galaxy-count number is also an acceptable model for the cosmological ...... a. D. (A3) is the total eâ ective energy density and. S \. 3G eff. G. 0. C p ]. /52. 2.
THE ASTROPHYSICAL JOURNAL, 488 : 14È26, 1997 October 10 ( 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.
THE OSCILLATING G MODEL : A POSSIBLE EXPLANATION FOR THE NATURE OF COSMOLOGICAL NONBARYONIC MATTER HERNANDO QUEVEDO,1 MARCELO SALGADO,2 AND DANIEL SUDARSKY3 Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, A.P. 70-543 Mexico 04510 D.F., Mexico Received 1997 March 3 ; accepted 1997 May 6
ABSTRACT We show that the oscillating G model proposed as an explanation of the periodic distribution in the galaxy-count number is also an acceptable model for the cosmological nonbaryonic matter when taken together with the standard inÑationary scenario () \ 1). The model, and in particular the value of the galactic redshift oscillation amplitude, is compatible with the values of the baryonic energy density ) bar allowed by the primordial abundance of light elements (0.010 ¹ ) h2 ¹ 0.020). Remarkably, these bar values also result in a value for the age of the universe compatible with recent observations. SpeciÐcally, the lower bound of 11.5 Gyr is satisÐed provided the Hubble parameter is given by H \ h 100 km s~1 0 Mpc~1 with h \ 0.68. Subject headings : cosmology : theory È dark matter È distance scale È gravitation 1.
INTRODUCTION
Srednicki 1996). This is the galactic dark matter problem. The dynamical behavior of clusters also indicates that there exists much more mass than observed in the form of stars, gases, etc. (Van Albada et al. 1987 ; Ostriker & Thompson 1987 ; Tremane & Lee 1987 ; Turner 1993 ; Srednicki 1996) ; this is the galactic cluster dark matter problem. The relation between these three dark matter problems is not clear. In fact, each of the three proposes a di†erent order of magnitude for the ratio of dark matter to ordinary matter energy densities. In this paper we will focus on the cosmological dark matter problem, the only one that clearly indicates a nonbaryonic nature for the dark matter (Turner 1993 ; Srednicki 1996). In order to avoid confusion we will hereafter refer to the dark matter implied by the cosmological dark matter problem as the cosmological hidden matter (CHM) or the cosmological nonbaryonic matter (CNBM). We will not deal with the possible connections between the CHM problem and the other two dark matter problems, as this would require a study of the evolutionary dynamics of inhomogeneities as well as certain suppositions about the primordial density Ñuctuation spectrum, problems which are outside the scope of this work. Moreover, such a treatment would necessarily test a mixture of the basic tenets of this paper, and test the postulates regarding the initial data for perturbations in a joint fashion, but not test each one separately. The speciÐc models that have been proposed to describe CHM are again inÑuenced by the inÑationary scenario. Given the fact that the values of the relative primordial abundances of light elements in big bang nucleosynthesis restrict the value of the baryonic energy density to the range [0.01 h~2, 0.02 h~2], it follows that the CHM must consist mostly of exotic particles : massive light neutrinos and similar species (hot dark matter) or weakly interacting massive particles like neutralinos, axions, superheavy monopoles, primordial black holes, etc. (cold dark matter). However, so far there is no independent evidence of the existence of these exotic particles. An apparently independent problem is presented by the deep pencil beam survey of the redshift distributions of galaxies at the galactic poles discovered by Broadhurst et al. (1990 ; see also Szalay et al. 1991), where a statistical analysis shows a periodicity in the galaxy-count number of
A remarkable feature of the standard big bang model, based on the Friedmann-Robertson-Walker (FRW) cosmology and hereafter referred to as the FRW model, is that the prediction of the primordial abundance of 4He (which is compatible with observations) does not depend on the present values of the baryonic ) and radiation ) rad energy densities. That is, we recoverbarthe correct primordial abundance of 4He (which corresponds to a freeze-out temperature of D0.7 MeV) regardless of the present value of the baryonic or radiation content of the universe. The same does not hold for other light elements : the FRW model correctly predicts the primordial abundance of light elements like 7Li, 3He, and D only if ) is in the range [0.01 bar h~2, 0.02 h~2] (Copi, Schramm, & Turner 1995). (The radiation energy density today, ) , is Ðxed by the observed rad 2.725 K cosmic microwave background radiation [CRB].) Obviously, with such a set of values and no other type of matter, it is not possible to satisfy the condition ) \ 1 predicted by the standard inÑationary models. In other words, ) \ ) ] ) > 1, that is, ordinary matter (baryons FRW photons) bar rad and cannot account for the energy density required for a conformally Ñat universe predicted by the standard inÑationary scenario. This is roughly speaking the cosmological dark matter problem. In fact, the observed ordinary matter is of the order of 1% of the critical value () \ 1), and thus there must exist a large amount of dark matter. The dynamics of stars in galaxies also suggests the existence of dark matter. It is well known that the observed rotation curves of spiral galaxies are inconsistent with the standard laws of gravitation if one considers only the observed ““ luminous ÏÏ matter as the source of gravitation. In order to explain the rotational velocities of stars within Newtonian gravity, it is necessary to suppose that the amount of matter contained in the galaxies must be about 10 times larger than the sum of all the counterpart luminous mass contained in them (Van Albada et al. 1987 ; Ostriker & Thompson 1987 ; Tremane & Lee 1987 ; Turner 1993 ; 1 quevedo=nuclecu.unam.mx. 2 marcelo=nuclecu.unam.mx. 3 sudarsky=nuclecu.unam.mx.
