Large-scale magnetic fields, non- Gaussianity, and gravitational - PowerPoint PPT Presentation

Large-scale magnetic fields, non- Gaussianity, and gravitational waves from inflation Reference: Physical Review D 91, 043509 (2015) [arXiv:1411.4335 [astro-ph.CO]] Sakata Memorial KMI Workshop on “Origin of Mass and Strong Coupling Gauge Theories” (SCGT15) Sakata-Hirata Hall, Nagoya University, Nagoya, Japan 5th March, 2015 Presenter : Kazuharu Bamba ( LGSPC, Ochanomizu University )

2 I. Introduction ・ Galactic magnetic fields ø ö G [Sofue et al ., Annu. Rev. Astron. Astrophys. 24, 459 (1986)] ・ Magnetic fields in clusters of galaxies 0 : 1 à 10 ö G 10kpc à 1Mpc [Clarke et al ., Astrophys. J. 547, L111 (2001)] Recent reviews (examples) * [Kandus et al ., Phys. Rep. 505, 1 (2011)] [Yamazaki et al ., Phys. Rep. 517, 141 (2012)] [Maleknejad et al ., Phys. Rep. (2013)] [Durrer and Neronov, Astron. Astrophys. Rev. 21, 62 (2013)]

3 Origin of cosmic magnetic fields 1 ． Astrophysical process: Plasma instability [Biermann and Schlüter, Phys. Rev. 82, 863(1951)] [Weibel, Phys. Rev. Lett. 2, 83 (1959)] [Hanayama et al ., Astrophys. J. 633, 941 (2005)] [Fujita and Kato, Mon. Not. Roy. Astron. Soc. 364, 247 (2005)] 2 ． Cosmolosical processes: ・ Electroweak/Quark-hadron phase transitions [Baym, Bödeker and McLerran, Phys. Rev. D 53, 662 (1996)] [Quashnock, Loeb and Spergel, Astrophys. J. 344, L49 (1989)] ・ Density perturbations [Ichiki et al ., Science 311, 827 (2006)] [Kobayashi, et al , Phys. Rev. D 75, 103501 (2007)]

4 Origin of cosmic magnetic fields (2) ・ Coherence scale ・ Strength It is difficult to obtain the results to explain the observations. The most natural origin of large-scale magnetic fields: Electromagnetic quantum fluctuations generated at the inflationary stage

5 Inflationary cosmology [Sato, Mon. Not. Roy. Astron. Soc. 195, 467 (1981)] [Guth, Phys. Rev. D 23, 347 (1981)] [Starobinsky, Phys. Lett. B 91, 99 (1980)] [Linde, Phys. Lett. B 108, 389 (1982)] [Albrecht and Steinhardt, Phys. Rev. Lett. 48, 1220 (1982)] Inflation: Exponential cosmic acceleration in the early universe 1. Homogeneity, Isotropy, and Flatness 2. Primordial density perturbations

6 Obstacle ・ Friedmann-Lema tre-Robertson-Walker (FLRW) metric is conformally flat. ・ The Maxwell theory is conformally invariat. The conformal invariance has to be broken at the inflationary stage.

7 Breaking mechanisms 1. Coupling of a scalar field to electromagnetic fields [Ratra, Astrophys. J. 391, L1 (1992)] [KB and Yokoyama, Phys. Rev. D 69, 043507 (2004); 70, 083508 (2004)] Non-minimal coupling of electromagnetic 2. fields to gravity [Turner and Widrow, Phys. Rev. D 37, 2743 (1988)] [KB and Sasaki, JCAP 0702, 030 (2007)] 3. Trace anomaly [Dolgov, Phys. Rev. D 48, 2499 (1993)]

8 Motivation and Purpose (1) By comparing the theoretical predictions of a toy model with observations, we obtain phenomenological implications on moduli inflation.

9 Motivation and Purpose (2) (a) Coupling of the electromagnetic field Model to a scalar field (b) That to a pseudo scalar field (inflaton) Both couplings: Novel point In the past works: Only (b) * E.g.: [Barnaby and Peloso, Phys. Rev. Lett. 106, 181301 (2011)] [Barnaby, Namba and Peloso, JCAP 1104, 009 (2011)] Observables (i) Large-scale magnetic fields (ii) Non-Gaussianity (iii) Tensor-to-scalar ratio

10 II. Model Gravity term Breaking of the conformal invariance : Reduced Planck mass R : Scalar curvature

11 Model (2) Ð Coupling of a scalar field to the electromagnetic field õ : Normalization constant ; U (Ð) : Potential of Ð F ö U(1) Y ; : gauge field

12 Model (3) Y Coupling of a pseudo scalar field (inflaton) to the electromagnetic field g ps : Dimensionless constant à ö÷ F : Dual tensor of F ö÷ M : Mass scale ö V : Normalization Y : Potential of constant (Feature of moduli inflation) m Y : Mass of

13 Equations of motion (EoM) (1) Flat FLRW space-time a ( t ) : Scale factor Ð EoM for ： Hubble parameter Y EoM for (inflaton)

14 Equations of motion (EoM) (1) U(1) Y EoM for the gauge field F 0 = * : Coulomb gauge Breaking of the conformal invariance : Totally antisymmetric tensor

