Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 148) Bled, Slovenia, June 23-July 1, 2018 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity S.V. Ketov1'2'3 * andM.Yu. Khlopov4 ** 1 Physics Department, Tokyo Metropolitan University, Minami-ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan 2 Kavli Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo, Chiba 277-8568, Japan 3 Research School of High Energy Physics, Tomsk Polytechnic University, Lenin ave. 30, Tomsk 634050, Russia 4 Institute of Physics, Southern Federal University Stachki 194, Rostov on Don 344090, Russia Abstract. We review some recent trends in the inflationary model building, the super-symmetry (SUSY) breaking, the gravitino Dark Matter (DM) and the Primordial Black Holes (PBHs) production in supergravity. The Starobinsky inflation can be embedded into supergravity when the inflaton belongs to the massive vector multiplet associated with a (spontaneously broken) U(1 ) gauge symmetry. The SUSY and R-symmetry can be also spontaneously broken after inflation by the (standard) Polonyi mechanism. Polonyi particles and gravitinos are super heavy and can be copiously produced during inflation via the Schwinger mechanism sourced by the Universe expansion. The overproduction and instability problems can be avoided, and the positive cosmological constant (dark energy) can also be introduced. The observed abundance of the Cold Dark Matter (CDM) composed of gravitinos can be achieved in our supergravity model too, thus providing the unifying framework for inflation, supersymmetry breaking, dark energy and dark matter genesis. Our supergravity approach may also lead to a formation of primordial non-linear structures like stellar-mass-type black holes, and may include the SUSY GUTs inspired by heterotic string compactifications, unifying particle physics with quantum gravity. Povzetek. Avtorja obravnavata nekaj novejsih modelov inflacije, zlomitve supersimetrije, temne snovi, ki jo sestavljajo gravitini in nastajanja prvotnih črnih lukenj v supergravitaciji. Inflacija Starobinskega se pojavi v supergravitaciji, ce je inflaton del masivnega vektorskga multipleta, ki spontano zlomi umeritveno simetrijo U(1). Supersimetrijo in simetrijo R lahko po inflaciji spontano zlomi tudi mehanizem Polonyija. Izredno masivni delci Polony-ija in gravitini, lahko nastanejov dovolj velikih kolocinah med inflacijo z mehanizmom Schwingerja. S tem se avtorja izogneta problemu prevelike produkcije tezkih delcev in nestabilnosti ter pojasnita tudi pozitivno kozmolosko konstanto (temno energijo). Njun model s supergravitacijo razlozi opazeno pogostost hladne temne snovi (CDM), ce jo sestavljajo gravitini in ponudi razlago za nastanek in potek inflacije, zlomitev supersimetrije, temno energijo in temno snov. Njun model lahko pojasni tudi nastanek prvotnih nelin-eranih struktur, kot so crne luknje, ki imajo maso enake masi obicajnih zvezd, in morda * E-mail: ketov@tmu.ac.jp ** E-mail: khlopov@apc.in2p3.fr 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity 149 vkljucuje supersimetricne teorije velikega poenotenja (GUT), izvirajoce iz kompaktifikacije heterotskih strun, kar bi poenotilo fiziko delcev in kvantno gravitacijo. Keywords: inflation, modified gravity, supergravity, cold dark matter, dark energy, supersymmetry breaking, primordial black holes 7.1 Introduction The Cosmic Microwave Background (CMB) data collected by the Planck collaboration [1-3] favours the slow-roll single-field inflationary scenarios, with an approximately flat scalar potential. The celebrated Starobinsky model [4] does provide such scenario, and relates its inflaton (called scalaron in this context) to the particular extension of Einstein-Hilbert gravity with the extra higher derivative term given by the scalar curvature squared, R2. However, a theoretical explanation of fundamental origin of the Starobinsky model is still missing. The viable inflationary