Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/AstroStat/Stat310_1011/ad_20110419.pdf Дата изменения: Wed Apr 20 01:32:14 2011 Дата индексирования: Tue Oct 2 04:18:30 2012 Кодировка: Поисковые слова: big rip

The Expansion History of the Universe: Myths and Facts
Antonaldo Diaferio
Universita' degli Studi di Torino - Dip. Fisica Generale " Amedeo Avogadro" Istituto Nazionale di Fisica Nucleare - Sezione di Torino Harvard-Smithsonian Center for Astrophysics

with Luisa Ostorero and Vincenzo Cardone
memb ers of the CAUSTIC Group www.dfg.unito.it/ricerca/caustic

Cambridge, April 19th 2011


SCIENTIFIC BACKGROUND
Observations of SNae: Interpretation in the CDM model Alternative cosmological models: Conformal gravity and kinematic conformal gravity GRBs as cosmological probes: Bayesian approach

BAYESIAN ANALYSIS
Parameter forecasts (posterior probability): Likelihood and priors Bayesian evidence: Parallel tempering


High-redshift type Ia Supernovae (SNae)


Hubble Diagram of SNae in CDM
Cosmological parameters 0 (and )

B , z i mi- M =5log 10 d L , z - 5
Observables
Intrinsic luminosity

(Constitution set of Hicken et al 2009)


The expansion history of the CDM model

z~1

Can we probe the deceleration phase at redshift larger than z1?


Conformal gravity: No deceleration phase!

Conformal cosmology (CG) (Mannheim 1990)

Kinematic conformal cosmology (KCG) (Varieschi 2010)


Cosmology in conformal gravity - CG
Action:

I W =-



d x - g C

4



C



g x x g x

"Friedmann" equation:

a2 a 2= H 2 a 4k a2 -nr a -r 0
Deceleration parameter:

nr q =- -r - 2
(Same as CDM model with nr,r -nr,r) Always accelerated expansion!




Kinematic conformal cosmology - KCG
Conformal gravity Schwarzschild solution: 2 dr 2 -3 2 2 2 2 -3 r - r ds =- B r c dt r d with B r =1- r B r Redshift
2

a 0 2 1 z = = 1 - kr - r a r

New inversesquare law:

F d L =

L

0 2 L

4d


d d
rs L

a

V

viable
Pioneer anomaly


Distance modulus
CDM and CG

, z = 5 log 10 d L , z -5
=

{

0 CDM q0 CG

KCG
, z = 2.5 2 aV log
10

[

0 1 z 1 z 2 - 1 -2 0 2 0

]

= a V , 0


Gamma-ray bursts (GRBs) as cosmological probes
Light curve

GRBs are currently observed up to redshift z~8


GRB light curves

Counts/second

Time


GRB distance indicators
lag



RT

Luminosity

V

E

pe a k

Light-curve parameter
(GRB sample from Schaefer 2007)


Distance indicator relations

l og 1 0 P

bolo

= a b log 10 X - f , z

X = light-curve parameter

flux

f , z = log 10 [ 4 d L , z ]
f , z = log 10 [ 4 d , z ] aV log
2 L

2

(CDM, CG)

10


d d
L rs

(KCG)

NO NEARBY GRBs NO CALIBRATION (unlike SNae)


Bayesian parameter estimation

p D , M p M p D , M = p D M
Likelihood

1 p D , M ~ exp - Y T C -1 Y 2
Y = measures ­ expected values()





D = {all the observables, including uncertainties} C = covariance matrix

Analysis is performed over the 4 relations at the same time!
Analysis performed with APEMoST (Automated Parameter Estimation and Model Selection Toolkit) by Buchner and Gruberbauer: apemost.sourceforge.net


Parameter probability density functions: p( | D, M)
a vs b int vs b int vs
0

(similar PDFs for the 4 relations)

l og 1 0 P

bolo

= a b log 1 0 X - f , z
i olo

W i =a b log 10 X i- f , z i is the mean of log10Pb

with variance

2 int


GRBs distance indicators

la g



RT

Luminosity

V

E

pe a k

Light-curve parameter


Parameter probability density functions: p( | D, M)
a vs b int vs b int vs
0

(similar PDFs for the 4 relations)

l og 1 0 P

bolo

= a b log 1 0 X - f , z
i olo

W i =a b log 10 X i- f , z i is the mean of log10Pb

with variance

2 int


Combining with other probes to obtain tighter constraints? e.g. SNae


Supernova Light Curves

Shape parameter

Colour parameter

B , z i = mi - M s i -1 - c
Cosmological parameters

i

Free parameters


Observed and derived quantities

B , z i = mi - M s i -1 - c
Observables

i

This is NOT a plot of directly observed quantities!

(Constitution set of Hicken et al 2009)


Bayesian parameter estimation again...


Parameter probability density functions: p( | D, M) derived for the SNae alone
vs int vs M vs 0


Combining GRBs with SNae
a vs b GRBs alone int vs b int vs
0

GRBs + SNae


What happens in CG and KCG?


CG
a vs b GRBs alone int vs b int vs q
0

GRBs + SNae


KCG
a vs b GRBs alone int vs b int vs
0

GRBs + SNae


GRBs distance indicators in CG

la g



RT

Luminosity

V

E

pe a k

Light-curve parameter


GRBs distance indicators in KCG

la g



RT

Luminosity

V

E

pe a k

Light-curve parameter


Models without an early decelerated expansion can clearly describe the GRB data


For completeness: SNae in CG and KCG?


PDFs of SNae parameters in CG
vs
int

vs

M vs q

0


PDFs of SNae parameters in KCG
vs
int

vs

M vs

0


Hubble diagram of SNae in the three models the distance modulus is indeed a model-dependent quantity


Hubble diagram of GRBs in the three models the distance modulus is indeed a model-dependent quantity
GRBs alone

GRBs + SNae


Two issues:

find the parameters that can describe the data

done

compare the models

?


The Bayesian Evidence
p DM = p D , M p M d
Model posterior probability

p D M p M p M D = p D
Comparing two models:

p M 1 D p DM 1 p M 1 = p M 2 D p D M 2 p M 2

B12 = Bayes factor

Estimate performed with APEMoST (Automated Parameter Estimation and Model Selection Toolkit) by Buchner and Gruberbauer: apemost.sourceforge.net


Parallel tempering algorithm

[ p D , M ] p M p D , M = p D M

ln p D M = ln p D , M



ln p D M =0 p D , M d

1

20 chains values of [ 0,1 ]


Values of ln B

12

sample

M1 / M

2

CDM/CG 37.9 6.6 1.5

CDM/KCG 12.0 7.2 24.3

GRBs SNae GRBs + SNae

B12 > 1

M1 favoured over M

2


Conclusions

CDM, CG, and KCG can describe the observational data

The Bayes factor favours CDM over CG and KCG

But CDM has dark matter, dark energy...