Secondary CMB temperature anisotropies from magnetic reheating
Shohei Saga, Atsuhisa Ota, Hiroyuki Tashiro, Shuichiro Yokoyama
MMNRAS , 1–10 (2019) Preprint 28 November 2019 Compiled using MNRAS L A TEX style file v3.0
Secondary CMB temperature anisotropies from magneticreheating
Shohei Saga, (cid:63) Atsuhisa Ota, , Hiroyuki Tashiro and Shuichiro Yokoyama , Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,Utrecht University, Princetonplein 5, NL-3584 CC Utrecht, The Netherlands Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK Department of Physics and Astrophysics, Nagoya University, Nagoya, 464-8602, Japan Kobayashi Maskawa Institute, Nagoya University, Aichi 464-8602, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institute for Advanced Study,University of Tokyo, Kashiwa, Chiba 277-8568, Japan
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Spatially fluctuating primordial magnetic fields (PMFs) inhomogeneously reheat theUniverse when they dissipate deep inside the horizon before recombination. Such anenergy injection turns into an additional photon temperature perturbation. We in-vestigate secondary cosmic microwave background (CMB) temperature anisotropiesoriginated from this mechanism, which we call inhomogeneous magnetic reheating .We find that it can bring us information about non-linear coupling between PMFsand primordial curvature perturbations parametrized by b NL , which should be im-portant for probing the generation mechanism of PMFs. In fact, by using cur-rent CMB observations, we obtain an upper bound on the non-linear parameter as log ( b NL ( B λ / nG ) ) (cid:46) − . n B − . with B λ and n B being a magnetic field amplitudesmoothed over λ = scale and a spectral index of the PMF power spectrum, re-spectively. Our constraints are far stronger than a previous forecast based on the futureCMB spectral distortion anisotropy measurements because inhomogeneous magneticreheating covers a much wider range of scales, i.e., − (cid:46) k (cid:46) Mpc − . Key words: cosmic background radiation — cosmology: theory — early Universe
Intergalactic magnetic fields have recently become one ofthe most interesting and important topics in cosmologyas several groups have reported lower bounds on them, B (cid:38) − – − G, by observing γ -ray emissions fromblazars (see e.g. Tavecchio et al. 2011; Vovk et al. 2012;Neronov & Vovk 2010; Essey et al. 2011; Chen et al. 2015). Itis a puzzle that there exist such magnetic fields in intergalac-tic space, where there are few astrophysical objects. Onemay answer the reason why magnetic fields exist there byintroducing primordial magnetic fields (PMFs), which mightbe generated in the early Universe. Indeed, many authorshave studied the mechanisms of generating PMFs from theviewpoint of high-energy physics in the early Universe (forreviews see e.g. Widrow 2002; Grasso & Rubinstein 2001;Durrer & Neronov 2013; Subramanian 2016), and the prop-erty of resultant PMFs highly depends on models of magne- (cid:63) E-mail: [email protected] togenesis. Hence, it would be interesting that we may testhigh-energy physics through high-precision measurements ofcosmological magnetic fields.Among such observations, recent precise measurementsof cosmic microwave background (CMB) anisotropies haveprovided rich information about PMFs (see e.g. Lewis 2004;Shaw & Lewis 2010). PMFs imprint their signatures on theCMB anisotropies in various ways. On superhorizon scales,their anisotropic stress affects the geometry of the space-time and leads to the additional curvature perturbationand gravitational waves, which are called passive modes.On sub-horizon scales, magnetohydrodynamical (MHD) ef-fects would be significant. Through these effects, the energydensity and anisotropic stress of PMFs can set isocurvaturescalar, vector, and tensor initial conditions, so-called, com-pensated modes (Lewis 2004; Shaw & Lewis 2012, 2010),and resultantly the MHD modes of tangled magnetic fieldsinduce the CMB temperature and polarization anisotropiesas discussed in Subramanian & Barrow (1998a). Moreover,PMFs also modify the thermal history of the intergalac- © a r X i v : . [ a s t r o - ph . C O ] N ov S.Saga et al. tic medium gas, which will also be imprinted in the CMBtemperature anisotropies through the thermal Sunyaev–Zel’dovich effect (Minoda et al. 2017).In this paper, we point out another effect on the CMBtemperature anisotropies induced by PMFs. In Saga et al.(2018), the authors studied a late-time reheating mechanismthrough the diffusion of PMFs and estimated upper boundson the PMF power spectrum on small scales. Indeed, thisidea was inspired by the works on acoustic reheating (Jeonget al. 2014; Nakama et al. 2014). Roughly speaking, acous-tic reheating is heating due to acoustic damping. Soundwaves of a primordial baryon-photon plasma are dissipateddue to shear viscosity on small scales. This process leavesnothing in linear perturbation theory; however, the aver-age component of the Universe would be slightly reheatedat second order in the fluctuations. Thus, this effect is ofsecond order in temperature perturbations, and hence, itis sensitive to the amplitude of primordial fluctuations onthe corresponding damping scales. Jeong et al. (2014) andNakama et al. (2014) derived a constraint on the amplitudeof short-wavelength density perturbations by comparing theradiation temperature at BBN and that at recombination.Then, Saga et al. (2018) applied this method for energy in-jection due to the dissipation of PMFs. Naruko et al. (2015)extended the framework of acoustic reheating to spatiallyfluctuated one. If primordial fluctuations are non-Gaussian,the cross-correlation between the primary CMB tempera-ture anisotropies and the secondary one induced by acousticreheating would be non-zero. Naruko et al. (2015) showedthat the observed CMB temperature anisotropies implic-itly put upper bounds on extremely squeezed shapes of pri-mordial non-Gaussianity because the secondary anisotropyshould be subdominant. On the analogy of acoustic reheat-ing, we can extend global magnetic reheating proposed inSaga et al. (2018) to spatially fluctuated one, which we call inhomogeneous magnetic reheating . The dissipation of spa-tially fluctuated PMFs leads to inhomogeneous energy in-jections into the CMB photons, which would be seen as ad-ditional anisotropies in the CMB. Moreover, if PMFs arenon-Gaussian, it might be a novel probe to explore suchprimordial non-Gaussianity in PMFs similar to anisotropicacoustic reheating.The non-Gaussianity in PMFs is recently well studiedin the context of inflationary magnetogenesis by several au-thors (Caldwell et al. 2011; Motta & Caldwell 2012; Barnabyet al. 2012; Jain & Sloth 2012, 2013; Nurmi & Sloth 2014;Ferreira et al. 2014; Chowdhury et al. 2018, 2019). In par-ticular, if there is a coupling between an inflaton φ and agauge field strength F µν during inflation, generated PMFswould become non-Gaussian through the non-linear inter-action. For example, f ( φ ) F µν F µν coupling, where f ( φ ) isa model-dependent function, is discussed in Caldwell et al.(2011); Motta & Caldwell (2012); Barnaby et al. (2012),and such a primordial non-Gaussianity would be a key ob-servable to distinguish models of magnetogenesis. Shiraishiet al. (2012) gave a constraint on this type of primordialnon-Gaussianity through the two-point function of the CMBanisotropy originated from the passive mode. Ganc & Sloth(2014) has also investigated a potential role of future CMBspectral distortion measurements for constraining the non-Gaussianity of PMFs. In this paper, we will show that in- homogeneous magnetic reheating can put a much strongerlimit on it.This paper is organized as follows. In the next section,we introduce the dissipation mechanism of PMFs based onthe MHD analysis and formulate inhomogeneous magneticreheating as a straightforward extension of global magneticreheating. Then we discuss possible corrections to the CMBtemperature anisotropies induced by PMFs in section 3. Insection 4, we compute the angular power spectrum and dis-cuss the constraints on PMFs in section 5. Finally, we willdevote section 6 to summarize this paper. First of all, we review the dissipation mechanism of cosmo-logical magnetic fields, following papers by Jedamzik et al.