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...6.16) where we have taken Nk = 55, and neglected a factor raised to the power α, as α ∼ 10−2. The data we we use consists of the WMAP7 best-fit values for Ps(k0) and ns in the standard ΛCDM model are =-=[33]-=- Ps(k0) = (2.43 ± 0.11) × 10−9, ns = 0.963 ± 0.012(68%CL) (6.17) with an upper limit on r of [34] r < 0.36(95%CL). (6.18) The string contribution is small, so we can equate the F -term prediction for ...

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...rmer case also because of the novel nature of the inflationary regime in that case. 5 The scalar potential at large s As we have developed it, the theory naturally provides for F -term inflation [28]–=-=[30]-=-. Although we have a U(1)′, there is a crucial difference between our scenario and the original D-term inflation paradigm [31], which is that the SM fields are necessarily charged under U(1)′. Consequ...

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...ogenesis scenario described in the next section. 9.3 Baryogenesis Creation of the observed baryon asymmetry requires baryon number violation, departure from thermal equilibrium and C and CP-violation =-=[61]-=-. Our model has no conserved lepton numbers, and so it is natural to explore creation of the observed baryon number via leptogenesis. Now in AMSB, non-CKM CP-violating phases do not exist in the soft-...

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...g potential source (of CP violation) is, however, the Yukawa sector for right-handed neutrinos, which (with the standard see-saw mechanism for generating neutrino masses) is relevant for leptogenesis =-=[62]-=-. Successful supersymmetric leptogenesis requires that the lightest right-handed neutrino (and the post-inflation reheating temperature) be greater than 109GeV; note that because of the large gravitin...

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...he former case also because of the novel nature of the inflationary regime in that case. 5 The scalar potential at large s As we have developed it, the theory naturally provides for F -term inflation =-=[28]-=-–[30]. Although we have a U(1)′, there is a crucial difference between our scenario and the original D-term inflation paradigm [31], which is that the SM fields are necessarily charged under U(1)′. Co...

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...digm is not, however, founded on a compelling underlying theory and therefore it is worthwhile exploring other possibilities. – 1 – J H E P06(2011)115 In this paper we focus on Anomaly Mediation (AM) =-=[1]-=-–[3]. This is a framework in which a single mass parameter determines the φ∗φ, φ3 and λλ supersymmetry-breaking terms in terms of calculable and, moreover, renormalisation group (RG) invariant functio...

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... data we we use consists of the WMAP7 best-fit values for Ps(k0) and ns in the standard ΛCDM model are [33] Ps(k0) = (2.43 ± 0.11) × 10−9, ns = 0.963 ± 0.012(68%CL) (6.17) with an upper limit on r of =-=[34]-=- r < 0.36(95%CL). (6.18) The string contribution is small, so we can equate the F -term prediction for the scalar power spectrum to the WMAP measured value to find M2 λ1m2p ≃ 6× 10−6. (6.19) Assuming ...

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...ossibility of a Fayet-Iliopoulos term in our model, it behoves us to address issues raised by the recent papers on the connection of FI terms with supergravity, by Komargodski and Seiberg, and others =-=[17]-=-–[20]. The upshot of this work is the conclusion that a global theory with a FI term cannot be consistently embedded in supergravity. An exception would be allowed if the full supergravity theory has ...

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..., the scalar spectral index ns. The importance of the tensor power spectrum is often parametrised by r = 4Pt/Ps. In slow-roll single-field inflation these are given by the standard formulae (see e.g. =-=[32]-=-) Ps(k) ≃ 1 24π2 V m4p 1 ǫ ∣∣∣∣ Nk , Pt(k) ≃ 1 6π2 V m4p ∣∣∣∣ Nk , (6.1) ns ≃ (1− 6ǫ+ 2η)|Nk , r = 16ǫ|Nk , (6.2) where ǫ = m2p 2 ( V ′ V )2 , η = m2p ( V ′′ V ) , (6.3) and Nk is the e-fold at which ...

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...rio a detailed modelling of the decay cascades is required as the bounds on the mass scale are sensitive to the primary Standard Model decay products of the “X” particles into which the string decays =-=[48]-=-. Cosmic strings constitute a p = 1 TD (topological defect or top-down) [48] model, for which there is a upper bound on the energy injection rate from the low energy diffuse γ-ray background [48] Q0 ....

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...estigate this further in conjunction with the dark matter and cosmic ray constraints on strings. Finally, in the ξ 6= 0 case, our AMSB model also has the right conditions for AffleckDine baryogenesis =-=[68]-=-, in that inflation naturally generates large vevs for fields with baryon and lepton number through the minimisation of theD-term. A detailed investigation would involve numerical simulations of the d...

