Finslerian dipolar modulation of the CMB power spectra at scales \(2<l<600\)
Eur. Phys. J. C
Finslerian dipolar modulation of the CMB power spectra at scales 2 < l < 600
Xin Li 0 1
HaiNan Lin 1
0 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190 , China
1 Department of Physics, Chongqing University , Chongqing 401331 , China
A common explanation for the CMB power asymmetry is to introduce a dipolar modulation at the stage of inflation, where the primordial power spectrum is spatially varying. If the universe in the stage of inflation is Finslerian, and if the Finsler spacetime is nonreversible under parity flip, x → −x , then a three dimensional spectrum which is a function of wave vector and direction is valid. In this paper, a three dimensional primordial power spectrum with preferred direction is derived in the framework of Finsler spacetime. It is found that the amplitude of dipolar modulation is related to the Finslerian parameter, which in turn is a function of wave vector. The angular correlation coefficients are presented, and the numerical results for the anisotropic correlation coefficients over the multipole range 2 < l < 600 are given.

The cosmological principle states that the space is
homogeneous and isotropic on large scales. It is the foundation of the
standard cosmological model, i.e., the CDM model [1,2].
Although the primordial energy density perturbation
generated by the vacuum fluctuation of the inflation field [3–7]
will cause a small perturbative cosmic anisotropy, most
inflation models [8] still preserve the homogeneity and isotropy
of the cosmic microwave background (CMB) [9,10].
However, the recent released data of Planck satellite show that the
CMB temperature map possesses power asymmetry [11–13],
which has also been observed independently by the WMAP
satellite [14–18]. This implies that the statistical isotropy of
CMB may be violated.
The power asymmetry can be described as a dipolar
modulation of the isotropic power [19], i.e., T (nˆ ) =
a email:
b email:
Tiso(nˆ )(1 + A pˆ · nˆ ), where Tiso denotes the statistically
isotropic temperature map, nˆ is the direction of the
temperature fluctuation, pˆ is the preferred direction of the universe,
and A is the dipole amplitude. One possible explanation
for the power asymmetry is the modulation of the
primordial power spectrum, i.e., P(k, r) = P(k)(1 + 2 A pˆ · r/rls)
[20,21], where rls is the distance to the last scattering
surface. One should notice that the primordial power spectrum
P(k, r) is a spatially varying power spectrum, but not a
three dimensional spectrum P(k) with preferred direction
[22]. This is due to the fact that the background space for a
Fourier transformation in the CDM model is a Euclidean
space, while the Euclidean space is reversible under parity
flip, x → −x. This property guarantees that the Fourier
component δ(k) satisfies the relation δ(−k) = δ∗(k).
The violation of statistical isotropy of CMB implies the
violation of rotational symmetry. Several anisotropic
inflationary models have been built to solve the power
asymmetry problem [23–30]. Usually, the background spacetime
of the anisotropic inflation model is described by Bianchi
spacetime [31]. However, the widely discussed Bianchi type
I spacetime [32,33] is reversible under parity flip, x → −x.
It means that inflationary models based on Bianchi type I
spacetime could not generate a primordial dipolar
modulation. Randers space [34], as a special type of Finsler space
[35], is nonreversible under parity flip. In Ref. [36], we have
studied anisotropic inflation in which the background
spacetime is taken to be Finslerian. This Finslerian background
spacetime breaks rotational symmetry and induces parity
violation. The dipolar modulation of the primordial power
spectrum is given in Ref. [36]. At large scales (l < 100),
this model could approximately explain the released data of
Planck satellite for the power asymmetry [11,12]. However,
the amplitude of the dipole modulation in this model is a
constant, which is in contradiction to the fact that the amplitude
of the dipole modulation is scaledependent at intermediate
scales 100 < l < 600 [11–13,37,38]. Recently, Ref. [39]
used the CMB temperature maps released by Planck
satellite to constrain the scale dependence of the amplitude of the
dipole modulation over the multipole range 2 < l < 600.
In this paper, we try to build an inflation model in Finsler
spacetime such that it generates the dipolar modulation of
the primordial power spectrum, and at the same time the
amplitude of the dipolar modulation is scaledependent.
The rest of the paper is arranged as follows: In Sect. 2,
we briefly introduce the basic concepts of Finsler spacetime,
and derive the primordial power spectrum for the quantum
fluctuation of inflation field in Finsler spacetime. In Sect. 3,
we derive the gravitational field equations in the perturbed
Finslerian background spacetime, and obtain a conserved
quantity outside the Hubble horizon. In Sect. 4, we derive
the angular correlation coefficients in our anisotropic
inflation model, and we plot the numerical results of the angular
correlation coefficients that describe the anisotropic effect.
