21 cm line signal from magnetic modes
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21 cm line signal from magnetic modes
Kerstin E. Kunze ∗ Departamento de F´ısica Fundamental, Universidad de Salamanca,Plaza de la Merced s/n, 37008 Salamanca, Spain (Dated: May 29, 2018)The Lorentz term raises the linear matter power on small scale which leads to interestingsignatures in the 21 cm signal. Numerical simulations of the resuting nonlinear densityfield, the distribution of ionized hydrogen and the 21 cm signal at different values of redshiftare presented for magnetic fields with field strength B=5 nG, and spectral indices n B = − . , − . I. INTRODUCTION
Magnetic fields come in different shapes and sizes in the universe. Observed magnetic fieldsrange from those associated with stars and planets upto to cluster and super cluster scales (cf.,e.g., [1–3]). Observations of the energy spectra of a number of blazars in the GeV range withFermi/LAT and in the TeV range with telescopes such as H.E.S.S., MAGIC or VERITAS havebeen interpreted as evidence for the existence of truly cosmological magnetic fields. These are notassociated with virialized structures but rather permeate the universe. Limits on the field strengthsof these void magnetic fields are of the order 10 − − − G (e.g, [4–6])which is considerably belowthose of galactic magnetic fields which are of the order of 10 − G (e.g., [7]).Cosmological magnetic fields present from before decoupling influence the cosmic plasma indifferent ways. Before recombination photons are strongly coupled to the baryon fluid via Thomsonscattering off the electrons. As the observed high degree of isotropy on large scales limits themagnitude of a homogeneous magnetic field the contribution of a putative, primordial magneticfield is modeled as a gaussian, random field. As such it actively contributes to the total energy ∗ [email protected] density perturbations as well as to the anisotropic stress perturbation. Furthermore the Lorentzterm changes the baryon velocity. This has important implications for the linear matter powerspectrum which will be the focus of this work. The linear matter power spectrum provides the initialdistribution of the density field from which nonlinear structure evolves. It determines implicitly thedistribution of neutral hydrogen in the post recombination universe during the cosmic dark agesand later on ionized hydrogen. Cosmic dawn starts with the formation of the first star forminggalaxies within dark matter halos. These are sources of high energetic UV photons and at laterepochs X-ray photons from quasars that ionize and heat matter. These high energetic photons canredshift down to the corresponding Lyman α wave length which can be absorbed by an hydrogenatom and emitted spontaneously allowing for the atom to change from, say, the hyperfine singlet tothe triplet state which is the Wouthuysen-Field mechanism coupling the spin and gas temperature(e.g. [8]). At some point the Ly α coupling saturates, by which time the gas has heated above thetemperature of the cosmic microwave background (CMB). The 21 cm line signal is the change inthe brightness temperature of the CMB as seen by an observer today. It is necessary for a non zerosignal that the spin temperature which determines the equilibrium of the ratio of the occupationnumbers in the ground state hyperfine states of neutral hydrogen and the CMB temperature aredifferent. At large redshifts the gas is still cold. Thus the 21 cm line signal is seen in absorption.Once the heating of the gas due to the high energetic photons in the UV and X-ray range becomesefficient the matter temperature is well above the CMB temperature and the 21 cm line signalis seen in emission. Moreover, in this case the change in brightness temperature will saturate.Magnetic fields can also influence the 21cm line signal by additional heating of matter because ofdissipative processes (cf. [9–11]). However, here the focus will be on the effects due to the changein the linear matter power spectrum.The magnetic field is assumed to be a non helical, gaussian random field determined by its twopoint function in k -space, h B ∗ i ( ~k ) B j ( ~q ) i = (2 π ) δ ( ~k − ~q ) P B ( k ) (cid:18) δ ij − k i k j k (cid:19) , (1.1)where the power spectrum, P B ( k ) is given by [12] P B ( k, k m , k L ) = A B (cid:18) kk L (cid:19) n B W ( k, k m ) (1.2)where k L is a pivot wave number chosen to be 1 Mpc − and W ( k, k m ) = π − / k − m e − ( k/k m ) is agaussian window function. k m corresponds to the largest scale damped due to radiative viscositybefore decoupling [13, 14]. k m has its largest value at recombination k m = 299 . (cid:18) B nG (cid:19) − Mpc − (1.3)for the bestfit parameters of Planck13+WP data [15, 16]. II. THE LINEAR MATTER POWER SPECTRUM
