PHYSICAL REVIEW LETTERS week ending PRL 100, 191303 (2008) 16 MAY 2008
Time Drift of Cosmological Redshifts as a Test of the Copernican Principle
Jean-Philippe Uzan* Institut d’Astrophysique de Paris, Universite´ Pierre and Marie Curie-Paris VI, CNRS-UMR 7095, 98 bis, Bd Arago, 75014 Paris, France
Chris Clarkson† and George F. R. Ellis‡ Cosmology and Gravity Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa (Received 10 January 2008; published 16 May 2008) We present the time drift of the cosmological redshift in a general spherically symmetric spacetime. We demonstrate that its observation would allow us to test the Copernican principle and so determine if our Universe is radially inhomogeneous, an important issue in our understanding of dark energy. In particular, when combined with distance data, this extra observable allows one to fully reconstruct the geometry of a spacetime describing a spherically symmetric underdense region around us, purely from background observations.
DOI: 10.1103/PhysRevLett.100.191303 PACS numbers: 98.80.Es, 98.62.Py
I. INTRODUCTION of the Lemaıˆtre-Tolman-Bondi (LTB) family (that is, spherically symmetric solution of Einstein equations Cosmological data is usually interpreted under the as- sourced by pressureless matter and no cosmological con- sumption that the Universe is spatially homogeneous and stant). Unfortunately, this simple extension of the RW isotropic. This is justified by the Copernican principle, stating that we are not located at a favored position in universes depends on two free functions (see below for space. Combined with the observed isotropy, this leads to details) so that the reconstruction is underdetermined and a Robertson-Walker (RW) geometry [1], at least on the one must fix one function by hand. Thus, one needs at least scale of the observable Universe. one extra independent observation to reconstruct the ge- This implies that the spacetime metric reduces to a ometry of an LTB universe. A limitation to this reconstruc- single function of the cosmic time, the scale factor a t . tion arises because most data lie on our past light cone. This function can be Taylor expanded as a t This takes us back to the observational cosmology program a H t t 1 q H2 t t 2 , where H is the [10] and the question [11] of how to extract as much 0 0 0 2 0 0 0 0 information as possible about our spacetime from cosmo- Hubble parameter and q0 the deceleration parameter. Low redshift observations [2] combined with the assumption of logical data alone. Among many results, it was demon- almost flatness of the spatial sections, justified mainly by strated [12] that the two free functions of a LTB spacetime the cosmic microwave background data [3,4], lead to the can be reconstructed from the angular distance and number counts, even though evolution effects make it impossible to conclusion that q0 < 0: the expansion is accelerating. This conclusion involves no hypothesis about the theory of be conclusive [13]. gravity or the matter content of the Universe [5], as long Recently, two new ideas were proposed. First, it was as the Copernican principle holds. This has stimulated a realized [14] that the distortion of the Planck spectrum of growing interest in possible explanations [5,6], ranging the cosmic microwave background allows one to test the from new matter fields dominating the dynamics at late Copernican principle. Second, a consistency relation be- times to modifications of general relativity. tween distances on the null cone and Hubble rate measure- While many tests of general relativity on astrophysical ments in RW universes was derived [15], based on the fact scales have been designed [7], the verification of the that the curvature is constant; this also serves as an obser- Copernican principle has attracted little attention, despite vational test of the Copernican principle. the fact that relaxing this assumption may be the most In this Letter, we reconsider the time drift of cosmologi- conservative way, from a theoretical perspective, of ex- cal redshift in spacetimes with less symmetries than the plaining the recent dynamics of the Universe without in- RW universe and we demonstrate how, when combined troducing new physical degrees of freedom [8]. with distance data, it can be used to test the Copernican This possibility that we may be living close to the center principle, mainly because observing the thickening of our (because isotropy around us seems well established past light cone brings new information. As pointed out by observationally) of a large underdense region has attracted Sandage and then McVittie [16], one should expect to considerable interest. In particular, the low redshift (back- observe such a time drift in any expanding spacetime. ground) observations such as the magnitude-redshift rela- This may lead to a better understanding of the physical tion can be matched [9] by a nonhomogeneous spacetime origin of the recent acceleration [17,18], or to tests of the
