Primordial magnetic fields in the $f^{2}FF$ model in large field inflation under de Sitter and power law expansion

4.9 MB·

by  Anwar AlMuhammad

Primordial magnetic fields in the f2FF model in large field inflation under de Sitter
and power law expansion
Anwar AlMuhammad1,∗ and Rafael Lopez-Mobilia1,†
1Department of Physics and Astronomy, The University of Texas at San Antonio,
One UTSA Circle, San Antonio Texas 78249, USA.
(Dated: January 12, 2016)
Weusethef2FF modeltostudythegenerationofprimordialmagneticfields(PMF)inthecontext
of large field inflation (LFI), described by the potential, V ∼Mφp. We compute the magnetic and
electric spectra for all possible values of the model parameters under de Sitter and power law
expansion. WeshowthatscaleinvariantPMFarenotobtainedinLFItofirstorderintheslowroll
approximation, if we impose the constraint V(φ = 0) ∼ 0. Alternatively, if these constraints are
relaxed,thescaleinvariantPMFcanbegenerated. Theassociatedelectricfieldenergycanfallbelow
6
the energy density of inflation, ρ for the ranges of comoving wavenumbers, k > 8×10−7Mpc−1
1 Inf
0 andk>4×10−6Mpc−1 indeSitterandpowerlaw(PL)expansion. Further,itcandropbelowρInf
2 on the ranges, e-foldings N >51, p<1.66, p>2.03, l0 >3×105MPl−1(Hi <3.3×10−6MPl), and
M >2.8×10−3MPl. All of the above ranges fit with the observational constraints.
n
a
PACSnumbers:
J
Keywords: cosmology;theory;earlyuniverse

9

]
O
C I. INTRODUCTION
.
h
p Inflationary cosmology is one of the most brilliant ideas proposed to resolve some of the fundamental shortcomings
- of the old Big Bang model ([1–4]). It solves the flatness and horizon problems, it explains the absence of exotic relics
o
(predictedbygrandunifiedtheories)anditevenprovidesanexplanationforthelarge-scalestructureoftheUniverse,
r
t which is linked to the quantum fluctuations of the field of inflation φ. The implications of inflation have been found
s
to be consistent with observations, in particular the large-scale homogeneity and isotropy of the universe. Another
a
[ impressive success of inflation has been the observation of a scale invariant spectrum of the anisotropies of the CMB.

The generation of primordial gravitational waves (PGW) during inflation is a key prediction that may be proven true
1
in the near future.
v
There are other outstanding problems in astrophysics and cosmology that could potentially have a solution in the
4
0 context of inflation. One of them is the presence of large-scale, very weak magnetic fields, observed in all kinds
1 of galaxies and cluster of galaxies at wide range of redshifts. There is even some evidence for the presence of the
2 magneticfieldsintheverylowdensityintergalacticmedium([5–11]). Thegalacticdynamoisaknownmechanismfor
0 the amplification of magnetic fields to the observed levels in galaxies, but it requires a seed field ([12, 13]). So far, no
.
1 convincing argument has been found for the origin of these magnetic fields, and a distinct possibility is that they are
0 of cosmological (primordial) origin. The mechanism for generating these so-called primordial magnetic fields (PMF)
6 is still under intense debate ([14–20]). The latest constraints on the strength of PMF (∼nG) have been set recently
1
by Planck 2015 ([67]).
:
v In the context of PMF, one of the most interesting and simplest models is the f2FF([21–25]). Part of its attrac-
i tivenessisthatitisstableunderperturbations. (See[26]andreferencestherein.) Also,itcanleadtoascaleinvariant
X
spectrum of PMF ([22–25]), an important characteristic that would explain why magnetic fields are detected nearly
r
at all scales in the universe.
a
Themaindifficultieswiththismodelarethebackreactionproblem,wherethescaleoftheenergyoftheelectricfield
associated with the PMF can exceed the scale of inflation itself ([24, 25, 27–29]), and the strong coupling between
electromagnetic fields and charged matter at the beginning of inflation ([28, 29]). A few different avenues have
been proposed to overcome these two problems, but they need more investigations in order to form a robust model,
self-consistent in both classical and quantum regimes.
