Simulated differential observations of the Sunyaev-Zel'dovich Effect: Probing the Dark Ages and Epoch of Reionization
Charles Mpho Takalana, Paolo Marchegiani, Geoff Beck, Sergio Colafrancesco
SSimulated differential observations of theSunyaev-Zel’dovich Effect: Probing the Dark Ages andEpoch of Reionization
C.M. Takalana • P. Marchegiani • G. Beck • S. Colafrancesco † Abstract
This work presents an analytical approachfor studying the cosmological 21cm background sig-nal from the Dark Ages (DA) and subsequent Epochof Reionization (EoR). We simulate differential obser-vations of a galaxy cluster to demonstrate how theseepochs can be studied with a specific form of theSunyaev-Zel’dovich Effect called the SZE-21cm. Thiswork produces simulated maps of the SZE-21cm andshows that the SZE-21cm can be extracted from fu-ture observations with low-frequency radio interferom-eters such as the Hydrogen Epoch of Reionization Ar-ray (HERA) and the Square Kilometre Array (SKA).In order to simulate near realistic scenarios, we lookinto cosmic variance noise, incorporate and take intoaccount the effects of foregrounds, thermal noise, andangular resolution for our simulated observations. Wefurther extend this exploration by averaging over a sam-ple of galaxy clusters to mitigate the effects of cosmicvariance and instrumental noise. The impact of pointsource contamination is also studied. Lastly, we applythis technique to the results of the EDGES collabora-tion, which in 2018 reported an absorption feature ofthe global 21cm background signal centred at 78 MHz.
C.M. [email protected]. [email protected]. BeckGeoff[email protected]. Colafrancesco † Deceased: 30 September 2018 School of Physics, University of the Witwatersrand, 1 Jan SmutsAvenue, Braamfontein, Johannesburg, 2000, South Africa South African Radio Astronomy Observatory, 2 Fir Street, Ob-servatory 7925, South Africa Dipartimento di Fisica, Universit`a La Sapienza, P. le A. Moro2, Roma, Italy
The challenges to be addressed in order to achieve theobjectives of this work include errors that arise due tocosmic variation, instrumental noise and point sourcecontamination.Our approach demonstrates the poten-tial of the SZE-21cm as an indirect probe for the DAand EoR, and we conclude that the spectral features ofthe SZE-21cm from our simulated observations yield re-sults that are close to prior theoretical predictions andthat the SZE-21cm can be used to test the validity ofthe EDGES detection.
The cosmic Dark Ages (DA) is an era of darkness inthe Universe and the Cosmic Microwave Background(CMB) does not provide information about this periodas baryonic matter and radiation have already decou-pled prior to this epoch. Subsequently neutral hydrogengas in the intergalactic medium (IGM) makes up themajority of baryonic matter in the Universe (Zaroubi2013). The DA end with the formation of the first gen-eration of stars, that are believed to have had signifi-cant effects on the Universe and subsequently initiatethe Epoch of Reionization (EoR) (Bromm et al. 2009;Wise 2019). The first generation of stars and galax-ies emit high-energy photons that permeate the IGMand gradually ionize the hydrogen, until at around z ∼ ν = ν (1+ z ) with ν = 1420.405 MHz), and by using thisproperty we can isolate every phase in the history of the a r X i v : . [ a s t r o - ph . C O ] D ec IGM. The signal is difficult to measure due to contam-ination by astronomical foregrounds (Furlanetto 2006).Numerous efforts are nevertheless under-way to observeit and the experiments designed to measure it come intwo different varieties namely power spectrum (Pacigaet al. 2013; Jacobs et al. 2015, 2016; DeBoer 2017; Mon-dal et al. 2019) and global signal (Burns et al. 2011;Bowman et al. 2009, 2018).The collaboration of the Experiment for the Detec-tion of the Global EoR Signature (EDGES) recentlyreported the first observational results on the global 21cm spectrum (Bowman et al. 2018). The results showan absorption feature with a remarkably large ampli-tude of about 500 mK at the redshift range of z ∼ , Low Frequency Array (LOFAR) ,Hydrogen Epoch of Reionization Array (HERA) , andthe Square Kilometre Array (SKA) .Colafrancesco et al. (2016) derived the SZE-21cm in ageneral relativistic approach and studied how its spec-tral shape can be used to establish the global features inthe mean 21 cm spectrum generated during, and priorto, the EoR. Additionally, the aforementioned work alsooutlined how the effect depends on the properties ofplasma in cosmic structures. Takalana et al. (2018) de-scribed the first version of a procedure to observe theSZE-21cm with upcoming interferometers. While thepast work on this topic is mainly theoretical, in thiswork we perform new simulations of the background sig-nal and study instrumental effects for HERA and SKA.We simulate differential observations of a low redshiftgalaxy cluster as a function of frequency for a theoreti-cal 21 cm background model to show that by measuringthe SZE-21cm we can derive the properties of the global21 cm background spectrum generated during the DAand EoR. Our simulated observations explore the im-pact of cosmic variance (or sample variance) and pointsources and take into account the effects of foregrounds,thermal noise, and angular resolution for upcoming in-terferometers. We also consider the observation andcomputation of the SZE-21cm towards multiple clus-ters as a method to ultimately reduce the uncertaintyand improve the signal-to-noise. This work takes intoconsideration newer models of the 21 cm background,both theoretical and observational.The EDGES experiment reported a 21 cm backgroundsignal that challenges