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...till purely cubic, and nevertheless dispense with the FI term. The U(1)′ breaking scale was then generated by dimensional transmutation in a way reminiscent of the geometric hierarchy model of Witten =-=[13]-=-. Here we return to an improved version of the original model of ref. [11], augmented by the introduction in the superpotential of a linear term for the gauge singlet field S. We will see that this in...

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...B models in general, and our model in particular, could produce the required neutralino abundance from the decays of thermally produced gravitinos, provided that the reheat temperature is high enough =-=[54]-=-–[56]. For m 3 2 ≃ 40TeV this is around 2 × 1010GeV. (For a Bayesian analysis of how the minimal AMSB scenario is constrained by other observables, see [57]). Our model also has cosmic strings, and ne...

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... is not, however, founded on a compelling underlying theory and therefore it is worthwhile exploring other possibilities. – 1 – J H E P06(2011)115 In this paper we focus on Anomaly Mediation (AM) [1]–=-=[3]-=-. This is a framework in which a single mass parameter determines the φ∗φ, φ3 and λλ supersymmetry-breaking terms in terms of calculable and, moreover, renormalisation group (RG) invariant functions o...

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... Moreover, this R-symmetry forbids the quartic superpotential terms QQQL and UUDE, which are allowed by the U(1)′ symmetry, and give rise to dimension 5 operators capable of causing proton decay [23]–=-=[25]-=-. It is easy to see, in fact, that the charges in eq. (4.5) disallow B-violating operators in the superpotential of arbitrary dimension. Of course this R-symmetry is broken by the soft supersymmetry b...

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...1H2). Moreover, this R-symmetry forbids the quartic superpotential terms QQQL and UUDE, which are allowed by the U(1)′ symmetry, and give rise to dimension 5 operators capable of causing proton decay =-=[23]-=-–[25]. It is easy to see, in fact, that the charges in eq. (4.5) disallow B-violating operators in the superpotential of arbitrary dimension. Of course this R-symmetry is broken by the soft supersymme...

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...he tilt is about 2σ away from the best-fit value for single-field inflation. A small string contribution to the CMB power spectrum at the level of 5-10% restores or even slightly improves the CMB fit =-=[14, 35, 36]-=- although at the cost of a higher baryon fraction and a less steep dark matter power spectrum, putting the model into tension with other data [14]. 6.1 The ξ = 0 case We can perform a similar analysis...

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... that the reheat temperature is high enough [54]–[56]. For m 3 2 ≃ 40TeV this is around 2 × 1010GeV. (For a Bayesian analysis of how the minimal AMSB scenario is constrained by other observables, see =-=[57]-=-). Our model also has cosmic strings, and neutralinos will generically be produced by decays of particles radiated from them [58]. We note that this and other works (see for example [59, 60]) studying...

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...nown as scaling [37, 38], which means that they maintain a constant density parameter Ωs = ρs/ρc, where ρs is the string energy density and ρc = 3m2PH 2 is the critical density. Numerical simulations =-=[43]-=-–[45] indicate that there are O(1) or a few Hubble lengths of string per Hubble volume, so that ρs ∼ µ/t2 and Ωs ∼ Gµ. The string energy density is therefore decaying at a rate ρ̇s ∼ µ/t3. There are t...

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...onal radiation or into high energy particles of the fields from which the string is made. The first scenario is constrained by bounds on the stochastic background of gravitational radiation (see e.g. =-=[46]-=-), and the second by the flux of cosmic rays [47]. The first depends on the typical loop size relative to the Hubble length, α, and the second on the complex decay processes of the massive scalars, ga...

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...he tilt is about 2σ away from the best-fit value for single-field inflation. A small string contribution to the CMB power spectrum at the level of 5-10% restores or even slightly improves the CMB fit =-=[14, 35, 36]-=- although at the cost of a higher baryon fraction and a less steep dark matter power spectrum, putting the model into tension with other data [14]. 6.1 The ξ = 0 case We can perform a similar analysis...

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...10−7. (7.1) where µ is the string tension (not to be confused with the renormalisation scale µ appearing in eq. (2.3) or eq. (5.8), for example). The Unconnected Segment Model of string perturbations =-=[39]-=- gives a similar upper bound [40]. The simple F -term hybrid inflation model is more tightly constrained [14], as the string tension is related to the inflation scale. However, the AMSB model has more...