dynamics is driven by the R2 term dominating over the (Einstein-Hilbert) R term. This is related to a missing UV completion of the non-renormalizable (R+R2) gravity. The interesting and ambitious project for string phenomenology would be to provide a derivation of the Starobinsky model from the first principles. A first step towards this is an embedding of the Starobinsky model into four-dimensional N = 1 supergravity. In the supergravity framework, the inflaton (scalaron) can mix with other scalars, and this mixing may ruin any initially successful inflationary mechanism. The inflationary model building based on supergravity in the literature usually assumes that inflaton belongs to a chiral (scalar) supermultiplet [5-7]. However, there is the alternative to this assumption: inflaton can also belong to a massive N = 1 vector multiplet. The vector multiplet-based approach avoids stabilization problems related to the inflaton (scalar) superpartner, as the way-out of the standard n-problem. The scalar potential of a vector multiplet is given by the D-term instead of the F-term. The minimal supergravity models, with inflaton belonging to a massive vector multiplet, were proposed in Refs. [8,9]. Then any desired values of the CMB observables (the scalar perturbations tilt ns and the tensor-to-scalar perturbations ratio r) can be recast from the single-field (inflaton) scalar potential proportional to the derivative squared of arbitrary real function J. However, in these models, the vacuum energy is vanishing after inflation, thus restoring supersymmetry, and only a Minkowski vacuum is allowed. The way-out of this problem was proposed in [10,11] by adding a Polonyi (chiral) superfield with a linear superpotential [12], leading to a spontaneous SUSY breaking and allowing a de-Sitter vacuum after inflation. A successful model of inflation in supergravity should also be consistent with the Cold Dark Matter (CDM) constraints and the Big Bang Nucleosynthesis (BBN). For example, many supergravity scenarios are plagued by the so-called gravitino problem. Gravitinos can decay, injecting hadrons and photons during the BBN epoch, which may jeopardize the good Standard Model prediction of nuclei ratios [13-16]. In very much the same way, the Polonyi (overproduction) problem and 150 S.V. Ketov and M.Yu. Khlopov its relation to the BBN results were extensively discussed in the literature [17-22]. In addressing these issues, the mass spectrum and the soft SUSY parameters are important. The leading (WIMP-like) dark matter production mechanisms and decay channels are selected from the mass pattern, and have either thermal or non-thermal origin. In this paper, we review a class of the minimalistic Polonyi-Starobinsky (PS) N = 1 supergravity models for inflation, with the inflaton belonging to a (massive) vector multiplet. These models can avoid the overproduction and BBN problems, while accounting for the right amount of CDM composed of gravitinos. In our analysis, we assume that the Polonyi field, inducing a spontaneous SUSY breaking at a high energy scale, and the gravitino, as the Dark Matter (DM) particle, are both super-heavy. The main mechanism producing DM is given by the Schwinger-type production sourced by inflationary expansion. After inflation, Polonyi particles rapidly decay into gravitinos. We find that gravitinos produced directly from Schwinger's production and from Polonyi particles decays, can account for the correct abundance of Cold Dark Matter. Another aspect is an inclusion of the (mini) Primordial Black Holes (PBHs) that may have been copiously produced in the early Universe, and later may have evaporated into gravitinos and other Standard Model particles [23-27]. A large amount of mini PBHs cannot be produced in our model when the other scalar and pseudo-scalar partners of inflaton are not participating in the inflationary dynamics. The Starobinsky inflaton entails a scalar potential