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Rotation Curves of Spiral Galaxies and Large Scale

arXiv:hep-ph/9203226v2 6 Apr 19924September 3, 2018Rotation Curves of Spiral Galaxies and Large ScaleStructure of Universe under Generalized Einstein ActionM. Kenmoku, E. Kitajima, Y. OkamotoDepartment of Physics, Nara Women’s UniversityNara 630, JapanandK.

Shigemoto⋆Department of Physics, Tezukayama UniversityNara 631, JapanABSTRACTWe consider an addition of the term which is a square of the scalar curvature tothe Einstein-Hilbert action. Under this generalized action, we attempt to explaini) the flat rotation curves observed in spiral galaxies, which is usually attributedto the existence of dark matter, and ii) the contradicting observations of uniformcosmic microwave background and non-uniform galaxy distributions against red-shift.

For the former, we attain the flatness of velocities, although the magnitudesremain about half of the observations. For the latter, we obtain a solution with os-cillating Hubble parameter under uniform mass distributions.

This solution leadsto several peaks of galaxy number counts as a function of redshift with the firstpeak corresponding to the Great Wall.PACS number(s): 04.20.-q, 98.80.Dr, 98.60.Eg⋆E-mail address: shigemot@jpnyitp.bitnet

1. IntroductionOne of the most straightforward generalizations of the Einstein-Hilbert actionis the addition of a square terms of the scalar curvature and Ricci tensor.

Thisaction was introduced as counter terms to regulate ultraviolet divergences of theEinstein theory. [1] It was also used to obtain a bounce universe, which avoids theinitial singularity of the big bang cosmology.

[2]In the present work we consider this action in order to overcome the difficultiesencountered by the standard Einstein theory in explaining certain astrophysicaland cosmological observations. The famous astrophysical observations that appearto be in conflict with our expectations from the Newton’s theory of gravity is theflat rotation curves of spiral galaxies.

[3] The rotation velocities are constant as afunction of the distance from the center of galaxy, while a naive application of the1/r2 force law implies a decline in the velocity function. This observation is usuallyaccounted for by the incorporation of dark matter.

[4] In the present work we attemptto obtain flat rotation curves from the generalized Einstein action without relyingupon dark matter. As for the cosmological observations, the recent technologicaldevelopments in observations have yielded contradictory results : highly uniformcosmic microwave background[5] and non-uniform galaxy distributions in the scalesof hundreds of Mpc.

[6,7] In order to resolve this difficulty, Morikawa introduced anon-conformal scalar field in the action and obtained a theory with oscillatingHubble constant. [8] According to this theory, the mass distribution of the universeis uniform at any moment (hence, uniform cosmic microwave background), but thegalaxiy distributions as a function of redshift become periodic (hence, non-uniformlarge scale structure of the universe) due to the oscillation in the expansion rate of2

the universe. In the present work, we follow the same idea under the generalizedEinstein action without introducing the scalar matter.The paper is organized as follows.In sect.

2 we discuss rotation curves ofspiral galaxies under the generalized Einstein action. In sect.

3 we consider galaxynumber counts under the generalized Einstein action. A concluding remark is givenin sect.

4.2. Rotation Curves of Spiral Galaxiesunder the Generalized Einstein ActionIn this section, we consider a modification of Newtonian theory of gravitationunder the generalized Einstein action and apply this force law to the rotation curvesof spiral galaxies.

The generalized Einstein action is given byIg = −116πGZd4x√g(R + 2Λ + c1R2 + c2RµνRµν) ,(2.1)where G is the gravitational constant, R is the scalar curvature, Rµν is the Riccitensor, Λ is the cosmological constants, and g is the negative determinant of themetric tensor. Throughout this paper we follow the conventions of Ref.

