# Presentation: 519488

**     ****The**** Darkness ****	****  ****of the Universe:**
***Acceleration and Deceleration***

**Eric Linder **
**Lawrence Berkeley National Laboratory**

## Discovery!  Acceleration

## Cosmic Concordance

**Cosmic Concordance**

** *****Supernovae alone*****   ****	****       ******** Accelerating expansion**
** ******** ****> 0**
** *****CMB (plus LSS)***
****** ****Flat universe **
****** ******** ****> 0 **
** *****Any two of SN, CMB, LSS***
****** ****Dark energy ~75%**

## Acceleration and Particle Physics

**The conformal horizon scale ****(aH)****-1**** tells us when a comoving scale (e.g. perturbation mode) leaves or enters the horizon. **

## Acceleration = Curvature

**The Principle of Equivalence teaches that **
**Acceleration = Gravity = Curvature**

**Acceleration**** ******** over time will get ****v=gh****, ****	****	****so ****z = v = gh**** (****gravitational redshift****). **
**But, ****t********t****0**** ******** parallel lines not parallel (****curvature****)!**

## Equations of Motion

**Expansion rate of the universe**** a(t) **
**ds****2 ****= ********dt****2****+a****2****(t)[dr****2****/(1-kr****2****)+r****2****d********2****]**
**   ****Friedmann equations***** ***
**(å/a)****2**** = H****2**** = (8********/3M****p****2****) [ ********m**** + ************** ]**
**  ****/a = -(4********/3M****p****2****) [ ********m**** + *************** *****+3p***********  *****]**
**	****	**
**Einstein-Hilbert action**
**S = ********d****4****x********-g [ R/2******** + ****L**********+ ****L****m**** ]**

## Spacetime Curvature

**Ricci scalar curvature**
**	****	****R = R************ ****= 6 [ ****a****/a + ****(å/a)****2**** ] ****	****	****	****	****    = 6 ( a/a + ****H****2****) **
**Define reduced scalar curvature**** **
**	****	****R**** = R/(12H****2****) = (1/2) [1 + aa/ ****å****2****] ****	****	****	****    = (1/2)(1-q) **
**Note that division between acceleration and deceleration occurs for**** ****R**** =1/2 ****(q=0)****.**** **
**Superacceleration (phantom models) is not**** (a) > 0****,**** ****but**** (a/a) > 0****,**** ****i.e.**** ****R**** > 1. **

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## Today’s Inflation

**To learn about the physics behind dark energy we need to map the expansion history. **

## Equations of Motion

**Expansion rate of the universe**** a(t) **
**ds****2 ****= ********dt****2****+a****2****(t)[dr****2****/(1-kr****2****)+r****2****d********2****]**
**   ****Friedmann equations***** ***
**(å/a)****2**** = H****2**** = (8********/3M****p****2****) [ ********m**** + ************** ]**
**  ****/a = -(4********/3M****p****2****) [ ********m**** + *************** *****+3p***********  *****]**
**	****	**
**Einstein-Hilbert action**
**S = ********d****4****x********-g [ R/2******** + ****L**********+ ****L****m**** ]**

## Scalar Field Theory

**Scalar field Lagrangian - **
**	****canonical, minimally coupled**
**L*********** *****= (1/2)(******************)****2 ****- V(**********)**** **
**Noether prescription ******** Energy-momentum tensor**
**T********=(2/********-g) [ ********(********-g ****L**** )/********g******** ****]**
**Perfect fluid form (from RW metric)**

**Energy density****  ************** = (1/2) *********** *****2**** + V(**********) ****+ (1/2)(**************)****2**** **
**Pressure**** ****	****	****p********** = (1/2) *********** *****2**** - V(**********) ****- (1/6)(**************)****2**** **

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## Scalar Field Equation of State

**Continuity equation follows KG equation**
**[(1/2)*********** *****2****]  + 6H ****[(1/2)*********** *****2 ****] = -V**
****** ****- V + 3H (********+p) = -V**
**d********/d*****ln***** a = -3(********+p) = -3******** (1+w)**

**Equation of state ratio **
**w = p/******
**Klein-Gordon equation ****(Lagrange equation of motion) **

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## Equation of State

**Reconstruction from EOS:**** **
******(a) = *************** *********c**** exp{ 3 ********d*****ln***** a [1+****w(z)****] }**
********(a) = ********d*****ln***** a H****-1**** sqrt{ ********(a) [1+****w(z)****] }**
**V(a) = (1/2) ********(a) [1-****w(z)****] **
**K(a) = (1/2)*********** *****2**** = (1/2) ********(a) [1+****w(z)****] **