At the epochs of interest here close to reionization the universe is matter dominated. The initiallinear matter power spectrum is assumed to be given by the contributions from the primordialcurvature mode as well as the magnetic mode. For modes inside the horizon the linear matterpower spectrum of the adiabatic curvature mode is given by (cf., e.g.,[17–19]) P ( ad ) m ( k ) = 2 π k (cid:18) ka H (cid:19) A s (cid:18) kk p (cid:19) n s − T ( k ) , (2.1)where the transfer function T ( k ) is given by [20, 21] T ( k ) = ln(1 + 2 . q )2 . q (cid:2) . q + (16 . q ) + (5 . q ) + (6 . q ) (cid:3) − (2.2)where q = k Ω m, h Mpc − .For the magnetic mode the matter power spectrum is found to be [19] P ( B ) m ( k ) = 2 π k (cid:18) ka H (cid:19) z dec ) (cid:18) Ω γ, Ω m, (cid:19) P L ( k ) , (2.3)where P L ( k ) is the dimensionless power spectrum determining the two point function of the Lorentzterm h L ∗ ( k ) L ( k ′ ) i = π k P L ( k ) given by [12] P L ( k ) = 9 (cid:2) Γ (cid:0) n B +32 (cid:1)(cid:3) (cid:18) ρ B, ρ γ, (cid:19) (cid:18) kk m (cid:19) n B +3) e − (cid:16) kkm (cid:17) × Z ∞ dzz n B +2 e − (cid:16) kkm (cid:17) z Z − dxe (cid:16) kkm (cid:17) zx (1 − zx + z ) nB − × (cid:2) z + (1 − z ) x − zx + 4 z x (cid:3) , (2.4)and x ≡ k · q kq and z ≡ qk where q is the wave number over which the resulting convolution integralis calculated.The resulting linear matter power spectrum for the magnetic plus adiabatic mode is shownin figure 1. Below the magnetic Jeans scale pressure supports against collapse and prevents any −1 k [(h/Mpc)] −2 −1 P m ( k )[( M p c / h ) ] B = 5 nG, n B =-1.5B = 5 nG, n B =-2.2B = 5 nG, n B =-2.9B =0 FIG. 1. Total linear matter power spectrum at the present epoch for different values of the magnetic fieldparameters and the best fit Planck13+WP parameters [16]. further growth of the density perturbation. Therefore the linear matter power spectrum of themagnetic mode is cut-off at the wave number corresponding to the magnetic Jeans scale k J , [22] (cid:18) k J Mpc − (cid:19) = . (cid:18) Ω m . (cid:19) (cid:18) h . (cid:19) (cid:18) B − G (cid:19) − (cid:18) k L Mpc − (cid:19) nB +32 nB +5 . (2.5)The linear matter power spectrum is normalized to σ of the best fit Planck13+WP parameters[16]. III. THE 21 CM LINE SIGNAL
For the simulations the
Simfast21 code [23, 24] is adapted to allow for reading in the modi-fied linear matter power spectra. Simfast21 calculates the change in the brightness temperaturefollowing a similar algorithm as the code [25]. The initial Gaussian, random densityfield is determined by the linear matter power spectrum. The subsequent evolution in time leadsto gravitational collapse and nonlinear structure and formation of dark matter halos. The halodistribution is found by using the excursion formalism whereby a given region is considered toundergo gravitational collapse if its mean overdensity is larger than a certain critical value δ c ( M, z )depending on the halo mass M and redshift z . As the halo positions are based on the linear densityfield these have to be corrected for the effects of the non linear dynamics. This is done using the https://github.com/mariogrs/Simfast21 http://homepage.sns.it/mesinger/DexM 21cmFAST.html Zel’dovich approximation. A source for reionization of matter in the universe are galaxies whichform inside dark matter haloes. Thus the corrected halo distribution allows to determine the ion-ization regions. In the version of
Simfast21 [24] used in this work the criterion to decide whethera given region is ionized is determined by the local ionization rate R ion and the recombination rate R rec . These are implemented using a numerical fitting formula which was obtained from numericalsimulations. In addition there is a free paramter which is the assumed escape fraction of ionizingphotons from star forming regions f esc . A bubble cell is defined to be completely ionized if thecondition f esc R ion ≥ R rec (3.1)is satisfied. In this work the value of the escape rate is set to f esc = 0 .