14
OSCILLATING G MODEL characteristic length 128 h~1 Mpc (see Faraoni 1997 for a review). Although the original analysis of Broadhurst et al. (1990) has been criticized by some astronomers (see Kaiser & Peacock 1991 ; Park & Gott 1991 ; Dekel et al. 1992), a more recent analysis (Szalay et al. 1993 ; Willmer 1994) conÐrmed the results of Broadhurst et al. However, a deÐnite answer as to the reality of this periodicity in the visible wavelength spectrum, as well as in the Lya absorption lines (Chu & Zhu 1989 ; Scott 1991 ; Tytler 1993) and in the Mg II quasar absorption systems, could be given only by further observations and analysis. (In the model discussed below, all of the above are expected to exhibit periodicities of 128 h~1 Mpc in all the spatial directions.) We will be concerned with the oscillating G model put forward as an explanation of these observations, and its possible relation to the CHM problem. The key point is that the oscillations in an e†ective gravitational constant G induce oscillations in the Hubble parameter and thereeff modulate the observed redshift z. For a uniform galaxy fore density per comoving volume n6 , the number of galaxies dN in a solid angle d) and with redshift (in presence of oscillations) between z and z ] dz, compared to the number in the absence of oscillations (with redshift z6 ), is given by dN dN dz6 \ , z2 dz d) z6 2 dz6 d) dz
(1)
where z2 \ z6 2 to the lowest order. Thus, a temporal periodic modulation factor dz6 /dz \ H1 /H (where H1 is the Hubble parameter in the absence of oscillations) would inÑuence our observations of distant points, giving rise to an apparent variation in the density of galaxies n(z) \ n6 /(dz/dz6 ), which is mistakenly interpreted as a real periodicity in the spatial distribution (Hill, Steinhardt, & Turner 1990). As we will describe in ° 2, the central feature of this model is a cosmological massive scalar Ðeld nonminimally coupled to gravity, which oscillates in cosmic time and thus results in an e†ective gravitational constant which also oscillates in cosmic time. The behavior of the e†ective gravitational constant G is determined by the expectation value of / ; eff oscillations of / induce oscillations in G and therefore, eff thus in the modulation factor dz6 /dz \ H1 /H (Morikawa 1990 ; Hill et al. 1990 ; Crittenden & Steinhardt 1992 ; Sisterna & Vucetich 1994 ; Salgado, Sudarsky, & Quevedo 1996). In this way the spatial periodicity reported by Broadhurst et al. (1990 ; see also Szalay et al. 1991, 1993) is explained as just an illusion that results from a true temporal periodicity induced by the oscillation in cosmic time of the e†ective gravitational constant. We take the modulation of the redshift to be roughly of the form (Hill et al. 1990 ; Crittenden & Steinhardt 1992)
C
D
2n dz (t [ t ) ] t , \ 1 ] A cos 0 0 L dz6
(2)
where L represents the characteristic length of 128 Mpc h~1 expressed in units of time, t is the cosmic time, t is its present value, and A stands for the amplitude 0of the 0 modulation factor responsible for the galactic periodicity. It has been argued by Hill, Steinhardt, & Turner (1990, 1991) that A º O(0.5) is required in order to reproduce the 0 sharp peaks observed by Broadhurst et al. (1990), although it is not clear what uncertainties are associated with this lower bound on A . In this paper we consider small varia0
15
tions in the value of A and their consequences for the 0 model. It has been also argued (Crittenden & Steinhardt 1992) that the phase t should be Ðxed in order to pass the BransDicke and Viking tests. In fact, the Viking experiments imply that G0 /(GH) ¹ 0.3 h~1. These can be passed by choosing convenient initial conditions for / (see ° 2), which is equivalent to Ðxing t. On the other hand, it has been discussed by Crittenden & Steinhardt (1992) that ““ Ðfth force ÏÏ tests impose much more severe constraints. For instance, a model where the nonminimal coupling between gravity and the scalar Ðeld is linear on / is ruled out, since for such a case the value of A D0.5 conÑicts with the 0 required Brans-Dicke parameter u º 500 (see Will 1981 BD for a review of experiments in general relativity), which for such a linear coupling is almost constant. However, as discussed by those authors, for a quadratic nonminimal coupling (such as the one we use) the phase t can also be used to pass the Ðfth force tests. In this case u P 1//2. So by BD in the neighchoosing t conveniently, / can be made small borhood of the solar system. In fact, however, the study of the e†ects on solar system scales associated with the delicate value of the phase of / should be carried out in the context of the analysis of local inhomogeneities, as the matter content of the galaxy will a†ect the behavior of the scalar Ðeld in the interior. In the model studied in this paper, the cosmological hidden matter needed for total energy density ) \ 1 will be represented by the energy density of the scalar Ðeld, ) , and Õ of we argue that the observed periodicity in the distribution galaxies can be considered as an indication of the existence of that scalar Ðeld. In fact, the scalar Ðeld plays a role similar to that of a cosmological constant, but with distinct observational consequences (see ° 4). One of the main problems faced by this type of model is related to the primordial nucleosynthesis of 4He. This is determined by the temperature T at which the rate of pn neutrons to protons, weak interactions ! , which convert pn equals the value of the Hubble parameter H. According to the standard cosmology, the value of this temperature is about 0.7 MeV, corresponding to a 4He abundance that is in agreement with observational data (see Kolb & Turner 1990, for a review). The condition ! \ H involves the value of G, so in principle, variations of pn the e†ective value of NewtonÏs constant can upset the successful prediction of the 4He abundance. A numerical analysis of the equations of motion (see ° 2) shows that when we integrate the Ðeld equations for a homogeneous and isotropic oscillating G model backward in cosmic time, the scalar Ðeld goes to ^O (see Fig. 1), depending on the initial data (i.e., data corresponding to todayÏs universe). This suggests the existence of ““ intermediate initial data ÏÏ for which the scalar Ðeld / remains steady and close to zero in an early epoch. The situation is represented by a kind of plateau in the behavior of the scalar Ðeld (see Fig. 1) during which the e†ective gravitational constant of the model approaches its Newtonian value, a situation where we expect to recover the correct prediction of the 4He abundance. Our analysis shows a correlation between the length of this plateau and the accuracy in the recovery of the standard freeze-out temperature. The larger the plateau, the closer the freeze-out temperature approached the value 0.7 MeV. Thus the correct value of the 4He abundance will be recovered if we
16
QUEVEDO, SALGADO, & SUDARSKY
FIG. 1.ÈBehavior of the scalar Ðeld for three di†erent values of ) bar within the ) \ 1 scenario and A \ 0.5. The solid line represents the best 0 [0.020, 0.022]. The dashed and dash““ adjustment ÏÏ of ) in the interval bar dotted lines correspond to the upper and lower values of the interval, respectively. Here a \ ln (a/a ), where a and a are the scale factors at time 0 0 t and at the present time, respectively.
