Stephen Hsu- Dark energy and the Future of the Universe

8/3/2019 Stephen Hsu- Dark energy and the Future of the Universe

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Dark energy and the Future of the Universe

Stephen Hsu

ITS, University of Oregon

November 1, 2006 /University of Illinois at Chicago


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Outline

1 Introduction to cosmology

2 Dark energy

3 NEC and instability

4 Summary


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2 Dark energy


4 Summary


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Einsteins equations

Spacetime-matter relation:G = 8GT

g: metricG: the Einstein tensor (built from g and its derivatives)

G: Newtons constant

T: the energy-momentum tensor


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Einsteins equations


g: metric

G: the Einstein tensor (built from g and its derivatives)

G: Newtons constant



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Einsteins equations


g: metric

G: the Einstein tensor (built from g and its derivatives)

G: Newtons constant



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Spacetime

A homogeneous, isotropic spacetime: at every moment t, themetric is the same at every point and in every direction.

The Friedman-Robertson-Walker (FRW)metric (ds2 = gdx

dx):

ds2 = dt2 R(t)2 dr2

1 kr2+ r2d2

k = 1, closed

0, flat1, open


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Spacetime



dx):

ds2 = dt2 R(t)2 dr2

1 kr2+ r2d2

k = 1, closed

0, flat1, open

Compare to the Minkowskian metric:

ds2 = dt2 dr2 r2d2.


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Spacetime



dx):

ds2 = dt2 R(t)2 dr2

1 kr2+ r2d2

k = 1, closed

0, flat1, open


ds2 = dt2 dr2 r2d2.


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Spacetime



dx):

ds2 = dt2 R(t)2 dr2

1 kr2+ r2d2

k = 1, closed

0, flat1, open


ds2 = dt2 dr2 r2d2.

S i


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Spacetime



dx):

ds2 = dt2 R(t)2 dr2

1 kr2+ r2d2

k = 1, closed

0, flat1, open


ds2 = dt2 dr2 r2d2.

M


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Matter

The energy-momentum tensor in a comoving frame:

T =

0 0 00 p 0 00 0 p 00 0 0 p

: the energy density

p: the pressure

= (p): the equation of state

w = p/: the equation of state parameter

examples:

cosmological constant: w = 1

radiation: w = 1/3dust: w = 0

M tt


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Matter


T =

0 0 00 p 0 00 0 p 00 0 0 p


p: the pressure



examples:



w < 1?

M tt


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Matter


T =

0 0 00 p 0 00 0 p 00 0 0 p


p: the pressure



examples:



w < 1?

M tt


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Matter


T =

0 0 00 p 0 00 0 p 00 0 0 p


p: the pressure



examples:



w < 1?

M tt


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Matter


T =

0 0 00 p 0 00 0 p 00 0 0 p


p: the pressure



examples:



w < 1?

Dynamics


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Dynamics

The Einstein equations:

R2

R2=

8

3G

k

R2,

R

R=

4

3G( + 3p).

A particle in 1D:

R2 = 83

GR2 k

12 mx2 = V(x) + E

k = 1, 0, expansion

Dynamics


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Dynamics


R2

R2=

8

3G

k

R2,

R

R=

4

3G( + 3p).

A particle in 1D:

R2 = 83

GR2 k

12 mx2 = V(x) + E

k = 1, 0, expansionk = 1, collapse possible

Dynamics


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Dynamics


R2

R2=

8

3G

k

R2,

R

R=

4

3G( + 3p).

A particle in 1D:

R2 = 83

GR2 k

12 mx2 = V(x) + E


Dynamics


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Dynamics


R2

R2=

8

3G

k

R2,

R

R=

4

3G( + 3p).

A particle in 1D:

R2 = 83

GR2 k

12 mx2 = V(x) + E

V

0

R

k = 1

k = 0

dS


Dynamics


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Dynamics


R2

R2=

8

3G

k

R2,

R

R=

4

3G( + 3p).

