Some models of modified gravity and their observational manifestations are
considered. It is shown, that gravitating systems with mass density rising with
time evolve to a singular state with infinite curvature scalar. The universe
evolution during the radiation dominated epoch is studied in $R^2$-extended
gravity theory. Particle production rate by the oscillating curvature is
calculated. Possible implications of the model for cosmological creation of
non-thermal dark matter are discussed.Comment: 8 pages, 4 figures, talk presented at Les Rencontres de Physique de
  la Vallee d'Aoste, 26 February - 3 March, 201

Arbuzova, E. V.

English

arXiv

Some models of modified gravity and their observational manifestations are considered. It is shown, that gravitating systems with mass density rising with time evolve to a singular state with infinite curvature scalar. The universe evolution during the radiation-dominated epoch is studied in R2-extended gravity theory. Particle production rate by the oscillating curvature is calculated. Possible implications of the model for cosmological creation of non-thermal dark matter are discussed

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DOI 10.1393/ncc/i2012-11348-8Colloquia: LaThuile12IL NUOVO CIMENTO Vol. 35 C, N. 6 Novembre-Dicembre 2012Modified gravity: Problems and observational manifestationsE. V. ArbuzovaDepartment of Higher Mathematics, University “Dubna” - 141980 Dubna, Russiaricevuto il 7 Settembre 2012Summary. — Some models of modified gravity and their observational manifesta-tions are considered. It is shown, that gravitating systems with mass density risingwith time evolve to a singular state with infinite curvature scalar. The universeevolution during the radiation-dominated epoch is studied in R2-extended gravitytheory. Particle production rate by the oscillating curvature is calculated. Possibleimplications of the model for cosmological creation of non-thermal dark matter arediscussed.PACS 98.80.Cq – Particle-theory and field-theory models of the early Universe (in-cluding cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe,etc.).PACS 95.35.+d – Dark matter (stellar, interstellar, galactic, and cosmological).PACS 95.30.Sf – Relativity and gravitation.1. – IntroductionDiscovering of the cosmic antigravity based on the accumulated astronomical data,such as observation of the large-scale structure of the universe, measurements of theangular fluctuations of the cosmic microwave background radiation, determination of theuniverse age (for a review see [1]), and especially discovery of the dimming of distantSupernovae [2], is the most attractive event in cosmology of the last quarter of century.It was established and unambiguously proved that the universe expansion is accelerated,but the driving force behind this accelerated expansion in still unknown.Among possible explanations, the most popular is probably the assumption of a new(unknown) form of cosmological energy density with large negative pressure, P < −ρ/3,the so-called dark energy, for a review see, e.g., [3].A competing mechanism to describe the accelerated expansion is represented by grav-ity modifications at small curvature, the so-called f(R)-gravity theories, as suggested inref. [4]. In these theories the standard Einstein-Hilbert Lagrangian density, proportionalto the scalar curvature R, is replaced by a function f(R), so the usual action of Generalc© Societa` Italiana di Fisica 3940 E. V. ARBUZOVARelativity acquires an additional term:S = −m2Pl16π∫d4x√−g f(R) + Sm = −m2Pl16π∫d4x√−g [R + F (R)] + Sm,(1)where mPl = 1.22 · 1019 GeV is the Planck mass and Sm is the matter action.The original version of such models [4] suffers from a strong instability in the presenceof gravitating bodies [5] and because of that more complicated functions F (R) have beenproposed [6-9], which are free from the mentioned exponential instability.Though free of instability [5], the models proposed in [6-8] possess another troublesomefeature, namely in a cosmological situation they should evolve from a singular state in thepast [10]. Moreover, it was found in refs. [11,12] that in presence of matter, a singularitymay arise in the future if the matter density rises with time; such future singularity isunavoidable, regardless of the initial conditions, and is reached in a time which is muchshorter than the cosmological one.2. – Explosive phenomena in modified gravityIn paper [12] the model of modified gravity with F (R) function suggested in ref. [6]was considered:F (R) = λR0[(1 +R2R20)−n− 1].