work under the ““ plateau hypothesis,ÏÏ i.e., the assumption that there is some mechanism that ensures the existence of such a plateau. The role of this hypothetical mechanism is to ensure that for arbitrary values of / in the preinÑationary era, the scalar Ðeld is driven to values near zero in the postinÑationary era. We must stress that while the plateau hypothesis seems completely unnatural when regarding the evolution equations as taking us from the present to the past, when looked at from the opposite and more natural direction, the situation is quite di†erent. In fact, all that seems to be required is for some mechanism to drive the scalar Ðeld to an extremely low value before the era of nucleosynthesis. Then, as our calculations show, the Ðeld will remain at that value up to and beyond that era, and the success of big bang nucleosynthesis will be recovered naturally. The plateau hypothesis thus ensures the recovery of the predictions of nucleosynthesis in a natural way. The explicit investigation of that mechanism will require the consideration of other e†ects present at earlier stages of the evolution of the universe, such as those of the inÑationary era. Perhaps it will be necessary to introduce a coupling of the inÑaton (the scalar Ðeld responsible for inÑation) with the scalar Ðeld of the present model. The speciÐc form of the coupling term must be chosen to fulÐll the expectations of the plateau hypothesis. However, the speciÐc treatment of this problem will require a more detailed analysis and is, therefore, beyond the scope of this work. After nucleosynthesis, the value of the Ðeld / will be ampliÐed by the curvature coupling just before the onset of the oscillatory behavior that occurs when H decreases below the value of the mass of the scalar Ðeld. The search for the values of the unknown physical constants and for values of the parameters describing todayÏs
Vol. 488
universe that are compatible with the plateau hypothesis is what we call ““ Ðne tuning.ÏÏ Let us stress that this is not the kind of Ðne tuning that implies the dismissal of the model, and therefore it will be referred hereafter as ““ adjustment ÏÏ) ; it is rather a procedure that becomes necessary in order to recover data describing our universe today when working under the plateau hypothesis. Of course, a nontrivial question is whether data so extracted are compatible with observations or not. In fact such questions can be regarded as a test of the whole model (i.e., the oscillating G model together with the plateau hypothesis). This paper will be concerned precisely with these tests. That is, we will consider how the model fares in view of the most important cosmological constraints : (1) the nucleosynthesis constraint, (2) the present content of baryons ) , bar and (3) the age of the universe. It turns out that the adjustment (i.e., the process of extracting predictions from the plateau hypothesis) can be done by Ðxing either the amplitude A of the galaxy count number, the baryonic energy density0 ) , or the total energy density ). We will assume also thebarstandard inÑationary scenario, so once and for all, we will Ðx ) \ 1. This program was initiated in Salgado et al. (1996, hereafter SSQ1), where we studied the compatibility of the oscillating G model with some of the quoted bounds but did not analyze the sensitivity of the model to the variation of the parameters allowed by the observational uncertainties. In Salgado, Sudarsky, & Quevedo (1997, hereafter SSQ2), we completed such an analysis and presented the boundaries of the region of the parameter space inside which the model satisÐes all the constraints mentioned above. The aim of the present work is to present in a more detailed fashion the results of SSQ2, and to show that, unlike the FRW model, the oscillating G model gives values for ) (when A is Ðxed and under the plateau and inÑabar hypotheses) 0 that constitute not a continuum but tionary rather a discrete set that resembles the energy spectrum of a quantum mechanical system, such as a harmonic oscillator. The mechanism that generates the allowed values is implemented by putting ) \ 1, ) \ 4.2 ] 10~5 h~2, choosing a value for the Hubble rad parameter h, and choosing a value for the redshiftÈgalaxy-count amplitude A [observationally there is a large uncertainty, and we only0 have A º O(0.5)] as needed in order to explain the data of 0 et al. 1990 ; see also Hill et al. 1990 ; Crittenden Broadhurst & Steinhardt 1992). Among the discrete set of values for ) so obtained, only a few lie in the range [0.01, 0.02] h~2. bar Moreover, if h is small enough, the model predicts an age of the universe that is consistent with observations. This was the strategy used in the original analysis (SSQ1), in which we only explored the cases with h \ 1 and A \ 0.5, and 0 unacceptwhich, under a subsequent analysis, resulted in an able value for the age of the universe (see ° 4). A di†erent approach consists of starting from given values of h and of the baryonic energy density ) . The bar value integration of the Ðeld equations results in a speciÐc for the amplitude A . This procedure is carried out for several values of h, 0contained in the range allowed by observational data, and for each h, ) is varied within the barrange of A is cominterval [0.01, 0.02] h~2. The resulting pared with the observational constraint A º (0.5).0 Finally, we calculate the age of the universe for the0resulting cosmological model, and compare it with recent determinations of its lower bounds (Jimenez et al. 1996).
OSCILLATING G MODEL
No. 1, 1997
The paper is organized as follows. Section 2 contains a review of the basic equations describing the oscillating G model in an isotropic and homogeneous spacetime. A more detailed derivation of these equations is given in Appendix A. Section 3 deals with the initial conditions (here deÐned as the conditions of the universe at the present time), the constraints, and the strategy we use to integrate the Ðeld equations backward and forward in cosmic time. In ° 4 we present the numerical results, and as an illustration we compare the numerical procedure of generating the set of allowed values of the parameters of the oscillating G model with another procedure that Ðnds the energy spectrum of the one-dimensional quantum harmonic oscillator (Appendix B). Finally, ° 5 contains some concluding remarks. 2.
THE OSCILLATING G MODEL
The oscillating G model is described by the Lagrangian L\
A
B
1 ] m/2 J[gR 16nG 0 1 (+ /)2 ] V (/) ] L , [ J[g mat 2
C
D
(3)
where G is NewtonÏs gravitational constant, R is the space0 time curvature, m stands for the nonminimal coupling constant, / is a massive scalar Ðeld, and V \ m2/2 is the scalar potential. Here L represents the Lagrangian for ordimat nary matter. This model leads to a general theory of relativity with an e†ective gravitational constant given by G 0 G \ , eff 1 ] 16nG m/2 0 and Ðeld equations Rkl [ 1 gklR \ 8nG T kl , 2 0 eff LV (/) K/ ] 2m/R \ , L/
(4)
(5) (6)
where the e†ective energy-momentum tensor is given by G T kl 4 eff (4mT kl ] T kl ] T kl ) , eff m sf mat G 0
(7)
with T kl 4 +k(/ +l /) [ gkl + (/ +j/) , (8) m j T kl 4 +k / +l/ [ gkl[1 (+ /)2 ] V (/)] , (9) sf 2 and T kl is the energy-momentum tensor of ordinary matter.mat In addition, by using the scalar Ðeld equation, the conservation equation for the e†ective energy-momentum tensor, + T kl \ 0 , (10) k eff reduces to the usual energy-momentum conservation law of matter, + T kl \ 0 . (11) k mat We consider now the dynamics of an isotropic and homogeneous universe described by the FRW metric g \ diag [[1, a2(t)/(1 [ kr2), a2(t)r2, a2(t)r2 sin2 h], wherekla(t) is the scale factor. We take k \ 0 as dictated by the standard
17
inÑationary scenarios. The matter-energy-momentum tensor is modeled by a combination of two noninteracting perfect Ñuids, baryonic matter (p \ 0) and photons 1 (p \ e /3), which lead to T kl \ T kl ] T kl \ ; 2 2 mat rad thati/1,2 [(p ] e )UkUl ] p gkl]. Furthermore, itbar is assumed the i i i matter Ðelds and the scalar Ðeld possess the same symmetries as the FRW spacetime. The energy of the Ñuid is taken as the sum of the corresponding components (baryonic matter plus radiation). These can be integrated by quadratures by assuming that the two components do not interact with each other (see ° 3). The remaining equations of motion, one for the scale factor and another for the scalar Ðeld, can be written in Hamiltonian form as a system of ordinary Ðrst-order coupled nonlinear di†erential equations (see Appendix A for details). These are solved as an initial-value problem by performing the integration with respect to the cosmic time or, more suitably, with respect to the parameter a deÐned as a 4 ln [a(t)/a ] . (12) 0 Here a 4 a(t ) represents the scale factor at the present 0 time. 0 3.
