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 JainObservations
• 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 CMBA 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 scalesStrong 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-reactionThe 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
casesThe 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 problemThe 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=21012
109
f
106
1000
1
0 10 20 30 40 50 60
E-Folds
➡Magnetic fields of order 10-16 Gauss on
Mpc scales todaySee 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
à ; (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 à , (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 defineLocal 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)
5Two 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 withThe 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 findsConsistency 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 JainYou can also read