A particle in 1D:

R2 = 83

GR2 k

12 mx2 = V(x) + E

V

0

R

k = 1

k = 0

dS

V

0

R

k = 1

k = 0

k = 1

dS

V

0

R

k = 1

k = 0

k = 1

dS

RD


Parameters


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Parameters

H=R

R: the Hubble parameter

c =3H2

8G: the critical density

(= |k=0 10 GeV m3)

=

c: the density parameter

Parameters


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Parameters

H=R


c =3H2


(= |k=0 10 GeV m3)

=


k

R2= H2( 1)

Parameters


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Parameters

H=R


c =3H2


(= |k=0 10 GeV m3)

=


k

R2= H2( 1)

k = 1 closed > 1k = 0 flat = 1

k = 1 open < 1

Parameters


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Parameters

H=R


c = 3H2


(= |k=0 10 GeV m3)

=


k

R2= H2( 1)


k = 1 open < 1

Parameters


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Parameters

H=R


c = 3H2


(= |k=0 10 GeV m3)

=


k

R2= H2( 1)


k = 1 open < 1

Parameters


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Parameters

H=R


c = 3H2


(= |k=0 10 GeV m3)

=


k

R2= H2( 1)


k = 1 open < 1

Fate of the Universe


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energy conservation: / = 3(1 + w)R/Rsolution: R3(1+w)

for k = 0: R t2/[3(1+w)], t (w > 1)



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for k = 0: R t2/[3(1+w)], t (w > 1)

dominant component w Rradiation 1/3 t1/2

dust 0 t2/3

cosmological constant 1 exp (/3)1/2t



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for k = 0: R t2/[3(1+w)], t (w > 1)


dust 0 t2/3

cosmological constant 1 exp (/3)1/2tAt present, both and M, later only .



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for k = 0: R t2/[3(1+w)], t (w > 1)


dust 0 t2/3




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for k = 0: R t2/[3(1+w)], t (w > 1)


dust 0 t2/3




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for k = 0: R t2/[3(1+w)], t (w > 1)


dust 0 t2/3


The Big Rip


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g p

R = R0 exp (/3)1/2(t t0)R = R0

1 + 3

2

12

0(t t0)

2/[3(1+w)]

The Big Rip


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g p

t

lnR

w=1 R = R0 exp (/3)1/2(t t0)

R = R0

1 + 3

2

12

0(t t0)

2/[3(1+w)]

For w < 1/3, the Universeaccelerates, R > 0.

The Big Rip


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t

lnR

w=1

t

lnR

w=1

w =1/3

R = R0 exp (/3)1/2(t t0)R = R0

1 + 3

2

12

0(t t0)

2/[3(1+w)]


The observational data slightly favors w < 1.

The Big Rip


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t

lnR

w=1

t

lnR

w=1

w =1/3

t

lnR

w=1

w =1/3

w > 1/3

R = R0 exp (/3)1/2(t t0)R = R0

1 + 3

2

12

0(t t0)

2/[3(1+w)]



w < 1: after a finite cosmological time, the Universe hits aBig Rip singularity, with infinite acceleration at each point inspace.

The Big Rip


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t

lnR

w=1

t

lnR

w=1

w =1/3

t

lnR

w=1

w =1/3

w > 1/3

R = R0 exp (/3)1/2(t t0)R = R0

1 + 3

2

12

0(t t0)

2/[3(1+w)]




The Big Rip


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t

lnR

w=1

t

lnR

w=1

w =1/3

t

lnR

w=1

w =1/3

w > 1/3

R = R0 exp (/3)1/2(t t0)R = R0

1 + 3

2

12

0(t t0)

2/[3(1+w)]




The Big Rip


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t

lnR

w=1

t

lnR

w=1

w =1/3

t

lnR

w=1

w =1/3

w > 1/3

ttrip

lnR

w=1

w =1/3

w < 1

w > 1/3

R = R0 exp (/3)1/2(t t0)R = R0

1 + 3

2

12

0(t t0)

2/[3(1+w)]




History of the Universe


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History of the Universe


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Comments


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CMB observations: 1.

Geometry is fixed: the Universe is almost flat.

remaining question:

matter content: cosmological constant?

w =?

Comments


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remaining question:


w =?

Comments


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remaining question:


w =?

Comments


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remaining question:


w =?

Cosmological constant


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is a vacuum energy.

Arises from quantum corrections to T. Each field theory

mode (k) contributes a zero point energy (k) = |k| to .



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is a vacuum energy.



Quantum contributions to are

infinite, unless we cut offthemomentum modes k at somescale M.



