Thermal Sunyaev-Zel'dovich Effect in the IGM due to Primordial Magnetic Fields
Teppei Minoda, Kenji Hasegawa, Hiroyuki Tashiro, Kiyotomo Ichiki, Naoshi Sugiyama
CConference Report
Thermal Sunyaev–Zel’dovich Effect in the IGM dueto Primordial Magnetic Fields
Teppei Minoda * ,† ID , Kenji Hasegawa , Hiroyuki Tashiro , Kiyotomo Ichiki andNaoshi Sugiyama Department of Physics and Astrophysics, Nagoya University, Nagoya 464-8602, Japan;[email protected] (K.H.); [email protected] (H.T.);[email protected] (K.I.); [email protected] (N.S.) Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Chikusa-ku,Nagoya 464-8602, Japan Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), The University of Tokyo,Chiba 277-8582, Japan * Correspondence: [email protected]† These authors contributed equally to this work.Received: date; Accepted: date; Published: date
Abstract:
In the present universe, magnetic fields exist with various strengths and on various scales.One possible origin of these cosmic magnetic fields is the primordial magnetic fields (PMFs) generatedin the early universe. PMFs are considered to contribute to matter density evolution via Lorentz forceand the thermal history of intergalactic medium (IGM) gas due to ambipolar diffusion. Therefore,information about PMFs should be included in the temperature anisotropy of the Cosmic MicrowaveBackground through the thermal Sunyaev–Zel’dovich (tSZ) effect in IGM. In this article, given aninitial power spectrum of PMFs, we show the spatial fluctuation of mass density and temperature ofthe IGM and tSZ angular power spectrum created by the PMFs. Finally, we find that the tSZ angularpower spectrum induced by PMFs becomes significant on small scales, even with PMFs below theobservational upper limit. Therefore, we conclude that the measurement of tSZ anisotropy on smallscales will provide the most stringent constraint on PMFs.
Keywords: magnetic fields; large-scale structure; magnetic turbulence
1. Introduction
According to many astronomical observations, we can find the magnetic fields in the universe ona very wide range of spatial scales. The magnetic fields associate with different types of astrophysicalobjects, such as planets [1], ordinary stars [2], compact objects, star clusters, galaxies, clusters ofgalaxies [3], and so on. Additionally, some papers have suggested the existence of magnetic fieldsin the intergalactic region, also known as the cosmic voids, based on the blazars’ γ -ray observations[4–7]. The origin of these cosmic magnetic fields is an open question, especially on large scales. Manyscenarios are considered to explain the generation of such magnetic fields, and roughly speaking,such scenarios are divided into two groups, primordial origin and astrophysical origin. The firstone indicates the mechanisms taking place in the primordial universe, and this scenario predicts tinymagnetic fields on cosmological scales. These weak seed fields are called “primordial magnetic fields”(PMFs). The latter entails that the cosmic magnetic fields were generated during or after large scalestructure formation via mechanisms such as Biermann battery or Weibel instability.The purpose of this study is to investigate how the PMFs affect structure formation after therecombination epoch, especially during the so-called Dark Ages. In addition, we estimate the CosmicMicrowave Background (CMB) temperature anisotropy from thermal Sunyaev–Zel’dovich (tSZ) effect,in order to constrain the strength of PMFs with the cosmological observable. The calculation methoddescribed in this paper is based on our previous work [8]. a r X i v : . [ a s t r o - ph . C O ] D ec of 8
2. Materials and Methods
We treat the PMFs as the statistically-isotropic and homogeneous fields, so we can determine thecharacteristics of such fields from the power spectrum defined as: (cid:104) B ∗ i ( k ) B j ( k (cid:48) ) (cid:105) = ( π ) δ D ( k − k (cid:48) )( δ ij − ˆ k i ˆ k j ) P B ( k ) . (1)In this work, we assume P B ( k ) as a single power-law function of k with the spectral index n B as: P B ( k ) = n B + ( π ) B n k n B + n k n B , (2)where B n is the field strength at the normalized scale k n ≡ π Mpc − . By taking the top-hat windowfunction in Fourier space, we can find the relation between B n and B λ , which is the amplitude of thePMF smoothed on any other spatial scale λ as: B λ = (cid:90) k λ P B ( k ) d k ( π ) = B n (cid:32) k λ k n (cid:33) n B + , (3)with k λ = π / λ .According to some previous works [9,10], we assume the cut-off scale of the PMFs caused by thedamping of Alfvén waves. This cut-off wave number, k c , is given by: k − c = B λ c ( t rec ) πρ γ ( t rec ) (cid:90) t rec l γ ( t (cid:48) ) a ( t (cid:48) ) dt (cid:48) , (4)where t rec , ρ γ , l γ , and a are the recombination time, the energy density of CMB photon, the mean-freepath of CMB photon, and the cosmic scale factor, respectively. Furthermore, λ c = π / k c shows thecut-off wavelength of PMFs.For k ≤ k c , the power spectrum of PMFs is given by Equation (2), and PMFs are completelydamped for k > k c . Thus, we put the cut-off scale only in the ultraviolet regime. We discuss the effectof the large-scale fluctuation of PMFs in Section 4. Furthermore, the time evolution of PMFs is assumedas B ( t , x ) = B ( t now , x ) / a ( t ) , and we neglect the helicity of PMFs for simplicity.When we fix the values of B n and n B , the amplitude of the power spectrum of the PMFs can beobtained by Equation (2), and the cut-off scale is calculated through Equations (3) and (4). Therefore,in this model, the parameters B n and n B determine the statistical property of PMFs. We study the tSZeffect induced by PMFs in the case of B n = n B = −
1, which are not excluded from thePlanck constraint on PMFs [11]. In this work, we perform the numerical calculation of matter densityevolution and the thermal evolution of the intergalactic medium (IGM) gas including the PMFs. Wethus obtain the CMB angular power spectrum with the effect of PMFs via tSZ taken into account. Inthe next three subsections, we explain the basic equations we used. of 8
When the density contrast δ is smaller than unity, the time evolutional equations of overdensitiescan be written in terms of the background baryon density ρ b and the cold dark matter density ρ c , asthe following [12]: ∂ δ c ∂ t + H ( t ) ∂δ c ∂ t − π G ( ρ c δ c + ρ b δ b ) =
0, (5) ∂ δ b ∂ t + H ( t ) ∂δ b ∂ t − π G ( ρ c δ c + ρ b δ b ) = S ( t ) , (6)where H ( t ) is the Hubble parameter and subscripts “b” and “c” represent the values for the baryonand the cold dark matter, respectively. S ( t ) is the source term for the baryon density fluctuation δ b dueto Lorentz force, given by: S ( t , x ) = ∇ · ( ∇ × B ( t , x )) × B ( t , x ) πρ b ( t ) a ( t ) , (7)where B ( t , x ) denotes the PMFs in a comoving three-dimensional space x and time t and ∇ is also takenin comoving coordinates. Assuming a matter dominated era and δ b = δ b = S ( t ) H ( t ) (cid:34)(cid:40) (cid:32) aa rec (cid:33) + (cid:32) aa rec (cid:33) − −
15 ln (cid:32) aa rec (cid:33)(cid:41) Ω b Ω m +
15 ln (cid:32) aa rec (cid:33) + (cid:32) − Ω b Ω m (cid:33) (cid:32) aa rec (cid:33) − − (cid:32) − Ω b Ω m (cid:33)(cid:35) , (8)where Ω m is the density parameter for total matter and Ω b is the one for baryonic matter. Furthermore, a rec ≡ a ( t rec ) is the scale factor at the recombination time t rec . In this work, we set the recombinationepoch at z rec ≡ δ b (cid:28) According to [12], PMFs not only induce the density fluctuation of IGM, but also affect the thermalhistory of IGM via an energy dissipation mechanism, so-called ambipolar diffusion. The equation ofIGM gas temperature T gas evolution with heating due to the ambipolar diffusion is given by [13]: dT gas dt = − H ( t ) T gas + ˙ δ b + δ b T gas + x e + x e ρ γ σ T m e c ( T γ − T gas ) + Γ ( t ) k B n b − x e n b k B [ Θ x e + Ψ ( − x e ) + η x e + ζ ( − x e )] , (9)where x e , m e , σ T , T γ , k B , and n b are the ionization fraction of hydrogen atoms, the rest mass of anelectron, the Thomson cross-section, the CMB temperature, the Boltzmann constant, and the numberdensity of the baryon gas, respectively. The first term in the right-hand side of Equation (9) isadiabatic cooling due to the cosmic expansion, and the second is the effect of adiabatic compression (orexpansion) from local density fluctuation. The third one is the energy transfer between CMB photonsand thermal electrons, the so-called Compton heating (or cooling). The forth term with Γ ( t ) representsthe extra heating source for the IGM, which is injected by the PMF dissipation through ambipolar of 8 diffusion. We adopt the expression for Γ ( t ) , which means the heating energy per unit time per unitvolume, as [12]: Γ ( t , x ) = | ( ∇ × B ( t , x )) × B ( t , x ) | π ξρ ( t ) ( − x e ) x e , (10)where ξ is the drag coefficient for H and H + , and here, the value of ξ = × cm g − s − [14].The last term of Equation (9) with the coefficients