Finally, conclusions and remarks are given in Sect. 5.
2 Inflationary field in Finsler spacetime
Finsler geometry [35] is a generalization of Riemann
geometry without quadratic restriction on the metric. The basic
quantity in Finsler geometry is the Finsler structure F , which
is defined on the tangent bundle of a manifold M , with the
property F (x , λy) = λF (x , y) for any λ > 0, where x ∈ M
represents position and y represents velocity. The
secondorder derivative of the squared Finsler structure with respect
to y gives the Finslerian metric [35]
Hereafter, Greek indices are lowered and raised by gμν and
its inverse matrix gμν , respectively. The Finslerian metric
reduces to Riemannian metric if and only if F 2 is quadratic in
y. The Finslerian spacetime is fully described by the Finsler
structure F .
In this paper, we propose that the background spacetime
of the universe is the Randers spacetime during the stage of
inflation. The Randers spacetime is a special kind of Finsler
spacetime, which has been widely discussed elsewhere [36,
40–43]. The Finsler structure of Randers spacetime takes the
form
F 2 = yt yt − a2(t )FR2a,
where FRa is the Randers space [34]
FRa =
δi j yi y j + b(x) · y, b · y ≡ δi j bi y j .
Now, we focus on finding the equation of motion for the
inflationary field. The action for the singlefield “slowroll”
scalar field is given by
S =
d4x √−g
Noticing that the determinant of the background spacetime
(2) is given by g = −a3 FR2a/(δi j yi y j ), we find from the
action (4) that the equation of motion for the quantum
fluctuation [44,45] field δφ is given by
δφ¨ + 3H δφ˙ − a−2g¯i j ∂i ∂ j δφ + 2∂i δφ
∂(b · y)
where the dot denotes the derivative with respect to time,
H ≡ a˙ /a, and g¯i j is the Finslerian metric on Randers space
FRa. Throughout this paper, the indices labeled by a Latin
letter are lowered and raised by g¯i j and its inverse matrix
g¯i j , respectively. In Finslerian gravity, the direction of y will
appear in the field equation and affect the gravitational field.
During the stage of inflation, following our former treatment
[36,46], we take the direction of y to be parallel to the wave
vector k in the momentum space. Therefore, in the
momentum space, Eq. (5) can be simplified to
δφ¨ + 3H δφ˙ − a−2k2(1 + 5(b · kˆ ))δφ = 0,
where kˆ denote the unit vector along k. We have neglected
the b2 term in deriving Eq. (6). Then, following the
standard quantization process in the inflation model [47], we can
obtain the primordial power spectrum from the solution of
Eq. (6). It is of the form
where P0 is the isotropic power spectrum for δφ, which
depends only on the magnitude of wave vector k.
3 The conserved quantity at horizon crossing
In the standard inflation model [47], the primordial power
spectrum that links to observed quantities of CMB is the
spectrum of the comoving curvature perturbation, which is
conserved outside the Hubble horizon. In this section, we
focus on finding a counterpart to the comoving curvature
perturbation in Finsler spacetime. In the standard process
[47], the conserved quantity at horizon crossing, i.e., the
comoving curvature perturbation is derived by using
gravitational field equations and the long wavelength
approximation, namely, we ignore the terms that are proportional to k2.
This approximation is valid since we are interested in the
modes of quantum fluctuations that are outside the Hubble
horizon during the stage of inflation. In Finslerian inflation,
one should notice that the Finslerian parameter b is only
a function of spatial coordinates, and the long wavelength
approximation means that the secondorder derivative of b
can be neglected. Thus, we can expect from the qualitative
analysis that the conserved quantity at horizon crossing in
Finslerian inflation is the same as the comoving curvature
perturbation in standard inflation model.
