The primordial magnetic field in our cosmic backyard

We reconstruct the 3D structure of magnetic fields, which were seeded by density perturbations during the radiation dominated epoch of the Universe and later on were evolved by structure formation. To achieve this goal, we rely on three dimensional initial density fields inferred from the 2M++ galaxy compilation via the Bayesian $\texttt{BORG}$ algorithm. Using those, we estimate the magnetogenesis by the so called Harrison mechanism. This effect produced magnetic fields exploiting the different photon drag on electrons and ions in vortical motions, which are exited due to second order perturbation effects in the Early Universe. Subsequently we study the evolution of these seed fields through the non-linear cosmic structure formation by virtue of a MHD simulation to obtain a 3D estimate for the structure of this primordial magnetic field component today. At recombination we obtain a reliable lower limit on the large scale magnetic field strength around $10^{-23} \mathrm{G}$, with a power spectrum peaking at about $ 2\, \mathrm{Mpc}^{-1}h$ in comoving scales. At present we expect this evolved primordial field to have strengthts above $\approx 10^{-27}\, \mathrm{G}$ and $\approx 10^{-29}\, \mathrm{G}$ in clusters of galaxies and voids, respectively. We also calculate the corresponding Faraday rotation measure map and show the magnetic field morphology and strength for specific objects of the Local Universe.


I. INTRODUCTION
Inference of primordial magnetic fields opens a unique window into the Early Universe between inflation and recombination. Although a variety of different astrophysical processes may generate magnetic fields, the primordial magnetic seed may very well be the origin of observed magnetic fields in galaxies and clusters. Primordial magnetic fields are a viable candidate for the 10 −16 G to ≈ 10 −15 G [8,27,34] fields expected in cosmic voids due to the non-observation of GeV emission from TeV blazars among other explanations [29]. In any case, they represent by definition the minimal amount of magnetic fields present in the Universe. Literature provides a variety of very diverse effects for the generation of primordial magnetic fields coherent on a large range of scales. A incomplete list of possible magnetogenesis effects may include mechanisms at the end of inflation (e.g. during the reheating phase or exploiting the electroweak phase transitions), during QCD phase transitions or effects that make use of speculative non-standard model physics such as gravitational coupling of the gauge potential or string theory effects. For a further discussion we refer the reader to comprehensive review articles [10,20,33,40].
A more conservative Ansatz solely relying on the assumption of a ΛCDM Universe and conventional plasma physics was proposed by Matarrese et al. [25]. This approach is based on a mechanism initially proposed by Harrison [16] in 1970. During the later phases of the radiation dominated epoch of the Universe a two fluid battery effect occurred between the proton fluid and the tightly coupled electron-photon fluid. The densities ρ (α) , with α ∈ {m, γ} for baryons and the electron-photon fluid respectively, of the two components scale with the scale factor a(t) as ρ (m) ∼ a(t) −3 and ρ (γ) ∼ a(t) −4 , respectively. Therefore, the separately conserved angular momenta L (α) ∼ ρ (α) ω (α) r 5 in a rotational setup with radius r(t) ∼ a(t) requires the angular velocities ω (α) to depend on a(t) with ω (m) ∼ a −2 and ω (γ) ∼ a −1 , respectively. In other words, protons spin down faster than electrons, as the latter are carried by the still dominant photons. This difference in rotation then leads to currents that induce magnetic fields [16]. The necessary vortical motion of both proton and radiation fluid are caused by effects that can be expressed as second order perturbations of the fluid equation [25].
The recent progress on the inference of the actual 3D realization of the large-scale dark matter structure and its formation history in the Local Universe and the fact that the Harrison mechanism is solely founded on well established plasma physics allows us to calculate the seed magnetic fields that had to be generated by this effect as well as their present day morphology and strength. Since these fields have to exist today in combination with fields of other sources, we are therefore able to provide credible lower bounds on the primordial magnetic field strength in the Nearby Universe. We structure this article as following: Section II first describes the outcome of the Matarrese paper [25] and then presents the computational steps that take us from dark matter over densities to magnetic fields. Section III gives a short overview on the dark matter density reconstruction used in this work. Section V provides the intermediate results on magnetic field configuration and power spectrum at radiation matter equality. Section IV shows the results of the subsequent MHD simulation. Section VI contains a summary and an outlook on potential improvements.

