cosmic acceleration from the basics to the frontiers

Cosmic Acceleration

from the basics to the frontiers

Je-An Gu ( 顧哲安 )

National Center for Theoretical Sciences (NCTS)

2007/04/27 @ Academia Sinica

Accelerating Expansion

(homogeneous & isotropic)

Based on FRW Cosmology

Concordance: = 0.73 , M = 0.27

Supernova Observations

Supernova (SN) : mapping out the evolution herstory

Type Ia Supernova (SN Ia) : (standard candle)

– thermonulear explosion of carbon-oxide white dwarfs –

Correlation between the peak luminosity and the decline rate

absolute magnitude M

luminosity distance dL

(distance precision: mag = 0.15 mag dL/dL ~ 7%)

Spectral information redshift zSN Ia Data: dL(z) [ i.e, dL,i(zi) ] [ ~ x(t) ~ position (time) ]

2 4 LdLF F: flux (energy/areatime)

L: luminosity (energy/time)

Distance Modulus Mm 25)pc/( 5 10 MdlogMm L

(z)

history

3.0 ,7.0

model fiducial

)()()(

m

zmzmzm

(can hardly distinguish different models)

SCP(Perlmutter et. al.)

Distance Modulus Mm 25)pc/( 5 10 MdlogMm L

1998

Fig.4 in astro-ph/0402512 [Riess et al., ApJ 607 (2004) 665]Gold Sample (data set) [MLCS2k2 SN Ia Hubble diagram]

- Diamonds: ground based discoveries - Filled symbols: HST-discovered SNe Ia

- Dashed line: best fit for a flat cosmology: M=0.29 =0.71

2004

Riess et al. astro-ph/0611572200625)pc/( 5 10 Mdlog L

2006 Riess et al. astro-ph/0611572

Supernova / Acceleration Probe (SNAP)

z 0~0.2 0.2~1.2 1.2~1.4 1.4~1.7

# of SN 50 1800 50 15

observe ~2000 SNe in 2 years

statistical uncertainty mag = 0.15 mag 7% uncertainty in dL

sys = 0.02 mag at z =1.5

z = 0.002 mag (negligible)

Definition of Acceleration

Accelerating Expansion: Definition

00

00

LL

LL

:ration Accele; :onDecelerati

:Collapse ; :Expansion

22 L

LL

H

LLq

LLH

:parameter onDecelerati

length) pervelocity (~ :rate Expansion

Distance L

E.g. 1. Proper distance (Line Acceleration)

E.g. 2. L = VD1/3 (Domain Acceleration)

, at time t

0 :ration Accele; 0 :onDecelerati

0 :Collapse ; 0 :Expansion

qq

HH

Accelerating Expansion : H > 0 , q < 0

VD

(Volume)

a large domain D (e.g. size ~ H01)

Friedmann-Lemaitre-Robertson-Walker Cosmology

Homogeneous & Isotropic Universe :

22

2

222

2222

;

1)( : distance proper

; 1

)( :metric Walker-Robertson

2

1

aaa

L

LLq

aa

LL

H

rk

drtaL

drrk

drtadtds

LL

r

r

0 , 0 :ration Accele; 0 , 0 :onDecelerati

0, :Collapse ; 0, :Expansion

qaqa

aHaH

Accelerating Expansion : H > 0 , q < 0 )0 , 0( aa

Friedmann-Lemaitre-Robertson-Walker (FLRW) Cosmology

22

2

2222

1 )( :metric (RW) Walker-Robertson dr

rk

drtadtds

Homogeneous & Isotropic Universe :

)( pG

a

a

G

a

k

a

a

33

4

3

82

2

)(

, costant with for

:onconservati momentum-energy

iwi

iiii

a

wwp

apdad

13

33

03

10 ap :onaccelerati ,

(Dark Energy)

Accelerating Expansion: Definition

Gauge Dependence of Acceleration ??

independent of gauge choice (coordinate choice) (frame choice)

Issues

Space Expansion or Particle Motion ??