The strong coupling problem is of quantum origin and may lead to a huge coupling between the electromagnetic
fields and charged particles. For example, for inflation with N =60 e-foldings, the electron effective charge would be
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
q ∝e120 ([29]). The theory is no longer valid in such an excessively strong coupling regime([28]).
The Lagrangian of a scalar (inflaton) field φ coupled to the gauged electromagnetic vector field A can be written
µ
([21, 22, 24]) as,
√ (cid:104)1
L=− −g (∂ φ)(∂µφ)+V (φ)+
2 µ
(1)
1 (cid:105)
gαβgµνf2(φ,t)F F ,
4 µα νβ
where, F =∂ A −∂ A is the electromagnetic field tensor and g is the determinant of the spacetime metric g .
νβ ν β β ν µν
The first term in the Lagrangian is the standard kinetic part of the scalar field, and the second term,V (φ), is the
potential.
Whereas the Lagrangian of a pure electromagnetic field is of the form −1F Fµν, in Eq.(1) we couple it to the
4 µν
scalar field through the unspecified function f(φ,t). The main reason behind this coupling is to break the conformal
symmetry of electromagnetism and hence prevent the dilution of the seed of magnetic field as it is generated in the
inflation era. At the end of inflation, the coupling function, f(φ,t)→1, to retrieve the conformal electromagnetism.
ThelastconditionisimportanttodecidetheformofcouplingfunctionandtostudyPMFinpost-inflationaryphases.
The standard model of inflation makes use of a single scalar field and a simple potential, such as a quadratic
(cid:16) (cid:16) (cid:112) (cid:17)(cid:17)2
function, V (φ) ∼ φ2, quartic, V (φ) ∼ φ4 ([30]), Higgs potential, V (φ) ∼ M4 1−exp − 2/3φ/M , ([31])
PL
√
(cid:0) (cid:1)
and the exponential potential, V (φ) ∼ exp − 2(cid:15) (φ−φ ) ([32]). The last one is used in ([22–25]) to find the
1 0
magnetic and electric spectrum in the f2FF model. These models became more interesting after WMAP ([33, 34])
and Planck ([35]). As a result, the preferred potential class is the so called plateau inflation, at which V (0)(cid:54)=0.
In this paper, the f2FF model is investigated for large field inflation LFI, for all possible values of the model
parameters, in the same way as done by ([24]). We adopt natural units, [c=(cid:126)=k = 1], the signature (−1,1,1,1),
B
and flat universe. The reduced Planck mass, M = (8πG)−1/2, will be taken as M = 1 for the computations.
Pl Pl
Hence, the potential of LFI can be written ([30, 41]) as
(cid:18) φ (cid:19)p
V (φ)=M4 , (2)
M
Pl
where,pisthemodelparameterandM isthenormalizationofthepotential. FromtheamplitudeofCMBanisotropies,
M/M (cid:39)3×10−3 ([48]).
Pl
Insection2wepresenttheslowrollinflationformulationforbothsimpledeSittermodelofexpansionandthemore
general power law expansion in the context of LFI. In section 3, the PMF and associated electric fields are computed
in LFI for all possible values of p. In section 4, a summary and discussion of the results is presented.
II. SLOW ROLL ANALYSIS OF LFI
During inflation we assume the electromagnetic field to be negligible compared to the scalar field, φ ([24]). Hence,
the equation of motion derived from (1) for the scalar field can be written as,
φ¨+3Hφ˙+V =0, (3)
φ
where, H(t)= a˙(t)/a(t), is the Hubble parameter as a function of cosmic time, t, and a(t) is the cosmological scale
factor. The over dot indicates differentiation with respect to cosmic time, and V = ∂ V. The Friedman equation
φ φ
can be obtained from the Einstein field equations (assuming a Friedmann-Robertson-Walker universe), which yields
(cid:20) (cid:21)
1 1 K Λ
H 2 = φ˙2+V (φ) − + , (4)
3M 2 2 a2 3
Pl
where, K =0,±1, is the 3-curvature index for a flat, closed or an open universe, and Λ, is the cosmological constant.