shape that cannot lead to a large number of PBHs, because it does not allow for amplifying instabilities and has no exit out of inflation with a first order phase transition. It is still possible that dynamics of other scalar fields changes this picture. In this case, the extra moduli can exit from inflation via ending in false minima. The tunneling process from a false minimum to the true one sources the production of bubbles related to the first order phase transition. As regards the (solar mass type) PBHs, their production in the early Universe is possible in our supergravity approach after a certain deformation of the Starobinsky scalar potential. We envisage a unification of the inflaton in a vector multiplet and the Supersymmetric Grand Unified Theories (SUSY GUTs), whose gauge group has at least one abelian factor, such as the flipped SU(5)xU(1) model arising from the compactified heterotic superstrings or the intersecting D-branes. 7.2 Starobinsky model of (R + R2) gravity Starobinsky model of inflation is defined by the action [4] where we have introduced the reduced Planck mass MP[ = W8nGN ~ 2.4 x 1018 GeV, and the scalaron (inflaton) mass m as the only parameter. We use the spacetime signature (—, +, +, +,). The (R + R2) gravity model (7.1) can be considered as the simplest extension of the standard Einstein-Hilbert action in the (7.1) 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity 151 context of (modified) F(R) gravity theories with an action M2 C ■ • LPl Sf = "2" d4^v/-gF(R) , (7.2) in terms of the function F(R) of the scalar curvature R. The F(R) gravity action (7.2) is classically equivalent to s[g,v,x] = M2 d4^v/-g [F'(x)(R - x)+ F(x)] (7.3) with the real scalar field x, provided that F " = 0 that we always assume. Here the primes denote the derivatives with respect to the argument. The equivalence is easy to verify because the x-field equation implies x = R. In turn, the factor F' in front of the R in (7.3) can be (generically) eliminated by a Weyl transformation of metric that transforms the action (7.3) into the action of the scalar field x minimally coupled to Einstein gravity and having the scalar potential V = (Ml 1 xIM_F(x . (7.4) (7.5) 2 ) F'(x)2 Differentiating this scalar potential yields dV ( M2l ^ F''(x)[2F(x)- xF'(x)] dx \2 (F '(x))3 The kinetic term of x becomes canonically normalized after the field redefinition x(0 and F''(R) >0, (7.8) and they are obviously satisfied for Starobinsky model (7.1) for R > 0. Differentiating the scalar potential V in Eq. (7.4) with respect to 9 yields 2 dV = dV dx = Mpl d9 dx d9 2 xF'' + F' - F' 2xF' - FF„ :/2 c/3 where we have dx dx dF ' dF ' /dF ' V2 F ' d9 dF' d9 d9 / dx V3MPl F'' £ , (7.9) d9 (7.10) 152 S.V. Ketov and M.Yu. Khlopov This implies ^ = Mpi dp 11 a/6F '2 (7.11) Combining Eqs. (7.4) and (7.11) yields R and F in terms of the scalar potential V, R = F = VédV 4V Mpi dip M2l V6 dV 2V MP1 dp m2 exp 3|/Mpi 12 exp I 2^ 3i/Mpi (7.12) (7.13) These equations define the function F(R) in the parametric form, in terms of a scalar potential V (<), i.e. the inverse transformation to (7.4). This is known [28] as the classical equivalence (duality) between the F(R) gravity theories (7.2) and the scalar-tensor (quintessence) theories of gravity (7.7). In the case of Starobinsky model (7.1), one gets the famous potential 41 V (p) = 7 M21m2 exp | -\l 3i/Mpi (7.14) This scalar potential is bounded from below (non-negative and stable), and it has the absolute minimum at < = 0 corresponding to a Minkowski vacuum. The scalar potential (7.14) also has a plateau of positive height (related to inflationary energy density), that gives rise to slow roll of inflaton in the inflationary era. The Starobinsky model (7.1) is the particular case of the so-called a-attractor inflationary models [29], and is also a member of the close family of viable inflationary models of F(R) gravity, originating from higher dimensions [30]. A