Constraints on primordial magnetic fields from magnetically-induced perturbations: current status and future perspectives with LiteBIRD and future ground based experiments
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Constraints on primordial magneticfields from magnetically-inducedperturbations: current status andfuture perspectives with LiteBIRDand future ground basedexperiments.
D. Paoletti, a , b , F. Finelli, a , b a Osservatorio di Astrofisica e Scienza dello Spazio di Bologna/Istituto Nazionale di Astrofisica, viaGobetti 101, I-40129 Bologna, Italy b Istituto Nazionale Di Fisica Nucleare, Sezione di Bologna,Viale Berti Pichat, 6/2, I-40127 Bologna,ItalyE-mail: [email protected], fabio.fi[email protected]
Abstract.
We present the constraints on the amplitude of magnetic fields generated prior to therecombination using CMB temperature and polarization anisotropy data from Planck 2018 release,alone and in combination with those from BICEP2/Keck array and the South Pole Telescope. Wemodel the fields with a generic parametrization and we make no assumptions on their origin in orderto provide general constraints on their characteristics. The analysis updates the former correspondingPlanck 2015 results both on data and numerical implementation. We then perform forecasts for thenext generation of CMB experiments such as LiteBIRD satellite alone and in combination with futureground based experiments. Corresponding author. a r X i v : . [ a s t r o - ph . C O ] O c t ontents The origin and evolution of cosmic magnetism is one of the most intriguing topics in modern cos-mology. In recent years there has been an impressive development from both data and theoreticalstudies, and future experiments at different wavelenghts, from the radio to the gamma rays, hold thepromise to disclose the many still open issues in the field and provide a complete scenario of cosmicmagnetism in the next decade. The possibility to generate magnetic fields in the early Universe rep-resents a very promising scenario, from the first ideas in the far past [1], up to the present days [2–13]and it has twofold implications for cosmology.On one side these primordial fields can provide the initial seeds for amplification by structureformation and self-induced dynamos. Therefore they can represent the progenitors (or co-progenitorsif we consider also astrophysical mechanisms) of the magnetic fields observed in large scale structureslike galaxy clusters [14–16] and galaxies, the latter up to very high redshift [17, 18]. A primordialorigin might also be a natural explanation for those large scale magnetic fields diffuse in voids, thatcan be responsible of the anomalous GeV photon deficit in high energy blazar observations [19–24].In such a context, future gamma-ray experiments as the Cherenkov Telescope Array, that will alsoprovide time delay and halo measurements of the blazar emission, will strongly improve the lowerlimits on the amplitude of magnetic fields in voids and at the same time provide hints on a possiblehelical components [25–27]. On the other end of the observational frequency range, future highsensitivity radio observations will improve significantly the measurements of the magnetic fields inthe cosmic structures [28–32] and will also have the capability to probe the presence of weak fieldsin the filaments of the large scale structure [33, 34].On the other side, primordial magnetic fields would provide an unconventional glimpse of thephysics of the early Universe. Several mechanisms could be responsible for the generation of the– 1 –anetic seeds and we describe the main ones without pretending to be exhaustive. The breaking ofconformal invariance during inflation [35–45] can provide seeds with a large coherence lenght butin general a small amplitude. Although the amplitude of the seeds can be further amplified duringthe preheating process after inflation [46–48], issues as back reaction or strong coupling problemneed to be addressed during inflation [41, 42, 49]. Note that large scale magnetic seeds are alsopredicted in models which are alternative to inflation [50]. PMF could be an imprint of a first orderphase transition [51–62] that through bubble nucleation provides the turbulent enviroment necessaryto generate and amplify