II. THEORY
This paper strongly relies on the theoretical framework outlined by Matarrese et al. [25]. This approach describes primordial density perturbations in the Early Universe before the recombination epoch as sources of magnetic fields via the Harrison mechanism [16].
In the first part of this section we will summarize their assumptions and results. The second part describes the implemented reconstruction approach to translate our knowledge on dark matter over-densities into magnetic field estimates, first described by Dorn [9].

A. Basics
All calculations here are performed using the standard ΛCDM model assuming the cosmological parameters described in Tab.1 following the 2015 results of the Planck mission [31]. Following [25] we further assume that the dominant constituents of the Universe in the relevant time frame behave as perfect fluids of dark matter, electrons, protons and photons. All equations and calculations are performed in Poisson gauge with the following line element: a is the scale factor depending on conformal time η. φ and ψ are the Bardeen potentials, χ i and χ ij are vector and tensor perturbations, respectively. The perfect fluid assumption results in a vanishing anisotropic stress tensor, which yields φ = ψ ≡ ϕ to first order in perturbation theory.
As perfect fluids are assumed, the energy momentum tensor simplifies to with P (α) the pressure, ρ α the density and u µ α the bulk velocity for each component α. Pressure and density of a component are related via an equation of state We define the energy over-density with respect to the mean energy density ρ (α) of a component as All quantities (δ (α) , u (α) , ϕ, χ i , χ ij ) can now be perturbed up to second order and related via their respective momentum equations including source terms to describe interactions. The coupling between the baryonic and radiation fluids is assessed by a tight coupling approximation to zeroth order which implies v i . This sets Thomson and Coulomb interaction terms to zero in this order. The curl of the momentum equations for the proton and photon fluids gives evolution equations for the respective vorticities. The photon vorticity equation has a source term in second order which, combined with Maxwells equations and the proton vorticity equation, gives a source term for magnetic fields. Matarrese et al. [25] expressed this equation in terms of the first order scalar perturbations of the metric ϕ (1) which in the Newtonian limit gives the gravitational potential. By assuming negligible resistivity and therefore omitting magnetic diffusion terms they get [9,25] for the magnetic field at time η, assuming some initial field B I at time η I . A prime denotes derivation by conformal time. m p is the proton mass, e the elementary charge, a the scale factor and H = a /a the comoving Hubble constant. The second last term was omitted by Mataresse in their analysis on the correlation structure of the field. The initial field B I can safely be set to zero due to the a 2 I /a 2 factor. In the following we discuss the implementation of equation (5).

B. Implementation
We now need to calculate ϕ and its spatial and temporal derivatives with respect to conformal time. We begin our calculation by translating the CDM density perturbations measured 1 shortly before the last scattering surface δ cdm (z ≈ 1000) into primordial initial conditions deep inside the radiation epoch at z p . Magnetogenesis can only take place on scales which have entered the cosmic horizon at the corresponding epoch. Therefore it makes sense to make this the criterion for z p . We know that the horizon condition can be roughly written as with the conformal time measured in units of one over length, the speed of light set to one and with η indicating conformal time. Knowing that the smallest scales of the grid correspond to k 256 ≈ 2.39 h Mpc −1 and k 512 ≈ 4.78 h Mpc −1 with respective grid sizes of 256 and 512 points respectively, we know that the initial times must be on the order of the grid resolution η 256 ≈ 0.42 Mpc h −1 and η 512 ≈ 0.21 Mpc h −1 with the speed of light set to one. The equivalent redshifts are z 256 ≈ 9.7 · 10 5 and z 512 ≈ 1.9 · 10 6 . Finally, z p = 10 7 was adopted in this work, as it safely satisfies the aforementioned condition. We obtain δ cdm (z p ) by using linear cosmological transfer functions and calculate the total energy over-densities δ tot (z p ) from it, The 4/3 factor comes from the adiabatic super-horizon solutions for the density perturbations (see e.g. [24]). The transfer functions T (k, z p , z rec ) in Poissonian gauge were calculated by the CLASS code [3]. They contain all relevant physics at linear order. The peculiar gravitational potential ϕ(k, η) in the radiation epoch (implying ω cdm ≈ 1/3) evolves as in Fourier space with initial conditions ϕ 0 at redshift z = 10 7 , j 1 is the spherical Bessel function of first order and Furthermore, the linearised Einstein equations relate the total energy perturbations to the potential by With this equation we can calculate the initial ϕ 0 in Fourier representation as 1 For further comments on the data see section III and [19] From there on we can use Eq. (8) again to calculate the potential and its derivatives at any time up to z eq . The potential ϕ will tend to a constant after radiationmatter equality, as pressure becomes negligible. This means that the first and third term in Eq. (5) do not contribute to magnetogenesis from the epoch of radiationmatter equality to recombination. We therefore evaluate these terms at radiation matter equality (z ≈ 3371). From there on, these magnetic field terms are then propagated to recombination at redshift z = 1088 via the induction equation of magneto-hydrodynamics (MHD) (assuming again perfect conductivity): The fluid velocity v is also calculated in first order perturbation theory. The second term of equation (5) contains no time derivative of the potential and is therefore evaluated at recombination (z = 1088).
We illustrate the steps of the calculation in Fig. 2.