duration of inflation is measured in the slow roll approximation by the e-foldings number i rv* A/ (7.15) Ne 1 MH V . V7 dp where p* is the inflaton value at the reference scale (horizon crossing), and pend is the inflaton value at the end of inflation when one of the slow roll parameters Mp2l V' and nv(i) = Mpl ( V ' V (7.16) is no longer small (close to 1). The amplitude of scalar perturbations at horizon crossing is given by [31] A = V3 12npM6l(V*')2 3m2 8n2Mpl sinh4 (v^M; p* (7.17) The Starobinsky model (7.1) is the excellent model of cosmological inflation, in very good agreement with the planck data [1-3]. The planck satellite mission measurements of the Cosmic Microwave Background (CMB) radiation [1-3] give the scalar perturbations tilt as ns « 1 + 2nv — 6ev « 0.968 ± 0.006 and restrict 2 2 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity 153 the tensor-to-scalar ratio as r « 16eV < 0.08. The Starobinsky inflation yields r « 12/N « 0.004 and ns « 1 — 2/Ne, where Ne is the e-foldings number between 50 and 60, with the best fit at Ne « 55 [32,33]. The Starobinsky model (7.1) is geometrical (based on gravity only), while its (mass) parameter m is fixed by the observed CMB amplitude (COBE, WMAP) as m « 3 • 1013 GeV or m « 1.3 • 10-5 . Mpi A numerical analysis of (7.15) with the potential (7.14) yields [31] 3^/Mpl ln ( 3Ne 5.5 , 3^end/Mpl ln 2 jj (4 + 3V3) (7.18) 0.5 , (7.19) where Ne « 55 has been used. 7.3 Starobinsky inflation in supergravity Let us introduce a set of two chiral superfields H) and a real vector superfield V coupled to the supergravity sector, with the following Lagrangian:1 £ = d292£ j 3 (DD - 8R)e-3 (K+2J) + 1waWa + W(®) j + h.c. , (7.20) where R is the chiral scalar curvature superfield, E is the chiral density superfield, (Da, Da) are the superspace covariant spinor derivatives, K = K(®, is the Kahler potential, W(®) is the superpotential, Wa = — 4 (DD — 8R)DaV is the abelian (chiral) superfield strength, and J = J(He2gVH) is a real function with the coupling constant g. The Lagrangian (7.20) is invariant under the supersymmetric U(1) gauge transformations H —> H' = e-igZH , H->H' = eigZH , (7.21) V^V' = V + 2(Z — Z) , (7.22) the gauge parameter of which, Z, is itself a chiral superfield. The chiral superfield H can be gauged away via the gauge fixing of these transformations by imposing the gauge condition H = 1. Then the Lagrangian (7.20) gets simplified to C = d292f j 3(DD - 8R)e-1 (K+2J) + 1waWa + wj + h.c. (7.23) After eliminating the auxiliary fields and moving from the initial (Jordan) frame to the Einstein frame, the bosonic part of the Lagrangian (7.23) reads [10]2 e-1 L = - 1r-KaAamAamA-lFmnFmn-2J"3mC3mC-1J"BmBm-V, (7.24) 1 We use the standard notation [34] for supergravity in superspace. 2 The primes and capital latin subscripts denote the derivatives with respect to the corre- sponding fields. 154 S.V. Ketov and M.Yu. Khlopov with the scalar potential V = g2 J '2 + eK+2J j Ka\ (Wa + KaW )(Wa + Ka W ) - - 2^ WW j (7.25) in terms of the physical fields (A, C, Bm), the auxiliary fields (F, X, D) and the vector field strength Fmn = DmBn — Bm. As is clear from Eq. (7.24), the absence of ghosts requires J "(C) >0, where the primes denote the differentiations with respect to the given argument. We restrict ourselves to the Kahler potential and the superpotential of the Polonyi model [12]: K = O® , W = + (3) , (7.26) with the parameters | and (. Our model includes the single-field (C) inflationary model, whose D-type scalar potential is given by V (C) = g2 (J ' )2 (7.27) in terms of arbitrary function J(C), with the real inflaton field C belonging to a massive vector supermultiplet. The Minkowski vacuum conditions (after inflation) can be easily satisfied when J ' = 0, which implies [12] (A) = V3 — 1 and ( = 2 — V3 . (7.28) This solution describes a stable Minkowski vacuum with spontaneous SUSY breaking at arbitrary scale (F) = The related