Gravitational waves from the early Universe Part 2 Sachiko Kuroyanagi (Nagoya University) 26 Aug 2017 Summer Institute 2017
GWs from inflation
Inflation Accelerated expansion in the early Universe Solves horizon/flatness/monopole problem
Horizon problem H -1 : Hubble horizon (region of causality = the distance light can travel ) aλ: physical scale past aλ H -1 aλ H -1 today no causal relation now radiation matter-dominated H -1 a 3/2 or a 2 horizon grows faster than physical scale log(a)
Horizon problem log(scale) Inflation phase H -1 const a exp(ht) exponential growth of the scale factor aλ H -1 today inside the causal region radiation matter-dominated H -1 a 3/2 or a 2 log(a)
Standard scenario What drives inflation? Inflation is driven by a scalar field slowly rolling down in its potential new particle? modification of gravity? V slow- roll ϕ energy density of a scalar field Equation of Motion Friedmann equation slow-roll approximation H = const. a exp(ht) Exponential expansion
Standard scenario Reheating The scalar field decays into standard model particles matter-dominated like Universe for φ 2 potential radiation-dominated for φ 4 potential V Oscillation ϕ Equation of Motion scalar field radiation :decay rate Friedmann equation Reheating temperature
GWs from inflation GW Power H 2 inf Vinf wide range
log(scale) Inflation H -1 const Generation mechanism stretched over the horizon Today becomes GWs Hubble horizon H -1 wavelength of GW a/k quantum fluctuation in space-time scale factor of the Universe log(a)
Generation mechanism log(scale) Inflation H -1 const long wavelength CMB Direct detection Hubble horizon H -1 wavelength of GW a/k short wavelength GWs are produced for all wavelength log(a)
Equation for GWs in the expanding Universe oscillation 1 ḧ ij +3Hḣij expansion of the Universe a 2 2 h ij = 16 G ij effect from matter Let us neglect the matter contribution and perform Fourier transform H>k/a H<k/a behavior is determined by the balance between H (Hubble) and k (wavenumber = f/2π)
Hubble expansion history after inflation log(scale) Inflation H -1 const V driven by a scalar field slow-roll ϕ in the conventional cosmological scenario RD Reheating H -1 a 2 MD H -1 a 3/2 H -1 a 3/2 V decay of the scalar field Hubble horizon size H -1 wavelength of GW a/k oscillations ϕ log(a)
Hubble expansion history after inflation log(scale) Inflation H -1 const in the conventional cosmological scenario RD Reheating H -1 a 2 MD H -1 a 3/2 H -1 a 3/2 2nd term Hubble horizon size H -1 wavelength of GW a/k 3rd term log(a)
Evolution of GWs log(scale) Inflation H -1 const Reheating H -1 a 3/2 outside the horizon RD H -1 a 2 MD H -1 a 3/2 Hubble horizon size H -1 wavelength of GW inside the horizon decay a/k log(a)
Hubble expansion history after inflation log(scale) Inflation H -1 const Reheating H -1 a 3/2 outside the horizon RD H -1 a 2 CMB Direct detection Hubble horizon size H -1 wavelength of GW a/k inside the horizon Each mode experiences different evolution = different amplitude for different frequency log(a)
GW amplitude log(scale) Inflation H -1 const Today Hubble horizon size H -1 wavelength of GW a/k quantum fluctuation in space-time initial condition 1 h k = a p km pl decay scale factor of the Universe log(a)
GW amplitude log(scale) Inflation H -1 const Today H -1 wavelength of GW a/k low energy scale inflation quantum fluctuation in space-time initial condition 1 h k = a p km pl decay scale factor of the Universe log(a)
GW amplitude log(scale) Inflation H -1 const Today Horizon out at k=ah a -1 H aout,1 < aout,2 H -1 wavelength of GW a/k quantum fluctuation in space-time initial condition 1 h k = a p km pl decay scale factor of the Universe log(a)
log(scale) Inflation H -1 const GW amplitude GW Power H 2 inf Today hk,high aout,high > < hk,low aout,low low-scale inflation H -1 wavelength of GW a/k quantum fluctuation in space-time initial condition 1 h k = a p km pl decay scale factor of the Universe log(a)
Amplitude of GW background Spectral shape Outside the horizon Inside the horizon Ω GW scale invariant = ΩGW k 2 P T k 0 Inflation reheating a exp(ht) scale invariant spectrum a t 2/3 k -2 k 0 k - 2 small scale modes begin to enter the horizon and damp with a -1 radiation dominant matter dominant a t 1/2 a t 2/3 k 0 the expansion decelerates = damping a -1 becomes smaller k -2 k -2 k
