Maximum Absorption of the Global 21 cm Spectrum in the Standard Cosmological Model
MMaximum Absorption of the Global 21 cm Spectrumin the Standard Cosmological Model
Yidong Xu , Bin Yue , ∗ , Xuelei Chen , , , ∗ ABSTRACT
The absorption feature in the global spectrum is likely the first 21cm observable from thecosmic dawn, which provides valuable insights into the earliest history of structure formation.We run a set of high-resolution hydrodynamic simulations of early structure formation to assessthe effect of non-linear structure formation on the maximum absorption level (i.e. assuming thespin temperature coupling is saturated) of the global 21 cm spectrum in the standard cosmologicalframework. We ignore the star formation and feedbacks, which also tends to reduce the absorptionsignal, but take into account the inevitable non-linear density fluctuations in the intergalacticmedium (IGM), shock heating and Compton heating which can reduce the absorption level.We found that the combination of these reduced the maximum absorption signal by ∼ Subject headings:
Cosmology: theory — dark ages, reionization, first stars — intergalactic medium
1. Introduction
The spectrum of the sky-averaged 21 cm bright-ness temperature, or the so-called global 21 cmsignal, provides valuable information on the earlyhistory of structure formation, from the dark agesto the cosmic reionization (e.g. Pritchard & Loeb2012). The EDGES experiment has reported thedetection of an absorption trough with a depth of δT = − +200 − mK (99% confidence level) cor-responding to the cosmic dawn (Bowman et al.2018), which is unexpectedly large as comparedwith theoretical predictions from the standardmodel. Although the claimed signal may be af-fected by instrumental effects (e.g. Bradley et al.2019), mis-modeled foregrounds (e.g. Hills et al. National Astronomical Observatories, ChineseAcademy of Sciences, Beijing 100101, China School of Astronomy and Space Science, University ofChinese Academy of Sciences, Beijing 100049, China Center for High Energy Physics, Peking University,Beijing 100871, China ∗ Corresponding author: Bin Yue ([email protected]),Xuelei Chen ([email protected]) a r X i v : . [ a s t r o - ph . C O ] F e b ARAS-2 has already put some constraints on the21 cm spectrum, and disfavors models that featureweak X-ray heating along with rapid reionization(Singh et al. 2018b).To correctly interpret the observations, it is im-portant to calculate the 21 cm absorption levelprecisely, taken into account of various effects. Al-though the absorption feature in the global 21cmspectrum is produced by the gas which is still quitehomogeneous during the cosmic dawn, where thevolume fraction of collapsed halos are still small,the budding inhomogeneity can still affect the re-sult. Xu et al. (2018) investigated the effect of gasinhomogeneity on the maximum absorption levelof the global 21 cm spectrum. We found thatthe non-linearity of the gas density fluctuationsinduced by the structure formation, and the asso-ciated adiabatic heating, suppress the signal levelsubstantially.Note that the 21cm absorption feature is pro-duced by the neutral hydrogen with spin temper-ature lower than the CMB temperature at thatepoch (Chen & Miralda-Escud´e 2004, 2008). Theneutral hydrogen spin temperature would gener-ally fall somewhere between the gas kinetic tem-perature and CMB temperature, depending on theintensity of the Lyman alpha (Ly- α ) background,which couples the spin and kinetic temperature ofthe gas. By maximum, we are referring to the casethat the spin-kinetic coupling of the gas is satu-rated, such that the spin temperature is equal tothe kinetic temperature, and the largest amountof absorption is produced. In realistic models,by the time a strong Ly- α background is set upby star formation and black hole accretion, someamount of ionization and heating would have al-ready taken place. The ionized gas would not con-tribute to the 21cm signal, while the neutral gasheated above the CMB temperature would appearin 21cm emission, reducing the total amount of21cm absorption. The ionization and radiationinduced heating would however depend on manymodeling details, which results in a variety of spec-trums (Cohen et al. 2017). However, to assess thenecessity of new physics, we can focus on the maxi-mum absorption case. As all of these effects reducethe absorption, one can obtain a very conservativelimit if we ignore them.However, the analytic estimates in Xu et al.