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is a vacuum energy.





The natural size of M4 is

determined by short-distancephysics.



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is a vacuum energy.



V

x





M MPl is tremendously large.



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is a vacuum energy.



V

x








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is a vacuum energy.



V

x








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is a vacuum energy.



V

x






Negative pressure


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For ordinary equations of state, theenergy density falls as the expansiondoes work on the piston the earlyUniverse gets colder and less denseas it expands.

Cosmological constant: p = = , (negative pressure).Negative work pdVdone by expanding Universe is exactlyenough to create a volume dVwith energy density . Theenergy density and pressure remain constant as the universeexpands. The process is self-sustaining.

The asymptotic evolution of the universe appears to bedetermined by the details of physics at the shortest scales.

Negative pressure


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Cosmological constant: p = = , (negative pressure).Negative work pdVdone by expanding Universe is exactlyenough to create a volume dVwith energy density . Theenergy density and pressure remain constant as the universe

expands. The process is self-sustaining.


Negative pressure


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Negative pressure


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2 Dark energy


4 Summary

Dark energy


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Cosmological constant is an example of dark energy.

From observations of type Ia supernovae:the expansion rate is increasing, R > 0.

Einsteins equation: RR

= 43

G

a

(a + 3pa)

Dark energy


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= 43

G

a

(a + 3pa)

The dominant component must have + 3p < 0 or w < 1/3.

Dark energy


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= 43

G

a

(a + 3pa)


Any such component is called dark energy.

Dark energy


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= 43

G

a

(a + 3pa)



Dark energy


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= 43

G

a

(a + 3pa)



Dark energy


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= 43

G

a

(a + 3pa)



Type Ia Supernovae


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SNe Ia are standardizablecandles.

By measuring the spectraand light curves of a largenumber of SNe Ia, one candetermine R(t) and inferinformation about M and.

Conclusion: dark energy causes distant SNe Ia to be dimmerthan they would be if there were only matter and radiation.

Type Ia Supernovae


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SNe Ia are standardizablecandles.

By measuring the spectra

and light curves of a largenumber of SNe Ia, one candetermine R(t) and inferinformation about M and

.

Conclusion: dark energy causes distant SNe Ia to be dimmerthan they would be if there were only matter and radiation.

CMB anisotropy


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Launched in 1989 COBE(Cosmic Background Explorer)found small anisotropies in thetemperature of the TCMB 2.7 KCosmic Microwave Backgroundradiation.

Launched in 2001 WMAP(Wilkinson Microwave

Anisotropy Probe) had 45 thesensitivity and 33 the angularresolution of COBE.

CMB anisotropy

h d CO


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Launched in 1989 COBE(Cosmic Background Explorer)found small anisotropies in thetemperature of the TCMB 2.7 KCosmic Microwave Backgroundradiation.

Launched in 2001 WMAP(Wilkinson Microwave

Anisotropy Probe) had 45 thesensitivity and 33 the angularresolution of COBE.

CMB anisotropy


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Combined CMB anisotropy

data from WMAP andground-based telescopesagree very well with thestandard CDM

cosmological model.The position of the firstacoustic peak tightlyconstrains the geometry

(curvature) of the universe( 1).

CMB anisotropy


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Combined CMB anisotropy

data from WMAP andground-based telescopesagree very well with thestandard CDM

cosmological model.The position of the firstacoustic peak tightlyconstrains the geometry

(curvature) of the universe( 1).

Future prospectsCombined with galaxy clusterd SN I d CMB


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data, SNe Ia and CMBanisotropy observations tightlyconstrain universe contents.It appears we cannot avoidsome form of dark energy.Future CMB probes (e.g., Planck

to be launched in 2007) andsupernova surveys (SNAP -Supernova/Acceleration Probestill in planning stage) will

allow us to go beyond theCDM model.

Future prospectsCombined with galaxy clusterd t SN I d CMB


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allow us to go beyond theCDM model.Possibilities: dark energy is not, i.e., w 1, time-varying w,etc.

