1. Constraining the Lattice Fluid Dark Energy from SNe Ia, BAO and OHD 报告人： 段效贤 中国科学院国家天文台 2012 年两岸粒子物理与宇宙学研讨会

2. 2 Outline Lattice Fluid Dark Energy Model Constraint from SNe Ia Constraint from BAO Constraint from OHD Result Conclusions and Discussions

3. 3 Lattice Fluid Dark Energy Model The equation of state of Lattice Fluid Dark Energy can be written as where ρ X and p X are the energy density and pressure of the fluid, respectively. A is a dimensionless constant, ρ 0 is the present-day cosmic energy density defined by ρ 0 =3H 0 2 /8π with H 0 the present- day Hubble parameter.

4. 4 "Statistical Thermodynamics of Polymer Solutions", by Isaac C. Sanchez, Robert H. Lacombe, Macromolecules (1978), Volume: 11, Issue: 6, Pages: 1145-1156.

5. 5 Now we study the evolution of the Universe filled with matter (includes baryon matter and dark matter), radiation and LFDE. The metric of FRW Universe is given by: where a(t) is the scale factor and K= +1, 0, -1 describe the topology of the Universe which correspond to closed, flat and open Universe, respectively.

6. 6 The scale factor evolves according to the Friedmann equation: whereρ total is the total energy density of the Universe. Observations reveal that the Universe is highly flat in space. So we put K= 0 in the following. Then the Friedmann equation can be written as where ρ m /a 3 and ρ r /a 4 are the energy density of matter and radiation, respectively; ρ m andρ r are respectively the energy density of matter and radiation in the present-day Universe.

7. 7 In order to obtain ρ X, we substitute the equation of state for LFDE Eq. (1) into the energy conservation equation: Keeping in mind the relation 1/a = 1 +z and defining Ω X ≡ρ X /ρ 0, we obtain the energy conservation equation as follows This ordinary differential equation could not be solved analytically. So we are going to study the numerical solution by using the observational data from SNe Ia, BAO and OHD.

8. 8 Constraint from SNe Ia The luminosity distance d L of SNe Ia is defined by where c is the speed of light. The function E(z) is defined by where Ω r ≡ρ r /ρ 0, is the sum of photons and relativistic neutrinos where N eff is the effective number of neutrino species (the standard value of N eff is 3.04), Ω γ = 2.469×10 −5 h −2 for T CMB = 2.725K.

9. 9 The theoretical distance modulus is defined by The observed distance modulus is listed in the dataset of Union 2.1. Then we can calculate theχ 2 SNeIa where C SN is the contravariant matrix which includes the systematical errors for the SNe Ia data, could be found in the dataset of Union 2.1.

10. 10 Constraint from BAO We used the measurements derived from observation data of the 6dFGS, the distribution of galaxies from the Sloan Digital Sky Survey Data(SDSS) Release 7 Galaxy Sample and the WiggleZ Dark Energy Survey. The BAO relevant distance measure is modelled by volume distance, which is defined by The comoving sound horizon is defined by where c s (z) is the sound speed, c s (z)= and = 3Ω b /4Ω γ 。

11. 11 We could get the distance ratio where z d is the drag redshift defined in D.J. Eisenstein et. al., Astrophys. J. 496,605 (1998). The acoustic parameter is defined by Beutler et al. derived the measurement from 6dFGS that the observed d obs z=0.106 = 0.336±0.015. Percival et al. measured the distance ratio at two redshifts:d obs z=0.2 = 0.1905±0.0061,d obs z=0.35 = 0.1097±0.0036.

12. 12 From the WiggleZ Dark Energy Survey, Blake et al. measured the acoustic parameter at three redshifts, A obs (z = 0.44) = 0.474±0.034, A obs (z = 0.6) = 0.442±0.020, A obs (z = 0.73) =0.424±0.021. Then we could get theχ 2 BAO

13. 13 Relative Galaxy Ages can also be used to constrain cos-mological parameters. Given the measurement of the age difference of two passively-evolving galaxies formed nearly at the same time,δt, and the small redshift intervalδz by which they are separated, the ratio δz/δt could be calculated. Then we can infer the derivative: dz/dt. The quantity measured in the method above is directly related to the Hubble parameter: Constraint from OHD

14. 14 we take the observational data at 15 different redshifts in J. Xu, and Y. Wang, JCAP. 1006, 002 (2010): 12 of them are from Gemini Deep Survey (GDDS), SPICES, VVDS, and Keck Observations; 3 more available data in where the authors obtained : H(z = 0.24) =79.69±2.32, H(z = 0.34) = 83.8±2.96, and H(z =0.43) = 86.45±3.27, by using the BAO peak positionas a standard ruler in the radial direction. The theoretical value of H(z) could be obtained from Eq. (8). Therefore χ2 for Hubble data is Then the totalχ 2 is:

15. 15 Result The marginalized probability contours at 1σ, 2σand 3σCL in the A- Ω m plane of the result of combined constraint of SN, BAO and OHD:

16. 16 The evolution of the equation of state w=p X /ρ X. When z >2.0,w is approaching -1. As the redshift decreases to 0,w increase slightly to w= -0.937.

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18. 18 Age of the Universe For our model, the age of the Universe is found to be 13.35 Gyr.

19. 19 Conclusions and Discussions We constrain the model with current cosmological observational data, find the best fit value of parameters:A=-0.3, Ω m = 0.30. Taking the best values of A and Ω m, we investigate the comic implications of the model. We find the equation of state is almost the same as the ΛCDM model at the redshifts greater than 2.0. For the present-day Universe, we have w=-0.937 which is consistent with many other observations. The age of Universe is estimated to be 13.35 Gyr in our LFDE model. Finally, the statefinder r and s of the LFDE model evolve from r = 1.0, s = 0 (which is the value of ΛCDM Universe) at high redshifts to the present values:r= 0.572,s= 0.144.

20. Thank you!