(1998); Subramanian & Barrow (1998b). In the early uni-verse, it was filled with highly conductive plasma. There-fore, we may apply ideal MHD to analyze the evolution ofthe fluid dynamics and magnetic fields. It is also known thatone can generalize MHD equations in flat space-time to thosein the expanding Universe (Brandenburg et al. 1996; Sub-ramanian & Barrow 1998b). In the cosmological setup, wedecompose PMFs into two scales: long-wavelength homoge-nous magnetic fields B and short-wavelength tangled ones B . We usually assume that we can take | B | (cid:28) | B | and lin-earize the Euler equation and ideal MHD equations to inves-tigate subhorizon dynamics of the comoving magnetic fields b ( τ, x ) = a ( τ ) B ( τ, x ) (Jedamzik et al. 1998). It should benoticed that this assumption may not be easily justified fora given configuration of PMFs, but we avoid full non-linearanalysis, which is beyond the scope of this paper.In the linearized MHD, there are three types of modes:the fast and slow magnetosonic modes, and the Alfv´en mode.The fast and slow modes are literally sound waves in themagnetized plasma, and the Alfv´en modes are incompress-ible waves. Then, which MHD mode is excited by stochas-tic PMFs? Subramanian & Barrow (1998b) and Jedamziket al. (1998) showed that excitation of MHD modes dependson the angle between B and propagation direction k . Forexample, if B (cid:107) k , the magnetosonic modes become stan-dard sound waves without magnetic pressure, and the tan-gled magnetic fields excite only the Alfv´en modes, whichsatisfy b ⊥ B . On the other hand, if B ⊥ k , the Alfv´enmodes are not induced, and the MHD modes become the fastmagnetosonic modes with b (cid:107) B . In this case, the Fouriercomponent of the tangled magnetic field associated withthe fast mode satisfies the following equation (Subramanian& Barrow 1998b): ∂ b ( τ, k ) ∂τ + η a ¯ ρ r k ∂ b ( τ, k ) ∂τ + k ( c + V A ) b ( τ, k ) = , (1) The Fourier component of the comoving amplitude of PMFs isdefined as b ( τ, k ) ≡ ∫ d x b ( τ, x ) e − i k · x . MNRAS000
Intergalactic magnetic fields have recently become one ofthe most interesting and important topics in cosmologyas several groups have reported lower bounds on them, B (cid:38) − – − G, by observing γ -ray emissions fromblazars (see e.g. Tavecchio et al. 2011; Vovk et al. 2012;Neronov & Vovk 2010; Essey et al. 2011; Chen et al. 2015). Itis a puzzle that there exist such magnetic fields in intergalac-tic space, where there are few astrophysical objects. Onemay answer the reason why magnetic fields exist there byintroducing primordial magnetic fields (PMFs), which mightbe generated in the early Universe. Indeed, many authorshave studied the mechanisms of generating PMFs from theviewpoint of high-energy physics in the early Universe (forreviews see e.g. Widrow 2002; Grasso & Rubinstein 2001;Durrer & Neronov 2013; Subramanian 2016), and the prop-erty of resultant PMFs highly depends on models of magne- (cid:63) E-mail: [email protected] togenesis. Hence, it would be interesting that we may testhigh-energy physics through high-precision measurements ofcosmological magnetic fields.Among such observations, recent precise measurementsof cosmic microwave background (CMB) anisotropies haveprovided rich information about PMFs (see e.g. Lewis 2004;Shaw & Lewis 2010). PMFs imprint their signatures on theCMB anisotropies in various ways. On superhorizon scales,their anisotropic stress affects the geometry of the space-time and leads to the additional curvature perturbationand gravitational waves, which are called passive modes.On sub-horizon scales, magnetohydrodynamical (MHD) ef-fects would be significant. Through these effects, the energydensity and anisotropic stress of PMFs can set isocurvaturescalar, vector, and tensor initial conditions, so-called, com-pensated modes (Lewis 2004; Shaw & Lewis 2012, 2010),and resultantly the MHD modes of tangled magnetic fieldsinduce the CMB temperature and polarization anisotropiesas discussed in Subramanian & Barrow (1998a). Moreover,PMFs also modify the thermal history of the intergalac- © a r X i v : . [ a s t r o - ph . C O ] N ov S.Saga et al. tic medium gas, which will also be imprinted in the CMBtemperature anisotropies through the thermal Sunyaev–Zel’dovich effect (Minoda et al. 2017).In this paper, we point out another effect on the CMBtemperature anisotropies induced by PMFs. In Saga et al.(2018), the authors studied a late-time reheating mechanismthrough the diffusion of PMFs and estimated upper boundson the PMF power spectrum on small scales. Indeed, thisidea was inspired by the works on acoustic reheating (Jeonget al. 2014; Nakama et al. 2014). Roughly speaking, acous-tic reheating is heating due to acoustic damping. Soundwaves of a primordial baryon-photon plasma are dissipateddue to shear viscosity on small scales. This process leavesnothing in linear perturbation theory; however, the aver-age component of the Universe would be slightly reheatedat second order in the fluctuations. Thus, this effect is ofsecond order in temperature perturbations, and hence, itis sensitive to the amplitude of primordial fluctuations onthe corresponding damping scales. Jeong et al. (2014) andNakama et al. (2014) derived a constraint on the amplitudeof short-wavelength density perturbations by comparing theradiation temperature at BBN and that at recombination.Then, Saga et al. (2018) applied this method for energy in-jection due to the dissipation of PMFs. Naruko et al. (2015)extended the framework of acoustic reheating to spatiallyfluctuated one. If primordial fluctuations are non-Gaussian,the cross-correlation between the primary CMB tempera-ture anisotropies and the secondary one induced by acousticreheating would be non-zero. Naruko et al. (2015) showedthat the observed CMB temperature anisotropies implic-itly put upper bounds on extremely squeezed shapes of pri-mordial non-Gaussianity because the secondary anisotropyshould be subdominant. On the analogy of acoustic reheat-ing, we can extend global magnetic reheating proposed inSaga et al. (2018) to spatially fluctuated one, which we call inhomogeneous magnetic reheating . The dissipation of spa-tially fluctuated PMFs leads to inhomogeneous energy in-jections into the CMB photons, which would be seen as ad-ditional anisotropies in the CMB. Moreover, if PMFs arenon-Gaussian, it might be a novel probe to explore suchprimordial non-Gaussianity in PMFs similar to anisotropicacoustic reheating.The non-Gaussianity in PMFs is recently well studiedin the context of inflationary magnetogenesis by several au-thors (Caldwell et al. 2011; Motta & Caldwell 2012; Barnabyet al. 2012; Jain & Sloth 2012, 2013; Nurmi & Sloth 2014;Ferreira et al. 2014; Chowdhury et al. 2018, 2019). In par-ticular, if there is a coupling between an inflaton φ and agauge field strength F µν during inflation, generated PMFswould become non-Gaussian through the non-linear inter-action. For example, f ( φ ) F µν F µν coupling, where f ( φ ) isa model-dependent function, is discussed in Caldwell et al.