figure courtesy of AEI

Gravitational wave Gravitational wave standard sirens as standard sirens as

cosmological probescosmological probes

Neal Dalal (CITA)with D. Holz, S. Hughes, B. Jain

Outline

1. overview of gravitational waves & detection

2. GW’s from inspiraling binaries

3. constraining cosmology

What are gravitational waves?

Consider metric perturbation ga2(t) [+h].h is a symmetric 4£4 tensor, so 10 components:

• 4 scalar (spin 0) • 4 vortical (spin 1)• 2 shear (spin 2)

For ||h|| 1, linearized vacuum Einstein equations !

0)( 2

2

212

htc

so h satisfies a wave equation. The two spin-2 modes are transverse shear waves propagating at v=c.

What are gravitational waves?

think of GW’s as waves of tidal gravity.

change distance between free-falling observers

L ¼ L h(t)generated by moving masses, with amplitude

So need large m, v to be interesting!

intext22 ~~ 22

2

4 cv

rcGm

dtd

rcGh I

e.g. NS pair with v/c~0.3, observed at r 1000 km, has h~3¢10-4. So for a person of height 2m, L~1mm!

What are gravitational waves?

• essentially non-interacting with matter, once produced.

• act transverse to propagation direction.

• seem wimpy, but are dominant mechanism of energy loss for highly relativistic binaries!

Hulse-Taylor binary pulsarPSR 1913+16

GWologyGW generated by bulk motion of matter, unlike EM waves which are generated by many incoherent patches. GW are coherent, with a characteristic frequency of order the dynamical frequency, f / (G)1/2.

Therefore GW wavelength exceeds the size of the emitting region, ' R c/v. GW cannot resolve their sources and so cannot be used for imaging.

Also – important to remember that strain h is the observable, not the power. So the observable falls off like 1/r, not 1/r2 !

GWology

Schutz (1999)Schutz (1999)

ground vs. space-based

h 2L

L

Absence of a GW:Armlengths are arranged so that the light destructively interferes – no signal is measured.

How to detect?Laser interferometry!

Split laser beam, send light down long paths, with mirrors at each end. Bounce back,

recombine.

Presence of a GW:Positioning of mirrors changes, so armlengths change!Interference is no longer perfect, and we measure an output signal.

How to detect?Laser interferometry!

Split laser beam, send light down long paths, with mirrors at each end. Bounce back,

recombine.

The network of gravitational wave The network of gravitational wave detectorsdetectors

LIGO HanfordLIGO Hanford

LIGO LivingstonLIGO Livingston

ground based laser interferometersground based laser interferometersLIGO/VIRGO/GEO/TAMALIGO/VIRGO/GEO/TAMA

space-based laser interferometer (hopefully space-based laser interferometer (hopefully with get funded for a 201? Lauch)with get funded for a 201? Lauch)

LISALISA

ALLEGRO/NAUTILUS/AURIGA/…ALLEGRO/NAUTILUS/AURIGA/…resonant bar detectorsresonant bar detectors

ALLEGROALLEGROAURIGAAURIGA

Pulsar timing network, CMB anisotropyPulsar timing network, CMB anisotropy

The Crab nebula … a supernovae The Crab nebula … a supernovae remnant harboring a pulsar remnant harboring a pulsar

Segment of the CMB Segment of the CMB from WMAP from WMAP

Images from Patrick BradyImages from Patrick Brady

LIGO GW channel (as of ~ year ago) + injected waveform

Simulated waveform from a binary black hole merger (M1=M2 ~ 10 M๏, at ~ 15 Mpc)

Detection of the inspiral with a SNR~16 after application of the matched filtering algorithm

How do we observe sources?How do we observe sources?the gravitational wave strain is too small by the time the the gravitational wave strain is too small by the time the wave reaches earth to directly “see” the signalwave reaches earth to directly “see” the signal

How do we observe sources?How do we observe sources?

• For the majority of sources, some knowledge of the nature of the source is required for detection of a signal

• Matched filtering will be the primary tool for extracting small, quasi-periodic signals from the data stream

– But because many templates must be run, the SNR threshold for detection must be set high, typically SNR>8.5

• Techniques such as the excess power method can be used for other sources, or if less is known about the exact nature of the source

How well do we know the expected waveforms?

For some sources, well enough!For some sources, well enough!

Survey of some sourcesSurvey of some sourcesWaves from the early universe:

Initial state fluctuationsPhase transitionsCosmic strings

Rotating and vibrating compact objects:Rotating neutron starsModes of neutron star fluidModes of black holes (defer to binaries)

Binaries:Combinations of white dwarfs, neutron stars,and black holes.

