Dipole of the Luminosity Distance: A Direct Measure of H(z )

1. Dipole of the Luminosity Distance: A Direct Measure of H(z) Wu Yukai 2013.11.1 Camille Bonvin, Ruth Durrer, and Martin Kunz

2. Background • Accelerated expansion of the universe • Homogeneous and isotropic universe Contributions to energy momentum tensor are described by energy density ρ(z) and pressure P(z) • Dark energy: equation of state • Cosmological constant Friedmann equations

3. Measurement of w(z) • Luminosity distances to supernovae(monopole) • Angular diameter distance to the last scattering surface (CMB) • Problems • Use double integration: insensitive to rapid variations • Model-dependent: strong biases(difficult to detect and quantify)

4. Solution • A direct measurement of the Hubble parameter H(z) • E.g. in a flat universe H0=H(0), Ωm: the fraction of mass (From Friedmann equations) • Methods to get H(z) • Numerical derivative of the distance data: noisy • Radial baryon oscillation measurements(future)

5. alternative method to measure H(z) • Dipole of the luminosity distance • Luminosity distance Where F is flux, and L is luminosity. Where a(t0) is the scale factor at time t0(when receiving the light), r is the coordinate distance, and z is the source redshift.

6. Luminosity distance • a(t0) comes from the FLRW metric Where K=0 for a flat universe. • 1+z comes from two part: • Frequency decreases to 1/(1+z) and therefore energy per photon decreases. • The rate of receiving photons is 1/(1+z) of that of emission Therefore F decreases to 1/(1+z)2 and DL increases to (1+z).

7. Direction-averaged luminosity distance Where n is the direction of the source. • Equivalent to the former definition, noting that • Dipole of the luminosity distance Where e represents the direction of the dipole. • Origin of the dipole • Doppler effect of Earth’s peculiar motion (dominate for z>0.02) • Lensing(dominate in small scale but vanish when integrating)

8. Dipole of the luminosity distance • From observation • From theoretical deduction(See the article for more details) • Given H(z), we can fit the velocity of the peculiar motion and compare it with the result of CMB. • Given v0 from CMB, we can get H(z).

9. Compatible with the CMB dipole • 44 low-redshift supernovae • Estimate the error: • Peculiar velocity of the source: 300 km/s • Dispersion of magnitude m: Δm = 0.12 The relationship between m and dL • Fitting result: in agreement with the result of CMB, 368km/s

10. Accuracy of the method • Assuming Δm is independent of z • For one supernova • Observation of N independent supernovae • To decide if dark energy is a cosmological constant • Compare measured values of H(z) with prediction of ΛCDM • should be larger than the error • Difference between a flat pure CDM universe and a flat ΛCDM universe is 10% at z=0.1, 19% at z=0.2, and 27% at z=0.3

11. Benefits • Dipole: more resistant to some effects which cause systematic uncertainties in monopole • Any deviation in H(z) from theoretical predictions can be directly detected. Easily be smeared out by using only monopole. • Enhance the measurement of monopole(dipole is considered as systematical error now; increasing N) • Future • Measurement of a large number of supernovae with low redshift(0.04~0.5) • Cover a large part of the sky to eliminate influence of lensing(dominate for l > 100 and z>1), cover the regions aligned and antialigned with the CMB dipole

12. Summary • An alternative way to measure H(z): dipole of luminosity distance • A sample of nearby supernovae: consistent with CMB • Estimate the number of SN needed for a given precision