(2018) should only be taken qualitatively, as the large-scale clustering of halos and the structureformation shocks that are inevitable during thecosmic dawn are not easy to model analytically.Also, the adopted density profile for the intergalac-tic medium (IGM) around collapsed halos fromthe “infall model” only applies to density peaks(Barkana 2004). Applying it to the whole IGM ofany environments requires an artificial normaliza-tion, which may result in an inaccurate level of gasdensity fluctuations and the resultant 21 cm sig-nal. Recently, Villanueva-Domingo et al. (2020)has also investigated analytically the maximumamplitude of the high-redshift 21-cm absorptionfeature, accounting for 21 cm heating, Lyman- α heating, and the density fluctuations. Adoptingthe non-linear density distribution of the MHR00model (Miralda-Escud´e et al. 2000), they find thatthe density fluctuations result in a decrement of ∼
10% in the maximum 21 cm absorption.In this work, we focus on the maximum sig-nal level of 21 cm brightness from cosmic dawnwithin the standard framework, i.e. assuming noextra cooling or extra radio background from newphysics, a fully neutral IGM before reionization,and saturated coupling between the spin temper-ature of neutral hydrogen and the kinetic tem-perature of gas. However, we do consider the in-evitable standard model evolutions, such as non-linear structure formation and Compton heating.To incorporate more reliable density profiles ofgas in the IGM, for both over-dense and under-dense regions, and to include the clustering ef-fect of non-linear structures, we use a set of high-resolution hydrodynamic simulations to computethe expected global 21 cm signal at high redshifts,and discuss various effects that impact the signallevel.This paper is organized as follows. We describeour simulations set and present some basic resultsin section 2, and then we discuss the effects of den-sity profiles, the shock heating and Compton heat-ing, and the large-scale clustering, in section 3. Weconclude in section 4. Throughout this paper, weadopt the ΛCDM model with the Planck 2018 cos-mological parameters (Planck Collaboration et al.2020).2 . Simulation and the maximum signal
We use hydrodynamical simulations to investi-gate the the effects of non-linear structure forma-tion on the global 21 cm signal from cosmic dawn.In this section we will first describe our simulationset up, and make some checks.
We carry out our cosmological simulations bythe publicly available code GADGET-2 (Springel2005; Springel et al. 2001) † , which uses thesmoothed particle hydrodynamics (SPH) methodto solve the gas dynamics equations. The publiclyavailable version does not involve radiative heat-ing/cooling and the chemistries that are necessaryfor correctly modeling the gas temperature evolu-tion. We add the evolution of the free electrons, Hand He ions, the associated cooling/heating pro-cesses. Free electrons are essential in the globalIGM temperature evolution. The initial free elec-tron abundance and gas temperature are com-puted from the epoch of recombination using theRECFAST code (Seager et al. 1999) ‡ . We ig-nore here the formation of H and the cooling itinduced, as we are focusing on the nonlinear struc-tures that have not yet experienced star formationprocesses. The homogeneous gas temperature andionization state is evolved by solving the equations dTdt = − H ( z ) T − net ( T )]3 k B n tot ,dn HII dt = − n HII H ( z ) + γ HI ( T ) n HI n e − α HII ( T ) n HII n e ,dn HeII dt = − n HeII H ( z ) + γ HeI ( T ) n HeI n e − α HeII ( T ) n HeII n e ,dn e dt = dn HII dt + dn HeII dt , (1)where n HI , n HII , n HeI , n HeII and n e are the physi-cal number densities of neutral hydrogen, ionizedhydrogen, neutral helium, singly ionized heliumand electron, respectively, He III is neglected here. α HII and α HeII are the recombination rates, γ HI and γ HeI are the collisional ionization rates which † ‡ are in fact negligible, and Λ net is the net coolingrate. In the gas temperature evolution we includethe Compton scattering and Bremsstrahlung, asdetailed in Maselli et al. (2003). z T [ K ] CMBhomogeneous Gas T gas ∝ (1 + z ) GADGET-2, h − Mpc , all densitiesGADGET-2, h − Mpc , | δ | ≤ . Fig. 1.— The evolution of the mean gas temper-ature in our fiducial simulation. The filled circlesshow the evolution of the mean temperature ofall gas particles, and the dashes line shows theaverage over gas particles with density contrast | δ | < .
