# INFLATIONARY MAGNETOGENESIS & NON-GAUSSIANITY - MARTIN S. SLOTH - BASED ON ARXIV:1305.7151, ARXIV:1210.3461, ARXIV:1207.4187

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Inflationary Magnetogenesis & Non-Gaussianity Martin S. Sloth Based on arXiv:1305.7151, arXiv:1210.3461, arXiv:1207.4187 w. Ricardo J.Z. Ferreira, Rajeev Kumar Jain

Observations • For explaining micro-Gauss Galactic magnetic fields, primordial seeds larger than 10-20 Gauss required • Recent claims of a lower bound on magnetic field in the intergalactic space of 10-15 Gauss [Neronov,Vovk 2010] ➡ Indication of inflationary magnetogenesis • Upper bound on primordial magnetic fields of order nano-Gauss from CMB

A little bit of history... 1p µ⌫ L= gFµ⌫ F 4 in FRW space is conformal inv. doesn’t feel expansion ➡Electromagnetic fields are not amplified by inflation ➡Breaking of conformal invariance needed • Consider coupling of EM fields to other fields, which may couple to gravity in a non-conformal invariant way ➡Production of magnetic fields? [Tuner, Widrow 1988]

Different models • Dynamical gauge coupling 1p L= g ( )Fµ⌫ F µ⌫ [Ratra 1992] 4 • Couplingpto gravity 1 ↵n n µ⌫ L= g Fµ⌫ F R Fµ⌫ F µ⌫ 4 4 same as above, when Φ is the inflaton • Axial coupling p 1 1 L= g Fµ⌫ F µ⌫ ( )Fµ⌫ F̃ µ⌫ 4 4 strong constraints from NG and backreaction

• Mass term p 1 L= g Fµ⌫ F µ⌫ + m2 Aµ Aµ 4 • Negative mass-squared needed for generating enough magnetic fields • Generating neg. mass-squared from Higgs mech. one needs ghost scalar field with neg. kinetic energy [Dvali et. al. 2007, Himmetoglu, Contaldi, Peloso 2009]

Magnetogenesis in Ratra-type models • In Coulomb gauge we have ( , ) • With the magnetic field given by • Defining the magnetic power spectrum it can be computed from

• Define pump field • and a canonically normalized vector field • Such that the quadratic action takes the simple form • The EOM for the mode function is • With the solution normalized to Bunch- Davis vacuum is • which leads to

➡power For n> -1/2 the spectral index of the magnetic spectrum is nB = (4 2n) • For a scale invariant spectrum, n=2, back-reaction remains small • In this case, with H≃1014 GeV, a magnetic field strength of order nano-Gauss an be achieved on Mpc scales

Strong coupling problem • Adding the EM coupling to the SM fermions p 1 ¯ L= g ( )Fµ⌫ F µ⌫ µ (@µ + ieAµ ) 4 • The physical electric coupling is p ephys = e/ ( ) • p Since / an then for n > 0 the electric coupling decreases by a lot during inflation, and must have been very large at the beginning ➡QFT out of control initially [Demozzi, Mukhanov, Rubinstein 2009] • Solutions??? Speculations [Bonvin, Caprini, Durrer 2011, Caldwell, Motta 2012, Bartolo, Matarrese, Peloso, Ricciardone 2012, and more...] ➡ More work required! [Ferreira, Jain, MSS 2013]

The Sawtooth Model

The Sawtooth Model • Relax assumption of monotonic coupling function

The Sawtooth Model • Relax assumption of monotonic coupling function ➡Patch together piecewise scale invariant sections of n=2 and n=-3

The Sawtooth Model • Relax assumption of monotonic coupling function ➡Patch together piecewise scale invariant sections of n=2 and n=-3 • Each section of n=-3 can be no-longer than N=20 to avoid back-reaction

The Sawtooth Model • Relax assumption of monotonic coupling function ➡Patch together piecewise scale invariant sections of n=2 and n=-3 • Each section of n=-3 can be no-longer than N=20 to avoid back-reaction ➡One might be worried that adding more sections of n=2 and n=-3, the energy density of the n=-3 pieces will add up and prohibit this!

The Sawtooth Model • Relax assumption of monotonic coupling function ➡Patch together piecewise scale invariant sections of n=2 and n=-3 • Each section of n=-3 can be no-longer than N=20 to avoid back-reaction ➡One might be worried that adding more sections of n=2 and n=-3, the energy density of the n=-3 pieces will add up and prohibit this! • This turns out not to be true!

The Sawtooth Model

The Sawtooth Model • By using the appropriate matching conditions, the dominant solution before the transition matches to the decaying solution after the transition.

