Gravitation Newton’s Law of Gravitation; Kepler’s Laws of Planetary Motion.
Eugenio Bianchi Institute for Gravitation and the Cosmos ... · Eugenio Bianchi Institute for...
Transcript of Eugenio Bianchi Institute for Gravitation and the Cosmos ... · Eugenio Bianchi Institute for...
Entanglement in the sky
Frankfurt Institute for Advanced Studies - Astrocoffee, March 2016
Eugenio Bianchi
Institute for Gravitation and the Cosmos, Penn State
Plan:
* The vacuum
* Inflation and pre-inflationary initial conditions
- entanglement, squeezing and the CMB
Entanglement in the sky
- origin of primordial entanglement
- PLANCK constraints on models
- gravity-driven inflation, initial conditions and entanglement
The Vacuum State of a Quantum Field
The Vacuum State of a Quantum Field
The Vacuum State of a Quantum Field
No particles a(~k) |0i = 0
Vanishing expectation value h0|'(~x) |0i = 0
Non-vanishing correlations
h0|'(~x) '(~y) |0i =
Zd
3~
k
(2⇡)
3
1
|~k|cos
�~
k ·(~x� ~y)
�=
1
2⇡
2
1
|~x� ~y |2
Fluctuations of the field averaged over a region of size R
(�'R)2 ⌘ h0|'R 'R|0i � (h0|'R|0i)2 ⇠ 1
R2
The Vacuum State of a Quantum Field
The Vacuum State of a Quantum Field
The Vacuum State of a Quantum Field
x
z
x
z
N
S
N
S
e� e+
z z
|si = 1p2
⇣| " i| # i � | # i| " i
⌘
⇡0 ! e� + e+EPR pair:
⇡0
Einstein-Podolsky-Rosen (1935)
Entanglement
Entangled oscillators
0 1 2 3 4 5
P (n1) P (n2)
0 1 2 3 4 5
Ground state
|0i|0i
Entangled oscillators
|0i|0iwithout the spring
with the spring
Ground state
!
1 2 3 4 5
P (n1) P (n2)
0 1 2 3 4 50
Entangled oscillators
0 1 2 3 4 5
P (n1) P (n2)
0 1 2 3 4 5
|0i|0i
|⌦i = 1
cosh r
1X
n=0
(tanh r)n |ni|ni
without the spring
with the spring
Ground state
!
Entangled oscillators
0 1 2 3 4 5
P (n1) P (n2)
0 1 2 3 4 5
|0i|0i
|⌦i = 1
cosh r
1X
n=0
(tanh r)n |ni|ni
without the spring
with the spring
Ground state
!
⇢2 = Tr1|⌦ih⌦| =1
Z
1X
n=0
e��n|nihn|
Thermal ensemble from entanglement
Entanglement entropy
S2 = �Tr(⇢2 log ⇢2)
The vacuum state of a quantum field is highly entangled
Squeezing the vacuum
examples:
- Schwinger effect
- Hawking radiation
- Primordial density fluctuations
Squeezing the vacuum and entanglement entropy growthTwo new results: (1) Entanglement growth at instabilities
SA(t) ⌘ �TrA
�⇢(t) log ⇢(t)
�⇠
⇣Pi �i
⌘t
linear growth: rate given by the sum of the positive Lyapunov exponentsBianchi-Hackl-Yokomizo Phys.Rev D (2015)
Squeezing the vacuum and entanglement entropy growthTwo new results: (2) Entanglement growth at parametric resonances
linear growth: rate given by the sum of the positive Floquet exponentsBianchi-Hackl-Yokomizo to appear (2016)
SA(t) ⌘ �TrA
�⇢(t) log ⇢(t)
�⇠
⇣Pi µi
⌘t
Squeezing the vacuum
examples:
- Schwinger effect
- Hawking radiation
- Primordial density fluctuations
Vacuum fluctuations are unstable in the presence of a strong electric field
Schwinger effect (1951)
~c eE0
(mc2)2> 1
E0
Extraction of entangled pairs of electrons and positrons from vacuum fluctuations
� � � � � ��
�
�
�
�
�
�
�
t
S(t) = Entanglement Entropy of Positrons
EB, Hackl and Yokomizo (2016)
e+e�
Hawking radiation (1974)
Starr