Future prospectsCombined with galaxy clusterd t SN I d CMB


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Future prospectsCombined with galaxy clusterdata SNe Ia and CMB


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Future prospectsCombined with galaxy clusterdata SNe Ia and CMB


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to be launched in 2007) andsupernova surveys (SNAP -Supernova/Acceleration Probestill in planning stage) willallow us to go beyond theCDM model.Possibilities: dark energy is not, i.e., w 1, time-varying w,etc.

A mystery?

Why is so small and yet not zero? Theorists long believed


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Why is so small and yet not zero? Theorists long believedthat would be exactly zero for some magical reason.

Now that observations tell us it is non-zero, we struggle tounderstand why it is non-zero and yet so small:

1/4obs 10

3

eV.

A new fundamental scale of physics?

Maybe is zero (for some unknown reason), and the darkenergy is due to some dynamical field Q, called Quintessence.

All we can say for now is that dark energy is a mystery.

Why is so small? And yet non-zero?

Weinbergs anthropic argument (1987)


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Weinberg s anthropic argument (1987)

Suppose many universes with different cosmologicalconstants. (The string theory Landscape?) How likely is ourvalue obs, given that life exists?

Assume that structure formation (galaxies, stars, etc.) isnecessary for life. (Otherwise, uniform soup of particles!)

For

> 200

obs, the universe becomes

-dominated beforedensity perturbations can grow, and hence no galaxies form.

Perhaps this explains the value of?



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Suppose that we fix all other parameters and vary only . A

flat prior-probability distribution is plausible, since isdetermined by short-distance physics, and the range of viablevalues is quite narrow.

P()|obs constant P( = 0) + O(/M4)

Result: In a Bayesian sense, obs is about 10% probable!

P( < obs us) = obs

0d P(us|)P(),

and assume P(us|) baryon fraction in galaxies.



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A prediction of the observed cosmological constant?

Alas, no. Weinbergs result assumes that all other parametersare held fixed. If one also varies (for example) the amplitude ofprimordial density perturbations (arising, e.g., from inflation),the probability ofobs is reduced substantially to values as lowas 104 (Graesser, Hsu, Jenkins and Wise, 2004).


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2 Dark energy


4 Summary

The null energy condition


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wi = Tii/T00: not Lorentz-invariant +pi 0: from Lorentz-invariantnull energy condition

NEC: Tnn 0

for null n (gnn = 0)

The energy density measured byan observer with the velocityv = n is non-negative.



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wi=

Tii/T00: not Lorentz-invariant +pi 0: from Lorentz-invariantnull energy condition

NEC: Tnn 0

for null n (gnn = 0)

The energy density measured byan observer with the velocity

v = n is non-negative.


x0


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x

x1

x2

n


NEC: Tn

n

0for null n (gn

n = 0)

The energy density measured by

an observer with the velocityv = n is non-negative.


x0


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x

x1

x2

n


NEC: Tn

n

0for null n (gn

n = 0)

The energy density measured by

an observer with the velocityv = n is non-negative.

Energy conditions and w 1


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p p p


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weak null dominant

null dominant strong w 1

p

p

p

p

p

p


p p p

p p p


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weak null dominant


p

p

p

weak null dominant


p

p

p

Perfect fluid

ltypical lmean


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Current: j

= Ju

(uu

= 1)Energy-momentum: T = ( +p)uu pg

Perfect fluid

ltypical lmean


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Current: j

= Ju

(uu


Invariant: J= (jj)

12

(matter density in the rest frame)

Perfect fluid

ltypical lmean


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Current: j

= Ju

(uu


Invariant: J= (jj)

12


Energy: = (J)

Pressure: p = J

Perfect fluid

ltypical lmean


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Current: j

= Ju

(uu


Invariant: J= (jj)

12


Energy: = (J)

Pressure: p = J

Generality: arbitrary equation of state given by (J)

(example: (J)=

Jfor free fluid)

Perfect fluid

ltypical lmean


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Current: j

= Ju

(uu


Invariant: J= (jj)

12


Energy: = (J)

Pressure: p = J


(example: (J)=

Jfor free fluid)NEC: Tnn = 2(un

)(un)

0

0

Perfect fluid

ltypical lmean


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Current: j

= Ju

(uu


Invariant: J= (jj)

12


Energy: = (J)

Pressure: p = J


(example: (J

)= J

for free fluid)NEC: Tnn = 2(un

)(un)

0

0

The NEC requires 0.

