The accelerated expansion of the universe and the cosmological

constant problem

Spring Summer School on Strings Cosmology and Particles

31 March – 4 April 2009, Belgrade-Niš, Serbia

Hrvoje Štefančić, Theoretical Physics Division,

Ruđer Bošković Institute, Zagreb, Croatia

Big issues - observational and theoretical

• Present accelerated expansion of the universe – observational discovery

• The cosmological constant (vacuum energy) problem – theoretical challenge

Our concept of the (present) universe

• Evolution dominated by gravity – the interactions governing the evolution of the

universe have to have long range to be effective at cosmological distances

– matter is neutral at cosmological (and much smaller) scales

• General relativity

• Known forms of matter (radiation, nonrelativistic matter)

• Four dimensional universe

The observed universe

• Isotropic (CMB, averaged galaxy distribution at scales > 50-100 Mpc)

• Homogeneous – less evidence (indirect) – Copernican principle

• Homogeneous and isotropic – Cosmological principle

• Robertson-Walker metric !

Expansion of the universe

• Hubble (1929) – dynamical universe • Cosmological redshift• Standard forms of matter lead to decelerated

expansion

• Inflation – early epoch of the accelerated expansion

• 1998 – universe accelerated (decelerated universe expected)

FRW model – theoretical description of the expansion

• Contents: cosmic fluids (general EOS)

• General relativity in 4D

• Friedmann equation

• Continuity equation (Bianchi identity - covariant conservation of energy-momentum tensor)

• Acceleration

FRW model

• Critical density

• Omega parameters

• Cosmic sum rules

Cosmological observations – mapping the expansion

• Standard candles (luminosity distance)• Supernovae Ia, GRB

• Standard rulers• CMB (cosmic microwave background)• BAO (baryonic acoustic oscillations)

• Others (gravitational lensing...)

Supernovae of the type Ia

• Standard candles – known luminosity• Binary stars – physics of SNIa understood • Light curve fitting• Luminosity distance – can be determined both

observationally and theoretically

• SNIa dimming – signal of the accelerated expansion

Cosmological observations - SNIa http://imagine.gsfc.nasa.gov/docs/science/know_l2/supern

ovae.html

http://www.astro.uiuc.edu/~pmricker/research/type1a/

Cosmological observations - CMB

http://map.gsfc.nasa.gov/

Cosmological observations - LSS

structure at cosmological scales (LSS)

http://cas.sdss.org/dr5/en/tools/places/

Standard cosmological model (up to 1998)

• Destiny determined by geometry

• Interplay of spatial curvature and matter content (Ω

m + Ω

k=1)

• Even EdS model advocated (Ωm=1)

Spatial curvature

• COBE – spatial curvature is small.

• EdS must do the job (models with considerable Ω

k are ruled out by the

observation of CMB temperature anisotropies

SNIa observations (1998)

• Observations by two teams– High z SN Search Team, Riess et al.,

http://cfa-www.harvard.edu/supernova//home.html– Supernova Cosmology Project, Perlmutter et al.,

http://supernova.lbl.gov/

• ΛCDM model – fits the data very well

• Measurement in the redshift range where the expansion of the universe is really accelerated or there is the transition from decelerated to accelerated expansion – “direct measurement”

CMB and BAO

• Influence to the determination of the acceleration – indirectly

• CMB – mainly through the distance to the surface of last scattering

• BAO – similarly

Combining observational data

• Degeneracies of cosmic parameters - different combinations of cosmic parameters may produce the same observed phenomena

• Removal of degeneracies – using different observations at different redshifts (redshift intervals)

• SNIa + WMAP + BAO – precision cosmology

Observational constraints to the DE EOS

E. Komatsu et al., Five-Year Wilkinson Microwave Anisotropy

Probe (WMAP) Observations: Cosmological Interpretation

http://arxiv.org/abs/0803.0547

Accelerated expansion

• In a FRW universe the observed signals strongly favor a presently accelerated expansion of the universe (and reject EdS model)

• Do we interpret the observational data correctly?