05. For comparison, the evolutions ofthe CMB temperature and the homogeneous gastemperature are plotted with the green and bluesolid lines respectively, and the thin black solidline shows a purely adiabatic evolution.In Fig. 1 we plot the evolution of the mean gastemperature in a simulation that has a box sizeof 4 Mpc/ h and 800 dark matter particles and800 gas particles respectively. The filled circlesshow the average temperature of all gas particles,while the dashed line corresponds to the averageover gas particles with density contrast | δ | < . z (cid:46)
25. Notethat, however, the shock-heated gas only occupiesa small volume fraction.For monoatomic gas that experiences only adi-abatic compression without shock-heating, radia-tive heating/cooling, or any change of chemicalspecies, a relation between the gas temperature3 . . . . . . . . . . l og ( T / K ) -2.0-1.00.01.02.03.04.05.06.07.08.09.010.0 l og ( d N / d ∆ d T ) Fig. 2.— The T − ∆ relation at z = 17 fromour simulation. The color denotes the number ofparticles at a certain position on the T − ∆ plane.As a comparison, we plot a curve of T = 6 . / K with the dashed line.and the density builds: T K ( δ ) = T ∆ / , (2)where T is the temperature of the mean-densitygas. This adiabatic relation is widely adoptedwhen estimating the IGM temperature analyti-cally. However, during the cosmic evolution, evenbefore the formation of any luminous objects,the Compton heating that arise from scatteringwith the CMB photons and the shock-heating inover-dense regions that are undergoing non-linearstructure formation can affect the gas tempera-ture and break this relation. Fig. 2 shows theprobability distribution of particles on the T − ∆plane from our simulation at z = 17. The colordenotes the number density of particles in T − ∆space, the power law relation is clearly broken inthe simulation. For ∆ (cid:46) (cid:38) We convert the particle field into the densityfield with the Cloud-in-Cell (CIC) method, andcalculate the global 21 cm signal from the simu-lation. The 21 cm brightness temperature from a
60 65 70 75 80 85 ν [MHz] − − − − − − − − δ T [ m K ] /h, × particles /h, × particles /h, × particles /h, × particles . /h, × particleshomogeneous IGM z Fig. 3.— The global 21 cm spectrum from cos-mic dawn assuming saturated coupling betweenthe spin temperature of neutral hydrogen and thegas kinetic temperature. The different thick linesshow results from simulations of different box sizesand resolutions, while the thin solid line representsthe maximum signal level expected from the ho-mogeneous IGM. The shaded regions of the samecolor as the corresponding lines indicates the jack-knife error.uniform cell in the simulated box isd T = T S − T γ z (cid:0) − e − τ (cid:1) , (3)where T S is the spin temperature of the neutralhydrogen, T γ is the brightness temperature of thebackground radiation, and τ is the 21 cm opticaldepth. In the absence of any extra radio back-ground at cosmic dawn (e.g. Ewall-Wice et al.2018), the only radio background is the cosmicmicrowave background (CMB), so that T γ ( z ) = T CMB ( z ). As the peculiar velocity has only negli-gible effect on the sky-averaged 21 cm signal (Xuet al. 2018), the optical depth can be written as τ = 316 (cid:126) c A k B ν n HI T S H ( z ) . (4)where A = 2 . × − s − is the Einstein coeffi-cient for the spontaneous decay of the 21 cm tran-sition, ν = 1420 . n HI and H ( z ) are the local neutralhydrogen number density and the Hubble parame-ter, respectively. In the present work, we focus on4he maximum absorption signal of 21 cm that isachievable in the standard ΛCDM model. There-fore, in all the following calculations, we assumesaturated coupling between the spin temperatureof hydrogen and the kinetic temperature of thegas, so that T S = T K . The 21 cm global signal iscomputed by averaging the 21 cm brightness tem-perature over all the cells in the simulation.The spectra of the maximum 21 cm absorptionfrom several simulations are plotted with in Fig. 3.The shaded regions of the same color as the cor-responding lines indicates the jackknife error inthe corresponding spectra. The expected spec-trum from the homogeneous IGM is plotted withthe thin solid line for comparison. It is seen thatthe non-linear structure formation affects the 21cm absorption level obviously; the homogeneousassumption of the IGM would over-estimates theabsorption. The effect gets more and more signif-icant for lower redshifts, as more non-linear struc-tures form.The different thick lines in Fig. 3 show resultsfrom simulations of different box sizes and reso-lutions. As the redshift decreases, the large-scaleperturbations becomes more and more important,and a limited box size would under-estimate theeffect of non-linear structure formation because ofthe delayed structure formation. This is seen fromgreen dotted line predicted by the simulation witha box size of 0 . /h , though it has the highestresolution. It significantly over-estimates the ab-sorption signal at (cid:38)
70 MHz. At the same time,the influence of non-linear structure formation be-comes more significant, and the IGM density fluc-tuations become more non-linear as the redshiftdecreases. An insufficient resolution would alsounder-estimate the effect of non-linear structureformation by losing small structures, and this isshown by the blue dot-dashed line in the figurefrom a simulation with a box size of 8 Mpc /h anda particle number of 2 × . By computing thejackknife errors on the 21 cm spectrum, we findthat the convergency can be achieved with a sim-ulation with a box size larger than 4 Mpc /h anda particle number larger than 2 × . The sys-tematic error is within 1%.In the following analysis, we will take the sim-ulation with 4 Mpc /h size and 2 × particlesas the fiducial simulation. From the fiducial sim-ulation, at z = 17, where the EDGES absorp- tion trough locates, the 21 cm absorption signalis −
190 mK. The absorption amplitude is reducedby 15% w.r.t. the homogeneous IGM case at thisredshift, when the non-linear structure formationis taken into account. The effects are more signif-icant as the IGM becomes more non-linear.
