# Presentation: 237923

**Measuring the Acceleration of the Universe with Radio Galaxies and Supernovae**** **
**Ruth A. Daly**
** ****Collaborator: George Djorgovski**

| Measuring the Acceration of the Universe with Supernovae and Radio Galaxies |
| --- |

## Slide 2

##  Dimensionless Luminosity Distances to SN and RG  

** ****Dimensionless Luminosity Distances to SN and RG**** **** **

- From Daly & Djorgovski 2003

**The dimensionless coordinate distance y(z)=H****o****(a****o****r) = D****L****/(1+z)**
**to 20 RG and 78 SN**

**Dimensionless Coordinate Distances Relative to a Standard Cosmological Model**

## FRW Metric +  a/ao = (1+z)-1 obtain:
Luminosity L = 4π (aor)2(1+z)2 f = 4π(dL)2 f
	where dL = (aor) (1+z) = luminosity distance
Size  S = θ (aor) (1+z)-1 = θ dA, 
	where dA=(aor) (1+z)-1 = angular size distance
And, for k = 0, (aor) = ∫ dt/a(t) = ∫ (a/á) dz

**Luminosity L = 4****π**** (a****o****r)****2****(1+z)****2**** f = ****4****π****(d****L****)****2**** f**
**	****where d****L**** = (a****o****r) (1+z) = luminosity distance**
**Size  S = ****θ**** (a****o****r) (1+z)****-1**** = ****θ**** d****A****, **
**	****where d****A****=(a****o****r) (1+z)****-1**** = angular size distance**
**And, for k = 0, (a****o****r) = ∫ dt/a(t) = ∫ (a/á) dz**

**For k = 0, (a****o****r) = ∫ dt/a(t) = ∫ (a/á) dz**
**Einstein Equations (for k=0)**
**(á/a)****2**** = (8****π****G/3)  ∑ ****ρ****i****  **
**	****	****   ****= H****o****2**** [****Ω****om****(1+z)****3**** + (1- ****Ω****om****) f****E****(z)],**
**	****	**** ****where ****ρ****E****(z) = ****ρ****Eo**** f****E****(z)**
** ****Current Method: select f****E****(z), use data to determine best fit values of the model.**
**Alternative Method: Differentiate a****o****r (z) **
**determined from data, and solve for **
**q(z) = - äa/(á)****2****,  p****E****(z),  f****E****(z),  etc.   **

**Rather than integrating an assumed model for f****E****(z), **
**and comparing with the data to obtain best fit model **
**parameters, we solve for y(z) to each source, smooth, **
**and differentiate the data to obtain dy/dz, and ****2****y/dz****2****.  **
**	****E(z) = (dy/dz)****-1****, where E(z) = (****á/a)/H****o**** **
**     ****_ ****q(z) = ****äa/(á)****2**** ****= 1 + (1+z) (dy/dz)****-1 ****(d****2****y/dz****2****) **
**	****O****nly assume FRW + a****o****/a = (1+z)!**
**	**** ****E****2****(z) = [****Ω****om****(1+z)****3**** + (1- ****Ω****om****) f****E****(z)]   ****(GR)**
**	**** ****where ****ρ****E****(z) = ****ρ****Eo**** f****E****(z); **
**	**** ****so ****	****f****E****(z) = [E****2****(z) - ****Ω****om****(1+z)****3****]/(1- ****Ω****om****)**
**	**** ****and p****E****(z) = P****E****(z)/ ****ρ****oc**** = E****2****(z) [2q(z)-1]/3   ****(GR)**

**SN:  *****M******B***** = m****Beff ****[********] - 5 log [(a****o****r)(1+z)] **
**RG: <D>/D********* = R* (********, y(z), obs) =  ********,**
**where     R********* = k****o****y****(6****β****-1)/7**** (k****1****y****-4/7**** + k****2****)****β****/3-1**** , **
**k****o****, k****1****, and k****2**** are observed quantities.**** **
**y(z) = H****0****d****L****/(1+z) = H****0 ****(a****o****r) is the Hubble-constant-free coordinate distance**
**(1) SN: *****M’******B****** *****= m****Beff ****[********] - 5 log [ y (z) (1+z)]**** **
**(2) RG:   <D>/D********* = R*(********, y [z], obs) =  ******
**Study 20 RG from GDW00 & GD98 + 78 SN from R98 & P99 **
**Use (1) and (2) to solve for y(z) to each source **