C. Simplifications
This calculation contains simplifying assumptions to keep the evolution equation for the magnetic field anaytically solvable. For completeness, those shall be discussed here.
The evolution of the potential via Eq. (8) is performed for a radiation dominated Universe with equation of state (3). The transition to the matter dominated era is modelled in an abrupt way with w = 1 3 before equality and w = 0 afterwards. As the real transition is smooth, scales in the order of the equality horizon maybe affected by the modelling and magnetogenesis may even take place even after recombination. A heuristic modelling via e.g. hyperbolic functions was not performed as the additional time dependence makes the evolution equations for the potential not analytically solvable. This could be incorporated in the model if needed for the price of more contrieved calculations. Related to that, the coupling between electrons and photons is modelled via the tight coupling approximation as mentioned earlier. Thomson scattering is very efficient for scales larger than the mean free path of the photons, which at recombination can be estimated via in comoving scales. As we will see in the next section, our calculation is performed on a ≈ 1.3Mpc h −1 grid. Therefore the tight coupling should ideally be expanded to higher order in case of the Thomson coupling. The above mentioned shortcomings where overcome by more detailed studies on the the generation of primordial magnetic fields via the Harrison mechanism, which have been conducted by several authors in the past 15 years.
Gopal et al. [14] showed that differences between electron and photon velocities lead to source terms for magnetic field generation. Saga et al. [32] and Fenu et al. [12] have refined this calculation by including anisotropic stresses stemming from the imperfect Thomson coupling. A similar calculation was done by Christopherson et al. [6] first on the generation of vorticity in second order and later on the subsequent magnetic field generation [26] via the introduction of non-adiabatic pressure terms.
All of these approaches give source terms on which the Harrison mechanism can operate. The corresponding equations can in principle all be solved given suitable initial conditions. Only the approach shown above, however, gives an analytically integrable expression. All other models require the iterative solution of the respective magnetic field evolution equation in combination with all relevant quantities throughout the whole plasma era of the Universe up until recombination. This requires considerable computational effort to be implemented on the 512 3 voxel grid used in this work. For this reason, and as the field strengths which were found in [12-14, 26, 32] are comparable to the ones found by Mataresse [25], we find the above mentioned simplifications acceptable. Fidler et al. [13] show that the exact treatment of the Thomson coupling gives rise to significant magnetogenesis even after last scattering, which highlights that a better modelling around recombination would be desirable, given that one can shoulder the resulting computational complications.