INITIAL CONDITIONS, COSMOLOGICAL CONSTRAINTS, AND NUMERICAL STRATEGY
The model contains a set of parameters which together with the initial data must be speciÐed numerically in order to integrate the di†erential equations. Let us focus Ðrst on the initial conditions (which we consider as the conditions of our universe today). We consider the values of the scale factor and the Hubble parameter at the present time t as given by a and H , 0 choice of gravitational 0 0 respectively. This means that our Ðeld variables,
C D
a(t) , (13) a 0 da H(t) P3 4 [ 4 [ , (14) a dt8 H 0 with t8 \ t/H~1, take the following trivial values at t \ t : 0 0 40 , (15) ao t/t0 a5 P3 o 4 4 [1 . (16) a t/t0 H 0 t/t0 On the other hand, as mentioned in Appendix A, the equations of motion of matter reduce to the conservation of energy density of photons and baryons : a(t) \ ln
K
ev \ 3(e8 ] p8 )P3 , (17) i i i a where e8 and p8 stand for the photon and baryon energy i densitiesi and their corresponding pressures in units of the critical energy density today. This equation is solved straightforwardly using the hypothesis that baryons and photons do not interact with each other. The result is well known :
C D
C D
a(t) ~4 a(t) ~3 ]) , (18) \) bar a rad a 0 0 where ) and ) represent the contributions of photons rad to the bar present total energy density of the uniand baryons e8 4 e8
rad
] e8
bar
18
QUEVEDO, SALGADO, & SUDARSKY
verse, ). ) is Ðxed in terms of he CBR of 2.725 K, which rad corresponds to ) D4.2 ] 10~5 h~2 . (19) rad We will Ðx the value of ) by one of two di†erent strabar tegies, which we explain later on. We turn now to the initial data of the scalar Ðeld. First we impose the so-called fortunate phase condition \0 (20) /5 o t/t0 for simplicity, as we argued that it cannot truly be inferred from bounds on the variation of G resulting from solar system experiments (Crittenden & Steinhardt 1992) because such an analysis would have to take into account the e†ect of local inhomogeneities on the dynamics of the scalar Ðeld. The values of the initial scalar Ðeld amplitude / and the 0 coupling constant m are given in terms of the parameters constrained by the observed periodicity of the counting number of galaxies, and ) , as described before. For bar potential we will assume instance, for the harmonic scalar V (/) \ m2/2, where the mass m deÐnes the frequency of oscillation of the scalar Ðeld, which in our conventions is u \ m[3/(4n)]1@2. This frequency is determined by the 128 Mpc h~1 period observed in the pencil beam survey (Broadhurst et al. 1990), and corresponds to the value u D 147 H . It is worthwhile to point out that in regions with large 0matter content, the e†ective mass is given by m2 \ m2 ] mR(i.e., the coefficient of /2 in eq. [3]. So, for eff example, inside our Galaxy we have m2 \ m2 eff This ] 8nG om B m2[1 ] 12m h~2(o /10~23 g cm~3)]. 0 galaxy e†ect illustrates how local inhomogeneities would a†ect the analysis of the behavior of the scalar Ðeld, and in particular its phase in the vicinity of the solar system. Next we note that one can express the present values / and m in terms of the redshiftÈgalaxy-count amplitude A 0, 0 deÐned as (see Hill et al. 1990 ; Crittenden & Steinhardt 1992) G@ u/ 0, A \ [ eff (21) 0 2H G 0 eff and the Hamiltonian constraint at t (that is, the Fried0 mann equation relating H and ) [see Appendix A]) )[) [) rad bar , /2 \ 0 m2 [ 16nm)
(22)
obtaining expressions of the form / \ / (), ) , ) , 0 rad such bar A , u) and m \ m(), ) , ) , A , 0u). SpeciÐcally, 0 rad bar 0 expressions for / and m yield 0 A m2 0 , (23) m\ 16n[A e8 ] u() [ e8 )] 0 0 0 A 0 , (24) /2 \ 0 16nm(u [ A ) 0 where we recall e8 \ ) ] ) . 0 before, rad some bar of the quantities appearing As we mentioned above are Ðxed straightforwardly by observations. Among these are u and ) (for a given h). Moreover ), which corresponds to theradtotal energy density of the universe (including the contribution of the scalar Ðeld), is chosen to be 1 in accordance with the standard inÑationary predic-
Vol. 488
tion. Thus, the values of / and m are determined by the 0 values of ) and A . In order to Ðx both parameters, we bar 0 will follow two alternative strategies, whose main di†erence is of a practical nature and results from the di†erent degrees of sensitivity in the model with respect to uncertainties in the two unknowns. Strategy 1.ÈIt has been argued that the model will explain the observed galaxy distribution if A º O(0.5) 0 (Hill et al. 1990 ; Crittenden & Steinhardt 1992). Hence, our Ðrst strategy consists of imposing the value A \ 0.5 and 0 then using ) as a shooting parameter (see ° 4). That is, we bar search by trial and error for the value of ) and thus of / bar 0 and m, which gives a plateau behavior on /(a). We followed such a strategy in SSQ1 and reported that for h \ 1 a remarkable value of ) D 0.021 resulted, implying that the bar compatible with some of the most oscillating G model was important cosmological bounds, while also explaining the observed galaxy periodicity. However, the resulting age of the universe was not compatible with such a large value of h. Moreover, we found the results extremely sensitive to variations of A . 0 Strategy 2.ÈThis approach consists of choosing Ðrst a value of h, then a value of ) in the range [0.01, 0.02] h~2 barlight elements other than 4He given by the abundance of the (Copi et al. 1995) and then using A as the shooting parameter. This is a more natural strategy0in the sense that observations leave very small uncertainties in the value of ) , but bar uncertainties in the value of A are not even quantiÐed 0 (Broadhurst et al. 1990 ; Szalay et al. 1991, 1993 ; Hill et al. 1990, 1991 ; Crittenden & Steinhardt 1992). This strategy produces a range of values of A that can be contrasted 0 In this paper we have with the observation that A D 0.5. 0 followed this strategy, repeating it for several values of h and evaluating the corresponding value of the age of the universe in each case (see ° 4). 4.