Spectral shape k -2 spectral amplitude k 0 k -2 frequency = k/2π
Spectral shape V slow- roll Power spectrum PT(k) k 0 ϕ spectral amplitude For slow-roll inflation GW Power H 2 inf Vinf frequency = k/2π
Effect of reheating V Reheating after inflation for Φ 2 potential, H a -3/2 matter-dominated Universe ϕ spectral amplitude reheating temperature TRH low ΩGW, today k -2 high frequency = k/2π
Effect of reheating V Reheating after inflation for Φ 4 potential, H a -2 radiation-dominated Universe ϕ spectral amplitude ΩGW, today k 0 frequency = k/2π
Effect of reheating Reheating after inflation kinatic energy > potential energy H a -6 kination-dominated Universe spectral amplitude ΩGW, today k 1 reheating temperature TRH low high frequency = k/2π
Spectral shape from numerical calculation matter dominated Inflation (m 2 Φ 2 potential) radiation dominated reheating V(Φ) MD ϕ RD tilt of the spectrum + deviation from the slow-roll MD S. Kuroyanagi, T. Chiba and N. Sugiyama, Phys. Rev. D 79, 103501 (2009)
Spectral shape from numerical calculation S. Kuroyanagi, T. Chiba and N. Sugiyama, Phys. Rev. D 79, 103501 (2009)
Effective number of degrees of freedom Damping due to the changes in effective number of degrees of freedom g * As the temperature of the universe decreases, relativistic matter particles become non-relativistic. log( g * ) becomes non-relativistic when T~m log( T [MeV] ) primordial spectrum with tilt temperature decreases contribution to ρ and s decreases step-like changes in H step shape in ΩGW
Damping due to the neutrino anisotropic stress Neutrino anisotropic stress anisotropic stress term =0 Before neutrino decoupling (T>2MeV) Anisotropic stress is suppressed by the coupling with matter (e ± )
Damping due to the neutrino anisotropic stress Neutrino anisotropic stress initially =0 gives energy Damping only when H~k/a Before neutrino decoupling (T>2MeV) Anisotropic stress is suppressed by the coupling with matter (e ± ) After neutrino decoupling (T<2MeV) Neutrino anisotropic stress affects GWs as a viscosity when they enter the horizon
Damping due to the neutrino anisotropic stress Neutrino anisotropic stress anisotropic stress term ρ ν ~0 Before neutrino decoupling (T>2MeV) Anisotropic stress is suppressed by the coupling with matter (e ± ) After neutrino decoupling (T<2MeV) Neutrino anisotropic stress affects GWs as a viscosity when they enter the horizon After the Universe becomes matter-dominated The energy density of radiation becomes negligible
Neutrino anisotropic stress Damping due to the neutrino anisotropic stress Start of matter domination Neutrino decoupling anisotropic stress term Before neutrino decoupling (T>2MeV) Anisotropic stress is suppressed by the coupling with matter (e ± ) After neutrino decoupling (T<2MeV) Neutrino anisotropic stress affects GWs as a viscosity when they enter the horizon After the Universe becomes matter-dominated The energy density of radiation becomes negligible
Dependence on inflation models Constraint on inflation from Planck Planck 2015 results. XX. Constraints on inflation, Planck Collaboration, A&A, 594 (2016) A20
Dependence on inflation models -11-12 -13 DECIGO Upgraded DECIGO Ultimate DECIGO T RH =10 9 GeV Kuroyanagi et al. PRD 79, 103501 (2009) Kuroyanagi et al. PRD 90, 063513 (2014) log 10 Ω GW -14-15 -16 no MD -17 λφ 4 /4-18 -19 m 2 φ 2 /2 Natural inflation: f=7m pl R 2 inflation quantum limit -15-10 -5 0 5 log 10 f[hz]
GWs from cosmic strings
Cosmological history Log(time) cosmic superstring inflation reheating cosmic phase cosmic string now transitions We are here Theories
Phase transition in the Universe potential V Φ2 low energy Φ1 low energy Φ height = energy Φ=0 higher energy vacuum Φ=Φ2 vacuum Φ=Φ1 In 3-dimension space, it looks like a wall
Phase transition in the Universe Φ=0 high energy in a string 2- dimensional potential in 3-dimension space becomes a string Φ=σ Cosmic string Z Z 2 d 2 x 2 Line Density: (@ i ) 2 d 2 x r
Important notes 1: Phase transition In all acceptable spontaneous symmetry breaking schemes, cosmic string formation is unavoidable 2: Cosmic superstrings Jeannerot et al., PRD 68 103514 (2003) The remarkable fact is that, although many possibilities remain, every one of them predicts the formation of topological or embedded cosmic strings at the end of inflation. So it seems that cosmic strings are almost unavoidable. Kibble, Lecture at COSLAB 2004, arxiv:astro-ph/0410073
