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Fluctuations of the luminosity distanceCopenhagen, December 16, 2005

Ruth Durrer

Départment de physique théorique, Université de Genève

Work in collaboration with CamilleBonvin and Alice Gasparini

(astro-ph/0511183)

- Introduction
- The luminosity distance in a general spacetime
- The luminosity distance in a perturbed Friedman universe
- The luminosity distance power spectrum
- A simple example

- Parameter estimation
- The dipole

- Conclusion

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

this spacetime.

geodesic deviation

The redshift is easy.

The Jacobi map is a bit more tedious. One obtains finally

(Sasaki 1987, Pyne & Birkinshaw 1996)

Doppler term

redshift

‘ISW’

lensing

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

harmonics,

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)

z=0.1

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.

Cl total

The variance

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

universe.

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)

Dependence on

- The fluctuations in the luminosity distance represent an interesting new tool to measure cosmological parameter.
- Its utility, complementarity have yet to be explored.
- The dipole allows to measure directly the history of the Hubble parameter, H(z).
- The higher multipoles can be calculated with a ‘CMBfast-like’ algorithm which we are now developing.
- Even if the fluctuations are not studied, they have to be taken into account as correlated noise in present and upcoming Supernova searches. (Especially low l contrubutions are important.)