III. DATA
This work builds upon three dimensional dark matter density fields previously inferred from the 2M++ galaxy compilation [22] via the BORG algorithm [23]. The BORG algorithm is a full scale Bayesian inference framework aiming at the analysis of the linear and mildly-non-linear regime of the cosmic large scale structure (LSS) [18,19]. In particular it performs dynamical LSS inference from galaxy redshift surveys employing a second order Lagrangian perturbation model. As such the BORG algorithm naturally accounts for the filamentary structure of the cosmic web typically associated to higher order statistics as induced by non-linear gravitational structure formation processes. A particular feature, relevant to this work, is the ability of the BORG algorithm to infer Lagrangian initial conditions from present observations of the galaxy distribution [18,19,23]. More specifically the algorithm explores a LSS posterior distribution consisting of a Gaussian prior for the initial density field at a initial scale factor of a = 0.001 linked to a Poissonian likelihood model of galaxy formation at redshift z = 0 via a second order Lagrangian perturbation theory (2LPT) model [for details see [18,19,23]], that is conditioned to the 2M++ galaxy compilation [23]. Besides typical observational systematics and uncertainties, such as survey geometries, selection functions and noise this algorithm further accounts for luminosity dependent galaxy bias and performs automatic noise calibration [23]. The BORG algorithm accounts for all joint and correlated uncertainties in inferred quantities by performing a Markov Monte Carlo chain in multi-million dimensional parameter spaces via an efficient implementation of a Hamiltonian Monte Carlo sampler [18]. As a result the algorithm provides a numerical representation of the LSS posterior in the form of data constrained realizations of the present three dimensional dark matter distribution and corresponding initial conditions from which it formed. It is important to remark that each individual Markov sample qualifies for a plausible realisation of the LSS. Each sample of the dark matter distribution consists of a box with 256 3 grid points and 677.7 Mpc h −1 edge length, resulting in a resolution of approximately 2.5 Mpc h −1 . For one sample of BORG we increase the resolution of the grid to 512 3 by augmenting the large scale modes with random fluctuations consistent with the known dark matter power spectrum. This sample is then propagated into todays configuration via a MHD simulation as explained in the following section. As described above we now apply the Harrison mechanism on data constrained initial conditions of the Nearby Universe.

IV. MHD SIMULATIONS
The MHD computation is started from the magnetic field generated at z = 1088 and is evolved to z = 0 using the cosmological code ENZO [4]. ENZO is a grid based code that follows the dynamics of dark matter with a particle-mesh N-body method, and a combination of several possible shock-capturing Riemann solvers to evolve the gas component [4]. The MHD method employed in this paper is the Dedner "cleaning" method [7], which makes use of hyperbolic divergence cleaning to keep the (spurious) divergence of the magnetic field as low as possible during the computation. The magnetic fluxes across the cells are computed with a piecewise linear interpolation method and the fluxes are combined with a Lax-Friedrichs Riemann solver, with a time integration based on the total variation diminishing second order Runge-Kutta scheme [39]. Thanks to the capabilities of ENZO of selectively refining interesting patches in the domain at higher resolution, we used adaptive mesh refinement (AMR) to selectively increase the dynamical resolution in the formation region of galaxy clusters and groups, which is necessary to properly resolve structure formation and overcome the effect of magnetic field dissipation in converging flows at low resolution [35].
In detail, we apply AMR only in the innermost (120 Mpc h −1 ) 3 region of the simulation, centred on the Milky Way location, and allowed for 5 levels of refinement (by increasing the resolution of a factor 2 at each level, therefore up to a 2 5 = 32 refinement) whenever the local gas/dark matter density exceeded the mean density of surrounding cells by a factor of 3; the procedure is recursively repeated at each AMR level. This ensures that the magnetic field evolution in the innermost clusters regions is typically followed with a spatial resolution of 61 − 122 kpc h −1 (comoving) within the innermost AMR region of our volume. To confirm the consistency of our result, we show a slice through the gas density resulting from the simulation in Fig. 3. This plot indicates that we reproduce the large scale structure consistently with observations.

V. RECONSTRUCTING PRIMORDIAL MAGNETIC FIELDS
We will present the results of our work in two steps. First we focus on the statistical properties of the field at recombination. By applying the procedure described in the previous sections to an ensemble of data constrained initial conditions we can propagate observational uncertainties of the matter distribution as traced by the 2M++ survey to the derived magnetic fields. In doing so we arrive at an ensemble of initial magnetic fields which constitutes a numerical description of the magnetic field posterior distribution at redshift z = 1088 conditional on 2M++ galaxy data. The goal here is to show how these uncertainties translate onto the calculated primor- dial magnetic field and to give scale dependent estimates on correlations and field strengths at this epoch.
The second part will show the results after the MHD run at redshift z = 0. Here we will also turn our face on one particular realisation of the primordial magnetic field. We will show the large scale primordial magnetic field of some clusters of galaxies as well as the field in the close proximity to Earth. The resulting fields are available for download at 2 .
A. Recombination