cosmological models and instru-mental precision. We use the EDGES detection resultsto make predictions for the SZE-21cm and make com-parisons with the simulated theoretical 21 cm back-ground model results to show that differential obser-vation techniques for the SZE-21cm can be used toconfirm detections made by experiments similar toEDGES. Lastly, we explore the feasibility of detectingthe SZE-21cm with the upcoming SKA telescope.In section 2 we present the theoretical background andmathematical formalism for studies of the 21 cm tran-sition signal and the SZE-21cm. In section 3 we sim-ulate differential observations of a galaxy cluster withupcoming radio interferometers and from this establishthe relationship between the SZE-21cm spectrum andthe global 21 cm background spectrum using a theo-retical 21 cm model. In section 4 we use the global 21 http://lofar.org http://reionization.org cm background signal measured by EDGES to makepredictions for the SZE-21cm and make comparisonswith the signal obtained through our simulations usingthe theoretical 21 cm model. In section 5 we discussthe simulation results and look at possible challengesfor observations of the SZE-21cm presented by instru-mental effects, cosmic variance, and point sources. Wesummarize our findings and conclude in section 6. Weassume a ΛCDM cosmology throughout this work withthe following Planck 2015 parameters: h = 0.673,Ω m = 0.315 , Ω b = 0.0491, Ω Λ = 0.685, σ = 0.815 and n s = 0.968 (Planck Collaboration 2015). In this section, we present theoretical background andformalism for studies of the 21 cm transition signaland discuss the detection of the signal claimed by theEDGES collaboration in 2018, and look at the concernsraised about the results. We also provide the theoreticalbackground of the SZE-21cm and revisit the formalismderived by Colafrancesco et al. (2016). Lastly, we dis-cuss observations of the 21 cm to probe the DA andEoR.2.1
The cosmological 21 cm background signal
Neutral hydrogen atoms undergo a quantum mechani-cal transition between the hyperfine energy levels (spin-flip transition) resulting from magnetic interaction be-tween the electron and the spin of the proton. Whenthe spins change from parallel to anti-parallel, a photonwith a wavelength of 21 cm is emitted, producing a sig-nal with a line center frequency of ν = 1420.405 MHz(van de Hulst 1945; Ewen & Purcell 1951; Muller &Oort 1957). The intensity of the 21 cm radiation from apopulation of neutral hydrogen is dependent on the spintemperature ( T S ) defined through the ratio between thenumber densities of hydrogen atoms in the 1 S tripletand 1 S singlet levels (Pritchard & Leob 2012). Thequantity of interest that radio interferometers will mea-sure is the offset of the 21 cm brightness temperaturefrom the CMB temperature, T CMB = 2 . z ) K,along a line of sight at the observed frequency (Furlan-etto 2006), which is written as δT b (cid:39) x HI (cid:20)(cid:18) . m (cid:19) (cid:18) z (cid:19)(cid:21) / (cid:18) Ω b h . (cid:19) (cid:20) − T R T s (cid:21) mK , (1) where x HI is the neutral fraction of hydrogen, h is theHubble parameter in units of 100 km s − Mpc − , andΩ b and Ω m are the baryon and matter densities respec-tively. T R is the temperature of the background radi-ation, which this work assumes to be from the CMB,hence T R = T CMB . Predicted by Shaver et al (1999),this global 21 cm background signal is sensitive to bothastrophysical and cosmological processes in the earlyUniverse, which makes it a good probe of the physicsbetween the CMB decoupling, the DA and the EoR(see Field 1958; Shaver et al 1999; Barkana & Leob2005; Furlanetto 2006; Pritchard & Leob 2012, for de-tailed discussions and formalism derivations for the 21cm background signal).This work focuses on a theoretical model of the 21 cmbackground signal that we simulate under standard cos-mological assumptions using a semi-numeric modelingtool Mesinger et al. (2010) and one based on the best-fitting 21 cm profile model from recent EDGES results(Bowman et al. 2018). For both cases, the brightnesstemperature offset from the CMB is shown in Figure 1as a function of frequency. Equation 1 tells us that theamplitude of the absorption signal can be increased byreducing T s . The claimed EDGES detection exhibits amuch larger flat-bottomed amplitude of − +200 − mKwith a Full-Width at Half-Maximum of 19 +4 − MHz cen-tred at around 78 MHz (corresponding to z ∼
17) Bow-man et al. (2018). Various proposals have been made toexplain the larger than expected amplitude observed byEDGES. One of the proposals argues that the anomalycould be a new signal of baryon-dark matter interac-tion as scattering processes that may cool neutral hy-drogen with respect to the standard expectations fromthe CMB measurements (see Barkana 2018; Fialkov etal. 2018). Another suggestion argues that the anomalymay be related to an additional radio background withlikely causes of this excess ranging from instrumentalsystematics to new astrophysics as may have been ob-served by ARCADE 2 (see Fixsen 2011; Ewall-Wice etal. 2018).Motivated by the unexpected nature of the EDGESclaim, concerns were raised regarding the modelling ofthe EDGES data. These concerns arise from the fol-lowing findings: • Hills et al. (2018); Singh & Subrahmanyan (2019);Sims & Pober (2019) suggest the potential presenceof an uncalibrated sinusoidal-form systematic in thedata. • Hills et al. (2018) also highlight the use of a mod-eling process that implies unphysical parameters inthe data for foreground emission, where the degener-acy between signal and foreground modeling allows awide range of signals to be consistent with the data.