(2011); Motta & Caldwell (2012); Barnaby et al. (2012),and such a primordial non-Gaussianity would be a key ob-servable to distinguish models of magnetogenesis. Shiraishiet al. (2012) gave a constraint on this type of primordialnon-Gaussianity through the two-point function of the CMBanisotropy originated from the passive mode. Ganc & Sloth(2014) has also investigated a potential role of future CMBspectral distortion measurements for constraining the non-Gaussianity of PMFs. In this paper, we will show that in- homogeneous magnetic reheating can put a much strongerlimit on it.This paper is organized as follows. In the next section,we introduce the dissipation mechanism of PMFs based onthe MHD analysis and formulate inhomogeneous magneticreheating as a straightforward extension of global magneticreheating. Then we discuss possible corrections to the CMBtemperature anisotropies induced by PMFs in section 3. Insection 4, we compute the angular power spectrum and dis-cuss the constraints on PMFs in section 5. Finally, we willdevote section 6 to summarize this paper. First of all, we review the dissipation mechanism of cosmo-logical magnetic fields, following papers by Jedamzik et al.(1998); Subramanian & Barrow (1998b). In the early uni-verse, it was filled with highly conductive plasma. There-fore, we may apply ideal MHD to analyze the evolution ofthe fluid dynamics and magnetic fields. It is also known thatone can generalize MHD equations in flat space-time to thosein the expanding Universe (Brandenburg et al. 1996; Sub-ramanian & Barrow 1998b). In the cosmological setup, wedecompose PMFs into two scales: long-wavelength homoge-nous magnetic fields B and short-wavelength tangled ones B . We usually assume that we can take | B | (cid:28) | B | and lin-earize the Euler equation and ideal MHD equations to inves-tigate subhorizon dynamics of the comoving magnetic fields b ( τ, x ) = a ( τ ) B ( τ, x ) (Jedamzik et al. 1998). It should benoticed that this assumption may not be easily justified fora given configuration of PMFs, but we avoid full non-linearanalysis, which is beyond the scope of this paper.In the linearized MHD, there are three types of modes:the fast and slow magnetosonic modes, and the Alfv´en mode.The fast and slow modes are literally sound waves in themagnetized plasma, and the Alfv´en modes are incompress-ible waves. Then, which MHD mode is excited by stochas-tic PMFs? Subramanian & Barrow (1998b) and Jedamziket al. (1998) showed that excitation of MHD modes dependson the angle between B and propagation direction k . Forexample, if B (cid:107) k , the magnetosonic modes become stan-dard sound waves without magnetic pressure, and the tan-gled magnetic fields excite only the Alfv´en modes, whichsatisfy b ⊥ B . On the other hand, if B ⊥ k , the Alfv´enmodes are not induced, and the MHD modes become the fastmagnetosonic modes with b (cid:107) B . In this case, the Fouriercomponent of the tangled magnetic field associated withthe fast mode satisfies the following equation (Subramanian& Barrow 1998b): ∂ b ( τ, k ) ∂τ + η a ¯ ρ r k ∂ b ( τ, k ) ∂τ + k ( c + V A ) b ( τ, k ) = , (1) The Fourier component of the comoving amplitude of PMFs isdefined as b ( τ, k ) ≡ ∫ d x b ( τ, x ) e − i k · x . MNRAS000 , 1–10 (2019) nhomogeneous magnetic reheating where τ , ¯ ρ r , c s , and V A are the conformal time, the back-ground energy density of radiation, the sound speed, andthe Alfv´en velocity, respectively. Here η = ¯ ρ γ l γ is shearviscosity emerges in photon-baryon fluids with l γ = ( σ T n e ) − being the photon mean-free path. Then, WKB solutions ofEq. (1) become b ( z , k ) = ˜ b ( k ) e −( k / k D ( z )) e ± i k (cid:113) c + V A τ , (2)where z is a cosmological redshift, ˜ b ( k ) denote the initialmagnetic fields, and k D expresses the damping scale of mag-netic fields given by k − = ∫ l γ ( t ) a ( t ) d t , (3)which is the same with the usual acoustic damping scale.The time dependence of k D can be read from Eq. (3) as k − ∝ ( + z ) − . Since the mean free path of photons becomeslarger as the Universe evolves, the dissipation scale also in-creases. Note that we only considered photons for shear vis-cosity, and this is valid for z (cid:46) . In order to calculate thedamping scale in z (cid:38) , we need to consider neutrino freestreaming. Moreover, anisotropic shear of relativistic weakbosons would be significant for z (cid:38) . Here we accountfor these effects, following Jeong et al. (2014).The above two cases, i.e., B (cid:107) k and B ⊥ k are specificchoices of propagation directions. These MHD modes aremixed in the case of the intermediate configurations, andit is not straightforward to analyse the dispersion relations.Thus, the excitation of MHD modes depends on realizationsof b on top of B and their propagation directions. Sincethe initial conditions of MHD modes would be randomlygiven for stochastic PMFs, we may, therefore, assume theequipartition of PMFs for the initial conditions of each MHDmode.As we have mentioned, the diffusion scale of the fastmode is determined by the damping scale of sound waves.This implies that the fast modes always travel with soundspeed if Alfv´en velocity is negligible, and the dissipative fea-ture does not depend on the size of background magneticfields. In contrast, the dissipations of the slow and Alfv´enmodes which propagate with Alfv´en velocity, are more non-trivial. If the B is tiny, the slow and Alfv´en modes propagateso slowly that they are overdamped. In this case, a drivingforce of oscillations is balanced with viscosity, and then dis-sipation of these modes becomes inefficient (Subramanian& Barrow 1998b; Jedamzik et al. 1998). Thus, the proper-ties of slow modes and Alfv´en modes highly depend on thesignificance of background magnetic fields. Hence we ignorethe slow and Alfv´en modes, and we assume that the back-ground magnetic fields are weak enough to have V A (cid:28) c s forsimplicity. Dropping these two modes forces us to underes-timate the effects of magnetic reheating. However, it wouldonly change an O ( ) factor to our results. As we have discussed in the previous subsection, a randomlygiven fast magnetosonic mode is damping due to shear vis-cosity of relativistic particles on small scales, regardless ofthe size of background magnetic fields. Then the energy ofsuch a magnetic field in a comoving volume is decreasing. Suppose the Universe is dominated by radiation, the releasedenergy would be redistributed to photons by the Comptonscattering. Below, we compute temperature variations dueto inhomogeneous magnetic reheating, solving the conserva-tion laws in the total system of radiations and PMFs.First of all, we write the comoving energy density ofPMFs ρ B ( z , x ) = π | b ( z , x )| . (4)The damping effect is taken into account in the evolution of b ( z , x ) , as seen in Eq. (2). According to Eqs. (2) and (4), wecan write the energy density in Fourier space as ρ B ( z , k ) = π ∫ d k d k ( π ) δ ( k − k − k )× ˜ b i ( k ) ˜ b i ( k ) e − ( k + k ) / k ( z ) . (5)As can be seen in Eqs. (4) and (5), the energy density ofPMFs would spatially fluctuate and hence the magnetic re-heating ratio, i.e., energy injection into the CMB photonsshould also fluctuate.For z (cid:38) z µ = . × , the Universe is in chemicalequilibrium by the Compton scattering, bremsstrahlung, thedouble Compton scattering, etc. (e.g., Danese & de Zotti1982; Sunyaev & Zeldovich 1970; Burigana et al. 1991; Il-larionov & Siuniaev 1975), so that the photon distributionfunction becomes a Planck distribution function of the form [ exp ( p / T ( + ∆ B )) − ] − , with a comoving temperature T and a dimensionless magnetic reheating temperature rise ∆ B . Note that we dropped the temperature perturbationsoriginated from primordial density perturbations, but itwould