Classification of theoretical

approaches ll

R. Bean, S. Caroll, M. Trodden, Insights into dark energy: interplay between theory and observation.

Rachel Bean (Cornell U., Astron. Dept.) , Sean M. Carroll (Chicago U., EFI & KICP, Chicago) , Mark Trodden (Syracuse U.) . Oct 2005. 5pp.

White paper submitted to Dark Energy Task Force.

http://arxiv.org/abs/astro-ph/0510059

Distorted signals and unjustified assumptions?

• Photons from SNIa convert to axions in the intergalactic magnetic field – light signal dissipated – Reduction in intensity confused for the effects of

acceleration

• C. Csaki, N. Kaloper, J. Terning, Phys. Rev. Lett. 88 (2002) 161302– does not work (very interesting attempt –

invokes more or less standard (or at least already known physics)

– connection with the phantom “mirage”

Distorted signals and unjustified assumptions?

• The influence of inhomogeneities (below 50-100 Mpc)• Nonlinearity of GR in its fundamental form• Solving Einstein equations in an inhomogeneous

universe and averaging the solutions is not equivalent to averaging sources and solving Einsteins equations in a homogeneous universe

• No additional components (just NR matter)• The acceleration is apparent • The perceived acceleration begins with the onset of

structure formation – very convenient for the cosmic coincidence problem

• The effect is not sufficient to account for acceleration, but is should be taken into considerations in precise determination of cosmic parameters

Distorted signals and unjustified assumptions?

• Inhomogeneities at scales above the Hubble horizon

• Underdense region

• Relinquishing the Copernican principle?

• Falsifiability?

• No additional components

• The effect of “super large scale structure”

Mechanism of the acceleration

• No acceleration in the “old standard cosmological model”

• Our (pre)concepts of the universe have to be modified– Modifying contents – dark energy (+ DM)– Modyfing gravity – modified (dark) gravity– Modifying dimensionality – new (large) dimesions –

braneworld models– ...– and combinations

Dark energy

• Acceleration by adding a new component – a dark energy component

• Key property – sufficiently negative pressure

• Physical realization of a negative pressure?– Geometric effect (Lambda from the left side of Einstein eq.)– Dynamics of scalar field - domination of potential energy

over kinetic energy

– Corpuscular interpretation – unusual dispersion relation – energy decreasing with the size of momentum

Dark energy

• DE equation of state

• Dynamics of ρd in terms of a

– w > -1: quintessence– w = -1: cosmological term– w < -1: phantom energy

• Multiple DE components

• Crossing of the cosmological constant barrier

Dark sector

• DE interacting with other cosmic components

• Interaction with dark matter

• Unification of dark matter and dark energy

• Chaplygin gas – EOS

– scaling with a

DE models

• Cosmic fluid

• Scalar fields (quintessence, phantom)

• ...

• Effective description of other acceleration mechanisms (at least at the level of global expansion)

ΛCDM

• Benchmark model

• Only known concepts (CC, NR matter, radiation)

• small number of parameters

• The size of Λ not understood – cosmological constant problem(s)

• Problems with ΛCDM cosmology

Quintessence

• Dynamics of a scalar field in a potential

• Freezing vs. thawing models

• “tracker field” models

• k-essence (noncanonical kinetic terms)

Phantom energy

• Energy density growing with time

• Big rip

• Stability

• Problems with microscopic formulation

• Instability to formation of gradients

• Effective description

Singularities

• New types of singularities

• Finite time (finite scale factor) singularities

• Sudden singularities

Modified gravity

• Modification of gravity at cosmological scales

• Dark gravity (effective dark energy)

• F(R) gravity – various formulations (metric, Palatini, metric-affine)

• Conditions for stability

• Stringent precision tests in Solar system and astrophysical systems

Braneworlds

• Matter confined to a 4D brane• Gravity also exists in the bulk• Dvali-Gabadadaze-Poratti (DGP)• Different DGP models – discussion of the status!