Fig. 1
The mean 21 cm background brightness temperature offset from the CMB spectrum as a function of frequency.The
Blue curve is for a theoretical 21 cm model simulated for this work and the
Red curve for the best-fitting 21 cm profilemodel derived from the EDGES observations (Bowman et al. 2018). • Work by Singh & Subrahmanyan (2019) enforcesa maximally smooth polynomial foreground model,constant amplitude sinusoidal systematic model,white noise and a Gaussian parametrisation of theglobal 21 cm signal, and found evidence for a dif-ferent 21 cm signal, substantially more in agreementwith the standard predictions. • Bradley et al. (2019) report on a possible systematicartifact within the ground plane due to resonancemodes beneath the antenna that may produce spuri-ous broad absorption features in the spectra.While the above ideas may not perfectly describe thedetection, they illustrate the efforts of the communityto use the results of Bowman et al. (2018) to explorepossible exotic physics and new source populations asextensions to standard models for cosmology and par-ticle physics. They also explore possible unmodeledsystematic effects present in the data as we have seenabove. Later in this work, we make predictions for theSZE-21cm using a 21 cm background signal describedby the Bowman et al. (2018) data to test the usefulnessof the SZE-21cm as a technique that can potentiallybe used to verify the EDGES detection and other suchclaims in the future.2.2
The SZE-21cm
The interaction between photons and free electronsby inverse Compton scattering results in the SZE (Zel’dovich & Sunyaev 1969; Sunyaev & Zel’dovich1970, 1972, 1980). The SZE is produced when thefree electrons in the dense cores of galaxy clustersthat contain hot ionized gas ( ∼ K) (Cavaliere &Fusco-Femiano 1976) up-scatter CMB photons caus-ing a change in the apparent brightness of the CMBphotons, which modifies the incident Planck spectrum(see Birkinshaw 1999, for a review). This scatteringresults in a systematic shift of the CMB photons fromthe Rayleigh-Jeans to the Wien side of the spectrum(Birkinshaw 1999; Carlstrom et al. 2002; Colafrancescoet al. 2002).This work examines the use of low-frequency SZE ob-servations ( <
200 MHz) to study the global propertiesof the universe during the DA and EoR with the 21cm line of neutral hydrogen. In this section, we sum-marise the formalism for the SZE-21cm as derived byColafrancesco et al. (2016) as this forms the basis of thesimulated observations we present in the sections thatfollow. For the full treatment of the formalism in therelativistic limit, the reader is referred to Colafrancescoet al. (2002) for the standard SZE and Colafrancescoet al. (2016) for the SZE-21cm. Here we consider theSZE due only to the thermal electrons in a hot cluster,which requires the appropriate relativistic correctionsto be taken into account.Any energy injecting process occurring during the EoRand DA modifies the CMB spectrum as described in
Section 2.1 and subsequently modifies the SZE. Theincident CMB radiation field modified during the DAand EoR is given by the specific intensity as: I ,mod ( ν ) = I ,st ( ν ) + δI ( ν ) , (2)where δI ( ν ) is the radiation intensity corresponding tothe differential brightness temperature ( δT b ) in equa-tion 1 obtained by applying the Rayleigh-Jeans law, δI ( ν ) = K B ν c δT b ( ν ), and the unperturbed or stan-dard CMB spectrum ( I ,st ) is given by I ,st ( x ) = 2 ( k B T CMB ) ( hc ) x e x − , (3)where c is the speed of light, h is the Planck constant, k B is the Boltzmann constant, and x = hνk B T CMB is thea-dimensional frequency. The spectral distortion due tothe SZE of the CMB radiation field as modified duringthe DA and EoR is given by I mod ( x ) = (cid:90) + ∞−∞ I ,mod ( xe − s ) P ( s ) ds, (4)where P ( s ) is the photon redistribution function thatis dependent on the electrons spectrum producing theCMB Comptonization (Birkinshaw 1999; Enßlin &Kaiser 2000; Colafrancesco et al. 2002, 2016), and isessentially the scattering kernel giving the probabilityof a scattering to shift a photon from a frequency ν to ν (cid:48) (with s = ln( ν (cid:48) ν )). The SZE-21cm is then given asthe difference between the scattered spectrum ( I mod )and the incoming radiation ( I ,mod ):∆ I mod ( ν ) = I mod ( ν ) − I ,mod ( ν ) . (5)This equation can then be transformed in terms ofbrightness temperature by applying the Rayleigh-Jeanslaw and be rewritten as∆ T mod ( ν ) = T mod ( ν ) − T ,mod ( ν ) . (6)2.3 Observations of the 21 cm signal from theDA and EoR
The 21 cm background signal from the DA and EoRis weak and may be observed as a faint, diffuse back-ground detectable at frequencies below 200 MHz. Pre-vious and current generation interferometer experi-ments attempt to measure a statistical power spectrum of the signal over the sky, rather than to image the sig-nal directly, by measuring a range of aggregate spatialscales on the sky. There are several challenges associ-ated with observations of the 21 cm background. Themain problem is that the signal is weak, at a character-istic scale of ∼
10 to 100 mK, and the task is furthercomplicated by the much stronger foreground contin-uum emission due to sources such as galactic diffusesynchrotron, free-free emission, and emissions from ex-tragalactic radio sources (Santos et al. 2005; Sims &Pober 2019). These contaminating signals are five or-ders of magnitude stronger than the 21 cm emission,making its observation extremely complex to study,and requiring very precise calibration and knowledgeof various foregrounds (see e.g., Furlanetto 2006). Sev-eral ideas have been presented to overcome the variousproblems identified above, including foreground avoid-ance and removal techniques. These strategies rely onthe spectral smoothness and spatial characteristics offoreground sources; we do not discuss the various ap-proaches here but refer the reader to other texts (seee.g., Barkana & Leob 2005; Furlanetto 2006; McQuinnet al. 2006).All observations that are sufficiently sensitive to detectthe SZE are differential (Carlstrom et al. 2002). Con-sequently, differential observations of the SZE-21cm to-wards galaxy clusters have been proposed as one wayto overcome the above-mentioned problems by Cooray(2006) and Colafrancesco et al. (2016). Unlike an ex-periment to directly determine the cosmological 21 cmbackground spectrum involving a total intensity mea-surement on the sky, differential observations with aninterferometer are less affected by the confusion fromgalactic foregrounds that are smooth over angular scaleslarger than a typical cluster and by issues such as theexact calibration of the observed intensity using an ex-ternal source (Cooray 2006; Colafrancesco et al. 2016).In the next section, we simulate differential observa-tions of a galaxy cluster to compute the SZE-21cm.