Three phases of Three phases of coalescencecoalescence

figure from K. Thorne

1. Inspiral Members are widely separated, distinct bodies.

“Post-Newtonian” expansionworks well.

“Chirping” gravitational waveform

Post-Newtonian expansion• iterative approximation to fully dynamical spacetime

• expansion in (v/c)2.

• For 2-body problem, an accuracy of 3PN has been achieved by several independent methods; all approaches agree.[Blanchet, Damour, Esposito-Farèse, Iyer; Damour, Jarownowski, Schaefer; Itoh]

• reliable up to v/c ' 0.3-0.5

• expect orbits to be circularized quickly if GW emission is dominant energy loss

EC decomposition (in STT form) CTS decomposition (in Dirac coordinates)

•

•

•

•

•

•

•

•

slide from Samaya Nissanke

2. Merger Spacetime transition: From two distinct bodies to a single body.

NO expansion works well!

Modeling requires tackling full nastiness of nonlinear field equations, properties of stars.

Waveform unknown!

Image credit: Teviet Creighton, Caltech

Recent progress in numerical GR!

within past 1-2 yrs, several groups have successfully calculated mergers of comparable-mass BH’s!

lapse function in orbital plane

courtesy F. Pretorius

Recent progress in numerical GR!

within past 1-2 yrs, several groups have successfully calculated mergers of comparable-mass BH’s!

Newman-Penrose scalar 4 (like h+)

courtesy F. Pretorius

3. Ringdown If final state is a black hole, last waves come a system a distorted Kerr black hole.

Black hole perturbation theory describes thesystem.Waveform: Damped harmonic

oscillator.

Three phases of Three phases of coalescencecoalescence

figure from K. Thorne

only rely upon well-understood inspiral phase!

GW from inspirals• can get useful insight from quadrupole approximation

3/113/5

3/23/5

fMf

fD

Mh

Chirp

L

Chirp

• if we observe how fast the frequency chirps, we know how much energy is being radiated in GW. By comparing to the measured strain amplitude, this tells us how distant the source is!(Schutz 1986)

GW from inspirals• the phase evolution is just determined by time until coalescence, tct, and by a combination of masses called the (redshifted) chirp mass

5121

5321

)(

)(chirp )1(

mm

mmzM

• the strain amplitude also depends on same combination!

)](cos[)angles()(

)(3235

chirp tD

tfMth

L

F

• but – emission is not isotropic: depends on inclination • can measure inclination if polarization is measured!

• measured amplitude depends on source direction • can measure this from timing of received signals

GW standard sirens

so the gravitational radiation from inspiraling binaries provides a self-calibrating distance indicator. Just need detectors with different locations and different orientations, to measure polarization and timing.

LIGO Hanford

LIGO Livingston

can achieve this with a network of detectors on Earth …

… while LISA can do both in space!

Cosmology with standard sirens

GW observatories can measure precise distances to sources at cosmological distances.

can be useful for cosmology!H2(z)=8G/3 [m(z)+(z)+K(z)+…]

and dL(z)=(1+z)s (c/H) dz

One problem: distances but no redshifts!So we need merger events that have some sort of EM counterpart to use them as standard sirens.

Binary neutron stars

• known to exist and radiate in GW.

• Galactic merger rate about 10-4 yr-1.

• very plausible that merger could have optical / X-ray counterpart, esp. if it produces BH with accretion disk.

movie courtesy M. Shibata

are short GRBs from NS coalescence??

Short GRBs• origin of short GRBs is still unknown, but NS mergers are a leading candidate!

•if NS merger ! GRB, they are ideal

•afterglow/host galaxy gives z

•known direction decreases distance errors

•known time reduces required SNR threshold!

Cosmology with GW from GRBs

• assume 4-element network of detectors (LIGO-H, LIGO-L, Virgo, AIGO) of comparable sensitivity

• double NS merger detectable out to 600 Mpc.

• distance errors improveif sources are collimated

• GRB trigger may not be necessary! Can get minutesto hrs warning from GW, ~degree localization, good enough for follow-up?

20± beaming

unbeamed

Cosmology with GW from GRBs

unbeamed100 GRBs

unbeamed

20± beaming

How well does this constrain cosmological parameters e.g. dark energy equation of state parameter w?

But how is this possible? We’re using GRBs only out to 600 Mpc, z . 0.15. How can or w be constrained?

Absolute distances!• this works because GW measure absolute distances to sources, in Mpc, not h-1 Mpc. The CMB tells us distance to LSS also in Mpc, so combining the two can measure DE parameters!

• put another way: for flat universe, only 3 parameters: {h, m, w}. CMB provides 2 constraints, on m h2, and on lA= dA(LSS)/rs. A 3rd constraint, like a measurement of H0, determines all three.