gravitino mass is given by m3/2 = |e2-73+J . (7.29) There is also a complex (Polonyi) scalar of mass Ma = 2|ie2-V3 > 2m3/2 (7.30) and a massless fermion in the physical spectrum. The inequality in Eq. (7.30) is saturated in the original Polonyi model [12] but it is not the case in our model when (J) < 0. As regards the early Universe phenomenology, our model has the following theoretically appealing features: • there is no need to "stabilize" the single-field inflationary trajectory against scalar superpartners of inflaton, because our inflaton is the only real scalar in a massive vector multiplet, • any values of CMB observables ns and r are possible by choosing the J-function, • a spontaneous SUSY breaking after inflation occurs at arbitrary scale • there are only a few parameters relevant for inflation and SUSY breaking: the coupling constant g defining the inflaton mass, g ~ minf., the coupling constant I defining the scale of SUSY breaking, | ~ m3/2, and the parameter ( in the 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity 155 constant term of the superpotential. Actually, the inflaton mass is constrained by CMB observations as minf. ~ 0(10-6), while |3 is fixed by the vacuum solution, so that we have only one free parameter ^ defining the scale of SUSY breaking in our model (before studying reheating and phenomenology). The D-type scalar potential associated with the Starobinsky inflationary model of (R + R2) gravity arises when [9] 3 J(C) = 2 (C - ln C) (7.31) that implies J'(C) = - (1 - C-1) and J"(C) = - (C-2) >0. (7.32) According to (7.24), a canonical inflaton field ^ (with the canonical kinetic term) is related to the field C by the field redefinition C = exp . (7.33) Therefore, we arrive at the (Starobinsky) scalar potential VStar.W = ^f2 (l - e-^*)2 with m2f. = 9g2/2 . (7.34) The full action (7.20) of this PS supergravity in curved superspace can be transformed into a supergravity extension of the (R + R2) gravity action by using the (inverse) duality procedure described in Ref. [9]. However, the dual super-gravity model is described by a complicated higher-derivative field theory that is inconvenient for studying particle production. Another nice feature of our model is that it can be rewritten as a supersym-metric (abelian and non-minimal) gauge theory coupled to supergravity in the presence of a Higgs superfield H, resulting in the super-Higgs effect with simultaneous spontaneous breaking of the gauge symmetry and SUSY. Indeed, the U(1) gauge symmetry of the original Lagrangian (7.20) allows us to choose a different (Wess-Zumino) supersymmetric gauge by "gauging away" the chiral and anti-chiral parts of the general superfield V via the appropriate choice of the su-perfield parameters Z and Z. Then the bosonic part of the Lagrangian in terms of the superfield components in the Einstein frame, after elimination of the auxiliary fields and Weyl rescaling, reads [11] e-1 L = - 1r - Kaa* 3mA3mA - 4FmnFmn - 2Jhfv3mh3mh - 1 Jy? BmBm + iBm(Jvh3mH - Jvh3mH) - V , (7.35) where h, h are the Higgs field and its conjugate. The standard U(1) Higgs mechanism arises with the canonical function J = 2 he2Vh, where we have chosen g = 1 for simplicity. As regards the Higgs sector, it leads to e-1 LHiggs = -3mh3mh + iBm(h3mh - h3mh) - hhBmBm - V . (7.36) 156 S.V. Ketov and M.Yu. Khlopov After changing the variables h and h as h = — (p + v)eiZ, h = — (p + v)e-iZ , (7.37) where p is the (real) Higgs boson, v = (h) = (h) is the Higgs VEV, and Z is the Goldstone boson, the unitary gauge fixing of h —» h' = e-iZh and Bm —» Bm = Bm + 3mZ, leads to the standard result 11 e-1LHiggs = - = 3mp3mp - = (p + v)2BmBm - V . (7.38) The Minkowski vacuum after inflation can be easily lifted to a de Sitter vacuum (Dark Energy) in our model by the simple modification of the Polonyi sector and its parameters as [11] (A) =(-3-,)+3T—=-T)6+O(62), 15 =(2--3)+Jt7)6+O(62), (739) where 6 is a very small