Simulations of the 21 cm background
We simulate the 21 cm cosmological background usingthe publicly available semi-numerical code 21cmFAST .21cmFAST takes into account the necessary physicalprocesses and produces large-scale simulations utilizingperturbation theory, the excursion set formalism (which https://github.com/andreimesinger/21cmFAST specifies the location and mass of collapsed dark matterhaloes), and analytic prescriptions to produce evolved3D realizations of density, spin temperature fields, ion-ization, and peculiar velocity, which when combined,can be used to deduce the global 21 cm brightness tem-perature at a given redshift as described by equation 1.Each simulation cube in our runs is 30 Mpc on a sidedown-sampled from the high-resolution initial condi-tions generated on a 460 grid and computed on a 115 grid corresponding to a cell resolution of 0.26 Mpc, inthe frequency range ν = 30 - 160 MHz. Our modelassumes that Population II stars are responsible forearly heating with parameters in Table 1. The parame-ters are the ionizing photons per stellar baryon N γ , theionizing efficiency (the number of photons per baryonper dark matter halo) ζ , the fraction of baryons con-verted to stars f ∗ , the minimum virial temperature forall sources T vir,min , the maximum horizon for ionizingphotons R max , the efficiency parameter correspondingto the number of photons per solar mass in stars ζ X ,and the clumping factor on the scale of the simulationcell C . We refer the reader to Mesinger et al. (2010)for detailed information and discussions on the code,relevant parameters, and the formalism. Table 1
Summary of parameters used in our run of the21cmFAST simulation run.
Parameter N γ ζ f ∗ T vir,min K R max
50 Mpc ζ X M (cid:12) C Procedure for simulated observations of theSZE-21cm
This work makes use of only the 21 cm brightness tem-perature δT b ( ν ) cubes from the 21cmFAST simulations.We use equation 3 to compute unperturbed CMB in-tensity cubes ( I ,st ( ν )), and Rayleigh-Jeans law to cal-culate it in terms of brightness temperature ( T ,st ( ν )),and add this signal to each cube of δT b ( ν ), which givesus the CMB radiation field modified during the DA andEoR ( T ,mod ( ν )). To conduct our mock observations ofthe SZE-21cm we insert a typical rich cluster with atemperature of 10 keV, an optical depth of 10 − , anda radius of 2 Mpc. The incoming background radiationfield ( T ,mod ( ν )) is scattered by energetic electrons in Fig. 2 Top : The mean 21 cm background brightness tem-perature offset from the CMB spectrum as a function of fre-quency incident on each of the cluster-sized regions (
Black ),along with the simulation global mean (
Red ). The shadedareas correspond to 2 σ and 3 σ standard deviations in thecluster-sized regions. Bottom : Noise on the 21 cm Globalsignal as a function of frequency (y-axis in log-scale). The(
Red )-dashed lines represent the cosmic-variance. For com-parison, we also have the instrumental noise for the HERAand SKA1-Low experiments calculated using equation 13. the cluster, resulting in a modified signal ( T mod ( ν )) cal-culated according to equations 4 - 6.One of the issues that we will face measuring the SZE-21cm is that the 21 cm background has variations onangular scales comparable to that of the galaxy clus-ters and their intrinsic level presents a potentially sig-nificant contaminant to SZE-21cm observations as thevariations may be comparable to the expected signalof the SZE-21cm as computed by Colafrancesco et al.(2016). The variations ( ∆ TT ) in our cubes are computedto be of orders between 10 − - 10 − . They arise fromsignificant degeneracies between parameters that con-trol the average 21 cm global temperature spectrum,which lead to inhomogeneities (McQuinn et al. 2006),and are greater than the CMB primordial fluctuationsthat are of the order of 10 − . To quantify the cos- mic variance noise that arises from the finite volume ofthe universe accessible to 21 cm experiments we plotthe 21 cm signal (versus frequency) incident on eachof the cluster-sized angular regions in the simulations,along with the global average (see Figure 2). The fig-ure illustrates how the global signal measured over thecluster-sized region can significantly depart from theaverage signal. Each of the cluster-sized regions pro-vides an estimator for the δT b ( ν ), which varies fromone to another, showing variance produced to the sig-nal in a finite region. We note that cosmic variancenoise increases the error budget resulting in a decreasein the significance of any detection. Hence, by allowingobservations of many more samples we will reduce theoverall error budget and effects of cosmic variance.Observations of the SZE at microwave frequencies willaid in distinguishing the SZE-21cm fluctuations fromintrinsic degeneracies between parameters that controlthe mean 21 cm brightness temperature spectrum andCMB fluctuations, and we can use information fromthese observations as a way to normalise low-frequencysignals that will be crucial to establishing signaturesrelated to the 21cm signal. Contamination from pointsources is also a major concern and for our mock obser-vations, it is, therefore, useful that we detect and dis-card point-source contaminated pixels. To do this weapply a threshold approach to avoid contamination inthe SZE-21cm computation, identifying possible pointsources in these fields above a given threshold and re-placing those pixels with the mean background value;this also helps us account for the cube fluctuations.Later in the section 5, we will focus on how possiblesources of contamination can be handled. To include realistic 21 cm instrumental effects we fol-low the methodology of Hassan et al. (2018) that ac-counts for the finite angular resolution of the instru-ment, foreground cleaning, and the presence of in-strumental noise. This methodology produces mockobservations using the 21cmSense package (Pober etal. 2013, 2014), which creates realisations of thermalnoise taking into account the antenna configuration,system temperature, and station/dish size to calculateexpected sensitivities of 21 cm experiments. For thiswork, we produce mock observations with HERA andthe low-frequency SKA1 (SKA1-low) instruments. Thesummary of our assumed HERA and SKA array designsare in Table 2. https://github.com/jpober/21cmSense We take a 3D-Fourier transform of each cube, this givesus the magnitude of the cube as a function of wavevector k ; k , the magnitude of k vector, is k = | k | = (cid:113) k ⊥ + k (cid:107) , where k ⊥ is the co-moving component onthe plane of the sky and k (cid:107) is the co-moving compo-nent along the line of sight. It is expected that fore-ground contamination of 21 cm measurements will