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


- origin of primordial entanglement

- PLANCK constraints on models

- gravity-driven inflation, initial conditions and entanglement

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

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


|0i|0iwithout the spring

with the spring

Ground state

!

1 2 3 4 5

P (n1) P (n2)

0 1 2 3 4 50


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

!

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







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







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]



0.0

0.1

0.2

0.3

0.4

Ten

sor-

to-s

cala

rra

tio

(r)




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


0.0

0.1

0.2

Ten

sor-

to-s

cala

rra

tio

(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]


0.0

0.1

0.2

0.3

0.4

Ten

sor-

to-s

cala

rra

tio

(r)




0.00.1

0.2Ten

sor-to-s

calarra

tio(r)





0.0

0.1

0.2

Ten

sor-

to-s

cala

rra

tio

(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

↳•


Rµ⌫⇢�Rµ⌫⇢� = 1/`40

`P < `0 ⌧ 105`P Planckian phase outside

EB-Fernandez (2016)


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


- entanglement, squeezing and instabilities




- gravity-driven inflation and entanglement

' '

¥t

¥l ' t -

-

*-

¥ 1.Di.t@€÷#1 I T

Pr=

n

i

i••

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