9 and, inparticular, the speed of light is set equal to unity. The Einstein equation in thistheory is then written as,[2]Rµν −12Rgµν −Λgµν + c1Jµν + c2Kµν = −8πGTµν ,(2.2)where Jµν and Kµν are defined byJµν = 2R(Rµν −14Rgµν) + 2(R;µν −gµνR) ,(2.3)3

Kµν = R;µν −Rµν −12( R + RαβRαβ)gµν + 2RαβRµανβ ,(2.4)withR;µν ≡∇µ∇νR,R ≡gαβR;αβ . (2.5)In order to obtain the modification to the Newton’s 1/r2 law under this theory,we consider the following static, weak field limit:g00 ∼= −(1 + 2φ) ,(2.6)gij ∼= δij(1 + 2ψ) ,(2.7)where φ and ψ are functions of spatial coordinates only.

Note that φ correspondsto the gravitational potential field. The Ricci tensor and scalar curvature are thengiven byR00 ∼= −△φ ,(2.8)Rij ∼= (∂2i φ + ∂2i ψ + △ψ)δij ,(2.9)R ∼= 2 △φ + 4 △ψ .

(2.10)Substituting these into the time-time component of Eq. (2.2), we obtain the equa-tion for φ and ψ:△[φ + 2c1 △φ + (4c1 + 2c2) △ψ] = 4πGρ ,(2.11)where ρ is the mass density.

The trace of Eq. (2.2), on the other hand , gives4

another equation for φ and ψ:[1 + (6c1 + 2c2)△] △(φ + 2ψ) = −4πGρ . (2.12)We have to solve (2.11) and (2.12) simultaneously.

For the point source withρ = Mδ(3)(⃗r) ,(2.13)we obtain the following solution:[10,11]φ = GM−1r −13e−m1rr+ 43e−m2rr,(2.14)ψ = GM1r −13e−m1rr−23e−m2rr,(2.15)wherem21 = −16c1 + 2c2,m22 = 1c2. (2.16)Note that the first term in (2.14) is the usual Newton potential and that the secondand third terms in (2.14) correspond to its corrections.

The third term in (2.14)is, however, undesirable, since its existence does not yield the usual attractiveNewton’s force law in the limit r →0. We thus setc2 = 0 .

(2.17)As for the constant c1, a positive c1 givesφ = −GMh1r + 13cos µrr+ ysin µrri,(2.18)5

where y is an arbitrary constant andµ =1√6c1,(2.19)while a negative c1 givesφ = −GMh1r + 13e−m1rri. (2.20)Both cases give the correct r →0 limit of the observed gravitational constant G0withG0 = 4G3.(2.21)Eq.

(2.20), however, implies more decline of force than the 1/r2 law as a function ofr, and this is opposite to what is expected from the rotation curves of galaxies. Wethus adopt the solution of Eq.

(2.18) and calculate rotational velocities of typicalspiral galaxies. Taking into account the exponential mass distribution implied bythe surface photometry data,[12] we approximate a spiral galaxy by a truncated diskwith mass densityρ(⃗r) =(σ0 e−ra δ(z) ,(r ≤rg)0 ,(r > rg)(2.22)where we takerg = 4.3a [kpc] ,a = 2, 3, 4, 5, 6 ,(2.23)for the truncation radius.

[13] The value of σ0 was also taken from Ref.11. Namely,we impose the condition that the total mass is 2.2 × 1010M⊙for a = 2, where M⊙6

is the solar mass. The free parameter µ and y in Eq.

(2.18) were chosen so thatthe velocity curve becomes flat. Our choice isµ = 150h 1kpci,(2.24)y = 3 .

(2.25)We thus obtain the square of the rotation velocity as a function of the distance rfrom the center of galaxy byv2(r) = σ04πrrgZ0dr′2πZ0dθ′r′e−r′/a ∂∂rφ0(|⃗r −⃗r′|) ,(2.26)where φ0 is the Green’s function of Eq. (2.18) with M = 1, and we only consider⃗r on the plane of the galaxy disk.In Fig.

1, we show the results of numerical integration of Eq. (2.26) for eachvalue of a with r in the range of typical galaxy radius, i.e., 10—100 kpc.

For acomparison, the result for a = 2 under usual Newtonian gravity is also shown bythe broken curve in the figure. As is clear from the figure, the rotation velocitiesbecome flat for r >∼50 kpc, while that of Newtonian gravity steadily decreasesas r increases.