In order to survey a large parameter space andinvestigate the various effects on the global 21cm spectrum, a set of semi-numerical simulationsare usually used to compute the signal (e.g. Co-hen et al. 2017). This kind of simulations usuallycover a sufficiently large volume while not having ahigh enough resolution to resolve non-linear struc-tures, such as halos and their ambient gas, thoughthe shock heating effects could be implementedwith a sub-grid algorithm. Therefore, it is nec-essary to see the effect of losing small-scale struc-tures while keeping large-scale fluctuations just asa semi-numerical simulation does.
50 55 60 65 70 75 80 85 90 ν [MHz] − − − − − − δ T [ m K ] / h , × particles / h , × particleshomogeneous IGM z Fig. 4.— The maximum absorption signal of the21 cm global spectrum from cosmic dawn. Thethick solid line shows the spectrum from our fidu-cial simulation, the thick dashed line shows the re-sults from a low-resolution simulation, which hasa box size of 600 Mpc /h and 2 × particles, andthe thin solid line is the maximum expectation forthe homogeneous IGM.In Fig. 4 we compare our high resolution hydro-dynamic simulation with the low resolution onestypically used for semi-numerical simulation. Thethick dashed line shows the expected signal from5 ν [MHz] . . . . . . . . . τ All pixelsOver-dense pixelsUnder-dense pixelsHomogeneous IGM z
45 50 55 60 65 70 75 80 85 90 ν [MHz] − − − − − − δ T [ m K ] All cellsOver-dense cellsUnder-dense cellshomogeneous IGM z Fig. 5.— The 21 cm optical depth ( left panel ) and the 21 cm global spectrum ( right panel ) averaged overover-dense pixels (blue dashed line) and that averaged over under-dense pixels (red dot-dashed line). Thethick solid line shows the averaged values over all pixels in the simulation, and the thin solid line in eachpanel represents the spectrum expected from the homogeneous IGM.a simulation with a box size of 600 Mpc /h and aparticle number of 2 × , a typical resolutionof a semi-numerical simulation. It shows that inthe low-resolution simulation with only linear den-sity perturbations would over-estimate the global21 cm signal significantly, predicting an absorp-tion level similar to the homogeneous IGM case.Therefore, one needs to achieve a resolution ofnon-linear scales, or to implement a sub-grid al-gorithm for the shock effects (e.g. Furlanetto &Loeb 2004), to account for the small-scale effectson the sky-averaged signal.
3. Dependence on different effects
We now look more closely at the various as-pects, including the density and temperaturedependence, and the large-scale clustering, thatwould have impacts on the maximum signal levelof the global 21 cm spectrum.