t
Event Horizon
Sing
ular
ity
Temperature
Energy flux
* emission of radiation
* energy conservation
Mass loss
-1.0 -0.5 0.0 0.5 1.0 1.5
0.0
0.1
0.2
0.3
0.4
0.5
Vacuum fluctuations are unstable in the presence of a strong gravitational field
S(t) = Entanglement Entropy of Radiation
t
S(t) ⇠ t
48M
EB, Smerlak, Phys.Rev.D (2014)EB, De Lorenzo and Smerlak, JHEP (2015)
Black Hole
Hot Big BangInflation
Planck Scale
CMB
The vacuum of a quantum field before inflation
P (k) =1
k
The vacuum of a quantum field after inflation
P (k) =(aH)2
k 3ns
ns ' 0.96
ns . 1
Mukhanov-Chibisov (1981)
H(t) =1
2p2 +
1
2
�k2 � f(t)
�q2
Mechanism: amplification of vacuum fluctuations by instabilities
Harmonic oscillator with time-dependent frequency
P (k) =1
k
The vacuum of a quantum field before inflation
P (k) =(aH)2
k 3ns
ns ' 0.96
ns . 1
Mukhanov-Chibisov (1981)
The vacuum of a quantum field after inflation
The anisotropies of the Cosmic Microwave Background as observed by Planck
ns = 0.965 ± 0.005
Planck Collaboration, arxiv.org/abs/1502.02114``Planck 2015 results. XX. Constraints on inflation''
`
`ns�1
ns = 0.965 ± 0.005
`Planck Collaboration, arxiv.org/abs/1502.02114``Planck 2015 results. XX. Constraints on inflation''
Hot Big BangInflation
Planck Scale
CMB
Plan:
* The vacuum
* Inflation and pre-inflationary initial conditions
- entanglement, squeezing and the CMB
Entanglement in the sky
- origin of primordial entanglement
- PLANCK constraints on models
- gravity-driven inflation, initial conditions and entanglement
Scenario: correlations present at the beginning of inflation are produced during a pre-inflationary BKL phase via vacuum squeezing
Origin of primordial entanglement
BKL conjecture (Belinsky-Khalatnikov-Lifshitz 1970)
In classical General Relativity, the spatial coupling of degrees of freedom is suppressed in the approach to a space-like singularity
Quantum BKL conjecture (Bianchi-Hackl-Yokomizo 1512.08959)
In quantum gravity, correlations between spatially separated degrees of freedom are suppressed in the approach to a Planck curvature phase
H [gij(x),'(x)] = 0{ limK!1/`4P
[a,�, �gij
(x), �'(x)] =Y
~x
��, �g
ij
(x), �'(x)�
Motivated by the “architecture conjecture” (Bianchi-Myers, CQG 2013)
Origin of primordial entanglement
Pre-inflationary scenario: BKL phase with entanglement entropy growing from a zero law to an area law
� � � � ���
�
�
�
�
�
����� ������ �
��������������������(�)
EB, Hackl and Yokomizo 1512.08959
Plan:
* The vacuum
* Inflation and pre-inflationary initial conditions
- entanglement, squeezing and the CMB
Entanglement in the sky
- origin of primordial entanglement
- PLANCK constraints on models
- gravity-driven inflation, initial conditions and entanglement
0.94 0.96 0.98 1.00Primordial tilt (ns)
0.0
0.1
0.2
0.3
0.4
Ten
sor-
to-s
cala
rra
tio
(r)
Planck+WP+BAO: ⇤CDM + r
Planck+WP+BAO: ⇤CDM + r + (dns/d ln k)
[1303.5082]
PLANCK 2013 constraints on inflationary models
quadratic potential
plateau potential
[Starobinsky 1979]
PLANCK 2015 constraints on inflationary models
0.94 0.96 0.98 1.00Primordial tilt (ns)
0.0
0.1
0.2
0.3
0.4
