The flatness problem and ¤ Kayll Lake [*] Department of Physics, Queen's University, Kingston, Ontario, Canada, K7L 3N6 (Dated: May 24, 2005) By way of a complete integration of the Friedmann equations, in terms of observables, it is shown that for the cosmological constant ¤ > 0 there exist non-flat FLRW models for which the total density parameter ­ remains » 1 throughout the entire history of the universe. Further, it is shown that in a precise quantitative sense these models are not ¯nely tuned. When observations are brought to bear on the theory, and in particular the WMAP observations, they con¯rm that we live in just such a universe. The conclusion holds when the classical notion of ¤ is extended to dark energy. The flatness problem is often considered to be the most In this letter we point out that if ¤ > 0 then there impressive issue in standard cosmology that is addressed exist standard models for which ­ » 1 throughout their by the inflation paradigm [1]. Let us start by summariz- entire evolution even though they are not spatially flat. ing the flatness problem for zero cosmological constant. Moreover, in a precise quantitative sense, we show that With ¤ = 0 the density parameter of a FLRW model is these models are not ¯nely tuned. When current obser- given by [2] vations are brought to bear on the theory, they con¯rm that we live in just such a universe [6]. 8¼G½ ­ = (1) 3H2c2 To include ¤ with dust de¯ne, in the usual way, and for a single fluid the state space is summarized for c2¤ c2k ­ ´ ; ­ ´ ¡ ; (4) standard models in FIG. 1 [3]. ¤ 3H2 k H2R2 so that the Friedmann equations reduce to ­M + ­¤ + ­k = 1 (5) where we have written ­ = ­M for convenience. To complement FIG. 1, which is the state space for ­¤ = 0, the state space for ­M = 0 is shown in FIG. 2. FIG. 1: The state space for a single fluid FLRW model with p = (γ¡1)½ and γ > 2=3. M is the Milne universe (Minkowski space). Bb is the spatially flat FLRW model and the Big Bang. Turning points (H = 0) are sent to in¯nity by the def- inition (1). The diagram reflects the evolution for expanding universes. The essential point is that except for the spatially flat FIG. 2: The state space for ­ = 0 (here rotated left ¼=2 to case M save space). Again M is the Milne universe (Minkowski space [7]). dS is the spatially flat representation of de Sitter space ­ = ­(t): (2) and acts as an attractor for the two other representations of de Sitter space. Observations show that ­ » 1: (3) Evolution in the full ­¤¡­M plane was ¯rst considered 0 by Stabell and Refsdal [8] for the case of dust (γ = 1). The flatness problem involves the explanation of (3) given Writing Z = 1 + z (1 + z the standard redshift), they (2). The problem can be viewed in two ways. First, one populated the phase plane from relations equivalent to can take the view that there is a tuning problem in the ­ Z3 sense that at early times ­ must be ¯nely tuned to 1 [4]. ­ = Mo ; (6) M ­ Z3 + (1 ¡ ­ ¡ ­ )Z2 + ­ However, this argument is not entirely convincing since Mo Mo ¤o ¤o all standard models necessarily start with ­ exactly 1. which is a restatement of the conservation law for dust More convincing is the view that except for the spatially 2 3 (the constancy of ­M H R ), and flat case the probability that ­ » 1 is strongly dependent on the time of observation [5] and so there is an epoch ­ ­ = ¤o ; (7) problem: why should (3) hold? ¤ 3 2 ­Mo Z + (1 ¡ ­Mo ¡ ­¤o )Z + ­¤o 2 which is a restatement of the constancy of ¤. They dis- tinguishedR trajectories via (half) the associated absolute 1 cdt horizon, 0 R . The same technique has been applied more recently and in more general situations [9]. Equivalently, from a dynamical systems point of view [10], we can, again using dust as an example, consider the system of di®erential equations 0 ­¤ = (­M ¡ 2­¤ + 2)­¤; (8) and 0 ­M = (­M ¡ 2­¤ ¡ 1)­M ; (9) 0 where ´ d=d´ and ´ ´ ln(1=Z). We now recognize the critical points in FIG. 1 and FIG. 2: Bb is a repulsor, dS an attractor and M a saddle point for the full ­¤ ¡ ­M plane. The approach used here is somewhat di®erent. Al- though what now follows can be generalized [11], we con- FIG. 3: The loci ® = constant and Ht = constant [13] tinue with dust as it presents an uncluttered relevant ex- in the ­¤ ¡ ­M plane. The loci shown are (bottom up) ample. We observe a constant of the motion (®) for the Ht = 0:6; 2=3; 3=4; 0:826; 1; 1:5; 1 (´ ® = 1) system (8)-(9). The constant is given by [12] (thick) and j®j = 0:01; 0:05; 0:1; 1=4; 1=2; 1 (thick), 2; 3; 4; 6; 8; 12; 20; 40; 100; 500. Note that for ¤ > 0 the delimiter ­¤ = 1 ¡ ­M (k = 0) is approached on either ­M 2 ­k 3 ( ) § ®( ) = 0 (10) side (k = §1) as ® ! 