be straightforward to add such a contribution. Duringthis period, variations of photon temperature are given bythe conservation law of energy. The photon energy densityis given by ρ γ ≈ ρ γ, ( + ∆ B ) , (6)where ρ γ, ∝ T is the comoving energy density of photonswithout magnetic reheating. Then, the conservation law ofthe total energy in a local diffusion patch around x is d ( ρ γ + ρ B ) = . (7)Note that the number of photons are not conserved whenthe system is in chemical equilibrium. Solving Eq. (7), weimmediately find ∆ B ≈ − d ρ B ρ γ, . (8)Thus, an additional temperature perturbation would begiven by 1/4 of local energy injection at leading order.On the other hand, bremsstrahlung and the doubleCompton scattering become less efficient below z µ . Then,photon number cannot be changed, but photons are, still, In the early Universe, the effective relativistic degrees of free-dom are larger than those of photons. Therefore, it should benoticed that photons’ share of energy injection depends on whenmagnetic reheating happens. However, we ignore this effect forsimplicity, because the number of relativistic species in the veryearly Universe is still an open question.MNRAS , 1–10 (2019)
S.Saga et al. in kinetic equilibrium due to the efficient Compton scat-tering. Hence energy injection not only raises the temper-ature but also produces the chemical potential called µ -distortion to respect the conservation laws of both energyand number (Chluba et al. 2012). In contrast to a Planckdistribution function, it should be noticed that the temper-ature of a Bose–Einstein distribution function is not thefourth root of energy. Given a Bose–Einstein distribution [ exp ( p / T ( + ∆ B ) + µ ) − ] − , with a dimensionless chemicalpotential µ , we get the following comoving energy and num-ber density at leading order in µ and ∆ B : ρ γ ≈ ρ γ, (cid:18) + ∆ B − ζ ( ) π µ (cid:19) , (9) n γ ≈ n γ, (cid:18) + ∆ B − π ζ ( ) µ (cid:19) , (10)where n γ, ∝ T is number density of photons without mag-netic reheating, and ζ ( n ) are the Riemann zeta functions.Then, the conservation laws of the comoving energy andnumber density in a local diffusion patch around x would begiven as d ( ρ γ + ρ B ) = , d n γ = , (11)which can be recast into ∆ B − ζ ( ) π d µ = − d ρ B ρ γ, , (12) ∆ B − π ζ ( ) d µ = . (13)Eliminating µ , one finds ∆ B ≈ − α d ρ B ρ γ, , (14)where α = (cid:16) − ( ζ ( )) − / π (cid:17) − ≈ . .After z f = × the Universe is out of kinetic equilib-rium; therefore, energy injection is not trivially transferredto photons, and hence we ignore this period for simplicity(Sunyaev & Zeldovich 1970). To summarize, we can describethe local heating rate due to the PMF dissipation as d ∆ B ( z , x ) d z = M( z ) (cid:20) − ρ γ, d ρ B d z ( z , x ) (cid:21) , (15)where we introduced M( z ) as M( z ) = (cid:40) ( z µ ≤ z ≤ z i ) α ( z f ≤ z ≤ z µ ) . (16)Here z i is the redshift of generation of PMFs, which dependson the magnetogenesis scenario. Throughout this paper, weset z i = (corresponding to the electroweak phase tran-sition era). The PMF dissipation does not depend on CMBphoton’s direction ˆ n . Therefore, inhomogeneous magneticreheating acts as a monopole source term in the Boltzmannhierarchical equations for CMB photons. In this section, we discuss corrections to the CMB temper-ature anisotropy in the presence of PMFs. Let us decom- pose the temperature harmonic coefficients into the follow-ing three parts: a (cid:96), m = a Θ (cid:96), m + a Θ B (cid:96), m + a ∆ B (cid:96), m . (17)The first term is the standard linear temperatureanisotropies given as a Θ (cid:96), m = π i (cid:96) ∫ d k ( π ) Y ∗ (cid:96), m ( ˆ k )T (cid:96) ( k ) (cid:18) ζ ( k ) (cid:19) , (18)where ζ is the primordial curvature perturbation on the uni-form density slice and T (cid:96) are the transfer functions of thetemperature perturbations. On the other hand, a Θ B (cid:96), m are theadditional anisotropies due to the passive mode and a ∆ B (cid:96), m are the new corrections of inhomogeneous magnetic reheat-ing which we will define later. We observe the total angularpower spectrum C (cid:96) = (cid:96) + (cid:213) m (cid:10) a (cid:96), m (cid:0) a (cid:96), m (cid:1) ∗ (cid:11) , (19)which contains the auto- and cross-correlations C XY (cid:96) = (cid:96) + (cid:213) m (cid:68) a X (cid:96), m (cid:16) a Y (cid:96), m (cid:17) ∗ (cid:69) , (20)where X / Y = Θ , ∆ B , or Θ B . The observed temperatureanisotropies are consistent with C ΘΘ (cid:96) . Therefore, the correc-tions due to PMFs should be subdominant so that we canput upper bounds on both the passive modes and magneticreheating. Let us see the explicit expressions of the harmoniccoefficients in the following subsections. PMFs intrinsically have anisotropic stress, which can inducean additional curvature perturbation ζ B on superhorizonscales. It would subsequently imprint on the CMB temper-ature anisotropy. Note that neutrino anisotropic stress can-cels that of the magnetic fields; the therefore generation of ζ B stops after neutrino decoupling. The multipole coefficientof the passive mode is then given as (Shaw & Lewis 2010) a Θ B (cid:96), m = π i (cid:96) ∫ d k ( π ) Y ∗ (cid:96), m ( ˆ k )T (cid:96) ( k ) ζ B ( k ) . (21)Here ζ B is the curvature perturbation generated by PMFs, ζ B ( k ) = ξ R γ Π B ( k ) , (22)where we define ξ = ln ( η ν / η B ) + /( R ν ) − , R γ = ρ γ /( ρ γ + ρ ν ) , R ν = − R γ , η ν is the conformal time at the neutrinodecoupling epoch, η B is the conformal time when PMFs are On scales smaller than the horizon scale at the recombinationepoch, PMFs might induce isocurvature-like perturbations andgenerate the additional CMB anisotropies which are called ascompensated magnetic modes (Lewis 2004; Shaw & Lewis 2012,2010). However, these anisotropies depend on the initial condi-tions, in particular, the relation between matter and the modelof magnetogenesis. Therefore, we neglect these types of CMBanisotropies. MNRAS000
S.Saga et al. in kinetic equilibrium due to the efficient Compton scat-tering. Hence energy injection not only raises the temper-ature but also produces the chemical potential called µ -distortion to respect the conservation laws of both energyand number (Chluba et al. 2012). In contrast to a Planckdistribution function, it should be noticed that the temper-ature of a Bose–Einstein distribution function is not thefourth root of energy. Given a Bose–Einstein distribution [ exp ( p / T ( + ∆ B ) + µ ) − ] − , with a dimensionless chemicalpotential µ , we get the following comoving energy and num-ber density at leading order in µ and ∆ B : ρ γ ≈ ρ γ, (cid:18) + ∆ B − ζ ( ) π µ (cid:19) , (9) n γ ≈ n γ, (cid:18) + ∆ B − π ζ ( ) µ (cid:19) , (10)where n γ, ∝ T is number density of photons without mag-netic reheating, and ζ ( n ) are the Riemann zeta functions.Then, the conservation laws of the comoving energy andnumber density in a local diffusion patch around x would begiven as d ( ρ γ + ρ B ) = , d n γ = , (11)which can be recast into ∆ B − ζ ( ) π d µ = − d ρ B ρ γ, , (12) ∆ B − π ζ ( ) d µ = . (13)Eliminating µ , one finds ∆ B ≈ − α d ρ B ρ γ, , (14)where α = (cid:16) − ( ζ ( )) − / π (cid:17) − ≈ . .After z f = × the Universe is out of kinetic equilib-rium; therefore, energy injection is not trivially transferredto photons, and hence we ignore this period for simplicity(Sunyaev & Zeldovich 1970). To summarize, we can describethe local heating rate due to the PMF dissipation as d ∆ B ( z , x ) d z = M( z ) (cid:20) − ρ γ, d ρ B d z ( z , x ) (cid:21) , (15)where we introduced M( z ) as M( z ) = (cid:40) ( z µ ≤ z ≤ z i ) α ( z f ≤ z ≤ z µ ) . (16)Here z i is the redshift of generation of PMFs, which dependson the magnetogenesis scenario. Throughout this