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Gravitational wave standard sirens as cosmological probes. Neal Dalal (CITA) with D. Holz, S. Hughes, B. Jain. figure courtesy of AEI. Outline. overview of gravitational waves & detection GW’s from inspiraling binaries constraining cosmology. What are gravitational waves?.

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Gravitational wave standard sirens as cosmological probes

Gravitational wave standard sirens as cosmological probes

Neal Dalal (CITA)

with D. Holz, S. Hughes, B. Jain

figure courtesy of AEI


  • overview of gravitational waves & detection

  • GW’s from inspiraling binaries

  • constraining cosmology

What are gravitational waves
What are gravitational waves?

  • Consider metricperturbation gmn=a2(t) [hmn+hmn].

  • 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 !

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

What are gravitational waves1
What are gravitational waves?

think of GW’s as waves of tidal gravity.

change distance between free-falling observers

DL¼L h(t)

generated by moving masses, with amplitude

So need large m, v to be interesting!

e.g. NS pair withv/c~0.3, observed atr = 1000 km, hash~3¢10-4. So for a person of height 2m, DL~1mm!

What are gravitational waves2
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 pulsar

PSR 1913+16


GW 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 !


Schutz (1999)

How to detect
How to detect?

Laser interferometry!

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

Absence of a GW:

Armlengths are arranged so that the light destructively interferes – no signal is measured.

How to detect1
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.

The network of gravitational wave detectors
The network of gravitational wave detectors



ground based laser interferometers

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

LIGO Hanford

LIGO Livingston


Pulsar timing network, CMB anisotropy

resonant bar detectors

Segment of the CMB from WMAP


The Crab nebula … a supernovae remnant harboring a pulsar


How do we observe sources
How do we observe sources?

  • the gravitational wave strain is too small by the time the wave reaches earth to directly “see” the signal

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

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

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

Images from Patrick Brady

How do we observe sources1
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 filteringwill 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 theexcess power methodcan 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
How well do we know the expected waveforms?

For some sources, well enough!

Survey of some sources
Survey of some sources

Waves from the early universe:

Initial state fluctuations

Phase transitions

Cosmic strings

Rotating and vibrating compact objects:

Rotating neutron stars

Modes of neutron star fluid

Modes of black holes (defer to binaries)


Combinations of white dwarfs, neutron stars,

and black holes.

Three phases of coalescence
Three phases of coalescence

figure from K. Thorne

1. Inspiral

Members are widely separated, distinct bodies.

“Post-Newtonian” expansion

works well.

“Chirping” gravitational waveform

Post newtonian expansion
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


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.

Image credit: Teviet Creighton, Caltech

Waveform unknown!

Recent progress in numerical gr
Recent progress in numerical GR!

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

lapse function a in orbital plane

courtesy F. Pretorius

Recent progress in numerical gr1
Recent progress in numerical GR!

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

Newman-Penrose scalar y4 (like h+)

courtesy F. Pretorius


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

Black hole perturbation theory describes the


Waveform: Damped harmonic oscillator.

Three phases of coalescence1
Three phases of coalescence

only rely upon well-understood inspiral phase!

figure from K. Thorne

Gw from inspirals
GW from inspirals

  • can get useful insight from quadrupole approximation

  • 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 inspirals1
GW from inspirals

  • the phase evolution is just determined by time until coalescence, tc-t, and by a combination of masses called the (redshifted) chirp mass

  • the strain amplitude also depends on same combination!

  • 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

LIGO Hanford

LIGO Livingston

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.

can achieve this with a network of detectors on Earth …

… while LISA can do both in space!

Cosmology with standard sirens
Cosmology with standard sirens

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

 can be useful for cosmology!

H2(z)=8G/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
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
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


20± beaming

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 improve

  • if sources are collimated

  • GRB trigger may not be

  • necessary! Can get minutes

  • to hrs warning from GW,

  • ~degree localization, good

  • enough for follow-up?

Cosmology with gw from grbs1
Cosmology with GW from GRBs

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


100 GRBs


20± beaming

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

  • what about LISA?

Overview of expected gravitational wave sources

Pulsar timing



Bar detectors

CMB anisotropy

>106 M๏ BH/BH mergers

102-106 M๏ BH/BH mergers

source “strength”

1-10 M๏ BH/BH mergers

NS/BH mergers

NS/NS mergers

pulsars, supernovae

EMR inspiral

NS binaries

WD binaries

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

relics from the big bang, inflation






source frequency (Hz)

Binary black holes in the universe
Binary 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
Cosmology with LISA

100 GW sources, 0<z<2

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

  • merger rates, EM counterparts still uncertain!


  • 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
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: ???

Ligo ii

  • Collaboration 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 the overview
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


  • 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

Determining source directions
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)

Determining source distances
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

Determining polarization
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






(notation from Cutler,Phinney)

Lisa sensitivity

2-arm “Michelson” sensitivity

Acceleration Noise

(Disturbance Level)

Short- Limit

Shot Noise

(Measurement Sensitivity)

1 Hz

0.1 mHz


LISA Sensitivity

2-arm “Michelson” sensitivity +

White Dwarf binary background

White Dwarf Background

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

Rate estimates for massive black hole mergers
Rate Estimates for Massive Black Hole Mergers

  • 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

[Sesana et al, astro-ph/0401543]

courtesy T. Prince

Can LISA Detect MassiveBlack Holes Mergers?

courtesy T. Prince

Is there an optical counterpart
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

  • Much more work is warranted

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

MacFadyen & Milosavljevic (2006)

Distance determination with optical counterpart
Distance determination with optical counterpart

  • Typical luminosity distance to much better than 1%

    Ultimate standard candle!

Holz & Hughes (2005)

Cosmology with lisa standard sirens
Cosmology with LISA standard sirens

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

Tremendously powerful probe of dark energy

3,000 SNe, 0.7<z<1.7

2 GWs, z=1.5, z=3.

3,000 SNe + 2 GWs


Gravitational lensing

Gravitational lensing
Gravitational Lensing

  • uniform, isotropic Robertson-Walker universe is generally assumed

    • Key assumption: the Universe is filled with homogeneous matter

  • 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 magnification distributions
Gravitational LensingMagnification Distributions


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

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

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

Every source at high z has been gravitationally lensed

Cosmology with smbh standard sirens
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 sirens1
Cosmology with SMBH standard sirens

Including the effects of gravitational lensing

3,000 SNe, 0.7<z<1.7

3,000 SNe + 2 GWs + lensing

3,000 SNe + 2 GWs

Lensing seriously compromises the use of

SMBH standard sirens!