The last two terms can be neglected in the inflation era. Also, under slow roll approximation, one can neglect the
second derivative in (3), which leads to the attractor condition,
V
φ˙ (cid:39)− φ . (5)
3H
3
Defining the slow roll parameters of inflation in terms of the potential ([48–50]), of a single inflation field for LFI (2),
1 (cid:18)V (cid:19)2 1 (cid:18)p(cid:19)2
(cid:15) (φ)= M 2 φ = M 2 , (6)
1V 2 Pl V 2 Pl φ
(cid:18) (cid:19)
V p(p−1)
(cid:15) (φ)=M 2 φφ =M 2 . (7)
2V Pl V Pl φ2
These can also be written in terms of the Hubble parameter,
(cid:18)H (cid:19)2 (cid:18)H (cid:19)
(cid:15) (φ)=2M 2 φ ,(cid:15) (φ)=2M 2 φφ . (8)
1H Pl H 2H Pl H
The relation between the two formalisms ([48–50]) can be written as
(cid:18)3−(cid:15) (cid:19)2
(cid:15) =(cid:15) 2H . (9)
1V 1H 3−(cid:15)
1H
All of the above parameters are assumed to be very small during the slow roll inflation,((cid:15) ,(cid:15) ,(cid:15) ,(cid:15) ) (cid:28) 1.
1V 2V 1H 2H
Furthermore, inflation ends when the values of ((cid:15) ,(cid:15) ) → 1. Hence, in the first order of approximation, one can
1V 1H
neglect (cid:15) and (cid:15) compared with 3, to obtain (cid:15) (cid:39)(cid:15) . Therefore, using (8) and the relation between the cosmic
1H 2H 1V 1H
time, t, and the conformal time, η, dt = a(η)dη, one can write the relation between conformal time and slow roll
parameter, (cid:15) , ([48–50]) as,
1H
1 (cid:90) (cid:15)
η =− + 1H da. (10)
aH a2H
Assuming, (cid:15) ≈const, and then integrating (10) yields the power law expansion of the universe during inflation,
1H
a(η)=l |η|−1−(cid:15)1H. (11)
0
where, l is an integration constant.
0
In the simplest form of inflationary expansion (de Sitter), the universe expands exponentially during inflation at a
very high but constant rate, H ,
i
a˙
H = (cid:39)const, (12)
i a
a(t)=a(t ) exp[H t]. (13)
1 i
where t is the start time of inflation. In a conformal time, Eq.(13) can be written as,
1
1
a(η)=− . (14)
H η
i
Plugging (14) into the relation between cosmic and conformal time and integrating implies that η → (−∞,0−) as
t→(0,∞).
Also, the relation between slow roll parameters and the scalar power spectrum amplitude, A , the tensor power
s
spectrum amplitude, A , the scalar spectral index, n , and tensor-to-scalar ratio, r, can be written as follows,
t s
V
A = , (15)
s 24π2M 4(cid:15)
Pl 1V
2V
A = , (16)
t 3π2M 4
Pl
n =1−6(cid:15) +2(cid:15) , (17)
s 1V 2V
4
A
r = t =16(cid:15) . (18)
A 1V
s
Using the LFI potential, (2), into (6)-(7) yields,
p(2p+1)
n =1−M 2 , (19)
s Pl φ2
p2
r =8M 2 . (20)
Pl φ2
One can find the relation between r and n which depends on the number of e-folds of inflation, N. To first order,
s
N can be written as
(cid:115)
1 φf 1
N (cid:39)− ∫ √ dφ, (21)
2MPl2 φ (cid:15)1
where φ is the initial field and φ is the field at the end of inflation. Assuming φ (cid:28) φ, and solving for φ from (21)
f f
and plugging it into (19)-(20), yields,
(cid:18) (cid:19)
4p 2p+1
r (cid:39) n + . (22)
N s 2N
The relation (22) is graphed for different values of the number N of e-folds, and observational constraints of r. For
example, the constraints of Planck 2015 ([68]) are included with N =50,60,70 are shown in Fig.1.