(2)Here constant λ is chosen to be positive to produce an accelerated cosmological expansion,n is a positive integer, and R0 is a constant with dimension of the curvature scalar. Thelatter is assumed to be of the order of the present day average curvature of the universe,i.e. R0 ∼ 1/t2U , where tU ≈ 4 · 1017 s is the universe age.The corresponding equations of motion have the form(1 + F ′)Rμν − 12 (R + F ) gμν + (gμνDαDα −DμDν)F ′ = 8πT(m)μνm2Pl,(3)where F ′ = dF/dR, Dμ is the covariant derivative, and T(m)μν is the energy-momentumtensor of matter.By taking trace over μ and ν in eq. (3) we obtain the equation of motion whichcontains only the curvature scalar R and the trace of the energy-momentum tensor ofmatter:3D2F ′ −R + RF ′ − 2F = T,(4)where T = 8πTμμ /m2Pl.We analyze the evolution of R in massive objects with time-varying mass density,ρm  ρc. The cosmological energy density at the present time is ρc ≈ 10−29 g/cm3,while matter density of, say, a dust cloud in a galaxy could be about ρm ∼ 10−24 g/cm3.Since the magnitude of the curvature scalar is proportional to the mass density of anon-relativistic system, we find R  R0. In this limit:F (R) ≈ −λR0[1−(R0R)2n].(5)MODIFIED GRAVITY: PROBLEMS AND OBSERVATIONAL MANIFESTATIONS 411 2 3 4-0.2-0.10. 10. 2Fig. 1. – Potential U(z) = z(1 + κτ)− z1−ν/(1− ν), ν = 15, τ = 0.The equation of motion is very much simplified if we introduce the new notationw = −F ′ = 2nλ(R0R)2n+1.(6)Evolution of w is governed by a simple equation of unharmonic oscillator:(∂2t −Δ)w + U ′(w) = 0.(7)Potential U(w) is equal toU(w) =13(T − 2λR0)w + R03[qν2nνw2nν +(qν +2λq2nν)w1+2nν1 + 2nν],(8)where ν = 1/(2n + 1), q = 2nλ, and in eq. (7) U ′(w) = dU/dw.Notice that infinite R corresponds to w = −F ′ = 0, so if F ′ reaches zero, it wouldmean that R becomes infinitely large.Potential U would depend upon time, if the mass density of the object under scrutinychanges with time, T = T (t). If only the dominant terms are retained and if the spacederivatives are neglected, eqs. (7), (8) simplify toz′′ − z−ν + (1 + κτ) = 0.(9)Here we introduced dimensionless quantitiest = γτ, γ2 =3q(−R0)(−R0T0)2(n+1),(10)w = βz, β = γ2T0/3 = q(−R0T0)2n+1.The minimum of the potential U(z) (fig. 1) sits at zmin = (1 + κτ)−1/ν . When themass density rises, the minimum moves towards zero and becomes less deep. If at theprocess of “lifting” of the potential z(τ) happens to be at U > 0 it would overjumppotential which is equal to zero at z = 0. In other words, z(τ) would reach zero, whichcorresponds to infinite R, and so the singularity can be reached in finite time (see fig. 2).42 E. V. ARBUZOVAFig. 2. – Ratio z(τ)/zmin(τ) (left) and functions z(τ) and zmin(τ) (right) for n = 2, κ = 0.01,ρm/ρc = 105. Initial conditions: z(0) = 1 and z′(0) = 0.The simplest way to avoid singularity is to introduce the R2-term into the gravitationalaction,δF (R) = −R2/6m2 ,(11)where m is a constant parameter with dimension of mass.In the homogeneous case and in the limit of large ratio R/R0 equation of motion forR is modified as[1− R2n+26λn(2n + 1)R2n+10 m2]R¨− (2n + 2) R˙2R− R2n+2(R + T )6λn(2n + 1)R2n+10= 0.(12)With dimensionless curvature and timey = − RT0, τ1 = t[− T2n+206λn(2n + 1)R2n+10]1/2(13)the equation for R is transformed into(1 + gy2n+2)y′′ − 2(n + 1)(y′)2y+ y2n+2 [y − (1 + κ1τ1)] = 0,(14)where the prime means derivative with respect to τ1.We introduced here the new parameter, g, which can prevent from the approach toinfinity and is equal tog = − T2n+206λn(2n + 1)m2R2n+10> 0.