Numerical simulations of gravitational waves from early-universe turbulence APS April meeting (April 18–21 2020) Alberto Roper Pol (PhD candidate) Research Advisors: Brian Argrow & Axel Brandenburg Collaborators: Tina Kahniashvili, Arthur Kosowsky & Sayan Mandal University of Colorado at Boulder Laboratory for Atmospheric and Space Physics (LASP) April 20, 2020 A. Roper Pol et al., Geophys. Astrophys. Fluid Dyn. 114 , 130. arXiv:1807.05479 (2019) A. Roper Pol et al., submitted to Phys. Rev. D arXiv:1903.08585 (2019) Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 1 / 22
Introduction and Motivation Generation of cosmological gravitational waves (GWs) during phase transitions and inflation Electroweak phase transition ∼ 100 GeV Quantum chromodynamic (QCD) phase transition ∼ 100 MeV Inflation Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 2 / 22
Introduction and Motivation Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 3 / 22
Introduction and Motivation Generation of cosmological gravitational waves (GWs) during phase transitions and inflation Electroweak phase transition ∼ 100 GeV Quantum chromodynamic (QCD) phase transition ∼ 100 MeV Inflation GW radiation as a probe of early universe physics Possibility of GWs detection with Space-based GW detector LISA Pulsar Timing Arrays (PTA) B -mode of CMB polarization Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 4 / 22
Introduction and Motivation Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 5 / 22
Introduction and Motivation Generation of cosmological gravitational waves (GWs) during phase transitions and inflation Electroweak phase transition ∼ 100 GeV Quantum chromodynamic (QCD) phase transition ∼ 100 MeV Inflation GW radiation as a probe of early universe physics Possibility of GWs detection with Space-based GW detector LISA Pulsar Timing Arrays (PTA) B -mode of CMB polarization Magnetohydrodynamic (MHD) sources of GWs: Hydrodynamic turbulence from phase transition bubbles nucleation Primordial magnetic fields Numerical simulations using Pencil Code to solve: Relativistic MHD equations Gravitational waves equation Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 6 / 22
Gravitational waves equation GWs equation for an expanding flat Universe Assumptions: isotropic and homogeneous Universe ıtre–Robertson–Walker (FLRW) metric γ ij = a 2 δ ij Friedmann–Lemaˆ Tensor-mode perturbations above the FLRW model: g ij = a 2 � � δ ij + h phys ij GWs equation is 1 h ij = 16 π G � t − c 2 ∇ 2 � ∂ 2 a c 2 T TT ij h ij are rescaled h ij = ah phys ij Comoving spatial coordinates ∇ = a ∇ phys Conformal time d t = a d t phys Comoving stress-energy tensor components T ij = a 4 T phys ij Radiation-dominated epoch such that a ′′ = 0 1L. P. Grishchuk, Sov. Phys. JETP , 40, 409-415 (1974) Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 7 / 22
Normalized GW equation 2 � t − ∇ 2 � ∂ 2 h ij = 6 T TT / t ij Properties All variables are normalized and non-dimensional Conformal time is normalized with t ∗ Comoving coordinates are normalized with c / H ∗ rad = 3 H 2 ∗ c 2 / (8 π G ) Stress-energy tensor is normalized with E ∗ Scale factor is a ∗ = 1, such that a = t 2A. Roper Pol et al., Geophys. Astrophys. Fluid Dyn. 114 , 130. arXiv:1807.05479 (2019) Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 8 / 22
Gravitational waves equation Properties Tensor-mode perturbations are gauge invariant h ij has only two degrees of freedom: h + , h × The metric tensor is traceless and transverse (TT gauge) Contributions to the stress-energy tensor γ − 1 T µν = p / c 2 + ρ U µ U ν + pg µν + F µγ F ν 4 g µν F λγ F λγ � � From fluid motions From magnetic fields: p / c 2 + ρ � � γ 2 u i u j + p δ ij T ij = − B i B j + δ ij B 2 / 2 T ij = Relativistic equation of state: p = ρ c 2 / 3 Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 9 / 22
