Matter power spectrum in f(R) gravity with massive neutrinos
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Matter Power Spectrum in f ( R ) Gravity with Massive Neutrinos
Hayato Motohashi , , Alexei A. Starobinsky , , and Jun’ichi Yokoyama , Department of Physics, Graduate School of Science,The University of Tokyo, Tokyo 113-0033, Japan Research Center for the Early Universe (RESCEU),Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan L. D. Landau Institute for Theoretical Physics, Moscow 119334, Russia Institute for the Physics and Mathematics of the Universe(IPMU),The University of Tokyo, Kashiwa, Chiba, 277-8568, Japan
Abstract
The effect of massive neutrinos on matter power spectrum is discussed in the context of f ( R )gravity. It is shown that the anomalous growth of density fluctuations on small scales due to thescalaron force can be compensated by free streaming of neutrinos. As a result, models which predictobservable deviation of the equation-of-state parameter w DE from w DE = − O (0 . ( R ) gravity is a simple and nontrivial extension of General Relativity (GR) which hasrecently received much attention as an alternative to the Λ-Cold-Dark-Matter (ΛCDM)model that can explain the present cosmic acceleration. The basic idea is to modify theEinstein-Hilbert action by using the general function of Ricci scalar R as S = Z d x √− gf ( R ) + S m , (1)where S m indicates a matter action (see Ref. [1] for a recent review and an extensive listof references). If f ′′ ( R ) is not zero identically, this modified gravity contains an additionaldegree of freedom corresponding to a new scalar field, dubbed scalaron in Ref. [2] wherethe simplest, f ( R ) = R + R / M , variant of such theory (with small one-loop quantumgravitational corrections) was used to construct an internally self-consistent inflationarymodel of the early Universe with a graceful exit from inflation to the subsequent radiation-dominated Friedmann-Robertson-Walker (FRW) stage after the gravitational creation ofmatter and reheating. However, f ( R ) gravity can be used to describe dark energy in thepresent Universe as well, although with the use of more complicated functions f ( R ) havinga nontrivial structure for low values of R .To have the correct Newtonian limit for R ≫ R ≡ R ( t ) ∼ H , where t is the presentmoment and H is the Hubble constant, as well as the standard matter-dominated FRWstage with the scale factor behaviour a ( t ) ∝ t / driven by cold dark matter and baryons,the following conditions should be fulfilled: | f ( R ) − R | ≪ R, | f ′ ( R ) − | ≪ , Rf ′′ ( R ) ≪ , R ≫ R , (2)where the prime denotes a derivative with respect to the argument R . The same conditionsguarantee smallness of non-GR corrections to a space-time metric for more general back-grounds for which GR has to be used in full. In addition, the stability condition f ′′ ( R ) > f ( R ) gravity. In quantum language, this condition means that the scalaron isnot a tachyon. Note that the other stability condition, f ′ ( R ) >
0, which means that gravityis attractive and the graviton is not a ghost, is automatically fulfilled in this regime. Spe-cific functional forms that satisfy these and other necessary conditions have been proposedin Refs. [3–5], and much work has been conducted on their cosmological consequences.2e can express field equations derived from action (1) in the following Einsteinian form, R µν − δ µν R = − πG (cid:16) T µν ( m ) + T µν (DE) (cid:17) , (3)where 8 πGT µν (DE) ≡ ( F − R µν −
12 ( f − R ) δ µν + ( ∇ µ ∇ ν − δ µν (cid:3) ) F, (4)and we have also defined F ( R ) ≡ f ′ ( R ) = df /dR . Working in the spatially flat Friedmann-Robertson-Walker (FRW) space-time with the scale factor a ( t ), we find3 H = 8 πGρ − F − H + 12 ( F R − f ) − H ˙ F , (5)2 ˙ H = − πGρ − F −
1) ˙ H − ¨ F + H ˙ F , (6)where H is the Hubble parameter and ρ is the energy density of the material content, whichwe assume to consist of nonrelativistic matter. From these expressions, the effective DEequation-of-state parameter w DE ≡ P DE /ρ DE is given by1 + w DE = − H − πGρ H − πGρ = 2(1 − F )( − ¨ a/a + H ) + ¨ F − H ˙ F − F )( − ¨ a/a ) + ( R − f ) / − H ˙ F . (7)Note that the phantom crossing is naturally realized in viable f ( R ) theories as shown inRefs. [3, 6–8].We specifically adopt the following functional form that satisfies all the aboverequirements[5], f ( R ) = R + λR s "(cid:18) R R s (cid:19) − n − , (8)where n, λ , and R s are model parameters. It is sufficient to describe the recent evolutionof the Universe, although it has to be modified both for R < R (including the region R <
