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<< Cosmo.WebHome %TOC% ---- %INCLUDE{"GrassmannCosmoFun2009Head"}% ---+ Fri 18 Sep 2009 ---++ STAROBINSKY - f(R) models * particle content: graviton + * massive scalar particle ( M^2 = 1/{3 f''(R)} ) (called "scalaron" in Starobinsky 1980) * stability conditions: * f' > 0 graviton is not a ghost * f'' > 0 scalaron is not a tachyon * imposed for R \ge R_{now} at least (i.e. during the whole evolution of the Universe) * possible microscopic origin of f(R) gravity * vacuum polarisation in curved space-time * reduction to 4D from curved (4+n)-D space-time * limiting case of scalar-tensor gravity * emergent gravity Klinkhamer & Volovik (2008) ArXiv:0807.3896 * violation of these conditions is undesirable also from the classical point of view * f'(R_*) = 0 instant loss of homogeneity and isotropy * f''(R_*) = 0 weak singularity * %$ R(t) = R_* + O(\sqrt{t}) $% * %$ a(t) = a_0 + a_1 t + a_2 t^2 + O(t^{5/2}) $% * existence of the Newtonian regime %$ \Delta(\phi) = 4 \pi G \rho $% * %$ |F(R)| \ll R, |F'(R)| \ll 1, R |F''(R)| \ll 1 $% for %$ R_{now} \ll R $% (up to some very large R) * de Sitter regime %$ R f' = 2f $% stable if %$ f'(R_*) > R_1 f''(R_*) $% * equivalent to %$ \omega_{BD} = 0 $% in scalar-tensor gravity * use for inflation Starobinsky 1980: %$ f(R) = R + \frac{R^2}{6 M^2} $% * internally consistent infolationary model with slow-roll decay, a graceful exit to the subsequent RD (radiation-dominated) FRW stage ... and sufficiently effective reheating * model remains viable, e.g. %$ N \sim 50, n_s = 1 - 2/N = 0.96, r = 12/N^2 = 4.8 \times 10^{-4} ... $% * oher viable f(R) inflationary models * 1. chaotic type - inflation over a large range of R * 2. new inflationary type - inflation around R =R_0 * both cases: f(R) close to R^2/{6M^2} * viable for DE??? * %$F(R) \propto R^{-n} $% for %$ R \rightarrow 0 $% does *not* work for many reasons * viable model - must be regular at $=0 * Starobinsky 2007, ArXiv:0706.2041 : %$ f(R) = R + \lambda R_* \left[\frac{1}{ \left( 1 + R^2/R_*^(?) \right)^n} -1 \right] $% * or: %$ f(R) = R - \lambda R_* \mathrm{tanh}^2 (R/R_*) $% * f(0) = 0 * observational constraints * cosmology - anomalous growth of non-relativistic matter perturbations in the regime %$ k \gg M(R) $% * lab and Solar System tests %$ M(R) L \gg 1 $% with R = 8 pi G T_m = 8 \pi G \rho_m; otherwise, \gamma_{(?)} = 1/2 and 'fifth' force appears * both OK if n \ge 2 * ... "scalaron production" problems * "big boost" singularity with %$ R \rightarrow \infty $% and its elimination * elimination: add R^2/{6M^2} to F(R) * conclusions: * viable models of DE, distinguishable from LambdaCDM, exist, given certain conditions: * with a regular f(R) satisfying: %$ f'(R) > 0, f''(R) > 0 \forall R $% * %$ |f-R| \ll R, |f'-1| \ll 1, R|f''| \ll 1 $% * ... * unification of primordial DE (i.e. inflation) and present DE is possible for %$ M = 3\times 10^{-6} M_{(Pl???)} $% * anomalous growth of scalar perturbations at recent times ( z about 1-3 for L = 8 /h Mpc ) would be the most critical test of the f(R) DE models satisfying these conditions ---++ David POLARSKI - DE * scalar-tensor theories AstroPh:0701650, AstroPh:0507290 * LambdaCDM problems? * nbDM halo density profile - no cups seen? * large scale peculiar flows * unexpected brightness of SNe Ia at z > 1 (??) * void problem * w = -1 constant is an Achilles' heel * quintessence * Lagrangian L = fn(F(phi), R, Z U(phi), g_munu * Brans-Dicke parametrisation %$ F(\phi) = \phi, Z(\phi) = \frac{\omega_{BD}(\phi)}{\phi} $% ... * %$ G_{\mathrm{eff}}, G_N, G_* $% * effects on structure formation ---++ Roman JUSZKIEWICZ * non-linear perturbation theory for sigma_8 ---++ Jerzy KROL * exotic R^4, QG and QFT - ArXiv:0904.1276 and references therein * R^n with n=4 case is particularly difficult ---++ Babak VAKILI * Noether approach ... -- Main.BoudRoukema - 18 Sep 2009
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Topic revision: r2 - 18 Sep 2009,
BoudRoukema
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