Gauge-Independent Definition ?

Models

(from vacuum energy)

• Quintessence

Candidates: Dark Geometry vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Dark Geometry

↑Dark Matter / Energy

↑

Gμν ＝ 8πGNTμν

• Modification of Gravity

• Averaging Einstein Equations

• Extra Dimensions

(Non-FLRW)

for an inhomogeneous universe(based on FLRW)

FLRW + CDM

22

2

2222

1dr

rk

drtadtds

)( :metric RW

)( mm

m

pG

a

a

G

a

k

a

a

33

4

3

3

8

32

2

0 a :onaccelerati enough large

1

88

00

pw

Gp

G

pwpm

mmm

,

, i.e. ,

Fine-tuning problems: cosmological constant () problem, coincidence problem

FLRW + CDM : fine-tuning problems

Coincidence problem

Cosmological constant problem

How to make vanish ?Pre-Dark-Energy

very huge if EWSUSYplcutoff MMMM ,,~4~ cutoffvac M

SSB Phase Transition:

Latent heat = vacuum energy (change) ~ TPT (eg. MEW)

How to make slightly deviate from 0 ?Post-Dark-Energy

411energydark eV103

+

Why ~ m NOW ?

Figure 1.1

3aMM

• Ratio changes rapidly with scale factor

• But at present time, M ~

• Why???

Why ΩΛ ~ ΩM now ?

Why accelerating now ?

Ωi ρi / ρc

Problem & Coincidence Problem

Why so small initially ?

FLRW + Quintessence

Quintessence: dynamical scalar field

VggdxS2

14Action :

01

3 222

2

Vat

Ht

Field equation:

Vat

p

Vat

2

2

2

2

2

2

6

1

2

1

2

1

2

1

energy densityand pressure :

Slow evolution and weak spatial dependence V() dominates w ~ 1 Acceleration

How to achieve it (naturally) ??

Non-Quinte: rapidly oscillating mode

: For 22

2

1 mV

22

22

0 ma

ktkx kk , :mode goscillatin an cos~

06

1

2

1 202

220

22

2

a

kpm

a

ktt

,

mode) -(small for ,

mode) -(large for ,

kma

k

kma

kp

wt

t

22

2

22

2

0

3

1

~ radiation

~ NR matter

time-averaged energy density and pressure :

Non-Quinte: ensemble of incoherent oscillators

: For 22

2

1 mV

)( , :soscillator of ensemble an 1210 Niiktkx ,,,cos~

06

1

2

1 202

220

22

2

a

kpm

a

kenen

,

mode) -(small for , 0

mode) -(large for , 3

1

22

2

22

2

kma

k

kma

kp

wen

en

~ radiation

~ NR matter

ensemble-averaged energy density and pressure :

(i : the phase of i-th oscillator)

Non-Quinte: oscillators

Thus, Oscillators

22

2

1 mV for least at

How about other potentials ??

?,,,, 643 V

Quinte: a slowly evolving mode or coherent state

: For 22

2

1 mV

22

22

0 ma

ktkx kk , :solution cos~

00

0

H

k

k ~

~

:evolutionslow

:dependence spatial weak

GeV480 10 ~Hm (unnaturally small !!)

G

HmV c

8

3

2

1 2022 ~~ [V() dominates.]

GeVenergy Planck 1921 10~~/G (unnaturally large !!)

Tracker Quintessence

V

n

nMV

4

MV exp

Inhomogeneous Cosmo. Model

(motivation & final goal: come to the reality)

-- Violating cosmological principle --

Is FLRW Cosmology a good approximation ??

Fundamental Question

If yes, then, WITHOUT DARK ENERGY,

there is NO WAY to generate Cosmic Acceleration.

Is FLRW Cosmology a good approximation ??