Θ , Ψ , η , and ζ is the radiative cooling effects forhydrogen atoms. We use the definitions and values from [13].In order to solve Equation (9), we have to follow the evolution of the ionization fraction, as well, dx e dt = + K α Λ n b ( − x e ) + K α ( Λ + β e ) n b ( − x e ) × (cid:34) − α e n b x e + β e ( − x e ) exp (cid:32) − E ion k B T γ (cid:33)(cid:35) + γ e n b x e , (11)where K α , Λ , α e , β e , and γ e are the parameters for the ionization and recombination processes, andthey are given as the functions of T gas in [15–17]. Therefore, we calculate Equations (9) and (11)simultaneously, and we take into account the fluctuations of the IGM density δ b , given by Equation (8).For simplicity, we neglect the presence of helium, heavier elements and, any astronomical objects. As explained above, the tangled PMFs could generate spatial fluctuations of the IGM numberdensity n b , ionization fraction x e , and temperature T gas . They can induce the CMB temperatureanisotropy via the inverse Compton scattering, which is called the tSZ effect. In this subsection, wederive the angular power spectrum of the CMB temperature caused by the tSZ effect.The strength of tSZ effect is represented by the Compton y -parameter integrated along aline-of-sight ˆ n [18], y ( ˆ n ) ≡ k B σ T m e c (cid:90) dz cw ( ˆ n , z ) H ( + z ) , (12)In Equation (12), w ( ˆ n , z ) is the convolutional function of n b , x i , and T gas as, w ( ˆ n , z ) = (cid:2) x e n b ( T gas − T γ ) (cid:3) ˆ n , z . (13)The CMB temperature anisotropies from the tSZ effect are related to the Compton y -parameter as: ∆ TT ( ˆ n ) = g ν y ( ˆ n ) , (14)where g ν is a function of the frequency where the CMB is observed, and is given by g ν = − + x / tanh ( x /2 ) with x ≡ h Pl ν / k B T . Therefore, we obtain the CMB angular power spectrum as: D (cid:96) = (cid:96) ( (cid:96) + ) π (cid:32) g ν k B σ T m e c (cid:33) (cid:90) d χ P w ( χ , (cid:96) / χ ) χ , (15)where (cid:96) is the multipole and P w ( χ , k ) is the power spectrum of the Compton y -parameter for two points,with a separation corresponding to wavenumber k at comoving distance χ . We can obtain P w ( χ , k ) from w via Equation (13). We have adopted Limber’s approximation when we derive Equation (15)because we are interested in only large (cid:96) modes, where the tSZ signal dominates the primary CMBanisotropy. of 8
3. Results
We performed our simulations for B n = n B = −
1. The left panel of Figure 1 showsthe two-dimensional structure of the x-component of the Lorentz force ( ∇ × B ) × B , which appearsin Equations (7) and (10). The middle and right panels show the baryon gas temperature and thebaryon number density at z =
10, respectively. The size of each panel is 2 × . Youcan see that the gas temperature rises to ∼ T gas and n b havea negative correlation. The reason for this anti-correlation is seen in Equation (10): the heating ratefrom the ambipolar diffusion is proportional to ρ − . We also have found that the density fluctuationof the baryon gas exceeds unity soon after the recombination epoch. Therefore, in order to avoid thenegative mass density, we put the lowest bound on density fluctuation of baryon gas as δ b = − y [ M p c ] rot B x B x /100 nG Mpc x [Mpc] T gas /1000 K 051015202530 0 1 2 x [Mpc] n b [cm ] 012345 Figure 1.
Illustration of ( ∇ × B ) × B x (left), the gas temperature (middle), and the gas numberdensity (right) when z =
10. You can see the spatial anti-correlation between T gas and n b . In Figure 2, we plot the CMB temperature angular power spectrum D (cid:96) caused by the tSZ effectin the IGM due to PMFs. The tSZ angular power spectrum has a sharp peak around the multipole (cid:96) ∼ . This angular scale corresponds to the PMF cut-off scale, which is of the order 100 comovingkpc. The current observation can reveal the CMB angular power spectrum only for (cid:96) (cid:46) , but thetSZ angular power spectrum from PMFs dominates the primary one on smaller scales. This is becausethe primary component of CMB anisotropy has experienced Silk damping, but the PMFs can continueto create fluctuations of physical variables of baryon gas after the recombination epoch. Therefore,the information about the PMF cut-off scale is included in the small-scale CMB anisotropy, due to thetSZ effect in the IGM. The future ground-based observations, such as “CMB-S4”, would give us muchinformation on such a small-scale CMB anisotropy. of 8 multipole 10 [ K ] CDMPMFsACTPol
Figure 2.
