Fluctuations of the luminosity distance Copenhagen, December 16, 2005. Ruth Durrer Départment de physique théorique, Université de Genève Work in collaboration with Camille Bonvin and Alice Gasparini (astro-ph/0511183). Contents. Introduction
Fluctuations of the luminosity distanceCopenhagen, December 16, 2005
Départment de physique théorique, Université de Genève
Work in collaboration with CamilleBonvin and Alice Gasparini
The luminosity distance is an observational tool which can be used
to determine the geometry of the universe. We denote it by
dL(n,z). Here n is a direction in the sky and z is the redshift of the
source. So far, only its angle average
has been used to determine cosmological parameters.
-> accelerated expansion.
It has been suggested lately, that actually 2nd order fluctuations
might mimic accelerated expansion.
This prompted us to study the fluctuations in the luminosity
distance in more detail. We did not find that they actually can
become of order one and change the sign of the deceleration
parameter, but that they represent a novel tool which can be used
to measure cosmological parameters.
Several papers in the literature have recently calculated
fluctuations in the luminosity distance (Dodelson & Vallonotto,
astro-ph/0511086 and Cooray et al. astro-ph/0509581), but they
have included only a part of the effect, the deflection by lensing.
Here I will present a completely general treatment which includes
besides lensing also a Doppler term, gravitational redshift and a
sort of integrated Sachs-Wolfe term. (This is also presented in
Lam & Greene, astro-ph/0512159, but also there, the fluctuations
are studied as noise which has to be taken into account in the error
budget of present and future SN surveys.)
Previous literature has mainly regarded this effects as a limitation
to the accuracy of the measurements. Here I’ll discuss that they
actually represent a new observational tool (This has been
mentioned but not worked out in astro-ph/0511086).
We consider an object emitting with luminosity L = ES/tS. Let the received flux at the observer be F = EO/AO/ tO
The luminosity distance of the source is given by
dL2(S,O) = L/(4F)
If spacetime is Minkowski, this clearly gives the distance betweensource and observer. In curved spacetime, it is simply a definition.
If dS denotes an infinitesimal solid angle at the source position and dAO is an infinitesimal surface at the observer position we have
where 1+z is the photon redshift 1+z = (nS¢ uS)/ (nO¢ uO).The factor dAO/dSis the determinant of the Jacobi map, mapping photon directions S at the source into distance vectors normal to the photon direction and normal to the 4-velocity of the observer xO .
xO = J.
Choosing a congruence of neighboring geodesics x(,y), one can choose the affine parameter such that xO = y x(O,y)y . One can show that
The variables () and x() satisfy the following system of ordinary differential equations
Solving them, applying the initial condition ((S), 0) to x(O), determines the Jacobi map. (This map looks 4d, we still have to project x(O) onto the plane normal to the observer 4-velocity.)
In this way, the Jacoby map for an arbitrary spacetime without caustics can be defined (no strong lensing).
We first note that for conformally related metrics,
The luminosity distances are related via
In a perturbed Friedmann universe with scalar metric
perturbations given by
We can therefore calculate the luminosity distance for the metric
And then simply use the above relation. We consider an energy
momentum tensor with vanishing anisotropic stresses so that =.
We now have to determine the redshift and the Jacobi map for
The redshift is easy.
The Jacobi map is a bit more tedious. One obtains finally
(Sasaki 1987, Pyne & Birkinshaw 1996)
We shall see that, depending on the value of l and z different
contributions dominate in the power spectrum. Apart from the dipole term due to the observer velocity, this
expression is gauge invariant. is the Bardeen potential.
Like for CMB anisotropies, the correlation function is given by
We now want to determine the power spectrum of dL.
dL(z,n) is a function on the sphere, hence its power spectrum is
given by the corresponding Cl’s. We expand dL(z,n) in spherical
dL(z,n)/dL(z)= lm alm(z)Ylm(n), Cl(z,z’) = halm(z) a*lm(z)i
With the help of the primordial power spectrum for the Bardeen
potential, the C_l’s can now be written as integrals over k-space:
Defining the transfer function by (,k) = Tk()(k)
and the primordial power spectrum by
k3h(k)*(k’)i = (2)33(k-k’)P(k)
We obtain integrals of the form
Which we may have then to integrate over time (4x)
We first neglect the dipole. We then split the power spectrum into
5 different contributions:
Cl(1) the contribution from redshift, never dominates, largest for small redshift, low l, then flat spectrum
Cl(2) correlation of redshift with ISW and Doppler term from source, never dominates, flat spectrum, negative
Cl(3) ISW and Doppler terms. Dominate at low redshift and low l
Cl(4) correlation of ISW with lensing. Changes sign at low l. Never dominates
Cl(5) lensing contribution. Dominates at z> 0.4 and l> 10.
hdL(n,z) dL(n,z)i= dL(z)2 (4)-1l(2l+1)Cl(z,z) » 10-5dL(z)2
Is far too small to effect dL(z) significantly. It is therefore very
improbable that 2nd order perturbations are relevant for the observed
behavior of the luminosity distance which implies an accelerating
The term due to the observer velocity gives
Measuring this term for different redshifts allows to determine
H(z), from which the eqn. of state for dark matter can be derived
Much easier than from the angle averaged luminosity distance
(which is proportional to s H-1dz).
It is also an advantage that we know (in principle) the direction
Of that dipole from the CMB.
Due to the motion of the observer wrt the source (for pure CDM).
In principle we know its direction and simply have to measure the amplitude.
C1(z)/C1CMB / 1/H(z)
To measure H(z)
with an accuracy to
10% in 4 redshift
bins, z=0.1, … 0.4
We need about 50’000 SN’s
( m » 0.15)