bogusław broda and michał szanecki- dark energy from quantum vacuum fluctuations
TRANSCRIPT
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8/3/2019 Bogusaw Broda and Micha Szanecki- Dark Energy from Quantum Vacuum Fluctuations
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Dark Energy from Quantum Vacuum Fluctuations
Dark Energy from Quantum Vacuum
Fluctuations
Bogusaw Broda and Micha Szanecki
Department of Theoretical Physics, University of dz
Warszawa, 7.IV.2009
Broda, B., Bronowski, P., Ostrowski, M., & Szanecki, M., Vacuum Driven Accelerated Expansion, Ann. Phys.
(Berlin), 17, 855863, 2008; [arXiv: 0708.0530].
Broda, B. & Szanecki, M., Quantum Vacuum and Accelerated Expansion, CRAL-IPNL conferenceDark
Energy and Dark Matter", Lyon 2008; [arXiv: 0812.4892].
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Dark Energy from Quantum Vacuum Fluctuations
Introduction
3 problems in modern physics and cosmology
accelerated expansion of the Universe
dark energy
cosmological constant
quantum vacuum energy density
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Dark Energy from Quantum Vacuum Fluctuations
Introduction
Dozens of candidates for the solution of the problem of the
accelerated expansion.
One of the possibility: quantum vacuum energy. But it does
not work well. Traditionally calculated, Casimir-like value ofquantum vacuum energy density is 2 orders of orders too big
than required!
One should solve the quantum vacuum energy problem
anyway.
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8/3/2019 Bogusaw Broda and Micha Szanecki- Dark Energy from Quantum Vacuum Fluctuations
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Dark Energy from Quantum Vacuum Fluctuations
Introduction
Potential resolutions:
lowering the ultraviolet cutoff scale UV using
supersymmetry arguments
quantum vacuum energy does not influence gravity
or zero-point energy does not gravitate in vacuum
Do not work!
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Dark Energy from Quantum Vacuum Fluctuations
Introduction
But
Possible to reasonably estimate the value of quantum
vacuum energy obtaining an experimentally acceptable
result
Approach does not appeal to any clever, arbitrary or exotic
assumption
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Dark Energy from Quantum Vacuum Fluctuations
Quantum vacuum energy
Standard estimation of the quantum vacuum energy densityvac for a bosonic mode
vac =1
2
UV
0
4
(2)3
c(mc)2 + k2 k2dk. (1)
For a large UV cutoff UV
vac 1
(4)2UV
4
3c . (2)
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D k E f Q V Fl i
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Dark Energy from Quantum Vacuum Fluctuations
Quantum vacuum energy
Setting UV = P (Planck momentum)
P =
c3
G 6.5 kgm/s, (3)
Pvac c5(4)2G2 3.4 1094 kg/m3. (4)
Experimentally estimated value
crit = 3H02/8G( 1026 kg/m3).
More than 120 orders less!
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D k E f Q t V Fl t ti
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Dark Energy from Quantum Vacuum Fluctuations
Quantum vacuum energy
Lowering UV to SUSY 1TeV/c yields
SUSYvac 1.5 1030 kg/m3.
Idea that gravitational field is insensitive to quantum vacuum
fluctuations yields
0vac = 0.
Incorrect results. Obvious also from theoretical point of view.
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Dark Energy from Quantum Vacuum Fluctuations
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8/3/2019 Bogusaw Broda and Micha Szanecki- Dark Energy from Quantum Vacuum Fluctuations
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Dark Energy from Quantum Vacuum Fluctuations
Quantum vacuum energy
Casimir-like calculation should not give any contribution to
gravitational (or any other) field by construction: No
contributions coming from closed loops without external lines,
Above, no classical external lines.
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Dark Energy from Quantum Vacuum Fluctuations
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Dark Energy from Quantum Vacuum Fluctuations
Quantum vacuum energy
One should consider loops with classical external lines.
Symbolically
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Dark Energy from Quantum Vacuum Fluctuations
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Dark Energy from Quantum Vacuum Fluctuations
Implementation of the estimation
Full quantum contribution
Seff =
2
log det
D, (5)
+ bosonic statistics
fermionic statistics
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Dark Energy from Quantum Vacuum Fluctuations
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Dark Energy from Quantum Vacuum Fluctuations
Implementation of the estimation
Formally
Seff = 2Tr logD =
2
0
ds
sTr esD. (6)
UV regularized
Seff =
2
ds
sTr esD
2
ds
sTD(s). (7)
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Dark Energy from Quantum Vacuum Fluctuations
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Dark Energy from Quantum Vacuum Fluctuations
Implementation of the estimation
SeeleyDeWitt expansion
TD(s)
t(s; x)
gd4x , (8)
t(s; x) =1
(4s)2
n=0
an(x)sn. (9)
Other an(x) (for n > 0) contain powers and derivatives of
curvature.
