LETTER TO THE EDITOR Anisotropic universes with isotropic cosmic microwave background radiation

We show the existence of spatially homogeneous but anisotropic cosmological models whose cosmic microwave background temperature is exactly isotropic at one instant of time but whose rate of expansion is highly anisotropic. The existence of these models shows that the observation of a highly isotropic cosmic microwave background temperature cannot alone be used to infer that the universe is close to a Friedmann-Lemaitre model. PACS numbers: 04.20.-q, 98.80.Dr Submitted to: Class. Quantum Grav. Letter to the Editor 2 It is widely believed by cosmologists that the universe can be accurately described by a Friedmann-Lemaitre (FL) model on sufficiently large scales. This belief stems from the fact that the temperature of the cosmic microwave background (CMB) is measured to be highly isotropic over the celestial sphere. In addition, certain theoretical results lend credence to this notion. The fundamental theorem due to Ehlers, Geren & Sachs (1968), the so-called EGS theorem, states that if all fundamental observers measure an exactly isotropic CMB temperature during some time interval in an expanding universe, then the universe is exactly an FL model during this time interval. This theorem has been generalized by Stoeger et al (1995), and subsequently by Maartens et al (1995a, 1995b), to the “almost EGS theorem”, which states that if all fundamental observers measure the CMB temperature to be almost isotropic during some time interval in an expanding universe, then the universe is described by an almost FL model during this time interval. In particular, the theorem implies that the overall expansion of the universe must be highly isotropic in the following sense. The anisotropy in the expansion is described by the shear scalar Σ, defined by Σ = σabσ ab 6H2 , (1) where σab is the rate of shear tensor and H is the Hubble scalar (see Wainwright & Ellis, p 18–19). The statement that the expansion is highly isotropic means that Σ 1. For our purposes, the detailed form of the primary assumption of each theorem is of importance. The EGS theorem requires the following hypothesis: I1(the isotropy condition): All fundamental observers measure the CMB temperature to be exactly isotropic during a time interval I. The almost EGS theorem requires the analogous hypothesis with “exactly” replaced by “almost”: AI1(the almost-isotropy condition): All fundamental observers measure the CMB temperature to be almost isotropic during a time interval I. In the application of these theorems, the time interval I is usually identified with te ≤ t ≤ to, where te is the time of last scattering and to is the time of observation (see, for example, Stoeger et al (1995), p 1–2). Both theorems make use of kinetic theory for describing the photons of the CMB (see, for example, Sachs & Ehlers (1971) for an introduction to relativistic kinetic theory), and the proofs follow from the EinsteinLiouville equations. The distribution function for the photon fluid is assumed to be either exactly isotropic (the EGS theorem) or almost isotropic (the almost EGS theorem). The multipoles of the CMB temperature anisotropy can subsequently be written in terms of the anisotropies of the distribution function (see equation (40) and (41) in Maartens et al (1995a)). It is also important to note that a number of technical assumptions about the behaviour of the derivatives of the multipoles of the CMB temperature anisotropy must be made in order to prove the almost EGS theorem (see, for example, equation (1), (2), (3), and (4) in Maartens et al (1995b)). Although the EGS theorem and the almost EGS theorem shed considerable light on the relation between idealized CMB temperature observations and the geometry Letter to the Editor 3 of cosmological models, they cannot be used to conclude that the physical universe is close to an FL model. The reason for this limitation is basically that the hypotheses I1 and AI1 above include the stipulation “during a time interval I”, whereas we can only observe the CMB temperature at one instant of time on a cosmological scale. This observational limitation also implies that the technical assumptions needed for the almost EGS theorem are not observationally testable. In view of this situation, it is of interest to ask what restrictions, if any, can be placed on the anisotropy in the rate of expansion, using the following much weaker hypotheses: I2(the weak isotropy condition): All fundamental observers measure the CMB temperature to be isotropic at some instant of time to. AI2(the weak almost-isotropy condition): All fundamental observers measure the CMB temperature to be almost isotropic at some instant of time to. On the basis of continuity, we can argue that either of assumptions I2 and AI2 imply that all fundamental observers will measure the CMB temperature to be almost isotropic in some time interval Iδ of length δ centered on to. This time interval could, however, be much shorter than the time interval I referred to in I1 and AI1. Nevertheless, if the technical assumptions mentioned above concerning the almost EGS theorem were satisfied, this theorem would imply that during the time interval Iδ the model is close to an FL model, and hence that the rate of expansion is highly isotropic (i.e. that Σ 1). Thus, it is reasonable to ask whether I2 and AI2 imply that the overall expansion of the universe must be almost isotropic at time to, without imposing any of the technical assumptions about the multipoles. In this Letter we answer this question in the negative. In particular we show that for a given time to, there are spatially homogeneous cosmological models such that at to the CMB temperature is measured to be isotropic by all fundamental observers, even though the overall expansion of the universe is highly anisotropic at to. We consider the class of non-tilted spatially homogeneous models whose threeparameter isometry group is of Bianchi type VIII, and which are also locally rotationally symmetric (LRS). Since we are modeling the universe since the time of last scattering, it is reasonable to consider models with dust as the source of the gravitational field. In order to calculate the temperature of the CMB in these models, we adopt the approach of Nilsson et al (1999)‡. This approach is based on the orthonormal frame formalism and makes use of expansion-normalized variables (see Wainwright & Ellis (1997), p 112, for a motivation of this normalization). We will use the combined gravitational and geodesic evolution equations for models of Bianchi class A (which includes Bianchi VIII), which are given by equations (24)–(30) in NUW. This approach, which is purely geometric, differs from the kinetic approach used in the EGS and almost EGS theorems in that effects of the CMB photons on the gravitational field are neglected. These effects, however, will not change the results in any significant way. The temperature of the CMB in a non-tilted spatially homogeneous cosmological ‡ From now on we will refer to this reference as NUW. Letter to the Editor 4 model as function of angular position on the celestial sphere is given by§ T (θ, φ) = Te exp [ − ∫ τo