Antisymmetric Cross-correlation between H I and CO Line Intensity Maps as a New Probe of Cosmic Reionization
AA Model-independent Indicator for the Speed of Cosmic Reionization
Meng Zhou, Yi Mao, ∗ and Jianrong Tan
1, 2 Department of Astronomy, Tsinghua University, Beijing 100084, China Department of Physics & Astronomy, University of Pennsylvania,209 South 33rd Street, Philadelphia, PA 19104, USA (Dated: Submitted to Phys. Rev. Lett. September 6, 2020)The antisymmetric part of two-point cross-correlation between intensity maps of the HI 21 cmline and the CO 2.61 mm line has emerged as a new probe of cosmic reionization. In this Letterwe demonstrate that the slope of the dipole of HI-CO cross-power spectrum at large scales is linearto the rate of change of global neutral fraction of hydrogen in a model-independent way, until theslope levels out near the end of reionization. The HI-CO dipole, therefore, can be a smoking-gunprobe for the speed of reionization, or “standard speedometer”. Observations of this new signal willunveil the global reionization history from the midpoint to near the completion of reionization.
In a companion Letter, we proposed a new inten-sity mapping analysis method, the antisymmetric cross-correlation between the H I
21 cm line and the CO2.61 mm line intensity maps from the epoch of reion-ization (EOR). From the observational point of view,the most interesting advantage of this new analysis isthat this signal is unbiased by foregrounds and thus canbe measured directly from the foreground-contaminateddata. It arises because the statistical fluctuations of the21 cm field have much more rapid evolution in time thanthe CO(1-0) line field, and therefore the H I -CO antisym-metric cross-correlation contains additional informationof the progressing of cosmic reionization, complementaryto the symmetric component of cross-correlation. For ex-ample, in the companion Letter we show that the sign ofthis signal can generically tell whether inside-out reion-ization happens during some time interval, regardless ofthe detail of reionization model.How the H I -CO antisymmetric cross-correlation, as anew probe of cosmic reionization, depends on reioniza-tion models calls for thorough investigations. As an at-tempt in this regard, [1] focused on a special reionizationmodel and showed that including the H I -CO antisym-metric cross-correlation can improve the constraint onthe model parameters significantly. Instead, this Letteris devoted to investigate the generic feature regardingmodel dependence and independence. Simulations .— We perform semi-numerical simula-tions of reionization with the publicly available code [2][3]. This code quickly generates the fields ofdensity, velocity, ionized fraction, spin temperature and21 cm brightness temperature on a grid. It is based onthe semi-numerical treatment of cosmic reionization withthe excursion-set approach[4] to identify ionized regions.Specifically, cells inside a spherical region are identifiedas ionized, if the number of ionizing photons in that re-gion is larger than that of neutral hydrogen atoms. Oursimulations were performed on a cubic box of 768 co-moving Mpc on each side, with 512 grid cells. Our EORmodel is parametrized with three parameters: ζ (the ion-izing efficiency), T vir (the minimum virial temperature of halos that host ionizing sources), and R mfp (the meanfree path of ionizing photons). For the purpose of illus-tration, we choose a reference case with the parametervalues ζ = 25, T vir = 3 × K, R mfp = 50 Mpc. Thisyields a global reionization history in which the reion-ization starts at z (cid:39)
16 and ends at z (cid:39) .
5, with thecosmic microwave background (CMB) Thomson scatter-ing optical depth τ = 0 . I -CO dipole, we fur-ther consider a wider range of astrophysical parametersin the EOR model as listed in the legend of Fig. 1. In allcases, R mfp = 50 Mpc is fixed because the results dependon this parameter very weakly. Fig. 1 shows that thesemodels result in very different reionization histories aswell as different rates of change of mean neutral fraction, d ¯ x HI /dz , or the “speed” of reionization.In this Letter, we adopt the standard ΛCDM cosmol-ogy with fixed values of cosmological parameters basedon the Planck 2016 results[6], ( h, Ω m , Ω b , Ω Λ , σ , n s ) =(0 . , . , . , . , . , . Modelling the brightness temperature .— The 21 cmbrightness temperature at position x relative to the CMBtemperature can be written[7] as T ( x , z ) = ˜ T ( z ) x HI ( x ) [1 + δ ( x )] (1 − T CMB T S ) , (1)where ˜ T ( z ) = 27 (cid:112) [(1 + z ) / . / Ω m h )(Ω b h / . x HI ( x ) is the neutral fraction, and δ ( x ) is the matter overdensity, at position x . We assumethe baryon distribution traces the cold dark matter onlarge scales, so δ ρ H = δ . In this Letter, we focus on thelimit where spin temperature T S (cid:29) T CMB , valid soonafter reionization begins. As such, we can neglect thedependence on spin temperature. Also, for simplicity,we ignore the effect of peculiar velocity, because it onlyweakly affects the light-cone effect.The CO(1-0) line specific intensity can be written[8]as I CO ( x , z ) = ¯ I CO ( z ) [1 + b CO ( z ) δ ( x )] with the meanintensity ¯ I CO ( z ) and the bias b CO ( z ) at redshift z . Theequivalent brightness temperature is computed using the a r X i v : . [ a s t r o - ph . C O ] S e p FIG. 1. Global reionization history for different reionizationmodels as marked in the legend: (top) mean neutral fraction¯ x HI vs redshift z , and (bottom) its redshift derivative d ¯ x HI /dz vs z . Generically, reionization has two stages, the acceleration(“accel.”) and deceleration (“decel.”) stages.FIG. 2. Evolution of the CO bias. We show the CO bias b CO vs redshift z in our fiducial EOR model. Rayleigh-Jeans Law, T CO = c I CO / (2 k B ν ). There-fore, it can be written as T CO ( x , z ) = ¯ T CO ( z ) [1 + b CO ( z ) δ ( x )] . (2)Here the observed frequency ν obs = ν CO / (1 + z ) for gasat redshift z emitting in the CO(1-0) line with the rest-frame ν CO = c/λ CO = 115 GHz and λ CO = 2 .