Means and Variances
To illustrate the uncertainties we show slice plots of the input data and the resulting magnetic field strength at recombination in Figs. 4 and 5. All plots are slices through the (677.7 Mpc h −1 ) 3 cube. The first plot shows the field resulting form one particular sample of the BORG algorithm. The comoving root mean square (rms) field strength is around 10 −23 G. The uncertainties are rather large compared to the mean. This is a consequence of he sparse data, which rather constrains the large scales than the small ones. Structures in the field appear to be rather small, typically with Mpc-scale (see Section V A 2). our magnetic field realisations. They show the posterior mean and variance field of the magnetic field strength generated from 351 samples from the BORG posterior distributions. Areas which are highly constrained by data have well distinguishable structures in the mean, and have low uncertainty variance. The outer regions are less constrained, structures which are well visible in one particular sample are averaged out in the mean, and the variance is high.

Power spectra
Information on the correlation structure of a scalar field s(x) can be gained from the corresponding scalar power spectrum defined as: where the asterisk denotes the complex conjugate. In case of a magnetic field B, the statistically isotropic and homogeneous correlation tensor is defined as where the tensor M ij is defined as (see e.g. [11]): The helical part P H (k) is assumed to be zero in this work. The magnetic field power spectra are therefore just the trace component of the magnetic field power spectrum tensor.
As a consistency check we first show the power spectrum of the initial CDM field and the scalar perturbations ϕ through some of the time steps of the algorithm in Figs. 6, 7 and 8. These plots show the averaged spectra from 351 samples from BORG together with the corresponding uncertainties. We also show the magnetic field power power spectrum in Fig. 9.
Despite some deviations on very large scales reflecting the uncertainties mentioned in the previous section, the spectra agree to a very good level with our expectations. These deviations can be noted in our intitial matter fields coming from the BORG algorithm (Fig. 6) and further on in all the other averaged power spectra. We note a clear k −3 dependence in the primordial potential and matter power spectrum corresponding to an approximately unity spectral index as expected for uncorrelated and scale invariant structures [17,41]. We can compare the spectrum with the Planck results as a consistency check, see Tab. 10 and the dashed line in Fig. 7.
The potential power spectrum at matter-radiation equality drops shortly above k = 0.1 Mpc −1 h, indicating the size of the horizon at that time. At small scales, the spectrum shows oscillations in Fourier space, which stem from the functional form of the potential evolution equation in Eq. (8). Physically speaking, these are the Baryon-accoustic oscillations (BAO's) induced by horizon crossing during the radiation epoch. The uncertainties again agree with the initial dark matter spectrum. The resulting power spectrum of the magnetic field is plotted in Fig. 9. It rises for small k-values with approximately k 2 as expected for a uncorrelated source-free vector field [11] and peaks at k peak ≈ 2 · 10 −1 Mpc −1 h. The plot shows little 'bumps' on small scales, which are remnants of the oscillating potential in the radiation epoch. At this point it shall also be noted that in the time frame of our calculation any turbulence due to primordial velocity perturbations is not relevant. In [38] the authors show that given these perturbations the very Early Universe has Reynolds numbers in the range of 10 3 . This then gives the perfect framework for a small-scale dynamo to amplify the magnetic seed fields originating from the Harrison effect to fields with strengths of approximately 10 −15 G, but with typical correlation lengths of the order of parsecs. Given the Mpc resolution of our grid, this is not relevant for this work.

Scale dependent mean field
To give a more intuitive picture of the expected magnetic field strengths, we convolve the magnetic field power spectrum with a Gaussian kernel in position space to get an estimate for B given a scale of reference λ.
The result of this operation is shown in Fig. 11. For scales reaching from 2.65 Mpc h −1 to ≈ 10 Mpc h −1 the magnetic field strength weakly declines and has a typical strength of approximately 10 −23 G. For scales larger than 10 Mpc h −1 , B λ roughly scales as The field strength reaches from 10 −23 G at the smallest scales just over 1 Mpc h −1 to less than 10 −27 G at scales over a 100 Mpc h −1 . This information is of course closely related to the magnetic field power spectrum.
B. Today

