Gravitational wave generation in a viable scenario of inflationary magnetogenesis Ramkishor Sharma Department of Physics and Astrophysics University of Delhi, Delhi Sandhya Jagannathan , T. R. Seshadri (Delhi University, Delhi) Kandaswamy Subramanian (IUCAA, Pune) “Gravitational Waves from the Early Universe” Nordita, Stockholm, 26 Aug. − 20 Sep. 2019
Outline of the Talk Part 1 : A viable model for the generation of large scale magnetic field in the early universe Based on: ◮ R. Sharma, S. Jagannathan, T. R. Seshadri, and K. Subramanian, Phys. Rev. D 96, 083511 (2017), arXiv:1708.08119 ◮ R. Sharma, K. Subramanian, and T. R. Seshadri, Phys. Rev. D97, 083503 (2018), arXiv:1802.04847 Part 2 : Stochastic background of gravitational wave from the anisotropic stress due to these fields Based on: ◮ R. Sharma, K. Subramanian, and T. R. Seshadri (In preparation)
◮ Observational evidences of magnetic fields
Observational evidences of Magnetic Fields ◮ � B over galactic scales (ordered on kpc) ∼ order of 10 µ G : Both coherent and stochastic [Beck 2001; Beck and Wielebinski 2013] ◮ Observed in clusters with a few µ G strength, coherence length of the order of 10-20 kpc [Clarke et al. 2001, Govoni and Feretti 2004] ◮ Evidence for equally strong � B in high redshift ( z ∼ 1 . 3) galaxies [Bernet et al. 08] ◮ FERMI/ LAT observations of GeV photons from Blazars B ≥ 10 − 16 G on intergalactic � ◮ Lower limit: � B at scale above 1 Mpc [Neronov & Vovk, Science 10]
Summary of Observational Constraints [Neronov & Vovk, Science 10]
◮ Observational evidences ◮ Generation mechanism of the magnetic fields
Origin and Growth: Broad Picture ◮ Amplification − → growth (flux freezing, Dynamo mechanism) Governing equation for these mechanisms is magnetic induction equation, ∂� B ∂τ = � ∇ × ( � V × � B − η� ∇ × � B ) Here τ and η are the time parameter and plasma registivity, respectively. ◮ However dynamo requires an initial seed field ∼ 10 − 20 G. ◮ Origin of seed field − → Astrophysical or Primordial
Generation Mechanism of magnetic field Generation Mechanism Astrophysical Scenario Primordial Scenario During phase transition During inflation Generate field of Generate coherent Generate field only coherence length B- field on in collapsed object smaller than the size of cosmological scales horizon Astrophysical origin of seeds may not be able to explain the presence of magnetic field in voids Worth considering primordial origin possibly during inflationary process. ( Durrer and Neronov 2013; K. Subramamnian, 2010, 2016 )
◮ Observational evidences ◮ Generation Mechanism of the magnetic fields ◮ Inflationary Magnetogenesis
Inflation ◮ An era of exponential expansion of space in the early Universe. ◮ Introduced to solve Horizon and Flatness problems. ◮ Also provides a natural explanation to initial density fluctuations. ◮ These initial density fluctuations arise due to the quantum mechanical nature of the field which causes inflation or some other field present during inflation. ◮ As different modes cross the horizon, the nature of fluctuations over these modes becomes classical.
Scalar field vs EM field fluctuations during inflation 10 4 Scalar and EM fluctualtions scalar 10 vector 10 - 2 10 - 5 10 - 8 10 - 11 1 10 100 1000 scale factor ( a ) Scalar field fluctuations � 0 | ˆ x , η ) ˆ δφ ( � δφ ( � y , η ) | 0 � ≈ ∆ φ ( k ) | k ∼ 1 / L x , η ) B i ( � EM field fluctuations � 0 | B i ( � y , η ) | 0 � ≈ ∆ B ( k ) | k ∼ 1 / L For inflationary scale H f = 10 14 GeV, the value of ∆ B for 1 Mpc mode at horizon crossing ≈ 10 − 10 G. However this value at the end of inflation becomes ≈ 10 − 10 × 10 − 46 G
Scalar field vs EM field fluctuations during inflation ◮ Scalar fluctuations: d 4 x √− g ( ∂ ν φ∂ ν φ − V ( φ )) S φ = − 1 � 2 k 2 − a ′′ � � ( a δφ ( k , η )) ′′ + a δφ ( k , η ) = 0 a ◮ EM fluctuations: � √− gd 4 x 1 16 π F µν F µν S EM = − ( aA ( k , η )) ′′ + k 2 aA ( k , η ) = 0 This implies B ∝ 1 a 2 . ◮ This happens due to the conformal invariance of the EM action and the conformal flatness of the background spacetime.
