Cosmic Magnetic Fields, their effects on the CMB and Gravity Waves COSMO07 Conference Brighton, August 23, 2007 Ruth Durrer Départment de Physique Théorique Université de Genève

Content Introduction Effects of magnetic fields on the CMB –a constant magnetic field –effects on CMB anisotropies –polarization –helical magnetic fields Causality and the magnetic field spectrum Limiting primordial magnetic fields with gravity waves –limits from causal spectra –ways out A remark on GW frequency Conclusions

In observational cosmology we try to constrain the history of the Universe by the observation of relics. The best example of this is the CMB which represents not only a relic of the time of recombination, t' 10 5 years after the big bang, but probably also of a much earlier moment, t» sec, when inflation took place. Another such relic is the abundance of light elements which comes from nucleosynthesis, t' 100 sec. But there are other very interesting events which may have left observable traces, relics, in the universe. –Most notably confinement at t ' sec or –the electroweak transition at t ' sec which may have led to the observed baryon asymmetry in the Universe. It has been proposed that confinement and, especially the electroweak phase transition might lead to the formation of primordial magnetic fields which represent seeds for the magnetic fields observed in galaxies and clusters. Cosmology is a search for relics...

Observed magnetic fields Magnetic fields a ubiquitous in the universe. Most galaxies, like the milky way, are permeated by a magnetic field of the order of a few  Gauss. Also galaxies at high redshift, z¸ 2, seem to have  Gauss magnetic fields Even clusters of galaxies have magnetic fields of the same order. If these fields are generated simply by the amplification of seed fields due to the contraction of the cosmic plasma during the process of galaxy formation, seed fields of the order of G are needed. However if they are amplified via a non linear dynamo mechanism, seed fields as little as about Gauss might be sufficient. It is not clear whether seed fields are really needed or charge separation processes during the process of structure formation can lead to currents which generate the magnetic fields. It is still a matter of debate whether 2 nd order perturbations can induce sufficient charge separation, i.e. currents to generate the initial fields.

In this talk I assume that primordial magnetic fields have been generated with some spectrum and I first discuss their effects on the CMB. Then I shall derive the spectrum of ‘causal’ magnetic fields (i.e. fields generated during a non-inflationary phase of the universe). We shall see that magnetic fields, especially causal magnetic fields, are very strongly constrained by the gravity wave background which they induce. I shall then discuss some possible ways out of this stringent constraints.

A constant magnetic field affects the geometry of the universe by introducing shear. It generates an anisotropic stress  ij ' B i B j  0. This leads to a well studied homogeneous model (Bianchi VII). Due to its effect on the CMB quadrupole it is limited to B< 6.8£ (  m h 2 ) 1/2 Gauss (Barrow et al. ’97) It also induces correlations ha l-1,m a * l+1,m i  0. Limiting such off- diagonal correlations with the COBE data also leads to limits of the order of B< 3£10 -9 Gauss (RD, Kahniashvili, Yates ’98). It is not surprising that these limits are comparable, since  B =   (B/10 -8 Gauss) 2 Magnetic fields of the order 3£10 -9 Gauss (on CMB scales) will leave 10% effects on the CMB anisotropies while Gauss will leave 1% effects. It is thus clear that we can never detect magnetic fields of the order of Gauss (on galactic scales) in the CMB, except if the magnetic field spectrum is very red. A constant magnetic field & CMB

The magnetic field energy momentum tensor contains scalar vector and tensor components which all modify the spacetime geometry and therefore affect the propagation of photons. For tensor perturbations this is all there is ( RD, Ferreira & Kahniashvili, ‘00 ). In addition, the magnetic field generates vector perturbations in the cosmic plasma via the induced Alfvén waves ( RD, Kahniashvili & Yates ‘98 ). There are also two scalar types of waves (fast and slow magnetosonic waves) induced if the cosmic plasma by the magnetic fields which affect the scalar perturbations ( Adams et al. ’96 ). Fast magnetosonic waves are simply the ordinary sound waves with a somewhat higher sound speed c s 2 ! c s 2 + (k¢B) 2 /(4  ) leading to a slight shift of the acoustic peaks. CMB anisotropies from magnetic fields

Adams et al. ’96 (B = 3£ G)

Magnetic field limits from CMB anisotropies We assume a magnetic field spectrum of the form statistical homogeneity the field value on scale the damping scale The power n is the spectral index and n=-3 characterizes a scale invariant spectrum. For the field not to diverge at large scales we require n¸ -3. transversality

Magnetic field limits from CMB anisotropies RD, Ferreira & Kahniashvili ‘00  = 0.1Mpc

Polarisation The CMB polarisation is affected by magnetic fields mainly through the following two mechanisms: –the modified gravitational field –Faraday rotation: the magnetic field polarizes the electrons in the plasma, which leads to a rotation of the polarisation of CMB photons scattering off them. This can turn E-polarisation into B-polarisation! The effect is frequency dependent / 1/ 2 and can therefore, in principle be distinguished spectrally from intrinsic B-polarisation. At l ' 1000, we expect a B-polarisation of »  K(B/10 -9 G)( /10GHz) -2 (Takada et al. ’01) This may provide an excellent way to detect an intergalactic magnetic field.