**E(z) = (dy/dz)****-1**

**Good agreement between RG and SN**

**The numerical smoothing and differentiation **
**technique was tested by creating a mock **
**catalogue of 2000 supernovae observations **
**with z from 0.1 to 1.7, following what is **
**expected from the SNAP satellite.****  **
**The smoothed data y(z) is within 2% of the **
**true value for all z, E(z) is within 3% to 5% of **
**the true value for z from 0.1 to 1.5, and q(z) **
**remains close to the true value out to a **
**redshift of about 1 (see DD03).  **

**Model Independent**
**q(z) = - [1 + (1+z) (dy/dz)****-1**** d****2****y/d****2****z]**
**Transitions to deceleration at z ****≈ ****0.45**

**Let the energy density of the dark energy evolve as **
**ρ****E**** (z) = ****ρ****Eo****  f****E**** (z)      ****then **
**f****E**** (z) = [E****2****(z) – ****Ω****om****(1+z)****3****]/(1- ****Ω****om****)**
**f****E**** (z) is shown for ****Ω****om**** = 0.3**

- From DD03

**Model Independent                                      q(z) = -[1 + (1+z) (dy/dz)****-1**** d****2****y/d****2****z]**

**Conclusions **
**The value of -q(z) = [1 + (1+z) (dy/dz)****-1 ****(d****2****y/dz****2****)] **
**depends only upon the FRW metric and a/a****o**** = (1+z)****-1****.**
**We obtain a transition from acceleration to **
**deceleration at a redshift of about 0.45 (model ind.).**** **** **
**We obtain the model independent result that the **
**pressure of the dark energy is constant at **
**P****E**** ****≈ - 0.6 ****ρ****oc**** out to z ≈ 0.6, and the ****energy density of **
**the dark energy is non-evolving to this redshift.  **
**Radio Galaxies are observed out to a redshift of 2; we **
**(O’Dea, Daly, Donahue, & Guerra) just obtained **
**VLA radio data for 10+ new sources!**

## Slide 18

- The work presented above comes from the paper of Daly & Djorgovski, 2003, ApJ, 597, 9, and from work prepared by Daly & Djorgovski and presented in a series of talks over the past year.
- The work that follows was published in a series of papers by Daly and collaborators.

**3C 55 is a FRIIb Radio Galaxy**** **
**D = largest linear size **

**The Radio Galaxy Method**

**<D> for the parent population of 70 3C Radio Galaxies with 178 MHz powers > 3 h****-2**** x 10****26**** W/Hz/sr**

## Radio Image of Cygnus A 

**Radio Image of Cygnus A **
**Obtained by Chris Carilli et al. at the VLA.**

_**Simple Model for Average Source Size D* = <D>**_** **
**D* = v****L****t* for v****L**** ****≈**** **** const; t* = total lifetime.****  **
**For a strong driven shock, ****v****L**** ******** (L****j**** n****a****-1**** a****L****2****)****1/3**
******  ****D* ******** t* L****j****1/3**** (n****a****-1**** a****L****-2****).  **
**Small dispersion in <D> ******** a cancellation between t* and L****j ****(both intrinsic to the BH/AGN).****   **
**Fact: <D> ******** with z for z > 0.5; **
**Fact: v****L**** (and L****j****) ******** with z****  ******** **_**t* **__****__** as L**__**j **__****__** **_
**write t* ******** L****j****-****β****/3**** ****so D* ******** L****j****(1-****β****)/3**** ****(n****a****-1**** a****L****-2****) **
**SS: **_**n**__**a**__****__**(B**__**L**__**/v**__**L**__**)**__**2**__** , **_**L****j**** ********(n****a**** a****L****2**** v****L****3****)********(B****L****2 ****a****L****2**** v****L****)**
**D*********(B****L**** a****L****)****-2****β****/3**** v****L****1-****β****/3**** ****about ******** ****(obs)(a****o****r)****-2****β****/3+4/7**

