CMB Hemispherical Power Asymmetry & its relation with - - PowerPoint PPT Presentation

CMB Hemispherical Power Asymmetry & its relation with Noncommutative Geometry

CosPA 2016, University of Sydney, Australia

Rahul Kothari Indian Institute of Technology, Kanpur, India

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Part 1 — Rudimentary Ideas Part 2 — The Connection

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Cosmic Microwave Radiation

• CMB is a field over sphere, spherical harmonic decomposition can be done
• Satisfies statistical isotropy aka Cosmological Principle
• Two point correlation depends upon angle between points of observation and

not location (or does it?)

∆T(ˆ n) = ∑almYlm(ˆ n)

h∆T ( ˆ m)∆T (ˆ n)i = f ( ˆ m· ˆ n) , halma?

l0m0i = δll0δmm0Cl

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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)
• Effect is absent at high l values (Donoghue 2005)
• Thus this is a large scale anisotropy

∆T (ˆ n) = ∆Tiso (ˆ n) ⇣ 1+Aˆ λ · ˆ n ⌘

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Modelling — The idea

• Isotropic and homogeneous power spectra leads to isotropy and homogeneity
• Dipole modulation might be related to an early phase of inhomogeneous and/
• r 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

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• R. Kothari et.al. Imprint of Inhomogeneous and Anisotropic Primordial Power Spectrum on CMB Polarisation, MNRAS 460, 1577-1587

The Algorithm

Before going any further, I would like to discuss the algorithm to calculate final correlations starting from two point density correlations Calculate Involved Integrals Using CAMB

Take Fourier transform

Model Real space Correlations hδ (x)δ (y)i Obtain required correlations

Integration measure two point density correlations transfer function and geometric quantities 6

Fourier space two Point Correlations D ˜ δ (k) ˜ δ ? k0E Spherical harmonic Correlations halma?

l0m0i =

ZZ

G (l,l0,k,k0) D ˜ δ (k) ˜ δ ? k0E d(k,k0)

Example

• For the anisotropic model real space density correlations
• Then the two point density correlations in Fourier space
• Harmonic coefficients correlations
• Evaluated with the help of CAMB

D ˜ δ (k) ˜ δ ? k0E = h P

iso (k)iˆ

k · ˆ λg(k) i δ

• kk0

halma?

l0m0ianiso = δmm0δl,l+1 (4πT0)2

s (l +1)2 m2 (2l +1)(2l +3)

Z ∞

0 k2dk∆(l,k)∆

• l0,k
• g(k)

Evaluated using CAMB

hδ (x)δ (y)i = f1 (R)+BiRi f2 (R), Bi 2 R, R = xy

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Part 1 — Rudimentary Ideas Part 2 — The Connection

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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)
• I’ll outline a proof that such a form isn’t possible in commutative field theory
• Noncommutative spacetime gives us a way out

hδ (x)δ (y)i = f1 (R)+BiRi f2 (R), Bi 2 R, R = xy

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The Proof

Proposition: Anisotropic form of the power spectrum isn’t possible in commutative spacetimes. Proof: Let us assume this to be true so that . In commutative regime fields commute, so that thus the above condition implies

• r in other words

which is a contradiction. Hence this form isn’t possible in commutative regime. F(R,X) = hδ (x)δ (y)i = f1 (R)+BiRi f2 (R), Bi 2 R, X = (x+y)/2 f2(R) 6= 0 hδ (x)δ (y)i = hδ (y)δ (x)i ) F (R,X) = F (R,X) f1 (|x−y|)+Bi (xi −yi) f2 (|x−y|) = f1 (|y−x|)+Bi (yi −xi) f2 (|y−x|) f2 (|x−y|) = f2 (R) = 0

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Generalised Moyal Product

• The power spectrum is defined to be the Fourier transform of
• The ‘simple’ product between the fields must be changed to Moyal star

product

• Here
• This is general analysis, functions f are to be taken as scalar fields later and F

is the form factor

Ωj1, j2,...,jm

i1,i2,...,im ?Ωl1,l2,...,ln k1,k2,...,kn =

" 1+ i 2Θµν

m

∑

p=1 n

∑

q=1

Fjplq ∂ ∂xµ

jp

∂ ∂xν

lq

+... # Ωj1, j2,...,jm

i1,i2,...,im Ωl1,l2,...,ln k1,k2,...,kn

h0|φ (x,t)?φ (y,t)|0i Ωj1, j2,...,jm

i1,i2,...,im = fi1 (xj1)?

• fi2 (x j2)?...?
• fim−1
• xjm−1
• ? fim (xjm)
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Rahul Kothari, Pranati Rath & Pankaj Jain, Cosmological Power Spectrum in Non-commutative Spacetime, Physical Review D, 94, 063531

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

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Can tell about associativity of real field

Results & Conclusion

• After all calculations are done the following correction to the standard power

spectrum is obtained

• Thus although other correlations are present, it has been shown that it is

possible to have such a form using noncommutative geometry ibΘ0iH3 k5 ⇣ 15ki ~ α · ˆ k 2 −3ki |~ α|2 −6kαi~ α · ˆ k ⌘ required form

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/kU;okn%

Thank You

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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

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