Ten
sor-
to-s
cala
rra
tio
(r)
Planck+WP+BAO: ⇤CDM + r
Planck+WP+BAO: ⇤CDM + r + (dns/d ln k)
0.94 0.96 0.98 1.00Primordial tilt (ns)
0.00.1
0.2Ten
sor-to-s
calarra
tio(r)
Planck TT,TE,EE+lowP:⇤CDM+r+Neff + meff⌫, sterile
Planck TT,TE,EE+lowP:⇤CDM+r+NeffPlanck TT,TE,EE+lowP:⇤CDM+r
0.94 0.96 0.98 1.00Primordial tilt (ns)
0.0
0.1
0.2
Ten
sor-
to-s
cala
rra
tio
(r)
Planck TT,TE,EE+lowP:⇤CDM+r+Neff + meff⌫, sterile
Planck TT,TE,EE+lowP:⇤CDM+r+NeffPlanck TT,TE,EE+lowP:⇤CDM+r
[1502.02114]
As = (2.196± 0.05)⇥ 10�9{ns = 0.9603± 0.0073
r < 0.11 at k⇤ = (20Mpc)�1
quadratic potential
plateau potential
[Starobinsky 1979]
0.94 0.96 0.98 1.00Primordial tilt (ns)
0.0
0.1
0.2
0.3
0.4
Ten
sor-
to-s
cala
rra
tio
(r)
Planck+WP+BAO: ⇤CDM + r
Planck+WP+BAO: ⇤CDM + r + (dns/d ln k)
0.94 0.96 0.98 1.00Primordial tilt (ns)
0.00.1
0.2Ten
sor-to-s
calarra
tio(r)
Planck TT,TE,EE+lowP:⇤CDM+r+Neff + meff⌫, sterile
Planck TT,TE,EE+lowP:⇤CDM+r+NeffPlanck TT,TE,EE+lowP:⇤CDM+r
PLANCK 2015 constraints on inflationary models
0.94 0.96 0.98 1.00Primordial tilt (ns)
0.0
0.1
0.2
Ten
sor-
to-s
cala
rra
tio
(r)
Planck TT,TE,EE+lowP:⇤CDM+r+Neff + meff⌫, sterile
Planck TT,TE,EE+lowP:⇤CDM+r+NeffPlanck TT,TE,EE+lowP:⇤CDM+r
[1502.02114]
As = (2.196± 0.05)⇥ 10�9{ns = 0.9603± 0.0073
r < 0.11 at k⇤ = (20Mpc)�1
quadratic potential
plateau potential
[Starobinsky 1979]
derived from General Relativity with quantum corrections
S =1
16⇡G
Z �R+ ↵R
2�p
�g d
4x
Slow-roll initial conditions in gravity R+ ↵R2
Action S =1
16⇡G
Z �R+ ↵R
2�p
�g d
4x
Classical FLRW background + quantum perturbation
gµ⌫(~x, t) = gµ⌫(t) + �gµ⌫(~x, t)
Einstein equationsGµ⌫ + ↵Hµ⌫ = 0
Gravity-driven inflation
As =1
18⇡2
`2P↵N2
ns = 1� 2
N
r =12
N2
As = (2.196± 0.05)⇥ 10�9{ns = 0.9603± 0.0073
r < 0.11 at k⇤ = (20Mpc)�1
Planck 2015
{Compatible with data with and N ' 60 ↵ ' (105 `P )
2
Pre-inflationary initial conditions in gravity R+ ↵R2
EB-Fernandez-Satz, in progress
Initial condition in post-Planckian phase
Rµ⌫⇢�Rµ⌫⇢� = 1/`40 with `P < `0 ⌧ 105 `P
Background: select initial conditions s.t. e-folds of slow-roll N � 60
Quantum perturbation: scalar and tensor modes of the geometry local vacuum has low entanglement entropy at the curvature scale
Pre-slowroll phase produces Minkowskian correlations + squeezing
Expected enhancement of power in tensor modes at low `
' '
¥t
li t -
¥=>g⇒t 1.⇒.t€±.
t"su r
Pra
n
i
↳•
Pre-inflationary initial conditions in gravity R+ ↵R2
Rµ⌫⇢�Rµ⌫⇢� = 1/`40
`P < `0 ⌧ 105`P Planckian phase outside
EB-Fernandez (2016)
Pre-inflationary initial conditions in gravity R+ ↵R2
Rµ⌫⇢�Rµ⌫⇢� = 1/`40
`P < `0 ⌧ 105`P Planckian phase outside
' '
¥t
¥l ' t -
-
*-
¥ 1.Di.t@€÷#1 I T
Pr=
n
i
i••N ' 60
- reheating- slow-roll inflation- post-planckian BKL
EB-Fernandez (2016) Entanglement growth in the pre-inflationary phase
Summary:
* The vacuum
* Inflation and pre-inflationary initial conditions
- entanglement, squeezing and instabilities
Entanglement in the sky
- origin of primordial entanglement
- PLANCK constraints on models
- gravity-driven inflation and entanglement
' '
¥t
¥l ' t -
-
*-
¥ 1.Di.t@€÷#1 I T
Pr=
n
i
i••