bespectrally smooth and should only affect the lowest k (cid:107) modes (Bowman et al. 2009). However, due to the fre-quency dependence of an interferometer’s response, thecontamination will leak into higher k modes through“mode mixing”, which is stronger for long baselines(high values of k ⊥ ) that have higher fringe rates. Thismode mixing causes the foregrounds to become confinedto a wedge-shaped region in the k ⊥ − k (cid:107) space (Liu etal. 2009, 2014).First, we remove the foreground contaminated k modesfrom the cubes assuming that foregrounds are con-fined to some regions in Fourier space along the line-of-sight. This confinement is expressed in terms of spatialFourier wave-numbers for Fourier modes k ⊥ and k (cid:107) .Foreground contaminants are expected to be mostly inmodes that satisfy the following condition: k (cid:107) < k (cid:107) H D c θ [Ω m (1 + z ) + Ω Λ ] c (1 + z ) k ⊥ , (7)where D c is the co-moving line-of-sight distance, θ is theangle between the line of sight and the direction of theFourier mode k , and k (cid:107) is an offset constant (detailedderivations of this may be found in Liu et al. 2014,2016). The modes of the signal that are contaminatedby foregrounds lie inside a foreground wedge in the k ⊥ − k (cid:107) plane. The foreground wedge slope is given by: m = H D c [Ω m (1 + z ) + Ω Λ ] sin θc (1 + z ) . (8)For our cubes, we zero out all the modes that are insidethe wedge to clean the foregrounds. More modes are re-moved at lower frequencies since m increases with de-creasing frequency. To account for the angular resolu-tion we compute the uv-coverage antennae distributionfor the HERA and SKA1-Low designs using the 21cm-Sense package. The uv-coverage gives the total numberof baselines that observe a given u-v pixel. The angulardirection is given by k x = 2 πk y , where the sky along xand y coordinates can be converted in terms of baselinelength in u and v coordinates. We compute the u-vcoverage for HERA and SKA1-Low from the antennadistribution and convert the uv coordinates into corre-sponding k x and k y modes. We then obtain a Fourier Fig. 3
Simulated SZE-21cm cluster maps from our mock differential observations at 59.43 MHz, 81.59 MHz, 112.02 MHz,and 176.67 MHz for a cluster with a plasma temperature of 10 keV, a radius of 2 Mpc, and optical depth of 10 − . Themaps are shown for simulated differential observations without instrumental effects and simulated observations with HERAand SKA1-Low. Table 2
Summary of parameters used in this work for HERA and SKA.
HERA SKA1-low
Antennae design 350 hexagonally packed dishes 512 stationsAntenna/station diameter (m) 14 35Collecting area ( m ) 53,878 416,595 A eff ( m ) 153.93 962.11 T sys ( K )(= T sky + T rcvr ) T sky + 100 1 . T sky + 40transform of T mod ( ν ) and T ,mod ( ν ) maps, and zero outthe Fourier modes that correspond to null u-v coverageat k x and k y . We smooth down the maps using a Gaus-sian filter with a full width half maximum of FWHM= cm (1+ z ) B , where B is the maximum baseline lengthfor HERA and SKA1-Low. Thereafter, we obtain theangular resolution limited T mod ( ν ) and T ,mod ( ν ) mapsand we inverse Fourier transform back to real space.We simulate a noise map whose pixel values are ob-tained from a Gaussian distribution with a zero meanand standard deviation that is set to thermal noise forHERA and SKA1-Low (see, Zaldarriaga 2004; Hassanet al. 2018): (cid:112) (cid:104)| N | (cid:105) [ Jy ] = 2 k B T sys A eff √ ∆ ν t int , (9)where A eff is the effective area of a single antenna,∆ ν is the frequency resolution, t int is the integrationtime, and T sys = T sky + T rcvr is the system tempera-ture, where T rcvr is the receiver temperature and thesky temperature is, T sky = 120 (cid:0) ν (cid:1) − . K, see Table 2.Our simulated experiments assume t int = 1000 hours,and ∆ ν = 97.8 kHz for HERA and ∆ ν = 50 kHz forSKA. The noise is suppressed by our u-v coverage ( N uv )by a factor of √ N uv , we inverse Fourier transform thenoise map to real space. This is then added to our fore-ground and angular resolution corrected T mod ( ν ) and T ,mod ( ν ) maps. We perform mock observations in the direction of thecluster T mod ( ν ) and compare this with an empty skyregion located away from the cluster region T ,mod ( ν )to obtain SZE-21cm maps with HERA and SKA1-Low.The empty sky region away from the cluster has thesame size as the region of the cluster to allow for pixel-by-pixel operations. The differential observation thatgives us the SZE-21cm as in equation 6 can be writtenas: ∆ T SZE − cm = T ON − T OF F , (10) where T ON is the value obtained when observing thecluster signal (hereafter titled signal ON), from whichwe subtract the background (empty sky), T OF F (here-after titled signal OFF). To account for correlation andposition-dependent offsets, we use two background re-gions and let T OF F = T OFF + T OFF . We perform thedifferential observation as set out by equation 10, whichgives us the SZE-21cm at the cluster as ∆ T mod =∆ T SZE − cm , which is the modification of the initialbackground signal. Assuming that the mock cluster’sgas is isothermal, we use the isothermal β -model tomodel its density profile (see e.g., Cavaliere & Fusco-Femiano 1976), which varies as a function of clusterradius, r : n e ( r ) = n e, (cid:18) r r c (cid:19) − β , (11)where n e, is the electron number density at the centerof the spherical symmetric cluster gas distribution; β isthe power-law index and r c is the core radius. This elec-tron density distribution leads to the following analyticform of SZE-21cm brightness temperature profile:∆ T mod ( ν ) = ∆ T mod, (cid:18) r r c (cid:19) − β , (12)∆ T mod, is the central temperature decrement. Wemodel the SZE-21cm brightness temperature, ∆ T mod ,at the cluster using the model in equation 12, with β = and r c = 0.248 Mpc for a typical rich cluster.We generate cluster maps and measure the SZE-21cmat the cluster, and use the measured signal to con-struct the SZE-21cm spectrum. The simulated mapsare shown in Figure 3 and the cluster radial profiles arein Figure 4. The mock observations are made in thefrequency range between 30 and 160 MHz, but here weonly show the simulated maps and cluster radial profilesat at 59.43 MHz, 81.59 MHz, 112.02 MHz, and 176.67MHz as these are the frequencies where we can see thestrongest decrements or increments in the SZE-21cmsignal. The SZE-21cm spectrum is shown in Figure 5and is discussed at length in section 3.3. Fig. 4
The SZE-21cm temperature decrement radial pro-file from a cluster of radius of 2 Mpc, a core radius ( r c ) of0.248 Mpc and β = at 59.43 MHz, 81.59 MHz, 112.02MHz, and 176.67 MHz. Top : simulation without instru-mental effects and simulated observations with HERA.