1. 2­¤ 3­¤ for k = §1 respectively. Since ­¤ = 1 ¡ ­M for k = 0, the system can be considered solved and the Friedmann throughout the entire evolution of the associated universe equations, in terms of observables, in e®ect integrated. if ® 2 (a; 1) where a is a matter of choice but of the order A phase portrait with trajectories distinguished by ® is >» 500. In any event, it is clear that ® need not be ¯nely shown in FIG. 3. tuned to produce (14) [15]. The physical meaning of ® is not hard to ¯nd. In terms Fortunate we are to live in an era of unprecedented of the bare Friedmann equation advances in observational cosmology. We now bring some of theses observations to bear on the foregoing dis- C ¤R2 R_ 2 + k = + ; (11) cussion. The Wilkinson Microwave Anisotropy Probe R 3 (WMAP) has enabled accurate testing of cosmological with (k; ¤) considered given, each value of the constant models based on anisotropies of the background radia- C > 0 determines a unique expanding universe (k 6= 0). tion [16]. Independently, the recent Hubble Space Tele- For each C, if k = 1, there is a special value of ¤ that scope (HST ) type Ia supernova observations [17] not only gives a static solution (the Einstein static universe or con¯rm earlier reports that we live in an accelerating uni- asymptotic thereto), verse [18] [19], but also explicitly sample the transition from deceleration to acceleration. A comparison of these 4 results in a partial phase plane is shown in FIG. 4 [20]. ¤E = : (12) 9C2 Clearly the (WMAP) results, and to a somewhat lesser extent the (HST ) results [21], support the view that we In terms of ¤E we have live in \large" ® universe for which (14) has held through- ¤ ® = (13) out its entire evolution. ¤E and so ® is a measure of ¤ relative to the \Einstein" We have argued that there are an in¯nity of standard value [14]. The state spaces shown in FIG. 1 and FIG. 2 non-flat FLRW dust models for which ­ » 1 through- correspond to ® = 0: ¤ = 0 in the ¯rst case and C = 0 out their entire evolution as long as ¤ > 0. Further, we in the second. have shown that the WMAP observations con¯rm that we live in just such a universe. The idea can be gen- As FIG. 3 makes clear, if ¤ > 0 and k 6= 0 then eralized in a straightforward way to more sophisticated multi-component models wherein ­ » 1 (­ = §i­i for ­ ´ ­M + ­¤ » 1 (14) all species i but not ­k) from Bb to dS within a phase 3 examined here is an excellent approximation now and the model relevant to a comparison with current obser- vations. With ­ » 1 there is no tuning or epoch prob- lem and so no flatness problem in the traditional sense. However, our presence in this hyper-tube is presumably favored and an explanation of this probability about the k = 0 trajectory in a sense presents a re¯nement of the classical flatness problem. Since the analysis here has been based on the classical notion of ¤, it is appropriate to conclude with a query as to whether or not this analysis is stable under perturba- tions in the de¯nition of ¤ itself (that is \dark energy" as opposed to the classical notion of ¤). Since ¤ can be introduced by way of a component pw = w½w where w ´ ¡1, consider w unspeci¯ed (but < ¡1=3). It can then be shown that the analysis given here generalizes in a straightforward way [22] and that for perturbations about w = ¡1 the phase portrait given in FIG. 3 is sta- ble, but the state space for ­ = 0 shown in FIG. 2 FIG. 4: WMAP [16] and HST results [17] superimposed on M part of the phase portrait of FIG. 3. In the former case the is not [23]. This latter point in no way alters the fact \WMAP only" results are used and in the latter case earlier that there remain an in¯nity of non-flat FLRW models con¯dence levels have been removed for clarity. The values of for which ­ » 1 throughout their entire evolution as long 3(1+w) H0t0 shown are (top down) 1.5, 1, 0.826, 3/4 and 2/3. Above as W (´ 8¼½wR ) > 0. the delimiter ­¤0 = 1¡­M0 , k = 1 and the values of ® shown are (bottom up) 500, 100, 40, 20, 12, 8, 6 and 5.4. Below the Acknowledgments delimiter k = ¡1 and the values of ® shown are (top down) 500, 100, 40, 20, 12, 8, 6 and 5.4.