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Dark Energy from Quantum Vacuum Fluctuations
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gy
Implementation of the estimation
Example:a1(x) =
16 R induces classical HilbertEinstein gravity
Sind = 2
1
1
(4)2 1
6R
gd4x = 112
c3
16GR
gd4x,
(10)
where = LP2 = G
c3.
Induced coupling constant is 12 times less than the
standard classical value!
a2(x) and further an(x) yield quantum corrections.
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gy
Implementation of the estimation
Cosmological constant or dark energy induced by a0,
SCas = 2
1
221
(4)2
gd4x
=
4
1
LP4
1
(4)2gd4x
= 14
c6
(4)2G2
gd4x.
(11)
Extract part corresponding to gravitational field!
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Dark Energy from Quantum Vacuum Fluctuations
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Implementation of the estimation
Simplification.
Take the flat FRWL metric
g =
1 0 0 0
0 a2(t) 0 0
0 0 a2(t) 00 0 0 a2(t)
. (12)
t = 0 present moment
Normalize
a(0) = 1. (13)
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Dark Energy from Quantum Vacuum Fluctuations
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Implementation of the estimation
Expanding around t = 0
a(t) = a(0) + a(0)t + a(0)t2 + O(t3)
= 1 + H0t 12
q0H02t2 + O(t3),
(14)
where
H0 a(0)a(0)
= a(0),
q0 deceleration parameter
q0 H02a1(0)a(0) = H02a(0).
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Dark Energy from Quantum Vacuum Fluctuations
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Implementation of the estimation
Hence
g = a
2(t)32
= 1 + 2H0t + (1 q0)H02t2 + O(t3)
32
. (15)
Now discard term linear in t gauge symmetry.
[I. L. Shapiro, Effective action of vacuum: semiclassical approach, Class. Quant.
Grav. 25, 103001 (2008); Arxiv: 0801.0216 [gr-qc], Chapt. 3.1.]
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Dark Energy from Quantum Vacuum Fluctuations
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Implementation of the estimation
Proof: Infinitesimal gauge transformations
g = + , (16)
= (
0
, i) gauge parameters.
1st Eq.
g00 = 2 = 0, (17)
g00 = 1 should be unchanged.
General solution: = (x).
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Implementation of the estimation
2nd Eq.
g0i gi0 = i + i = 0, (18)
g0i = 0 should be unchanged.
Hence i = i(x), and consequently
i = t i(x) + i(x). (19)
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Implementation of the estimation
For spatial indices
gij =ij + ji
= 2t ij(x) + ij(x) + ji(x).(20)
Put
gij =
0, for i= jf(t), for i = j.
(21)
Linear in t function can be gauged away.
Solution
(x) =1
2
3i,j=1
ijxixj, (x) = 0. (22)
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Implementation of the estimation
Constant matrix ij, diagonal
ij =12 H0ij.
Solution
=
1
4H0x
2,12
H0txi
. (23)
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Implementation of the estimation
Now
g = 1 +3
2 (1 q0)H02
t2
+ O(t3
). (24)
Divide by the spatial volumed3x.
Dividing by time is time averaging.
Analysis perturbative in t
the longer t the smaller the reliability.
Shortest time t = TP (the Planck time).
Averaging t around present moment (t = 0)
t limTTP
1
T
T0
dt ( ). (25)
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Dark Energy from Quantum Vacuum Fluctuations
I l t ti f th ti ti
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Implementation of the estimation
Estimated
= 14
c5
(4)2G2lim
TTP
1
T
T
0
dt (
g 1)
14
c5
(4)2G21
2(1 q0)H02TP2.
(26)
No ad hoc subtraction!
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Dark Energy from Quantum Vacuum Fluctuations
Concl sions
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Conclusions
TP2
= G/c5
,
H02 =
8
3Gcrit,
1
48
(1
q0)crit. (27)
exp 0.72 crit,
q0 0.7
0.01exp,
per mode.
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Dark Energy from Quantum Vacuum Fluctuations
Conclusions
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Conclusions
Thanks for your attention
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