**<D>/D********* = R* [********, y(z), obs] =  ******** = const,**
**where     R********* = k****o****y****(6****β****-1)/7**** (k****1****y****-4/7**** + k****2****)****β****/3-1**** , **
**k****o****, k****1****, and k****2**** are observed quantities. **
_**from D**__*****__** = v**__**L**__** t**__*****__**,  t**__*****__** **__****__** L**__**j**__**-**__**β**__**/3**__**,  + strong shock equations**_
**	****t********* ******** L****j****-****β****/3**** is expected in standard magnetic breaking models of spin energy extraction from a rotating black hole; the empirical relation implies a maximum**** ****or limiting magnetic field**** ****strength**** **
**	****	****	****B ******** M****(2****β****-3)/2(3-****β****)**** (a/m)****β****/(3-****β****)**** **
**	****	****	**** ****B ******** (a/m) for ****β**** = 1.5.****  **
**The best fit value of ****β**** = 1.75 +/- 0.25**
**For ****β**** = 1.75, ****t********* ******** L****j****-****β****/3**** ****** **B ******** (a/m)****1.4**** M****0.2****  **

**D* has been computed for 20 FRIIb Radio Galaxies**** **
**shown for ******** = 1.75, ********m**** = 0.1, and ************ = 0. **
**D*=v****L****t*, t*********(L****j****)****-****β****/3 **** ******** D* ******** (B****L**** a****L****)****-2****β****/3**** ****v****L****1- ****β****/3**** **
****** ****f (obs, a****o****r)**** **** **** ****roughly** ****** (obs) (a****o****r)****-0.6**** **

**The average source size <D> ******** (a****o****r); the average source size D********* about ********  (a****o****r)****-0.6**** **

**Results from 20 RG for ********m****,**** ************, and k.**** ****Left: the lowest redshift bin is excluded (see GDW00); Right: all redshift bins included.****  **

**In any model it is important to consider covariance between cosmological parameters and model parameters.****  ****Clearly, the model parameter ******** has no covariance with ********m**** (or with ************)**** ****[shown for k=0 on left, and ********=0 on right].**

**In a quintessence model, there is no covariance between ******** and ********m**** or with w=P/****ρ****.**** **

## RG are able to constrain w because they are observed at high redshift. Results shown for 20 RG in a k=0 quintessence model with w=P/ρ.  

**RG are able to constrain w because they are observed at high redshift. Results shown for 20 RG in a k=0 quintessence model with w=P/****ρ****.****  **

## 54 SN in a k=0 Quintessence model, w=P/ρ

**54 SN in a k=0 Quintessence model, w=P/****ρ**

## 20 RG + 54 SN in a Quintessence Model

**20 RG + 54 SN in a Quintessence Model**

## Implications for Dark Energy

**E****2****(z) = ****Ω****m****(1+z)****3**** + (1- ****Ω****m****) f****E**** (z) , **
-    where **ρ****E  = ****ρ****E0**** ***** ****f****E**** (z)**
- For Quint. **E****2 ****(z) = (1+z)****3 ****[****Ω****m****+ (1-****Ω****m****)(1+z)****3w****] **
**q(z) = 0.5 [1+(****Ω****m****-1  ****- 1)(1+3w)(1+z)****-3 ****f****E****(z)]/              ****	****	****[1+(****Ω****m****-1  ****-1)(1+z)****-3**** f****E****(z)]**
-  For Quintessence
**q(z)= 0.5 [****Ω****m ****+ (1- ****Ω****m****)(1+3w)(1+z)****3w****]/      ****	****	****	****[****Ω****m****+(1- ****Ω****m****)(1+z)****3w****]**