0) and for very large positive R to avoid problems in the early Universe cosmology (seeRefs. [6, 9]).In f ( R ) gravity, the evolution equation for density fluctuation with comoving wave num-ber k in Fourier space, δ k , in the deeply subhorizon regime is given by [10, 11]¨ δ k + 2 H ˙ δ k − πG eff ρδ k = 0 , (9)where G eff = GF k a F ′ F k a F ′ F . (10)3n the high-curvature regime, the differential equation (9) can be solved analytically [9],and we analyzed it in the general regime in terms of numerical calculation in the previouspaper [8]. It is revealed that the effective gravitational coefficient G eff enhances the growthof density fluctuations compared with that in the standard ΛCDM model due to an extraforce mediated by the scalaron. To suppress the deviation of the matter power spectrumfrom that in the ΛCDM model within, say 10%, the parameter space for n and λ should berestricted. For example, λ should be greater than 8 in n = 2 [8].In this paper, we show that the anomalous growth of density fluctuations on small scalesdue to the scalaron is rectified if neutrino masses are taken into consideration. That is, thefree streaming of neutrinos partially erases small-scale density fluctuations, which can beharmful in Einstein gravity, to compensate their unwanted extra growth in f ( R ) gravity.To set up the initial condition, we calculate the evolution of density fluctuations for z ≥
10 using CAMB [12, 13], and obtained P ( k, z = 10), power spectrum at z = 10. Weassume one mass eigenstate for neutrino, and all three neutrinos have practically the samemasses. Note that the density parameter of neutrinos isΩ ν h ≃ . × − ( P m ν = 0) P m ν . P m ν = 0) , (11)and the free streaming of massive neutrinos suppresses fluctuations below the scale [14, 15] k fs ( z ) ≃ . z ) / (cid:16) m ν (cid:17) (cid:18) Ω m . (cid:19) / h Mpc − . (12)Therefore, δ k in Eq. (9) implies density fluctuations for CDM below that scale.At z &
10, the effect of modified gravity is sufficiently small. Therefore, we start to solvethe evolution of density fluctuations in f ( R ) gravity using Eq. (9) at z = 10. We have foundthat the physical wave number crossing the scalaron mass today is given by k s a ≃ . λ . H , (13)for n = 2[8], which is to be compared with Eq. (12). It takes 1 . × − h and 8 . × − h Mpc − for λ = 1 and 3, respectively.The present power spectrum is constructed as P ( k, z = 0) = P ( k, z = 10) (cid:18) δ ( z = 0) δ ( z = 10) (cid:19) . (14)4 -4 -3 -2 -1 ( P f R G - P Λ CD M ) / P Λ CD M k / hMpc -1 n=2 λ =1 -0.20-0.100.000.100.200.300.4010 -4 -3 -2 -1 ( P f R G - P Λ CD M ) / P Λ CD M k / hMpc -1 n=2 λ =3 massless0.1eV0.2eV0.3eV0.4eV0.5eV0.6eV0.7eV0.8eV FIG. 1: Deviation of power spectrum for n = 2 , λ = 1 ,
3. Captions indicate total neutrino mass,and it is assumed that three neutrinos have the same masses. -0.3-0.2-0.10.00.10.20.30.4 0 1 2 3 4 5 6 7 8 9 10 ( P f R G - P Λ CD M ) / P Λ CD M λ n = 2k = 0.174 h Mpc -1 massless0.1 eV0.2 eV0.3 eV0.4 eV0.5 eV0.6 eV0.7 eV0.8 eV Σ m ν / e V λ n = 2n = 3 FIG. 2: Left: Relative deviation of linear matter power spectrum from ΛCDM model at k =0 . h Mpc − as a function of λ for various values of neutrino mass. Right: The region betweentwo lines for each n corresponds to the parameter space where the deviation of power spectrum issmaller than 10%. Figure 1 shows cancellation of the contributions from f ( R ) gravity and neutrino masses.Equations (12) and (13) explain the threshold wave number. In particular, the transitionoccurs at a similar wave number for λ = 3 when we take n = 2. The deviation at k =0 . h Mpc − , which is the scale corresponding to σ normalization [8], is minimum whenthe total neutrino masses are 0 . . λ = 1 and 3, respectively.In the left panel of Fig. 2, the relative deviation of the power spectrum from that of theΛCDM model is depicted as a function of λ for various total neutrino masses with n = 2. Theright panel of Fig. 2 shows the range of total neutrino masses where the relative deviationof the power spectrum at k = 0 . h Mpc − remains smaller than 10%.As seen in the right panel, models with smaller values of λ , which tend to exhibit alarger deviation from the ΛCDM model, become compatible with the observation of fluc-5 w D E Σ m ν / eVn = 2n = 3n = 4 -1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 γ Σ m ν / eVk = 0.174 h Mpc -1 n = 2n = 3n = 4 FIG. 3: Possible range of time variation of w DE (left) and growth index γ (right) as a function oftotal neutrino mass. γ is measured at the comoving wave number k = 0 . h Mpc − . tuations. In other words, models with deviations of w DE and the growth index γ ( z ) =log (cid:16) ˙ δ/Hδ (cid:17) / log Ω m large enough to be observable become viable thanks to the suppressionof small-scale fluctuations due to finite neutrino masses. Figure 3 depicts the maximumrange of the time variation of w DE and γ that can be realized for each value of the totalneutrino mass. That is, if we choose a minimum possible value of λ for each total neutrinomass, w DE varies between the upper and lower edges of the spatula shape at the correspond-ing neutrino mass. Similarly, γ evolves between the upper and lower edges of the ax-shape.Note that γ takes an almost constant value, 0 .