Bot-tom : simulation without instrumental effects and simulatedobservations with SKA1-Low. The shaded errors regionsassume observations with HERA (Top) and SKA1-Low (Bottom) . Discussion: HERA and SKA simulatedobservations of the SZE-21cm for thetheoretical model
According to Cooray (2006) and Colafrancesco et al.(2016), detailed measurement of the SZE-21cm will al-low us to derive precise information on the epochs atwhich the CMB was modified (i.e., DA and EoR) and onthe physical mechanisms that imprint distinct featureson the global 21 cm background ( δT b ) that are presentin the SZE-21cm spectrum. Figure 3 in the work pre-sented here shows images resulting from the simulateddifferential observations of a galaxy cluster at differentfrequencies, and figure 5 gives us the SZE-21cm mea-sured at the cluster without instrumental effects andalso taking into account the effects of foregrounds, ther-mal noise, and angular resolution observing with HERA and SKA1-Low, which gives us a good idea of the im-pact that instrumental effects have on the differentialobservations. For our simulated observations using alow-frequency radio interferometer, we employ a simplestrategy to provide order of magnitude approximationsto the noise errors at the frequencies of relevance. Thesystem temperature ( T sys ) for HERA and SKA1-Lowcan be found in Table 2. The instrumental noise on thebrightness temperature, ∆ T N , measured by an inter-ferometer is given by Furlanetto (2006)∆ T N = T sys η f √ ∆ νt int . (13)The array filling factor is given by η f = A eff /D max ,where D max and A eff are the maximum baseline andtotal effective area of the array respectively.We realise that the SZE-21cm spectrum almost re-sembles an inverted δT b spectrum. As the SZE-21cmchanges as a function of frequency, its features corre-spond to some changes in the δT b spectrum. For exam-ple, the minimum point of the input spectrum at ap-proximately 80 MHz, Figure 1, corresponds to the max-imum point of the SZE-21cm spectrum, Figure 5. Thespectrum shows absorption of the SZE-21cm between40-60 MHz, as a result, we observe a decrement in theSZE-21cm map at 59.43 MHz, this followed by emissionin the SZE-21cm spectrum between 60-90 MHz and as aresult, we observe an increment on the SZE-21cm mapat 81.59 MHz. Between 80-110 MHz there is absorptionin the SZE-21cm spectrum and as a result, we observeanother decrement at 112.02 MHz. At ν >
110 MHzthe SZE-21cm spectrum starts to pick-up and this isevident in the SZE-21cm map at 176.67 MHz. Notably,we have a deviation between the simulated observationand the true signal at high frequencies in Figure 5, andthe deviation is more substantial for SKA1-Low thanfor HERA, contrary to expectations, and it is not clearwhat produced this effect.The cluster is easily identifiable in all the simulatedmaps thanks to the angular resolution of our simulatedHERA and SKA1-Low designs and the uv-coverage thatextends small scales at the frequencies of interest. Thenoise for SKA1-Low is less as compared to HERA andthe lower angular resolution for HERA makes it slightlymore challenging to resolve the cluster. Figure 4 alsoshows that we can recover the cluster radial profile withboth HERA and SKA as the simulated observation re-sults with instrumental effects are compatible with themodel. At higher frequencies, SKA1-Low will be able toproduce good results, however, at low frequencies, theerrors are larger. SKA1-Low may be an ideal instru-ment to conduct studies of the SZE-21cm as compared Fig. 5
The SZE-21cm spectrum from our mock differential observations of a galaxy cluster with a plasma temperature10 keV and τ = 10 − . The Black solid line is the theoretically expected value calculated for our simulated cluster. The
Green points and error bars are from the SZE-21cm measurement of the simulated observations for HERA, and the
Red points and error bars are from the SZE-21cm measurement of the simulated observations for SKA1-Low. to HERA, which promises to be instrumental in makingthe first detections of the effect as it will come onlineahead of SKA1-Low. Minimising the errors when ob-serving with HERA will require significant amount ofintegration time, this work simulate the observationsassuming 1000 hours with both instruments.
