to the 21cm line emission. ", | |
| "Table 1. Survey parameters for the SKA survey ", | |
| "The maximum mass Mmax(z) is set to the virial mass M∗(z) in Eq. ((<>)4.6) for which a virial temperature is 104K. Above M∗(z), most of the hydrogen in halos is ionized and do not contribute to 21cm line emission (This is also consistent with the discussion we presented in Section (<>)4). On the other hand, the minimum mass is set to the Jeans mass ", | |
| "(5.7) ", | |
| "Defining the beam-averaged “e ective” di erential antenna temperature δTb by dδF/dν= 2ν2kBδTb( )beam, we obtain ", | |
| "(5.8) ", | |
| "So far, we have assumed that the halo number density is homogeneous, following dn/dM. However, halos cluster, depending on the density fluctuations, ", | |
| "(5.9) ", | |
| "where δN (M) is the number density contrast of halos with mass Mand b(M) is bias. This clustering causes fluctuations in δTb(ν). Since the rms density fluctuations in a beam are given by ", | |
| "(5.10) ", | |
| "where Wis a pencil beam window function at frequency νwith band width νand beam width θ(See e.g. Ref. [(<>)29]), the rms halo number density fluctuations with Min a beam is obtained from ", | |
| "(5.11) ", | |
| "Therefore, the rms fluctuations of δTb(ν) are provided by ", | |
| "(5.12) ", | |
| "where β(z) is the e ective bias of the minihalos weighted by their 21cm line fluxes, defined as ", | |
| "(5.13) ", | |
| "where F(z,M) ≡d2bδTb(z,b,M) ∝Tbrt2σV [(<>)28] is the flux from a minihalo. ", | |
| "In Fig. (<>)7, we plot 3σTb as a function of 6 <z= ¯ erent parameterν/ν0 −1 <20 for di values for the isocurvature power spectrum (p,α). The adopted resolution and bandwidth ", | |
| "assume a SKA-like survey whose survey parameters are summarized in Table. (<>)1. As a reference, the noise level of the SKA-like survey is also depicted in the figure, and is given by [(<>)30] ", | |
| "(5.14) ", | |
| "where Ais the antenna collecting area and tis the integration time. From the figure, we first see that, for p= 2, the signal σTb monotonically increases as αdoes (though we cannot distinguish lines for α<10−2). On the other hand, for p= 3 and p= 4, we can see the dependence of σTb on αis more complicated; As αincreases from zero, σTb also grows. When αexceeds some value depending on redshift, increment of αmakes σTb decrease. However, keeping αincreasing, one can find that σTb starts to increase again. In particular, what may be remarkable is that the signal can even be smaller than the vanilla model at low redshifts. As we will show in the following, this complicated behavior arises mainly from the mass function which has a non-trivial dependence on αfor large p& 3 as shown in section (<>)3. We also note that the redshift evolution of σTb becomes quantitatively di erent for large values of α; while σTb increases as the universe ages for small α, it tends to evolve less for large α. ", | |
| "To understand the complicated dependence of σTb on αand p, we divide σTb into parts σ(ν,ν,θ), β(z) and δTb(z) and investigate each of them separately. In Fig. (<>)8, we plot the rms of smoothed matter fluctuations σ(ν,ν,θ), the e ective bias β(z) and the mean 21cm brightness temperature δTb(ν) as a function of z= ν0/ν−1 for di erent values of pand α. Here νand θare fixed to the values adopted in Fig. (<>)7. ", | |
| "As expected, σ(ν,ν,θ) is monotonically enhanced as αincreases. If we fix α, the enhancement is more significant for larger p. However, as long as αis not so large as to be excluded by current observations (See Figs. (<>)2), the enhancement is not significant and at most O(1). We can also see that β(z) is not significantly a ected by the isocurvature power spectrum; for 2 <p<4, β(z) changes at most 30 % as long as α<0.1. ", | |