55, in the ΛCDM model, which correspondsto the upper bound in f ( R ) gravity.Finally, we comment on how nonlinear effects may modify our results. So far, nonlineareffects on matter clustering in f ( R ) gravity have been analyzed in higher order perturbationtheory [16] and N-body simulations [17]. The latter has shown that nonlinear effects lowerthe relative deviations of the power spectrum, ( P fRG − P ΛCDM ) /P ΛCDM , by about 5% in aspecific case f ( R ) = R − − f R R /R with | f R | = 10 − , where R is the current value ofthe scalar curvature. It has also been shown that the nonlinear effects tend to suppress theanomalous growth of small-scale fluctuations observed in linear perturbation in f ( R ) theory.We may therefore conclude that nonlinear effects will not change our results qualitatively.In conclusion, neutrino rest-masses with P ν m ν = O (0 . f ( R ) gravity, which follows from the anomalous k -dependent growth of densityperturbations in the cold dark matter + baryon component at recent redshifts, makingpossible for w DE and γ ( k ) deviate from those of the ΛCDM model noticeably. One can6istinguish the effects of f ( R ) and massive neutrinos by analyzing the ratio of correlationbetween galaxies and curvature perturbation probed by weak lensing to that between galaxiesand velocity field, namely, an estimator E G = Ω m / ( F β ), where β is the growth rate[18].According to the current observational analysis, both Einstein gravity and f ( R ) gravity areconsistent with the data. However, in the future, one can break the degeneracy in principle.AS acknowledges RESCEU hospitality as a visiting professor. He was also partially sup-ported by the grant RFBR 08-02-00923 and by the Scientific Programme “Astronomy” ofthe Russian Academy of Sciences. This work was supported in part by JSPS Research Fel-lowships for Young Scientists (HM), JSPS Grant-in-Aid for Scientific Research No. 19340054(JY), Grant-in-Aid for Scientific Research on Innovative Areas No. 21111006 (JY), JSPSCore-to-Core program “International Research Network on Dark Energy”, and Global COEProgram “the Physical Sciences Frontier”, MEXT, Japan. [1] A. De Felice and S. Tsujikawa, Living Rev. Rel. , 3 (2010) [arXiv:1002.4928].[2] A. A. Starobinsky, Phys. Lett. B , 99 (1980).[3] W. Hu and I. Sawicki, Phys. Rev. D , 064004 (2007) [arXiv:0705.1158].[4] S. Appleby and R. Battye, Phys. Lett. B , 7 (2007) [arXiv:0705.3199].[5] A. A. Starobinsky, JETP Lett. , 157 (2007) [arXiv:0706.2041].[6] S. Appleby, R. Battye and A. Starobinsky, JCAP , 005 (2010) [arXiv:0909.1737].[7] A. Ali, R. Gannouji, M. Sami and A. A. Sen, Phys. Rev. D , 104029 (2010)[arXiv:1001.5384].[8] H. Motohashi, A. A. Starobinsky and J. Yokoyama, Prog. Theor. Phys. , 887 (2010)[arXiv:1002.1141].[9] H. Motohashi, A. A. Starobinsky and J. Yokoyama, Int. J. Mod. Phys. D , 1731 (2009)[arXiv:0905.0730].[10] P. Zhang, Phys. Rev. D , 123504 (2006) [arXiv:astro-ph/0511218].[11] S. Tsujikawa, Phys. Rev. D , 023514 (2007) [arXiv:0705.1032].[12] A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. , 473 (2000) [arXiv:astro-ph/9911177].[13] http://camb.info/
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