CMB temperature power spectrum for the multipole 10 < (cid:96) < . We plot the resultantangular power spectrum caused by the tSZ effect from the PMFs with the thick green line. Theresult is obtained by integrating Equation (15) from z = z =
10, and the parameters forthe PMFs are fixed to B n = n B = −
1. The primary component of the CMB angularpower spectrum predicted from the Λ CDM cosmology and the observational data with the AtacamaCosmology Telescope (two-season ACTPol) [19] are also shown with the blue solid line and the orangedots with error bars, respectively.
4. Discussions and Conclusions
In this section, we discuss the validity of the setup and assumptions in this study. In this work,we have fixed the box size as ( ) and the parameters of the PMF model as B n = n B = −
1. Although we do not show it in this paper, we have checked the dependence of ourresults on these parameters. When we enlarged the simulation box size and calculated the resultant tSZsignal, we obtained the same amplitude and shape of tSZ signal as those in the fiducial box size. Next,we followed the IGM evolution with various spectral indices of the PMFs in the range − ≤ n B ≤ B n and n B .Next, we discuss the treatment of the matter density evolution. As seen in Section 2.2, we haveassumed two important conditions to derive Equations (5) and (6). First of all, we neglect the thermalpressure of the baryon gas, and this condition is satisfied on scales larger than the Jeans scale of thefluid. We have confirmed that the minimum grid length of our calculation is always greater thanthe Jeans length. Next, we found that the local density perturbation is greater than unity soon afterrecombination, even for the sub-nano Gauss PMF model. However, the contribution to the tSZ powerspectrum is largely created by the low density region, because of the anti-correlation between theelectron density and temperature. Therefore, this overestimation of the local matter density is nota severe problem for our results. In order to confirm this point, we will use the cosmological MHDsimulation to estimate such a non-linear effect on the tSZ signal instead of linear perturbation theoryin our future work. Such a simulation is also thought to reveal highly non-linear MHD dissipation of 8 mechanisms, in addition to the ambipolar diffusion, as recently discussed for the recombinationepoch [20].In this work, we have investigated the tSZ signal caused by the PMFs in the so-called Dark Ages.By assuming the power spectrum of the PMFs as P B ( k ) ∝ B n k n B , with B n = n B = −
1, weconsistently solve the evolutionary equations for the IGM gas density and temperature. At last, weestimate the tSZ angular power spectrum caused by the fluctuations of IGM gas with sub-nano GaussPMFs. We found that the PMFs create an anti-correlation between the gas temperature and density. Wehave shown that tSZ signal has a sharp peak at (cid:96) ∼ , which corresponds to the cut-off scale of PMFsdue to the radiative viscosity in the early universe. Therefore we can conclude that the observation ofthe small scale CMB temperature anisotropy could give us nature of the cosmic magnetic fields. Author Contributions:
Conceptualization, H.T., K.I.; Methodology, T.M., K.H., H.T., K.I.; Validation, K.H., K.I.;Formal Analysis, T.M., K.I.; Investigation, T.M., K.H.; Writing—Original Draft Preparation, T.M.; Writing—Review& Editing, H.T., K.I., and N.S.; Visualization, T.M.; Supervision, N.S.; Project Administration, T.M., H.T., and N.S.;Funding Acquisition, H.T., K.I., and N.S.
Funding:
This research is supported by KAKEN Grant-in-Aid for Scientific Research, (B) No. 25287057 (K.I. andN.S.), (A) No. 17H01110 (N.S.), and 16H01543 (K.I.) and for Young Scientists (B) No. 15K17646 (H.T.).
Acknowledgments:
We thank the local and scientific organizing committee for the international conference,“THE POWER OF FARADAY TOMOGRAPHY — TOWARDS 3D MAPPING OF COSMIC MAGNETIC FIELDS—”. We also appreciate the anonymous referees and editor, and special thanks to Patel Teerthal for careful readingour paper.
Conflicts of Interest:
The authors declare no conflict of interest. The founding sponsors had no role in the designof the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; nor in thedecision to publish the results.
Abbreviations
The following abbreviations are used in this manuscript:MDPI Multidisciplinary Digital Publishing InstituteCMB Cosmic Microwave BackgroundPMF primordial magnetic fieldIGM intergalactic mediumtSZ thermal Sunyaev–Zel’dovich effect
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