paper, weset z i = (corresponding to the electroweak phase tran-sition era). The PMF dissipation does not depend on CMBphoton’s direction ˆ n . Therefore, inhomogeneous magneticreheating acts as a monopole source term in the Boltzmannhierarchical equations for CMB photons. In this section, we discuss corrections to the CMB temper-ature anisotropy in the presence of PMFs. Let us decom- pose the temperature harmonic coefficients into the follow-ing three parts: a (cid:96), m = a Θ (cid:96), m + a Θ B (cid:96), m + a ∆ B (cid:96), m . (17)The first term is the standard linear temperatureanisotropies given as a Θ (cid:96), m = π i (cid:96) ∫ d k ( π ) Y ∗ (cid:96), m ( ˆ k )T (cid:96) ( k ) (cid:18) ζ ( k ) (cid:19) , (18)where ζ is the primordial curvature perturbation on the uni-form density slice and T (cid:96) are the transfer functions of thetemperature perturbations. On the other hand, a Θ B (cid:96), m are theadditional anisotropies due to the passive mode and a ∆ B (cid:96), m are the new corrections of inhomogeneous magnetic reheat-ing which we will define later. We observe the total angularpower spectrum C (cid:96) = (cid:96) + (cid:213) m (cid:10) a (cid:96), m (cid:0) a (cid:96), m (cid:1) ∗ (cid:11) , (19)which contains the auto- and cross-correlations C XY (cid:96) = (cid:96) + (cid:213) m (cid:68) a X (cid:96), m (cid:16) a Y (cid:96), m (cid:17) ∗ (cid:69) , (20)where X / Y = Θ , ∆ B , or Θ B . The observed temperatureanisotropies are consistent with C ΘΘ (cid:96) . Therefore, the correc-tions due to PMFs should be subdominant so that we canput upper bounds on both the passive modes and magneticreheating. Let us see the explicit expressions of the harmoniccoefficients in the following subsections. PMFs intrinsically have anisotropic stress, which can inducean additional curvature perturbation ζ B on superhorizonscales. It would subsequently imprint on the CMB temper-ature anisotropy. Note that neutrino anisotropic stress can-cels that of the magnetic fields; the therefore generation of ζ B stops after neutrino decoupling. The multipole coefficientof the passive mode is then given as (Shaw & Lewis 2010) a Θ B (cid:96), m = π i (cid:96) ∫ d k ( π ) Y ∗ (cid:96), m ( ˆ k )T (cid:96) ( k ) ζ B ( k ) . (21)Here ζ B is the curvature perturbation generated by PMFs, ζ B ( k ) = ξ R γ Π B ( k ) , (22)where we define ξ = ln ( η ν / η B ) + /( R ν ) − , R γ = ρ γ /( ρ γ + ρ ν ) , R ν = − R γ , η ν is the conformal time at the neutrinodecoupling epoch, η B is the conformal time when PMFs are On scales smaller than the horizon scale at the recombinationepoch, PMFs might induce isocurvature-like perturbations andgenerate the additional CMB anisotropies which are called ascompensated magnetic modes (Lewis 2004; Shaw & Lewis 2012,2010). However, these anisotropies depend on the initial condi-tions, in particular, the relation between matter and the modelof magnetogenesis. Therefore, we neglect these types of CMBanisotropies. MNRAS000 , 1–10 (2019) nhomogeneous magnetic reheating generated, and Π B is the scalar part of the PMF anisotropicstress, which is obtained as Π B ( k ) = πρ γ, ∫ d k ( π ) ∫ d k ( π ) ( π ) δ ( k − k − k )× (cid:18) ˆ k i ˆ k j − δ ij (cid:19) C Θ B ( k , k ) ˜ b i ( k ) ˜ b j ( k ) . (23)In the above equation, we introduced the damping scale ofPMFs, following Shaw & Lewis (2010); Planck Collaborationet al. (2016) as C Θ B ( k , k ) = e − ( k + k ) / k ( z ν ) , (24)where z ν is the redshift at the neutrino decoupling epoch.Note that Shaw & Lewis (2010) dropped time dependence of Π B when they derived Eq. (22) by integrating the Einsteinequation, and they evaluated Π B at neutrino decoupling.This simplification makes us underestimate the contributionof the passive modes, but we justify this prescription becausethe relevant scales are covered by magnetic reheating as wewill show in Fig. 1. For magnetic reheating, integrating Eq. (15) with respect tothe redshift, we get ∆ B ( k ) = πρ γ, ∫ d k ( π ) ∫ d k ( π ) ( π ) δ ( k − k − k )× C ∆ B ( k , k ) ˜ b i ( k ) ˜ b i ( k ) , (25)where C ∆ B ( k , k ) is a kernel function related to the dissipa-tion efficiency of PMFs as C ∆ B ( k , k ) = exp (cid:34) − k + k k ( z i ) (cid:35) − exp (cid:34) − k + k k ( z µ ) (cid:35) + α (cid:32) exp (cid:34) − k + k k ( z µ ) (cid:35) − exp (cid:34) − k + k k ( z f ) (cid:35)(cid:33) . (26)Then we find that the resultant CMB angular power spec-trum induced by magnetic reheating can be expressed by thefollowing approximate formula for the multipole coefficient: a ∆ B (cid:96), m ≈ π i (cid:96) ∫ d k ( π ) Y ∗ (cid:96), m ( ˆ k )T (cid:96) ( k ) ∆ B ( k ) . (27) We are now in a position to calculate the PMF correc-tions to the CMB temperature angular power spectrum C (cid:96) .In this work, we compute C (cid:96) by modifying public Boltz-mann codes, e.g., CLASS (Lesgourgues 2011). According toEqs. (18), (21) and (27), it should be noticed that C XY (cid:96) cancontain two types of initial correlation functions: (cid:104) B (cid:105) and (cid:104) B ζ (cid:105) . The four-point function of B can be reduced to prod-ucts of (cid:104) B (cid:105) at the leading order. On the other hand, thelatter correlation is non-zero in the case where the couplingbetween PMFs and the primordial curvature perturbationsexists. We will see that the scale dependence is entirely dif-ferent for these two initial conditions in the following sub-sections. (cid:10) B (cid:11) When initial PMFs are statistically homogeneous andisotropic Gaussian random fields, the stochastic property ofPMFs is completely characterized by the power spectrum (cid:104) ˜ b i ( k ) ˜ b ∗ j ( k (cid:48) )(cid:105) = ( π ) δ ( k − k (cid:48) ) (cid:16) δ ij − ˆ k i ˆ k j (cid:17) P B ( k ) . (28)Then, the four-point function of B is reduced to the productof P B . Taking the ensemble averages of Eqs. (27) and (21),we obtain the following angular power spectrum of auto andcross-correlation C ∆ B ∆ B (cid:96) = ρ γ, ( π ) ∫ d k k T (cid:96) ( k ) ∫ d k k ∫ − d µ × F ( k , k , µ ) P B ( k ) P B (| k − k |) (cid:0) C ∆ B ( k , | k − k |) (cid:1) , (29) C Θ B ∆ B (cid:96) = R γ ξ ρ γ, ( π ) ∫ d k k T (cid:96) ( k ) ∫ d k k ∫ − d µ G( k , k , µ )× P B ( k ) P B (| k − k |) C Θ B ( k , | k − k |) C ∆ B ( k , | k − k |) , (30) C Θ B Θ B (cid:96) = R γ ξ ρ γ, ( π ) ∫ d k k T (cid:96) ( k ) ∫ d k k ∫ − d µ × I( k , k , µ ) P B ( k ) P B (| k − k |) (cid:0) C Θ B ( k , | k − k |) (cid:1) , (31)where µ = ˆ k · ˆ k and the configuration factor, F ( k , k , µ ) , G( k , k , µ ) , and I( k , k , µ ) are given by F ( k , k , µ ) = ( + µ ) k − k k µ + k k + k − k k µ , (32) G( k , k , µ ) = k ( − µ ) − k ( + µ ) + k k µ ( + µ ) k + k − k k µ , (33) I( k , k , µ ) = k ( + µ ) + k k ( µ − µ ) + k ( − µ + µ ) k + k − k k µ . (34)Let us substitute the following delta-function typepower spectrum of PMFs P B ( ln k ) = π k B δ D (cid:0) ln (cid:0) k / k p (cid:1)(cid:1) . (35)Using this power spectrum, we would find the angular scalewhich is sensitive to a given k p mode. We can proceed µ -and k -integrals appeared in Eqs. (29), (30), and (31) as C ∆ B ∆ B (cid:96) ≈ B πρ γ, (cid:0) C ∆ B ( k p , k p ) (cid:1) ∫ d kk T (cid:96) ( k ) k k , (36) C Θ B ∆ B (cid:96) ≈ B πρ γ, R γ ξ C Θ B ( k p , k p ) C ∆ B ( k p , k p ) ∫ d kk T (cid:96) ( k ) k k , (37) C Θ B Θ B (cid:96) ≈ B πρ γ, R γ ξ (cid:0) C Θ B ( k p , k p ) (cid:1) ∫ d kk T (cid:96) ( k ) k k , (38)where we use the fact that the transfer function can be ap-proximated as T (cid:96) ( k ) ∝ j (cid:96) ( k η ) with the present conformaltime η . The integration over k can pick up the contribution MNRAS , 1–10 (2019)
S.Saga et al. k p [Mpc ]10 C X ( k p , k p ) k D ( z ) k D ( z ) k D ( z i ) k D ( z f ) C B C B Figure 1.
