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

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 H 0 =H(0), Ω m : 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(t 0 ) is the scale factor at time t 0 (when receiving the light), r is the coordinate distance, and z is the source redshift.

6. Luminosity distance – a(t 0 ) 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 D L 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 v 0 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 d L – Fitting result: in agreement with the result of CMB, 368km/s

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