FIG. 1: The limits of LFI parameter, p, based on the constraint of of BKP 2015, r < 0.12 as shown in, p−r for different
e-foldngs, N, at n =0.960, which is favored by Planck.
s
Shortly after the onset of inflation the value of H becomes very high and is approximately constant, but later on
i
it decreases as the value of the field changes. For the zeroth approximation and after the first few e-foldings, we can
consider H as a constant ratio. That is basically the de Sitter expansion, which is exactly exponential expansion as
i
described by Eq.(13). Also, after few N , the spacetime (pivot scale, k ) exits from the Hubble horizon. We adopt
∗ ∗
Planck, 2015 pivot scale, k = 0.05Mpc−1, and in most of the cases we use the upper limit of Hubble parameter in
∗
the inflationary era, H (cid:39)3.6×10−5M , ([68]). So, it is worthwhile to investigate both cases; the de Sitter, and the
i Pl
more realistic power law model described by Eq.(11).
5
A. Slow roll of LFI on de Sitter expansion
The de Sitter model is the zeroth order approximation, and does not have graceful exit from inflation ([58]). But
it can be assumed as a valid approximation for most of the period of inflation. In conformal time, η, Eq.(5) can be
written as,
1 V
φ(cid:48) (cid:39)− φ , (23)
a(η) 3H
i
where, φ(cid:48) =∂ φ. Substituting of (2) and (14) into (23) and then integrating both sides yields,
η
dφ M dη
= . (24)
pφp−1 M p H
Pl i
By solving (24) for φ(η), we have two different solutions,
− 2M4 (cid:18) 2M4 c (cid:19)
φ(η)=(Hiη) 3Hi2MPl2.exp −3H 2M 2 ,p=2, (25)
i Pl
(cid:18)2(2−p)pM4 (cid:19)1/(2−p)
φ(η)= [ln(H η)+c] ,p(cid:54)=2, (26)
3H 2M p i
i Pl
where c is the integration constant. Assuming the inflation potential vanishes at the end of inflation, η ,
f
V (φ(η ))∝[φ(η )]p (cid:28)1. (27)
f f
Hence, for p=2, we have for the integration constant
3H 2M 2
c(cid:29) i Pl . (28)
2M4
However, for p<2, we have
c→−ln(H η ), (29)
i f
and for p>2,
3H 2M p
c(cid:29) i Pl . (30)
2(2−p)pM4
Ontheotherhand,ifwedonotassumeEq.(27),wecanchoosecsoastoleadtoascaleinvarianceconditionofPMF.
Both cases will be studied in section 3, to derive the coupling function f(η) and then calculate the electromagnetic
spectra in the de Sitter case of inflation.
B. Slow roll of LFI on a power law expansion
To have a more optimal slow roll analysis that has a smooth exit from inflation, the Hubble parameter can be
written as a function of φ , H (φ). If the field falls below a certain value, it starts to oscillate and then converts into
i
particles in the reheating era, right after inflation. The expansion of space-time during inflation can be described by
a power law function. Thus, plugging (6) into (11), yields
a(η)=l0|η|−1−21MPl2(φp)2. (31)
Solving for φ from (21) and then substituting it into (31) gives,
a(η)=l0|η|−1−4pN,
a(cid:48)(η) 1 (cid:16) p (cid:17) p (32)
H(η)= =− 1+ |η|4N.
a2(η) l 4N
0
6
Substitution of (32) into (5), yields,
dφ = pM4l02η−(1+2pN)dη. (33)
φp−1 3Mp (cid:0)1+ p (cid:1)
Pl 4N
Again, the solution of (33) will be φ(η) and depends on the model parameter, p. We have two different cases:
p=2, and p(cid:54)=2. In the case of p=2, one can write the solution of (33) as
(cid:32) (cid:33)
φ(η)=cexp −2 M4l02η−N1 , (34)
3M2 1 (cid:0)1+ 1 (cid:1)
PlN 2N
where, c is the integration constant. Adopting (27) yields, φ(η )(cid:28)1, and c(cid:28)1, where η (cid:28)−1.
end end
However, for p(cid:54)=2, the solution of (33) will be,
(cid:32) (cid:33) 1
φ(η)= −2NM4l02η−2pN(2−p) +c 2−p. (35)
3 Mp (cid:0)1+ p (cid:1)
Pl 4N
By adopting (27), for p<2, the integration constant c(cid:28)1. However, for p>2, the integration constant is
c(cid:29) 2NM4l02ηend−2pN(2−p). (36)
3 Mp (cid:0)1+ p (cid:1)
Pl 4N
III. THE PMF GENERATED IN LFI MODEL
This subject was investigated in ([24]), where they conclude that the LFI does not lead to sensible model building
in generating PMF. We use the same method to investigate the generation of PMF for all possible values of p.