61 mm.The mean intensity [8, 9] is ¯ I CO ( z ) =( λ CO / π H ( z )) (cid:82) ∞ M CO , min dM ( dn/dM )( M, z ) L CO ( M ),where dn/dM is the halo mass function[10] and L CO ( M )is the CO(1-0) luminosity which is assumed to be afunction of the halo mass M . We follow the modellingof [9] in which the CO luminosity is linear to halomass, L CO ( M ) = 2 . × L (cid:12) ( M/ M (cid:12) ). Under thisassumption, the mean brightness temperature is [9]¯ T CO ( z ) = 59 . µ K (1 + z ) / f coll ( M CO , min ; z ) , (3)where f coll ( M CO , min ; z ) is the mean collapse fraction at z with the lower mass cutoff at M CO , min . We assume FIG. 3. The dipole of the HI-CO cross-power spectrum P A vs wavenumber k for different reionization models (as markedin the legend of Fig. 1) at the fixed redshift z = 8 .
48 (corre-sponding to ¯ x HI = 0 .
50 in our fiducial model). The error barsare 1 σ standard deviation for cosmic variance correspondingto the simulation volume of 100 realizations. that M CO , min , the minimum mass of halos that can hostgalaxies, is the same as the mass scale of atomic hydrogencooling. In other words, M CO , min corresponds to T vir ,the minimum virial temperature of halos that can hostionizing sources.The bias b CO ( z ) describes how well the CO brightnesstemperature fluctuations trace the matter density fluctu-ations. It can be modelled [8, 9] as b CO ( z ) = (cid:82) ∞ M CO , min dM dndM ( M, z ) L CO ( M ) b ( M, z ) (cid:82) ∞ M CO , min dM dndM ( M, z ) L CO ( M ) , where b ( M, z ) is the halo bias[10]. The linear luminosityassumption can further simplify this expression. Insteadof evaluation by direct numerical integration, we find ananalytic form for the bias b CO ( z ). We take the formof halo bias, b ( M, z ) = 1 + ( ν − /δ c ( z ), where ν = δ c /σ ( M ), by setting a = 1 and p = 0 in equation (12) of[10]. In this case, the mass function is the Press-Schechterform. Thus we can obtain an analytical form, b CO ( z ) = 1 − δ c ( z ) + 2 √ πδ c ( z ) f coll ( M CO , min ; z ) × Γ inc (cid:18) . , δ c ( z )2 σ (cid:19) , (4)where the incomplete Gamma function Γ inc ( a, x ) ≡ (cid:82) ∞ x t a − e − t dt , and σ min ≡ σ ( M CO , min ). In Fig. 2, wefind that the CO bias decreases with time, a trend alsofound in [8]. Results .— We postprocess the reionization simulationswith the method described in detail in the companion
FIG. 4. The same as Fig. 3 but at the fixed rate d ¯ x HI /dz = 0 .
378 (top left) and 0 .
246 (top right) in the acceleration stage andat the fixed d ¯ x HI /dz = 0 .
378 (bottom left) and 0 .
246 (bottom right) in the deceleration stage, respectively. We fit the dipoledata of all models (dots) to a modified power law (red dashed lines).