Field strength and correlation structure
In Figs. 12 we depict the power spectrum of the magnetic field today for one sample of the BORG posterior. We show the complete spectrum as well as the void power spectrum inferred via negleting dens voxels with gas density ρ > 3 · ρ and the critical filter technique [21], which assumes that the unmasked regions are typical for the whole volume. The BAO signature and most small structures have been destroyed during structure formation, leading to a mostly red spectrum. For the complete spectrum and the voids, most power still lies on scales of about k peak ≈ 10 −1 Mpc −1 h. The morphology of the complete power spectrum is rather similar to the void power spectrum, which is expected, as they compromise the largest volume share of the Universe and calculating a power spectrum is effectively a volume averaging procedure. The decrease at large scales again reflects the solenoidality of magnetic fields (∇ · B = 0) for uncorrelated signals [11], as the large scale structure has a characteristic size and therefore larger scales are not strongly causally connected via gravity.
In Fig. 13 we show the joint probability function of matter density and magnetic field strength. Most of the probability mass lies on rather small densities, with varying magnetic field strengths. Large densities tend to be associated with large magnetic field strength. The lower bound of this plot follows a B ∝ ρ 2 3 proportional-ity, which was already found in previous simulations, e.g: [36]. All in all this leads to the picture that the magnetic field in the low density areas which are relatively little affected by structure formation mostly retain their correlation structure and morphology. After recombination, the field is frozen into the plasma. Therefore the field strength scales with a(t) −2 , explaining the field strengths somewhere around 10 −29 G. Within dense structures the field is at least amplified up to 10 −26.5 G. We underline this view with specific examples in the next chapter.

Field realisations
In Figs.14 and 15 we show structures in the magnetic field and the density field which belong to different morphological features of the Local Universe. We find that the magnetic field strength strongly correlates with the gas density in all of these structures, consistent with a frozen-in behaviour of magnetic fields. In very dense clusters such as Virgo and Perseus-Pisces in Fig. 14 the magnetic field morphology seems to be driven by the infall of matter on the cluster. Of course the simulation is too coarse to correctly cover the structure formation and magnetic field behaviour on small scales within these structures, for this reason any small scale structures in these plots are highly uncertain. Also the maximum magnetic field strength maybe higher, as we cannot resolve any potential dynamo mechanism during structure formation. In underdense regions as depicted in the upper image in Fig. 15, we observe a morphology similar to the initial conditions, with a charcteristic scale of a few Mpc/h. Apart from the aforementioned a(t) −2 dependence of the field strength, the morphology seems to relatively unaffected, which is consistent with our view of a 'frozen' magnetic field. In the lower image of Fig. 15 we show the magnetic field in a slice around our galaxy. The field here is slightly amplified up to field strengths of 10 −28 G, as a slight overdense structure seems to have formed in the region.

Full sky maps
We can use the results of the ENZO simulation to estimate the expected Faraday rotation of linear polarized light under the influence of a magnetic field. Faraday rotation measure (RM) is calculated via Rmax 0 n th B dr (19) in cgs units (see e.g. [28]). It is essentially a line of sight (LOS) integration up to a distance R max over the magnetic field B weighted with the electron number density n th . This can be computed using the publicly available Hammurabi Software [37], which performs the necessary LOS integration over a sphere around the Earth 3 . The result can be seen in Fig. 18. The same software is also able to calculate the dispersion measure of electrons (DM) and the LOS averaged absolute magnetic field strength, shown in Figs. 19 and 16. These maps nicely trace dense structures in the sphere over which we integrated. We also used the output of Hammurabi for the LOS perpendicular components of the magnetic to generate a polarization like plot in Fig. 17, which traces the magnetic field morphology in the sphere. Comparing the plots we see that areas of large RM correspond to large electron densities, as we expect given the linear n th dependence of RM. We also note again that the magnetic field strength and morphology correlates with the density. Of course the expected signal is beyond any chance of measurability, and in the realistic case we expect that the memory of any such tiny seed field within clusters is entirely lost due to the dynamo amplification process [2], which is expected to be much more efficient on scales smaller than what is resolved at our resolution here. The void signal, however, although a few order of magnitudes smaller, may be relatively undisturbed by such processes, at least away from other possible sources of magnetisation, like dwarf galaxies [1].