NUMERICAL RESULTS
In this section we will show the results of the numerical analysis obtained by focusing on the second strategy described in the last section. As we have explained in the Introduction, for some values of A and ) the scalar Ðeld goes to ^O, indicating that bar there0is a transition point (i.e., a particular set of values of the parameters) for which the scalar Ðeld satisÐes the plateau hypothesis (see Fig. 1). The numerical determination of the adjusted values of the initial data (todayÏs universe) is similar to the numerical determination of the energy eigenvalues corresponding to bound states of onedimensional mechanical systems (e.g., a quantum harmonic oscillator ; see Appendix B). In the latter example we need to Ðnd the values of the energy E such that the numerical integration of the time-independent Schrodinger equation will yield a vanishing wave function ((x) as x ] ^O, while in the former we need to Ðnd the values of A and ) 0 by other bar (depending upon which one we have chosen to Ðx means) such that the integration of the equations of motion will yield values of the scalar Ðeld that go to zero as a ] [O. The accuracy of the determination of E in the quantum mechanical analogy (see Appendix B) or of A (or ) ) in the oscillating G model will be associated with0 the bar of the region where ( and / stay close to zero as the length numerical integration is performed toward the inÐnite value of the integration variables x and a. In practice we can take values for x and a that lie in the region where the exact 0 0
No. 1, 1997
OSCILLATING G MODEL
solution should be very close to zero, and then look for the values of E or A (or ) ) that correspond to the numeri0 and / barhaving a value close to zero at x cally determined ( 0 or a . Figures 2 and 3 summarize the numerical procedure 0
19
of adjustment by showing the behavior of / at a point where oscillations cannot have started (a D [8), as a function of ) for a Ðxed A . The Ðgures 0correspond to two bar 0 values of h, and the three curves of each Ðgure correspond
FIG. 2.ÈScalar Ðeld valuated at a \ [8 for di†erent values of ) and A with h \ 0.60. The solid line corresponds to A \ 0.4865, while the dashed bar 0 The asterisks indicate the extreme values of ) 0 in the interval [0.01, 0.02] and dash-dotted lines correspond to A \ 0.49 and A \ 0.4973, respectively. 0 0 bar h~2.
FIG. 3.ÈSimilar to Fig. 2, with h \ 0.65 and A \ 0.49, A \ 0.495, A \ 0.499 for the solid, dashed, and dash-dotted lines, respectively 0 0 0
20
QUEVEDO, SALGADO, & SUDARSKY
to three values of A for conÐgurations with ) at the 0 bar extremes of the allowed interval and in the middle of it. The nodes of the curves represent the values of ) consistent bar with the plateau hypothesis. Table 1 shows the ranges of A 0 compatible with the plateau hypothesis and with the condition that ) ½ [0.01, 0.02] h~2. These values of A are bar 0 very close to the value A \ 0.5 needed to explain the 0 galactic periodicity. As suggested in Figure 4, by increasing or decreasing the values of A beyond the intervals of 0 Table 1 it is possible to Ðnd another interval compatible with the plateau hypothesis, but this will correspond to values of A very di†erent from 0.5. 0 Figure 5 shows the behavior of the scalar Ðeld (evaluated at a ) for di†erent values of the coupling constant m 0 obtained via equation (23) from the corresponding values of ) . The nodes mi of this curve represent the discrete set of bar of m that are 0 consistent with the plateau hypothesis. values The upper dashed line shows the corresponding value of TABLE 1 PARAMETER RANGESa H 0 (100 km s~1 Mpc~1) 0.75 0.70 0.65 0.60 0.55 0.50
................... ................... ................... ................... ................... ...................
)
bar [0.017, 0.035] [0.020, 0.041] [0.023, 0.047] [0.028, 0.055] [0.033, 0.066] [0.040, 0.080]
A
0 [0.495, 0.501] [0.492, 0.5] [0.490, 0.499] [0.486, 0.497] [0.482, 0.495] [0.477, 0.492]
Age (Gyr) [10.3, [10.99, [11.7, [12.7, [13.7, [15.0,
10.6] 11.26] 12.0] 13.0] 14.1] 15.4]
a Ranges are given for the present baryon content of the universe, the redshift-galaxy count amplitude, and the age of the universe for a given h obtained from the oscillating G model. Inside these ranges, it is always possible to Ðnd values of such quantities that enable us to recover the plateau hypothesis.
Vol. 488
) for each m. Note that the intersections of the lines m \ bar mi with the upper curve yield the corresponding values of 0 ) . In particular, the line associated with the smallest mi bar 0 corresponds to the value of ) for the dash-dotted line of bar Figure 2, which lies inside the allowed interval [0.028, 0.055] (cf. Table 1). Figure 6 shows the region of compatibility of A and h 0 with the plateau hypothesis for di†erent values of ) in the bar range [0.01, 0.02] h~2. These values also result in di†erent ages of the universe, which might or might not agree with observations (see ° 5). These results are summarized in Table 1. 4.1. T he Age of the Universe It has been argued that cosmology today might be facing a crisis. Among the main reasons for adopting such a position we can cite the present status of the values of h, ) mat (baryonic ] dark matter), and the age of the universe), which within the standard inÑationary value ) \ 1 (a generalized belief) can be reconciled only if one introduces a cosmological constant " (Ostriker & Steinhardt 1995 ; Frieman 1996 ; Primack 1997). That is, if one assumes that ) \ ) \ 1, then a small h is needed (h ¹ 0.57) in order to satisfy mat the lower bound (11.5 Gyr) of the age of the universe from the globular clusters (Jimenez et al. 1996) (see Fig. 7). However, recent observations compatible with measurements using Cepheids and Type I supernovae (Freedman et al. 1994 ; Riess, Press, & Kirshner 1995) point rather to the range h ½ [0.65, 0.75]. A standard way out of these contradictions is to introduce a cosmological constant that allows a reduction of ) while Ðlling the gap needed for a ) \ 1 mataddition predicts an old enough universe model, and that in for h ½ [0.65, 0.75] (Primack 1997). However, despite the fact that there seem to be fundamental reasons for assuming
FIG. 4.ÈSimilar to Fig. 2 with h \ 0.75 for Ðve di†erent values of A
0
No. 1, 1997
OSCILLATING G MODEL
21
FIG. 5.ÈThe oscillating dashed line depicts the scalar Ðeld valued at a \ [8 as a function of the coupling constant obtained with ) corresponding to bar the dash-dotted line of Fig. 2. The upper dashed line shows such ) as a function of m. bar
a nonvanishing cosmological constant, as well as theoretical predictions of its vanishing value (Hawking 1984), no independent determination of its value exists so far except for the very unrestrictive upper bound (Maoz & Rix 1993 ; Kochanek 1993, 1996). Our calculations show that one nice feature of the oscillating G model is that the resulting age of the universe is compatible wih the lower bound of 11.5 Gyr for values h ¹ 0.68 with ) \ 1, all without a cosmological constant. Even when considering the very restrictive range h ½ [0.65,
FIG. 6.ÈRedshift count oscillation amplitude A as a function of h for 0 the range of values values of ) in the interval [0.01, 0.02] h~2. Note that bar of A corresponds to values compatible with observations (shaded 0 The darker shaded region corresponds to values of A for h regions). within the intersection of the resulting ranges of recent observations.0
0.75], there is a ““ window ÏÏ of [0.65, 0.68] for which the model is compatible with observations. Figure 7 shows the age of the universe obtained from the oscillating G model as a function of h, and compares it with the resulting age of a ) \ 1 standard cosmology with " \ 0. We can understand the ““ aging ÏÏ of the universe in this model in the following way. A formal mechanism for introducing a cosmological constant in general relativity is to consider a perfect Ñuid energy-momentum tensor with an equation of state (EOS) of the form p \ [e and e P ". The negative pressure produces a [ 0 and, thus, an inÑationary
FIG. 7.ÈAge of the universe within the oscillating G model (solid line) as a function of h obtained with di†erent values of ) (see Table 1). The bar The dash-dotted dashed line corresponds to the lower limit of 11.5 Gyr. line depicts the age for a matter-dominated universe with ) \ 1 and " \ 0.