Perfect fluid

ltypical lmean


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Current: j

= Ju

(uu


Invariant: J= (jj)

12


Energy: = (J)

Pressure: p = J

Generality: arbitrary equation of state given by (J)(example: (J) = Jfor free fluid)

NEC: Tnn = 2(un

)(un)

0

0

The NEC requires 0.

Perfect fluid

ltypical lmean


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Current: j

= Ju

(uu


Invariant: J= (jj)

12


Energy: = (J)

Pressure: p = J


NEC: Tnn = 2(un

)(un)

0

0

The NEC requires 0.

Perfect fluid

ltypical lmean


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Current: j

= Ju

(uu


Invariant: J= (jj)

12


Energy: = (J)

Pressure: p = J


NEC: Tnn = 2(un

)(un)

0

0

The NEC requires 0.

Clumping instability

Speed of sound: s = (dp/d)12 = (J/)

12


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p ( p ) (J )

Real s: no exponentially-growing modesIf NEC is violated ((J) < 0) (J) < 0



12


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p p


What happens to fluid when (J) < 0 and (J) < 0 ?



12


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12


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12


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(J) = (J)J+12

(J)(J)2



12


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V V

(J) = (J)J+12

(J)(J)2

(J) = 12

(J)(J)2< 0



12


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V VV V

J

(J) = (J)J+12

(J)(J)2

(J) = 12

(J)(J)2< 0

Clumping is energetically favorable.



12


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V VV V

J

V V

J

+

(J) = (J)J+12

(J)(J)2

(J) = 12

(J)(J)2< 0


Perfect fluid is stable only if the null energy condition issatisfied.



12


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V VV V

J

V V

J

+

(J) = (J)J+12

(J)(J)2

(J) = 12

(J)(J)2< 0





12


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V VV V

J

V V

J

+

(J) = (J)J+12

(J)(J)2

(J) = 12

(J)(J)2< 0





12


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V VV V

J

V V

J

+

(J) = (J)J+12

(J)(J)2

(J) = 12

(J)(J)2< 0



Field theory: a simple example

A scalar field with the opposite sign kinetic energy:


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V = 12

m2




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V

L = 12V

V = 12

m2

eikx




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V

L = 12V

V V

L = 12

V L =+1

2+V

V = 12

m2

eikx

Instability: k C




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V

L = 12V

V V

L = 12

V L =+1

2+V

V = 12

m2

eikx

Instability: k C

Energy-momentum: T = Lg




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V

L = 12V

V V

L = 12

V L =+1

2+V

V = 12

m2

eikx

Instability: k C


NEC: Tnn =

n

2< 0




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V

L = 12V

V V

L = 12

V L =+1

2+V

V = 12

m2

eikx

Instability: k C


NEC: Tnn =

n

2< 0




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V

L = 12V

V V

L = 12

V L =+1

2+V

V = 12

m2

eikx

Instability: k C


NEC: Tnn =

n

2< 0




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V

L = 12V

V V

L = 12

V L =+1

2+V

V = 12

m2

eikx

Instability: k C


NEC: Tnn =

n

2< 0

Classical field theories

Background: space-time with fixed metric g

Variables: scalar fields a and gauge fields Aa


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Da= a, + CabcAbc

Fa= Aa; Aa; + CabcAbAc





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Da= a, + CabcAbc


Action: S =

ddx |g|12

L(a, Da, Fa) +

12

f(a)R





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Da= a, + CabcAbc


Action: S =

ddx |g|12

L(a, Da, Fa) +

12

f(a)R

L: an arbitrary Lorentz invariant function

f: an arbitrary function(f = 1 1

2

a a

2a for non-minimal coupling)