0031-9007=08=100(19)=191303(4) 191303-1 © 2008 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 100, 191303 (2008) 16 MAY 2008 variation of fundamental constants [19]. This measure- implies that the luminosity distance is given by DL y 2 ment, while challenging, may be achieved with extremely 1 z DA. The redshift is given by large telescopes (ELT) and, in particular, it is one of the a uak emission A w0; 0 main science drivers in design of the Cosmic Dynamics 1 z a ; (2) Experiment (CODEX) spectrograph [20]. Our result may uak observer A w0;y strengthen the scientific case for this project. where the matter velocity and photon wave vector are a 1 a a 1 a We start by introducing observational coordinates, given by u A w and k AB y, respectively. which allow us to derive the general expression for the We deduce that the isotropic expansion rate, defined by time drift in a spherically symmetric, but not necessarily a 3H rau , is given by spatially homogeneous, universe [see Eq. (6)]. We show that observation of both the luminosity distance and the 1 @wB w; y @wC w; y H w; y 2 : (3) redshift drift allows one to probe the Copernican principle 3A B w; y C w; y at low redshifts, when ‘‘dark energy’’ dominates [see the For the central observer, who sees the Universe isotropic, consistency relation (8)]. We demonstrate that this expres- H is simply the Hubble expansion rate. At small redshifts, sion may be used with distance data to fully reconstruct the H w; y @wB w;0 O y , so the Hubble constant is geometry of a LTB spacetime [see Eq. (9)]. B w;0 A w;0 H0 @wB w0; 0 =B w0; 0 A w0; 0 . In the particular case of a dust dominated universe, the II. z_ IN A SPHERICALLY SYMMETRIC UNIVERSE acceleration and vorticity vanish and the fluid four-velocity Observational coordinates.—We consider a spherically can be expressed as the gradient of the proper time along symmetric spacetime in observational coordinates the matter worldlines: ua @at. Since we also have fw; y; #; ’g, where w labels the past light cones of events ua A@aw B@ay we deduce that dt Adw Bdy along the wordline C of the observer, assumed to lie at the so that A @wt and B @yt. The surfaces of simultaneity center so that w is constant on each past light cone, with are thus given by Adw Bdy and we have the integra- a a bility condition @ A @ B 0. u @aw>0, u being the four-velocity of the cosmic fluid, y w a uau 1. y is a comoving radial distance coordinate The covariant derivative of ua is therefore of the form specified down the past light cone of an event O on C. raub H gab uaub ab, where the shear ab is a Many choices are possible, such as the affine parameter symmetric, traceless, and satisfies u ab 0. The scalar 2 ab down the null geodesics from O, the area distance, or the shear ab=2 is consequently the only nonvanish- redshift; whatever choice we assume on the past light cone ing kinematical variable and is given by of O, it is specified on other past light cones through being 2 1 @ B @ C a w w comoving with the cosmic fluid, i.e., y u 0.(#, ’) are w; y p ; (4) ;a 3 A B C angular coordinates based at C and propagated parallelly along the past light cone. The metric in these observational where an arbitrary sign has been chosen. The regularity coordinates is conditions imply w; 0 0, which is expected since the
2 2 2 expansion is observed to be isotropic about the central ds A w; y dw 2A w; y B w; y dydw worldline. C2 w; y d 2; (1) Expression of the redshift drift.—From the expression z (2), it is straightforward to deduce that z_ w w0;y is which is clearly spherically symmetric around the world- given by line C defined by y 0. The requirement that the 2-spheres fw; yg behave regularly around when y ! @wA w0; 0 @wA w0;y const 0 C z_ w0;y 1 z : (5) implies [10] that A w; y !A w; 0 Þ 0, B w; y ! A w0; 0 A w0;y 2 B w; 0 Þ 0, and C w; y B w; 0 y O y . Now, we can choose w such that A w0; 0 1. Then, on There remain two coordinate freedoms in possible re- our past light cone, we can choose y such that scalings of w and y. However, once specified on C, w is @w lnB w0;y @w lnA w0;y . Note however that such a determined on the other worldlines by the condition that choice is not possible for all w. It follows that fw constg are past light cones of events on C. This allows 1 us to arbitrarily choose A w; 0 . Also, once specified on z_ w0;y 1 z H0 H w0;y p w0;y : (6) one past light cone, y is determined on all the others 3 because it is a coordinate comoving with the fluid. This This is the general expression for the time drift of the allows us to choose B w0;y for a given value of w w0. redshift as it would be measured by an observer at the On each past light cone, the cross-sectional area of a center of a spherically symmetric universe. source is related to the solid angle d 2 under which it is Robertson-Walker case.—Since 0 for a RW space- 2 2 observed by an observer on C at w w0 by C w0;y d . time, Eq. (6) reduces to the standard Sandage-McVittie This implies that C is the angular distance, DA, i.e., formula. Let us consider the RW metric in conformal DA y C w0;y . The distance duality relation [21] then coordinates 191303-2 PHYSICAL REVIEW LETTERS week ending PRL 100, 191303 (2008) 16 MAY 2008