The first step is to find the equation of motion for the electromagnetic field, A , from the Lagrangian (1),
µ
√
∂ (cid:2) −ggµνgαβf2(φ,t)F (cid:3)=0. (37)
µ νβ
In conformal time, (37) can be written as
f(cid:48)
A(cid:48)(cid:48) (η,x)+2 A(cid:48) (η,x)−a2(η) ∂ ∂jA (η,x)=0 (38)
i f i j i
Define A¯ (η,x) = f(η)A (η,x), and write its quantum version in terms of creation and annihilation operators, b†
i i λ
and b (k) as
λ
A¯ (η,x)=(cid:90) d3k (cid:88)2 ε (k)×[b (k)A(η,k)eik.x
i (2π)3/2 iλ λ (39)
λ=1
+b† (k)A∗(η,k)e−ik.x],
λ
where, ε is the transverse polarization vector, and k = 2π, is the commoving wave number. Hence, Eq.(38) can be
iλ λ
written as,
A(cid:48)(cid:48)(η,k)+(cid:0)k2−Y (η)(cid:1)A(η,k)=0, (40)
where, Y (η)= f(cid:48)(cid:48).
f
The magnetic and electric spectra can be calculated ([24]) respectively by,
dρ 1 (cid:18)k(cid:19)4
B = k|A(η,k)|2. (41)
dlnk 2π2 a
dρ f2 k3(cid:12)(cid:12)(cid:20)A(η,k)(cid:21)(cid:48)(cid:12)(cid:12)2
E = (cid:12) (cid:12) . (42)
dlnk 2π2 a4(cid:12) f (cid:12)
(cid:12) (cid:12)
7
Therefore, we need first to define the coupling function, f(η), in order to solve for the electromagnetic vector field,
A . We will assume that the relation between the coupling function and scale factor is of the power law form ([24]),
µ
f(η) ∝ aα. Then, by combining (3) and (4) in the slow roll limit,
(cid:20) (cid:21)
α φ V (φ)
f(φ) ∝exp − ∫ dφ . (43)
3M 2 V(cid:48)(φ)
Pl
Substitution of (2) into (43) gives
(cid:20) α φ2(cid:21)
f(φ(η)) =D exp − , (44)
6M 2 p
Pl
where D is a coupling constant. In the next two sections we substitute (44) into (40) for the two cases of inflationary
expansion, de Sitter and power law.
A. the PMF generated in LFI in a simple de Sitter model of expansion
The de Sitter approximation was used by ([28]) to investigate PMF. One can investigate PMF in the de Sitter case
by substituting (25-26) into (44), and applying the limits, (28-30) for the selected values of model parameter, p. The
value p=2 is the most interesting one, because it fits well with spectrum index, n detected by Planck ([34]). Also,
s
it is the closest case to the standard inflationary models. Substituting of (25) into (40) and (44) yields,
Y (η)= e−3M8cP2l2MH4i2M8α2(η Hi)−2−3M8PMl24Hi2 . (45)
81M 8H 2
Pl i
During inflation H 2 (cid:29)1, hence,
i
8M4
(cid:28)1. (46)
3M 2H 2
Pl i
Therefore, (40) becomes,
A(cid:48)(cid:48)(η,k)+
k2− e−3M8cP2l2MH4i2M8α2(η Hi)−2A(η,k)=0. (47)
81M 8H 2
Pl i
The solution of (47) is a Bessel function ([59]). Hence the solution of A(η,k), can be written as,
A(η,k)=(kη)1/2[C (k) J (kη)+C (k) J (kη)], (48)
1 χ 2 −χ
where χ is written as
(cid:118)
χ= 1 (cid:117)(cid:117)(cid:116)81+ 4e−3M8cP2lM2H4i2M8α2. (49)
18 M 8H 4
Pl i
In the long wavelength regime, kη (cid:28)1 (outside Hubble radius), Eq.(48) can be written as
A (η,k)=(k)−1/2[D (χ) (−kη)χ+1/2
kη(cid:28)1 1
(50)
+D (χ)(−kη)1/2−χ].
