Observational Constraints on Sudden Future Singularity Models

1. Observational Constraints on Sudden Future Singularity Models Hoda Ghodsi – Supervisor: Dr Martin Hendry Glasgow University, UK Grassmannian Conference in Fundamental Cosmology Szczecin, Poland, September 2009

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

3. Concordance Cosmology Cosmological observations of most importantly the Cosmic Microwave Background Radiation (CMBR), the Large Scale Structure and SNeIa 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/

4. 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: ,

5. Sudden Future Singularity Model • Occurrence regardless of curvature, homogeneity or isotropy of the universe • Pressure behaviour satisfies observation: current acceleration possible Note that no explicit Dark Energy component has been assumed to exist. Dabrowski calls the cause some “pressure driven dark energy” . log (pressure) time

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

7. SNIa redshift-magnitude relation Luminosity distance is given by: where The distance modulus is defined as: • Previously it was shown by Dabrowski et al. (2007) that the SFS SNIaredshift-magnitude relation matches observations and the Concordance model. • Test redone with 182 SNeIa as compiled by Riess et al. (2007) In the Gold data set  same results were achieved. 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).

8. Deceleration parameter Concordance Model parameters: SFS Model q(z) SFS Model parameters: Concordance Model z

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

10. 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) Courtesy: http://www.sdss3.org/ Courtesy: http://cmb.as.arizona.edu/

11. 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

12. 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

13. Results δ n

14. SN R A Age All

15. BAO SNIa CMBR δ n

16. BAO SNIa δ CMBR n

17. SNIa BAO δ CMBR n

18. SNIa BAO δ CMBR n

19. SNIa BAO δ CMBR n

20. BAO SNIa CMBR δ n

21. BAO SNIa Results CMBR δ n

22. BAO SNIa Results CMBR δ n

23. BAO SNIa CMBR δ n

24. 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).