The kernel functions C ∆ B ( k p , k p ) (blue, solid) and C Θ B ( k p , k p ) (orange, dashed) related to inhomogeneous magneticreheating and passive mode, respectively, as a function of k p , de-fined in Eqs. (26) and (24), respectively. The arrows show the dif-fusion scales of each redshift, i.e., k D ( z f ) ≈ × Mpc − , k D ( z µ ) ≈ × Mpc − , k D ( z ν ) ≈ . × Mpc − , and k D ( z i ) ≈ Mpc − based on Jeong et al. (2014). only k p (cid:29) k ∼ (cid:96) / η with the configuration factor F ≈ , G ≈ / , and I ≈ / .We find that k p dependence is summarised in C ∆ B ( k p , k p ) and C Θ B ( k p , k p ) , which are shown in Fig. 1. We can see thatthe shape of C ∆ B ( k p , k p ) appears like a rectangular func-tion (blue, solid), which can explore much smaller scalesthan the passive mode shown in the orange dashed line.Fig. 1 shows that both inhomogeneous magnetic reheat-ing and the passive mode can probe to the PMFs on thescales, − < k p < Mpc − . On the other hand, onsmaller scales, i.e., k p (cid:29) O ( ) Mpc − , contrary to the pas-sive mode, inhomogeneous magnetic reheating is possiblyavailable to constrain the amplitude of PMFs.Note that, from the integrand in Eq. (36), we find thatthe shape of the angular power spectrum C ∆ B ∆ B (cid:96) correspondsto the one induced by the adiabatic initial condition with ascalar spectral index n s = . (cid:10) B ζ (cid:11) Several magnetogenesis models, for example, f ( φ ) F µν F µν interaction, produce non-zero cross-correlation between theenergy density of PMFs and the curvature perturbations(e.g., Caldwell et al. 2011; Motta & Caldwell 2012; Shiraishiet al. 2012). In such a case, we can parametrize the cross-correlation in the squeezed limit ( k ∼ k (cid:29) k ) following inJain & Sloth (2012); Ganc & Sloth (2014) as (cid:10) ˜ b i ( k ) ˜ b j ( k ) ζ ∗ ( k ) (cid:11) k (cid:28) k ∼ k = ( π ) δ ( k + k − k ) b NL δ ij − ˆ k i ˆ k j P B ( k ) P ζ ( k ) , (39) where P ζ ( k ) is a power spectrum of the primordial curvatureperturbations, which is simply parametrized as k π P ζ ( k ) = A ζ (cid:18) kk (cid:19) n s − . (40)In such a non-Gaussian case, superhorizon curvature pertur-bations couple to subhorizon magnetic fields. Hence, inho-mogeneous magnetic reheating on tiny scales leads to addi-tional superhorizon temperature perturbations. Inhomoge-neous magnetic reheating is more sensitive to smaller scalesthan passive modes, as seen in Fig. 1; therefore C Θ∆ B (cid:96) wouldbe more significant than C ΘΘ B (cid:96) . Thus inhomogeneous mag-netic reheating can become a novel probe of the correlationfunction (39).The cross-correlations are given by using Eqs. (27), (21),and (18) as C Θ B Θ (cid:96) = − b NL πξ R γ ρ γ, ( π ) ∫ d k k T (cid:96) ( k )× ∫ d k k ˜ C Θ B ( k , k ) P B ( k ) P ζ ( k ) , (41)for the cross-correlation between the passive mode and adi-abatic curvature perturbation, and C ∆ B Θ (cid:96) = b NL π ρ γ, ( π ) ∫ d k k T (cid:96) ( k )× ∫ d k k ˜ C ∆ B ( k , k ) P B ( k ) P ζ ( k ) , (42)for the cross-correlation between inhomogeneous magneticreheating and the adiabatic curvature perturbation. In theabove equations, we define ˜ C Θ B ( k , k ) = ∫ − d µ C Θ B ( k , | k − k |) (cid:18) µ − (cid:19) = − k ( z ν ) k k (cid:34) (cid:169)(cid:173)(cid:171) e − k + k ( z ν ) − e − k − k ( z ν ) (cid:170)(cid:174)(cid:172) k k + (cid:169)(cid:173)(cid:171) e − k + k ( z ν ) + e − k − k ( z ν ) (cid:170)(cid:174)(cid:172) k k k ( z ν ) + (cid:169)(cid:173)(cid:171) e − k + k ( z ν ) − e − k − k ( z ν ) (cid:170)(cid:174)(cid:172) k ( z ν ) (cid:35) , (43) ˜ C ∆ B ( k , k ) = ∫ − d µ C ∆ B ( k , | k − k |) = k k (cid:34) α (cid:169)(cid:173)(cid:171) e − k + k ( z f ) − e − k − k ( z f ) (cid:170)(cid:174)(cid:172) k ( z f )− (cid:169)(cid:173)(cid:171) e − k + k ( z i ) − e − k − k ( z i ) (cid:170)(cid:174)(cid:172) k ( z i ) + ( − α ) (cid:169)(cid:173)(cid:171) e − k + k ( z µ ) − e − k − k ( z µ ) (cid:170)(cid:174)(cid:172) k ( z µ ) (cid:35) , (44)where we define k ± = k ± k k + k .In the case of the power-law type, the power spectrumof PMFs is given as k π P B ( k ) = ( π ) n B + Γ (cid:16) n B + (cid:17) B λ (cid:18) kk λ (cid:19) n B + , (45) MNRAS000
The kernel functions C ∆ B ( k p , k p ) (blue, solid) and C Θ B ( k p , k p ) (orange, dashed) related to inhomogeneous magneticreheating and passive mode, respectively, as a function of k p , de-fined in Eqs. (26) and (24), respectively. The arrows show the dif-fusion scales of each redshift, i.e., k D ( z f ) ≈ × Mpc − , k D ( z µ ) ≈ × Mpc − , k D ( z ν ) ≈ . × Mpc − , and k D ( z i ) ≈ Mpc − based on Jeong et al. (2014). only k p (cid:29) k ∼ (cid:96) / η with the configuration factor F ≈ , G ≈ / , and I ≈ / .We find that k p dependence is summarised in C ∆ B ( k p , k p ) and C Θ B ( k p , k p ) , which are shown in Fig. 1. We can see thatthe shape of C ∆ B ( k p , k p ) appears like a rectangular func-tion (blue, solid), which can explore much smaller scalesthan the passive mode shown in the orange dashed line.Fig. 1 shows that both inhomogeneous magnetic reheat-ing and the passive mode can probe to the PMFs on thescales, − < k p < Mpc − . On the other hand, onsmaller scales, i.e., k p (cid:29) O ( ) Mpc − , contrary to the pas-sive mode, inhomogeneous magnetic reheating is possiblyavailable to constrain the amplitude of PMFs.Note that, from the integrand in Eq. (36), we find thatthe shape of the angular power spectrum C ∆ B ∆ B (cid:96) correspondsto the one induced by the adiabatic initial condition with ascalar spectral index n s = . (cid:10) B ζ (cid:11) Several magnetogenesis models, for example, f ( φ ) F µν F µν interaction, produce non-zero cross-correlation between theenergy density of PMFs and the curvature perturbations(e.g., Caldwell et al. 2011; Motta & Caldwell 2012; Shiraishiet al. 2012). In such a case, we can parametrize the cross-correlation in the squeezed limit ( k ∼ k (cid:29) k ) following inJain & Sloth (2012); Ganc & Sloth (2014) as (cid:10) ˜ b i ( k ) ˜ b j ( k ) ζ ∗ ( k ) (cid:11) k (cid:28) k ∼ k = ( π ) δ ( k + k − k ) b NL δ ij − ˆ k i ˆ k j P B ( k ) P ζ ( k ) , (39) where P ζ ( k ) is a power spectrum of the primordial curvatureperturbations, which is simply parametrized as k π P ζ ( k ) = A ζ (cid:18) kk (cid:19) n s − . (40)In such a non-Gaussian case, superhorizon curvature pertur-bations couple to subhorizon magnetic fields. Hence, inho-mogeneous magnetic reheating on tiny scales leads to addi-tional superhorizon temperature perturbations. Inhomoge-neous magnetic reheating is more sensitive to smaller scalesthan passive modes, as seen in Fig. 1; therefore C Θ∆ B (cid:96) wouldbe more significant than C ΘΘ B (cid:96) . Thus inhomogeneous mag-netic reheating can become a novel probe of the correlationfunction (39).The cross-correlations are given by using Eqs. (27), (21),and (18) as C Θ B Θ (cid:96) = − b NL πξ R γ ρ γ, ( π ) ∫ d k k T (cid:96) ( k )× ∫ d k k ˜ C Θ B ( k , k ) P B ( k ) P ζ ( k ) , (41)for the cross-correlation between the passive mode and adi-abatic curvature perturbation, and C ∆ B Θ (cid:96) = b NL π ρ γ, ( π ) ∫ d k k T (cid:96) ( k )× ∫ d k k ˜ C ∆ B ( k , k ) P B ( k ) P ζ ( k ) , (42)for the cross-correlation between inhomogeneous magneticreheating and the adiabatic curvature perturbation. In theabove equations, we define ˜ C Θ B ( k , k ) = ∫ − d µ C Θ B ( k , | k − k |) (cid:18) µ − (cid:19) = − k ( z ν ) k k (cid:34) (cid:169)(cid:173)(cid:171) e − k + k ( z ν ) − e − k − k ( z ν ) (cid:170)(cid:174)(cid:172) k k + (cid:169)(cid:173)(cid:171) e − k + k ( z ν ) + e − k − k ( z ν ) (cid:170)(cid:174)(cid:172) k k k ( z ν ) + (cid:169)(cid:173)(cid:171) e − k + k ( z ν ) − e − k − k ( z ν ) (cid:170)(cid:174)(cid:172) k ( z ν ) (cid:35) , (43) ˜ C ∆ B ( k , k ) = ∫ − d µ C ∆ B ( k , | k − k |) = k k (cid:34) α (cid:169)(cid:173)(cid:171) e − k + k ( z f ) − e − k − k ( z f ) (cid:170)(cid:174)(cid:172) k ( z f )− (cid:169)(cid:173)(cid:171) e − k + k ( z i ) − e − k − k ( z i ) (cid:170)(cid:174)(cid:172) k ( z i ) + ( − α ) (cid:169)(cid:173)(cid:171) e − k + k ( z µ ) − e − k − k ( z µ ) (cid:170)(cid:174)(cid:172) k ( z µ ) (cid:35) , (44)where we define k ± = k ± k k + k .In the case of the power-law type, the power spectrumof PMFs is given as k π P B ( k ) = ( π ) n B + Γ (cid:16) n B + (cid:17) B λ (cid:18) kk λ (cid:19) n B + , (45) MNRAS000 , 1–10 (2019) nhomogeneous magnetic reheating where k λ = π / λ and in this paper, we fix λ = asfollowing Planck Collaboration et al. (2016). Note that B λ corresponds to the amplitude of the magnetic field aftersmoothing over the pivot scale λ . Then, Eqs. (41) and (42)are rewritten as C Θ B Θ (cid:96) = − . × − (cid:18) A ζ . × − (cid:19) ( π ) n B + Γ (cid:16) n B + (cid:17) × b NL (cid:18) B λ (cid:19) ∫ d kk (cid:18) kk (cid:19) n s − T (cid:96) ( k )D Θ B ( k , n B ) , (46) C ∆ B Θ (cid:96) = . × − (cid:18) A ζ . × − (cid:19) ( π ) n B + Γ (cid:16) n B + (cid:17) × b NL (cid:18) B λ (cid:19) ∫ d kk (cid:18) kk (cid:19) n s − T (cid:96) ( k )D ∆ B ( k , n B ) , (47)where we define D X ( k , n B ) ≡ ∫ d k k (cid:18) k k λ (cid:19) n B + ˜ C X ( k , k ) , (48)with X = ∆ B or Θ B . The function D X ( k , n B ) generally de-pends on both k and n B and we show these functions inFig. 2. As we discussed in the previous section, the domi-nant contribution in the k -integral of Eq. (47) is coming from k ∼ (cid:96) / η (cid:28) O ( ) Mpc − . Then we find the most dominantparts are given as D Θ B ( k → , n B ) ≈ −( + n B )/ (cid:18) kk λ (cid:19) (cid:18) k D ( z ν ) k λ (cid:19) n B + × ( n B + n B + ) Γ (cid:18) n B + (cid:19) , (49) D ∆ B ( k → , n B ) ≈ − nB + (cid:18) k D ( z i ) k λ (cid:19) n B + Γ (cid:18) n B + (cid:19) . (50)Here, we use the fact that k D ( z i ) (cid:29) k D ( z ν ) > k D ( z µ ) > k D ( z f ) .These D X determine the amplitude of the angular powerspectra of the CMB temperature anisotropy. The func-tion of the passive mode (49), is sensitive to the damp-ing scale k D ( z ν ) while that of magnetic reheating (50),is to k D ( z i ) . Therefore, since k D ( z i ) (cid:29) k D ( z ν ) , the cross-correlation between Θ B and Θ is strongly suppressed, com-pared to that between ∆ B and Θ . The above features are alsofound in Fig. 2, where the cross-correlation with the passivemodes is plotted with multiplication by although thatof magnetic reheating is shown without any multiplication.Finally, the angular power spectra of the CMB temper-ature anisotropy are given by C Θ B Θ (cid:96) ∝ ∫ d kk (cid:18) kk (cid:19) n s − + T (cid:96) ( k ) , (51) C ∆ B Θ (cid:96) ∝ ∫ d kk (cid:18) kk (cid:19) n s − T (cid:96) ( k ) . (52)From Eqs. (51) and (52), the shapes of the angular powerspectra induced by non-Gaussian PMFs are the same as theprimary CMB spectrum of n s + (the passive mode) and n s (inhomogeneous magnetic reheating), respectively. Thisfeature can be shown in Fig. 2.In this paper, we set the initial redshift z i to that ofthe electroweak phase transition. However, the correlation k [Mpc ]10 D X ( k , n B ) n B = 2.9 n B = 2.7 n B = 2.5 n B = 2.9 n B = 2.7 n B = 2.5 D B : D B : ( + ) C B / ( ) , × ( + ) C B / ( ) n B = 2.9 n B = 2.7 n B = 2.5FiducialPlanckACT Figure 2. ( Top ) The functions D ∆ B ( k , n B ) and D Θ B ( k , n B ) re-lated to inhomogeneous magnetic reheating and passive mode,respectively, which determine the amplitude of the angular powerspectrum, as a function of k for various values of n B . ( Bot-tom ): Dimensionless angular power spectra obtained by Eqs. (46)(dashed) and (47) (solid). In this figure, we set b NL = , B λ = , A ζ = . × − , and n s = . . For comparison, we presentplots of the passive mode multiplied by . We also show theobserved data from Planck (Planck Collaboration et al. 2018) andACT (Louis et al. 2017). Note that we plot the absolute value of C (cid:96) . (cid:10) B ζ (cid:11) can be initiated when the PMFs are generated dur-ing inflation. In this case, the initial redshift z i should goback to reheating epoch, corresponding to the conformaltime η ν / η B ≈ . When we choose this redshift as theinitial time, the effect of inhomogeneous magnetic reheatingon the CMB temperature anisotropy would be enhanced.In this sense, we will give conservative results in the nextsection. MNRAS , 1–10 (2019)
S.Saga et al.
First, we investigate a constraint on the amplitude of PMFsfrom the Gaussian part discussed in Sec. 4.1. As shown inthe previous section, we found that the disconnected part of (cid:104) B (cid:105) can contribute to C ∆ B ∆ B (cid:96) , C Θ B ∆ B (cid:96) , and C Θ B Θ B (cid:96) . Then,the angular power spectra of them show the scale depen-dence similar to those from the blue-tilted adiabatic initialcondition with a scalar spectral index n s = . Hence, let ustake a closer look at the angular power spectrum on smallscales, where the corrections would be relatively enhanced.The CMB measurement by ACT (Louis et al. 2017)gives the precise data of the small-scale CMB angular powerspectrum and the minimum value of the observed ampli-tude of the CMB temperature fluctuations can be read as C ( obs ) (cid:96) ACT ≈ . × − around (cid:96) ACT = , as shown in Fig. 2.In the standard cosmology, it is interpreted that the pri-mordial CMB anisotropy dominates on scales larger thanACT-scale ( ∼ (cid:96) ACT ) while the secondary CMB anisotropy in-cluding the Sunyaev–Zel’dovich effect of galaxy clusters andforeground contributions would dominate the anisotropy onsmaller scales. However, as shown in the previous section, C ∆ B ∆ B potentially have a significant contribution on smallscales. Therefore, from the condition C ( obs ) (cid:96) ACT > C ∆ B ∆ B (cid:96) ACT , in-homogeneous magnetic reheating put an upper bound onPMFs as B (cid:46) . × nG ( k p = O ( ) Mpc − ) . (53)This constraint is much weaker than that obtained from theCMB spectral distortion, i.e., B <
10 nG on
10 Mpc − (cid:46) k p (cid:46) Mpc − (Jedamzik et al. 2000). Although C Θ B ∆ B (cid:96) and C Θ B Θ B (cid:96) lead to constraints similar to Eq. (53) on thesame scales, that from the CMB spectral distortions aremuch tighter. Thus, inhomogeneous magnetic reheating isnot useful to constrain Gaussian PMFs. Indeed, Narukoet al. (2015) also showed that the anisotropic acoustic re-heating is not useful to discuss upper bounds on small-scaleGaussian curvature perturbations. b NL Let us derive a constraint on the non-linear parameter ofPMFs, which gives mode mixing of large and small scales.In the previous section, we explored the consequence ofthe primordial cross-correlation (39). As seen in the bot-tom of Fig. 2 and Eq. (52), we found that the scale de-pendence of the angular power spectrum is the same withthat of the scale-invariant adiabatic fluctuations, namely, C ∆ B Θ (cid:96) ∝ C Fiducial (cid:96) . The amplitude of C Fiducial (cid:96) seen in Fig. 2can be fixed by the current observations, and therefore weconstrain the signal of inhomogeneous magnetic reheatingby C ∆ B Θ (cid:96) < C Fiducial (cid:96) for an arbitrary (cid:96) . Here, we comparedthe values of (cid:96) = to put the constraint on b NL B λ , andthe result is also shown in Fig. 3. In this figure, we alsoshow the upper bound forecasted by using the CMB spec-tral distortion in Eq. (6.11) of Ganc & Sloth (2014). It isknown that dissipating PMFs can reheat the CMB pho-tons and create µ -distortion during z f < z < z µ . Ganc &Sloth (2014) used the cross-correlation between the inhomo- n B b N L B [ n G ] E x c l ud e d PIXIECMBPolInhomogeneous magnetic reheating
Figure 3.