the fields. As an alternative scenario, fields generated through second orderperturbations via the Harrison mechanism and at recombination [63–65] are generally too weak togenerate relevant magnetic seeds but are anyway interesting [66] for their evolution up to presentdays in the local Universe. Another possibility is that primordial magnetic fields could have beengenerated at later times during reionization by Biermann battery effect [67].Current and future data will have the possibility to shed light on PMFs characteristics and ontheir connection to the physics of the early Universe. PMFs characteristics are indeed related totheir generation mechanism, e.g. causal fields are bonded to positive spectral indices, greater orequal two [68], in the power law formalism, whereas for inflationary fields the spectral index isconnected to the coupling and can also assume negative values (with a natural lower bound n B > − We assume a stochastic background of PMFs described by the two point correlation function: (cid:68) B a ( (cid:126) k ) B ∗ B ( (cid:126) q ) (cid:69) = (2 π ) δ (3) ( (cid:126) k − (cid:126) q ) (cid:104) ( δ ab − ˆ k a ˆ k B ) P B ( k ) + i (cid:15) abc ˆ k c P H ( k ) (cid:105) . (2.1)We defer the analysis of the helical component for a following work, and therefore we retain onlythe non-helical term (i.e. P H = P B ( k ) = A B k n B and, as usual, we adopt the smoothed amplitude of the fields on a Mpcscale: B
21 Mpc = (cid:90) ∞ dk k π e − k (1 Mpc) P B ( k ) = A B π λ n B + Γ (cid:32) n B + (cid:33) . (2.2)Magnetic perturbations survive Silk damping thanks to the overdamped mode induced by Alfvenvelocity: the fields are damped at smaller scales, with the exact scale depending on the Alfven velocityand therefore on the amplitude and configuration of the fields. We model this damping with a sharpcut off in the PMFs power spectrum at the scale k D defined as [150, 164]: k D = (5 . × ) n B + (cid:32) B nG (cid:33) − n B + (2 π ) n B + n B + h n B + (cid:32) Ω b h . (cid:33) n B + Mpc − , (2.3) We use the Fourier convention: Y ( (cid:126) k ,τ ) = (cid:82) d x e i (cid:126) k · (cid:126) x Y ( (cid:126) x ,τ ), and its inverse Y ( (cid:126) x ,τ ) = (cid:82) d k (2 π ) e − i (cid:126) k · (cid:126) x Y ( (cid:126) k ,τ ). – 3 –here h is the reduced Hubble constant, H = h km s − Mpc − , and Ω b is the baryon densityparameter. The magnetic energy momentum tensor is given by: T = − ρ B = − B ( (cid:126) x )8 π a ( τ ) , (2.4) T i = , (2.5) T ij = π a ( τ ) (cid:32) B ( (cid:126) x )2 δ ij − B j ( (cid:126) x ) B i ( (cid:126) x ) (cid:33) . (2.6)It contributes as an additional source term in the perturbed Einstein equations δ G µν = π G ( δ T F luid µν + T µν ). Decomposing into scalar, vector and tensor we obtain the different source terms for magneti-cally induced perturbations. | ρ B ( k ) | = π (cid:90) Ω d p P B ( p ) P B ( | (cid:126) k − (cid:126) p | ) (1 + µ ) , (2.7) (cid:12)(cid:12)(cid:12) L (S) ( k ) (cid:12)(cid:12)(cid:12) = π (cid:90) Ω d p P B ( p ) P B ( | (cid:126) k − (cid:126) p | ) (cid:104) + µ + γβ ( γβ − µ ) (cid:105) , (2.8) (cid:12)(cid:12)(cid:12) σ ( k ) (cid:12)(cid:12)(cid:12) = π (cid:90) Ω d pP B ( p ) P B ( | (cid:126) k − (cid:126) p | ) (cid:34)
49 (4 + µ − γ − β + γ β − γβµ ) (cid:35)(cid:12)(cid:12)(cid:12) Π (V) ( k ) (cid:12)(cid:12)(cid:12) = π (cid:90) Ω d p P B ( p ) P B ( | (cid:126) k − (cid:126) p | ) (cid:104) (1 + β )(1 − γ ) + γβ ( µ − γβ ) (cid:105) , (2.9) (cid:12)(cid:12)(cid:12) Π (T) ( k ) (cid:12)(cid:12)(cid:12) = π (cid:90) Ω d p P B ( p ) P B ( | (cid:126) k − (cid:126) p | )(1 + γ + γ β ) , (2.10)where µ = ˆ (cid:126) p · ( (cid:126) k − (cid:126) p ) / | (cid:126) k − (cid:126) p | , γ = ˆ (cid:126) k · ˆ (cid:126) p , β = ˆ (cid:126) k · ( (cid:126) k − (cid:126) p ) / | (cid:126) k − (cid:126) p | , and Ω denotes the volume with p < k D .The scalar mode has three different source terms: the magnetic energy density, the anisotropic stressand the Lorentz force induced on baryons. Only two of three are independent imposing the energyconservation of the fields. This relation allows a freedom of basis for the initial conditions, namelythe choice of which two quantities are computed directly from the field configuration for the analysis.We choose