04/22/23 1

Attempts to explain the nature of dark energy -- How desperate can we get?

Ed Copeland University of Sussex

1. Quintessence – tracking solutions.

2. Models (some inspired by particle physics).

3. K-essence v Quintessence

4. Evidence for evolving dark energy (enter WMAP) ?

5. Problems facing models of dark energy.

Cochin, India, Jan 5th, 2004

04/22/23 2

``Disks represent an aging and expanding universe.

Work this year confirmed a bizarre story of how the cosmos was born and what it is

made of.

Dark energy is the primary ingredient in a universe whose expansion rate and age are

now known with unprecedented precision.’’

Science Magazine -- Breakthrough of the year -Dec 2003

04/22/23 3

Big result in cosmology: still there in 2004

Lum dist to over 170Type 1a SN -- Universe is accelerating from z of order 1.

New sort of matter driving acceleration.-- Where from?

04/22/23 4

The problem with the cosmological constantEinstein (1917) -- static universe

with dust

Not easy to get rid of it, once universe found to be expanding.

Anything that contributes to energy density of vacuum acts like a cosmological constant

Lorentz inv

or

Effective cosm const Effective vac energy

Age Flat Non-vac matter

04/22/23 5

Hence:

Problem: expect <> of empty space to be much larger. Consider summing zero-point energies of all normal modes of some field of mass m up to wave number cut off >>m:

Planck scale:

But:

Must cancel to better than 118 decimal places.

Even at QCD scale require 41 decimal places!

04/22/23 6

Different approaches to Dark Energy include amongst many:

A true cosmological constant -- if so, why this value? Many possible cosmological constants (false vacua) A time-dependent cosmological constant. Solid –dark energy such as arising from frustrated

network of domain walls. Time dependent solutions arising out of evolving

scalar fields -- Quintessence/K-essence. Modifications of Einstein gravity leading to

acceleration today.

Over 300 papers on archives since 1998 with dark energy in title.

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Key equations :

Friedmann eqn

Fluid eqn.

Acceleration eqn

G82 π≡κwhereNote:

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Universe dom by Quintessence at:

If:

Univ accelerates at:

Coincidence problem – why now?

04/22/23 9

Tracker solutionsWetterich,

Peebles and Ratra,

Zlatev, Wang and Steinhardt

Scalar field: )(V2

p;)(V2

:22

φ−φ

=φ+φ

=φ

•

φ

•

φ

EoM:)(

3H B

22 +

κ= φ

φ−φ−=φ

γ−=

γ+φκ

−=

•••

•

••

ddVH3

H3

)(2

H

BB

B2

2+ constraint:

Intro:H6

x•

φκ= H3

Vy κ=

φκ−

≡λddV

V1 ⎟⎠

⎞⎜⎝⎛κλφ

≡−Γ1

dd1

04/22/23 10

Eff eqn of state: 22

2

2

2

yxx2

2V

+=

φ+

φ=γ

•

•

φ 222

2

yxH3

+=κ

=Ω φφ

Friedmann eqns and fluid eqns become:

( )[ ]2222 yx1x2x23y

23x3'x −−γ++λ+−=

( )[ ]222 yx1x2y23xy

23'y −−γ++λ−=

1yxH3

222

2

=++κ γ where )a(lnd/d/ ≡

Note: 10:20 ≤Ω≤≤γ≤ φφ

04/22/23 11

Generic behaviour Ng, Nunes and Rosati

1. PE KE

2. KE dom scalar field energy den.

3. Const field.

4. Attractor solution: almost const ratio KE/PE.

5. PE dom.

Attractors make initial conditions less important

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Example of late time domination of energy density -- 20 seconds left!

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Original Quintessence model

κφα−

=λα

≡−Γ11( ) α

α+

φ=φ

4MV

α+−α

=φ 22ww B

Peebles and Ratra;

Zlatev, Wang and Steinhardt

)2()w1(3

ii

B

aa α+

+

⎟⎟⎠⎞

⎜⎜⎝⎛

φ=φFind: and

6=α

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Fine Tuning in QuintessenceNeed to match energy density in

Quintessence field to current critical energy density.