Fundamental Question

Acceleration from Inhomogeneity ??

Is FLRW Cosmology

a good approximation ??

…………

YES !YES !YES !

FLRW Cosmology

homogeneous & isotropic

Robertson-Walker (RW) metric

Friedmann-Lemaitre-Robertson-Walker (FLRW) Cosmology

jiij dxdxtadtds 222

Einstein equations: G = 8 G T

Representing the “real” situation of the energy contents of our universe

Is FLRW Cosmology a good approximation ??

(homogeneous & isotropic)

Apparently, our universe isNOT

homogeneous & isotropic.

NO

At large scales, after averaging,

the universe IS homogeneous & isotropic.)( , )(

spacespacetgt

In general, averaging/coarse graining is

NOT VALIDfor Einstein equations.

(due to the non-linearity)

YES

Einstein equations

abmnab GTgG 8

abmnabmnab TGgGgG 8

abmn Tg ,

abmn Tg ,

For which satisfy Einstein equations,

in general

DO NOT.

Is FLRW Cosmology a good approximation ??

NO YES

Contributions from metric perturbations are negligible.

Ishibashi & Wald [gr-qc/0509108]

jiij dxdxtadtds 2121 222

ji

jii

ii

i DDDDDDDDat

2

2

2

, 1

, 1

perturbed metric, non-perturb T

eff

peff

Averaged Einstein equations:

Gphhak

aa

aa

Ghhak

aa

881

81

2

883

815

3

2222

2222

kzkykxth coscoscos

Toy Model[ h(t) << 1 ]

Issues :

(1) Do these requirements fit the real situation of our universe ?

(2) (How much) Can we trust the perturbative analysis ?

Is FLRW Cosmology a good approximation ??

NO YES

jiij dxdxtadtds 2121 222

Newtonianly perturbed metric

terms keeping 22 xt ,

8G peff8G eff

23 t 2t

weff peff / eff

31 /

21

15

xa

21

xa

151 /

cannot generate acceleration

might be significant

Acceleration

from

Inhomogeneity ??

Acceleration from reality ?? -- Don’t know.(i.e. from the inhomogeneities of our universe)

General possibility ?? -- To be discussed

Do we really need

Dark Energy ??

NO YES

FLRW Cosmology:

Acceleration Dark Energy

homogeneous & isotropic

RW metric : jiij dxdxtadtds 222

Einstein equations: (G = 8 G T )

pG

a

a

G

a

k

a

a

33

43

82

2

3

103 pp i.e., , whenonaccelerati

Cosmic acceleration requires negative pressure (repulsive/anti gravity).

based on FLRW cosmology

could be model-dependent

Need Dark Energy ?? YES

Intuitively,

Normal matter

attractive gravity

slow down the expansion

CommonIntuition /

Consensus

We found

line/domain accel. Examples(generated by inhomog)(not by DE)

based on the LTB solution. (Lemaitre-Tolman-Bondi)

(exact solution)

(dust fluid) (spherical symmetry)

Chuang, Gu & Hwang [astro-ph/0512651]

( need dark energy )

Join the dark.

concentrate, balance.…

Do we really need Dark Energy ??

NO YES

Examples of Acceleration : q < 0

)( 0660.

Over-density

Under-density

Acceleration

Deceleration

Deceleration

Acceleration

Inhomogeneity

Examples of Line (Radial) Acceleration : qL < 0

)( 0660.

Acceleration from Inhomogeneity ??

Warning!! Be careful (!!)when connecting two regions.

E.g. FLRWdecel.

FLRWdecel.

Domain Acceleration !!

No physically observable effects of acceleration [regarding,e.g., dL(z)]

There could exist singularity which leads to strange pheno.

E.g. a lesson from Nambu & Tanimoto(incorrect accel. example) [gr-qc/0507057] (perhaps NOT exist at all !!)

Mr. Anderson, …

NOYES

Fake!? Illusion!?