However, the magnitudes of velocities are about half of what areobserved for each value of a in Eq. (2.23) for r >∼50 kpc.7

3. Large Scale Structure of Universeunder the Generalized Einstein ActionIn this section we study the cosmology under the generalized Einstein action.The action we consider is given by Eq.

(2.1) with c2 = 0. Assuming the homoge-neous and isotropic universe, we employ the Robertson-Walker metric:ds2 = −dt2 + a2(t)dr21 −kr2 + r2dθ2 + r2 sin2 θdϕ2,(3.1)and the energy-momentum tensor T µν of the formT µν = pgµν + (p + ρ)UµUν .

(3.2)The time-time component and space-space component of the Einstein equation,Eq. (2.2), are then respectively written asH2 + K −Λ3 −6c1(−˙H + K2 + 6 ˙HH2 −2KH2 + 2H ¨H) = 8πG3 ρ ,(3.3)and−2 ˙H −3H2 −K + Λ + 6c1(2∂3t H + 12H ¨H + 9 ˙H2 + 18 ˙HH2 −4K ˙H−2KH2 −K2) = 8πGp ,(3.4)whereH = ˙aa ,(3.5)and the dots stand for time derivatives.

We now introduce a few dimensionless8

parameters as follows;τ ≡H0t ,(3.6)Ω0 ≡ρ0ρc,(3.7)k0 ≡kH20,(3.8)λ0 ≡Λ3H20,(3.9)where H0 is the Hubble parameter at the present time, and the critical density ρcis defined byρc ≡38πGH20 . (3.10)The density parameter Ω0 have two components:Ω0 ≡ΩN0 + ΩR0 ,(3.11)where ΩN0 and ΩR0 are respectively non-relativistic and relativistic components.We can then write8πG3H20ρ = Ω0ρρ0= ΩN0a3 + ΩR0a4,(3.12)where we have set the present value of the scale factor a0 to unity:a0 ≡a(0) = 1 .

(3.13)By (3.6) ∼(3.9), (3.12) and the energy-momentum conservation˙ρ = −3 ˙aa(ρ + p) ,(3.14)9

we finally reduce (3.3) and (3.4) toa′′′a = −112c0 aa′"ΩN0a3 + ΩR0a4 −a′a2−k0a2 + λ0#−12 aa′"2a′2a′′a3−a′′a2−3a′a4+ k0a2 k0a2 −2a′a2!#,(3.15)anda′′′′a= 112c0"ΩR0a4 + 2a′′a +a′a2+ k0a2 −3λ0#−12"4a′a′′′a2−12a′2a′′a3+ 3a′′a2+ 3a′a4−k0a2 k0a2 + 4a′′a −2a′a2!#,(3.16)respectively, wherec0 ≡c1H20 ,(3.17)and the prime stands for derivative with respect to τ.These are third-orderand fourth-order differential equations for a(τ). In particular, we consider solv-ing Eq.

(3.16) numerically. For this we need initial condition of a,a′, a′′, and a′′′.We have chosen a0 = 1 (Eq.

(3.13)). It then follows thata′0 = 1 ,(3.18)a′′0 = −q0 ,(3.19)10

where q0 is the deacceleration parameterq0 ≡−a′′0a0a′20. (3.20)By choosing values of Ω0,k0,λ0, and q0 which are observationally acceptable, wecan determine a′′′0 for each value of c0 from Eq.

(3.15) evaluated at the presenttime:Ω0 −k0 + λ0 + 6c02a′′′0 −(a′′0)2 + 2a′′0 −3 + k20 −2k0= 1 . (3.21)In the present work, we setk0 = 0 ,(3.22)λ0 = 0 ,(3.23)ΩR0 = 10−5 ,(3.24)for simplicity and use only positive q0.

(Non-zero values of k0 and λ0 or negativeq0 gave similar results unless their values are taken to be unnaturally large.) Thisleaves ΩN0, q0(= −a′′0), and c0 as free parameters.

In Fig. 2 we show the results ofnumerical integration of Eq.

(3.16) for several sets of values of these parameters.For c0 > 0, we have a de Sitter-like expansion (Fig. 2(a)), a bounce (Curve f inFig.