deformation parameter, 0 < 5 C ,. It leads to a positive cosmological constant Vo = 6 = m2/26 (7.40) and the superpotential VEV (W) = ^((A) + 5) = ^a + b - =6) , (7.41) where a = ( %/3 -, ) and b = (2 - %/3) provide the SUSY breaking vacuum solution to the Polonyi parameters in the absence of a cosmological constant. The full scalar potential (7.25) is a sum of the D- and F-type terms, while there is a mix of the inflaton - and Polonyi-dependent terms in the F-type contribution. This mixing leads to instability of the (Starobinsky) inflationary trajectory that is supposed to be driven by the D-term only. This issue was resolved in Ref. [35] where a modification of the original PS supergravity action (7.20) was proposed via adding the generalized Fayet-Iliopoulos term and modifying the J-function (7.31). 7.4 Super heavy gravitino dark matter The complete set of equations of motion in our supergravity model (Sec. 3) is very complicated. In this section, we consider only the leading order with respect to the inverse Planck mass. In addition, we neglect the coupling of Polonyi and gravitino particles to the inflaton, and introduce the effective action of the Polonyi field in the Friedmann-Lemaitre-Robertson-Walker (FLRW) background (in comoving coordinates) as I[A] = dt d3Xy (a2 - (VA)2 - MAa2 - ZRA2 ) , (7.42) 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity 157 where the non-minimal coupling constant of the Polonyi field to gravity is equal to Z = 1, A is the Polonyi field, MA stands for its mass, R is the Ricci scalar, and a is the FLRW scale factor. The mode decomposition of the Polonyi field reads A(x) = d3k(2n)-3/2a-1(n) bkhk(n)eikx + bkhk(n)e-ikx , (7.43) where the conformal time coordinate n is introduced, b,b are the (standard) creation/annihilation operators, and the coefficient functions h, h+ are normalized as follows: hkhk* - hkhk = i . (7.44) Because of Eqs. (7.42) and (7.43), the equation of motion of the modes is a " hk(n) + ^k(n)hkin) = 0, where w2 = 5 — + k2 + Mla2 , (7.45) and h" = d2h/dn2. Equation (7.45) can be conveniently rescaled by using some reference scales a(n*) = a* and H(n*) = H* as follows: hk'(n ) + (k2 + b2 a2)hk (n )= 0, (7.46) in terms of the rescaled quantities n = na*H* , a = a/a* , k = k/(H*a*) . The leading order of the gravitino action coincides with the massive RaritaSchwinger action, IM = d4xei|> ffRff(M}, (7.47) where the gravitino kinetic operator has been introduced as = m3/2YffVMv + iYffVpDvMp , (7.48) and the supercovariant derivative is D^v = -rPvMp + + Mv , (7.49) in the Y-notation yMi '''Mn = Y[mi ....YMn]. Since the supergravity torsion is of the second order with respect to the inverse Planck mass, we ignore it in the leading order approximation. The f£v can be represented by the standard symmetric Christoffel symbols that are actually cancelled from the Rarita-Schwinger action (7.47). The Rarita-Schwinger action leads to the gravitino equation of motion, (iD - m3/2)M M - (iDM. + m2/2Y M) Y ■ M = 0 . (7.50) In the flat FLRW background, Eq. (7.50) reduces to iYmn3mMn = - (m3/2 + i^Y°) Ym3mM , (7.51) 158 S.V. Ketov and M.Yu. Khlopov where w^b = 2aa-1e^[aeb] , = a(n)6£ , m.3/2 = m3/2(n) . (7.52) A solution to Eq. (7.51) is ^ (x)=J d3p(2n)-3(2p0)-1 (n,A)akA(n) + e-ikxbC(n, A)akA(n)} . A (7.53) We find that the equations of motion for the 3/2-helicity gravitino modes have the same form as that of Eq. (7.45), namely, b"(n,A) + C(k,a)b^ (n,A) + w2(k,a)b^(n,A) = 0, (7.54) where we have introduced the notation a ' Ô(k,a)b^(n,A) = -2iYVikiYvn3nb^ - 2Yv(m3/2 + i-Y^V^b^ , (7.55) w2 (k, a)/2 = k2 + m3/2 + y0ma/2 - (. (7.56) Following a procedure similar to the standard one in the case of Dirac and KleinGordon equations, we can reformulate the mode equations of motion in our case as Pv Pvb^(n,A) = 0, (7.57) where we have introduced the projector operator Pv = iYvn - Yviki - (ma/2 + i^y^ Yv = 0. (7.58) The dynamics of the gravitino and Polonyi