Ⅰ. HIGHLIGHTS OF THE YEARRIKEN Accel. Prog. Rep. 48 (2015)

Entanglement entropy of de Sitter space α-vacua†
N. Iizuka,∗1 T. Noumi,∗2 and N. Ogawa∗2The de Sitter space is a very interesting space-time.
It is a solution of Einstein equation when cosmologicalconstant dominates, and it is related to the inflationary
stage of our universe, as well as to the current stageof accelerating universe. A peculiar property of the deSitter space is that the de Sitter invariant vacuum isnot unique; it has a one-parameter family of invariantvacuum states |α, called α-vacua1,2) .
The α-vacua give very peculiar behavior for the twopoint functions in the de Sitter space; the two-point
functions on α-vacua between points x and y containnot only the usual short distance singularity δ(|x − y|),where |x − y| is the de Sitter invariant distance between x and y, but also contain very strange singularity such as δ(|x − ȳ|) and δ(|x̄ − y|), where x̄ and ȳ
represent the antipodal points of x and y, respectively.
Since antipodal points in the de Sitter space are notphysically accessible due to the separation by a horizon, one cannot have an immediate reason to discard
two-point functions containing such an antipodal singularity. It is therefore unclear which vacuum should
be realized in our universe. As a result, a number ofstudies have been done on phenomenological aspects ofthe α-vacua (e.g. primordial perturbations generated
during inflation).
Since which vacuum one should choose is always avery important question, one is motivated to calculate
physical quantities not only in a particular vacuum but
also in others, and see if there is a big reason to choose
or discard a particular vacuum.

SEE
V
0.20

0.15

0.10

0.05

0.2

0.4

0.6

0.8

1.0

1.2

1.4

SEE
V

0.10
0.08
0.06
0.04
0.02

1.0

0.5

0.5

1.0

Α

Fig. 2. Plot of SEE /V against α, for ν = 0 (red), 0.25(orange), νc = 0.4062 . . . (green), 0.43 (blue) and 0.5(purple).

in de Sitter α-vacua. By generalizing the recent calculation by Maldacena and Pimentel3) in the Euclidean(or Bunch-Davies) vacuum for free scalar fields, we derived how entanglement entropy depends on α. The
results are shown in Fig. 1 and Fig. 2.
As is seen in Fig. 2, the entanglement entropy increases significantly as we take α very large for genericvalues of ν. However only for ν = 1/2 and 3/2, thistendency disappears. Note that ν = 1/2 is the conformal mass and ν = 3/2 is massless. It is interesting tounderstand more physically why such a mass dependence occurs.
Our calculation is done for the free scalar field.
Therefore direct comparison with the holographic calculation for the Euclidean vacuum3) is difficult. It
must be interesting to ask how the calculation of entanglement entropy on the α-vacua can be done in
the strong coupling limit via holography, a la RyuTakayanagi formula4) . Understanding these will hopefully shed more light on the question of which vacuum
one should choose in the de Sitter space. We hope tocome back to this question in near future.
References

ΝFig. 1. Plot of SEE /V against ν, for α = 0 (red), 0.1 (orange), 0.25 (yellow), 1 (green) and 2 (blue). Notice theperiodicity and reflection symmetries.

1) E. Mottola, Phys.Rev. D31 (1985) 754.
2) B. Allen, Phys.Rev. D32 (1985) 3136.
3) J. Maldacena and G. L. Pimentel, JHEP 1302 (2013)
038, [arXiv:1210.7244].
4) S. Ryu and T. Takayanagi, Phys.Rev.Lett. 96 (2006)
181602, [hep-th/0603001].

In this work, we computed the entanglement entropy†∗1∗2

Condensed from the article arXiv:1404.7487 [hep-th].
Department of Physics, Osaka University
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