• Phenomenological modifications of the Friedmann equation – Cardassian expansion

The cosmological constant

• Formally allowed – a part of geometry • Introduced by Einstein in 1917 – a needed element

for a static universe

• Pauli – first diagnosis of a problem with zero point energies

• Identification with vacuum energy – Zeldovich 1967• Frequently used “patch”

The expansion with the

cosmological constant

J. Solà, hep-ph/0101134v2

The expansion with the cosmological constant

Contributions to vacuum energy

• Zero point energies – radiative corrections– Bosonic– Fermionic

• Condensates – classical contributions– Higgs condensate– QCD condensates– ...

Zero point energies

• QFT estimates– real scalar field

– spin j

Condensates

• Phase transitions leave contributions to the vacuum energy

• Higgs potential

• minimum at

• contribution to vacuum energy

The size of the CC

• Many disparate contributions

• Virtually all many orders of magnitude larger than the observed value

• ZPE - Planck scale “cutoff” ≈ 1074 GeV4

• ZPE - TeV scale “cutoff” ≈ 1057 GeV4

• ZPE - ΛQCD scale “cutoff” ≈ 10-5 GeV4

• Higgs condensate ≈ -108 GeV4

• melectron4 ≈ 10-14 GeV4

• The observed value

The “old” cosmological constant problem – the problem of size

• Discrepancy by many orders of magnitude (first noticed by Pauli for the ZPE of the electromagnetic field)

• Huge fine-tuning implied• How huge and of which nature

– Numerical example: 10120

1-

0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

=0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

– Financial example

Instability to variation of a single contribution (parameter)

The “old” cosmological constant problem

• Fundamental theoretical problem – the problem of the vacuum energy density

• All proposed solutions assume that the “old” CC problem is somehow solved– ΛCDM model – CC relaxed to the observed value– DE models and other models – CC is zero or much smaller in

absolute value compared to the observed DE energy density

• Even should the future observations confirm the dynamical nature of DE or some other alternative acceleration mechanism, the “old” CC problem must be resolved

DE vs CC

• Raphael Bousso, “TASI lectures on the cosmological constant”– “If a poet sees something that walks like a duck

and swims like a duck and quacks like a duck, we will forgive him for entertaining more fanciful possibilities. It could be a unicorn in a duck suit – who's to say! But we know that more likely, it's a duck.”

• Conditions for a mechanism solving the CC problem

Proposed solutions of the “old” CC problem

• Classification (closely following S. Nobbenhuis, gr-qc/0609011)– Symmetry– Back-reaction mechanisms– Violation of the equivalence principle– Statistical approaches

Symmetry

• Supersymmetry

• Scale invariance

• Conformal symmetry

• Imaginary space

• Energy → - Energy

• Antipodal symmetry

Back-reaction mechanisms

• Scalar

• Gravitons

• Running CC from Renormalization group

• Screening caused by trace anomaly

Violation of the equivalence principle

• Non-local Gravity, Massive gravitons

• Ghost condensation

• Fat gravitons

• Composite gravitons as Goldstone bosons

Statistical approaches

• Hawking statistics

• Wormholes

• Anthropic Principle

The cosmic coincidence problem – the problem of timing

• Why the CC (DE) energy density and the energy density of (NR) matter are comparable (of the same order of magnitude) at the present epoch?

• A problem in a DE (CC) approach to the problem of accelerated expansion:– DE (CC) energy density scale very differently with the

expansion (if presently comparable they were very different in the past and will be very different in the future

• NR: ρ ~ a-3

• DE: ρ ~ a-3(1+w) , slower than a-2 , CC: ~ 1

• Also present in many approaches not based on DE

Possible solutions of the cosmic coincidence problem

• Naturally solved in (matter) back-reaction approaches

• “tracker field”• Oscillating DE model• DE-DM interaction models (although problem

still present in e.g. Chaplygin gas model)• Composite DE model (LambdaXCDM model)

– Two interacting DE components: a (dynamical) cosmological term and an additional DE component (cosmon X)

• ….