In this paper, we just consider the scalar perturbation of
Finsler structure (2). It is of the form
F 2 = (1 + 2 (t, x))yt yt − a2(t )(1 + 2 (t, x))FR2a,
where and are scalar perturbations. There are two
important geometric quantities in Finslerian gravity. One is the
geodesic spray coefficient Gμ, which is related to the
firstorder variation of the Finslerian length. The firstorder
variation of the Finslerian length gives the geodesic equation,
which reads
1 gμν
= 4
In Finsler geometry, the Ricci scalar is given by [35]
The Ricci scalar is a geometric invariant quantity and is
insensitive to the connections. It only depends on the Finsler
structure F . Substituting Eqs. (11) and (12) into Eq. (13), we
obtain
F 2Ric = −3 aa¨ yt yt + (aa¨ + 2a˙ 2)FR2a
+ yt yt a−2g¯i j ,i, j − 3 ¨ + 3H ( ˙ − 2 ˙ )
+ a2 ¨ − g¯i j ,i, j
− FRa g¯i j (b · y),i, j
− 2 ) + aa˙ (6 ˙ − ˙ )
From the relation ∂∂g¯yiij y j = 0, and noticing that the direction
of y is parallel to the wave vector k in momentum space, one
can infer that the term proportional to ∂∂g¯yiij in the righthand
side of Eq. (14) should be vanishing. In Refs. [43,49], it has
been proven that the gravitational field equation in Finsler
spacetime is of the form
where Tνμ is the energymomentum tensor. Here the Ricci
tensor is defined by [50]
and the scalar curvature in Finsler spacetime is given by
S = gμν Ricμν . Combining Eqs. (14)–(16), we obtain the
background field equations
It can be proven that the Finsler structure F is a constant
along the geodesic [35]. Substituting the Finsler structure
(8) into Eq. (10), we obtain
Gi =
− 2 ) + a2 ˙ FR2a,
where the comma denotes the derivative with respect to
spatial coordinate x . Another important geometric quantity is
the Ricci scalar, which is related to the secondorder
variation of Finslerian length. In physics, the secondorder
variation of Finslerian length gives the geodesic deviation
equation which describes the gravitational effect between two
particle moving along the geodesics. The analogy between
geodesic deviation equations in Finsler spacetime and
Riemannian spacetime gives the vacuum field equation in Finsler
gravity [48,49], namely, the vanishing of the Ricci scalar.
and the perturbed field equations in the momentum space
− 2 ˙ ki ,
8π GδTii /3 = 2 ¨ + H 6 ˙ − 2 ˙ −
2 2
8π GδTi j (kˆi kˆ j − g¯i j /3) = 3 k (
) (1 + 2(b · kˆ ))
)(1 + 2(b · kˆ ))
The term δTi j (kˆi kˆ j − g¯i j /3) on the lefthandside of Eq.
(22) denotes the anisotropic stress. The scalar perturbation of
the energymomentum tensor of perfect fluid does not have
an anisotropic stress [47], hence δTi j (kˆi kˆ j − g¯i j /3) = 0.
Therefore, Eq. (22) reduces to
From Eq. (23) we can see that Finsler spacetime (2) will
affect the anisotropic part of the gravitational field equation.
Here, the energymomentum tensor in the above field
equations is derived by the variation of action (4) with respect
to the metric. It is of the form
where T(00)0 and T(i0)i in Eqs. (17, 18) correspond to the
zerothorder part of inflation field, i.e. φ0(t ), and δTνμ in Eqs. (19)–
(22) corresponds to the firstorder part of inflation field, i.e.
δφ (t, x). At horizon crossing, by making use of the long
wavelength approximation, we can neglect the terms
proportional to k2 in Eqs. (19)–(22). Thus, following the approach
of standard inflation model [47], we find from the field Eqs.
(17)–(22) that
= 0,
where the prime denotes the derivative with respect to
confortmhealctoimmeovηin≡g curdavt a,taunrde pHer≡turbaaa.tiEoqnuRatcio≡n H(25δφφ)0 i−mpliiessctohnat
served outside the Hubble horizon. The comoving curvature
perturbation Rc is the same as that in the standard inflation
model. This is expected from the qualitative analysis at the
beginning of this section. Now, we can use Eq. (7) to obtain
the primordial power spectrum for Rc. It is of the form
where Piso is the isotropic power spectrum for Rc.