2
Theconstants, D (χ)andD (χ),canbefixedbyusingthenormalizationofA(η,k)andotherlimit,A (η,k)→
(cid:46)√ 1 2 kη(cid:29)1
e−kη 2k.
The magnetic spectra can be obtained by substituting of (50) into (41). It can be written ([24]) as,
dρ F(n) (cid:18) k (cid:19)4+2n
B = H4 , (51)
dlnk 2π2 aH
8
where, γ =χ+1/2, then n=γ if γ ≤1/2 and n=1−γ for γ ≥1/2. The function F(n) can be written as,
π
F(n)= . (52)
22n+1Γ2(n+ 1)cos2(πn)
2
Similarly, the electric field spectrum can be written as,
dρ G(m) (cid:18) k (cid:19)4+2m
E = H4 , (53)
dlnk 2π2 aH
where, m=γ+1 if γ ≤−1/2 and m=−γ for γ ≥−1/2. The function G(m) can be written as,
π
G(m)= . (54)
23m+3Γ2(m+ 3)cos2(πm)
2
The scale invariant PMF can be achieved if the magnetic spectrum, dρB = constant. Hence, from Eq.(51), the
dlnk
valuesofγ ={−2,3}. ThefirstvalueismoreacceptableingeneratingthePMFwithouthugeamountofbackreaction
and without assuming a small scale of inflation ( as in the case of γ =3) . Also, for α=−3, the reheating period will
be to long, such that the reheating temperature falls to few MeV. For α = 2, the problem of strong coupling exists.
One way to solve this problem is to assume that, the initial coupling function is much less than the coupling function
at the end of inflation, f(η ) (cid:28) f(η ) ∼ 1. This assumption, in turns, will create a weak coupling between the
0 end
gauge field and charges at the end of inflation ([24]). For these reasons, the solution α=2 is adopted in this paper.
Usingboth(28)and(46),onecanwrite,χ(cid:39)1/2. Comparingitwiththesolutionin([24]),itimpliesγ =0. Hence,
it cannot generate a scale invariant PMF.
In order to find the electromagnetic spectra, one can substitute (25) into (44), to get,
(cid:20) (cid:21)
α
f(η)=D exp − ∆ . (55)
6M 2
Pl
where,
(cid:18)(Hiη)−3Hi22MM4Pl2.exp(cid:16)−3H2iM2M4Pcl2(cid:17)(cid:19)2
∆= . (56)
2
Employing(28),theexponentof(55)willbeverysmall,hence,f(η)→D. Thus,theplotofthespectraofbothPMF
andelectricfieldshowsthattheyareofthesameorderofmagnitudeatlowvalueofkη,anddivergeatrelativelyhigh
kη, see Fig.2. In this case, the electromagnetic spectrum is not scale invariant.
FIG. 2: The magnetic and electric spectra, generated under LFI model in the simple de Sitter expansion, and by considering
the limit, Eq.(27), where p = 2, kη (cid:28) 1, and χ (cid:39) 1/2(γ=0). They are of the same order at low value of k. In the plot, we
assume, H = 3.6×10−5M and Planck pivot scale, η = −20. In this case the spectra are not scale invariant and for the
i Pl
observable scale, k, they are less than the energy of inflation, ρ .
Inf
9
On the other hand, if we relax the limits (28) and (46), and choose c to enforce χ=5/2, the case at which PMF is
scale invariant ([24]), we have
3M 2H 2 (cid:20)486M 8H 4(cid:21)
c=− Pl i ln Pl i . (57)
8M4 M8α2
We substitute (57) into (56) to find the coupling function, and use it to plot the electromagnetic spectra, see Fig.3.
It shows that a scale invariant PMF can be achieved without a backreaction problem as long as k >8×10−7Mpc−1,
see Fig.3. However, the electric spectra can go over the scale of the inflation, ρ , for k < 8×10−7Mpc−1. Hence,
Inf
thebackreactionproblemstillexistsatlowvaluesofkη. Intheabovecalculation, weuseα=2, H =3.6×10−5M ,
i Pl
M =3×10−3M and (M ,D)=1.