In Xu et al. (2018), by assuming the analyticalinfall model, we found that the weakly nonlineargas around collapsed halos is adiabatically heatedand could affect the global absorption signal. Nowwe study the dependence on the local over-densityand density profiles of the IGM in this section.Fig. 5 shows the averaged optical depth (leftpanel) and the 21 cm brightness temperature (right panel) of over-dense pixels (dashed lines)and those of under-dense pixels (dot-dashed lines),respectively. The averaged values over all pixelsare plotted with the thick solid line. Although thegas in the over-dense regions has a higher tem-perature, most still has a lower temperature thanthe CMB, so they still contribute to the total ab-sorption signal. As the dense regions have largeroptical depth, they actually contribute more ab-sorption signal than the gas in the under denseregion. However, due mainly to the adiabaticheating and the shock heating during the struc-ture formation (the large-scale clustering also hasa minor effect, as discussed in section 3.4), as morenon-linear structures form at lower redshifts, theaveraged absorption signal is weaker than the ex-pected signal for the homogeneous IGM, even ifone considers only the over-dense regions, as canbe seen at the right end of the right panel of Fig. 5.The gas in halos is shock-heated to a tempera-ture close to the halo virial temperature, suppress-ing substantially its contribution to the 21 cm ab-sorption, and the main contribution to the 21 cmabsorption signal during the cosmic dawn comesfrom the gas in the less-heated IGM. The forma-tion of dark matter halos, however, enhances thegas density surrounding them, resulting in non-linear density fluctuations in the vast IGM. Herewe investigate how the detailed density profiles af-fect the predicted global 21 cm signal.6 − − − r [ h − cMpc] − ∆ = ρ / ¯ ρ r vir Simulated IGM around halos with M ∼ M (cid:12) Infall model for M = 10 M (cid:12) Artificially normalized profile for M = 10 M (cid:12) Simulated mean profile: M = (0 . − . × M (cid:12) − − − r [ h − cMpc] − ∆ = ρ / ¯ ρ r vir Simulated IGM around halos with M ∼ M (cid:12) Infall model for M = 10 M (cid:12) Artificially normalized profile for M = 10 M (cid:12) Simulated mean profile: M = (0 . − . × M (cid:12) Fig. 6.—
Left: the density profiles of halos with M ∼ M (cid:12) at redshift z = 17. Right: the density profilesof halos with M ∼ M (cid:12) at z = 17.In Fig. 6, we plot the density profiles of the IGMsurrounding some of the individual halos from thehydro-dynamic simulation with light green lines,at redshift 17. The mean density profiles over allthe halos in the mass range (0 . − . × M (cid:12) (left panel), and (0 . − . × M (cid:12) (right panel),are plotted with black solid lines with the errorbars being the standard deviations. For compari-son, the blue solid lines show the profiles with thecorresponding halo mass and redshift predicted bythe infall model, which uses the excursion set the-ory (Bond et al. 1991; Lacey & Cole 1993) to pre-dict analytically the density profile around a halolocated at a density peak (Barkana 2004). Gen-erally, the density profile from the simulation isconsistent with the infall model, though there arelarge scatters in individual halos.To assess the effect of adopting an inaccuratedensity profile, we populate a mock simulationbox with the halo catalog, including the halo po-sition and mass information, from the hydrody-namic simulation, but assign an artificial densityprofile to the IGM around each halo as prescribedby the infall model, but further normalized it suchthat the minimum density is zero, and the meandensity equals the cosmic mean. Note that theinfall model is appropriate for halo surroundingsthat are located in density peaks, but may not beapplicable for under-dense environments. By in-troducing this artificial normalization, we mimicthe existence of under-dense regions while keepingthe cosmic mean density. However, we caution the readers that this artificial density profile, asshown by the blue dot-dashed lines in Fig. 6, isunphysical, and here we only use it to investigatethe effect of a steeper density profile on the global21 cm signal.A mock adiabatic temperature is assigned toeach pixel according to the local density using theadiabatic relation of T K ∝ ρ / for the ideal gas,with the mean-density gas having a temperatureof 6 .
46 K at redshift 17, consistent with the homo-geneous gas temperature calculated with Eq.(1)(i.e. the blue solid line in Fig. 1). We call thistemperature the mock adiabatic temperature asit accounts for the adiabatic heating or coolingaccording to the local density, but the Comptonheating is taken into account when determiningthe mean-density gas temperature, which is notpurely adiabatic.The resulting 21 cm signal is −
208 mK ( ∼ z = 17, as compared to −
213 mK ( ∼
