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

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