The Sawtooth Model • By using the appropriate matching conditions, the dominant solution before the transition matches to the decaying solution after the transition. ➡This leads to a very large k-dependent loss in the magnetic field spectrum in all the concave transitions and in the electric field in the opposite cases

The Sawtooth Model • By using the appropriate matching conditions, the dominant solution before the transition matches to the decaying solution after the transition. ➡This leads to a very large k-dependent loss in the magnetic field spectrum in all the concave transitions and in the electric field in the opposite cases ➡The loss in the electric field spectrum avoids the back reaction problem

The Sawtooth Model • By using the appropriate matching conditions, the dominant solution before the transition matches to the decaying solution after the transition. ➡This leads to a very large k-dependent loss in the magnetic field spectrum in all the concave transitions and in the electric field in the opposite cases ➡The loss in the electric field spectrum avoids the back reaction problem ➡The loss in the magnetic spectrum implies a smaller value of the magnetic field strength at the end of inflation -- it can however be compensated by having n=3 instead of n=2

1012 109 f 106 1000 1 0 10 20 30 40 50 60 E-Folds ➡Magnetic fields of order 10-16 Gauss on Mpc scales today

See Ricardo Ferreira’s poster for more Inflationary Magnetogenesis without the Strong Coupling Problem Ricardo Jose Zambujal Ferreira Inflationary Magnetogenesis Spectrum Observations show the existence of coherent magnetic fields in the The transitions in the coupling function lead to a loss which can intergalactic medium with a lower bound of 10≠16 G and coherent be explained by a proper matching of the vector field A, and not the lengths larger than the Mpc scale [1]. The most plausible origin for canonically normalized A, at the transition. From the matching one magnetic fields with such a large coherent length is some inflationary can see that, for transitions from – > ≠1/2 to – < ≠1/2 (and the mechanism. reverse for the electric spectrum), the leading behavior does not pick However, generating large scale magnetic fields during inflation is up immediately leading to a loss in the magnetic spectrum (see Fig. a hard task if one has in mind that the Electromagnetic (EM) ac- 2). tion is conformally invariant. Therefore, any magnetic field generated is highly diluted during inflation. One of the simplest gauge invari- k=0.01 Mpc -1 H2 10-9 ant ways of tackle this problem by breaking the conformal invariance k=0.66 Mpc -1 H4 through a coupling function between EM field and a time dependent 10-15 k=35.89 Mpc -1 background field „ (inflaton, curvaton, etc.) as follows fperiodic drB êdlogk ⁄ 1 10-21 Ô SEM = ≠ d4 x ≠g f 2 („) Fµ‹ F µ‹ , (1) 4 10-27 while restoring standard EM at the end of inflation f („) æ 1. After quantizing the EM field one obtains the canonically normalized vector 10-33 field A = f A which satisfies the following equation of motion 10-39 3 4 f ÕÕ 0 10 20 30 40 50 60 AÕÕk + k 2 ≠ Ak = 0. (2) E-Folds f Figure 2: Time evolution of dﬂB (÷, k)/dlogk for three different modes Strong Coupling Problem These interesting behaviors sum up to a final spectrum with k- dependence given by The strong coupling problem is related to the self-consistency of I the theory. One class of coupling functions which has receive great dﬂB k 8≠2–i , –i > 1/2 Ã ; (3) attention is the one where there is has a power law dependence on d log k k 6+2–i , ≠1/2 < –i < 1/2 the scale factor f (÷) Ã a– . Such a coupling function leads to a scale if the a given mode leaves the horizon in a stage preceding a loss and invariant magnetic spectrum for – = 2 and – = ≠3. While in the latest case the energy stored in the electric field back reacts after ≥ 10 I dﬂB k 6+2– , – Æ ≠1/2 e-folds of inflation, which is clearly too short, for – = 2 a different Ã , (4) problem appears. Assuming that EM is coupled to charged matter d log k k 4≠2– , – Ø ≠1/2 in the usual gauge invariant way, then, the physical charge is given in the other cases. However, the most interesting feature of this model by ephys = e/f . Now, having in mind that the coupling function is is the fact that one can generate magnetic fields of strength 10≠16 G on a monotonic increasing function, thus, if we go back in time we enter scales around the Mpc scale, depending on the post-inflationary model. rapidly in strongly coupled regimes where the theory is no longer valid. This is the so-called strong coupling problem [2]. 10-24 The sawtooth model 10-28 fperiodic k1.9 The model we present here solves the strong coupling problem by 10-32 d rB êdlogk k-0.1 gluing stages of different – (see Fig. 1) while generating magnetic fields k1.9 of strength 10≠16 G on scales around Mpc [3]. The transitions are an 10-36 k-0.1 indispensable ingredient in the model because they lead to a loss in the k1.9 spectrum which directly avoids back reaction. 10-40 k-0.1 10-44 1012 10-4 10 106 1011 1016 1021 k H Mpc-1 L 109 Figure 3: dﬂB (÷, k)/dlogk at the end of inflation f 106 References 1000 1 0 10 20 30 40 50 60 [1] A. Neronov and I. Vovk, Science 328 010 (2012), [arXiv:1006.3504] E-Folds [2] V. Demozzi, V. Mukhanov and H. Rubinstein, JCAP 0908 (2009) Figure 1: Time evolution of the coupling function during inflation 025, [arXiv:0907.1030]. [3] R. J. Z. Ferreira, R. K. Jain and M. S. Sloth, In preparation.