| "On the other hand, as seen in the figure, δTb(ν) has a complex response to αdepending on p. For a moderately blue-tilted isocurvature spectrum with p= 2, δTb(ν) monotonically increases as αincreases. On the other hand, for p= 3 and p= 4, the dependence of δTb(ν) on αdramatically changes. As αincreases from zero, while at first δTb(ν) increases, then it begins to decrease at some value of α, which depends on p. Since the isocurvature power spectrum a ects δTb(ν) only through the mass function, this behavior can be understood by looking at the mass function dn/dln Mwith MJ(z) <M<M∗(z). As we discussed in Section (<>)3, an increase in αdoes not necessarily increase dn/dln Mfor a highly blue-tilted isocurvature power spectrum and, in particular, dn/ln Mfor large αcan be below that for the vanilla model α= 0 at relatively small masses 103M⊙. M. 107M⊙(this roughly corresponds to MJ(z) <M<M∗(z) for z≃10). Since minihalos which contribute to the 21cm brightness temperature are limited to those with these small masses, δTb(ν) reflects this complicated behavior of the mass function. ", | |
| "To summarize, the complicated dependence of σTb on αis understood as follows. A blue-tilted isocurvature power spectrum a ects σTb mainly through the rms smoothed mass fluctuations σ(ν,ν,θ) and the mean di erential brightness temperature δTb(ν). For a moderately blue-tilted isocurvature spectrum p= 2, σ(ν,ν,θ) and δTb increase monotonically as αincreases. On the other hand, for a highly blue-tilted spectrum p= 3 or p= 4, while σ(ν,ν,θ) continuously monotonically increases as αincreases, δTb decreases at some α. As long as αis not so large as to be excluded by current observations, σTb first increases ", | |
| "Figure 7. Shown are the rms 21cm di erential brightness fluctuations 3σTb for models with blue-tilted isocurvature power spectra as a function of redshift. In order from left to right, the spectral index of isocurvature power spectrum pis varied from 2 to 4. In each panel, cases of di erent values of α= 10−1 (red solid), 10−3 (green short-dashed), 10−5 (blue dotted), 10−7 (magenta long-dashed), 10−9 (green dot-dashed), 10−11 (blue two-dot-chain), and the pure adiabatic case α= 0 (black three-dot-chain) are plotted. For each parameter set (p,α), we indicate the reionization optical depth computed in Section (<>)4 assuming fnet = 10−6 (former) and 10−4 (latter) in the legend with square bracket. As a reference, we also depict the noise level of a SKA-like survey with survey parameters summarized in Table (<>)1 (thin black solid). ", | |
| "and then decreases, mainly reflecting the dependence of δTb. When αis significantly large so that it is already excluded by current observations, the influence of σ(ν,ν,θ) dominates and σTb again starts to increase. ", | |
| "Finally, we compute χ2 , assuming that SKA can observe 21cm line fluctuations over 5 ≤z≤20. Here we simply assume that each band in the observed frequency range is independent and has the same survey parameters as in Table (<>)1. Then χ2 can be given as ", | |
| "(5.15) ", | |
| "where σT(fid) b denotes the rms 21cm line emission fluctuations in the case of the fiducial pure adiabatic model. In Fig. (<>)9, we plot the 2σconstraint expected from the SKA survey. As a reference, the 2σconstraints from the reionization optical depth with a conservative eÿciency parameter fnet = 10−6 and the CMB power spectrum are also shown here. In the figure, one can see that the 2σexcluded region from 21cm line fluctuations has a complex geometry and a region with relatively large α∼10−6-10−3 with p& 3.5 is not excluded by 21cm line observations alone. This reflects the complicated dependence of the 21cm line fluctuations on the isocurvature power spectrum. However, the constraint from the reionization optical depth covers the region. This shows that the 21cm line fluctuations and the reionization optical depth can be complementary as probes of a blue-tilted isocurvature power spectrum. Combining the constraint from the WMAP determination of τreion with fnet >10−6 , we found that the SKA is expected to give a constraint ", | |
| "(5.16) ", | |
| "at the 2σlevel. In the same figure, we also plot bands of regions corresponding to τreion = 0.09 ±0.01 (2σ), which is expected to be achieved by Planck observations of the CMB polar-", | |