the energy density and Lorentz force as independent quantities for the compensated mode,deriving the anisotropic stress. For the passive mode instead we consider the anisotropic stress as theindependent quantity, it being the only quantity required. We assume full anticorrelation between theLorentz force and the energy density as predicted to lowest order [155]. The exact solutions to theconvolutions Eq. 2.10 has been derived in [142, 143] and we refer to them.The vector mode is source by the anisotropic stress and the Lorentz force whereas the tensorperturbations are sourced solely by the tensor projection of the anisotropic pressure. As described in previous sections PMFs source all types of perturbations, with different initial condi-tions. Scalar and tensor modes are characterized by both compensated and passive initial conditions.The former is the standard particular solution to the inhomogeneous Einstein-Boltzmann equationssourced by PMFs. The term “compensated” refers to the compensation between the energy densityof the relativistic component of the fluid and the magnetic energy density, and to the compensationbetween the neutrino anisotropic stress and the magnetic fields’s one [142–144]. The latter, the pas-sive modes, derive from the homogeneous solutions before neutrino decoupling when PMFs providethe only anisotropic stress and source both scalar and tensor pre-neutrino-decoupling modes. Both– 4 –odes are suppressed after the neutrino decoupling thanks to the neutrino compensation describedabove but a trace of their existence remains as relic inflationary-like modes whose primordial powerspectrum is provided by the matching of initial conditions at the decoupling Eq.2.10 [144–146] (andis dependent on PMFs anisotropic stress Fourier spectrum). Contrary to scalar and tensors, the vectormodes excite only compensated initial conditions. We use an upgrade, described in the next section, of the camb code [165] extension that in-cludes all PMFs gravitational contributions developed in [79, 142, 143, 155, 157] to predict the an-gular power spectra of magnetically-induced perturbations. In Fig. 1 we show the different modes indifferent colours for a value of the field amplitude and spectral index of 3 nGauss and almost scaleinvariant spectrum ( n B = − . n B > − / k n B + for lower indices. Thisbehaviour is presented in Fig. 2 where are shown the temperature angular power spectra of vectorand passive tensor modes for different spectral indices. The different gravitational contributions of PMFs on the Einstein-Boltzmann equations have beenimplemented into camb [165] and cosmomc [166] and have been used in our previous analyses[79, 142, 143, 155, 157]. The magnetic part of the code is based on numerical fits to the largescale part (the only relevant for CMB data) of the analytic solutions to the convolution integrals inEq.2.10 [142, 143] that in their original form involve a large number of hypergeometric functions.The previous fits were computed using a grid for the full spectra expressions and were derived byfitting sets of parameters for each grid element. The fits used in the current work have been upgradedwith a new approach which introduces a multiparameter sampling not of the full formula, like inthe previous ones, but for each coefficient of the expression which is then fitted for the involvedparameters. This new method allows to correctly sample the regions near the multiple poles of thecoefficients of the expressions that in the previous analyses had a lower accuracy because of thefull single formula parametric fit. We have increased the number of fits computed reducing thespectral indices grid spacing for the sampling, increasing the accuracy over the whole range explored.Applying this new sampling technique the new analytical fits used in this work represent the analyticsolutions to the convolution integrals with a precison of one part over 10 for negative indices and10 for positive indices. The new fits have also lead to an increased numerical stability which addsto the generally improved numerical accuracy of the 2018 camb version. This new precision hasallowed to increase the range of multipoles considered for the passive tensor magnetically-induced A third kind of initial condition has been