06. Once the evolution ofthe ionization field has been determined the 21cm line signal can be calculated. In equilibrium theratio of the populations of the two hyperfine states, the less energetic singlet state and the moreenergetic triplet state, of neutral hydrogen is determined by the spin temperature T S , e.g. [26, 27], n n = (cid:18) g g (cid:19) exp (cid:18) − T ∗ T S (cid:19) , (3.2)where T ∗ = E /k B = 68 mK and the energy difference E corresponds to a wave length λ ∼ T b ( z ) = T CMB ( z ) e − τ ( z ) + (cid:16) − e − τ ( z ) (cid:17) T S ( z ) (3.3)where T CMB ( z ) is the brightness temperature of the CMB without absorption and τ ( z ) is thecorresponding optical depth along the ray through the medium. Thus the change in the brightnesstemperature of the CMB as measured by an observer today is given by, e.g. [26, 27], δT b = T b − T CMB = (cid:0) − e − τ ( z ) (cid:1) [ T S ( z ) − T CMB ( z )]1 + z . (3.4)With the approximations τ ≪ z ≫
1, the 21cm line signal is given by δT b = 28mK (cid:18) Ω b, h . (cid:19) (cid:18) Ω m, . (cid:19) − (cid:18) z (cid:19) ( T S − T CMB ) T S x HI . (3.5)As can be seen from equation (3.5) there is only a signal if the spin temperature is different fromthe CMB radiation temperature. Otherwise the hydrogen spin state is in thermal equilibrium withthe CMB and emission and absorption processes are compensated on average. The net emission orabsorption result from a higher or lower, respectively, spin temperature than the CMB radiationtemperature. There are several processes which can lead to the spin temperature being differentfrom the CMB temperature such as the presence of radiation sources or heating of the gas. Inaddition there are two processes which can change the spin temperature of the neutral hydrogengas. Firstly, collisional excitation and de-excitation of the spin states. Secondly the Wouthuysen-Field process which couples the two spin states. In the limit that the spin temperature T S is muchhigher than the temperature of the CMB photons T CMB the change in the brightness temperature δT b (cf. equation (3.5)) becomes saturated. This is the case for lower redshifts when UV photonsfrom star forming galaxies heat the IGM. For simplicity we will assume here that T S ≫ T CMB .The focus here is to study the effect of the presence of a primordial magnetic field on the 21 cmline signal induced by the change in the linear matter power spectrum.
IV. RESULTS
In the numerical simulations the total linear matter power spectrum as calculated in section IIis used. In figure 2 the density field at z = 0 is shown when a linear evolution is assumed. Thesimulation boxes with each side corresponding to 100 Mpc are shown for no magnetic field, and inthe presence of a magnetic field of field strength 5 nG and spectral indices n B = − . n B = − . n B = − . tocmfastpy has been adapted. The Simfast21 code uses the Zeldovich approximation to obtain the nonlinear density field from which the halodistribution is obtained. The nonlinear density fields are shown for redshifts z = 32 to z = 10 infigure 3. The effect of the feature in the initial linear matter power spectrum manifests itself by anincrease in structure and amplitude in the matter density field. In figure 4 the distribution of theionized hydrogen regions are shown at different redshifts with and without the primordial magneticfield. At the largest redshift shown, z = 32, reionization has not started yet and there is no trace J. Prichard, https://github.com/pritchardjr/tocmfastpy −40 −30 −20 −10 0 10 20 30 40 δ m P D F z=0.00 B=0B=5,n B = −1.5B=5,n B = −2.2B=5,n B = −2.9 FIG. 2. The density field at z = 0 when linear evolution upto the present is assumed. The upper panel showsfrom left to right the simulations for B = 5 nG and the spectral magnetic indices n B = − . n B = − . n B = − .
9. The lower panel shows the simulation of a pure adiabatic mode ( left ) and the PDF for allfour cases ( right ). of ionized hydrogen ( upper panel ). The epoch of reionization (EoR) starts before redshift z = 20.Ionized gas forms bubbles of increasing size. This corresponds to the classic inside-out topologywhere the densest regions are ionized first which is the underlying assumption of the Simfast21 code. This can be nicely seen when comparing, for example, the nonlinear matter density fieldsand the distributions of the ionized regions at a redshift z = 20 ( second panel from above in figures3 and 4). Moreover, the effect of the matter density field modified by the presence of the magneticfield varies visibly with the different choices of the the magnetic field spectral index n B . In figure5 the average ionization fraction of each simulation box is shown as a function of redshift z . Thebeginning of EoR varies with the parameters of the magnetic field, with the magnetic field withthe smallest spectral index, n B = − .
9, having the largest effect. This is a simplified vision of theevolution of the ionization fraction as shown in the simulations of figure 4. As can be appreciatedfrom figure 5 reionization is completed at a redshift below z = 10. From figure 5 it is also interestingto note that models including magnetic fields with the smallest spectral index, n B = − .