R: Ricci scalar





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Da= a, + CabcAbc


Action: S =

ddx |g|12

L(a, Da, Fa) +

12

f(a)R



2

a a


R: Ricci scalar





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Da= a, + CabcAbc


Action: S =

ddx |g|12

L(a, Da, Fa) +

12

f(a)R



2

a a


R: Ricci scalar





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Da= a, + CabcAbc


Action: S =

ddx |g|12

L(a, Da, Fa) +

12

f(a)R



2

a a


R: Ricci scalar

Strategy


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Strategy

L


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Strategy

L

LL


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Strategy

L

LL

LL


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EL

Strategy

L

LL

LL

LL 2L


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ELEL

Strategy

L

LL

LL

LL 2L

LL 2L


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ELELEL 2

H

Strategy

L

LL

LL

LL 2L

LL 2L

LL 2L


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ELELEL 2

HEL 2

H2

K

2V

Strategy

L

LL

LL

LL 2L

LL 2L

LL 2L

LL 2L


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ELELEL 2

HEL 2

H2

K

2V

T

EL 2

H2

K

2V

Strategy

L

LL

LL

LL 2L

LL 2L

LL 2L

LL 2L

LL 2L


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ELELEL 2

HEL 2

H2

K

2V

T

EL 2

H2

K

2V

T

NEC NEC

EL 2

H2

K

2V

Strategy

L

LL

LL

LL 2L

LL 2L

LL 2L

LL 2L

LL 2L

LL 2L


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ELELEL 2

HEL 2

H2

K

2V

T

EL 2

H2

K

2V

T

NEC NEC

EL 2

H2

K

2V

T

NEC NEC

stableunstable

unstableQM

unstable

EL 2

H2

K

2V

Stability

Theorem


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Theorem

For the theory given by the action

S = ddx |g| 12 L(a, Da, Fa) + 12 f(a)R ,only solutions satisfying the null energy condition can bestable.

Stability

Theorem


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Theorem

For the theory given by the action

S = ddx |g| 12 L(a, Da, Fa) + 12 f(a)R ,only solutions satisfying the null energy condition can bestable.

Fermions

The bosonic part as before: L(b) = L(a, Da, Fa) +12

f(a)R


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Add fermions: L(f) = i D m()

Conclusion

If the system with L(b) + L(f) does not satisfy the NEC, the

bosonic degrees of freedom are unstable.

Fermions


f(a)R


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Conclusion



Fermions


f(a)R


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Conclusion



Fermions


f(a)R


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Conclusion



w


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current bounds:1.62 < w < 0.74

(at 95% CL)

R. A. Knop et al., Astrophys. J. 598, 102 (2003) [astro-ph/0309368]


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w


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current boundsand stability:

1 w < 0.74


w


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current boundsand stability:

1 w < 0.74




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2 Dark energy


4 Summary

Summary

Dark energy:

The single largest component (by energy) of the current

and future universe.


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Determines the large time evolution of the Universe.

Summary

Dark energy:




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What is it?

Summary

Dark energy:




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What is it?

NEC and instability:

Summary

Dark energy:




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What is it?


For a very broad class of field theories and perfect fluids,violation of the null energy condition implies instability.


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Summary

Dark energy:


and future universe.D i h l i l i f h U i


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What is it?



For dark energy, w 1.Wormholes and time machines cannot be both stable andpredictable.

Summary

Dark energy:


and future universe.D i h l i l i f h U i


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What is it?




Summary

Dark energy:


and future universe.D t i th l ti l ti f th U i


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What is it?




Summary

Dark energy:


and future universe.D t i th l ti l ti f th U i


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What is it?




Wormholes and time machines

Geodesics:first converging, then diverging

Expansion of a hypersurface orthogonal

null congruence (Landau-Raychaudhuri):


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d

d= 1

22


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d

d= 1

22

0 Tn

n < 0

Exotic matter on the throat stabilizes the wormhole.






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d

d= 1

22

0 Tn

n < 0


To construct traversable wormholes one needs to violate theNEC.






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d

d= 1

22

0 Tn

n < 0








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d

d= 1

22

0 Tn

n < 0




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Two types of devices

Devices:


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Devices:

type A (classical)type B (quantum)

Two types of devices

Devices:


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type A (classical) type B (quantum)

Type A devices are unstable

semiclassical device (semiclassical g)


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semiclassical T

semiclassical ,




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semiclassical T

semiclassical ,

instability




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semiclassical T

semiclassical ,

instability




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semiclassical T

semiclassical ,

instability




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semiclassical T

semiclassical ,

instability


Type A devices are unstable.

Type B devices are unpredictable.


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Conclusion

Wormholes and time machines cannot be both stable andpredictable.





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Conclusion






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Conclusion






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Conclusion


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Stephen Hsu- Dark energy and the Future of the Universe

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