5% decre-ment) at the same redshift if we adopt the densityfield from the hydrodynamic simulation with thecorresponding mock adiabatic temperature. Wefind that the steeper density profile, or equiva-lently a higher level of density fluctuations, resultsin lower absorption level in the 21 cm signal, butthe effect is only moderate.7 − − − r [ h − cMpc]10 T [ K ] infall-mockno Compton & no shockno shockfiducial − − − r [ h − cMpc]10 T [ K ] − − − r [ h − cMpc]10 T [ K ] − − − r [ h − cMpc]10 T [ K ] Fig. 7.— The gas temperature profiles around a ∼ M (cid:12) halo (Top panels) and a ∼ M (cid:12) halo (Bottompanels) at redshift 17. In each row, the Left panel shows the profiles of the median gas temperature, and theRight panel shows the upper boundaries of the 90% probability in the temperature distribution. The red,green, and blue curves correspond to the fiducial case, the case without shock-heating, and the case withboth shock-heating and Compton-heating removed, respectively. Dashed line represents the mock-adiabatictemperature of the infall model, and the thin solid lines shows the CMB temperature. In section 2.1, we have seen that the shock-heating and Compton-heating can make the gastemperature deviate significantly from the T k ∝ ∆ / relation (Fig. 2). These effects may play asignificant role in determining the 21 cm signallevel during the cosmic dawn. Here we make a de-tailed comparison between the temperature pro-files from the hydrodynamic simulations with orwithout these heating effects and test their im-pacts on the 21 cm signal.Fig. 7 shows the gas temperature profiles of theIGM surrounding halos with mass ∼ M (cid:12) (toppanels) and ∼ M (cid:12) (bottom panels) at red- shift z = 17. For each panel we plot the tem-perature profiles from the fiducial simulation, thesimulation without shock-heating, and the simu-lation without either shock-heating or Compton-heating, respectively. In each row, the left panelshows the median gas temperature at the variousdistances with the error bars being the standarddeviation of the median temperature, while theright panel shows the 90% upper limit of the tem-perature distributions and the corresponding stan-dard deviation. As a reference, we also plot the mock adiabatic temperature profile derived fromthe infall model. Although the scatter among ha-los is quite large, it is clear that the shock heat-ing significantly increases the gas temperature in8 ν [MHz] − − − − − − δ T [ m K ] /h, × particles, real T /h, × particles, mock adiabatic Thomogeneous IGM z Fig. 8.— The 21 cm global spectrum, for thesimulated gas temperature ( thick solid line), mockadiabatic temperature ( thick dashed line), and thehomogeneous IGM ( thin solid line ).over-dense regions near halos, and the Comptonheating dominates the heating effect in the vastunder-dense regions. As a result, the adiabaticassumption for the IGM temperature would sub-stantially over-predict the 21 cm absorption level,as shown in Fig. 8 which compares the global 21cm spectrum from the fiducial simulation (thicksolid line) with the mock adiabatic gas temper-ature (dashed line). Because of the shock heat-ing and the Compton heating incorporated in thehydro-dynamic simulation, the 21 cm signal atredshift 17 is further suppressed to be −
190 mK( ∼
15% decrement w.r.t. the homogeneous IGMcase), as compared to −
213 mK ( ∼
5% decrement)in the case considering only the gas density fluc-tuations and the mock adiabatic temperature.Interestingly, in Fig. 7 we find that between ∼ . − h − Mpc, the temperature profile in thefiducial simulation has large scatters. It impliesthat the gas in these regions experienced com-plicated dynamical processes. The shock heatingmakes this over-dense gas optically thin for the 21cm signal. However, it is very hard to observa-tionally resolve these shocked regions. During thecosmic dawn the shock-heated regions only occupya small volume fraction, but they result in ∼ − . − . − . . . . . . . δ local − − − − − − δ T [ m K ] homogeneous IGM T ∝ ρ / Fiducial simulationNo shock heatingSimulated ρ with mock adiabatic T Fig. 9.— The mean 21 cm brightness temperatureof subboxes with different mean local densities atredshift z = 17. The black dots are the δT ofsubboxes from the fiducial simulation, the red cir-cles are from the simulation with no shock heating,and the blue triangles are the δT of the subboxesassuming densities from the simulation but withmock adiabatic temperatures. The black line in-dicates the δT expected from the homogeneousIGM, and the cyan line represents the δT ∝ ρ / scaling.To investigate the heating effect in regions withdifferent densities, we divide the box into 4 × × /h and a different local density δ local . InFig. 9 we plot the mean 21 cm brightness tem-perature of the subboxes as a function of theirlocal mean overdensity at z = 17, for our fidu-cial simulation (black dots), and for the case withthe fiducial density field and the mock adiabatictemperature (blue triangles). The stars of the cor-responding colors indicate the average values overthe whole box. The black solid line in the figureindicates the 21 cm brightness expected from ahomogeneous IGM for comparison, while the cyancurve shows the 21 cm signals with the scaling of δT ∝ ρ / , which is expected for the cold gas( T S (cid:28) T CMB ) in the linear regime. We find thatat under-dense regions, the scaling between the 21cm brightness and the local overdensity is closeto δT ∝ ρ / if we assume the mock adiabatictemperature, while at over-dense regions, the re-lation deviates from this scaling significantly. TheCompton heating and shock heating effects further9uppress the 21 cm signal, which is more promi-nent in over-dense regions.For the fiducial simulation and the mock adi-abatic case, we find that a relation between themaximum 21 cm signal and the local overdensityof the form δT = − α ∆ β local mK (5)holds between − . < δ local < .
