1 Title page Cosmological Models with No Big Bang Wun-Yi Shu (���������) Institute of Statistics National Tsing Hua University Hsinchu 30013, Taiwan E-mail: shu@stat.nthu.edu.tw

2 Abstract In the late 1990s, observations of Type Ia supernovae led to the astounding discovery that the universe is expanding at an accelerating rate. The explanation of this anomalous acceleration has been one of the great problems in physics since that discovery. In this article we propose cosmological models that can explain the cosmic acceleration without introducing a cosmological constant into the standard Einstein field equation, negating the necessity for the existence of dark energy. There are four distinguishing features of these models: 1) the speed of light and the gravitational ���constant��� are not constant, but vary with the evolution of the universe, 2) time has no beginning and no end, 3) the spatial section of the universe is a 3-sphere, and 4) the universe experiences phases of both acceleration and deceleration. One of these models is selected and tested against current cosmological observations of Type Ia supernovae, and is found to fit the redshift-luminosity distance data quite well.

3 I. INTRODUCTION In the late 1990s, observations of Type Ia supernovae made by two groups, the Supernova Cosmology Project [1] and the High-z Supernova Search Team [2], indicated that the universe appears to be expanding at an accelerating rate. The current mainstream explanation of the accelerating expansion of the universe is to introduce a mysterious form of energy���the so called dark energy that opposes the self-attraction of matter. Two proposed forms for dark energy are the cosmological constant, which can be viewed physically as the vacuum energy, and scalar fields, sometimes called quintessence, whose cosmic expectation values evolve with time. Currently, in the spatially flat �� CDM model of cosmology, dark energy accounts for nearly three-quarters of the total mass-energy of the universe [3]. The introduction of dark energy raises several theoretical difficulties, and understanding the anomalous cosmic acceleration has become one of the greatest challenges of theoretical physics. There are a number of excellent review papers on this issue [4-6]. In this article we propose cosmological models that can explain the accelerating universe without introducing a cosmological constant into the standard Einstein field equation, negating the necessity for the existence of dark energy. There are four distinguishing features of these models:

4 i The speed of light and the gravitational ���constant��� are not constant, but vary with the evolution of the universe. i Time has no beginning and no end i.e., there is neither a big bang nor a big crunch singularity. i The spatial section of the universe is a 3-sphere, ruling out the possibility of a flat or hyperboloid geometry. i The universe experiences phases of both acceleration and deceleration. One of these models is selected and tested against the current cosmological observations, and is found to fit the redshift-luminosity distance data quite well. This article has the following structure: In the next section, the cosmological models are developed, with the details of the calculations presented in the Appendix. In Sec. 3, the dynamical evolution of the universe is determined by solving the Einstein field equation under various conditions. In Sec. 4, a selected model is tested against the observations of Type Ia supernovae. Four data sets available in the literature are included in the test. Finally, the results are discussed in Sec. 5. Throughout this article we follow the sign conventions of Wald [7]. In particular, we use metric signature ��� + + + , define the Riemann and the Ricci tensors by equations (3.2.3) and (3.2.25) of Wald [7] respectively, and employ abstract index notation to denote tensors. Greek indices, running from 0 to 3, are used to denote

5 components of tensors while Latin indices are used to denote tensors. Einstein���s summation convention is assumed. II. COSMOLOGICAL MODELS A cosmological model is defined by specifying: 1) the spacetime geometry determined by a metric g ab , 2) the mass-energy distributions described in terms of a stress-energy-momentum tensor T ab , and 3) the interaction of the geometry and the mass-energy, which is depicted through a field equation. A. The spacetime metric Under the assumption that on the large scale the universe is homogeneous and isotropic and expressed in the synchronous time coordinate and co-moving spatially spherical/hyperbolic coordinates ( , , ) t, �� �� ��, the line element of the spacetime metric g ab takes the form [7] (d ( ) ( 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 sin sin ( ) sin , sinh sin d dt t d d d�� d ds2 c a2 �� �� �� ��)2��d �� �� �� �� �� �� �� ��)2��d ��� + + ��� ��� = ��� + + + ��� ��� + + ���d ��� (2.1) where c is the speed of light and the three options listed to the right of the left bracket correspond to the three possible spatial geometries: a 3-sphere, 3-dimensional