CMB Hemispherical Power Asymmetry & its relation with Noncommutative Geometry CosPA 2016, University of Sydney, Australia Rahul Kothari Indian Institute of Technology, Kanpur, India 1
Part 1 — Rudimentary Ideas Part 2 — The Connection 2
Cosmic Microwave Radiation • CMB is a field over sphere, spherical harmonic decomposition can be done n ) = ∑ a lm Y lm ( ˆ ∆ T ( ˆ n ) • Satisfies statistical isotropy aka Cosmological Principle h ∆ T ( ˆ n ) i = f ( ˆ n ) , h a lm a ? l 0 m 0 i = δ ll 0 δ mm 0 C l m ) ∆ T ( ˆ m · ˆ • Two point correlation depends upon angle between points of observation and not location (or does it?) 3
Hemispherical Power Asymmetry • Analysis of 2003 WMAP data revealed extra correlations (Eriksen et.al. 2004) • Coined as Hemispherical Power Asymmetry • Parametrisation(Gordon et.al. 2005, Prunet et.al. 2005, Bennett et.al. 2011) ⇣ ⌘ 1 + A ˆ ∆ T ( ˆ n ) = ∆ T iso ( ˆ n ) λ · ˆ n • Effect is absent at high l values (Donoghue 2005) • Thus this is a large scale anisotropy 4
Modelling — The idea R. Kothari et.al. Imprint of Inhomogeneous and Anisotropic Primordial Power Spectrum on CMB Polarisation, MNRAS 460 , 1577-1587 • Isotropic and homogeneous power spectra leads to isotropy and homogeneity • Dipole modulation might be related to an early phase of inhomogeneous and/ or isotropic phase • These modes generated during early phases of inflation may later re-enter the horizon and cause anisotropy • Thus by modifying primordial power spectra HPA can be explained 5
The Algorithm Before going any further, I would like to discuss the algorithm to calculate final correlations starting from two point density correlations Fourier space two Model Real space Take Fourier transform Point Correlations Correlations D k 0 �E δ ( k ) ˜ ˜ δ ? � h δ ( x ) δ ( y ) i Calculate Involved Spherical harmonic Correlations Integrals Using CAMB h a lm a ? l 0 m 0 i = ZZ D k 0 �E δ ( k ) ˜ ˜ δ ? � G ( l , l 0 , k , k 0 ) d ( k , k 0 ) Obtain required correlations two point transfer function Integration density correlations and geometric measure quantities 6
Example • For the anisotropic model real space density correlations h δ ( x ) δ ( y ) i = f 1 ( R )+ B i R i f 2 ( R ) , B i 2 R , R = x � y • Then the two point density correlations in Fourier space D k 0 �E h i δ ( k ) ˜ ˜ k · ˆ iso ( k ) � i ˆ δ ? � � k � k 0 � = λ g ( k ) δ P • Harmonic coefficients correlations • Evaluated with the help of CAMB s ( l + 1 ) 2 � m 2 Z ∞ l 0 m 0 i aniso = δ mm 0 δ l , l + 1 ( 4 π T 0 ) 2 l 0 , k 0 k 2 dk ∆ ( l , k ) ∆ � � h a lm a ? g ( k ) ( 2 l + 1 )( 2 l + 3 ) Evaluated using CAMB 7
Part 1 — Rudimentary Ideas Part 2 — The Connection 8
Derivation of Anisotropic Power Spectrum • Direction dependence introduces anisotropy so we want a correlation that depends upon direction • The simplest of such correlations is (from slide 7) h δ ( x ) δ ( y ) i = f 1 ( R )+ B i R i f 2 ( R ) , B i 2 R , R = x � y • I’ll outline a proof that such a form isn’t possible in commutative field theory • Noncommutative spacetime gives us a way out 9
The Proof Proposition: Anisotropic form of the power spectrum F ( R , X ) = h δ ( x ) δ ( y ) i = f 1 ( R )+ B i R i f 2 ( R ) , B i 2 R , X = ( x + y ) / 2 isn’t possible in commutative spacetimes. Proof: Let us assume this to be true so that . In commutative regime fields f 2 ( R ) 6 = 0 commute, so that h δ ( x ) δ ( y ) i = h δ ( y ) δ ( x ) i ) F ( R , X ) = F ( � R , X ) thus the above condition implies f 1 ( | x − y | )+ B i ( x i − y i ) f 2 ( | x − y | ) = f 1 ( | y − x | )+ B i ( y i − x i ) f 2 ( | y − x | ) or in other words f 2 ( | x − y | ) = f 2 ( R ) = 0 which is a contradiction. Hence this form isn’t possible in commutative regime. 10
Generalised Moyal Product Rahul Kothari, Pranati Rath & Pankaj Jain, Cosmological Power Spectrum in Non-commutative Spacetime, Physical Review D, 94 , 063531 • The power spectrum is defined to be the Fourier transform of h 0 | φ ( x , t ) ? φ ( y , t ) | 0 i • The ‘simple’ product between the fields must be changed to Moyal star product " # m n ∂ ∂ 1 + i Ω j 1 , j 2 ,..., j m i 1 , i 2 ,..., i m ? Ω l 1 , l 2 ,..., l n 2 Θ µ ν Ω j 1 , j 2 ,..., j m i 1 , i 2 ,..., i m Ω l 1 , l 2 ,..., l n ∑ ∑ F j p l q k 1 , k 2 ,..., k n = + ... ∂ x ν k 1 , k 2 ,..., k n ∂ x µ p = 1 q = 1 l q j p • Here Ω j 1 , j 2 ,..., j m � � � � �� i 1 , i 2 ,..., i m = f i 1 ( x j 1 ) ? f i 2 ( x j 2 ) ? ... ? ? f i m ( x j m ) f i m − 1 x j m − 1 • This is general analysis, functions f are to be taken as scalar fields later and F is the form factor 11
Generalised Moyal Product Properties & Features Calculation was done at the linear order • Definition having different spacetime points was absent in the literature, we • gave a recursive definition approach When all spacetime points become same this becomes standard star product • Generalised product is still — (a) Associativity & (b) cyclic • Our definition can be used to prove associativity of the standard star product • Can tell about associativity of real field 12
Results & Conclusion • After all calculations are done the following correction to the standard power spectrum is obtained ib Θ 0 i H 3 ⇣ � 2 − 3 k i | ~ ⌘ α | 2 − 6 k α i ~ α · ˆ α · ˆ � ~ 15 k i k k k 5 required form • Thus although other correlations are present, it has been shown that it is possible to have such a form using noncommutative geometry 13
/kU;okn% Thank You 14
References • Eriksen et.al. 2005 ApJ 605, 14 • Gordon et.al. 2005, PRD 72, 103002 • Prunet et.al. 2005, PRD 71, 083508 • Bennett et.al. 2011, ApJS 192, 17 • Donoghue 2005, PRD 71, 043002 15
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