2. At redshift17, we find log α = 2 .
28 and β = 0 .
091 forthe fiducial simulation, while for the mock adia-batic case, we have log α = 2 .
33 and β = 0 . α = 2 . − .
31 log (1 + z ), and β = − .
540 + 1 .
299 log (1 + z ), for redshifts from25 to 15. This fitted relation could be used insemi-numerical simulations with large box sizesbut low resolutions, which would not be able tocapture the shock-heating and non-linear densityfluctuations. Note, however, that this fitted re-lation applies only to the 1 Mpc /h cells. A differ-ent smoothing scale would have a different scaling,and that would require a separate simulation witha relevant box size and resolution. − − − − | ( δT shock21 − δT noshock21 ) /δT noshock21 | . . . . . P r obab ili t y Fig. 10.— The probability distribution of the frac-tional difference in the 21 cm brightness temper-ature between the simulations with and withoutshock heating.To distinguish the effect of shock heating and Compton heating, we also run a simulation withthe shock heating turned off, the results at z = 17is plotted as the red circle symbol in Fig. 9. Themean 21 cm brightness temperature averaged overthe whole simulation box is −
200 mK, which isabout 10% decrement w.r.t. the homogeneousIGM case. Comparing the various cases, we seethe shock heating and the Compton heating havecomparable effects in decreasing the 21 cm absorp-tion signal, and this is consistent with the pre-vious study by McQuinn & O’Leary (2012). Inthe absence of radiation sources, the shock heat-ing dominates the heating effects in over-dense re-gions, while the Compton heating dominates theheating effect in under-dense regions.Fig. 10 shows the probability distribution of thefractional difference of the 21 cm brightness tem-perature between the pixels in the default simula-tion and the corresponding pixels in the simulationwithout shock-heating. For most pixels, the shockheating results in a suppression of the 21 cm signalby a few percent, but there is a small fraction ofpixels that are shock-heated significantly, resultingin a long tail in the probability distribution.
The large scale clustering generated during thestructure formation may also affect the global 21cm signal. We investigate the effect of clusteringby comparing two mock simulations; one using thehalo catalog with both mass and position informa-tion from the hydrodynamic simulation (“Mock-clustering” simulation), and the other using onlythe halo mass catalog with random halo positions(“Mock-random” simulation). Both use the in-fall model with appropriate normalization to pre-dict the gas density distribution in the IGM, andthe mock adiabatic temperature is adopted. Themean 21 cm brightness temperatures of the 64 sub-boxes are plotted in the left panel for the “Mock-clustering” simulation and in the right panel forthe “Mock-random” simulation, respectively inFig. 11. Note the range of density fluctuations δ local are much smaller for the “Mock-random”case. With the clustered positions of halos, the av-eraged 21 cm brightness temperature is −
208 mK(the green star in the plot), which is about 7%decrement with respect to the homogeneous IGMcase, while if the halos are randomly distributed,then δT ∼ −
213 mK, which is only about 5%10 . − . . . . . δ local − − − − − − δ T [ m K ] homogeneous IGM δT ∝ ρ / Infall profile with mock adiabatic T K − . − . − .
01 0 .
00 0 .
01 0 .
02 0 . δ local − − − − − − δ T [ m K ] homogeneous IGM δT ∝ ρ / Random positions & infall-profiles & mock adiabatic T K Fig. 11.— The mean 21 cm brightness temperature of sub-boxes with different mean overdensity δ local at z = 17. The mock simulation has a box size of 4 Mpc /h and a resolution of 800 . Left panel:
Computed withthe halo catalog (masses and positions) from the hydrodynamic simulation;
Right panel:
Computed with thesame halo mass catalog but with randomized halo positions. Both panels assume the density profiles of theIGM predicted by the infall model. The black line indicates the expected value for a homogeneous IGM,and the cyan line shows the scaling of δT ∝ ρ / for cold optical-thin gas.decrement. Therefore, the clustering effect alsoreduces the absorption level of the 21 cm signal,by introducing higher level of density fluctuations,but the effect is only moderate.