Helical magnetic fields The magnetic field spectrum is generically of the form The second term is parity odd and can only be generated by parity violating interactions (e.g. at the electroweak phase transition, Vachaspati ’01). In the CMB such a term induces parity odd correlations between temperature anisotropies and B-polarisation and between E- and B-polariation (Caprini, RD & Kahniashvili ’04) in units of  A  S /  r 2 )ln 2 (z * /z eq ))

All effects on the CMB from vector and tensor perturbations due to non-helical magnetic fields have implemented numerically in CAMBcode (Lewis ’05) the vector mode, n=-2.9, B = 3£ G, no helical component, no Faraday rotation TT TE EE BB

Causality If the magnetic fields have been produced at some time  * when the universe was not inflating, the correlations vanish for sufficiently large distances, e.g |x-y|> . Hence the correlation function is a function with compact support and therefore its Fourier transform is analytic. From this ansatz with S(k) = S 0 k n s and A(k) = A 0 k n A we imply n s ¸ 2, an even integer and n A ¸ 1 an odd integer. Positivity of the magnetic field energy requires in addition n A ¸ n S hence n A ¸ 3. Causal magnetic field spectra are therefore very blue and might lead to better constraints on small scales than on large scales.

Gravity waves from magnetic fields This prompted us to look at the gravity waves produced from magnetic fields with a given spectral index n s and amplitude B normalized to today at the scale of =0.1Mpc. At horizon crossing the magnetic fields convert a sizable fraction of their energy into gravity waves. These gravity wave background remains and is not damped by subsequent interactions with the cosmic plasma. Comparing the produced intensity of gravity waves with the nucleosynthesis bound leads to very stringent limits on the causal production of cosmic magnetic fields (Caprini & RD ’02, ‘03, ‘05).  GW · 0.1  rad

Gravity waves from magnetic fields  ji = P ij mn T mn, T mn =B m B n /(8  )

Gravity waves from magnetic fields For -3/2 < n

Gravity waves from magnetic fields Magnetic field energy density limit Gravity wave limit from ew phase transition Gravity wave limit from inflation min caus Caprini & RD ’02

Ways out: 1) inverse cascade We have seen that magnetic fields causally generated at the electroweak phase transition cannot lead to sufficient seed fields if they simply follow the expansion of the cosmic plasma on large scales, B/ 1/a 2. ‘Normal’ magnetic fields cannot do better. Their spectra evolve by damping on small scales and cascade (moving power from larger into smaller scales) on large, but sub-horizon scales. However, analytical arguments and numerical simulations show that helical magnetic fields can invoke an inverse cascade: transporting power from smaller into larger scales. A bit like a cosmic string networks, flux lines which intersect can reconnect and produce larger scale coherence (Vachaspati ’01, Jedamzik et al. ’02, Hindmarsh et al.’03, Brandenburg ‘05 ) A trustable, quantitative evaluation of this inverse cascade is still missing.

2) inflation Another possibility would be that magnetic fields were generated during inflation. There the horizon can be much larger than the present Hubble scale and the causality bound on the spectrum does not apply. However, since the electromagnetic field is conformally coupled, it is not produced during ‘ordinary’ inflation. It is, however produced e.g. during a pre-big bang phase with a dynamical dilaton (Gasparini et al. ’95). There, large scale coherent electromagnetic fluctuations are generated. Due to the high conductivity of the cosmic plasma, the electric field is rapidly dissipated and it remains a magnetic field. Also non-standard couplings of the electromagnetic field during inflation etc. can lead to the generation of magnetic fields. Typically one finds spectra with n ' 0 (Turner & Widrow ’88) but also spectra with n ' -3 have been proposed for some very specific (albeit not very well motivated) situation(Bamba et al. ’04).

Remark on the frequency of gravity waves We have seen that magnetic fieds, even if they do not oscillate, lead to the generation of gravity waves. The gravity wave spectrum generated in this way carries an imprint of the wave number spectrum of the magnetic field. The same result is obtained e.g. if gravity waves are generated from a turbulent plasma or colliding bubbles which form during a cosmological (first order) phase transition. From ordinary (localized) astrophysical sources we are used that the gravity waves generated inherit the frequency of the source. This comes as a simple consequence of the wave zone approximation. E.g. for a monochromatic source we find

For an extended source this is no longer so simple and it has to be checked on a case by case basis, whether it inherits the frequency spectrum or the wave number spectrum or a combination of both. (Caprini, RD, Sturani, 2006). Remark on the frequency of gravity waves ‘eternal’, localized source Short lived, extended source

Conclusions Primordial magnetic fields leave an imprint on the CMB. Since  B = 10 5   (B/10 -8 G) 2, this is only detectable if B~10 -9 G on CMB scales. But if n>-3, this means that the magnetic fields on smaller scales are much larger and might be constrained better by other means, e.g. the induced gravity wave background. To generate the observed galactic or cluster magnetic fields by simple contraction, seed fields of B~10 -9 G on about 1Mpc scale are needed. Dynamo amplification requires seed fields of at least G. The induced gravity wave background limits causally produced (non-helical) fields to B< G on 1Mpc scale and fields from inflation with spectral index n~0 to B< G. Only scale invariant magnetic seed fields may be as large as G and therefore leave a detectable imprint on the CMB. Helical fields might induce an inverse cascade leading to larger fields on large scales. Currents induced by charge separation (2 nd order cosmological perturbation theory) may generate seed fields at much later times ( )G (Riotto et al. ’05, Ichiki et al. ’05).