(15)For very large m, or small g, when the second term in the coefficient of the secondderivatives in eqs. (12) and (14) can be neglected, the numerical solution demonstratesthat R would reach infinity in finite time in accordance with the results presented above(see fig. 3, left panel). Non-zero g would terminate the unbounded rise of R. To avoid toolarge deviation of R from the usual gravity coefficient g should be larger than or of theorder of unity. In the right panel of fig. 3 it is clearly seen, that for g = 1 the amplitudeMODIFIED GRAVITY: PROBLEMS AND OBSERVATIONAL MANIFESTATIONS 4320 40 60 80 1001.52.02.53.020 40 60 80 1001.21.41.61.82.0Fig. 3. – Numerical solutions of eq. (14) for n = 3, κ1 = 0.01, y(τin) = 1 + κ1τin, y′(τin) = 0.Left panel: g = 0. Right panel: g = 1.of oscillations remains constant whereas the average value of R increases with time. Asfollows from eq. (14), the frequency of small oscillations of y around y0 = 1 + κ1τ1 indimensionless time τ1 isω2τ =1ggy2n+201 + gy2n+20≤ 1g.(16)It means that in physical time the frequency would beω ∼ 1tU(T0R0)n+1yn+10√1 + gy2n+20≤ m.(17)In particular, for n = 5 and for a galactic gas cloud with T0/R0 = 105, the oscillationfrequency would be 1012 Hz ≈ 10−3 eV. Higher density objects, e.g., those with ρ =1g/cm3 would oscillate with much higher frequency, saturating bound (17), i.e. ω ∼ m.All kinds of particles with masses smaller than m might be created by such oscillatingfield.3. – Cosmological evolution and particle production in R2 gravityIn the present section we study the cosmological evolution of the Universe in a theorywith only an additional R2 term in the action, neglecting other terms which have beenintroduced to generate the accelerated expansion in the contemporary universe [13]. Theimpact of such terms is negligible in the limit of sufficiently large curvature, |R|  |R0|,where R0 is the cosmological curvature at the present time.In other words, we study here the cosmological evolution of the early and not so earlyuniverse in the model with the actionS = −m2Pl16π∫d4x√−g(R− R26m2)+ Sm.(18)The modified Einstein equations for this theory readRμν − 12gμνR−13m2(Rμν − 14Rgμν + gμνDαDα −DμDν)R =8πm2PlTμν .(19)44 E. V. ARBUZOVAExpressing the curvature scalar R through the Hubble parameter H = a˙/a as R =−6H˙ − 12H2, we get the time-time component of eq. (19):H¨ + 3HH˙ − H˙22H+m2H2=4πm23m2PlHρ,(20)where over-dots denote derivative with respect to the physical time t.Taking the trace of eq. (19) yieldsR¨ + 3HR˙ + m2(R +8πm2PlTμμ)= 0.(21)In what follows, we study the cosmological evolution in the R2-theory assuming rathergeneral initial conditions for R and H and dominance of relativistic matter with thefollowing equation for the matter content:ρ˙ + 4Hρ = 0.(22)It is convenient to rewrite the equations in terms of the dimensionless quantitiesτ = H0 t, h = H/H0, r = R/H20 , y = 8πρ/(3m2PlH20 ), and ω = m/H0,where H0 is thevalue of the Hubble parameter at some initial time t0. Thus the following system ofequations for dimensionless Hubble parameter is obtained:h′′ + 3hh′ − h′22h+ω22h2 − yh= 0,(23a)y′ + 4hy = 0.(23b)First we assume that the deviations from General Relativity (GR) are small andexpand h = 1/(2τ) + h1 and y = 1/(4τ2) + y1, assuming that h1/h  1 and y1/y  1,and solve the linearized system of equations.The complete asymptotic solution for h has the formh(τ)  12τ+c1 sin(ωτ + ϕ)τ3/4.(24)The Hubble parameter oscillates around GR value, h0 ∼ 1/(2τ) with rising amplitude,h1/h0 ∼ τ1/4, and for sufficiently large τ the second term would start to dominate andthe linear approximation would no longer hold. Using trancated Fourier expansion it ispossible to obtain approximate analytical solutions of the full non-linear system in thehigh-frequency limit ωτ  1. The same results are found numerically for the initialconditions h0 = 1 + δh0, h′0 = −2 + δh′0 y0 = 1 + δy0 (see fig. 4).Gravitational particle production may non-trivially affect the solutions of the aboveequations. Below we consider particle production by the external oscillating gravitationalfield and present the equation of motion for the evolution of R with the account of theback-reaction from particle