Cosmologists hope to "hear the shape of space", namely find its topology, by analyzing in detail the temperature fluctuations in the cosmic microwave background radiation (CMB). A group of cosmologists, including researchers from l’Observatoire de Paris, has recently developped a model for the vibrations of the universe. For the first time [1], they have simulated high resolution CMB maps containing the signatures of a wide class of topologies, for comparison with the forthcoming MAP satellite data in early 2003.

The shape of space

In recent years, cosmologists have become interested in the global shape of space [2]. Previously, most of them had neglected the fact that, even if space is flat on a large scale, it can take many different shapes, for instance that of a doughnut-like hypertorus. A space of a given curvature admits a number a topologies. Indeed, 18 flat topologies along with an infinite number of spherical and hyperbolic ones are theoretical candidates to describe the shape of physical space. Although recent CMB observations constrain the value of space curvature to a very narrow range about zero, they still leave open the question of whether the average curvature is exactly zero (corresponding to a flat universe) slightly positive (spherical universe) or slightly negative (hyperbolic universe), and above all whether the topology is simple (for instance an infinite flat space) or not (for instance a finite flat hypertorus). In a previous article [3], three authors have proved that the spherical topologies would be more easily detectable observationally than hyperbolic or flat ones. The reason is that, no matter how close space is to perfect flatness, only a finite number of spherical shapes are excluded by observational constraints. Due to the special structure of spherical spaces, topological imprints would be potentially detectable within the observable universe. Thus cosmologists are taking a renewed interest in spherical spaces as possible models for the physical universe. Now the main question is : how to detect the topology of space ?

The Universe as a drumhead

If you sprinkle fine sand uniformly over a drumhead and then make it vibrate, the grains of sand will collect in characteristic spots and figures, called Chladni patterns. These patterns reveal much information about the size and the shape of the drum and the elasticity of its membrane. In particular, the distribution of spots depends not only on the way the drum vibrated initially but also on the global shape of the drum, because the waves will be reflected differently according to whether the edge of the drumhead is a circle, an ellipse, a square, or some other shape. In cosmology, the early Universe was crossed by real acoustic waves generated soon after Big Bang. Such vibrations left their imprints 300 000 years later as tiny density fluctuations in the primordial plasma. Hot and cold spots in the present-day 2.7 K CMB radiation reveal those density fluctuations. Thus the CMB temperature fluctuations look like Chladni patterns resulting from a complicated three-dimensional drumhead that vibrated for 300 000 years. They yield a wealth of information about the physical conditions that prevailed in the early Universe, as well as present geometrical properties like space curvature and topology. More precisely, density fluctuations may be expressed as combinations of the vibrational modes of space, just as the vibration of a drumhead may be expressed as a combination of the drumhead’s harmonics. For the first time, a team of physicists has shown how the shape of a spherical space can be heard in a unique way. They calculated the harmonics (the so-called "eigenmodes of the Laplace operator") for most of the spherical topologies [4]. Next, starting from a set of initial conditions fixing how the universe originally vibrated (the so-called Harrison-Zeldovich spectrum), they evolved the harmonics forward in time to simulate CMB maps for a number of topologies, including flat and spherical ones [1]. riaz-fig1.jpg Figure 1. A multiconnected topology translates into the fact that any object in space may possess several copies of itself in the observable Universe. For an extended object like the region of emission of the CMB radiation we observe, the so-called last scattering surface, it can happen that it intersects with itself along pairs of circles [5]. In this case, this is equivalent to say that an observer (located at the center of the last scattering surface) will see the same region of the Universe from different directions. As a consequence, the temperature fluctuations will match along the intersection of the last scattering surface with itself, as illustrated in the above figure. This CMP map is simulated for a multi-connected flat space, namely a cubic hypertorus whose length is 3.17 times smaller than the diameter of the last scattering surface. Click on the picture to enlarge it. riaz-fig2a.jpg riaz-fig2b.jpg riaz-fig2c.jpg riaz-fig2d.jpg Figure 2. When space is curved, the signature of a multi-connected topology can take an unexpected form. For example when space is spherical, it is possible that the last scattering surface matches with itself after a rotation of one of its hemisphere as shown on the above 4 figures. These CMB maps are simulated for a multi-connected spherical space whose volume is 21 times smaller than the ordinary hypersphere, the so-called L(21,1) lens space, although with non realistic values of the cosmological parameters. Click on the pictures to enlarge them. Balloon-borne CMB experiments (Boomerang, DASI, Archeops) have put tight constraints on the curvature of space, but provide too little data to test the topology of the Universe because they cover only a small portion of the sky. The situation is about to change dramatically with the MAP (Microwave Anisotropy Probe) satellite mission. Launched by NASA in April, 2001, it will provide high resolution maps of CMB fluctuations on the whole sky, excluding the portion obscured by our own Milky Way galaxy. The 6-month MAP data will be released late January or early February 2003. A topological signal as predicted in [5], and simulated in the maps of [1], may be subtly encoded in these data, and may eventually answer the fascinating question whether space is finite. Peer reviewed publications and references [1] A. Riazuelo, J.-P. Uzan, R. Lehoucq and J. Weeks, "Simulating Cosmic microwave background maps in multi-connected universes" (e-print astro-ph/0212223). [2] J.- P. Luminet : "L’Univers chiffonné", Fayard, Paris, 2001, 369 p. R. Lehoucq : "L’univers a-t-il une forme ?", Flammarion, Paris 2002, 152 p. J. Weeks : "The Shape of Space", Dekker, 2nd edition, 2001, 328 p. [3] J. Weeks, R. Lehoucq and J.-P. Uzan : "Detecting topology in a Nearly Flat Spherical Universe", (e-print astro-ph/0209389). [4] R. Lehoucq, J. Weeks, J.-P. Uzan, E. Gausmann and J.-P. Luminet, "Eigenmodes of 3-dimensional spherical spaces and their application to cosmology", Classical and Quantum Gravity,(2002) 19, 4683-4708 (e-print gr-qc/0205009). [5] N. Cornish, D. Spergel and G. Starkman,"Circles in the sky : finding topology with the microwave background radiation", Classical and Quantum Gravity (1998), 15, 2657-2670 (e-print astro-ph/9801212). Contacts Alain Riazuelo, Service de Physique Théorique, CEA Saclay, France — riazuelo spht.saclay.cea.fr Jean-Philippe Uzan, Laboratoire de Physique Théorique, Orsay, and Institut d’Astrophysique de Paris, France — uzan iap.fr Jeffrey Weeks, Canton, USA — weeks northnet.org Roland Lehoucq, Service d’Astrophysique, CEA Saclay, France, and Observatoire de Paris (LUTH) — roller discovery.saclay.cea.fr Jean-Pierre Luminet, LUTH, CNRS-UMR 8102, Observatoire de Paris, France — Jean-Pierre.Luminet obspm.fr