22
QUEVEDO, SALGADO, & SUDARSKY
behavior of the scale factor. In this way, the time needed for reaching the point a D 0, starting from the present and evolving to the past, becomes larger, thus giving rise to an older universe. In the oscillating G model, the scalar Ðeld introduces a kind of cosmological ““ constant ÏÏ that varies in time. Actually, we can deÐne an e†ective pressure and an e†ective energy density of the oscillating G model by P 4 S/3 (with S 4 T (i) , i.e., the trace of the spatial part of the e†ective eff(i) energy-momentum tensor) and E 4 T (0) , respectively, eff(0) which results in an EOS with pressures oscillating between positive and negative values (see Fig. 8). Such an e†ective EOS with negative pressures results in phases of a [ 0 (““ mini-inÑations ÏÏ), as we can appreciate from Figure 9. The Ðnal result is that as we evolve from the present to the past (as we did in the numerical integration), the time needed to reach the value a D 0 is larger than when no mini-inÑations take place (as it happens in a matter-dominated universe with ) \ 1). In our case, those mini-inÑations start at the time of the oscillatory behavior of / (they do not last forever), and, obviously, before that time (corresponding to larger values of E for which the e†ective pressure is always positive) the plateau hypothesis ensures that we reach the value a D 0 in a Ðnite time. The di†erence between the oscillating G model and the standard cosmological constant models is that our scalar Ðeld has not been introduced in an ad hoc fashion, as in the case of a nonzero ", but rather was introduced to explain an independent phenomenon (the galactic periodicity), following which the data corresponding to our universe today under the plateau hypothesis lead to values of the di†erent parameters that result naturally in a reconciliation between them and the age of the universe. This is a highly nontrivial result. Moreover, independent evidence of the existence of this scalar Ðeld has been provided by Sisterna & Vucetich (1994), in an analysis of coral fossils and bivalve skeletal records that show that the duration of the solar year had been changing (oscillating in time) with periods of 370È680 Myr. The authors attribute this e†ect to a variation of an e†ective gravitational constant (presumably due to the scalar Ðeld). Remarkably, the characteristic scale of 128 Mpc h~1 observed in the apparent galactic periodicity reported by Broadhurst and coworkers (Broadhurst et al.
FIG. 8.ÈE†ective ““ equation of state ÏÏ for the oscillating G model. The Ðgure depicts the behavior of an e†ective pressure with respect to the e†ective energy density.
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FIG. 9.ÈScale factor of the oscillating model in units of its value today as a function of cosmic time (in units of H~1). The present time t has been 0 0 taken to be t \ 1 H~1. The Ðgure corresponds to a conÐguration calcu0 lated with ) \ 1, h \ 0.65, ) D 0.033, and A \ 0.495 (/ D 3.26 bar 0 0 ] 10~3 and m D 6.28). The resulting age of the universe corresponds to 0.79 H~1, which translates into D11.9 Gyr. 0
1990 ; Szalay et al. 1991, 1993) translates into a period of 596 Myr (with h \ 0.7), which lies well in the above range. 5.
CONCLUSION
We have shown that the oscillating G model is compatible with the most important cosmological bounds and at the same time provides a plausible explanation for the galactic periodicity discovered by Broadhurst and coworkers (Broadhurst et al. 1990 ; Szalay et al. 1991, 1993). Furthermore, the model can also be considered a viable candidate for explaining the nature of the missing mass needed to Ðt the standard inÑationary prediction ) \ 1. In particular, the model, together with the hypothesis that the scalar Ðeld goes through a plateau phase during which its amplitude is very small, results in a natural recovery of the prediction of the primordial abundance of 4He from nucleosynthesis, and also results in a value for the age of the universe compatible with recent observations (Jimenez et al. 1996). The parameters compatible with the plateau phase, namely, the present content of baryons ) and the redshift count oscillation amplitude A , are also bar in agreement with 0 the bound imposed by the primordial abundances of 7Li, 3He, D, Be, etc. ([0.01, 0.02] h~2), and with the bound A º O(0.5) needed to explain the observations of 0
OSCILLATING G MODEL
No. 1, 1997
Broadhurst et al. (1990) and Szalay et al. (1991, 1993), respectively. Despite these nice features, we stress that the model must be subjected to more tests in order to prove its full validity. Among these are the binary pulsar tests (Damour & Esposito-Fare`se 1993, 1996), the study of Ñuctuations giving rise to galaxy formation, and an analysis of the model in the context of the stellar dynamics in galaxies. Finally, we mention that any clear-cut observation deÐnitely indicating the non-existence of the galactic periodicity in most direc-
23
tions other than the south and north poles of our Galaxy would imply the dismissal of the oscillating G model. However, if this were so some puzzles would remain, namely, how did we happen to look at the only two directions in which the galactic periodicity is present and also, how can we explain a variation in the solar year with a period of 370È680 Myr revealed in the study of coral fossils and bivalve skeletons (Sisterna & Vucetich 1994), if not with the oscillating G model ?