2(b)) which is essentially the bounce solution of Ref. 2, an expansion followedby an eventual collapse (Fig.

2(c)), or an expansion, shrink, and eventual de Sitter-like expansion (Curve c in Fig. 2(d)).

For c0 < 0, on the other hand, we generallyhave oscillation: oscillating expansion or oscillating bounce (Fig. 2(e) and (f)).11

This last solution, Curve e in Fig. 2(e) in particular, is of interest, since it mightexplain the observed large scale structure of galaxy number counts as a function ofred shift.

[6,7] The number of galaxies dN which are located between the comovingdistance r and r + dr at a fixed solid angle dΩwith absolute luminosity betweenL and L + dL is given bydN = n(L, t)a3r2drdΩdL ,(3.25)where n(L, t) is the number density at time t with absolute luminosity L, satisfyingn(L, t) =a0a3n(L, 0) . (3.26)Since the redshift z and scale factor a are related byz = a0a −1 ,(3.27)we haver =0Ztdta =0Ztz(t′) + 1a0dt′ .

(3.28)By substituting (3.26) and (3.28) into (3.25), we have[8]dNdzdΩdL ∝hR 0t (z(t′) + 1)dt′i2H(t). (3.29)Note that Eq.

(3.27) gives a differential equation for z(t) :dzdt = −(1 + z)H(t) . (3.30)Hence, by solving Eqs.

(3.16) and (3.30) simultaneously, we obtain the number12

count in Eq. (3.29).

The results forΩN0 = 0.1 ,(3.31)a′′0 = −1.0 ,(3.32)c0 = −0.0002 ,(3.33)are shown in Fig. 3 together with the scale factor a(τ).Note that there existfour peaks in the galaxy count for z < 0.5 with the first one corresponding tothe location of Great Wall.

[6] Even though the peaks are not separated with equalinterval, overall features are remarkably similar to observations.[7]4. Concluding RemarkIn this paper we discussed a possible generalization of the Einstein theory byadding a square term, c1R2, of scalar curvature to the action.

Choosing appropriatevalues of the coefficients, we found successful explanation of the flat rotation curvesof spiral galaxies (c1H20 ≈5 × 10−11) and the large scale structure of universe(c1H20 ≈−2 × 10−4 ) at least at a qualitative level.Our theory, however, isnot all faultless. It turned out that the coefficient γ of the Robertson expansion is1/2, and this disagrees strongly with, for instance, the radar echo experiments fromMercury, which implies that γ is very close to 1 in accord with the Einstein theory.

[14]Originally R2 term was introduced as a result of quantum corrections,[1] so that thecoefficient c1 will have the logarithmic r dependence as c1 = (const.) × log(r/r0).This interpretation may lead to a consistent explanation of both the rotation curves13

of spiral galaxies and the large scale structure of the universe from the singleeffective Lagrangian with the running coefficient c1 that depends on the distancescale of interest. The difficulty of γ = 1/2 may also be avoided if r0 takes a valueof the order of solar radius.Acknowledgements: We are grateful to M. Fukugita, H. Kawai, K. Kume, and K.Uehara for useful discussions.REFERENCES1.

R. Utiyama and B.S. de Witt, J.

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3, 608 (1962).2. H. Nariai and K. Tomita, Prog.

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For example, see Ref. 9.FIGURE CAPTIONS1) Rotation curves of spiral galaxies under the generalized Einstein action.

Thenumbers on the curves indicate the values of a in Eq. (2.23).

The predictionof Newtonian theory for a = 2 is also given by a broken curve for comparison.2) Scale factor of universe as a function of τ for various values of parametersc0, ΩN0, and a′′0 . The parameter c0 is positive for (a) ∼(d) and negative for(e) and (f).

The values of |c0| are indicated in the figure by the letters a, b,c, d, e, f for 100, 10−1, 10−2, 10−3, 10−4, 10−5, respectively.3) (a) Scale factor of universe as a function of τ for the set of parameters in(3.31) ∼(3.33) and (b) number count of galaxies as a function of z for thesame set of parameters. The scale of the ordinate for (b) is arbitrary.15

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