2 2 2 2 2 2 ds a d d fK d ; (7) This solution depends on three arbitrary functions of r only, E r , M r , and T0 r . Their choice determines where is the radial comoving coordinate and f K the model completely. For instance, E; M; T0 sin ; ; sinh according to the curvature of the spatial 2 3 K0r ;M0r ; 0 corresponds to a RW universe. One can sections. Now, with w and y , this leads to further use the freedom in the radial coordinate to fix one of the form (1) for the metric (7) with A B a w y and the three functions at will so that one effectively has only C a w y fK y . H is thus constant on each constant two arbitrary independent functions. Assume we fix M r . time hypersurface and 0 everywhere. Since We want to determine fE r ;T0 r g to reproduce some @wAjy const @ a, Eq. (5)givesz_ 1 z H0 H z observables on our past light cone. This can be represented where we have shifted to cosmic time (dt ad ), using parametrically as fr z ;E z ;T0 z g. 1 z a0=a. Let us sketch the reconstruction and use r as the inte- We understand from this exercise why the observational gration coordinate, instead of z. Our past light cone is coordinates are adapted to the computation of z_ in an defined as t t^ r and we set R r R t^ r ;r . The arbitrary spacetime. They are just a generalization to less time derivative of R is given by R_ t^ r ;r R symmetric spacetimes of the conformal coordinates, where p 1 3 0 ^ the expression is easily obtained in the RW case. 2M0r =R r 2E r . Then we get R t r ;r ^ 0 We also conclude that since the three functions (A, B, C) R2 r fR r 3 t r T0 r R1 r =2gE =E 0 are expressed in terms of a unique function, all background R1 r T0 r R r =r. Finally, more algebra leads to _ 0 3 observations depend on some function of H, z_ being no R t^ r ;r R3 r fR1 r 3M0r t^ r T0 r = 2 0 3 0 2 exception. R r gE r =2E r M0r T0 r =R R1 r =r. Thus, 0 0 Consistency relation.—It follows that, in a RW universe, R_ , R , and R_ evaluated on the light cone are just functions we can determine a consistency relation between several of R r , E r , T0 r , and their first derivatives. Now, the observables. From the metric (7) and the relation for null geodesic equation gives that 1 0 z that follows, one deduces that H z D z 1 ^ r dz 1 z 2 2 1=2 dt R2 H D z , where a prime stands for @ and D z p ; p R3 r ; K0 0 z dr 1 2E r dr 1 2E r DL z = 1 z . This relation is the basis of the test of the RW structure, as recently proposed in Ref. [15], arguing and there that knowledge of H z at different redshifts, from, e.g., baryon acoustic oscillations or differential age esti- dR R1 r
p mates of passively evolving galaxies, could then be used to 1 R2 r : dr 1 2E r check this yields the same value of K0. Here, we argue that it can be implemented using z_ z as an observational These are three first-order differential equations relating input. Defining five functions R r , t^ r , z r E r , and T0 r . To recon- 2 2 struct the free functions we thus need two observational Cop DL z ; z_ z ;z 1 K0H0D z relations. R z DA z is the obvious choice. Then, from 2 0 2 H0 1 z z_ z D z ; (8) z_ z , we have the new relation 2 2 we must have Cop DL z ; z_ z ;z 0 whatever the matter 1 R3 R1 1 R3 R1 content of the Universe and the field equations, since it z_ 1 z H0 2 : 9 R2 R 3 R2 R derives from a purely kinematical relation that does not (9) rely on the dynamics (i.e., the Friedmann equations). This is a consistency relation between independent observables It was shown [9] that the observed DL z can be reproduced that holds in any Robertson-Walker spacetime. from the function T0 r assuming E 0, or from the Spherically symmetric spacetimes.—Writing the LTB function E r assuming T0 0. Here, we have shown metric in observational coordinates requires the solution that fE r ;T0 r g can be completely reconstructed from of the null geodesic equation, which is in general possible the data without assumptions. only numerically. Consider an LTB spacetime with metric 2 2 2 2 2 2 ds dt S r; t dr R r; t d ; III. DISCUSSION p where S r; t R0= 1 2E r and R_ 2 2M r =R r; t In this Letter, we have shown that observation of both 2E r , using a dot and prime to refer to derivatives with the luminosity distance and time drift of the redshift as a respect to t and r. The Einstein equations can be solved function of z allows one to construct a test of the M r 0 Copernican principle. We have derived the general expres- parametrically as fR r; ;t ; r g f ;T0 r E r sion for z_ z in a spherically symmetric spacetime [see M r g, where is defined by sinh E r 3=2 Eqs. (5) and (6)]. This extends the standard computation ; 3=6; sin , and E r 2E; 2; 2E according which was restricted to RW spacetimes and was extended to whether E is positive, null, or negative. to almost RW spacetimes only recently [22]. 191303-3 PHYSICAL REVIEW LETTERS week ending PRL 100, 191303 (2008) 16 MAY 2008
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