3/1 , 0 DDD VLL

You are illusion !!

Acceleration from Inhomogeneity ??

Issues gauge-dep of acceleration

frame acceleration !?

NOYES

Fake!? Illusion!?

definition of acceleration

A system consisting of freely moving particles

(interacting only through gravity)

Frame Acceleration

Distance L

E.g. 1. Proper distance (Line Acceleration)

E.g. 2. L = VD1/3 (Domain Acceleration)

, at time t

VD

(Volume)

a large domain D

00

00

LL

LL

:ration Accele; :onDecelerati

:Collapse ; :Expansion

(e.g. size ~ H01)

A system consisting of freely moving particles

(interacting only through gravity)

Frame Acceleration

Distance L

E.g. 1. Proper distance (Line Acceleration)

E.g. 2. L = VD1/3 (Domain Acceleration)

, at time t

VD

(Volume)

a large domain D

00

00

LL

LL

:ration Accele; :onDecelerati

:Collapse ; :Expansion

(e.g. size ~ H01)

Frame Acceleration ??

independent of gauge choice (coordinate/frame choice)

Issues

Space Expansion or Particle Motion ??

Definition of Acceleration (revisit)

Gauge-Independent Definition ??

Definition of Acceleration (revisit)

Gauge-independent definition of accelerating expansion ?(maybe no)

Distance L

0 :onDecelerati

0 :onAccelerati

0 :Collapse

0 :Expansion

L

L

L

L

E.g. 1. Proper distance (Line Acceleration)

E.g. 2. L = VD1/3 (Domain Acceleration)

, at time t

VD

(Volume)

a large domain D (e.g. size ~ H01)

L

proper distance between two freely moving particles

constant particle number inside

Consider a system consisting of freely moving particlesInteracting with each other only through gravity

Avoid confusion from particle motion & frame acceleration ?

Benefits of Comoving/Synchronous Gauge

Universal time (?)

Avoiding frame acceleration.

Avoiding confusion about particle motion and space expansion.

) ( jiij dxdxgdtds 22

Definition of Acceleration (revisit)

proper distance between two freely moving particles(line)proper distance between two points fixed in space

constant particle number inside(domain)size of a domain with its boundary fixed in space

Summary and Perspectives

Model: FLRW + CDM

Fine-tuning problems:

cosmological constant problem

coincidence problem?

?

Model: FLRW + Quintessence

Oscillators

22

2

1 mV for least at

Other potentials ?? ? ? gV ,,,, 643

? Other fields ?

Slow evolution and weak spatial dependence V() dominates w ~ 1 Acceleration

GeV480 10 ~Hm (unnaturally small !!)

GeVenergy Planck 1921 10~~/G (unnaturally large !!)

Observations(SN Ia & others) 0RWa

ion AcceleratFLRW

Inhomogeneity Cosmic Acceleration

?

?

?

Acceleration? from DE? from Inhomogeneity?

Do we really need Dark Energy (DE) ??

(definition ?)

(Chuang, Gu & Hwang: mathematical examples)

(data fitting in LTB models)

Reality ? -- Don’t know.

General possibility ?? -- Yes.

Acceleration? from DE? from Inhomogeneity?

Do we really need Dark Energy (DE) ??

Difficulties & limitation stemming from the complexities of :

the complicated energy distribution of our universe the non-linear Einstein equations

Current approaches

Perturbative analysis approach (not convincing)

Utilizing exact solutions of the Einstein equations (toy model, maybe far away from the real situation)

Cannot deal with the full Einstein equations describing our universe with complicated energy distribution

Is FLRW Cosmology a good approximation ??

NO YES

Acceleration from Inhomogeneity ??

whowillwin???

NO

DON’T KNOW

YES

Is FLRW Cosmology a good approximation ??

NO YES

Acceleration from Inhomogeneity ??

whowillwin???

NO

April 30 (Mon)(early morning)

YES