有效场论、全息原理暴胀宇宙与暗能量

Effective Field Theory & Holographic Principle

An effective field theory that can saturate the equation necessarily includes many states with Schwarzschild radius much larger than the box size.

An effective quantum field theory is expected to be capable of describing a system at a temperature T , provided that T ≤ Λ ， so long as T 1/L≫ .

Thermal energy

Entropy

The corresponding Schwarzschild radius

Entropy

To avoid these difficulties Cohen-Kaplan-Nelson propose a stronger constraint on the IR cutoff 1/L which excludes all states that lie within their Schwarzschild radius. Since the maximum energy density in the effective theory is Λ^4, the constraint on L is

Thermal energy ～ ～

Schwarzschild radius

Local quantum field theory appears unlikely to be a good effective low energy description of any system containing a black hole, and should probably not attempt to describe particle states whose volume is smaller than their corresponding Schwarzschild radius.

Holographic Principle: (Cohen-Kaplan-Nelson, PRL1999)

In Effective Field Theory, UV Cut-off is related to the IR Cut-off due to the limit set by the formation of a Black Hole

Effective Theory describes all states of system except those already collapsed to a Black Hole.

Vacuum energy density via quantum fluctuation

Effective Field Theory & Holographic Principle

Holographic Dark EnergyHolographic Dark Energy Model:

Dark energy density is given by the vacuum energy density caused via quantum fluctuation

Characteristic length scale of universe

Choosing different characteristic length scale L

Various Holographic Dark Energy Models

Review see: M. Li, X. -D. Li, S. Wang, Y. Wang, CTP. 56, 525-604 (2011) [arXiv:1103.5870].M. Li, Phys. Lett. B 603, 1 (2004) [arXiv:hep-th/0403127].R. -G. Cai, Phys. Lett. B 657, 228-231 (2007) [arXiv:0707.4049 [hep-th]].

Model parameter Reduced Planck mass

Holographic Dark Energy Characterized by Conformal-age-like Length (CHDE)

Z.P. Huang, YLW, arXiv:1202.2590,

Z.P. Huang, YLW, arXiv:1202.3517 [astro-ph.CO]

Conformal-age-like length scale of universe

Motivated from 4D space-time volume of FRW Universe

Holographic Dark Energy Characterized by Conformal-age-like Length (CHDE)

Fractional energy density of CHDE Friedman Equation

Equation of Motion of CHDEConservation of energy density Friedman equation

EoS for CHDE

Equation of motion for CHDE

Density with constant

CHDE

Solution of EoM for CHDE At early time of universe

Assuming: Dark energy is negligible

Equation of motion for CHDE in a good approximation

Solution of EoM for CHDE consistency

Inflationary Universe & Conformal-age-like Length of CHDE

At early time of universe with Universe with constant

Conformal-age-like Length of CHDE

= -1

= 1/3

Consistent check from L

EoS of Dark Energy

Epoch: Inflation Radiation Matter Today <

CHDE is a single parameter (d) model like

More General Analysis

Equation of motion for CHDE

Friedman Equation

Interaction With Background

General Equation of motion for CHDE

EoS for Dark energy

Holographic Dark Energy Characterized by Total Comoving Horizon (ηHDE)

Z.P. Huang, YLW, arXiv:1202.2590,

Holographic Dark Energy Characterized by Total Comoving Horizon (ηHDE)

Total comoving horizon of the universe

Characteristic Length Scale L of Universe from Causality

Energy density of holographic dark energy

Rescaled independent parameter & Fractional DE Density

Primordial part of comoving horizon generated by inflation

Comoving horizon in radiation- & matter-dominated epoch

grows

Total comoving horizon of the universe

Energy density & fractional energy density of dark energy

behaves like a cosmological constant

• 真实年龄大于哈勃年龄（这一情形在宇宙常数不为 0 时可能出现）

哈勃年龄（ 1/H0）减速 加速

等速

Fractional energy density of dark energy

Fraction of dark energy in matter-dominated epoch

New agegraphic dark energy (NADE) Avoid Divergence

C. -Y. Sun, R. -H. Yue, Phys. Rev. D 83, 107302 (2011) .

Equation of Motion of ηHDEConservation of energy density Friedman equation

EoS for ηHDE

Equation of motion for ηHDE

Density with constant

ηHDE

Best-Fit Analysis on HDE Models

Initial input:

Friedman Equation

Relevant Cosmological Observations• Union2 compilation of 557 supernova Ia (SNIa) data, • Baryon acoustic oscillation (BAO) results from the Sloan

Digital Sky Survey data release 7 (SSDS DR7) , • Cosmic microwave background radiation (CMB) data from

7-yr Wilkinson Microwave Anisotropy Probe (WMAP7)• Hubble constant H measurement from the Wide Field

Camera 3 on the Hubble SpaceTelescope (HSTWFC3)

Likelihood function and Minimal

Type Ia Supernovae (SN Ia) Theoretical Distance modulus

Hubble-free luminosity distance Minimal

Expand with respect to

Minimal with respect to

Baryon Acoustic Oscillations (BAO)Volume averaged distance Proper angular diameter distance

Comoving sound horizon

Fitting formula

Distance ratio of BAO Observation and analysis of BAO

Cosmic Microwave Background (CMB) RadiationAcoustic scale Shift parameter

Redshift of the decoupling epoch

WMAP7 observations and analysis of CMB

Hubble ConstantHubble constant and analysis

Best-Fit Results for CHDE Model at 1σ (68.3%) and 2σ (95.4%)

Best-Fit Results at 1σ (68.3%) & 2σ (95.4%)

Best-Fit Results at 1σ (68.3%) & 2σ (95.4%)

SYSTEMATIC ANALYSIS ON CHDE MODEL

Cosmic evolution of the fractional energy density of CHDE

Cosmic evolution of the EoS of CHDE

SYSTEMATIC ANALYSIS ON CHDE MODEL

The decelerating parameter

The statefinder pair { j; s}

SYSTEMATIC ANALYSIS ON CHDE MODEL

Eolution of the decelerating parameter

SYSTEMATIC ANALYSIS ON CHDE MODEL

SYSTEMATIC ANALYSIS ON CHDE MODEL

The statefinder parameter j− s contour evolves in redshift inteval z [−0.2; 15]∈ (The arrow indicatesthe evolution from high redshift to low redshift); Model parameters take the best-fit values, i.e.d = 0.235 r0 = 3.076 × 10−4

On CHDE MODEL

Best-Fit Results for ηHDE Model at 1σ (68.3%) and 2σ (95.4%)

Best-Fit Results for ηHDE Model at 1σ (68.3%) and 2σ (95.4%)

Fractional Energy Density of Dark Matter

Cosmic evolution of the fractional energy density of ηHDE

Cosmic evolution of the EoS of ηHDE

Cosmic evolution of the ratio η/d with different d

On ηHDE Model

Behave Like Cosmological Constant

General Discussion

n=m=0, ADE; n=0,m=-1, ηHDE; n=4,m=3, CHDE

The minimum of by using only the Union2 SNIa data; for comparison,

The best-fit results of some models with n - m = 1 by using only the Union2 SNIa data

The best-fit by using SNIa+BAO+CMB data sets; for comparison

The best-fit results at (68.3%) and (95.4%) confidence levels by using SNIa+BAO+CMB data sets;

Holographic Dark Energy Cosmological Constant

Understanding Fine-tuning Problem & Coincidence Problem

Inflationary Universe Accelerated Universe

Holographic Principle

Summary

THANKSTHANKS