• works for any H0 determination, e.g. using water masers. Measuring H0 = measuring w !

more precisely, measures integral constraint on w(z), assuming flat universe.

other GW sources

• focused on GRBs since they have afterglows and a (reasonably) known rate from BATSE, Swift.

• other stellar mass inspirals in LIGO bands, like NS-BH, BH-BH, could also serve as standard sirens, if they have EM counterparts.

• if -rays beamed, but afterglows less so, then even off-axis GRBs could be useful!

• what about LISA?

source frequency (Hz)source frequency (Hz)

sou

rce

sou

rce

“str

en

gth

”“s

tren

gth

”

1010441010-12-12 1010-8-8 1010-4-4 11

relics from the big bang, inflationrelics from the big bang, inflation

exotic physics in the early universe: phase transitions, cosmic strings, domain walls, …exotic physics in the early universe: phase transitions, cosmic strings, domain walls, …

1-10 M1-10 M๏๏ BH/BH BH/BH

mergersmergers

NS/BH mergersNS/BH mergers

NS/NS mergersNS/NS mergers

pulsars, pulsars, supernovaesupernovae

EMR inspiralEMR inspiral

NS binariesNS binaries

WD binariesWD binaries

101022-10-1066 M M๏๏ BH/BH BH/BH

mergersmergers

>10>1066 M M๏๏ BH/BH mergersBH/BH mergers

CMB CMB anisotropyanisotropy

Pulsar timingPulsar timing LISALISA LIGO/…LIGO/…Bar Bar detectordetectorss

Overview of expected gravitational wave Overview of expected gravitational wave sourcessources

Binary black holes in the UniverseBinary black holes in the Universe• Strong circumstantial evidence that black

holes are ubiquitous objects in the universe

– supermassive black holes (106 M๏ - 109 M๏) thought to exist at centers of most galaxies

• high stellar velocities near the centers of galaxies, jets in active galactic nuclei, x-ray emission, …

– more massive stars are expected to form BH’s at the end of their lives

VLA image of the galaxy NGC 326, with HST image of jets inset. CREDIT: NRAO/AUI, STScI (inset)

• Galaxy mergers are observed commonly, suggesting SMBH mergers may also be common.

• LISA can detect all SMBH mergers within the horizon (e.g out to z=10) !

Two merging galaxies in Abell 400. Credits: X-ray, NASA/CXC/ AIfA/D.Hudson & T.Reiprich et al.; Radio: NRAO/VLA/NRL)

Cosmology with LISA

• for LISA standard sirens to be useful, must have ~100 to average out lensing

• merger rates, EM counterparts still uncertain!

100 GW sources, 0<z<2

Conclusions!

• exciting times for GW astronomy• waveforms from inspirals of compact

object binaries are well-understood• these provide a self-calibrating

distance indicator• the number of sources detectable

with ground-based detectors is large enough to provide interesting constraints on cosmology!

upcoming experiments

LIGO • operating at target sensitivity• began science run Nov 2005, expect to continue through 2007• 2008, begin upgrade to LIGO-II (10£ increase in sensitivity!)• LIGO-II begins operations around 2009

Virgo• European observatory, similar sensitivity, expect to follow LIGO by 2-3 yrs

AIGO• Australian observatory, funding uncertain

LISA: ???

slide from Keith Riles

LIGO-IICollaboration between LIGO and GEO600, to upgrade to advanced sensitivity (10£ increase).

• increased laser power (10W ! 100W) • new test mass material (sapphire), lower internal thermal noise in bandwidth• increased test mass (10kg ! 40kg)• new suspension: single ! quadruple pendulum• improved seismic isolation (passive ! active)

10£ increase in sensitivity gives 1000£ in volume!

LISA

LISA - The Overview Mission Description

– 3 spacecraft in Earth-trailing solar orbit separated by 5 x106 km.

– Gravitational waves are detected by measuring changes in distance between fiducial masses in each spacecraft using laser interferometry

– Partnership between NASA and ESA

– Launch date ~2015+

Observational Targets

– Mergers of massive black holes

– Inspiral of stellar-mass compact objects into massive black holes

– Gravitational radiation from thousands of compact binary systems in our galaxy

– Possible gravitational radiation from the early universe

LISA

Orbits

Three spacecraft in triangular formation; separated by 5 million km

Spacecraft have constant solar illumination

Formation trails Earth by 20°; approximately constant arm-lengths

1 AU = 1.5x108 km

LISA

Determining Source Directions

Directions (to about 1 degree) : 2 methods: AM & FM

FM: Frequency modulation due to LISA orbital doppler shifts

– Analagous to pulsar timing over 1 year to get positions

– FM gives best resolution for f > 1 mHz

AM: Amplitude modulation due to change in orientation of array with respect to source over the LISA orbit