( ) α

α+

φ=φ

4MV 4472

20 GeV10H −

≈κ

≤

Find: 222

22c H3

Vy φκ∝κ

= so: pl02

22 MVH ≈φ⇒κ∝

φ= φ

Hence:

04/22/23 15

A few models1. Inverse polynomial – found in SUSY QCD - Binetruy

2. Multiple exponential potentials – SUGR and String compactification.

( )φκλ−φκλ− +=

+=φ21 eVeV

VVV

0201

21

Enters two scaling regimes depends on lambda, one tracking radiation and matter, second one dominating at

end. Must ensure do not violate nucleosynthesis constraints.

Barreiro, EC, Nunes

04/22/23 16

5.0;20 =β=α

Scaling for wide range of i.c.

Fine tuning: 434470 )eV10(GeV10V −−

φ ≈≈≈

eV10MVm 33

2pl

0 −≈≈Mass:Fifth

force !

04/22/23 17

Quintessential Inflation – Peebles and Vilenkin

€

V φ( ) = λ (φ4 + M 4 ) for φ < 0

= λM 4

1+ φ M( )αfor φ ≥ 0

Same field provides both initial inflaton and todays Quintessence – not tracker.

Reheating at end of inflation from grav particle production Ford

Avoids need for minima in inflaton potential,GeV10M;47.0:10 514

0==α⇒=Ω=λ φ

−

Need to be careful do not overproduce grav waves.

04/22/23 18

Quintessential Axion -- Kim and Nilles

Light CDM axion (solve strong CP problem) with decay const through hidden sector squark condensation:

Quintaxion (dark energy) with decay const as expected for model independent axion of string theory:

Linear combination of two axions together through hidden sector supergravity breaking.

Model works because of similarities in mass scales:

Scale of susy breaking and scale of QCD axion.

Scale of vacuum energy and mass of QCD axion.

Potential for quintaxion remains very flat, because of smallness of hidden sector quark masses, ideal for Quintessence.

Quintessence mass protected through existence of global symmetry associated with pseudo Nambu-Goldstone boson.

04/22/23 19

Acceleration from new Gravitational Physics? Starobinski 1980, Carroll et al 2003

Modify Einstein

Const curv vac solutions:

de Sitter or Anti de Sitter

Transform to EH action:

Scalar field min coupled to gravity and non minimally coupled to matter fields with potential:

04/22/23 20

Cosmological solutions:1. Eternal de Sitter - just reaches Vmax

and stays there. Fine tuned and unstable.

2. Power law inflation -- overshoots Vmax , universe asymptotes with wDE=-

2/3.

3. Future singularity-- doesn’t reach Vmax , and evolves back towards =0.

Fine tuning needed so acceleration only recently: ~10-33eV

Also, any modification of Einstein-Hilbert action needs to be consistent with classic solar system tests

of gravity. Not obvious these models are!

04/22/23 21

Quintessence and M-theory -- where are the realistic models?

`No go’ theorem: forbids cosmic acceleration in cosmological solutions arising from compactification of pure SUGR models where internal space is

time-independent, non-singular compact manifold without boundary --[Gibbons]

Why? : 1.acceleration requires violation of strong energy condition.

2. Strong energy condition not violated by either 11D SUGR or any of the 10D SUGR theories

3. For any compactification described above, if higher dim stress tensor satisfies SEC then so does the lower dimensional stress tensor.

i.e

Must avoid no-go theorem by relaxing conditions of the theorem.

04/22/23 22

1. Drop condition that internal space is compact, but not so realistic -- Townsend

2. Allow internal space to be time-dependent, analogue of time-dependent scalar fields -- Lukas

et al, Kaloper et al, Townsend & Wohlfarth, Emparan & Garriga.

