Cosmologia è Geometria

1. Cosmologia è Geometria an overview of the relevantcontributions of Peppe Marmo to Cosmology Cosimo Stornaiolo INFN – sezione di Napoli San Rufo 11-13 July 2016

2. Contributions to cosmology • Noethersymmetries in cosmology • Bianchi groups • Quantum cosmology in tomographicapproach • Discussions and suggestions

3. Noethersymmetries • Motivations • Inflation • Slow roll vs exactsolutions

4. Experimental status of inflationarymodels • Recentobservationalresults (Planckmission) rule out 91% of the slow rollinflationarymodels • Onlythose with a potential with a plateau • Need a similarcheck for exactmodels

5. see for example

6. 111 citations! • Elencare i membri principali della scuola (preside, bibliotecario e così via) • Includere un organigramma

8. The first twoequations are a dynamicsystemassociated to the vector

10. The potentialis

11. Constant of motion

13. We need to consider all the cases after imposing • For the first two cases (with sin and sinh) we have and • For the exp case we have

18. Solution with power law inflation

19. Solution with exponentialinflation

20. New approach to findexactsolutions for cosmologicalmodels with a scalar field. R. de Ritis,G. Marmo, G. Platania, C. Rubano, P. Scudellaro, C. Stornaiolo, Phys.Rev.D 42, 1091-1097 (1990)TOPCITE = 100+ • New exactsolutions of cosmologicalequations with considerationsupon the cosmologicalconstant. R. de Ritis, G. Marmo, G. Platania, C. Rubano, P. Scudellaro, C. Stornaiolo, Phys.Lett.A149, 79-83 (1990) • Scalar field, nonminimalcoupling, and cosmology, M. Demianski, R. de Ritis, G. Marmo, G. Platania, C. Rubano, P. Scudellaro, C. Stornaiolo Phys.Rev.D 44 3136-3146 (1991)

21. Bianchi groups

28. Papers • V.I Man’ko, G. Marmo and C.S.‘’Radon Transform of the Wheeler-De Witt equation and tomography of quantum states of the universe’’ Gen. Relativ. Gravit. (2005) 37: 99–114 • V.I Man’ko, G. Marmo and C.S. ‘’Cosmological dynamics in tomographic probability representation’’ Gen.Rel.Grav.37:2003-2014, 2005 • V.I Man’ko, G. Marmo and C.S. ‘‘Tomographic entropy and cosmology’’ Gen.Rel.Grav.40:1449-1465, 2008 • S. Capozziello, V.I Man’ko, G. Marmo and C.S. ‘ • Tomographic Representation of Minisuperspace Quantum Cosmology and Noether Symmetries’’. Gen. Rel. Grav. 40 (2008) 2627-2647 • S. Capozziello, V.I Man’ko, G. Marmo and C.S “A Tomographicdescription for classical and quantum cosmologicalperturbations. ”Phys.Scripta 80 (2009) 045901

29. Main purposes • New formulation of quantum cosmology • Reconstruction of the initialconditions • Study of the properties of the quantum potential in quantum cosmology • Tomographicformulation of classicalcosmology.

34. The Hartle-Hawking Wignerfunction

35. The Hartle-Hawking tomogram

40. Wheeler-De Witt tomographic equation • Similarlyto Quantum Mechanics , we can define a correspondingequationfor the Wheeler-DeWittequation

41. ClassicalTomograms Itispossibletodefineclassicaltomograms, asfunctions on phasespace, ifinsteadof the Wignerfunctiononeconsiders the classicalsolutionof the Liouvilleequation. .

42. Propagators • The evolution of tomograms is described by the following propagatore, i.e. probability transition functions in the following way

43. The equation for the probability transition function for the harmonic oscillator • The probability transition function Π satisfies the following equation

44. The homogeneous and isotropic metric • Let us consider the metric • Which, by defining the conformal time takes the form

47. Examples of constrainedFLRW with and withoutcosmologicalconstant

48. Tomogram for a constrainedsystem