Terms of use

Terms of use Sometimes it might even be useful to know the terms of use...

Terms of use Sometimes it might even be useful to know the terms of use... In the following, the U(1) vector field doesn’t need to be the electro-magnetic one, but it could be a general vector iso-curvature field during inflation!

Terms of use Sometimes it might even be useful to know the terms of use... In the following, the U(1) vector field doesn’t need to be the electro-magnetic one, but it could be a general vector iso-curvature field during inflation! We don’t need to worry about strong coupling problem when the vector field is not the electro-magnetic one!

Non-Gaussianity • Consider 1p L= g ( )Fµ⌫ F µ⌫ 4 w. direct coupling of magnetic field with the inflaton ➡NG correlation of magnetic field with inflaton field H h⇣(k1 )B(k2 ) · B(k3 )i ⇣= ˙ [Kamionkowski, Caldwell, Motta (2012), Jain, MSS (2012), Biagetti, Kehagias, Morgante, Perrier, Riotto (2013)]

(Ordinary) Non-Gaussianity • To leading order, the perturbations are encoded in the two- point function h⇣k ⇣k0 i = (2⇡) (~k + k~0 )P⇣ (k) 3 • A non vanishing three point function h⇣k1 ⇣k2 ⇣k3 i is a signal of non-Gaussianity • Introduce dimensionless fNL : fN L ⇠ h⇣k1 ⇣k2 ⇣k3 i /P⇣ (k1 )P⇣ (k2 ) + perm. as a measure of non-Gaussinity • Similarly ⌧N L ⇠ h⇣k1 ⇣k2 ⇣k3 ⇣k4 i /P⇣ (k1 )P⇣ (k2 )P⇣ (k14 ) + perm.

Non-Gaussianity: Single field slow-roll • Perturbations conserved on super-horizon scales: NG is computed at hoizon crossing • Bispectrum from 3-point interaction fN L ( 0.2) [Maldacena ’02, ] • Trispectrum from connected 4-point interaction and graviton exchange ⇥N L [Seery, Lidsey, Sloth ’06, Seery, Sloth,Vernizzi ’08]

Magnetic non-linearity parameter: bNL • Let’s come back to h⇣(k1 )B(k2 ) · B(k3 )i and parametrize it in a similar way ➡Introduce new magnetic non-linearity parameter: bNL • Define the cross-correlation bispectrum • We then define

Local bNL • In the case where b NL is momentum independent, it takes the local form: • Compare with local f NL , given ⇣ ⌘2 by 3 ⇣ = ⇣ (G) + fN local L ⇣ (G) 5

Two interesting shapes 1. The squeezed limit • We obtain a new magnetic consistency relation h⇣(k1 )B(k2 ) · B(k3 )i = (nB 4)(2⇡)3 (3) (k1 + k2 + k3 )P⇣ (k1 )PB (k) with bNLlocal = (nB - 4) • Compare with Maldacena consistency relation h⇣(k1 )⇣(k2 )⇣(k3 )i = (ns 1)(2⇡)3 (3) (k1 + k2 + k3 )P⇣ (k1 )P⇣ (k) with fNLlocal = - (ns - 1)

Two interesting shapes 2. The flattened shape • This is the shape where bNL turns out to be maximized with

The magnetic consistency relation • In terms of the vector field, we have • where the magnetic field power spectrum is

• Consider in the squeezed limit • The effect of the long wavelength mode is to shift the background of the short wavelength modes • Since the vector field only feels the background through the coupling λ, all the effect of the long wavelength mode is captured by

• Define pump field • and linear Gaussian canonical vector field • Such that the quadratic action takes the simple form

• Since all the effect of the long wavelength mode is in ➡One finds • Using • and ➡One finds

Consistency relation • Expressing it in terms of the magnetic fields ➡Magnetic consistency relation • With ➡One has b NL= (nB - 4)

The flattened shape • The correlation is maximal in flattened shape • In this case [Jain, MSS 2012] ➡See talk of Rajeev Kumar Jain!

Conclusions • The magnetic consistency relation is a probe of an important class of models • The consistency relation is an important theoretical tool for consistency check of calculations • The new b NLparameter can be very large in the flattened limit and might have interesting phenomenological implications

Inflationary Magnetogenesis & Non-Gaussianity Martin S. Sloth Based on arXiv:1305.7151, arXiv:1210.3461, arXiv:1207.4187 w. Ricardo J.Z. Ferreira, Rajeev Kumar Jain

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