Review for Final Also: Review homework problems Example: Binary - PowerPoint PPT Presentation

Review for Final Also: Review homework problems

Example: Binary star system in Virgo cluster (16.5 Mpc away) would produce h ~ 10 -21 . Over a distance of L = 1 AU, Δ L would be ~ 1 atomic diameter. Credit: LIGO

Gravitational-wave Sources for Ground-based Detectors

Gravitational-wave Spectrum

What is a tensor? Scalar - tensor rank 0, magnitude, ex: Temperature Vector - tensor rank 1, magnitude and direction, ex: Force Tensor - combination of vectors where there is a fixed relationship, independent of coordinate system; ex: Dot product, work T mn = A m B n Principle of relativity - “Physics equations should be covariant under coordinate transformation.” To ensure that this is automatically satisfied, write physics equations in terms of tensors.

Einstein Summation Convention Repeated indices imply summation. 3 X A µ B µ = A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 A µ B µ = µ =0   A 0 A 1   = [ B 0 B 1 B 2 B 3 ]   A 2   A 3 Dummy index - appears exactly Free index - appears twice in one given term of exactly once in every equation but only once in term of equation equation

The Metric of Curved Space: General Relativity General relativity as a geometric theory of gravity posits that matter and energy cause spacetime to warp so that g µ ν 6 = η µ ν Thus gravitational phenomena are just effects of a curved spacetime on a test particle. Source particle Field Test Particle Field Equation equation of motion Source Curved spacetime Test Particle Einstein Geodesic Field equation equation

Tensor Calculus: Covariant Derivative Ordinary derivatives of tensor components are not tensors. ∂ ν A µ The combination does not transform properly. ν A 0 µ 6 = ∂ x λ ∂ x 0 µ ∂ ν A µ ! ∂ 0 ∂ x ρ ∂ λ A ρ ∂ x 0 ν We seek a covariant derivative to be used in covariant r ν physics equations. Such a differentiation is constructed so that when acting on tensor components it still yields a tensor. ν A 0 µ = ∂ x λ ∂ x 0 µ r ν A µ ! r 0 ∂ x ρ r λ A ρ ∂ x 0 ν In order to produce the covariant derivative, the ordinary derivative must be supplemented by another term: r ν A µ = ∂ ν A µ + Γ µ r ν A µ = ∂ ν A µ � Γ λ νλ A λ ν µ A λ

The Geodesic Mathematical expression for parallel transport of vector components is r A µ = dA µ + Γ µ νλ A ν dx λ = 0 x µ ( σ ) A µ The process of parallel transporting a vector along a curve can be expressed according to: νλ A ν dx λ dA µ d σ + Γ µ d σ = 0 But the geodesic is a curve for which the tangent vector parallel transports itself, i.e.: A µ = dx µ d σ Thus, the geodesic equation is: d 2 x µ dx ν dx λ d σ 2 + Γ µ d σ = 0 νλ d σ

Curvature and the Riemann Tensor Local Lorentz Frame: effects of curvature become noticeable when taking second derivatives. [ r α , r β ] A µ = r α r β A µ � r β r α A µ ⌘ R µ λαβ A λ να Γ ν νβ Γ ν R µ λαβ = ∂ α Γ µ λβ − ∂ β Γ µ λβ − Γ µ λα + Γ µ λα In Local Lorentz Frame: R µ ναβ = 1 2 ( ∂ µ ∂ α g νβ − ∂ ν ∂ α g µ β + ∂ ν ∂ β g µ α − ∂ µ ∂ β g να ) ∂ 2 g + ( ∂ g ) 2 R = d Γ + ΓΓ Form of Riemann Tensor: In flat space , the first and second derivatives of the metric vanish. R µ λαβ = 0 implies flat space.

Motivating Einstein Equations ? We need the spacetime curvature term on the left. Einstein thought it should be the Ricci curvature tensor. But there is a problem. Due to energy conservation: But the derivative of Ricci tensor does not equal zero as can be seen with the Bianchi Identities. Instead, what is found is ✓ ◆ R µ ν � 1 G µ ν ≡ R µ ν − 1 r µ 2 g µ ν R = 0 2 g µ ν R Einstein tensor

Methods Solving Einstein’s equations is difficult. They’re non-linear. In fact, the equations of motion are impossible to solve unless there is some symmetry present. In the absence of symmetry, there are two methods: 1. Numerical relativity (next time) 2. Approximation techniques For the approximation technique, we consider a metric very close to flat space with a small perturbation. And we consider only first order perturbations.

