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

Camille Bonvin, Ruth Durrer, and Martin Kunz

Wu Yukai2013.11.1

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

1

3

P zw z

z

1w

Friedmann equations

• 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)

• 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)

• 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.

0 1LD a t r z

• 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 emissionTherefore F decreases to 1/(1+z)2 and DL increases to (1+z).

0 1LD a t r z

• 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)

0 '

' 1 and '' '

a t a tz H z

a t a t

• 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).

• 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 = 0.12The relationship between m and dL

– Fitting result:0 405 192km sv

in agreement with the result of CMB, 368km/s

• 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

mm

N

• 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

• 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