9, show
FIG. 3. The density field at z = 32 , , ,
10 with the highest redshift shown in the first row and the lowestredshift in the last row. In each panel from left to right are shown the simulations for B = 0 ( far left )and subsequently for the cases B = 5 nG and the spectral magnetic indices n B = − . n B = − . n B = − . the longest duration of EoR. It starts at larger values of redshift than in the other cases but it stillreaches completion only below redshifts z less than 10.The evolution of the ionization fraction of hydrogen is a key ingredient to determine the 21cmline signal. In figure 6 the simulation boxes of the 21cm line signal are shown for redshifts z = 32 FIG. 4. HII regions shown at z = 32 , , ,
10 with the highest redshift shown in the first row and thelowest redshift in the last row. In each panel from left to right are shown the simulations for B = 0 ( farleft ) and subsequently for the cases B = 5 nG and the spectral magnetic indices n B = − . n B = − . n B = − . to z = 10 for the standard ΛCDM model and in the presence of the stochastic magnetic field.At z = 32 hydrogen is neutral. At this epoch the 21 cm line signal is saturated and observed inemission, δT b >
0. This is an effect of not taking into account the details of Ly α coupling butrather assuming that the spin temperature is much larger than the temperature of the CMB. The0 z ̄ x H II B=0B=5̄n B = −1.5B=5̄n B = −2.2B=5̄n B = −2.9 FIG. 5. The average ionization fraction as a function of redshift obtained from the simulation boxes. spatial distribution of the δT b traces the underlying matter density field. This can be appreciatedwhen comparing the upper panels of figures 3 and 6 which correspond to redshift z = 32.In figure 7 the average 21 cm line signal is shown as a function of redshift together with theprojected sensitivity of SKA1-LOW of the Square Kilometre Array (SKA) [28]. The baseline designof SKA1 will cover a frequency range of 50-350 MHz. In calculating the sensitivity we assumed onebeam of bandwidth 300 MHz and an integration time of 1000h.In figure 8 the power spectra of the change in the CMB brightness temperature P ( k ) for allmagnetic fields models at different redshifts are shown for our simulations. It is interesting to notethat the feature introduced by the magnetic mode into the total linear matter power spectrum hasleft a mark on the power spectrum of the 21 cm line signal. Comparing curves for the magneticspectral indices n B = − . n B = − . z = 20 shows that whereas the former reaches a localmaximum the latter rises steadily for large values of k . The difference in amplitude of the powerspectra reflects the earlier beginning of EoR for smaller spectral indices, resulting in a suppressionof the 21 cm line signal. This is also observed in the average signal δT b in figure 7. In figure 9the evolution with redshift of the power spectra of the change in the CMB brightness temperature P ( k ) is shown for the two extreme cases, no magnetic mode, B = 0, and a magnetic mode with B = 5 nG and n B = − .
9. Whereas the change in amplitude is the dominant feature, the changein spectral index of P ( k ) is subleading in the evolution with z .1 FIG. 6. The 21 cm line signal at z = 32 , , ,
10 with the highest redshift shown in the first row and thelowest redshift in the last row. In each panel from left to right are shown the simulations for B = 0 ( farleft ) and subsequently for the cases B = 5 nG and the spectral magnetic indices n B = − . n B = − . n B = − . V. CONCLUSIONS
Primordial magnetic fields present since before decoupling add additional power on small scalesto the linear matter power spectrum. In using the modified linear matter power spectrum as initial2 z δ T b [ K ] B=0B=5,n B = −1.5B=5,n B = −2.2B=5,n B = −2.9SKA1-LOW236 129 88 67 54 45ν [MHz] FIG. 7. The average 21 cm line signal as a function of redshift including sensitivity for SKA1-LOW. −2 −1 k[Mpc −1 ] −3 −2 P ( k )[ m K ] z=20.00 B=0B=5,n B = −1.5B=5,n B = −2.2B=5,n B = −2.9 10 −2 −1 k[Mpc −1 ] −3 −2 P ( k )[ m K ] z=16.00 B=0B=5,n B = −1.5B=5,n B = −2.2B=5,n B = −2.9 FIG. 8. The estimated power spectrum of the 21 cm line signal P ( k ) at redshifts z = 20 ( left ) and z = 16( right ). condition for the simulation of the nonlinear density field clearly shows this effect. Moreover itsubsequently changes the distribution of ionized hydrogen as well as the distribution of the 21 cmline signal. Simulations have been reported for magnetic fields of B = 5 nG and different magneticfield indices, n B = − . n B = − . n B = − . n B = − . Simfast21 code.When comparing the average 21 cm line signal with the projected sensitivity of the plannedSKA1-LOW indicates that observations at frequencies above 120 MHz will be able to constrainparameters of a primordial magnetic fields.3 −2 −1 k[Mpc −1 ] −3 −2 −1 P ( k )[ m K ] B=0 z=32z=20z=16z=10 10 −2 −1 k[Mpc −1 ] −3 −2 −1 P ( k )[ m K ] B=5 nG, n B =-2.9 z=32z=20z=16z=10 FIG. 9. The estimated power spectrum of the 21 cm line signal P ( k ) comparing all magnetic field modelsat different redshifts, B = 0 ( left ) and B = 5 nG, n B = − . right ). VI. ACKNOWLEGEMENTS
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