Pl Pl
FIG. 3: The magnetic and electric spectra, generated under LFI model, in the simple de Sitter expansion, with, p(cid:39)2,k(cid:28)1,
and χ (cid:39) 5/2(γ = −2,3). The electric spectra can go over the scale of the inflation, ρ , for k < 8×10−7Mpc−1. However,
Inf
the backreaction problem can be avoided for k >8×10−7Mpc−1. In the plot, we use, η =−20, α=2, H =3.6×10−5M ,
i Pl
M =3×10−3M and (M ,D)=1.
Pl Pl
Ontheotherhand,plottingthespectraversustheHubblerateH showsthatchangingH willchangebothmagnetic
i i
and electric field almost in the same manner, see Fig.4. However, for H (cid:39)1.3×10−3M , the electric energy can go
i Pl
over the ρ which causes the backreaction problem. The value H ∼ 1.3×10−3M is well above the upper bound
Inf i Pl
of H reported by Planck (3.6×10−5M ).
i Pl
Similarly, plotting electromagnetic spectra as function of the parameter M, shows that the backreaction problem
can be avoided if M >8.5×10−5M , see Fig.5. It is below (3×10−3M ) the value calculated by the amplitude of
Pl Pl
CMB ([48]).
Asaresultoftheforegoingdiscussion,onecanconcludethatascaleinvariantPMFcannotbegeneratedinLFI,for
p=2, in the limit Eq.(27), which was the now-discredited BICEP2 favored shape of inflationary potential. However,
a scale invariant PMF can be generated if we relax that limit. Also, the backreaction problem can be avoided under
some conditions which fit with some observable scales of k.
For p < 2, there are some interesting cases, such as, p = 1,2/3. Following the same way as done in the previous
subsection, one has to substitute (26) into (44). In this case,
Y (η)= 2(2−4p) 3(−2−2−4p)M8MPl−4−2pα2Θ, (58)
H 4η2
i
where,
(cid:18)M4M −p(2−p)p(c +ln[H η])(cid:19)−2+2−4p
Θ= Pl 2 i . (59)
H 2
i
Using limits (29), (47), and the fact that, ln(η/η )(cid:39)C, since both, η , η (cid:28)−1. Hence, substituting (57) into (40)
f f
yields that,
A(η,k)=(kη)1/2[C (k) J (kη)+C (k) J (kη)], (60)
1 χ 2 −χ
10
FIG. 4: The magnetic and electric spectra, generated under LFI model, in the simple de Sitter expansion, with p=2,k (cid:28)1,
and χ (cid:39) 5/2(γ = −2,3), as a function of H . They both change in the same manner. For H > 1.3×10−3M , the electric
i i Pl
energycangoovertheρ whichcausesthebackreactionproblem. ButthevalueH ∼1.3×10−3M iswellabovetheupper
Inf i Pl
bound of H reported by Planck (3.6×10−5M ). In the plot, we use, k = 10−3, η = −20, α = 2, H = 3.6×10−5M ,
i Pl i Pl
M =3×10−3M and (M ,D)=1.
Pl Pl
FIG.5: Themagneticandelectricspectraandinflationarydensityofenergyρ ,generatedunderLFImodel,inthesimplede
Inf
Sitterexpansion,withp(cid:39)2,k(cid:28)1,andχ(cid:39)5/2(γ =−2,3),asafunctionofM. Theelectromagneticspectraareindependent
ofM. ForM >8.5×10−5M ,theelectricenergycangobelowtheρ whichavoidthebackreactionproblem. Butthevalue
Pl Inf
M (cid:39)8.5×10−5M is well below the one calculated one from the amplitude of CMB, M =3×10−3M . In the plot, we use,
Pl Pl
k=10−3, η=−20, α=2, H =3.6×10−5M , and (M ,D)=1.
i Pl Pl
where, χ can be written as,
√
3−1+2ζ Π
χ= , (61)
2CM 2(−2+p)p
Pl
where,
Π=32(−4+p)ζC2M 4ζ−2p2+
Pl
(62)
(−1)−4ζ22(−4+p)ζC−4ζH 8ζM−16ζM 4pζζ4ζp−4ζα2,
i Pl
and, ξ =(−2+p)−1.
Employing limit (47), the second term under the square root will vanish and the value of (61) reduces to χ(cid:39)1/2.
Therefore, PMF cannot be scale invariant when it is generated in LFI for p<2, under the limit, Eq.(27). In order to