Cosmic string network infinite string becomes a loop by reconnection loop strings emit gravitational waves especially from singular structures infinite string kink cusp loops lose energy and shrink by emitting GWs and eventually evaporate
GWs from cosmic strings cusps on loops kinks on infinite strings GW from cusps are dominant
What determines the GW amplitude? 3 main parameters to characterize cosmic string G tension = line density Generation mechanism initial loop size L H -1 Network evolution p reconnection probability Phase transition origin: p=1 Cosmic superstring: p<<1
Evolution of cosmic string network The energy density of cosmic strings (line density length)/volume a 1 a 3 a -2 X a: scale factor of the Universe energy density a -4 radiation Cosmic strings a -3 a -2 matter L a 1 V a 3 becomes dominant component of the Universe? a: scale factor
Evolution of cosmic string network The energy density of cosmic strings (line density length)/volume a 1 a 3 a -2 X a: scale factor of the Universe energy density a -4 radiation a X -2 Cosmic strings GWs L a 1 V a 3 follows the total energy density of the Universe a: scale factor
Evolution of cosmic string network Scaling law The Universe always has O(1-10) strings per horizon String network keeps producing loops Strings can communicate each other inside the horizon horizon reconnection Loops disappear after continuous GW emission
Evolution of loops depends on Gμ and α Initial loop length GW power ti time when the loop formed Γ: numerical constant 50-100 From the energy conservation law energy of loop at time t =μl initial energy of the loop =μαti enegry released to GWs =PΔt Loop length at time t Lifetime of the loop initial loop energy energy release rate per time i i
Gravitational waves from cosmic string loops Gravitational waves coming from different directions overlap each other and form gravitational wave background Search for bursts & stochastic background are both important
Wide range GW background spectrum Scaling law The Universe always has O(1-10) strings per horizon Early Universe Late Universe horizon GW frequency ~ 1 / (loop size) high frequency GWs low frequency GWs
Spectrum of the GWB dependence on Gμ 10-6 new Matter dominant Radiation dominant old large Gμ 10-8 10-10 ΩGW 10-12 small Gμ 10-14 10-16 10-18 Gμ=10-8, 10-10, 10-12, 10-14, 10-16 α=10-1, p=1 10-18 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 10 0 10 2 10 4 frequency [Hz] GW power from cusps h 2 (Gμ) 2 life time of loops (Gμ) -1
Spectrum of the GWB dependence on α loop size directly corresponds to the frequency of the GW 10-6 small α 10-7 10-8 10-9 ΩGW 10-10 10-11 10-12 10-13 Gμ=10-8, p=1 α=10-1, 10-5, 10-9, 10-13, 10-17 10-14 10-18 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 10 0 10 2 10 4 frequency [Hz]
Spectrum of the GWB dependence on p 10-2 small p increases the number density of loops small p 10-4 10-6 10-8 ΩGW 10-10 10-12 large p 10-14 10-16 Gμ=10-12, α=10-1 p=1,10-1, 10-2, 10-3 10-18 10-18 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 10 0 10 2 10 4 frequency [Hz]
Accessible cosmic string parameter space reconnection probability p=1 Line density Gμ G Dotted: burst Solid: background α Loop size α Kuroyanagi et al. PRD 87, 023522 (2013)
Accessible cosmic string parameter space reconnection probability p=10-2 Line density Gμ Dotted: burst Solid: background Loop size α Kuroyanagi et al. PRD 87, 023522 (2013)
Accessible cosmic string parameter space reconnection probability p=10-2 Line density Gμ Theoretical lower limit of cosmic superstring theory Gμ>10-12 Dotted: burst Solid: background Loop size α Kuroyanagi et al. PRD 87, 023522 (2013)
Summary GWs can become a powerful probe of the very early Universe GWs from inflation are promising (but the amplitude may be small) It may provide information not only on inflation but also on the thermal history after inflation (eg. reheating) GWs from cosmic strings has relatively large amplitude and testable by near-future experiments It will be useful constraint for the model building of the early Universe