VI. SUMMARY
We calculated the large scale primordial magnetic field originating from the Harrison effect [16] and second order vorticity generation in the radiation dominated era.
For that, we used our knowledge about the large scale structure in the Universe coming from the 2M++ galaxy survey and the BORG algorithm to infer the corresponding density distributions deep in the radiation epoch. Using an existing formalism for the magnetic field generation from these initial conditions, we then found at recombination a field coherent on comoving scales in the 10 Mpc h −1 regime, with a maximum field strength of Here we considered voxels with gas density ρ < 3 · ρ as void voxels. The void power spectrum was inferred using the critical filter technique [21], which assumes that the unmasked regions are typical for the whole volume.
FIG. 13. Joint histogramn of the magnetic field and matter density at redshift z = 0 with 512 2 bins. The dashed line indicates the B ∝ ρ 2/3 relation resulting from the flux freezing of the magnetic field lines. This relation is also observed in simulations starting with unconstrained magnetic field conditions, see e.g. [36]. about 10 −23 G at these scales. By means of a MHD simulation we evolved the magnetic field through structure formation and came up with field strengths higher than ≈ 10 −27 G and ≈ 10 −29 G in clusters of galaxies and voids, respectively. We specifically showed the structure of the field around well known structures in the Local Universe, such as the Virgo and Perseus Pisces cluster. The resulting fields are too wea to explain phenomena like the missing GeV observations in voids from blazars. We also expect the actual morphology of the magnetic field within dense structures to be different, as small scales not strongly constraint by the initial galaxy data. Nonetheless, considering the rather conservative assumptions made in these calculations, we can provide a credible lower bound on the strength of the large scale magnetic field today and an impression of its expected morphology that becomes more accurate on large scales. The logical next step building up on this work would be a refinement of the calculation via the implementation of more sophisticated formalisms for the generation of primordial magnetic fields, including a more accurate electron-photon interaction.

ACKNOWLEDGMENTS
This research was supported by the DFG Forschungsgruppe 1254 "Magnetisation of Interstellar and Intergalactic Media: The Prospects of Low-Frequency Radio Observations" and the DFG cluster of excellence "Origin and Structure of the Universe" (www.universecluster.de). This work made in the ILP LABEX (under reference ANR-10-LABX-63) was supported by French state funds managed by the ANR within the Investissements dAvenir programme under reference ANR-11-IDEX-0004-02. GL is supported in part by the French "Agence Nationale de la Recherche" grant ANR-16-CE23-0002.
The cosmological simulations described in this work were performed using the ENZO code (http://enzoproject.org), which is the product of a collaborative effort of scientists at many universities and national laboratories. We gratefully acknowledge the ENZO development group for providing extremely helpful and wellmaintained online documentation and tutorials. The computation presented in this work was produced on at Jülich Supercomputing Centre (JSC), under projects no. 10755 and 11764. F.V. acknowledges financial support from the European Union's Horizon 2020 research and innovation programme under the Marie-Sklodowska-Curie grant agreement no.664931, and from the ERC Starting Grant "MAGCOW", no.714196. DP acknowledges financial support by the ASI/INAF Agreement I/072/09/0 for the Planck LFI Activity of Phase E2 and by ASI Grant 2016-24-H.0. Some of the results in this paper have been derived using the HEALPix [15] package.
FIG. 14. The magnetic field and gas matter density in a slice trough the Virgo (above) and the Perseus-Pisces (below) cluster. The plots shows the gas matter density overplotted with the y − z components of the magnetic field vectors. All colorbars have a logarithmic scaling. The coordinates are defined via the equatorial plane with reference to the galactic centre. The choice of the slice is purely for artistic reasons.
FIG. 15. The magnetic field and gas matter density in an underdense region (above) and around the galactic center in the x − y plane. The plots shows the gas matter density overplotted with the x − y components of the magnetic field vectors. All colorbars have a logarithmic scaling. The coordinates are defined via the equatorial plane with reference to the galactic centre. FIG. 17. A polarization-like plot visualizing the magnetic field morphology perpendicular to the LOS. This plot was generated using the 'Alice' module of the HEALPix software a [15] and the linear integral convolution algorithm [5]. The plot is in galactic coordinate.
FIG. 18. The primordial magnetic field Faraday rotation measure for polarized sources located within a distance of 60 Mpch −1 from earth in units of radians per square metre. The plot is in galactic coordinates. The colormap is logarithmic on both the negative and the positve regime with a linear scaling between −10 −29 and 10 −29 rad·m −2 , connecting both parts of the scale. We used the rescaled gas mass density as an estimate for the electron number density.