Compactified spaces are hyperbolic and lead to cosmologies with transient accelerating phase. Four

dimensional picture, solutions correspond to bouncing the radion field off its exponential potential.

Acceleration occurs at the turning point where the radion stops and potential energy momentarily

dominates.

04/22/23 23

Emparan & Garriga

• Field starts at large positive values, with large kinetic energy.

• At turning point, energy is pot dominated and acceleration.

• Left picture, two positive potentials, right picture, sum of positive and negative potentials.

Problems:Difficult to obtain sustained period of inflation.

Current realistic potentials are too steep

These models have kinetic domination, not matter domination before entering accelerated phase.

04/22/23 24

However progress is being made to obtain inflation in string theory:

Metastable de Sitter string vacua in TypeIIB string theory, based on stable highly warped IIB

compactifications with NS and RR three-form fluxes. [Kachru et al 2003]

There remain fine tuning issues in these brane models concerning the method of volume stabilisation, the warping of the internal space and the source of the inflationary energy scale. [Kachru et al 2, 2003]

Still early days for inflation in string/M-theory.

04/22/23 25

Dymanical approach to cosmological constant [Mukohyama and Randall 2003]

with

and

Pot min at negative value.

As well as std kinetic term, non standard kinetic term-coeff diverges at zero curvature.

Causes lowest energy state never to be achieved.

Instead cosmological constant stalls at or near zero.

Model stable under radiative corrections, leads to stable dynamics.

Still requires reheating

Also issues over solar system tests as earlier.

R~0

04/22/23 26

K-essence v Quintessence

K-essence -- scalar fields with non-canonical kinetic terms. Advantage over Quintessence through solving the

coincidence model? -- Armendariz-Picon, Mukhanov, Steinhardt

Long period of perfect tracking, followed by domination of dark energy triggered by transition to

matter domination -- an epoch during which structures can form.

Eqn of state can be <-1

04/22/23 27

Fine tuning in K-essence as well: -- Malquarti, EJC, Liddle

Not so clear that K-essence solves the coincidence problem. The basin of attraction into the regime of tracker solutions is small compared to those

where it immediately goes into K-essence domination.

Shaded region is basin of attraction for stable

tracker solution at point R. All other

trajectories go to K-essence dom at point

K.

Based on K-essence model

astro-ph/0004134, Armendariz-Picon et

al.

04/22/23 28

Ω+Ω=Ω m0

Enter CMBR:

Provides clue. 1st angular peak in power spectrum.

0peak

220lΩ

≈

€

Ω0=1.095−0.144+0.094

WMAP-Depends on assumed priors

Tegmark et al 2003

Evidence for Dark Energy?

04/22/23 29

Concordat between SN1a and CMBR

SN1a measures Ω−Ωm)3/4(

Ω+Ω=Ω m0CMB measures Almost

orthogonal

3/2;3/1;1 m0 ≅Ω≅Ω≅Ω

Consistent with 2df LSS survey and clusters.

Efstathiou et al, astro-ph/0109152

04/22/23 30

Evidence for dynamical dark energy ?

Ideally look for evidence in evolution of equation of state as go back in time.

1. Precision CMB anisotropies – lots of models currently compatible.

2. Combined LSS , SN1a and CMB data – tend to give wQ<-0.85 difficult to tell from cosmological constant.

3. Look for more SN1a – SNAP will find over 2000 – can then start to constrain eqn of state.

4. Constraining eqn of state with SZ cluster surveys – compute number of clusters for given set of cosm parameters.

5. Probing the Dark Energy with Quasar clustering – redshift distortions constrain cosm parameters –sensitive to matter-lambda combination.

6. Reconstruct eqn of state from observation – offers hope of method indep of potentials – example is Statefinder method.

7. Look for evidence in variation of fine structure constant.

04/22/23 31

How much dark energy is there?

WMAP + SDSS: lots

flatclosedopenTegmark et al.

astro-ph/0310723

1km =Ω+Ω+Ω

04/22/23 32

Cmbgg OmOlCMBflat

closedopen

How much dark energy is there?