Linearized Theory of Metric Field And impose the harmonic gauge, then the last three terms in previous equation vanish and we end up with the Linearized Einstein Equations h µ ν = − 16 π G ⇤ ¯ T µ ν c 4

Solution in a Vacuum What happens outside the source, where ? T µ ν = 0 Then, the EFE reduces to ⇤ ¯ h µ ν = 0 ✓ ◆ � 1 c 2 ∂ t 2 + r 2 ¯ h µ ν = 0 Wave equation for waves propagating at speed of light c! Solutions to wave equation can be written as superpositions ~ of plane waves traveling with wave vectors and frequency k � � � ~ ! = c k � � �

Solution with Source Now allow for source. What would cause the waves to be generated? h µ ν = − 16 π G ⇤ ¯ T µ ν c 4 Solve using retarded Green’s function assuming no incoming radiation from infinity. The solution is ✓ ◆ x ) = 4 G 1 t − | ~ x 0 | Z x − ~ ¯ d 3 x 0 x 0 h µ ν ( t, ~ , ~ x 0 | T µ ν c 4 | ~ x − ~ c

Generation of Gravitational Waves To leading order in v/c , we can eliminate the multipole moments in favor of the mass moments to get a solution of the form: quad = 1 2 G M kl ( t − r/c ) n ) ¨ h TT ⇥ ⇤ ij ( t, ~ x ) c 4 Λ ij,kl (ˆ r S ij = 1 ¨ M ij where we have used: 2 Mass quadrupole radiation!

Effect of Gravitational Waves on Matter h + polarization δ x ( t ) = h + 2 x 0 cos( ω t ) δ y ( t ) = − h + 2 y 0 cos( ω t ) h x polarization δ x ( t ) = − h × 2 y 0 cos( ω t ) δ y ( t ) = − h × 2 x 0 cos( ω t )

Noise spectral density is the noise spectral density (aka noise spectral S n ( f ) sensitivity or noise power spectrum): Z ∞ n 2 ( t ) ⌦ ↵ = f S n ( f ) d 0

Interferometric GW Detector Pattern Functions F + ( θ , φ ; ψ = 0) = 1 1 + cos 2 θ � � cos 2 φ 2 F × ( θ , φ ; ψ = 0) = cos θ sin 2 φ Thus GW interferometers have blind directions. For instance, for a GW with plus polarization, y and φ = π / 4 F + = 0 x This wave produces the same displacement in the and arm. y x Differential phase shift vanishes!

Define the signal-to-noise ratio... Using this scalar product definition, we have: ( u | h ) S u ( f ) = 1 2 S n ( f ) ˜ where N = ˜ K ( f ) ( u | u ) 1 / 2 u/ ( u | u ) 1 / 2 We are searching for vector such that its scalar product with vector h is maximum. ˜ h ( f ) They should be parallel (i.e. proportional): ˜ K ( f ) = const . S n ( f ) This is the Wiener filter (aka matched filter).

Burst Signals • GW searches with minimal assumptions • Do not assume accurate source models - affected by noise • Limited ability to measure waveforms, sky positions, polarizations • Can detect the unexpected Cosmic string cusps Heavy stellar binary Supernovae Pulsar glitching black holes Eccentric binary black holes Credit: NASA, ESA, Sanskrit, Credit: NASA, CXC, PSU, Credit: B. Allen & E. P . Shellard Blair Pavlov

Continuous Gravitational Wave Sources Non-axisymmetric rotating neutron stars; asymmetry could arise from: • equatorial ellipticity (mm-high mountain) • free precession around rotation axis • excitation of long-lasting oscillations • deformation due to matter accretion Credit: NASA/CXC/PSU/ Credit: NASA/HST/ASU/ Pavlov, et al. CXC/Hester, et al. mpy Neutron Sta

What is a stochastic background? • Stochastic (random) background of gravitational radiation • Can arise from superposition of large number of unresolved GW sources 1. Cosmological origin 2. Astrophysical origin • Strength of background measured as gravitational wave energy density ρ GW

Detecting Stochastic Backgrounds The filter function has the form: Q ( f ) = N γ ( f ) Ω GW ( f ) H 2 ˜ 0 f 3 P 1 ( f ) P 2 ( f ) γ ( f ) overlap reduction function: Ω GW ( f ) = Ω α ( f/ 100 Hz) α power law template for GW spectrum: present value of Hubble parameter: H 0 P 1 ( f ) noise in detector 1: P 2 ( f ) noise in detector 2: Purpose: Enhance SNR at frequencies where signal is strong and suppress SNR at frequencies where detector noise is large.

Overlap Reduction Function Signal in two detectors will not be exactly the same because: i) time delay between detectors ii) non-alignment of detector

Michelson interferometer ETMY L y ETMX BS L x Laser Laser Photodiode Monochromatic light source ● 50/50 beam-splitter, note sign flip in reflection coefficient ● to conserve energy (see Stokes relations) Perfectly reflecting end-mirrors (End Test Mass): ● Light of arms interferes on photodiode, which measures power ● B. Swinkels – Experimental GW detection 11

Interferometric GW detection Michelson interferometer is a natural fit for measuring gravitational waves: GW cause a ● differential change of arm length: Idea first proposed by Braginsky, first technical feasibility study by R. Weiss (1972) ● Note: interferometers measure the amplitude of the GW and not the power, so ● dependency on source distance is 1/R instead of 1/R^2 A simple Michelson is not sensitive enough to detect GW, need several extra tricks ... ● B. Swinkels – Experimental GW detection 13