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Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Department of Mathematics Jadavpur University 10 th September, 2019

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Acceleration of the Universe Recent data [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] indicates that the expansion of our Universe is accelerating . To explain this phenomena either one has to modify matter or one has to modify geometry. To modify the matter term cosmologists introduced a compo- nent with negative pressure dubbed as Dark Energy .

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Bulk viscosity in the accelerating Universe Bulk viscosity play an important role in the early stage of the Universe. Bulk viscosity can also describe present accelerating phase [10, 11]. Origin of bulk viscosity: Interaction between different compo- nents or non-conservation of particle number.

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Gravitational collapse The gravitational collapse of a star follows as [12, 13, 14] Star White Dwarf White ( M < 1 . 4 M ⊙ ) Dwarf Neutron Star Neutron ( M < 2 . 17 M ⊙ ) Star Black Hole

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Laws and Hypothesis in gravitational collapse Singularity Theorem [15] Cosmic Censorship Conjecture [16]

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Trapped Horizon and Singularity In trapped horizon both incoming and outgoing null geodesic converges [53, 54, 55]. At singularity all the physical laws break down. Here pressure, density, curvature diverges. CCC [16] may be assumed to be related to the thermodynamic nature of the spacetime manifold near Naked Singularity (NS).

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee The basic set up The matter of the collapsing star is chosen in the form of perfect fluid with barotropic equation of state p = ( γ − 1) ρ . The thermodynamic system is chosen as adiabatic. The effec- tive bulk viscous pressure is determined by the particle creation rate [27, 28, 29, 30, 34] as Π = − Γ 3 H ( p + ρ ) . (1) The interior geometry is characterized by the flat Friedmann- Robertson-Walker (FRW) model − = dt 2 − a 2 ( t )( dr 2 + r 2 d Ω 2 ds 2 2 ) . (2) Gravitational collapse if ˙ a < 0 .

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Trapped Surfaces and Apparent Horizon The apparent horizon is a trapped surface lying in a boundary of a particular surface S . For the present FRW model, the apparent horizon is character- ized by [39, 40, 41] R , i R , j g ij ≡ ( r ˙ a ) 2 − 1 = 0 . (3) The comoving boundary surface of the star is spacelike: r | Σ = constant , say r Σ . Thus we have on Σ: R , i R , j g ij ≡ { r Σ ˙ a ( t ) } 2 − 1 < 0 . (4) Here r Σ denotes the boundary of the collapsing star and we have on Σ: Σ = d τ 2 − R 2 ( τ ) d Ω 2 ds 2 2 .

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee The Exterior Metric and the Mass of the Collapsing Cloud The metric outside the collapsing star in general can be written in the form [26, 42] + = A 2 ( T , R ) dT 2 − B 2 ( T , R )( dR 2 + R 2 d Ω 2 ds 2 2 ) . The mass function due to Cahill and McVittie [43] is defined as m ( r , t ) = R 2 (1 + R ,α R ,β g αβ ) = 1 2 R ˙ R 2 . Thus the total mass of the collapsing cloud is m ( τ ) = m ( r Σ , τ ) = 1 2 R ( τ ) ˙ R 2 ( τ ) . (5)

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Basic Equations The basic Friedmann equations for the present model are 3 H 2 = 8 π G ρ 2 ˙ and H = − 8 π G ( ρ + p + Π) . (6) Conservation equation ρ + 3 H ( ρ + p + Π) = 0 . ˙ (7)

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Collapse Dynamics The collapse dynamics is characterized by the particle creation rate as � � 2 ˙ H 1 − Γ 3 H 2 = − γ . (8) 3 H In the present work, we shall choose Γ as [30] Γ = Γ 3 + 3Γ 0 H + Γ 1 H . (Γ 0 , Γ 1 , Γ 3 ∈ R , Γ � = 0) (9) The evolution of scale factor of the collapsing core � ˙ � 3 γ � � a 2 ¨ a 2 − γ Γ 3 a − γ Γ 1 ˙ a a a + 1 − Γ 0 − 1 = 0 . (10) 2 2 2

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee The choices for Γ We shall consider five choices: Γ = Γ 3 + 3Γ 0 H + Γ 1 H . Γ = Γ 3 + 3 H + Γ 1 H . Γ = 3Γ 0 H . Γ = 3 H + Γ 1 / H . Γ = Γ 3 + 3Γ 0 H

