11

Polinômios de Legendre e Polinômios de Legendre e Harmônicos EsféricosHarmônicos Esféricos

Miguel QuartinMiguel QuartinInstituto de Física, UFRJInstituto de Física, UFRJ

Astrofísica, Relativ. e Cosmologia (ARCOS)Astrofísica, Relativ. e Cosmologia (ARCOS)

Curso de Eletro I – 2019/2Curso de Eletro I – 2019/2

22

As we have seen, a function As we have seen, a function ff in spherical coordinates with in spherical coordinates with azimuthal symmetry can be expanded in a sum of azimuthal symmetry can be expanded in a sum of Legendre Polinomials PLegendre Polinomials Pℓℓ(cos(cosθθ))

Without azimuthal symmetry, we need to expand Without azimuthal symmetry, we need to expand ff in in spherical harmonicsspherical harmonics Y Yℓmℓm

The PThe Pℓmℓm are called are called associated Legendre Polinomialsassociated Legendre Polinomials

Spherical Harmonics and theSpherical Harmonics and theCosmic Microwave BackgroundCosmic Microwave Background

33

In science, the best example of the use of spherical In science, the best example of the use of spherical harmonics is in the analysis of the Cosmic Microwave harmonics is in the analysis of the Cosmic Microwave Background (CMB)Background (CMB)

Summary of the CMB:Summary of the CMB: The oldest light in the universe, emitted when the universe The oldest light in the universe, emitted when the universe

was only 380000 years oldwas only 380000 years old Now it has 13.8 billion years of ageNow it has 13.8 billion years of age

almost isotropic (up to 1 part in 10almost isotropic (up to 1 part in 1055) radiation field) radiation field blackbodyblackbody spectrum (the best one we have ever seen), with spectrum (the best one we have ever seen), with

microwave radiation (~ 50 – 400 GHz)microwave radiation (~ 50 – 400 GHz)

Spherical Harmonics and theSpherical Harmonics and theCosmic Microwave BackgroundCosmic Microwave Background

44

55

Temperature FluctuationsTemperature Fluctuations

But CMB has small (1 in 10But CMB has small (1 in 10–5–5) ) temperature fluctuationstemperature fluctuations

These fluctuations tell us about how the early universe These fluctuations tell us about how the early universe departed from perfect homogeneity & isotropydeparted from perfect homogeneity & isotropy To analyze them spherical harmonic decomposition→To analyze them spherical harmonic decomposition→

66

CMB Satellites: COBE (1989 –1993)CMB Satellites: COBE (1989 –1993)

77

CMB Satellites: WMAP (2001 –2010)CMB Satellites: WMAP (2001 –2010)

88

CMB Satellites: Planck (2009 –2012)CMB Satellites: Planck (2009 –2012)

99

Temperature Fluctuations (2)Temperature Fluctuations (2) It is convenient to expand the temperature field in It is convenient to expand the temperature field in

spherical harmonicsspherical harmonics

ℓℓ → → defines a scale; defines a scale; mm defines just a phase→ defines just a phase→

All the information is in the (complex) All the information is in the (complex) aaℓℓmm set. set. Set is finite in practice because any instrument's finite Set is finite in practice because any instrument's finite

resolution sets an resolution sets an ℓℓmaxmax

1010

VisualizationVisualization

1111

Planck's Power SpectrumPlanck's Power Spectrum

1212

CMB Power SpectrumCMB Power Spectrum

1313

Spherical Harmonic Love (Spherical Harmonic Love (ℓℓmaxmax = 3) = 3)

1414

ℓℓmaxmax = 10 = 10

1515

ℓℓmaxmax = 20 = 20

1616

ℓℓmaxmax = 50 = 50

1717

ℓℓmaxmax = 100 = 100

1818

ℓℓmaxmax = 2000 = 2000

1919

Earth Topography Angular SpectrumEarth Topography Angular Spectrum