Letter, and extract the dipole, P A ( k ), of the cross-powerspectrum between the 21 cm and CO(1-0) line brightnesstemperature maps. Fig. 3 shows that the H I -CO dipoleat the fixed redshift of bandwidth center is very model-dependent. Thus one might use this model-dependenceas the basis of constraining reionization model parame-ters with the H I -CO dipole.Fig. 1 shows that the progressing of reionization gener-ically undergoes an acceleration stage ( d ¯ x HI /dz < d ¯ x HI /dz > z , if we comparethe dipole for different reionization models at the samespeed d ¯ x HI /dz = 0 . .
246 of bandwidth center in theacceleration stage, which correspond to ¯ x HI = 0 . . σ cosmic variance correspond-ing to the simulation volume of 100 realizations. Wefind the similar model-independence in the decelerationstage, too. This implies that the H I -CO dipole is to lead-ing order determined by the speed of cosmic reionization d ¯ x HI /dz of the bandwidth center, regardless of the detailof reionization models. This can be understood becausethe H I -CO dipole is dominated by the evolution effectdue to cosmic reionization, which is characterized by thedifference of ionization level between the front- and back-end on the lightcone, or d ¯ x HI /dz to leading order.At each fixed d ¯ x HI /dz (during either acceleration ordeceleration stage) , we fit the H I -CO dipole of all modelsto a modified power law, P A ( k ) = − A R ( k/k ∗ ) − n R exp [ − β R ( k/k ∗ ) α R ] , (5)with best-fit coefficients listed in Table I. The coefficients TABLE I. Best-fit coefficients of the HI-CO dipole to the ansatz P A ( k ) = − A R ( k/k ∗ ) − n R exp [ − β R ( k/k ∗ ) α R ], at the fixed speed d ¯ x HI /dz for the acceleration stage (“accel.”) and deceleration stage (“decel”), respectively. The slope n R is measured for therange of k = 0 . − . h Mpc − . Here we choose k ∗ = 1 h Mpc − . R is the coefficient of determination. d ¯ x HI /dz A R [( µ K) h − Mpc ] n R β R α R R a cc e l. .
378 (2 . ± . × . ± .
200 3 . ± .
976 2 . ± .
430 0 . .
312 (4 . ± . × . ± .
308 3 . ± .
084 2 . ± .
391 0 . .
246 (6 . ± . × . ± .
500 5 . ± .
298 2 . ± .
368 0 . d e c e l. .
378 (1 . ± . × . ± .
211 3 . ± .
738 3 . ± .
725 0 . .
312 (1 . ± . × . ± .
200 3 . ± .
189 2 . ± .
533 0 . .
246 (1 . ± . × . ± .
286 2 . ± .
152 2 . ± .
599 0 . n R vs the speed of reion-ization d ¯ x HI /dz during the acceleration (left) and deceler-ation (right) stage, respectively. We show the slope mea-sured for k = 0 . − . h Mpc − (black solid dots) and for k = 0 . − . h Mpc − (red open dots). We fit the data toa linear relation (dashed lines) in the acceleration stage anda constant (dashed lines with the shaded regions representingthe 1 σ error) in the deceleration stage. The arrows show thedirection of time flow or increasing the redshift z . A R and n R quantify the power law at large scales, while β R and α R quantify the exponential suppression at smallscales. We find that among these four coefficients, n R hasthe smallest error bars ( (cid:46)
15% in most cases) in fittingall model data, which indicates that the slope at largescales is most robust against the variation of models.Fig. 5 shows that the slope of H I -CO dipole increaseswith the speed d ¯ x HI /dz with an approximately linear re-lation, n R = a ( d ¯ x HI /dz ) + a , until n R reaches a max-imum and levels out near the end of reionization. Thelinear relation can be explained by the fact that the largerspeed d ¯ x HI /dz can result in larger magnitude of dipole atlarge scales while having less effect at small scales, which makes the H I -CO dipole steeper in k and increases theslope n R . On the other hand, if the slope is measured fora smaller scale, then Fig. 5 shows that the slope reachesthe maximum at earlier time. (We define this moment tobe when n R overlaps with the 1 σ region of the constant n R in the deceleration stage.) This implies that the slopereaches the maximum when the scale for which the slopeis measured is below the characteristic scale of H II re-gions; thereafter the changes in the speed of reionization,which mostly affect the dipole on scales above the char-acteristic scale of H II regions, do not considerably affectthe scale-dependence of the dipole on scales beneath. Discussions .— The linear relation between the slopeof H I -CO dipole at large scales and the speed of reion-ization can be exploited to infer d ¯ x HI /dz from the n R measurement in a model-independent way. In this sense,the H I -CO dipole is a smoking-gun probe for the speedof reionization, so we term it a “standard speedometer”for cosmic reionization. However, this approach is onlyvalid before n R levels out at the time that depends onthe scale of interest. For example, if the range of scale k = 0 . − . h Mpc − is considered, then the effectiverange for standard speedometer is 0 . (cid:46) ¯ x HI (cid:46) .