proposed, called the inflationary mode [147, 148], this is a mode character-izing only the inflationary produced fields and although with a spectral shape similar to the passive case the characteristicof the mode strongly depend on the coupling choice. Since we want to maintain the maximum generality concerning thepossible origin of the fields we are not considering this type of initial condition in the current analysis and leave it for afuture study. Two other different implementations based on similar assumptions but different numerical settings are done in [145,146, 160]. – 5 – ‘ − − − − − ‘ ( ‘ + ) C ‘ / π (cid:2) µ K (cid:3) TT ‘ − − − − − ‘ ( ‘ + ) C ‘ / π (cid:2) µ K (cid:3) EE ‘ − − − − − ‘ ( ‘ + ) C ‘ / π (cid:2) µ K (cid:3) TE ‘ − − − − − ‘ ( ‘ + ) C ‘ / π (cid:2) µ K (cid:3) BB Figure 1 . Magnetically-induced CMB anisotropies angular power spectra in temperature [upper left], E-mode po-larization [upper right], T-E cross correlation [lower left]. The colors represent: compensated scalar-pink, vector-blue,tensor-orange, passive scalar-yellow, tensor-green; black is the primary CMB. The B-mode polarization [lower right] colorcode is analogous but in this case black solid and dotted lines represent primary tensor modes for tensor to scalar ratios ofr=0.1 and 0.01 and the dashed line represents the lensing B-mode from primary scalar perturbations. The field amplitudeis 3 nG and the spectral index is the almost scale-invariant. mode to (cid:96) = ‘ − − − − ‘ ( ‘ + ) C ‘ / π (cid:2) µ K (cid:3) TT ‘ − − − − ‘ ( ‘ + ) C ‘ / π (cid:2) µ K (cid:3) TT Figure 2 . Magnetically-induced CMB anisotropies angular power spectra in temperature for vector [left] and passivetensor [right]. The colors represent the different spectral indices, darker corresponds to bigger indices, starting from 2,whereas lighter represent negative indices following the shading, ending in quasi scale invariant.
We perform a Bayesian analysis deriving the probability distributions of cosmological parametersplus the two parameters characterizing the PMFs configuration, i.e. the amplitude on the Mpc scaleand the spectral index . We perform the analysis through a dedicated extension of the Markov ChainMonteCarlo sampler cosmomc [166]. We vary the standard cosmological parameters: baryon den-sity ω b = Ω b h , the cold dark matter density ω c = Ω c h , the reionization optical depth τ , the acousticscale θ , and the primordial power spectrum amplitude ln(10 A S ) and spectral index n S . The magneticparameters are indicated by B and n B and are varied with flat priors [0,10] nG and [-2.9,3] re-spectively. We restrict our anaylsis to three massless neutrinos, since the small neutrino mass allowedby current observations leads to a small effect [146] without inserting additional degeneracies. Wesample the posterior using the Metropolis-Hastings algorithm [167] imposing the Gelman-Rubin con-vergence criterion to [168] of R − < .
01. We consider lensing effect on only primary anisotropies,since modelling the contribution of PMFs to the large scale structures would require non-linear MHDtreatments currently not available in the context of Einstein-Boltzmann codes. We therefore neglectthe impact of PMFs on the lensing effect and viceversa the lensing of magnetically induced angularpower spectra.
We first analyse the Planck 2018 data release using the combination of the high multipole baselinelikelihood in temperature and polarization (Plik TT,TE,EE), the low-multipole Commander likeli-hood in temperature (lowl) and the new cross spectra 100x143 GHz based likelihood in polarization(LowE) based on the first release of the High Frequency Instrument maps with the SROLL algorithm[163, 169]. To complete the Planck baseline likelihood, in addition to the primary anisotropies weinclude also the Planck lensing likelihood [170]. The baseline Plik TT,TE,EE+ lowl+LowE+lensinglensing provides a 95% C.L. on the amplitude of the fields on the Mpc scale: B < . In case of real data likelihood we add also the nuisance’s parameters – 7 –arameter/Data Planck 2018 Planck 2018+BK15 Planck 2018 + BK15+SPT Pol B [nG] <3.4 <3.5 <2.8 n B <-0.2 <-0.3 <-0.6 Table 1 . Constraints on the PMFs amplitude with CMB data marginalizing over the spectral index marginalized 95% C.L. n B < − .