| "Figure 8. The three components in Eq. ((<>)5.12) are plotted separately. In order from top to bottom, the rms smoothed matter fluctuations σ(¯ν,ν,θ), the e ective bias β(z) and the mean value of the 21cm di erential brightness temperature δTb(¯ν) are shown as a function of redshift z. Cases of the spectral index p= 2, 3 and 4 are shown in order from left to right. In each panel, cases of di erent values of α= 10−1 (red solid), 10−3 (green short-dashed), 10−5 (blue dotted), 10−7 (magenta long-dashed), 10−9 (green dot-dashed), 10−11 (blue two-dot-chain), and the pure adiabatic case α= 0 (black three-dot-chain) are plotted. ", | |
| "ization spectrum [(<>)31]7 (<>)for the values of fnet = 10−5 and 10−6 , which correspond to the cases where isocurvature perturbations at small scales are responsible for inducing cosmological reionization. From the figure, one can see that in these bands, 21cm line fluctuations can be detected with significance of more than 2σ, and thus by observing 21cm line emission, we will have a chance to test whether the isocurvature perturbations are responsible for reionization. " | |
| ], | |
| "6 Conclusions ": [ | |
| "In this paper, we have studied cosmological signatures of a blue-tilted isocurvature power spectrum with spectral index 2 . p. 4. Such an isocurvature power spectrum modifies the mass function of dark matter halos. This modification in mass function consequently a ects both the reionization history and 21cm line fluctuations from neutral hydrogen in ", | |
| "Figure 9. Forecast of constraints on the isocurvature power spectrum from observations of 21cm line fluctuations. The 2σconstraint expected from SKA is shown as red shaded region. The 2σconstraints from WMAP9+ACT2008 (region shaded with blue horizontal stripes) and reionization optical depth τreion >0.13 for fnet = 10−6 (region shaded with green vertical stripes) are also shown. Black and gray bands show the expected Planck measurements τreion = 0.09±0.01 (2σ) for fnet = 10 5 and 10−6 . ", | |
| "minihalos at high redshifts. The presence of a blue-tilted isocurvature spectrum enhances the number density of massive halos at high redshifts and hence promotes formation of galaxies and reionization of the universe. Given the reionization optical depth estimated from current CMB observations, we can obtain constraints on the amplitude and spectral index of the isocurvature power spectrum. Although the constraints strongly depend on the eÿciency of reionization, for reasonable parameter values it can surpass those obtained from the spectral shape of the CMB power spectrum. On the other hand, a blue-tilted isocurvature power spectrum changes the mass function at smaller masses in a complicated manner, which results in either an enhancement or even a suppression of 21cm line fluctuations depending on the parameters of the isocurvature power spectrum. Future surveys such as SKA can probe the parameter space of isocurvature power spectra which cannot be explored directly from the CMB power spectrum. We emphasize that the reionization optical depth and 21cm line fluctuations can provide complementary probes of blue-tilted isocurvature power spectra. ", | |
| "Note added: While we were finishing the present work, we noticed that Ref. [(<>)33], which appears on arXiv around the similar time as ours, has some overlap with our analysis on 21cm line fluctuations from a blue-tilted power spectrum. " | |
| ], | |
| "Acknowledgments ": [ | |
| "We would like to thank Yoshitaka Takeuchi and Sirichai Chongchitnan for helpful discussion. HT is supported by the DOE at ASU. TS is supported by Japan Society for the promotion of Science and the Academy of Finland grant 1263714. NS is supported by the Grant-in-Aid for the Scientific Research Fund under Grant No. 25287057. This work is also supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The research of JS has been supported at IAP by the ERC project 267117 (DARK) hosted by Universit´e Pierre et Marie Curie - Paris 6 and at JHU by NSF grant OIA-1124403. " | |
| ] | |
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