The SZE-21cm, which is our quantity of interest, is afaint signal and the best way to deal with such a sig-nal would be to average it over as many clusters aspossible. We demonstrate the feasibility of reducingthe error budget that arises due to instrumental effectsby averaging the SZE-21cm observed towards multiplegalaxy clusters with the low-frequency SKA telescope,we estimate the expected error for cluster observationsof the SZE-21cm by scaling and combining observationstowards multiple clusters to improve the signal-to-noisewhich is given by equation 13. The expected errors inFigure 6 are determined assuming observations towardsone cluster and the improvement by computing the av-erage over signals towards ten and then one hundredclusters. To give a sense of the impact this may haveon reducing the error budget, we assume an improve-ment in ∆ T with √ N cl of the number of clusters ( N cl )used (Cooray 2006; Clanton et al. 2012). In each ofthe clusters, the SZE-21cm signal is present, thereby adding to it its contribution. Whereas, noise is ran-dom and therefore does not add but begins to cancel asthe number of clusters we average increases. For thisreason, we can expect that the signal-to-noise improve-ment obtained by averaging the SZE-21cm from mul-tiple galaxy clusters to be proportional to the squareroot of the number of clusters averaged. Measuringthe SZE-21cm observed towards multiple galaxy clus-ters and obtaining an average as described here will alsobe beneficial to reduce the uncertainty that arises dueto cosmic variance on the 21 cm background signal inci-dent on each of the cluster-sized angular regions in thesimulations, see Figure 2, which we begin to suppressstacking up to a few hundred clusters.The SZE-21cm computation requires prior knowledgeof the cluster’s electron temperature and optical depthprofile, which can be obtained from observations ofthe SZE related to the CMB spectrum alone at muchhigher radio and microwave frequencies, possible withthe Planck telescope (see e.g., Hansen et al. 2002). Co-lafrancesco et al. (2016) pointed out that the SZE-21cmas a function of frequency is dependent on the temper-ature of the intracluster plasma, mathematically thiscan be written in terms of the Compton parameter as y ∝ k B T · τ . We propose two solutions to overcomethe challenge that arises from the SZE-21cm varyingfor clusters with different plasma temperatures. To av-erage the SZE-21cm of clusters with different plasma Fig. 6
The SZE-21cm spectrum from our mock differential observations of a galaxy cluster with a plasma temperature10 keV and τ = 10 − . The shaded errors regions assume observations with an interferometer similar to SKA. The Green error region is for differential observations towards a single cluster,
Grey area towards 10, and the
Gold area towards 100clusters (see e.g., Cooray 2006). The
Black solid line is the best-fit to theoretically calculated data points, the
Magenta dots are points measured from our simulated mock observations and the
Blue solid line is the standard SZE (∆ T st ) forcomparison. temperatures, we can study and measure the Compton y parameter at higher frequencies and factor it in ourcomputations, alternatively we can average the SZE-21cm for clusters with the same plasma temperaturesonly. In this regard, we expect the low-frequency SKAarray to have large fields-of-view of approximately 2.5 –10 degree from 200 MHz down to 50 MHz (Mellema etal. 2012), which will enable us to identify large clustersamples to study the SZE-21cm by selecting the oneshaving similar plasma temperatures as can be derivedfrom X-ray observations. Detecting a larger number ofsources will also strengthen the statistical studies of theDA and EoR and the cosmological radio backgrounds.To better derive information on the SZE-21cm pro-duced by low redshift galaxy clusters, multi-frequencymeasurements of the SZE at X-ray and microwavebands are highly desirable for the extraction of addi-tional information from its spectral shape, and for sep-arating the signal from the various sources of contam-ination and confusion. Such multi-frequency measure-ments will also be useful in finding suitable cluster can-didates for low-frequency studies based on the strengthof the SZE signal at high-frequencies and will be in-strumental in estimating the profile of the optical depthand the temperature of the cluster gas, which must beknown when computing the SZE-21cm. In this section, we compare our results for the SZE-21cm obtained using the fiducial 21 cm backgroundmodel in section 3 to the results we calculate using the21 cm background spectrum claimed by the EDGESexperiment (Bowman et al. 2018). To calculate theSZE-21cm with the input spectrum being the 21 cmbackground signal claimed by the EDGES collabora-tion we used the formalism in section 2.2. We showthe spectra in Figure 7. We note that the SZE-21cmspectrum computed from the EDGES input spectrumis accurate only in the frequency range (70-90 MHz)due to the edge effects of the frequency range.The EDGES 21 cm background spectrum in Figure1 implies that gas temperatures during DA and EoRwere cooler by orders of 2 to 3 as compared to pre-dictions by any prior models. The peak of the SZE-21cm for the EDGES detected background signal andthe troughs of the SZE-21cm are also amplified by or-ders 2 to 3 compared to the theoretical model. Figure7 therefore, demonstrates that the SZE-21cm spectrumcan be used to establish the amplitude of the inputbackground spectrum and features that probe the cor-responding epochs.In Figure 8 we plot the low-frequency SKA telescopesensitivity compared to the standard SZE and the SZE-21cm. We note that the EGDES background is reliableonly in the band 70-90 MHz. We list our findings below: Fig. 7
The SZE-21cm computed for the simulated fiducial 21 cm background model from Figure 1 (
Blue ) compared tothe SZE-21cm computed using the 21cm background spectrum detection claimed by the EDGES experiment (
Red ). Theshaded errors regions assume observations with an interferometer similar to SKA for differential observations towards asingle cluster. The SZE-21cm spectrum computed from the EDGES input spectrum is only reliable between 70-90 MHzdue to the edge effects of the frequency range of the input spectrum. For both cases, we assume a cluster with a plasmatemperature of 10 keV and an optical depth of τ = 10 − . • With SKA1-low-50% the SZE-21cm can be detectedfor the fiducial model at ν (cid:38)
90 MHz and the EDGE-like background at ν (cid:38)
75 MHz. Good frequencychannels to distinguish between the standard SZEand the SZE-21cm signals are 110 - 140 MHz for thefiducial model and 90 - 130 MHz for an EDGES-likebackground spectrum. • With SKA1-low the SZE-21cm can be detected forthe fiducial model at ν (cid:38)
70 MHz and the EDGES-like background at ν (cid:38)
60 MHz. Good frequencychannels to distinguish between the standard SZEand the SZE-21cm signals is ν (cid:38)
75 MHz for thefiducial model and ν (cid:38)
70 MHz for an EDGES-likebackground spectrum. • With SKA2 the SZE-21cm can be detected for boththe fiducial model and EDGES-like background at ν (cid:38)
50 MHz. Good frequency channels to distinguishbetween the standard SZE and the SZE-21cm signalsis ν (cid:38)
70 MHz for the fiducial model and ν (cid:38)
55 MHzfor an EDGES-like background spectrum.To match the unexpectedly large extent of the SZE-21cm computed using the EDGES signal without re-course to any form of new physics we may need tochange the conditions for the source of the backgroundradiation in our simulated models. Apart from con- cerns raised on the foreground model and systematics,the EDGES detection has been discussed extensivelyand potential explanations consider new physics, radiobackground in excess of the CMB at redshifts relatedto the DA and EoR, interaction between baryons anddark-matter particles (Barkana 2018), and new modesof star formation such as metal-free Pop III stars (seee.g., Mebane et al. 2019). While this detection has yetto be confirmed by other techniques and experiments,it exhibits features that are inconsistent with existingtheoretical predictions. Independent experiments andmethods are needed to verify this EDGES result, there-fore, we propose measuring the SZE-21cm with upcom-ing instruments like the SKA as this method probes thefeatures of the input 21 cm background spectrum.