15 Slow-roll inflation : Hubble parameter at the inflationary stage a k t = t k : Value of the scale factor at t k ： Time when a comoving wavelength of the U(1) Y gauge field first crosses the horizon during inflation

16 U(1) Y Quantization of gauge field ù 0 = 0 ; Canonical momenta : Comoving Canonical commutation relation wavenumber k = ฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀ ： ; Annihilation and Normalization condition creation operators

17 U(1) Y Circular polarization of gauge field ・ axis to lie along the spatial momentum direction Transverse directions ( ) ・ Circular polarization: : Normalized amplitude

18 C + ( k; t ) Evolution of (1) C + ( k; t ) ： H inf t ø 10 End of After , C + ( k; t ) inflation approaches a constant: For the case of (Dotted line) For the case of dynamical Y (Solid line) Fig. 1 H inf t

19 C + ( k; t ) Evolution of (2) Values of parameters for Fig.1 ; ; ; ;

20 C + ( k; t ) Evolution of (3) ・ COBE normalization (Amplitude of power spectrum of the curvature perturbations) k = 0 : 002Mpc à 1 at ・ Planck result (Spectral index of power spectrum of the curvature perturbations) k = 0 : 05Mpc à 1 (68% CL) at [Ade et al . [Planck Collaboration], Astron. Astrophys. 571, A22 (2014)] Slow-roll parameters: à ; ・ : Friedmann equation

21 III. Current strength of the magnetic fields Proper magnetic field t = t R ・ Instantaneous reheating at û c ý H ・ After reheating, . û c : Cosmic conductivity B proper / a à 2

22 Current strength of the magnetic fields (2) Tachyon instability / ( X ( t k )) à 1 ø k ñ ø ( t = t k ) ; : Comoving scale : After inflation, the Maxwell theory * is recovered.

23 Current strength of the magnetic fields (3) Current strength of magnetic fields on the H à 1 Hubble horizon scale : 0 t 0 : Present time 0 ;t 0 ) ø 10 à 64 G B ( H à 1 For Fig. 1

24 IV. Non-Gaussianity of curvature perturbations ・ Perturbed Einstein equation Metric perturbation = Matter density perturbation (1) Scalar mode (Curvature perturbation) Temperature perturbation of the cosmic microwave îT ' 10 à 5 background (CMB) radiation: T (2) Vector mode (3) Tensor mode (Primordial gravitational wave)

25 Non-Gaussianity of curvature perturbations (2) ・ U(1) Y F ö We suppose that the gauge field ' couples to another Higgs-like field , which develops a vacuum expectation value, through the kinetic term . [Meerburg and Pajer, JCAP 1302, 017 (2013)] : Covariant derivative for ' g 0 : Gauge coupling The gauge symmetry can spontaneously be broken, so that the gauge field acquires its mass.

26 Non-Gaussianity of curvature perturbations (3) ・ The number of e -folds of inflation could ' be changed by the perturbations of . ・ The curvature perturbations can be ' generated through the perturbations of . The local-type non-Gaussianity in terms of the curvature perturbations is produced. îN formalism * [Sasaki and Stewart, Prog. Theor. Phys. 95, 71 (1996)] [Starobinsky, JETP Lett. 42, 152(1985)] [Lyth and Rodriguez, Phys. Rev. Lett. 95, 121302 (2005)]

27 Non-Gaussianity of curvature perturbations (4) Quantity showing the non-Gaussianity [Ade et al . [Planck Collaboration], Astron. Astrophys. 571, A24 (2014)] : Gravitational potential with the Gaussian statistical property

28 Non-Gaussianity of curvature perturbations (5) g 0 : Gauge coupling [Meerburg and Pajer, JCAP 1302, 017 (2013)] ø k = 2 : 5590616 Planck result (68% CL) [Ade et al . [Planck Collaboration], arXiv:1502.01592]

29 V. Tensor-to-scalar ratio Power spectrum of the curvature perturbation / k n s à 1 n s : Spectral index Tensor-to-scalar ratio Power spectrum of the tensor mode r TS ñ Power spectrum of the scalar mode

30 Tensor-to-scalar ratio (2) ; [Barnaby and Peloso, Phys. Rev. Lett. 106, 181301 (2011)] [Barnaby, Namba and Peloso, JCAP 1104, 009 (2011)] Planck result k = 0 : 002Mpc à 1 r TS < 0 : 11 (95% CL) at [Ade et al . [Planck Collaboration], arXiv:1502.02114] Cf. [Ade et al . [Planck Collaboration], arXiv:1502.00612]

31 VI. Conclusions ･ We have studied the generation of the large- scale magnetic fields from a kind of moduli inflation. ･ We have first estimated the explicit values of three cosmological observables such as the current magnetic fields on the Hubble horizon, local non-Gaussianity, and the tensor-to-scalar ratio. ･ The local non-Gaussianity and tensor-to-scalar ratio obtained in this model are consistent with the Planck results.

32 Acknowledgments ･ The presenter would like to sincerely appreciate the significant discussions with Professor Tatsuo Kobayashi and Professor Osamu Seto and their kind important suggestions and comments.

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