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## Equation of State

**Limits of (canonical) Equations of State:**** **
**w = (K-V) / (K+V)**
**Potential energy dominates (slow roll) **
**V >> K ******** w = -1**
**Kinetic energy dominates (fast roll)**
**K >> V ******** w = +1**
**Oscillation about potential minimum ****	****	****(or coherent field, e.g. axion)**
******V******** = ********K******** ******** w = 0**

## Equation of State

**Examples of (canonical) Equations of State:**** **
**d********/d*****ln***** a = -3(********+p) = -3******** (1+w)**
**	******** ****= ****(Energy per particle)(Number of particles) / Volume**** ****	****   = E N a****-3**** **
**Constant w implies**** ******** ~ a****-3(1+w)**
**Matter:**** E~m~a****0****, N~a****0**** ******** ****w=0**
**Radiation:**** E~1/********~a****-1****, N~a****0**** ******** ****w=1/3**
**Curvature energy: ****E~1/R****2****~a****-2****, N~a****0**** ******** ****w=-1/3**
**Cosmological constant:**** E~V, N ~a****0**** ******** ****w=-1**
**Anisotropic shear: w=+1 ****	****	****	****	****	****Cosmic String network: w=-1/3  ;  Domain walls: w=-2/3**

## Expansion History

**Suppose we admit our ignorance:**
**	****H****2**** = (8********/3) ********m**** + ********H****2****(a)**
**Effective equation of state:**
**w(a) = -1 - (1/3) d*****ln *****(********H****2****) / d*****ln***** a**
**Modifications of the expansion history are equivalent to time variation w(a).****  ****Period.**** **

**Observations that map out expansion history a(t), or w(a), tell us about the ****fundamental physics of dark energy.**
**Alterations to Friedmann framework ******** *****w(a)***

- gravitational extensions or high energy physics

## Expansion History

**For modifications ********H****2****, define an effective scalar field with**
**  ****V = (3M****P****2****/8********) ********H****2**** + (M****P****2****H****0****2****/16********) [ d ********H****2****/d *****ln *****a] **
**  ****K = - (M****P****2****H****0****2****/16********) [ d ********H****2****/d *****ln *****a] **
**Example:**** ********H****2**** = A(********m****)****n**** **
**	****w = -1+n**
**Example:**** ********H****2**** = (8********/3) [g(********m****) - ********m****]**
**	****w= -1 + (g********-1)/[ g/********m**** - 1 ]**

## Weighing Dark Energy

## Exploring Dark Energy

**Exploring Dark Energy**

## Dark Energy Models

**Scalar fields can roll: **
**fast -- “kination” ****[Tracking models] **
**slow -- acceleration ****[Quintessence]**
**steadily -- acceleration deceleration ****	****	****[Linear potential] **
**oscillate -- potential minimum, pseudoscalar, PNGB ****[V~ ***********n*****]**

## Power law potential

**“****Normal” potentials don’t work: **
**V(**********) ~ ***********n  ***
**have minima (n even), and field just oscillates, leading to EOS**
**w = (n-2)/(n+2)**
**	****	****n****	****0****	****2****	****4****	****∞****	****	****	****	****	****w****	****-1****	****0****	****1/3****	****1    **

## Oscillations

**Oscillating field**
**w = (n-2)/(n+2)**
**Take osc. time << H****-1**** and ******** constant over osc.**
**  ****	***************2********** = *********dt ***********2****** / *********dt = ********d*********** *********** / ********d*********** *****/*********** ***
***	******	*****= 2******** ********d********** [1-V/V****max****]****1/2**** / [1-V/V****max****]****-1/2**
**If ****V = V****max****(********** /**********max****)****n**** then **
**	********w******** = -1 + 2**** ********0****1****dx (1-x****n****)****1/2**** / ********0****1****dx (1-x****n****)****-1/2**** **
**	****      ****= -1 +2n/(n+2)**

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- Turner 1983

## Linear Potential

**Linear potential ****[Linde 1986]**
**V(********)=V****0****+******
**leads to collapsing universe, can constrain t****c**

**curves of ******

## Tracking fields

**Criterion ******** = VV********/(V********)****2**** > 1****, ****d *****ln***** (********-1)/dt <<H****.  **
**However, generally only achieves ****w****0 ****> -0.7****. **
**Successful model requires fast-slow roll.**** **