APPENDIX A FIELD EQUATIONS FOR A FRW SPACETIME In this appendix we present the explicit expressions for the di†erential equations that are obtained by introducing the FRW line element into the general Ðeld equations (5) and (6). First, we consider the Einstein equation (5). We obtain two di†erential equations, the Ðrst of which corresponds to the Hamiltonian constraint a5 2 k 8 ] \ nG E . 0 a2 a2 3 The second one represents the dynamical equation for the scale factor
A
(A1)
B
1 a a5 2 k ] 2 ] 2 \ 4nG E[ S , 0 3 a a2 a2 where
C
(A2)
D
1 a5 G E \ eff e ] /5 2 ] V (/) [ 12m//5 2 a G 0 is the total e†ective energy density and
C
A
(A3)
BD
/5 2 a5 3G eff p ] [V (/) ] 4m /5 2 [ //5 [ / K/ (A4) 2 a G 0 plays the role of 3 times the e†ective ““ pressure.ÏÏ We remark that in this paper ) 4 E /e , where e \ 3c2H2/8nG , is the 0 c c 0 0 so-called critical energy-density. Finally, the equation for the scalar Ðeld (6) can be written explicitly as S\
a5 LV (/) / ] 3/5 ] \ 16nG m/(E[S) , 0 a L/
(A5)
where we have replaced the scalar of curvature in terms of the energy-momentum tensor quantities E and S. Since the contribution of the scalar Ðeld to the expression + T kl \ 0 vanishes identically, the energy momentum of the k eff Ñuid components (baryons and photons) do not interact ordinary matter satisÐes + T kl \ 0. Moreover, since the two perfect k mat among themselves, each of their corresponding energy-momentum tensors is separately conserved, leading to a5 e5 ] 3(e ] p ) \ 0 . i i i a
(A6)
Equation (A6) integrates immediately with respect to the scale factor, as in the standard cosmology case, and leads to equation (18). In order to transform the Ðeld equations into an initial-value problem consisting of a system of Ðrst-order di†erential equations, we shall rearrange these conveniently. Here we present the Ðnal form of the equations with source terms containing no second-order derivatives, and introduce a more convenient set of variables, a5 P \ [a5(t) \ [ , a a
P \ /5 , Õ
where a(t) \ ln [a(t)/a ]. The dynamic equation (A2) then takes the form 0 4 P0 [ P2 \ nG (E ] S) , a a 3 0
(A7)
(A8)
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QUEVEDO, SALGADO, & SUDARSKY
Vol. 488
where we have used the Hamiltonian constraint of equation (A1) in order to eliminate from equation (A2) the term proportional to k. Furthermore, introducing equations (A7) into the scalar Ðeld equation, we obtain LV (/) P0 [ 3P P ] \ 16nG m/(E[S) . Õ a Õ 0 L/ The source terms then take the following form :
C
(A9)
D
1 G E \ eff e ] P2 ] V (/) ] 12m/P P , Õ a 2 Õ G 0 and
G
C
(A10)
D
H
1 LV (/) G eff p ] P2 [ V (/) ] 4m /P P ] P2 [ / ] 64nG m2/2E . (A11) Õ a Õ 0 2 Õ L/ m2/2) G eff 0 In obtaining the source term in equation (A11) from equation (A4), we have used the scalar Ðeld equation (A9) in order to eliminate the term with K/. Thus, we have reduced the Ðeld equations to a set of four Ðrst-order di†erential equations that can be integrated numerically. The accuracy to which the Hamiltonian constraint is satisÐed has been used throughout the integration as a criterion for checking the consistency of calculations. S\
3 (1 ] 192nG
APPENDIX B THE QUANTUM-MECHANICAL ANALOGY Let us recall that a quantum-mechanical system described by the time-independent Schrodinger equation has either a discrete or a continuum energy spectrum depending on the boundary conditions imposed on the wave function (and of course on the shape of the potential interaction [see Landau & Lifshitz 1977]). In particular, discrete spectra are a landmark of bound systems in which the boundary conditions correspond to ( ] 0 and (@ ] 0 for distances much larger than the characteristic length of the system. Such conditions usually translate mathematically (at least for simple systems) into the fact that ( is a function represented by polynomials, trigonometric or special functions modulated by a decaying exponential factor.4 The polynomial representation is found by the requirement that the power series representation of ( terminates (which translates into the condition that the energy of the systems takes speciÐc discrete valuesÈthe eigenvalues). Although such features are found in a straightforward way when analytically studying the asymptotic behavior of the wave function, from a numerical point of view (at least when using a numerical approach based on Ðnite di†erences) the process for obtaining the energy eigenvalues is not as simple. The fact is that the conditions for the determination (integration) of bound states in quantum mechanics are not given in an explicit form. The boundary conditions corresponding to the asymptotic behavior of ( for bound states cannot be Ðxed numerically with sufficient precision (which would have to be inÐnite)5 and thus the numerical integration will result not in a wave function that goes to zero at x ] ^ O but rather in a wave function that stays close to zero over a large region but then diverges. In order to clarify the comparison between quantum-mechanical systems and the procedure we have employed to Ðnd the values of ) and A needed to establish the compatibility of the oscillating G model with the plateau hypothesis, let us consider thebar following0concrete example. Suppose that we solve numerically the stationary Schodinger equation for a simple one-dimensional problem, namely, the harmonic oscillator. Then the stationary Schrodinger equation for the one-dimensional harmonic oscillator reads [
1 +2 (A ] mu2x2( \ E( . 2m 2
(B1)
where ( \ ((x) and m, u are the mass and frequency of the oscillator, respectively. By introducing dimensionless quantities x8 4 x/a [a 4 (+/mu)1@2] and E3 4 2E/+u, we write equation (B1) as (A ] (E3 [ x8 2)( \ 0 .
(B2)
Now let us suppose in addition that we do not know the nature of the eigenvalues of the energy, but instead use the fact that a ““ physical ÏÏ wave function will be the one for which ((o x8 o ] O) ] 0 and (@(o x8 o ] O) ] 0. So for the numerical procedure we assume as ““ initial conditions ÏÏ (which are actually boundary conditions) and evanescent asymptotic behavior ((x8 ] O) ] 0 and (@(x8 ] O) ] 06. For instance, the boundary conditions imposed at x8 \ 7.05 are 0 4 If the interaction potential has surfaces of discontinuities beyond which the potential becomes inÐnite, the wave function can have discontinuous derivatives on the surfaces. 5 It is possible that by means of a spectral method approach (Canuto et al. 1988) the boundary conditions can be imposed exactly. 6 Here x8 ] O means values of x large in comparison to the characteristic length of the system.