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... tension (not to be confused with the renormalisation scale µ appearing in eq. (2.3) or eq. (5.8), for example). The Unconnected Segment Model of string perturbations [39] gives a similar upper bound =-=[40]-=-. The simple F -term hybrid inflation model is more tightly constrained [14], as the string tension is related to the inflation scale. However, the AMSB model has more freedom, and we shall use the mo...

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...he fields from which the string is made. The first scenario is constrained by bounds on the stochastic background of gravitational radiation (see e.g. [46]), and the second by the flux of cosmic rays =-=[47]-=-. The first depends on the typical loop size relative to the Hubble length, α, and the second on the complex decay processes of the massive scalars, gauge bosons, and fermions of the string fields, he...

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... from the U(1)′ D-term are naturally of the same order of magnitude as the soft breaking terms, and can eliminate the tachyonic slepton problem. This scenario was first explored in any detail in ref. =-=[4]-=-, and subsequently by a number of authors [5]–[12]. In ref. [11] it was shown how the characteristic low energy theory can arise in a natural way from a U(1)′ with a Fayet-Iliopoulos (FI) D-term. The ...

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...ame order of magnitude as the soft breaking terms, and can eliminate the tachyonic slepton problem. This scenario was first explored in any detail in ref. [4], and subsequently by a number of authors =-=[5]-=-–[12]. In ref. [11] it was shown how the characteristic low energy theory can arise in a natural way from a U(1)′ with a Fayet-Iliopoulos (FI) D-term. The scale of the U(1)′ breaking was associated wi...

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...e large gravitino mass (around 40TeV) associated with AMSB there is no danger that the – 22 – J H E P06(2011)115 decay of gravitinos produced in this reheating will pose a problem for nucleosynthesis =-=[63]-=-. (For a recent discussion of some other ways of evading the gravitino bound see ref. [64]). There is also a source of leptogenesis through out-of-equilibrium decays of particles radiated by the strin...

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...pplied by the neutrino sector. Gravitinos are very massive, and so decay early enough not to be in conflict with nucleosynthesis. The model has the same field content as the FD hybrid inflation model =-=[15, 16]-=-, but different charge assignments and couplings. FD hybrid inflation also has a singlet which is a natural inflaton candidate, but differs in other ways: for example, right-handed – 2 – J H E P06(201...

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...ility of a Fayet-Iliopoulos term in our model, it behoves us to address issues raised by the recent papers on the connection of FI terms with supergravity, by Komargodski and Seiberg, and others [17]–=-=[20]-=-. The upshot of this work is the conclusion that a global theory with a FI term cannot be consistently embedded in supergravity. An exception would be allowed if the full supergravity theory has a cer...

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.... Thus the right-handed neutrinos acquire large masses, and an appropriate µ-term for the Higgs doublets is generated in the manner of the NMSSM (for a review of and references for the NMSSM see ref. =-=[27]-=-). At the minimum of V we have λ1vφvφ +M 2 ≈ 0, (4.8) ξ − qφv2φ + qφv2φ ≈ 0, (4.9) vs ≈ − hλ1vφvφ + ρM 2m 3 2 v2φ + v 2 φ (4.10) and we see that vs is O(λ1m 3 2 /(16π2)) if M ≫ √λ1ξ or if M ∼ √ λ1ξ, a...

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...t & Moss [14]. One should note that their inflation superpotential isWBGM = κS(ΦΦ̄−M2BGM), so that κ = λ1 andMBGM =M/ √ λ1, and that they take g ′ = 0.7 and qφ = 1. They take the string tension to be =-=[42]-=- eq. (A.4), with v2 = M2BGM. The best fit models have MBGM ∼ 5 × 1015GeV and κ ∼ 10−3–10−2. Comparison is not straightforward, as our model’s string tension receives contributions from the D-term eq. ...

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...he string tension by eq. (A.13). – 16 – J H E P06(2011)115 7 Cosmic microwave background string constraints The symmetry-breaking of the U(1)′ symmetry at the end of inflation produces cosmic strings =-=[37, 38]-=-, which are a source of gravitational perturbations and contribute to the cosmic microwave background fluctuations. The exact constraint depends on details of the modelling of the strings, but simulat...

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...iffuse γ-ray background [48] Q0 . 4.4 × 10−23h eV cm−3 s−1. (9.2) Assuming the massive particles decay into Standard Model particles with a non-zero branching fraction f , one can derive a bound [49]–=-=[52]-=- Gµ . 10−9x2∗f −1, (9.3) where x2∗ = µ/ρst2 parametrises the Hubble lengths of string per Hubble volume, with x∗ ∼ 0.3 [43]–[45]. The string gauge field couples to all Standard Model particles, so the...