production. This leads to an exponential damping of theoscillating part of R, while the non-oscillating “Friedmann” part remains practicallyMODIFIED GRAVITY: PROBLEMS AND OBSERVATIONAL MANIFESTATIONS 4550 100 150ωτ0.499900.499950.500000.500050.50010h τ100 150 200 250 300 350ωτ0.20.40.60.81.01.2h τFig. 4. – Numerical solution of eqs. (23). Left panel: linear regime with δh0 = 10−4, δh′0 = 0,y0 = 1, ω = 10. Right panel: high-frequency limit with δh0 = 1.5, δh′0 = 0, y0 = 0, ω = 100.Initial conditions are different from GR, the central value hτ = 0.6 is shifted from GR value 0.5.undisturbed. The particle production influx into the cosmological plasma is estimatedin the case of a massless, minimally-coupled to gravity scalar field with the actionSφ =12∫d4x√−g gμν∂μφ∂νφ.(25)It terms of the conformally rescaled field, χ ≡ a(t)φ, and conformal time η, such thatadη = dt, we can rewrite the equations of motion asR′′ + 2a′aR′ + m2a2R = 8πm2m2Pl1a2[χ′2 − (∇χ)2 + a′2a2χ2 − a′a(χχ′ + χ′χ)],(26a)R = −6a′′/a3,(26b)χ′′ −Δχ + 16a2Rχ = 0,(26c)We derive a closed equation for R taking the average value of the χ-dependent quan-tum operators in the r.h.s. of eq. (26a) over vacuum, in presence of an external classicalgravitational field R following the procedure described in ref. [14], where such equationwas obtained in one-loop approximation.The dominant contribution of particle production is given by the equationR¨ + 3HR˙ + m2R  112πm2m2Pl∫ tt0dt′R¨(t′)t− t′ .(27)This equation is linear in R and naturally non-local in time since the impact of particleproduction depends upon all the history of the evolution of the system.Using again the procedure of truncated Fourier expansion including the back-reactioneffects in the form of eq. (27), we obtain the decay rateΓR =m348m2Pl.(28)46 E. V. ARBUZOVAThis result is in agreement with ref. [15]. The characteristic decay time of the oscillatingcurvature isτR =12ΓR=24m2Plm3 2(105 GeVm)3s.(29)The contribution of the produced particles into the total cosmological energy densityreaches its maximum value at approximately this time.The influx of energetic protons and antiprotons could have an impact on BBN. Thusthis would either allow to obtain some bounds on m or even to improve the agreementbetween the theoretical predictions for BBN and the measurements of the primordialabundances of light nuclei.The oscillating curvature might be a source of dark matter in the form of heavysupersymmetric (SUSY) particles. Since the expected light SUSY particles have not yetbeen discovered at LHC, to some people supersymmetry somewhat lost its attractiveness.The contribution of the stable lightest SUSY particle into the cosmological energy isproportional toΩ ∼ m2SUSY /mPl(30)and for mSUSY in the range 100–1000GeV the cosmological fraction of these particleswould be of order unity. It is exactly what is necessary for dark matter. However, itexcludes thermally produced LSP’s if they are much heavier. If LSP’s came from thedecay of R and their mass is larger than the “mass” of R, i.e. m, the LSP productioncould be sufficiently suppressed to make a reasonable contribution to dark matter.In contemporary astronomical objects oscillation frequency could vary from m downto very low frequency. The oscillations may produce radiation from high-energy cosmicrays down to radio waves.∗ ∗ ∗I thank my coauthors, A. D. Dolgov and L. Reverberi, for cooperation. I amespecially grateful to A. D. Dolgov for help and fruitful discussions during the entireperiod of collaboration. This work was supported by the Grant of the Government ofRussian Federation, No. 11.G34.31.0047.REFERENCES[1] Nakamura K. et al. (Particle Data Group), J. Phys. G, 37 (2010) 075021.[2] Riess A. G. et al., Astron. J., 116 (1998) 1009; Astrophys. J., 607 (2004) 665;Perlmutter S. et al., Nature, 391 (1998) 51; Astrophys. J., 517 (1999) 565; SchmidtB. P. et al., Astrophys. J., 507 (1998) 46.[3] Peebles P. J. 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Modified gravity. Problems and observational manifestations

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