21. We stress that the marginalized posterior probability for thespectral index only reflects the lower amplitude allowed for positive spectral indices compared tonegative ones. Figure 3 shows the comparison of the marginalized posteriors for parameters includ-ing PMFs with the standard Λ CDM. We note how the presence of PMFs does not significantly affectthe standard Λ CDM parameters: the induced shift in the Hubble parameter is simply given by theassumption of no neutrino mass with respect to the case of the minimal neutrino mass of Planckbaseline results for Λ CDM. Ω b h P / P m a x Ω c h ln(10 A s ) n s θ MC
66 67 68 69 70 H τ P / P m a x θ MC B Mpc − − n B D M( z ∗ ) / Gpc r ∗ LCDM+PMFs LCDM
Figure 3 . Marginalized posterior of the Planck 2018 results compared with the standard cosmological model.
We now complement Planck 2018 with B-mode data from ground based high sensitivity polarizationexperiments such as BICEP/Keck array [171] and South Pole Telescope [172].The inclusion of the B-mode polarization on large and intermediate angular scales from BI-CEP/Keck 2015 data (hence BK15) modifies the posterior marginalized probability distribution al-lowing for a slighter larger signal from PMFs. The limit in this case is B pc < . (cid:96) of 500. In Fig.4 we show the cor-relation of the magnetic field amplitude with the spectral index and a set of standard parameters andin particular the scalar spectral index and the amplitude of scalar perturbations. Beside the degener-acy between the amplitude and the spectral index we do not have any relevant degeneracy with thestandard cosmological parameters. In Table 1 we present the results for the magnetic parameters.As in previous works [79] we complement the previous results obtained allowing the spectralindex to vary with the constraints on the amplitude for physical fixed values for the spectral index. In– 8 – B Mpc n B B Mpc n s n B n s B Mpc l n ( A s ) Planck 2018 Planck 2018 + BK15 Planck 2018 + BK15 + SPT
Figure 4 . Two-dimensional marginalized posterior distributions for the magnetic parameters, first panel, and the compar-ison of the PMFs amplitude with some of the standard parameters, the scalar spectral index, the σ and finally the Hubbleconstant. particular we focus on the minumum allowed spectral index for causally-generated post-inflationaryfields, the case n B = B < B < n B = − .
9, this represents possible PMFs generated during inflation, although inflationarygeneration mechanisms are not limited to infrared indices but span the entire range from negative topositive depending on the specific type of mechanism considered. This case is particularly interest-ing because of the shape of the tensor passive magnetically-induced mode, as shown in Fig.1, thisis slightly similar to the angular power spectrum of primary tensor modes and may represent a nui-sance for the detection for primordial gravitational waves from inflation. In this case the bound onthe amplitude of the fields is given by B < . B = . + . − . nG when adding the SPT data, with a marginal detection which is not confirmedat 99% C.L. B < . n B = B < . B < . r < .
06 at 95% C.L. and B < . r < .
06 at 95% C.L., we note that thecase marginalized on the spectral index of the PMFs does not show a significant correlation betweenthe magnetic parameters and the tensor to scalar ratio. We perform the same analysis for the almostscale invariant case whose spectral shape is limited to be very similar to the primary tensor modes.The constraints on the two amplitudes for the cases without and with SPT are B < . r < .
06 at 95% C.L. and B < . r < .
055 at 95% C.L.. In the scale invariant casethere is a slight correlation between the amplitude and the tensor to scalar ratio but for the current in-struments sensitivity is not relevant for the results, the tensor to scalar ratio limit is almost unchangedwith respect to results without the PMFs contribution. In Fig.5 and Fig.6 we show a summary of thedifferent constraints. – 9 – B Mpc n B B Mpc r n B r Planck 2018 + BK15Planck 2018 + BK15 + SPT B Mpc r Planck 2018 + BK15Planck 2018 + BK15 + SPT
Figure 5 . Contour plot of the magnetic fields amplitude and spectral index on the left, the correlation between the PMFsamplitude and the tensor to scalar ratio in the middle and with the spectral index on the right. Far right panel shows thecorrelation between the fields amplitude and the tensor to scalar ratio for the almost scale-invariant case. B Mpc
Planck 2018 + BK15.. + SPTPlanck 2018 + BK15 n B = − . .. +SPT n B = − . B Mpc r Planck 2018 + BK15..+ SPTPlanck 2018 + BK15 n B = − . ..+SPT n B = − . Figure 6 . Marginalized posterior distribution of the magnetic fields amplitude for different data combination and settingsof the field for the LCDM+PMF case on the left and jointly with the tensor to scalar ratio for LCDM+r+PMF on the right.