4. Conclusions and discussions
In this work, we investigate the maximum sig-nal level of the global 21 cm spectrum from cos-mic dawn that could be achieved in the standardcosmology, and discuss various theoretical effectsthat could have impacts on the absorption level.By running a set of high resolution hydrodynamicsimulations, we find that the non-linear structureformation affects the IGM density and tempera-ture distribution significantly. The shock heatingand Compton heating, the non-linear density fluc-tuations, and the halo clustering, all have non-negligible effects reducing the 21 cm absorptionsignal. Under the assumption of saturated cou-pling between the spin temperature of hydrogenand the gas temperature, the maximum absorp-tion level that is achievable in the standard frame-work is reduce by about 15% at redshift 17, ascompare to the homogeneous IGM case.Among the various effects considered here, theshock heating during the non-linear structure for-mation and the Compton heating play a domi- nant role in reducing the maximum absorptionlevel. The non-linear density fluctuations withadiabatic heating can also reduce the contributionfrom over-dense regions, but the effect is moder-ate. The clustering of halos, on the other hand,also enhances the density fluctuations and reducesthe 21 cm signal mildly. By comparing the densityprofiles in the simulated IGM and those predictedby the infall model, we find that the infall modelprovides a fairly reasonable prediction for the den-sity distribution around density peaks in the IGM.We note that the heating effect of structureformation shocks during the cosmic dawn is stillsomewhat uncertain. The early work by Gnedin &Shaver (2004) shows that the shock heating has adramatic effect, dominating over Ly- α heating andX-ray heating at high redshifts, and reduces the 21cm global absorption substantially. Nevertheless,latter works (Furlanetto & Loeb 2004; Furlanettoet al. 2006; McQuinn & O’Leary 2012) show thatthe structure formation shocks have only modesteffect in heating the gas, being subdominant toX-rays, though the relative importance dependson the timing of the X-ray heating. Even if wedisregard the uncertainties in the X-ray produc-tion, there are still significant theoretical uncer-tainties on the structure formation shocks. Tocapture shocks, the smoothed particle hydrody-11amics (SPH) algorithm, which is adopted in theGADGET-2 used here, introduces an artificial vis-cosity to provide the entropy generated by micro-physics process. This introduces unphysical extraheating and broadens the shock front (Monaghan1992; Springel 2005). It may lead to the over-cooling problem (Creasey et al. 2011; Nelson et al.2013), and produce artificial cold blobs near thestar-forming regions (Hobbs et al. 2013). O’Sheaet al. (2005) did find that the gas properties in theSPH and the adaptive mesh refinement (AMR)based simulations generally agree with each other.In particular, McQuinn & O’Leary (2012) foundthat for the same resolution, the GADGET-2 andthe AMR-based Enzo code give quite similar evo-lution of the mean gas temperature. Our resultsare all based on the GADGET-2 simulations, andthey provide reasonable predictions for the scaleswe are interested in. Nevertheless, Jia et al. (2020)noted that there are still significant divergencein the number and strength of structure forma-tion shocks among different numerical schemes,such uncertainties could affect the results obtainedhere.In the present work, we have focused on themaximum absorption level, and include only theCompton heating and shock heating effects thatare inevitable during the cosmic dawn, isolatingthem from any other astrophysical heating relatedto radiation sources. Note that as more and moregalaxies form, various feedback effects includingphotoionization and X-ray heating would gradu-ally dominates over the shock heating and Comp-ton heating, the non-linear density fluctuations,and the clustering effects, reducing the global 21cm absorption more significantly. Our results pro-vide a modified signal base for any other feed-back processes to take further effects on, and in-dicate that one has to take into account the ef-fects of non-linear structure formation in order toaccurately interpret upcoming observational data,and/or inferring any requirement of new physics(e.g. Barkana 2018; Yang 2020).We thank Anastasia Fialkov, Rennan Barkana,Joe Silk, and Paul R. Shapiro for helpful discus-sions. This work is supported by National SKAProgram of China No. 2020SKA0110401, theMoST-BRICS Flagship Project No. 2018YFE0120800,the National Natural Science Foundation of China (NSFC) grant 11973047, 11633004, the ChineseAcademy of Sciences (CAS) Strategic Priority Re-search Program XDA15020200, the CAS Fron-tier Science Key Project QYZDJ-SSW-SLH017,and the NSFC-ISF joint research program No.11761141012. BY acknowledges the supportof the BR program from the CAS, the NSFCgrant 11653003 and the NSFC-CAS joint fundfor space scientific satellites No. U1738125. Thiswork used resources of China SKA RegionalCentre prototype (An et al. 2019) funded bythe National Key R&D Programme of China(2018YFA0404603) and Chinese Academy of Sci-ences (114231KYSB20170003). REFERENCES
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