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Choice I: Γ = Γ 3 + 3Γ 0 H + Γ 1 H We consider the evolution equation (10) � 3 γ � � � ˙ a 2 a ¨ a 2 − γ Γ 3 a − γ Γ 1 a ˙ a + 1 − Γ 0 − 1 = 0 . (11) 2 2 2 The solutions for rate of contraction and scale factor H = [ − H − 1 + µ tanh T ] − 1 (12) 2 � � �� m � � µα 1 = e lT Γ 3 a 2Γ 1 cosh T − µ sinh T . (13) H 2 a 0 µ 2 = { 12Γ 1 (1 − Γ 0 )+Γ 2 3 } , α 1 = γ Γ 1 µ 2 , m = 2Γ1 ) 2 ] , 4Γ 2 [ µ 2 − ( Γ3 1 H − 1 ] , T = µα 1 ( t − t 0 ), H 2 = ( Γ 3 2Γ 1 ) − 1 . l = 2 [ µ 2 − H − 2 t c = t 0 + ( µα 1 ) − 1 � tanh − 1 � �� 2 1 / H 2 µ .

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Time of formation of apparent horizon � n +1 � � n µα 1 ) T aH � 1 − tanh T aH l R 0 H 2 e ( cosh T aH = 1 . tanh T c m T aH = µα 1 ( t aH − t 0 ) , R 0 = a 0 r and n = µα 1 − 1. t c > t aH for any real value of n (except n to be a positive integer). t c < t aH or t c > t aH , if n is an even integer.

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Choice II: Γ = Γ 3 + 3 H + Γ 1 H The evolution equation (10) simplifies to H = γ ˙ 2(Γ 3 H + Γ 1 ) . (14) The solutions for scale factor and rate of contraction H = − δ + ( H 0 + δ ) e − γα 2 ( t − t 0 ) (15) � � �� − 2( H 0 + δ ) a = a 0 e − δ ( t − t 0 ) exp 2 ( t − t 0 ) − 1 e − γα . γα α = − Γ 3 , µ = − Γ 1 and δ = Γ 1 Γ 3 . t c = ∞ .

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Time of formation of apparent horizon � � R 0 e − δ � 2 � δ − ( H 0 + δ ) e − γα T aH T aH � � �� 2 � − 2( H 0 + δ ) e − γα T aH − 1 exp = 1 (16) γα � T aH = t aH − t 0 . t aH always has a finite solution.

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee The measure of acceleration is given by � � 2 � γ 2 α 2 � − γµ 2 a ¨ 2 ( t − t 0 ) − γα − δ + ( H 0 + δ ) e − γα a = − . (17) 4 16 2 � � δ + H 0 2 Accelerating if t > t 0 + γα ln or � γ 2 α 2 δ + γα 4 − 16 − γµ � � 2 2 δ + H 0 t < t 0 + γα ln � γ 2 α 2 δ + γα 16 − γµ 4 + 2 � � 2 δ + H 0 Decelerating if t 0 + < t < γα ln � γ 2 α 2 δ + γα 4 + 16 − γµ � � 2 2 δ + H 0 t 0 + γα ln . � γ 2 α 2 δ + γα 4 − 16 − γµ 2

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Figure: The figure on the left side represents accelerating collapsing process against time t given by (17) and the figure on the right side denotes evolution of the rate of contraction ( H ), which is given by (15) against t , respectively for Γ 0 = 1. In both the figures, the curves in the solid line represent ¨ a a and H , respectively for γ = 4 3 . The curves in the dashed line represent ¨ a a and H , respectively for γ = 2 3 and the curves in the dash-dotted line represent ¨ a a and H , respectively for γ = 1 3 . α = 3, δ = 1 .

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Choice III: Γ = 3Γ 0 H The evolution equation simplifies to (Γ 0 � = 1) � 3 γ � a ¨ H 2 = 0 , a + 2 (1 − Γ 0 ) − 1 (18) The solutions for scale factor and rate of contraction H 0 � � . H = (19) 1 + 3 γ H 0 2 (1 − Γ 0 )( t − t 0 ) � � 2 1 + 3 γ H 0 3 γ (1 − Γ0) a = a 0 (1 − Γ 0 )( t − t 0 ) . 2 2 t c = t 0 − 3 γ H 0 (1 − Γ 0 ) .

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee Time of formation of apparent horizon � � � 1 l � 2 1 2 t aH = t 0 + − 1+ − , l = 3 γ (1 − Γ 0 ) − 1 . 3 γ H 0 (1 − Γ 0 ) R 0 H 0 � � 1 1 1 l t aH − t c = − H 0 R 0 H 0 t aH < t c

Role of particle creation mechanism on the collapse of a massive star Sudipto Bhattacharjee The measure of acceleration is given by { 1 − 3 γ 2 (1 − Γ 0 ) } H 2 ¨ a 0 a = (20) � � 2 1 + 3 γ H 0 2 (1 − Γ 0 )( t − t 0 ) Accelerating if 3 γ 2 (1 − Γ 0 ) < 1 Decelerating if 3 γ 2 (1 − Γ 0 ) > 1.