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...els in general, and our model in particular, could produce the required neutralino abundance from the decays of thermally produced gravitinos, provided that the reheat temperature is high enough [54]–=-=[56]-=-. For m 3 2 ≃ 40TeV this is around 2 × 1010GeV. (For a Bayesian analysis of how the minimal AMSB scenario is constrained by other observables, see [57]). Our model also has cosmic strings, and neutral...

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...how the minimal AMSB scenario is constrained by other observables, see [57]). Our model also has cosmic strings, and neutralinos will generically be produced by decays of particles radiated from them =-=[58]-=-. We note that this and other works (see for example [59, 60]) studying dark matter production from strings make conservative assumptions about the amount of particle production by assuming that gravi...

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...ervables, see [57]). Our model also has cosmic strings, and neutralinos will generically be produced by decays of particles radiated from them [58]. We note that this and other works (see for example =-=[59, 60]-=-) studying dark matter production from strings make conservative assumptions about the amount of particle production by assuming that gravitational radiation dominates, so a re-calculation of the dark...

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...ted than the simple models in the literature. Inflation ends with a phase transition producing gauge cosmic strings, and hence tight constraints on the parameters from the Cosmic Microwave Background =-=[14]-=-. In common with other supersymmetry models with conserved R-parity, it naturally accommodates a Dark Matter candidate, which is mostly wino. Although it has been argued that in AMSB models the annihi...

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...rgy diffuse γ-ray background [48] Q0 . 4.4 × 10−23h eV cm−3 s−1. (9.2) Assuming the massive particles decay into Standard Model particles with a non-zero branching fraction f , one can derive a bound =-=[49]-=-–[52] Gµ . 10−9x2∗f −1, (9.3) where x2∗ = µ/ρst2 parametrises the Hubble lengths of string per Hubble volume, with x∗ ∼ 0.3 [43]–[45]. The string gauge field couples to all Standard Model particles, s...

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...he gravitino bound see ref. [64]). There is also a source of leptogenesis through out-of-equilibrium decays of particles radiated by the strings, along the lines of the scenario investigated in refs. =-=[65]-=-–[67] for B − L cosmic strings. It would be interesting to investigate this further in conjunction with the dark matter and cosmic ray constraints on strings. Finally, in the ξ 6= 0 case, our AMSB mod...

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...tude as the soft breaking terms, and can eliminate the tachyonic slepton problem. This scenario was first explored in any detail in ref. [4], and subsequently by a number of authors [5]–[12]. In ref. =-=[11]-=- it was shown how the characteristic low energy theory can arise in a natural way from a U(1)′ with a Fayet-Iliopoulos (FI) D-term. The scale of the U(1)′ breaking was associated with the scale of the...

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...order of magnitude as the soft breaking terms, and can eliminate the tachyonic slepton problem. This scenario was first explored in any detail in ref. [4], and subsequently by a number of authors [5]–=-=[12]-=-. In ref. [11] it was shown how the characteristic low energy theory can arise in a natural way from a U(1)′ with a Fayet-Iliopoulos (FI) D-term. The scale of the U(1)′ breaking was associated with th...

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... determined in accordance with eq. (2.2): hλ1 = −m 3 2 λ1 16π2 ( 3λ21 + 1 2 Trλ22 + 2λ 2 3 − 4q2φg′2 ) , (4.7) denoting the U(1)′ charge by g′. We also introduce a soft breaking term linear in s (see =-=[26]-=- for a discussion of linear terms in this context), and a Fayet-Iliopoulos term ξ for U(1)′. Our model thus has two explicit mass parameters M, √ ξ as well as the gravitino mass m 3 2 . It is easy to ...

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...e an approximate upper bound on B, B . 7× 10−7 ( 2λ1m 2 p M2 ) ≃ 0.2. (7.5) Hence the value of λ21/2q 2 φg ′2 has to be small in order for strings not to exceed the CMB bound. Using the approximation =-=[41]-=- B(β) ≃ 2.4/ ln(2/β) for β . 10−2, we find λ1√ 2qφg′ . 3× 10−3. (7.6) A Monte-Carlo fit in the simple F -term inflation model, using the numerically determined string tension, and taking into account ...

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...avitino bound see ref. [64]). There is also a source of leptogenesis through out-of-equilibrium decays of particles radiated by the strings, along the lines of the scenario investigated in refs. [65]–=-=[67]-=- for B − L cosmic strings. It would be interesting to investigate this further in conjunction with the dark matter and cosmic ray constraints on strings. Finally, in the ξ 6= 0 case, our AMSB model al...