No. 1, 1997
OSCILLATING G MODEL
25
((x8 ) \ 10~15 , (B3) 0 (@(x8 ) \ [10~15 . (B4) 0 Now, if we impose such conditions for the positive values of x, we cannot (in principle) impose another set of conditions for the negative values, since we are dealing with a second-order di†erential equation. However, by suitably adjusting the total energy of the system E3 (i.e., looking for the eigenvalues), it is possible to recover the same asymptotic behavior for the negative values of x. In order to determine numerically the eigenvalues of E, we must analyze numerically the behavior of ( in the space of the possible values of E3 and chose as eigenvalues those for which ((x8 ] [O) ] 0 and (@(x8 ] [O) ] 0. The numerical approach to accomplishing this is implemented by adjusting E3 . That is, we study the behavior of ( for some range of values of E3 , and then by focusing on the amplitude of ( at some point x8 \ 0 that is small enough so that there are no 0 8 ) D 0. In this way, we see that for E \ +u(1 oscillations of (, we Ðnd the eigenvalues of the energy as those for which ((x 0 n 2 ] n) (the actual eigenvalues with n an integer) there is this plateau region of x inside of which ( D 0 and (@ D 0. Figure 10 shows the behavior of ( for three di†erent values of the energy. The solid line represents the state with E \ 11+u/2, while the dashed and dash-dotted lines correspond to wave functions with 2n ] 1 \ 10.99 (( ] ]O) and 11.01 (( ] [O), respectively. We can see a plateau for the exact quantum value of the energy. However, we cannot extend the plateau regions for both the positive and negative values of x, because with a Ðnite-di†erenceÈbased numerical method it is impossible to adjust the asymptotic conditions with the inÐnite precision that this would require. Thus after the plateau region the numerically determined wave function diverges exponentially. Except for such behavior, the solid-line solution corresponds to the Ðfth excited state in which we recognize the oscillations due to the Hermite polynomial of degree 5 and the evanescent behavior (along the plateau) due to the Gaussian factor e~·2. Figure 11 depicts the trial wave function evaluated at x8 \ [5 as a function of E3 . The wave function oscillates around the point x8 \ [5 with a large amplitude (therefore we have employed a logarithmic 0
FIG. 10.ÈBehavior of the one-dimensional harmonic oscillatorÏs wave function (not normalized) for E3 \ 11 (solid line), E3 \ 10.99 (dashed line) and E3 \ 11.01 (dash-dotted line). The actual eigenvalues are E3 \ 2n ] 1 (see Appendix B). n
FIG. 11.ÈAbsolute value of the one-dimensional harmonic oscillatorÏs wave function valued at x8 \ [5 as a function of E3 in log linear scales. Peaks 10 correspond to the energies (eigenvalues) where ( is very small at x8 \ [5.
26
QUEVEDO, SALGADO, & SUDARSKY
scale on the absolute value of (), and the nodes in a linear-linear scale plot (corresponding to the peaks in Fig. 11) represent the values of the energies for which we Ðnd a plateau. Such values correspond to the actual values of the quantized energy for the Ðrst Ðve states E3 \ 2n ] 1 (with n an integer) of the quantum harmonic oscillator, in accordance with the analytical results. This work was partially supported by DGAPA-UNAM, project IN105496, and CONACYT, project 3567. REFERENCES Broadhurst, T., Ellis, R., Koo, D., & Szalay, A. 1990, Nature, 343, 726 Park, C., & Gott, J. R. 1991, MNRAS, 249, 288 Canuto, C., Hussaini, M. Y., Quarteroni, A., & Zang, T. A. 1988, Spectral Primack, J. R. 1997, Proc. International School of Astrophysics ““ D. Methods in Fluid Dynamics (Berlin : Springer) Chalonge ÏÏ 5th Course, ed. H. J. de Vega & N. Sanchez, (Erice, Italy), in Chu, Y., & Zhu, X. 1989, A&A, 222, 1 press Copi, C., Schramm, D. N., & Turner, M. S. 1995, Science, 267, 192 Riess, A. G., Press, W. H., & Kirshner, R. P. 1995, ApJ, 438, L17 Crittenden, R. G., & Steinhardt, P. J. 1992, ApJ, 395, 360 Salgado, M., Sudarsky, D., & Quevedo, H. 1996, Phys. Rev. D., 53, 6771 Damour, T., & Esposito-Fare`se, G. 1993, Phys. Rev. Lett., 15, 2220 (SSQ1) ÈÈÈ. 1996, Phys. Rev. D, 54, 1474 ÈÈÈ. 1997, Phys. Lett. B., in press (SSQ2) Dekel, A., Blumenthal, G. R., Primack, J. R., & Stanhill, D. 1992, MNRAS, Scott, D. 1991, A&A, 242, 1 257, 715 Sisterna, P. D., & Vucetich, H. 1994, Phys. Rev. Lett., 72, 454 Faraoni, V. 1997, Gen. Relativ. Gravitation, 29, 251 Srednicki, M. 1996, in Dark Matter, Review of Particle Physics, Phys. Rev. Freedman, W. L., et al. 1994, Nature, 371, 757 D, 54, 116 Frieman, J. 1996, in Proc. Third Paris Cosmology Symposium 1995, ed. Szalay, A., Ellis, R., Koo, D., & Broadhurst, T. 1991, in Primordial NucleoH. J. de Vega & N. Sanchez (Singapore : World ScientiÐc), 150 synthesis and Evolution of the Universe, ed. K. Sato & J. Audouze Hawking, S. 1989, Phys. Lett., 134B, 403 (Dordrecht : Kluwer), 435 Hill, T. C., Steinhardt, P. J., & Turner, M. S. Phys. Lett. B, 252, 343 Szalay, A., Broadhurst, T. J., Ellman, N., Koo, D. C., & Ellis, R. S. 1993, ÈÈÈ. 1991, ApJ, 366, L57 Proc. Natl. Acad. Sci 90, 4853 Jimenez, R., et al. 1996, MNRAS, in press Tremane, S., & Lee, H. M. 1987, in Proc. Jerusalem Winter School for Kaiser, N., & Peacock, J. A. 1991, AJ, 379, 482 Theoretical Physics, Vol. 4, Dark Matter in the Universe, ed. J. N. Kochanek, C. 1993, ApJ, 419, 12 Bahcall, T. Piran, & S. Weinberg (Singapore : World ScientiÐc), 103 ÈÈÈ. 1996, ApJ, 466, 638 Tytler, D., Sandoval, J., & Fan, X. M. 1993, ApJ, 405, 57 Kolb, E. W., & Turner, M. S. 1990, The Early Universe (Redwood City : Turner, M. S. 1993, Proc. Natl. Acad. Sci., 90, 4827 Addison-Wesley) Van Albada, T. S., Bahcall, J. N., Begeman, K., & Sanscisi, R. 1987, in Proc. Landau, L. D., & Lifshitz, E. M. 1977, Quantum Mechanics (3d ed. Jerusalem Winter School for Theoretical Physics Vol. 4, ed. J. N. Oxford : Pergamon) Bahcall, T. Piran, & S. Weinberg (Singapore : World ScientiÐc), 58 Maoz, D., & Rix, H. W. 1993, ApJ, 416, 425 Will, C. M. 1981, Theory and Experiment in Gravitational Physics Morikawa, M. 1990, ApJ, 362, L37 (Cambridge : Cambridge Univ. Press) Ostriker, J. P., & Steinhardt, P. J. 1995, Nature, 377, 600 Willmer, C. N. A., et al. 1994, ApJ, 437, 560 Ostriker, J. P., & Thompson, C. 1987, in Proc. Jerusalem Winter School for Theoretical Physics, Vol. 4, Dark Matter in the Universe, ed. J. N. Bahcall, T. Piran, & S. Weinberg (Singapore : World ScientiÐc), 69