52 forall models considered herein. The upper-bound is be-cause at even earlier time, the dipole is too small to bedistinguished from the cosmic variance.Standard speedometer is of important astrophysicalapplication — the global reionization history ¯ x HI ( z ),however relative to the value when the slope just be-gins to level out, can be reconstructed by integration of d ¯ x HI /dz . This approach of global history reconstructionis model-independent and unbiased by foregrounds. Mea-surements of the H I -CO dipole will motivate the greatersynergy between 21 cm observations using radio inter-ferometers, e.g. the low frequency array of the SquareKilometre Array[11] (SKA), and CO observations usingsingle dish arrays, e.g. the middle frequency array of SKAor the update of the CO Mapping Array Pathfinder[12](COMAP). Acknowledgements .— This work is supported bythe National Key R&D Program of China (GrantNo.2018YFA0404502, 2017YFB0203302), and the Na-tional Natural Science Foundation of China (NSFC GrantNo.11673014, 11761141012, 11821303). YM was also sup-ported in part by the Chinese National Thousand YouthTalents Program. We are grateful to Andrea Ferrara, BenWandelt, Le Zhang and Pengjie Zhang for comments anddiscussions. The simulations in this work were ran at theVenus cluster at the Tsinghua University. ∗ Corresponding author. Email: [email protected][1] G. Sato-Polito, J. L. Bernal, E. D. Kovetz, andM. Kamionkowski, Antisymmetric cross-correlation ofline-intensity maps as a probe of reionization, arXiv e-prints , arXiv:2005.08977 (2020).[2] https://github.com/21cmfast/21cmFAST .[3] A. Mesinger, S. Furlanetto, and R. Cen, 21cmfast: a fast,seminumerical simulation of the high-redshift 21-cm sig-nal, Mon. Not. R. Astron. Soc. , 955 (2011).[4] S. R. Furlanetto, M. Zaldarriaga, and L. Hernquist, TheGrowth of H II Regions During Reionization, Astrophys.J. , 1 (2004).[5] B. Greig and A. Mesinger, The global history of reion-ization, Mon. Not. R. Astron. Soc. , 4838 (2017).[6] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Ar-naud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J.Banday, R. B. Barreiro, J. G. Bartlett, and et al., Planck2015 results. XIII. Cosmological parameters, Astron. As-trophys. , A13 (2016).[7] S. R. Furlanetto, S. P. Oh, and F. H. Briggs, Cosmologyat low frequencies: The 21cm transition and the high-redshift Universe, Phys. Rep. , 181 (2006). [8] Y. Gong, A. Cooray, M. B. Silva, M. G. Santos, andP. Lubin, Probing reionization with intensity mappingof molecular and fine-structure lines, Astrophys. J. ,L46 (2011).[9] A. Lidz, S. R. Furlanetto, S. P. Oh, J. Aguirre, T.-C.Chang, O. Dor´e, and J. R. Pritchard, Intensity mappingwith carbon monoxide emission lines and the redshifted21 cm line, Astrophys. J. , 70 (2011).[10] R. K. Sheth and G. Tormen, Large-scale bias and thepeak background split, Mon. Not. R. Astron. Soc. ,119 (1999).[11] L. Koopmans, J. Pritchard, G. Mellema, J. Aguirre,K. Ahn, R. Barkana, I. van Bemmel, G. Bernardi,A. Bonaldi, F. Briggs, A. G. de Bruyn, T. C. Chang,E. Chapman, X. Chen, B. Ciardi, P. Dayal, A. Ferrara,A. Fialkov, F. Fiore, K. Ichiki, I. T. Illiev, S. Inoue,V. Jelic, M. Jones, J. Lazio, U. Maio, S. Majumdar, K. J.Mack, A. Mesinger, M. F. Morales, A. Parsons, U. L.Pen, M. Santos, R. Schneider, B. Semelin, R. S. de Souza,R. Subrahmanyan, T. Takeuchi, H. Vedantham, J. Wagg,R. Webster, S. Wyithe, K. K. Datta, and C. Trott, TheCosmic Dawn and Epoch of Reionisation with SKA, in
Advancing Astrophysics with the Square Kilometre Ar-ray (AASKA14) (2015) p. 1, arXiv:1505.07568 [astro-ph.CO].[12] K. Cleary, M.-A. Bigot-Sazy, D. Chung, S. E. Church,C. Dickinson, H. Eriksen, t. gaier, P. Goldsmith, J. O.Gundersen, S. Harper, A. I. Harris, J. Lamb, T. Li,R. Munroe, T. J. Pearson, A. C. S. Readhead, R. H.Wechsler, I. Kathrine Wehus, and D. Woody, The COMapping Array Pathfinder (COMAP), in