Composite dark energy – ΛXCDM models

ordinary matter (radiation and NR matter) separately conserved)

ΛXCDM11 : CT interacting with cosmon

J. Grande, J. Solà, H. Š., JCAP 0608 (2006) 011. ΛXCDM

2 : varaible CT i G, X concerved

J. Grande, J. Solà, H. Š., Phys. Lett . B645 (2007) 236.

Ratio of DE and matter energy density

Parameter constraints primordial nucelosynthesis:

Existence of a stopping point

height of the maximum of r :

Parameter constraints – cross sections

The CC relaxation mechansim

• Two component model (H.Š. Phys.Lett. B 670 (2009) 246)

• The inhomogeneous equation of state (S. Nojiri, S.D. Odintsov, Phys. Rev. D 72 (2005) 023003)

• The continuity equation

The model dynamics• The dynamics of the Hubble parameter

• Notation

• Dynamics in terms of dimensionless parameters

• with the initial condition

• HX and a

X in principle arbitrary

α < −1: the relaxation mechanism forthe large cosmological constant

• The α = −3 case

• Closed form solution

case

• Late-time symptotic behavior

• Ʌeff

is small because |Ʌ| is large!

Dependence on model parameters

Dependence on model parameters

Dependence on model parameters

Dependence on model parameters

case

• Asymptotic behavior

• Late-time asymptotic behavior

• Ʌeff

is small because Ʌ is large!

Dependence on model parameters

Dependence on model parameters

Dependence on model parameters

Dependence on model parameters

Other parameter regimes

• For α > −1 the behavior is different• The relaxation mechanism is not

automatic

Fixed points, approach to de Sitter regime

• general dynamics

• Fixed point ⇒

• Example: ⇒

General inhomogeneous EOS

• dynamics of the scaled Hubble parameter

• condition for the relaxation mechanism

• for a small h at late-time

Variable cosmological term

• Running CC

• Extended running CC

• Interaction with matter + put βn → 0

• Dynamics of the Hubble parameter

Variable cosmological term

• Late-time asymptotic behavior

f(R) modified gravity

• general dynamics

• specific example

• asymptotic de Sitter regime

• n=1

Important questions • Abruptness of the transition

• The onset of the transition

• The connection to other eras of (accelerated) expansion

• Addition of other components and other cosmological (RD,MD) eras

• Cosmological coincidence problem

• Stability of the mechanism to perturbations

• Precision tests and the comparisons with the observational data

– astrophysical scales (e.g. solar system tests)

– cosmological scales (growth of inhomogeneities)

Summary of the relaxation mechanism properties

• The solution of the CC problem without fine-tuning for both signs of the CC

• The universe with a large CC has a small positive positive effective CC

• Ʌeff

is small because |Ʌ| is large

• Ʌeff

~ 1/|Ʌ|

• candidate physical mechanisms: modified gravity, (nonlinear) viscosity, quantum effects

• Exchanging “unnatural” parameters for some new (not too complicated) dynamics

Relaxing a large cosmological constant - adding matter and radiation

• F. Bauer, J. Sola, H. Š. arXiv:0902.2215 • Components:

– variable cosmological term (containing a large constant term)

– dark matter– baryons – radiation

• Variable cosmological term and DM interact

The formalism

• The variable cosmological term• Constructing f from general coordinate

covariant terms

• Interaction with the DM component

½ Model

• f=R

• Radiation domination (controlled by 1-q)• transition to de Sitter regime (controlled by

small H2)

• abrupt transition removed• RD phase introduced

The model

• Two terms dominated by different values of q and different powers of H

• Sequence of a RD, MD and de Sitter phases

• Realistic cosmological model with a relaxed CC

The deceleration parameter

Normalized energy densities

Absolute energy densities

Effective DE EOS

Conclusions

• The question of the mechanism of the acceleration of the universe still open

• The cosmological constant problem(s) – many proposed approaches – decisive arguments still to come

• The nexus of physics at many very different distance/energy scales

• Testing ground of the future theoretical, observational and experimental efforts