fields during inflation necessary lead to their quantum production. The number density of produced particles can be calculated by using a Bogoliubov transformation, hk1 (n) = akhk° (n) + PkK110 (n). (7.59) This transformation is performed from the vacuum solution selected by the boundary conditions at n = nin, corresponding to the initial time of inflation, to the final time n = nf, when the particles creations process from inflation stops. In the inflationary epoch, the dynamical regime is a '/a2 C MPl and MPl ba/k C 1. This implies that we can consider the extremes as nin = -œ and nf = +00, performing a WKB semiclassical approximation. By assuming these boundary conditions, the energy density of the Polonyi particles produced during inflation reads 1 pA(n) = MAnA(n) = MAH3f ( an) ) Pa , (7.60) where pa = i diek2|pk|2 . (7.61) 0 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity 159 The inflaton mass sets the characteristic energy scale for the Hubble constant, calculated at fixed cosmological time t = tf : H2(tf) ~ m2, p(tf) ~ m2M2!. We propose the following formula for Polonyi particles (energy-density and Polonyi mass) produced during inflation [36]: t,2Ï 8n/MAA ATrehA nA(tf) (QA-2/Q«h2) = f^ -^fefj MPf, (7-62) where MA is the Polonyi mass, HrH2 ~ 4.31 x 10-5 is the radiation energy density at today's temperature To, HAh2 is the energy density of the produced Polonyi fields, all in the units of the critical energy density. There is about 8th-orders-of-magnitude suppression of the energy density. The normalized power spectrum PA cannot provide such suppression with our values for MA and Hinf. However, it comes from the dilution factor (a.)-3 = (af/at)-3 in Eq. (7.60). To get the gravitino and Polonyi masses, we have to add a few cosmologi-cal assumptions about the relevant parameters of the reheating process and, in particular, about the reheating temperature Treh. The cosmological parameters can be fixed by specifying the e-foldings number N e in the range between 50 and 60. For a more precise estimate of the CDM abundance, we choose Ne = 55, as in Sec. 2. This implies ns = 0.964, r = 0.004, minf = 3.2 • 1013 GeV and Hinf = nM^Pg/2 = 1.4 • 1014 GeV. In our scenario, well below the inflaton mass scale the low-energy effective field theory is given by the Standard Model (SM) that has the effective number of d.o.f. as g* = 106.75. It is reasonable to assume that all the SM particles originated from perturbative inflaton decay via the (Starobinsky) universal reheating mechanism, whose reheating temperature is known [37,38]: ( 90 \ 1/4 Treh = V^totMp = 3 • 109 GeV. (7.63) On the other hand, the reheating temperature for heavy gravitino is given by [39] Treh = 1.5 • 108 GeV (*0f y^f2. (7.64) Combining Eqs. (7.63) and (7.64) we get the gravitino and Polonyi masses as follows: m3/2 = (7.7 ± 0.8) • 1012 GeV and MA = 2e-(I>m3/2 > 2m3/2 . (7.65) 7.5 Primordial Black Holes in supergravity PBHs may be formed in the early Universe by collapse of primordial density perturbations resulting from inflation, when these perturbations re-enter the horizon and are large enough, i.e. when gravity forces are larger than pressure, in general. 160 S.V. Ketov and M.Yu. Khlopov Apart from being considered as another (non-particle) source for DM, some PBHs (of stellar mass type) are also considered as the candidates for the gravitational wave effects caused by the binary black hole mergers observed by LIGO/Virgo collaboration [40,41]. The PBH mass Mpbh is related to the perturbations scale k by Carr's formula [42] *„ 4nH-3 ( y \( g* \-1 ( k/(2n) V2 Mpbh = Yp — k M0 {^{f^ (JT^j , (7.66) whose coefficient y = 3-3/2 k 0.2, the (normalized) energy density is almost equal to the (normalized) entropy density g* k 3.36, and M0 stands for the Solar mass, M0 k 2 x 1033 g. The PBHs abundance f = nPBH/nc is proportional to the amplitude of the scalar perturbations Pz, while for the LIGO events one finds k/(2n) ~ 10-9 