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Geometric model of dark energy according to projected hyperconical universes

RBI-T-WINNING - Cosmology 2018 October 22-27, 2018. Dubrovnik, Croatia. Geometric model of dark energy according to projected hyperconical universes. Robert Monjo. Department of Algebra, Geometry and Topology, Complutense University of Madrid.

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Geometric model of dark energy according to projected hyperconical universes

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  1. RBI-T-WINNING - Cosmology 2018 October 22-27, 2018. Dubrovnik, Croatia Geometric model of dark energy according to projected hyperconical universes Robert Monjo Department of Algebra, Geometry and Topology, Complutense University of Madrid

  2. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes Contents I. Introduction II. General considerations III. Theoretical features IV. Compatibility V. Conclusions • A. Standard LCDM model • B. Arnowitt-Deser-Misner(ADM) formalism • C. Problems of the current theory & Motivation RESULTS & DISCUSSION

  3. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes I. Introduction A. Standard LCDM model Friedmann Equations Wr 8.4·10-5, Wm = 0.3089  0.0062, WL= 0.6911  0.0062, Wk 0

  4. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes I. Introduction B. Arnowitt-Deser-Misner (ADM) formalism ADM takes these as zero! Einstein-Hilbert action Field theories action ADM considers space-time of the universe is foliated into a family of space-like surfaces. Conjugate momenta lapse Hamiltonian constraint shift Auxiliar momenta spatial total det

  5. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes I. Introduction C. Problems of the current theory • Horizon problem: Why is the temperature isotropic in regions initially disconnected. • Flatness problem: Why is energy density approximately equal to the critical density. • Dark energy: What is the origin of dark energy • (Age of universe: Why is the age of the universe similar to the inverse of the Hubble parameter? Is the linear expansion compatible?) • Approximations: • General Relatity valid for cosmologic scales. -> Valid only for local scale? • Homogeneous and isotropic fluid, that is, Friedmann-Robertson-Walker (FRW). -> Are inhomogeneous metrics valid?

  6. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes I. Introduction C. Motivation The most simple manifold with linear expansion: Similar idea 1/H = t Dirac-Milne model (2012) [k < 0] Proper-time preservation: homogeneous case Local-time preservation: inhomogeneous case

  7. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes Contents I. Introduction II. General considerations III. Theoretical features IV. Compatibility V. Conclusions • Definition of hyperconicaluniverses • Hypothesis of local equivalences and projection • Considerations of compatibility

  8. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes II. General considerations A. Definition of hyperconical universes Linear expansion Coordinates: Choosing t := tX, the constraint is: If b0 < 1, St3 is a 3-spheroid If b0 > 1, St3 is a 3-hyperboloid Comoving observer (assuming b0 < 1) Wick rotation of u (complexification) or change of metric as: diag(-1, 1, 1, 1, -1)

  9. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes II. General considerations B. Hypothesis of local equivalences and projection Equivalence in time The local time of an observer in H4 is the same that in R1,3 Let be x0, x R1,3 two static points, x0 = (t0,0), x =(t,0), with t > t0> 0 Let be x0’, x’ H4 their extension, x0 = (t0,0, nt0), x =(t,0, nt) x0’’ = (t0,0, t/t0·nt0) k :=n– 2 Curvature

  10. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes II. General considerations B. Hypothesis of local equivalences and projection Spatial projection The local space of an observer in H4 is the same that in R1,3 Local-time preservation (Expansion Operator). Hh4 M  Rh1,4  Projection (locally conformal but global distortion) Rg1,3 Local extension

  11. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes II. General considerations B. Hypothesis of local equivalences and projection Spatial projection The local space of an observer in H4 is the same that in R1,3 That is, the spatial distance is given by a locally conformal projection Let be r’ the comoving distance in H4, the spatial distance measured by an observer is: where Distorted stereographic projection Inverse of distorted stereographic projection

  12. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes II. General considerations C. Considerations of compatibility Features of the model • Metric g • Ricci curvature R • Evolution: Arnowitt-Deser-Misner (ADM) equations Theoretical compatibility between the proposed model and LCDM • Equivalence between proper distance of both models. • Comparison of the Hubble parameter of both models. • Calculus with Mapple. Observational compatibility • 580 pairs of (z, mobs) from Type Ia supernovae (Sne Ia) • Supernova Cosmology Project (SCP) Union2.1 database • Minimisation of c2 , using R statistical language, for: • Location of the first CMB acoustic peak Obtaining value of a

  13. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes Contents I. Introduction II. General considerations III. Features of the model IV. Compatibility V. Conclusions • Metric • Redshift-distance relation • Ricci curvature • Arnowitt-Deser-Misner (ADM) equations

  14. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes III. Theoretical features A. Metric tensor Differential line element Metric tensor Change of coordinates

  15. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes III. Theoretical features B. Redshift-distance relation Redshift and comoving distance

  16. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes III. Theoretical features C. Ricci curvature locally k  1 K  0

  17. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes III. Theoretical features D. Arnowitt-Deser-Misner (ADM) equations ko = k = 1

  18. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes Contents I. Introduction II. General considerations III. Theoretical features IV. Compatibility V. Conclusions • Equivalent proper distance: Obtaining cosmological parameters • Observational compatibility from expansion

  19. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes IV. Compatibility A. Equivalent proper distance: Obtaining cosmological parameters

  20. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes IV. Compatibility A. Equivalent proper distance: Obtaining cosmological parameters

  21. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes IV. Compatibility A. Equivalent proper distance: Obtaining cosmological parameters

  22. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes IV. Compatibility A. Equivalent proper distance: Obtaining cosmological parameters Prediction for z <1.4 0.29 0.33 First CMB peak multipole =zmax Sound horizon Baryon-to-photon density ratio,

  23. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes IV. Compatibility Accelerates always B. Observational compatibility from expansion “Equilibrium” region: Decelerated in the past and accelerates now  Linear always ?? Decelerates always Observation for z < 1.4 k = 1 aa = 0.30 ± 0.01 (c02 = 562) ba = 0.36 ± 0.02 (c02 = 562)

  24. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes Contents I. Introduction II. General considerations III. Theoretical features IV. Compatibility V. Conclusions

  25. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes V. Conclusions • We have considered an inhomogeneous model with linear expansion that does not depends on the matter content. • The Hyperconical model is consistent with k1, and thus freedom for fitting is given by (locally conformal) spatial projections fa. • However, there exists a unique local projection fa(r << t)compatible with the LCDM model (that is, where model provides real values). • Thanks to this compatibility, the Hyperconical model predicts that  L = 0.6937181(2) and m = 0.306192(6) for to  1  k and r = (9.0 ± 0.5) ·10-5, which is compatible with the Planck Mission results (L= 0.6911 ± 0.0062 and m = 0.3089 ± 0.0062). • Prediction distortion parameter (a) is in agreement with the fitted local and regional values. Moreover, they obtain the same statistical significance that theLCDM model.

  26. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes V. References Description of the inhomogeneous model used: R. Monjo, Phys. Rev. D, 96, 103505 (2017). doi:10.1103/PhysRevD.96.103505. arXiv:1710.09697 Geometric interpretation of the dark energy: R. Monjo, Phys. Rev. D, 98, 043508 (2018). doi:10.1103/PhysRevD.98.043508. arXiv:1808.09793

  27. Robert Monjo: Geometric model of dark energy according to projected hyperconical universes Questions Thanks to Antonio López-Maroto and RutwigCampoamor for their feedback, and Thanks for your attention email: rmoncho@ucm.es

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