Radiation spectra for h.d. SdS black holes:
The effect of the temperature
Thomas Pappas
Theory Division
Physics Department
University of Ioannina
TP and Kanti Panagiota
arXiv: 1704.XXXXX
April 08, 2017
(Work supported by Alexander S. Onassis
Public Benefit Foundation)
Thomas Pappas (University of Ioannina)
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Outline
ä The gravitational background
ä Black hole radiation
ä The temperature definition
ä Effective temperatures
ä Power spectra
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The gravitational background
Consider the gravitational action in D = 4 + n dimensions
Z
SD =
√
d4+n x −g
RD
−Λ
2κ2D
(1)
The corresponding Einstein equations lead to the h.d. Schwarzschild de-Sitter
solution in the bulk
ds2 = −h(r)dt2 +
dr2
+ r2 dΩ22+n
h(r)
(2)
The induced metric on the brane is obtained by projection
ds2 = −h(r)dt2 +
dr2
+ r2 (dθ2 + dφ2 sin2 θ)
h(r)
(3)
where the metric function in both cases is
h(r) = 1 −
Thomas Pappas (University of Ioannina)
µ
rn+1
− Λ̃r2 , Λ̃ :=
2κ2D Λ
.
(n + 2)(n + 3)
(4)
April 08, 2017
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Depending on the values of µ and Λ̃ the equation
h(r) = 1 −
µ
rn+1
− Λ̃r2 = 0
(5)
will in general have n+3 roots corresponding to the horizons of this spacetime.
Constraining the parameter space in the following way 1
µ2 Λ(n+1) <
4(n + 1)(n+1)
(n + 3)(n+3)
(6)
ensures that only 2 real and positive horizons exist. The black hole horizon rh
and the cosmological horizon rc where always
rh 6 rc .
(7)
In the extreme case where the 2 horizons coincide we have the so called Nariai
limit (black hole).
1
C. Molina, Phys. Rev. D 68, 064007 (2003)
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The global maximum of h(r)
1.2
1.0
0.8
h(r)
0.6
0.4
0.2
0.0
0
Thomas Pappas (University of Ioannina)
50
100
r
150
200
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We assume a free, massless scalar field Φ with a non-minimal coupling to gravity:
S=−
1
2
Z
√ d4+n x −g ξΦ2 RD + g µν Φ,µ Φ,ν
(8)
The Hawking radiation spectrum can be calculated by
X Nl,n |Al,n (ω)|2 ω
d2 E
(2l + n + 1)(l + n)!
=
, Nl,n =
.
dtdω
2π exp(ω/TH ) − 1
l!(n + 1)!
(9)
l
The function |Al,n (ω)|2 is the greybody factor that modifies the blackbody
radiation spectrum to that of a black hole.
Analytic2 (valid only in the low-ω , low-Λ regime)
and numerical3 (valid for all Λ and ω )
calculations have been performed for |Al,n (ω)|2 in h.d. SdS.
2
3
P. Kanti, TP and N. Pappas, Phys. Rev. D 90, no. 12, 124077 (2014)
TP, P. Kanti and N. Pappas, Phys. Rev. D 94, no. 2, 024035 (2016)
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The black hole temperature
The standard definition of the temperature of the black hole is given in terms of
the surface gravity 4
Ti =
κi
,
2π
κ2i := −
1
lim Kµ;ν K ν;µ ,
2 r→ri
K = γt
∂
.
∂t
(10)
For spherically-symmetric gravitational backgrounds
1
κi = γt lim |h(r),r |
2 r→ri
(11)
The normalization constant γt for the asymptotically flat case
(Λ → 0 , rc → ∞) is taken to be
γt = 1 ,
4
K µ Kµ = −1 ,
lim |h(r)| = 1
(12)
r→∞
J.M. Bardeen, B. Carter and S.W. Hawking, Comm. Math. Phys. 31 (1973) 161
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Assuming rc rh each horizon can have its own independent
thermodynamics5 and so the corresponding temperatures are
i
1 h
κh
=
(n + 1) − (n + 3)Λ̃rh2
2π
4πrh
i
κc
1 h
Tc = −
=−
(n + 1) − (n + 3)Λ̃rc2
2π
4πrc
T0 =
(13)
(14)
Bousso and Hawking proposed 6 an alternative normalization to account for the
asymptotic non-flatness of SdS:
TBH =
i
h
κh
1
1
p
=
(n + 1) − (n + 3)Λ̃rh2
2π
4πrh h(r0 )
(15)
At r = r0 given by lim h0 (r) = 0 there is a preferred non-accelerated
r→r0
observer where the gravitational attraction and Λ-repulsion cancel out.
lim |h(r)| ' 1
(16)
r→r0
5
6
D. Kastor and J. H. Traschen, Phys. Rev. D 47 (1993) 5370
R. Bousso and S. W. Hawking, Phys. Rev. D 54 (1996) 6312
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Effective temperatures
Another approaches have emerged during the recent years through effective
temperatures involving both the black hole Th and the cosmological horizon Tc
temperatures.