Breaking conformal invariance Action: Modified electromagnetic action + interaction with charged particles/current � √− gd 4 x � f 2 ( φ ) 1 � 16 π F µν F µν + j µ A µ S EM = − i + 2 f ′ A ′′ f A ′ i − a 2 ∂ j ∂ j A i = 0 . In Coulomb Gauge , A ′′ + 2 f ′ A ′ + k 2 ¯ Define ¯ ¯ ¯ A ≡ aA ( k , η ) A = 0 . f k 2 − f ′′ � � A ≡ f ¯ A ′′ ( k , η ) + Define A ( k , η ) A ( k , η ) = 0 . f
Energy density of the EM field ◮ Energy momentum tensor F αβ F αβ T µν = f 2 � � g αβ F µα F νβ − g µν 4 ◮ Energy density ρ = � 0 | T µν u µ u ν | 0 � T µν u µ u ν = f 2 2 B i B i + f 2 2 E i E i � 0 | f 2 k 5 � d ln k 1 � d ln k d ρ B a 4 |A ( k , η ) | 2 ≡ 2 B i B i | 0 � = 2 π 2 d ln k � 0 | f 2 d ln k f 2 k 3 � � A ( k , η ) � ′ � 2 � d ln k d ρ E � 2 E i E i | 0 � = ≡ � � 2 π 2 a 4 � f � d ln k
Generated magnetic field ◮ For, f = f i a α a = − 1 and H f η , there are two possibility for a scale invariant magnetic field spectral energy density; α = 2 and α = − 3. d ρ B 9 ◮ For scale invariant spectrum 4 π 2 H 4 d ln k ≈ f ◮ After generation, magnetic energy density varies with time as ρ B ∝ 1 / a 4 . ◮ Corresponding magnetic field strength � 2 � d ρ B � a f � H f � � ∼ 5 × 10 − 10 G B 0 = 2 � 10 − 5 M pl d ln k a 0 � f
Back reaction and strong coupling problems ◮ Scale invariant spectral magnetic energy density: α = 2 and α = − 3 ◮ For α = − 3, Electric energy density spectrum ∝ ( k aH ) − 2 • Electric energy density diverges towards the end of inflation. • Electrical energy density dominates over inflation energy density. This is known as back reaction problem . ◮ In the usual approach with conformal breaking, the final value of f is made unity to match with the standard EM theory. ◮ Since f grows as a 2 = ⇒ initial value of f is very small. ◮ Effective coupling parameter e N = e / f 2 becomes very large. This is known as strong coupling problem .
◮ Observational evidences ◮ Generation Mechanism of the Magnetic fields ◮ Inflationary Magnetogenesis ◮ Viable model of magnetic field generation
Addressing the strong coupling problem ◮ In our model, we bring the system back to the standard form not at end of inflation but some time after it before reheating. f = a 2 > 1(during inflation) & f ≫ 1 at end of f i = 1 = ⇒ , inflation. Hence no strong coupling problem. ◮ As coupling parameter is very small at the end of inflation. Hence, f need to be brought back to unity post inflation. f = a 2 During Inflation, f = f f ( a / a f ) − β Post Inflation, ◮ Models are constrained by the requirement of how fast the factor f falls to 1 from a large value.
Post Inflationary era ◮ We assume a matter dominated universe after inflation till reheating. ◮ For f ∝ a − β , we solved vector potential by demanding the continuity of vector potential and its time derivative at the end of inflation. ◮ Energy density in magnetic and electric field can be calculated as before. d ln k ∝ k 4 for d ρ B ◮ At reheating, for super horizon modes α = 2
Constraints from Post Inflationary Pre-reheating phase Total energy in electric and magnetic field should be less that in inflation field at reheating. π 2 30 T 4 ρ E + ρ B < ρ φ | reheat = g r r
Post reheating evolution of magnetic field ◮ � − p � q � a m � a m B NL 0 [ L NL , L NL c 0 ] = B 0 [ L c 0 ] c 0 = L c 0 , a r a r where a m = ⇒ scale factor at radiation-matter equality, p ≡ ( n + 3) / ( n + 5) and q ≡ 2 / ( n + 5) here n is defined in such a way that d ρ B d ln k ∝ k n +3 (Banerjee and Jedamzik, 2004; Brandenburg et al. 2015) ◮ After incorporating the results of magnetic field evolution suggested by simulation, c 0 ] = B 0 [ L c 0 ] ( a m / a r ) − 0 . 5 , c 0 = L c 0 ( a m / a r ) 0 . 5 B S 0 [ L S L S (Brandenburg et al. 2015; Brandenburg and Kahniashvili 2016)
Results taking nonlinear effects into account Without inverse Transfer With inverse Transfer ◮ For T R = 100 GeV, B 0 ∼ 10 − 15 G and coherence length ∼ 10 − 5 Mpc. ◮ B 0 ∼ 10 − 13 G and coherence length ∼ 10 − 3 Mpc ( with inverse transfer).
◮ Observational evidences ◮ Generation Mechanism of the Magnetic fields ◮ Inflationary Magnetogenesis ◮ Viable model of magnetic field generation ◮ Helical magnetic field generation
EM action for the generation of helical magnetic field ◮ Action � √− gd 4 x � f 2 ( φ ) � F µν F µν + F µν ˜ F µν � � + j µ A µ S EM = − 16 π ◮ Modified Maxwell’s Equation i + 2 f ′ A ′′ A ′ − a 2 ∂ j ∂ j A i = 0 � � i + ǫ ijk ∂ j A k f ◮ In terms of circular polarisation basis h + 2 f ′ A ′′ ¯ � ¯ A ′ h + hk ¯ + k 2 ¯ � A h A h = 0 f here h = ± 1
Magnetic field energy spectrum k 5 d ρ B ( k , η ) 1 � |A + ( k , η ) | 2 + |A − ( k , η ) | 2 � = (2 π ) 2 a 4 d ln k Evolution of a mode k = 10 5 H f During inflation Post Inflation
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