Fluctuations of the luminosity distance Copenhagen, December 16, 2005

1. 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)

2. Contents • 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

3. Introduction 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.

4. 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).

5. The luminosity distance (I) 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/(4F) If spacetime is Minkowski, this clearly gives the distance betweensource and observer. In curved spacetime, it is simply a definition. If dS 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/dSis 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.

6. The luminosity distance (II) 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).

7. The luminosity distance in a perturbed Friedmann universe(I) 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.

8. geodesic deviation The luminosity distance in a perturbed Friedmann universe (II) The redshift is easy. The Jacobi map is a bit more tedious. One obtains finally (Sasaki 1987, Pyne & Birkinshaw 1996)

9. 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.

10. Like for CMB anisotropies, the correlation function is given by The luminosity distance power spectrum 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

11. The luminosity distance power spectrum (II) 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)33(k-k’)P(k) We obtain integrals of the form Which we may have then to integrate over time (4x)

12. z=0.1 A simple example, m=1 pure CDM with scale invariant scalar fluctuations 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.

13. Scale invariant m=1 pure CDM (II) Cl total

14. The variance hdL(n,z) dL(n,z)i= dL(z)2 (4)-1l(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.

15. The dipole 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.

16. The dipole (II) 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.

17. The dipole (III) 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 

18. Conclusion • 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.)