MHD equations Conservation laws T µν ; ν = 0 Relativistic MHD equations are reduced to 3 MHD equations ∂ ln ρ = − 4 ∇ ln ρ ) + 1 J 2 � � 3 ( ∇ ∇ ∇ · u u + u u u u · ∇ ∇ u · ( J u u J × B J B B ) + η J J ∂ t ρ Dt = 4 u u − 1 ∇ ln ρ + 3 B + 2 Du u ∇ ln ρ ) − u u � B ) + η J 2 � 3 ( ∇ ∇ · u u + u u u · ∇ u u u u · ( J J J × B B 4 ρ J J J × B B ∇ · ( ρν S S ) S ∇ ∇ 4 ∇ ∇ ρ ∇ ∇ ρ ∂ B B B ∂ t = ∇ ∇ ∇ × ( u u B − η J B J J ) u × B for a flat expanding universe with comoving and normalized p = a 4 p phys , ρ = a 4 ρ phys , B i = a 2 B i , phys , u i , and conformal time t . 3A. Brandenburg, K. Enqvist, and P. Olesen, Phys. Rev. D 54 , 1291 (1996) Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 10 / 22
Linear polarization modes + and × Linear polarization basis (defined in Fourier space) e + ij = ( e e 2 ) ij e e 1 × e e e 1 − e e e 2 × e e e × ij = ( e e 2 + e e 1 ) ij e e 1 × e e e e 2 × e e Orthogonality property e A ij e B ij = 2 δ AB , where A , B = + , × + and × modes h + = 1 T + = 1 ˜ 2 e + ij ˜ h TT ˜ 2 e + ij ˜ T TT ij , ij h × = 1 T × = 1 ˜ ij ˜ ˜ ij ˜ h TT T TT 2 e × ij , 2 e × ij Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 11 / 22
Solving the GW equation Compute Fourier transform of stress-energy tensor ˜ T ij � ˜ Project into TT gauge ˜ P il P jm − 1 T TT � T TT = 2 P ij P lm ij lm T + and ˜ T × modes Compute ˜ Discretize time using δ t from MHD simulations Assume ˜ T + , × / t to be constant between subsequent timesteps (robust as δ t → 0) GW equation solved analytically between subsequent timesteps in Fourier space 4 � ω ˜ � � ω ˜ h − 6 ω − 1 ˜ � t + δ t h − 6 ω − 1 ˜ � t � T / t cos ωδ t sin ωδ t T / t = ˜ ˜ h ′ − sin ωδ t cos ωδ t h ′ + , × + , × 4A. Roper Pol et al., Geophys. Astrophys. Fluid Dyn. , 114 , 130 arXiv:1807.05479 (2019) Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 12 / 22
Gravitational waves energy density GWs energy density: crit = 3 H 2 0 c 2 Ω GW = E GW / E 0 E 0 crit , 8 π G � ∞ Ω GW = Ω GW ( k ) d ln k −∞ Ω GW ( k ) = ( a ∗ / a 0 ) 4 k � �� 2 2 � � ˙ � � � ˙ � k 2 d Ω k h phys ˜ ˜ h phys + � � � � + 6 H 2 × � � 4 π 0 H 0 = 100 h 0 km s − 1 Mpc − 1 a 0 ≈ 1 . 254 · 10 15 ( T ∗ / 100 GeV ) ( g S / 100) 1 / 3 a ∗ Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 13 / 22
Characteristic amplitude of gravitational waves GWs amplitude: � ∞ h 2 h 2 c = c ( k ) d ln k −∞ �� 2 � � 2 � � � k 2 d Ω k � ˜ h phys � ˜ h phys h 2 c ( k ) = ( a ∗ / a 0 ) k + � � � � + × � � 4 π Frequency: f = H ∗ ( a ∗ / a 0 )( k / 2 π ) ≈ 1 . 6475 · 10 − 5 ( k / 2 π ) Hz for T ∗ = 100 GeV, g S ≈ g ∗ = 100. Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 14 / 22
Numerical results for decaying MHD turbulence 7 Initial conditions 8 Fully helical stochastic magnetic field Batchelor spectrum, i.e., E M ∝ k 4 for small k Kolmogorov spectrum for inertial range, i.e., E M ∝ k − 5 / 3 Total energy density at t ∗ is ∼ 10% to the radiation energy density Spectral peak at k M = 100 · 2 π , normalized with k H = 1 / ( cH ) Numerical parameters 1152 3 mesh gridpoints 1152 processors Wall-clock time of runs is ∼ 1 – 5 days 7 A. Roper Pol, et al. arXiv:1903.08585 8 A. Brandenburg, et al. Phys. Rev. D 96 , 123528 (2017) Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 15 / 22
Initial magnetic spectra k M = 15 � � � T ij ( k )˜ ˜ k 2 d Ω T ∗ E T ( k ) = ij ( k ) E M ( k ) = 1 � � � B ( k ) · ˜ ˜ k 2 d Ω B ∗ ( k ) 4 π 2 4 π E T ( k ) = 1 � � ∞ � � B 2 ( k )˜ ˜ B 2 , ∗ ( k ) k 2 d Ω Ω M = E M ( k ) dk 2 4 π 0 � ∞ Ω T = E T ( k ) dk 0 Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 16 / 22
Detectability with LISA LISA Laser Interferometer Space Antenna (LISA) is a space–based GW detector LISA is planned for 2034 LISA was approved in 2017 as one of the main research missions of ESA LISA is composed by three spacecrafts in a distance of Figure: Artist’s impression of LISA from 2.5M km Wikipedia Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 17 / 22
Orbit of LISA Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 18 / 22
Numerical results for decaying MHD turbulence Alberto Roper Pol (University of Colorado) Gravitational Waves from the early-universe April 20, 2020 19 / 22
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