After presenting the Planck 2018 update results alone and in combination with ground based experi-ments we proceed with the forecasts for future experiments.
We consider the LiteBIRD mission [173] among the various proposals for a fourth generationspace mission dedicated to CMB anisotropy measurements [173–175]. LiteBIRD is a proposal for asatellite downselected by the Japanese space agency JAXA, with the contributions of USA, Europeand Canada, with planned launch in 2027. Its main goal is the study and characterization of the CMBpolarization at large and intemediate angular scales. It has fifteen frequency channels spanning from40 GHz to 402 GHz, a range optimized for the foreground removal and it will have a payload-focalplane design to provide very high sentivity measurements of the E and B-mode polarization up to amultipole of several hundreds. We summarize the instrumental characteristics of the central frequencychannels we used for the analyses in Table 2 [173] which updates previous configurations [176, 177].We consider the highest and lowest frequency channels as foreground tracers and we assume they willbe used to clean the central ones. We assume only instrumental noise and ideal foreground cleaningwith a inverse-Wishart likelihood on mock datasets [177]. This choice represents a very optimisticsetting and therefore provide a snapshot of the limits on primordial magnetic fields which can bereached by future experiments. We consider the lensing signal as additional noise and simply add a http://litebird.jp/eng/ – 10 –requency GHz FHWM [arcmin] T Sensibility[ µ K arcmin] P Sensibility[ µ K arcmin]78 39 9.56 13.589 35 8.27 11.7100 29 6.50 9.2119 25 5.37 7.6140 23 4.17 5.9166 21 4.60 6.5195 20 4.10 5.8 Table 2 . LiteBIRD instrumental characteristics [173]
Parameter/Data LiteBIRD TE LiteBIRD TEB LiteBIRD TEB + primordial tensors B [nG] <3.8 <2.7 <2.7 n B <-0.26 <-0.19 <-0.16 Table 3 . Constraints on the PMFs amplitude with CMB data marginalizing over the spectral index lensing fiducial, derived with the same cosmological parameters as the primary CMB fiducial, to thenoise bias. We will also investigate the impact on the results of delensing possibilities.In Table 3 we present different cases for the LiteBIRD study. Being cosmic variance limited inthe E-mode polarization we would like to investigate the LiteBIRD capabilities to constraint magneticfields amplitude using only temperature and E-modes. When only temperature and E-mode polariza-tion are included we obtain B < . B < . B < . r < . n B = n B ∼ − . B < B < . The launch of LiteBIRD is planned for 2027 and will carry out the fourth generation of CMB obser-vations from space. In the timescale leading to the launch and also during the LiteBIRD observationaltime advancements from ground based experiments are expected. In particular, we refer to the future– 11 – B Mpc n B B Mpc r n B r LiteBIRD TEB+tensors LiteBIRD TEB LiteBIRD TE
Figure 7 . Marginalized posterior distribution of the magnetic fields amplitude for different data combination and settingsof the field for the LCDM+PMF case, together with the tensor to scalar ratio for LCDM+r+PMF on the right. Note that redand grey curves in the left panel are overimposed since the results do not change sensibly adding the tensor to scalar ratio. ground based experiments Simons Obsevatory [178, 179] and CMB Stage IV [180, 181]. The envi-roment offered from the ground allows for very high sensitivity observations thanks to the possibilityof large focal planes containing tens of thousands of receiver and the possibility to observe in greatdepth small region of the sky. On the other side ground based observatories due to their nature sufferfrom two limitations, namely, sky coverage and frequency range. The former is due to a limited skycoverage and therefore the impossibility to cover the largest angular scales, where most of the primaryinflationary B-mode signal resides, and a larger sampling variance on the intermediate scales. The lat-ter is due to the atmosphere which is opaque for frequencies larger than 300 GHz limiting the higher“foregrounds monitor” frequencies which are crucial for dust and cosmic infrared background con-taminations. The contemporaneity of a space borne experiment dedicated to the large angular scalesand ground based observatories on small angular scales represents a unique opportunity to exploit thepotential of CMB polarization in the context of cosmic magnetism. For this reason we investigate inthe following subsections the case of the combination of LiteBIRD with ground based future experi-ments. In order mitigate effects due to the cross correlation between maps we consider LiteBIRD upto a multipole of 600 and ground based experiments for higher multipoles. At multipoles around 600the signal to noise of LiteBIRD and ground based experiments are comparable.