04/22/23 33

Cmbgg OmOlCMB+LSS

How much dark energy is there?

WMAP + SDSS: lots

flatclosedopen

04/22/23 34

Cmbgg OmOlCMB+LSS

How much dark energy is there?

flatclosedopen

04/22/23 35

Cmbgg OmOlCMB+LSS

How much dark energy is there?

flatclosedopen

Note impact of SN 1a -- if excluded parameter

space opens up

04/22/23 36

Dynamical evolution of w?

SNAP as a discriminator

Weller and Albrecht; Kujat et al; Maor et al; Gerke and Efstathiou, Kratochvil et al

Write:∑=

=N

0i

iizw)z(w

or:

∑=

+=N

0i

ii )z1ln(w)z(w

Evaluate magnitude difference for each model and compare with Monte Carlo simulated data

sets.

04/22/23 37

Evolution of Fine Structure Constant

Non-trivial coupling to emg:

Olive and Pospelov

ννφ−= FF)(B

41L Fm

2FFF 2

11)(B φξ+φζ+=φExpand about current value of field:

Eff fine structure const depends on value of field

22FFF

F

20

)2(21

)(B4e)(

φζ−ξ+φζ=ααΔ

φπ=φα

510)5.35.0z( −≈−=ααΔ

Claim from analysing quasar absorption

spectra: Webb et al

04/22/23 38A way of constraining the eqn of state?

EJC, Nunes, Pospelov,

04/22/23 39

Model independent approachesSahni et al; Gerke&Efstathiou; Bassett et al; Corasiniti & EC

Idea: certain features common to many models, such as tracker behaviour

characteristic time scales for most models.

m,r

m,rcaam,r

3m2r1P

e1

1)a(f

F)a(fF)a(fF)a(w

Δ−

−

φ

+

=

++=

Constants: Δ;a c Scale factor at transition and width of transition to and from

tracker regimes.

04/22/23 40

Actual Best fitCorasiniti &EC

1. Allows for case where have rapid change

2. Physically motivated

3. Depends on tracker properties – otherwise model indep.

04/22/23 41

Probing Quintessence with the cmb through the ISW effect-- Corasaniti et al

Use parameterisation of w(z) to test whether we can see difference with Lambda CDM

Rapid transition

Slow transition

Need late rapid

transition to differentiate

04/22/23 42

Probing Quintessence with WMAP and SN1a -- Kunz et al

2D Likelihood shows that today w very close to - 1,

as expected.

This is not the same though as saying it always had to be -1.

04/22/23 43

Using

Black dots:

Red dots:

Pointing towards

lambda if sigma8 turns

out to be large.

04/22/23 44

Cmbgg OmOlCMB+LSS

Nature of the dark energy

Tegmark et al.astro-ph/0310723

Recent claim that w<-1

preferred with evolution from

w=0.

Alam et al.

04/22/23 45

Example showing rapid evolution of w(z) for different matter contributions, assuming no priors on remaining parameters, in particular allowing w<-1, and allowing

violtaion of weak energy condition. Note claimed best fit is for w<-1! [Alam et al, 2003]

What does it mean for cosmology to have unstable phantom energy (w<-1)? How do we then include

cosmological perturbations ?

04/22/23 46

Summary•Observations transforming field, especially CMBR and LSS. Constraining the cosmological parameters, even before Planck arrives on the scene.

•Why is the universe inflating today – can particle physics provide an answer through scaling solutions – see it through time varying constants?

•Brane inspired cosmology – so much going on in this area. Not discussed here [i.e. self tuning mechanisms -- Nilles et al].

•New Gravitational Physics [1/R terms in Action -- Carroll et al]. Not discussed here.

•Lots of models of dark energy but may yet prove too difficult to separate one from another such as cosmological const – need to try though!

•Finally -- could we all be wrong and we do not need a lambda term? -- [Break in primordial power spec -- Blanchard et al ]