– AM gives best resolution for f < 1 mHz

Summary: LISA will have degree level angular resolution for many sources (sub-degree resolution for strong, high-frequency sources)

– See e.g. Cutler (98), Cutler and Vecchio (98), Moore and Hellings (00), also Hughes (02)

(Cornish and Larson, ’01)(F+ & Fx)

LISA

Determining Source Distances

Distances(to about 1%) Binary systems with orbital evolution (df/dt)

– “Chirping” sources

– Determine the luminosity distance to the system by comparing amplitude, h, and period derivative, df/dt, of the gravitational wave emission

– Quadrupole approximation:

Luminosity distance (DL) can be estimated directly from the detected waveform

See e.g. work by Hughes, Vecchio for quantitative estimates

3/113/5

3/23/5

fMf

fD

Mh

Chirp

L

Chirp

LISA

Determining Polarization

LISA has 3 arms and thus can measure both polarizations

Gram-Schmidt orthogonalization of combinations that eliminate laser frequency noise yield polarization modes

– Paper by Prince et al. (2002)

– gr-qc/0209039

L2 L3

L1

(notation from Cutler,Phinney)

X

Y

hHHL

LLL

hHHL

LL

YXXY

YYXX

2

3

4

3)2(

2

3

4

3)(

132

13

LISA

LISA Sensitivity

2-arm “Michelson” sensitivity +White Dwarf binary background

2-arm “Michelson” sensitivity

frequency0.1 mHz 1 Hz

Acceleration Noise(Disturbance Level)

h Tobs

Shot Noise (Measurement Sensitivity)

Short- Limit

White Dwarf Background

(Includes gravitational wave transfer function averaged over sky position and polarization). Source sensitivities plotted as hSqrt(Tobs).

LISA

Rate Estimates for Massive Black Hole Mergers

[Sesana et al, astro-ph/0401543]

Use hierarchical merger trees Rate estimates depend on

several factors– In particular space density of

MBHs with MBH<106 M

– Depends on assumptions of formation of MBHs in lower mass structures at high-z

Some recent estimates– Sesana et al. (2004): about 1 per

month

– Menou (2003): few to hundreds per year depending on assumptions

– Haehnelt (2003): 0.1 to 100 per year depending on assumptions

3 year mission

courtesy T. Prince

LISA

Can LISA Detect Massive Black Holes Mergers?

courtesy T. Prince

Is there an optical counterpart?

• Some modeling suggests likely counterpart

e.g. delayed afterglow (Milosavljevic & Phinney 2004) • inspiral hollows out circumbinary gas• subsequent infall after merger

MacFadyen & Milosavljevic (2006)

• Much more work is warranted

• Regardless of what theorists have to say, error box will be scrutinized for counterparts

Distance determination with optical counterpart

• Typical luminosity distance to much better than 1%

Ultimate standard candle!

Holz & Hughes (2005)

Cosmology with LISA standard sirens

• Non-evolving equation-of-state: / a -3(1+w)

3,000 SNe, 0.7<z<1.7

2 GWs, z=1.5, z=3.

3,000 SNe + 2 GWs

Tremendously powerful probe of dark energy

BEWARE

Gravitational lensing

• The Universe is mostly vacuum, with occasional areas of high density– Photons do not experience a

Robertson-Walker Universe• Gravitational lensing due to

matter inhomogeneities causes a change in brightness of observed images– strong lensing: tremendous

increase in brightness, and multiple images

– weak lensing: slight increase or decrease in brightness

Gravitational Lensing• uniform, isotropic Robertson-Walker universe is

generally assumed– Key assumption: the Universe is filled with

homogeneous matter

Gravitational LensingMagnification Distributions

Probability distribution, P(), of magnification, , due to gravitational lensing

The average magnification is given by the Robertson-Walker value (normalized to 1)

The distributions are peaked at <1, with tails to high magnification

Every source at high z has been gravitationally lensed

z=1

Cosmology with SMBH standard sirens

Without the effects of gravitational lensing

3,000 SNe, 0.7<z<1.7

3,000 SNe + 2 GWs

When neglecting lensing, even a few SMBH standard sirens have a major impact on cosmology!

Cosmology with SMBH standard sirens

Including the effects of gravitational lensing

3,000 SNe, 0.7<z<1.7

3,000 SNe + 2 GWs

3,000 SNe + 2 GWs + lensing

Lensing seriously compromises the use ofSMBH standard sirens!