We study the combined analysis of LiteBIRD and the Simons Observatory that represents the nearfuture of ground based experiments. For Simons Observatory we consider the central frequency chan-nels as in Table.4 with a wide survey-like sky fraction of 40%. We analise several possible PMFs con-figurations for the combination LiteBIRD+SO. The standard case in which the PMFs are marginalizedover the spectral index and no primordial tensor modes are considered provides B < . n B < − .
6. We note a significant improvement with respect to current results reported in previoussection. Another aspect that we to stress is that the polarization sensitivity is high enough to provideconstraints even without considering the temperature. This aspect is particularly promising becausethe temperature signal is contaminated on small angular scales by astrophysical signals and secondaryanisotropies, and has been shown in [79, 157] that these may be correlated with the PMFs contribu-tions and lead to contamination of the PMFs limits. Therefore we tested a case where we exclude thetemperature from the analysis using only E-mode, the T-E cross correlation and B-mode polarizationobtaining B < . B < ‘ − − − − − ‘ ( ‘ + ) C BB ‘ / π (cid:2) µ K (cid:3) n B = 2 n B = − n B = − n B = − . n B = 1 LensingDelensing 80%Delensing 40%
Figure 8 . B-mode angular power spectrum for the magnetically-induced modes compared with delensing signal.
In fact we tested also an extreme value for the delensing that assume to be able to delens upto the 80% of the sky with the PMFs configuration with marginalization over the spectral indexobtaining the same results of 1 nG upper limit.The situation changes when we consider specific PMFs configuration, in particular for the al-most scale invariant case without delensing we obtain B < .
55 nG at 95% CL which representsa good improvement with respect to current data mainly provided by the large scale impact of thealmost scale invariant configuration. If we consider the delensing for this specific case as describedbefore we obtain B < .
50 nG and B < .
44 nG at 95% CL for respectively 40% and 80%– 13 –requency GHz FHWM [arcmin] T Sensibility[ µ K arcmin] P Sensibility[ µ K arcmin]93 2.2 5.8 8.2145 1.4 6.3 8.9 Table 4 . Simons Observatory instrumental characteristics [Simons]
Model Baseline n B = − . n B = B [nG] < 1.0 < 0.055 < 5 × − < 1.0 <0.99 <0.84 r - - - <0.0004 r = . ± . r = . ± . Table 5 . Constraints on the PMFs amplitude with the combination LiteBIRD+SO. All limits provided are at95% CL.
Frequency GHz FHWM [arcmin] T Sensibility[ µ K arcmin] P Sensibility[ µ K arcmin]85 4.9 1.77 2.595 4.4 1.41 2.0145 2.9 1.84 2.6155 2.7 1.91 2.7215 1.7 4.74 6.7 Table 6 . Stage IV instrumental characteristics [182] delensing showing that for the infrared tilted spectra the delensing becomes effective in reducing theupper limit.For PMFs configuration which allows a generation also through the causal mechanism channel, n B = B < . B < r < . r = . R model of inflation [184], that is at the centerof the region allowed by current data [185] and can be detected with an high significance by theLiteBIRD experiment. For this case we obtain B < .
99 nG at 95% CL and r = . ± . B < .
84 nG at 95% CL and r = . ± . B = . ± . .0 1.5 3.0 4.5 6.0 B Mpc − − n B LiteBIRD TELiteBIRD TEBLiteBIRD+SOLiteBIRD+SO+delensing 20%LiteBIRD+SO+delensing 40% B Mpc r LiteBIRD+SO r=0 0.0 0.4 0.8 1.2 B Mpc r LiteBIRD+SO r=0.004LiteBIRD+SO r=0.004 n B = − Figure 9 . Contour plot of the magnetic fields amplitude and spectral index on the left and tensor to scalar ratio on theright. Left panel shows the comparison between Planck 2018+BICEP KECK and the combination with SPT. The middlepanel considers the case with a null tensor to scalar ratio whereas the right panel the case where it is assumed an r=0.004
We investigate some of the main forecasts also for the combination of LiteBIRD and S4 to get anoverview of possible scenarios in a farther future. For the instrumental characteristics for a stage IVexperiment we use [182] as reported in Table 6, once combined in inverse noise weighting we obtainan overall few microK sensitivity and 1-2’ FWHM as adopted in S4 science book [180, 181]. Weagain assume a wide survey with the 40% of the sky. For the baseline case of PMFs with the marginal-ization on the spectral index we obtain B < .