The upper bound on the combination b NL B λ as thefunction of n B . The gray-shaded region is excluded by inhomo-geneous magnetic reheating. Here, we use the same model pa-rameters as Fig. 2. We also show the forecast from (cid:104) µ T (cid:105) cross-correlation of Eq. (6.11) in Ganc & Sloth (2014). geneous µ -distortion and passive mode in order to constrain b NL .In Fig. 3, we can obtain the approximate form of theupper bound on b NL B λ in n B - b NL B λ plane as followings. Bysubstituting Eq. (50) into Eq. (47), we obtain C ∆ B Θ ≈ b NL (cid:18) B λ (cid:19) × . × − (cid:32) k ( z i ) (cid:33) nB + × (cid:18) A ζ . × − (cid:19) ∫ d kk (cid:18) kk (cid:19) n s − T (cid:96) ( k ) = b NL (cid:18) B λ (cid:19) × . × − . × − (cid:32) k ( z i ) (cid:33) nB + C Fiducial (cid:96) , (54)where C Fiducial (cid:96) is the CMB temperature anisotropy from theadiabatic curvature perturbation defined by C Fiducial (cid:96) = A ζ ∫ d kk (cid:18) kk (cid:19) n s − T (cid:96) ( k ) . (55)Then we impose the condition C ∆ B Θ (cid:96) < C Fiducial (cid:96) and finallyobtain the approximate form on n B (cid:29) − of the upper limitas b NL ( B λ / nG ) . In the current setting z i = , the upperlimit can be given as b NL (cid:18) B λ (cid:19) < e − . n B − . . (56)We can find this approximate form in Fig. 3 in the case oflarge n B .Although the upper bound from inhomogeneous mag-netic reheating is coming from the current Planck observa-tion, it can put a tighter constraint on the primordial non-Gaussianity in PMFs than the forecasts based on the CMBspectral distortion in Ganc & Sloth (2014), especially in thecase of the bluer tilted magnetic fields. This is the advantage MNRAS000
The upper bound on the combination b NL B λ as thefunction of n B . The gray-shaded region is excluded by inhomo-geneous magnetic reheating. Here, we use the same model pa-rameters as Fig. 2. We also show the forecast from (cid:104) µ T (cid:105) cross-correlation of Eq. (6.11) in Ganc & Sloth (2014). geneous µ -distortion and passive mode in order to constrain b NL .In Fig. 3, we can obtain the approximate form of theupper bound on b NL B λ in n B - b NL B λ plane as followings. Bysubstituting Eq. (50) into Eq. (47), we obtain C ∆ B Θ ≈ b NL (cid:18) B λ (cid:19) × . × − (cid:32) k ( z i ) (cid:33) nB + × (cid:18) A ζ . × − (cid:19) ∫ d kk (cid:18) kk (cid:19) n s − T (cid:96) ( k ) = b NL (cid:18) B λ (cid:19) × . × − . × − (cid:32) k ( z i ) (cid:33) nB + C Fiducial (cid:96) , (54)where C Fiducial (cid:96) is the CMB temperature anisotropy from theadiabatic curvature perturbation defined by C Fiducial (cid:96) = A ζ ∫ d kk (cid:18) kk (cid:19) n s − T (cid:96) ( k ) . (55)Then we impose the condition C ∆ B Θ (cid:96) < C Fiducial (cid:96) and finallyobtain the approximate form on n B (cid:29) − of the upper limitas b NL ( B λ / nG ) . In the current setting z i = , the upperlimit can be given as b NL (cid:18) B λ (cid:19) < e − . n B − . . (56)We can find this approximate form in Fig. 3 in the case oflarge n B .Although the upper bound from inhomogeneous mag-netic reheating is coming from the current Planck observa-tion, it can put a tighter constraint on the primordial non-Gaussianity in PMFs than the forecasts based on the CMBspectral distortion in Ganc & Sloth (2014), especially in thecase of the bluer tilted magnetic fields. This is the advantage MNRAS000 , 1–10 (2019) nhomogeneous magnetic reheating of inhomogeneous magnetic reheating which can be sensitiveto smaller scales as shown in Fig. 1. In this paper, we explore additional CMB temperatureanisotropies from the dissipation of PMFs and derive someconstraints on the statistical quantities of PMFs.Fast magnetosonic modes originated from PMFs dissi-pate on small scales so that the energy of PMFs in a localcomoving volume is decreasing. Thanks to the energy con-servation laws of the total system of radiation and magneticfields, the released energy would be transferred to radiation.If PMFs are initially fluctuating, the energy injection into ra-diation should be also fluctuating, which leads to additionaltemperature perturbations. We call such a secondary CMBtemperature anisotropy inhomogeneous magnetic reheating ,which has never been considered in the literature. Inhomoge-neous magnetic reheating should be subdominant part of theCMB anisotropy; therefore, we can put some upper boundson the statistical properties of PMFs, imposing inhomoge-neous magnetic reheating not to exceed the observed CMBtemperature power spectrum.First of all, we give a formulation of the secondary CMBtemperature anisotropy from inhomogeneous magnetic re-heating as a straightforward extension of global magneticreheating proposed by Saga et al. (2018). Then, we evaluatethe corrections to the CMB temperature power spectrumthat originated from Gaussian PMFs. We find that inhomo-geneous magnetic reheating can have relatively huge contri-butions on the scales where the passive mode is not pro-duced: i.e., Mpc − (cid:46) k . However, the final expression ofthe secondary temperature anisotropies is highly suppressedon the CMB anisotropy scales so that we cannot get a mean-ingful constraint. This is because the secondary tempera-ture power spectrum from disconnected four-point PMFs ismainly produced on the scales where magnetic reheating ac-tually happened, which are far smaller than the dampingfeature in the temperature angular power spectrum. As wesee in Fig. 2, we observe the low (cid:96) tail of the blue-tiltedpower spectrum, which is suppressed on the CMB scales.On the other hand, we find that inhomogeneous mag-netic reheating has the advantage to probe non-GaussianPMFs on small scales, which can be another hint of pri-mordial magnetogenesis. Here, we assume local-type non-Gaussianity of PMFs parametrized in Eq. (39). This shapeof non-Gaussianity leads to the mode mixing betweensmall-scale magnetic fields and a large-scale curvature per-turbation. Hence, inhomogeneous magnetic reheating cre-ates additional long-wavelength temperature perturbationswhich are correlated with original long-wavelength curva-ture perturbations. In this case, we obtain the strongestupper limit on the amplitude of local-type PMF non-Gaussianity. Although we mainly focus on the phenomeno-logical parametrization throughout this paper, exploring theexplicit model of primordial non-Gaussianity in PMFs willbe left as future work. ACKNOWLEDGEMENTS
This work is supported by a Grant-in-Aid for Japan Societyfor Promotion of Science (JSPS) Research Fellow Number17J10553 (SS), JSPS Overseas Research Fellowships (AO),JSPS KAKENHI Grant Number 15K17646 (HT) and17H01110 (HT), and MEXT KAKENHI Grant Number15H05888 (SY) and 18H04356 (SY).
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