Physics Letters B 790 (2019) 243–247
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Statistical anisotropy in CMB spectral distortions
Atsuhisa Ota
a,b,∗
a
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK
b
Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht, the Netherlands
a r t i c l e i n f o a b s t r a c t
Article history:
Received
13 October 2018
Received
in revised form 16 January 2019
Accepted
16 January 2019
Available
online 21 January 2019
Editor:
H. Peiris
Measurements of the cosmic microwave background (CMB) spectral y-distortion anisotropy offer a test
for the statistical isotropy of the primordial density perturbations on 0.01 k Mpc 1. We compute
the 1-point ensemble averages of the y-distortion anisotropies which vanish for the statistically isotropic
perturbations. For the quadrupole statistical anisotropy, we find 4π y
2m
=−6.8A
2
× 10
−9
Y
2m
(d) with
the quadruple Legendre coefficient of the anisotropic powerspectrum A
2
and the = 2spherical
harmonics Y
2m
(d) for the preferred direction d. Also, we discuss the cosmic variance of the y-distortion
anisotropy in the statistically anisotropic Universe.
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
There exist 10
−5
of anisotropies in the cosmic microwave back-
ground (CMB),
the fossil of the radiation emitted about 380,000
years after the Big Bang [1–3]. These fluctuations are random
fields on top of the statistically isotropic background spacetime,
and cosmic inflation can explain their origin as the quantum fluc-
tuations
in the very early stage of the expanding Universe [4–6].
However, such a rotational invariance is not a mandatory require-
ment.
Indeed several inflationary models can break the rotational
symmetry in the early Universe [7–10]. For example, avector
field during inflation leads to a preferred direction and produces
the quadrupole asymmetry in the primordial powerspectrum of
the density perturbations [8–10]. More generally, spinning parti-
cles
imprint the multipole asymmetry of the primordial corre-
lators [11–13],
and hence the statistical asymmetries are sensi-
tive
to the matter contents during inflation. Thus, the statistical
isotropy is an assumption to be tested through observations, and
its probes have been discussed with the powerspectra of the CMB
anisotropies, the 21-cm lines and galaxies [14–24].
In
this Letter, we point out another way to link the pri-
mordial
statistical anisotropy with a cosmological observable, i.e.,
adeviation of the CMB energy spectrum from the blackbody
one. In particular, we discuss the sensitivity for the short wave-
length (
O(0.01) < k Mpc < O(1.)) statistical anisotropy. We focus
on the spectral y-distortion, a kinetic deviation from the Planck
distribution due to the Compton scattering in the late epoch of
the early Universe [25,26]. Atypical source of the y-distortion is
*
Correspondence to: Department of Applied Mathematics and Theoretical
Physics, University of Cambridge, Cambridge, CB3 0WA, UK.
E-mail
address: a.ota@damtp.cam.ac.uk.
energy release from dissipation of acoustic waves on small scales.
It produces the y-distortions at second-order in the cosmological
perturbations, and hence the ensemble average of them is related
to the short wavelength primordial powerspectrum [27–29]. In
particular, the isotropic spectral distortion has been studied for the
small-scale primordial powerspectrum. Here, we investigate im-
prints
of the primordial statistical anisotropy on the y-distortion
anisotropy. In contrast to Refs. [30–32], we do not discuss the 2- or
higher correlation functions of the spectral distortion anisotropy.
Instead, we compute a simple 1-point ensemble average of the
y-distortion anisotropy in an explicit way based on the second-
order
Boltzmann equation. Such a quantity is zero in the statis-
tically
isotropic Universe, but we show that it is sensitive to the
statistical anisotropy. Also, we comment on the cosmic variance in
the statistically anisotropic Universe.
Primordial
powerspectrum—. Let d be the preferred direction in the
Universe. The powerspectrum of the curvature perturbation on the
uniform density slice ζ depends on d, and we write it in Fourier
space as
ζ
k
1
ζ
k
2
=(2π)
3
δ
(3)
(k
1
+k
2
)P
d
ζ
(k
1
). (1)
The SO(3) rotational symmetry is broken to SO(2) in the presence
of d. The Universe is still statistically symmetric under the rotation
around the d axis; therefore, the anisotropy can be parameterized
by the angle between d and k. Then, we generally expand the
powerspectrum by the Legendre polynomial P
L
(d ·
ˆ
k) as
P
d
ζ
(k) = P
ζ
(k)
∞
L=0
(−i)
L
(2L +1) A
L
(k)P
L
(d ·
ˆ
k), (2)
https://doi.org/10.1016/j.physletb.2019.01.030
0370-2693/
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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