63 nG with n B < − .
63 showing an improvementwith respect to Simons Observatory thanks to the improved sensitivity. For the conservative delensingoptions considered we obtain again similar results to the unlensed case B < .
61 nG for 20%delensing, B < .
59 nG for 40% with only a minimal improvement for this case. For the spe-cific PMFs configurations we have that the almost scale invariant is constrained to B < .
49 nGwithout delensing, B < .
45 nG for 40% delensing and B < .
40 nG for the extreme 80%delensing case with again an improvement due to the delensing for the infrared indices configurations.We tested again also the minimal spectral index for causal generated PMFs configurations which islimited to B < . The cosmological magnetic fields observed on large scales might have their roots in primordial seedsgenerated in the early Universe.The generation mechanisms are several and range from a first orderphase transition to the breaking of conformal invariance during inflation opening a new observa-tional window on the physics of the early Universe. PMFs being a fully relativistic component withanisotropic pressure diffuse on cosmological scales have an impact on the entire history of the Uni-verse with several observational probes with an always increasing diversity thanks to the progress ofcosmological observations.In this paper we have focused on one of the best known effects of PMFs: the gravitational ef-fect on CMB anisotropies. We have upgraded our previous treatments using a completely new fittingtechnique for the exact solutions of the source terms of magnetically-induced perturbations reachingan unprecedented accuracy that improves theoretical predictions for the angular power spectra forcurrent and future CMB experiments. Thanks to the increased numerical stability due to both im-proved fits and improved basic camb version we could also increase the maximum multipole of the– 15 –ensor modes to 2000 taking more advantage of the high multipole part of the passive tensor modefor positive spectral indices.We have updated the constraints on PMFs to Planck 2018 data. Using the Planck 2018 baselinealone we obtain B < . B < . B < . n B = B < B = . + . − . nG at 95% C.L. The addition of primordial inflationary tensor mode doesnot impact the results, and both the constraints on the PMFs amplitude and tensor to scalar ratioare unchanged, showing a lack of correlation of the two signals for the precision of current data.A non-vanishing correlation is present for the almost scale invariant case due to the similarity ofcontributions on large angular scales but does not affect our results significantly.We have then performed the forecasts for future experiments. In particular, we have consideredthe latest LiteBIRD instrumental configuration [173] showing how forecasted constraints are at thePlanck level, i.e. B < . B < . below nanoGauss level. The effect ofdelensing is mitigated when the index of PMFs is allow to vary, but is important for specific spectralindices. The constraints on the amplitude with n B = n B ≥
2, improves the constraints by orders of magnitude, reaching the sub-picoGauss level for thefuture ground based experiments. This results will squeeze the allowed observational window givenby the upper limits from the CMB and the lower limits from gamma-ray observations. The almostscale invariant configuration is interesting for two reasons. On one side LiteBIRD will provide muchtighter constraints than current data, and on the other side this type of PMF contribution shows acorrelation with the primordial gravitational wave signal. Therefore, it represents a possible sourceof confusion to be taken in account for future high precision data.Our analysis considered semi-idealistic conditions with a perfect cleaning of the foregroundsbut ongoing studies for future experiments (e.g.[186]) show that there might still be some residualcontamination on large angular scales even after cleaning. Previous studies on small angular scaleshave already shown how the analogous residuals degrade the constraints on PMFs amplitude [79,157], the update of these studies on large angular scale polarization in the context future experimentsis in progress. It will be interesting to study the status and perspectives for another key signature from– 16 –MFs on CMB polarization as the one from the post-recombination heating.
Acknowledgments
DP and FF acknowledge financial support by ASI Grant 2016-24-H.0. This research used computa-tional resources provided by INAF OAS Bologna and by CINECA under the agreement with INFN.
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