Observational Constraints on Sudden Future Singularity Models

Hoda Ghodsi – Supervisor: Dr Martin HendryGlasgow University, UK

Grassmannian Conference in Fundamental CosmologySzczecin, Poland, September 2009

Outline•Concordance Cosmology Overview•Sudden Future Singularity Theory•Observational Constraints•Method•Results•Conclusions and Future Directions

Concordance CosmologyCosmological observations of most importantly the Cosmic Microwave Background Radiation (CMBR), the Large Scale Structure and SNe Ia have helped establish a standard Concordance Cosmology with the following characteristics:

•Evolution: Accelerating expansion driven by a form of Dark Energy•Geometry: Flat•Contents: 74% Dark Energy, 22% Dark Matter, 4% Baryonic Matter•Age: 13.7 Gyr old•Fate: Empty de-sitter type fate

Courtesy: http://map.gsfc.nasa.gov/

Courtesy: http://abyss.uoregon.edu/

Sudden Future Singularity Model• Barrow (Class. Quantum Grav. 21, L79) discovered a new type of

possible end to the Universe (assuming no equation of state) which violated the dominant energy condition only

• He called them Sudden Future Singularities• Pressure singularities• Barrow then constructed an example model which could

accommodate an SFS with scale factor of the form: , )y)δ(δ)y(+(δa=a(t) nm

s 11st

ty

Sudden Future Singularity Model• Occurrence regardless of curvature, homogeneity or isotropy of the

universe• Pressure behaviour satisfies observation: current acceleration possible

log

(p

ress

ure

)

time

Note that no explicit Dark Energy component has been assumed to exist. Dabrowski calls the cause some “pressure driven dark energy” .

Observational Constraints•SNIa redshift-magnitude relation•Deceleration parameter•The Location of the CMBR Acoustic Peaks•Baryon Acoustic Oscillations•Age of the Universe

SNIa redshift-magnitude relation

• Previously it was shown by Dabrowski et al. (2007) that the SFS SNIa redshift-magnitude relation matches observations and the Concordance model.

• Test redone with 182 SNe Ia as compiled by Riess et al. (2007) In the Gold data set same results were achieved.

rz)a+(=DL 01

ta

tdc=r

25log5 10 LDMm

Luminosity distance is given by: where

The distance modulus is defined as:

Distance modulus vs. log(redshift) for the SFS and Concordance models as compared with SNIa data from Tonry et al. (2003) ‘Gold’ sample and Astier et al. (2006) SNLS sample. Graph from Dabrowski et al. (2007).

Deceleration parameter

q(z)

z

SFS Model

Concordance Model

Concordance Model parameters:

SFS Model parameters:

74.0

26.0

m

99936.032

471.0

9999.1

0

y

m

n

CMBR acoustic peaks• Shift parameter, :

Angular diameter distance to the last scattering surface (LSS) divided by Hubble horizon at the decoupling epoch

The apparent size of the sound horizon at recombination

Can be found using the formula:

• Acoustic scale, :

Angular diameter distance to the LSS divided by sound horizon at the decoupling epoch

Can be calculated using the formula:

The observed values of these parameters are taken from Komatsu et al. (2008)

CMBz

'

'

m )E(z

dzΩ=R

0

1

2/1

)(1

)(1)()(ln151

eq

reqr

baaf

afafafR

a

R

Courtesy: http://map.gsfc.nasa.gov/

Baryon Acoustic Oscillations• Cosmological perturbations excite sound waves in the early

universe photon-baryon plasma competition between gravity and radiation pressure. These oscillations leave their imprint on matter distribution now

• Natural standard ruler useful distance indicators now• Can be used to constrain the quantity known as the distance

parameter, , well:

• Angular scale of oscillations• Observed value taken from Komatsu et al. (2008)

3/2

0

3/1

1

BAOz

'

'

BAOBAO

m

)E(z

dz

z)E(z

Ω=A

Courtesy: http://cmb.as.arizona.edu/

Courtesy: http://www.sdss3.org/

A

Age of the Universe• Using the standard Friedmann equation, the age of the Universe is

calculated from:

where and

• Corresponding age for the SFS model was calculated • Observed value for the age: from the globular cluster estimates as

Krauss and Chaboyer (2003) present, i.e. no cosmology assumed• Hubble constant from the HST Key Project as given by Freedman et

al. (2001) which assumes only local cosmology• Therefore Hubble constant constraint also included

)(

)()(

ta

tazH

0

0 1 ))E(zz+(

dzH=t

''

'1

0

0

)()(

H

zHzE

Method• Used statistics to fit model parameters to data• Theoretically to obtain an SFS: and a currently

accelerating universe: • To comply with early universe requirements:• For , was used as the fraction of the time to an SFS

elapsed• 2d parameter space of search while keeping constant),( n

21 n

3

2m

0

2

st

ty 00 st

0y

Results

n

δ

SN

AllAge

AR

a

BAO SNIa

CMBR

n

δ99936.00 y Myrt s 7.8

BAO SNIa

CMBR

n

δ99.00 y Myrt s 138

BAO

SNIa

CMBR

n

δ9.00 y Gyrt s 5.1

BAO

SNIa

CMBR

n

δ8.00 y Gyrt s 4.3

BAO

SNIa

CMBR

n

δ7.00 y Gyrt s 8.5

BAO SNIa

CMBR

n

δMyrt s 7.899936.00 y

Results

BAO SNIa

CMBR

Myrt s 3.1

n

δ9999.00 y

Results

BAO SNIa

CMBR

yrt s 000,137

n

δ99999.00 y

BAO SNIa

CMBR

n

δyrt s 700,13999999.00 y

Conclusions and Future Directions•The example SFS model (with kept constant)

investigated has been shown not to be compatible with current data.

•With the data analysis tools set up we are planning to continue our research by working on other non-standard models like the GR averaging model proposed by Wiltshire (2007).

m