Hz, Pz ~ 10-2 and f ~ 10-2, as the regards the orders of their magnitudes [40,41]. The value of 10-9 Hz corresponds to 106 Mpc-1. In a single-field inflation, relevant perturbations are controlled by inflaton 2 2 scalar potential, so that large fluctuations PR k K- (2^) are produced when the slow roll parameter £ = r/16 goes to zero, i.e. when the potential has a near-inflection point where V' k V" k 0 . (7.67) Since we want a copious PBH production along with observationally consistent CMB observables, we should "decouple" these events, and demand the existence of another ("short") plateau in the scalar potential after the inflationary plateau towards the end of inflation. This is not the case for the Starobinsky inflation with the scalar potential (7.14), however, it can be easily achieved in a more general framework. Our supergravity framework in Sect. 3 is an example of such framework, because it leads to a single-field inflation governed by arbitrary function J, so that the associated inflaton scalar potential is given by V = g- (J')2. As an example, let us consider the inflaton scalar potential r^ = (1 + £ - e-a* - £,e-13*2) 2 , (7.68) Vo V / which is a deformation of the Starobinsky potential (7.14) with a = \J2/3 and the new real parameters |3 > 0 and £ > 0. The Starobinsky potential (7.14) is recovered when £ = 0. The scalar potential (7.68) falls into our supergravity framework, has Minkowski minimum at ^ = 0 and the inflationary plateau for large positive But, in addition, it also has an inflection point in the "waterfall" region between the inflationary plateau and the Minkowski vacuum. Indeed, the conditions (7.67) result in two equations, ae-a* + 2£p^e-p*- = 0 (7.69) and a2e-a* - 2£pe-p*- + 4£pVe-**2 = 0 , (7.70) 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity 161 respectively. They imply a quadratic equation on a^ + 1 - 2p^2 = 0, (7.71) whose solution is given by a + Va2 + 4p =-4p-> 0 . (7.72) Then the remaining condition above is solved by 4 = 2M+ ' (773) Of course, there are many other possibilities to choose the scalar potential having the form of a real function squared. We just showed that it is possible to combine a viable (Starobinsky-like) inflation with a viable (stellar mass type) PBHs production in the context of supergravity. 7.6 Conclusion Our results lead to the intriguing unifying picture of CDM, dark energy (positive cosmological constant) and cosmological inflation, in which their parameter spaces are linked to each other. This scenario also suggests the interesting phenomenology in the ultra high energy cosmic rays: super heavy Polonyi particles may decay into the SM particles, as the secondaries, in top-bottom decays. Cosmological high energy neutrinos from the primary and secondary decay channels can be tested by IceCube and ANTARES experiments. Another interesting outcome is that some (stellar mass type) PBHs remnants produced from the supergravity fields can compose part of the CDM halo coexisting with gravitinos. In this scenario, gravitational wave signals from the PBHs mergers can be envisaged, with intriguing implications for LIGO/VIRGO experiments. In short, gravitational wave experiments may provide us with precious indirect information about the scalar sector of the inflationary supergravity. Finally, the intriguing possibility exists for a unification of the inflaton in the vector multiplet, and the SUSY GUTs such as the flipped SU(5) x U(1) model arising from (Calabi-Yau) compactified heterotic superstrings or the intersecting D-branes. Acknowledgements The work by SVK on gravity and supergravity is supported by the Competitiveness Enhancement Program of Tomsk Polytechnic University in Russia. This work is also supported by a Grant-in-Aid of the Japanese Society for Promotion of Science (JSPS) under No. 26400252, and the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. 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