4 CMB power spectra with dipolar modulation
The amplitude b(k) of the dipolar modulation of the
primordial power spectrum (26) can be constrained by the CMB
temperature map. Reference [39] has given the constraint on
the scale dependence of the amplitude A(l) of the dipole
modulation over the multipole range 2 < l < 600. To relate b(k)
to A(l), we note that they have the same effect on the CMB
correlation coefficients. In Ref. [39], the authors expanded
the dipolar modulation T (nˆ ) = Tiso(nˆ )(1+ A(l) pˆ ·nˆ ) into
the usual spherical harmonics, and one derived the correlation
coefficients of the spherical harmonics, i.e. C X X ,ll ,mm ≡
al∗m alm . The offdiagonal correlation corresponds to the
dipolar term A(l) pˆ · nˆ . In our paper, we derive the dipolar
modulation of primordial power spectrum in Finsler
spacetime. The dipolar modulation of primordial power spectrum
could also induce the offdiagonal correlation C X X ,ll ,mm ,
which will be discussed in the rest of this section. Reference
[39] has constrained A(l) from the Planck data by using the
correlation. We could choose the amplitude of the dipolar
modulation b(k) to be of the same form as A(l), if we want
to account for the correlation. Therefore, the relation between
A(l) and b(k) can be written as
where we have used the approximate relation l ∼ krls in
deriving the above equation, and the distance to the last
scattering surface rls is approximately 14 Gpc. The constraint
on the scale dependence of the amplitude A(l) in Ref. [39]
gives A0 = 0.031+−00..001112 and n = −0.64−+00..1149. In this paper,
we will use the central values of A0 and n given in Ref. [39]
to calculate the angular correlation coefficients.
The general angular correlation coefficients that describe
the anisotropic effect are given by C X X ,ll ,mm [51], where
X represents T or E . By making use of Eq. (26), we obtain
the CMB correlation coefficients for scalar perturbations,
C X X ,ll ,mm =
X,l0(k) ∗X ,l 0(k) Pll mm ,
Pll mm =
PRc Yl∗m Yl m
X,ls(k) denote the transfer functions, and ClLmMl m are the
Clebsch–Gordan coefficients. The CMB correlation
coefficients Cll ≡ C X X ,ll ,mm contain two parts. The one
proportional to δll denotes the statistical isotropy of the CMB.
The other, proportional to C1l0mlm C1l00l0, denotes the statistical
anisotropy of the CMB. The Clebsch–Gordan coefficients
do not vanish only if l = l ± 1. This property implies
that our Finslerian modification of primordial power
spectrum does not affect the isotropic part of correlation
coefficients Cll which describes the statistical isotropy of the
CMB. This is different from the anisotropic inflation models
that induce quadruple modulation of the primordial power
spectrum [51,52].
Reference [39] has used the Planck 2013 data to constrain
the amplitude A(l) of the dipole modulation. To be consistent
with their results, we adopt the bestfitting central value of
cosmological parameters released by the Planck 2013 data
and use the formula of angular correlation coefficients (28)
and (29) to plot numerical results for the anisotropic part
of Cll . One can find from Eqs. (28,29) that the anisotropic
part of Cll depends on m, and the T E and E T correlation
coefficients are different.
The anisotropic T T , T E , E T , E E correlation coefficients
D X X
ll ,mm are plotted in Fig. 1. Here, the coefficients D X X
ll ,mm
are defined as
DlXl X,mm ≡ (2π )−1 l(l + 1)l (l + 1) C X X ,ll ,mm .
The correlation coefficients in m = 0 and m = l cases are
plotted in the left panel and right panel, respectively. From
this figure we can see that the amplitude of the anisotropic
coefficients for m = l is relatively smaller than that for m =
0. This is due to fact that we have already chosen the dipolar
direction along zaxis.
5 Conclusions and remarks
In this paper, we investigated the dipolar modulation of CMB
power spectrum in the framework of Finsler spacetime.
Differing from most previous models in which the primordial
power spectrum P(k, r) is spatially varying, the model we
derived here allows a three dimensional spectrum P(k) with
a preferred direction. This is due to the fact that the
Randers spacetime, a special kind of Finsler spacetime, is
nonreversible under parity flip, x → −x. Such a property has the
consequence that the Fourier component δ(k) does not satisfy
the relation δ(−k) = δ∗(k). Thus, a three dimensional
spectrum P(k) with preferred direction appears if the background
spacetime is Randers spacetime during the stage of inflation.
Following a similar approach to the standard inflation model
[47], we found from the gravitational field equations that the
comoving curvature perturbation Rc is the same as the one
in the standard inflation model. Then we used the primordial
power spectrum of Rc to derive the general angular
correlation coefficients C X X ,ll ,mm . The correlation coefficients
contain two parts. One represents the statistical isotropy of
the CMB, which is similar to that in the CDM model. The
other represents the statistical anisotropy of the CMB, which
does not vanish only if l = l ± 1. This is the main feature
of the dipolar modulation. Finally, we used the mean value
of the amplitude of the dipole modulation in the multipole
range 2 < l < 600 given by Ref. [39] to obtain the numerical
results for the anisotropic part of correlation coefficients.
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