A thermodynamical first law7 for SdS black holes:
dM = T dS + P dV
(17)
with the mass of the black hole in the role of the enthalpy M = −H ,
P = Λ/8π and S = Sh + Sc leads to an effective temperature8 :
1
1 −1
T0 Tc
Tef f − =
−
=
T0 Tc
T0 − Tc
(18)
Good news: In the limit rh → 0 , Tef f − → Tc
Bad news: In the limit rc → ∞ , Tef f − → 0 ⇒ Effective temperatures
are invalid for Λ = 0
7
8
J.M. Bardeen, B. Carter and S.W. Hawking, Comm. Math. Phys. 31 (1973) 161
M. Urano, A. Tomimatsu and H. Saida, Class. Quant. Grav. 26 (2009) 105010
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Worse news: Tef f − exhibits some unphysical properties, especially for
RNdS black holes → non-positive, infinite jumps at the critical point9 .
To circumvent these issues, an "ad-hoc" effective temperature was proposed10
in analogy to Tef f − but under the assumption: S = Sc − Sh
Tef f + =
1
1
+
T0 Tc
−1
=
T0 Tc
T0 + Tc
(19)
Again for rh → 0 , Tef f + → Tc and for rc → ∞ , Tef f + → 0.
Inspired by the above we propose11 another expression for the effective
temperature:
Tef f BH =
9
10
11
1
TBH
1
−
Tc
−1
=
TBH Tc
TBH − Tc
(20)
For the lukewarm solution when Th = Tc
D. Kubiznak, R. B. Mann and M. Teo, Class. Quant. Grav. 34 (2017) no.6, 063001
In an upcoming paper that is to appear soon
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April 08, 2017
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0.5
0.5
0.4
0.4
T0
0.3
T BH
T eff+
0.2
T effT effBH
0.1
0.0
Temperature
Temperature
Temperatures as functions of Λ (in units of rh−2 ) and n
0
5
T0
T BH
T eff+
T eff-
0.3
T effBH
0.2
0.1
10
15
Λ (n = 5)
Thomas Pappas (University of Ioannina)
20
0.0
0
1
2
3
n (Λ = 0.8)
4
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11 / 17
Power spectra on the brane - ξ = 0
0.020
0.30
T0
TBH
0.25
TeffTeff+
0.010
TeffBH
TBH
0.15
0.
Teff-
0.05
0.1
Teff+
0.10
0.005
0.003
T0
0.20
EER
EER
0.015
0.007
TeffBH
0.05
0.000
0.0
0.5
1.0
ω
n = 2 , Λ = 0.8
Thomas Pappas (University of Ioannina)
1.5
2.0
0.00
0.0
0.5
1.0
1.5
2.0
2.5
ω
n=2,Λ=5
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Power spectra in the bulk - ξ = 0
1.2
0.007
Teff+
0.003
TeffBH
EER
Teff-
0.004
0.8
Teff+
0.6
TeffBH
0.02
0.
0.05
0.1
0.4
0.002
0.2
0.001
0.000
0.0
TBH
Teff-
TBH
0.005
EER
1.0
T0
0.006
0.05
T0
0.5
1.0
ω
n = 2 , Λ = 0.8
Thomas Pappas (University of Ioannina)
1.5
2.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
ω
n=2,Λ=5
April 08, 2017
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Power spectra on the brane - ξ = 1
0.008
T0
TBH
Teff-
TeffTeff+
0.004
TeffBH
0.002
0.000
0.0
TBH
0.20
EER
EER
0.006
0.25
T0
0.15
Teff+
TeffBH
0.10
0.05
0.5
1.0
1.5
ω
n = 2 , Λ = 0.8
Thomas Pappas (University of Ioannina)
2.0
2.5
0.00
0.0
0.5
1.0
ω
1.5
2.0
n=2,Λ=5
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Power spectra in the bulk - ξ = 1
0.8
T0
0.0020
TBH
TeffTeffBH
0.0010
Teff0.4
Teff+
TeffBH
0.2
0.0005
0.0000
0.0
TBH
Teff+
EER
EER
0.0015
T0
0.6
0.5
1.0
1.5
ω
n = 2 , Λ = 0.8
Thomas Pappas (University of Ioannina)
2.0
2.5
0.0
0.0
0.5
1.0
1.5
ω
2.0
2.5
3.0
n=2,Λ=5
April 08, 2017
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Conclusions
ä Various Tef f introduced to take in to account both horizons of SdS are
valid only for Λ 6= 0 and lead in general to significantly suppressed EERs
compared to the more "traditional" TH definitions.
ä Tef f get suppressed with n while T0 and TBH get enhanced.
ä The effect of Λ varies. In the low-Λ regime T0 and TBH dominate while for
the Nariai limit only TBH and Tef f − assume non vanishing values.
ä The inclusion of a non-minimal coupling (effective mass term for Φ) results
in general suppression of the EERs and affects Tef f the most since the
low-ω part of the spectrum is "flattened-out".
ä In all cases studied here the radiation spectrum for an SdS black hole with
TBH is the most dominant one both in the bulk and on the brane.
ä What about the total emissivity on the brane and in the bulk? (in progress)
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Thank you!
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April 08, 2017
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