Work by Cooray (2006) and Colafrancesco et al. (2016)showed that contamination from background and fore-ground emissions on scales larger than a typical galaxycluster has less impact on differential observationalmeasures of the SZE-21cm. However, radio pointsources have historically been the main source of con-tamination in the SZE measurements (Carlstrom et al. Fig. 8 Right:
The spectra of the SZE-21cm ∆ I mod . The blue solid line represents the SZE-21cm for our simulatedfiducial 21 cm background model and the Red line for a 21 cm background spectrum similar to the EDGES detection.The
Grey dotted and dashed line represents the SZE for a non-modified CMB ∆ I st . Left:
The absolute value of thedifference between the SZE-21cm and the standard SZE. The
Blue solid line for our simulated fiducial model and the
Red line for the SZE-21cm calculated from the spectrum similar to the EDGES detection. For both figures, we compared withthe SKA1-low-50% (
Black ), SKA1-low (
Green ), and SKA2 (
Magenta ) sensitivities for 100 kHz bandwidth, 1000 hrs ofintegration, 2 polarizations, no taper, and no weight. < z = 0.1 with point sources assuming the typical spectral index is about α = − .
7, where S ( Jy ) ∝ ν α at 176.67MHz with radio flux ranging from 30 to 290 mJy (seee.g., George et al. 2017), which is of the order of mag-nitude of some of brightest galaxies in clusters at fre-quencies relevant to this work. We make use of a codedesigned to decompose radio interferometry images intosources called PyBDSF (the Python Blob Detector andSource Finder, formerly PyBDSM) (Mohan & Rafferty2015). The code reads in the input image, calculatesbackground rms and mean images, finds regions of emis-sion, fits Gaussians to the regions, and groups them intosources. We left the PyBDSF parameters at their de-fault values during this source extraction. We obtain apercentage error of 6% for the SZE-21cm simulated forobservation with the SKA1-low beam after point sourcesubtraction compared to the SZE-21cm computed atthe same frequency in Section 3. The presence of thepoint source and the subtraction thus has an impact onthe SZE-21cm signal that we measure. The point sourcesubtraction removes flux within the point source sub-traction beam area from the cluster map at the positionof the point source. Figure 9 illustrates the simulatedpoint source and subtraction described here.Another possible issue is due to the presence of diffuseemission in the radio band (Radio halos, mini-halo andrelics) that have been detected and studied in galaxyclusters (see review, Feretti et al. 2012). This may beone order of magnitude higher at frequencies relevantto observations of the SZE-21cm. It is also worth not-ing that contamination due to synchrotron emission in Fig. 9
Simulated SZE-21cm cluster maps at 176.67 MHz: with point sources in the original image, the model image, andthe residual which is original minus model image. Boundaries of the regions/island of emission found by code are shown inlight blue and the fitted Gaussians the shaded areas shown for each region as small ellipses. the cluster decreases for objects at large distances, sinceradio synchrotron emission changes with luminosity dis-tance, whereas the SZE does not change as a function ofredshift and distance of the source (see., Colafrancescoet al. 2016). By using this property, Multi-frequencyand spatially resolved analysis with upcoming interfer-ometers will allow us to reduce the importance of in-trinsic radio emission from the clusters. It is important,however, that we study and analyse both contributionsto distinguish and separate them.
Further simulations and observations are necessary topinpoint the best strategy for observing and measuringthe SZE-21cm to probe the DA and EoR, which willrequire very deep, thorough observations and highly ac-curate theoretical analysis. The detection of the SZE-21cm with the upcoming HERA and SKA instrumentshas the potential to yield insightful information on the history and physical mechanisms occurring during theDA and the subsequent EoR as they are sensitive. TheSKA will particularly play a large role in the study ofcosmological radio backgrounds by providing SZE datawith high accuracy. Cosmic variance and instrumen-tal noise have potential to contaminate and decreasechances for a clean detection of the SZE-21cm signal.Averaging the signal from a number of clusters has beendemonstrated as a method to suppress the noise anddeal with the challenges we may face observing towardsa individual cluster. To identify ideal clusters and ob-tain cluster plasma temperatures, X-ray measurementsof the ICM properties will be of great importance andwill form a crucial part of our observational strategy forstudies of the SZE-21cm, particularly clusters that arein the Southern hemisphere, where HERA and the SKAwill observe. In this regard, the next step would be toidentify clusters in Microwave source catalogs from theSPT, Planck and, ACT. Furthermore, we have shownthat point source removal is crucial to address the is-sue of contamination that may fill the SZE-21cm signal. We also noted a deviation between the true signal andsimulated observation with HERA and SKA at highfrequencies, which is particularly contrary to expecta-tions for observations with SKA-low. The source ofthis deviation needs to be investigated and adequatelyaddressed. This work has shown that HERA and Low-frequency SKA will be able to obtain maps of the SZE-21cm. The outcomes from our simulations and calcu-lations make us optimistic that the SKA telescope willdetect the SZE-21cm spectrum and will be instrumentalin following-up on the EDGES result.
Acknowledgements
The South African Radio As-tronomy Observatory (SARAO), which is a facility ofthe National Research Foundation (NRF), a Depart-ment of Science and Innovation (DSI) agency, has spon-sored this research. C.M., P.M. & G. B. acknowledgethe encouragement, advice and supervision of Prof.Sergio Colafrancesco in this work, may his soul restin peace. References
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