**Can start from wide variety of initial conditions, then join attractor trajectory of tracking behavior. **

## Quintessence

**Interesting models have dark energy: **
**dynamically important, **
**accelerating, **
**not ******
**	****	****	****~ ********[(1+w)********] ~ ********(1+w) HM****p**
**Damped so**** H   ~ V********,     ****and timescale is H****-1****. **
**Therefore ************** ~ M****p****.  **
**Unless ****1+w << 1****, then ************** ****<< M****p**** and very hard to reconstruct potential. **

## Dark Energy Models

**Inverse power law ****V(**********) ~ ***********-******n  ***
**“****SUGRA” ****V(**********) ~ ***********-******n *****exp(***************2*****)**
**Running exponential ****V(**********) ~ exp[- *********(***********)**********]**
**PNGB or “axion” ****V(**********) ~ 1+cos(**********/f)**** **
**Albrecht-Skordis ****V(**********) ~ [1+c****1********** +c****2**********2****] exp(-**************)**
**“****Tachyon” ****V(**********) ~ [cosh(**************)-1]****n**
**Stochastic ****V(**********) ~ [1+sin(**********/f****)] exp(-**************)**
**...**** **

## Tying HEP to Cosmology

**Accurate to 3% in EOS back to *****z=1.7***** (vs. 27% for *****w******1*****). **
**Accurate to 0.2% in distance back to *****z******lss******=1100*****!**

***w(a) = w******0******+w******a******(1-a)***

## Scalar Field Dynamics

**The cosmological constant has ****w=-1=constant****. Essentially no other model does.**
**Dynamics in the form of ****w/H = w******** = dw/d*****ln***** a**** can be detected by cosmological observations. **
**Dynamics also implies spatial inhomogeneities. Scale is given by effective mass**
**m****eff**** = ********V ̋**
**This is of order ****H ~ 10****-33**** eV****, so clustering difficult on subhorizon scales. Vaguely detectable through full sky CMB-LSS crosscorrelation. **

**.**

## Growth History

**While dark energy itself does not cluster much, it affects the ****growth**** of matter structure. **
**Fractional density contrast**** ******** = ********m****/********m ****evolves as**
**	****	******** ****+ 2H**** ******** ****= 4********G********m ******
**Sourced by ****gravitational instability**** of density contrast, suppressed by ****Hubble drag****. **
**Matter domination case:**** **
**	******** ****~ a****-3**** ~ t****-2****, H ~ (2/3t)****.  Try ******** ****~ t****n****. **
**Characteristic equation n(n-1)+(4/3)n-(3/2)(4/9)=0.**** Growing mode ****n=+2/3****, i.e. ****	****	****	****	****	****	****	****	****	******** ****~ a**** **

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## Growth History

**Growth rate of density fluctuations***** *****g(a)**** ****= (********m****/********m****)/a **

## Gravitational Potential

**Poisson equation**** **
******2********(a)****=4********Ga****2 ********m****= 4********G********m****(0) g(a)**
**In matter dominated (hence decelerating) universe,**** ********m****/********m**** ~ a**** so ****g=const**** and ********=const****. **
**Photons don’t interact with structure growth: blueshift falling into well matched by redshift climbing out.  **
**Integrated Sachs-Wolfe (ISW) effect = 0. **

## Inflation, Structure, and Dark Energy

**Matter power spectrum**
**P****k**** = ******** (********m****/********m****)****2 ******** ~ k****n**
**Scale free**** ****(primordially, but then distorted since comoving wavelengths entering horizon in radiation epoch evolve differently - imprint ****z****eq****).**** **
**Potential power spectrum**
******2 ********L**** ~ L****4 ******** (********m****/********m****)****2 ********L**** ~ L****4 ****k****3****P****k ****~ L****1-n**
**Scale invariant**** ****for n=1 (Harrison-Zel’dovich).**** **
**CMB power spectrum**
**On large scales (low *****l*****), Sachs-Wolfe dominates and power *****l(l+1)C******l***** is flat. **

## Deceleration and Acceleration

**CMB power spectrum measures ****n-1**** and inflation.**
**Nonzero ****ISW**** measures breakdown of matter domination: at early times (radiation) and late times (dark energy). **
**Large scales (****low *****l*****) not precisely measurable due to cosmic variance.  So look for better way to probe decay of gravitational potentials. **
**Next:**** ****	****	****	****	****	****	****	****	****	****The Darkness of the Universe 3:**** ****	****	****	*****Mapping Expansion and Growth***