Cosmic Acceleration from the basics to the frontiers Je-An Gu ( 顧哲安 ) National Center for Theoretical Sciences (NCTS) 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 d L (distance precision:  mag = 0.15 mag   d L /d L ~ 7%)  Spectral information  redshift z SN Ia Data: d L (z) [ i.e, d L,i (z i ) ] [ ~ x(t) ~ position (time) ] F: flux (energy/area  time) L: luminosity (energy/time) Distance Modulus  (z) (z) history

(can hardly distinguish different models) SCP (Perlmutter et. al.) Distance Modulus 1998

Fig.4 in astro-ph/ [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   =

Riess et al. astro-ph/

Riess et al. astro-ph/

Supernova / Acceleration Probe (SNAP) z0~0.20.2~1.21.2~1.41.4~1.7 # of SN observe ~2000 SNe in 2 years statistical uncertainty  mag = 0.15 mag  7% uncertainty in d L  sys = 0.02 mag at z =1.5  z = mag (negligible)

Definition of Acceleration

Accelerating Expansion: Definition Distance L E.g. 1. Proper distance (Line Acceleration) E.g. 2. L = V D 1/3 (Domain Acceleration), at time t Accelerating Expansion : H > 0, q < 0 V D (Volume) a large domain D (e.g. size ~ H 0  1 )

Friedmann-Lemaitre-Robertson-Walker Cosmology Homogeneous & Isotropic Universe : Accelerating Expansion : H > 0, q < 0

Friedmann-Lemaitre-Robertson-Walker (FLRW) Cosmology Homogeneous & Isotropic Universe : (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 ↑ G μν ＝ 8πG N T μν Modification of Gravity Averaging Einstein Equations Extra Dimensions (Non-FLRW) for an inhomogeneous universe (based on FLRW)

FLRW +  CDM 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 SSB Phase Transition: Latent heat = vacuum energy (change) ~ T PT (eg. M EW ) How to make  slightly deviate from 0 ? Post-Dark-Energy + Why   ~  m NOW ?

Figure 1.1 Ratio changes rapidly with scale factor But at present time,  M ~   Why??? Why Ω Λ ~ Ω M now ? Why accelerating now ? Ω i  ρ i / ρ c  Problem & Coincidence Problem Why  so small initially ?

FLRW + Quintessence Quintessence: dynamical scalar field  Action : Field equation: energy density and pressure : Slow evolution and weak spatial dependence  V(  ) dominates  w  ~  1  Acceleration How to achieve it (naturally) ? ?

Non-Quinte: rapidly oscillating mode ~ radiation ~ NR matter time-averaged energy density and pressure :

Non-Quinte: ensemble of incoherent oscillators ~ radiation ~ NR matter ensemble-averaged energy density and pressure : (  i : the phase of i-th oscillator)

Non-Quinte: oscillators Thus, Oscillators How about other potentials ? ?

Quinte: a slowly evolving mode or coherent state (unnaturally small !!) [V(  ) dominates.] (unnaturally large !!)  

Tracker Quintessence  

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 !

FLRW Cosmology homogeneous & isotropic  Robertson-Walker (RW) metric Friedmann-Lemaitre-Robertson-Walker (FLRW) Cosmology 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 is NOT homogeneous & isotropic. NO At large scales, after averaging, the universe IS homogeneous & isotropic. In general, averaging/coarse graining is NOT VALID for Einstein equations. (due to the non-linearity) YES

Einstein equations 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/ ] perturbed metric, non-perturb T   eff  p eff Averaged Einstein equations: 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 Newtonianly perturbed metric 8  G  p eff 8  G  eff w eff   p eff /  eff 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 : Einstein equations: (G  = 8  G T  ) 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 Common Intuition / 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/ ] (  need dark energy ) Join the dark. concentrate, balance.… Do we really need Dark Energy ?? NO YES

Examples of Acceleration : q < 0 Over-density Under-density Acceleration Deceleration

Acceleration Inhomogeneity Examples of Line (Radial) Acceleration : q L < 0

Acceleration from Inhomogeneity ?? Warning!! Be careful (!!) when connecting two regions. E.g. FLRW decel. FLRW decel.  Domain Acceleration !! No physically observable effects of acceleration [regarding,e.g., d L (z)] There could exist singularity which leads to strange pheno. E.g. a lesson from Nambu & Tanimoto (incorrect accel. example) [gr-qc/ ] (perhaps NOT exist at all !!) Mr. Anderson, … NO YES Fake!? Illusion!? You are illusion !!

Acceleration from Inhomogeneity ?? Issues gauge-dep of acceleration frame acceleration !? NO YES 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 = V D 1/3 (Domain Acceleration), at time t V D (Volume) a large domain D (e.g. size ~ H 0  1 )

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 = V D 1/3 (Domain Acceleration), at time t V D (Volume) a large domain D (e.g. size ~ H 0  1 )

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 E.g. 1. Proper distance (Line Acceleration) E.g. 2. L = V D 1/3 (Domain Acceleration), at time t V D (Volume) a large domain D (e.g. size ~ H 0  1 ) L proper distance between two freely moving particles constant particle number inside Consider a system consisting of freely moving particles Interacting 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. 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 Other potentials ? ? ? Other fields ? Slow evolution and weak spatial dependence  V(  ) dominates  w  ~  1  Acceleration (unnaturally small !!) (unnaturally large !!)

Observations (SN Ia & others) 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 ?? who will win ??? NO DON’T KNOW YES

Is FLRW Cosmology a good approximation ?? NO YES Acceleration from Inhomogeneity ?? who will win ??? NO April 30 (Mon) (early morning) YES