Testing gravity: equivalence principle, and gravitational ... · i.e. photons bend in a...

Testing gravity: equivalence principle, and gravitational waves

Lam HuiColumbia University

Monday, May 14, 2012

Outline:

1. Equivalence principle testing. Collaborators: Phil Chang, Alberto Nicolis, Chris Stubbs.

2. Gravitational wave detection. Collaborators: Sean McWilliams, I-Sheng Yang.


Essentially all attempts to explain cosmic acceleration by modifying gravity involves introducing new long range forces. The simplest option is a scalar force i.e. these theories are some kind ofscalar-tensor theory (at least in appropriate limits).This includes theories that at first sight might not look like a scalar-tensor theory e.g. DGP, massive gravity.

Ubiquitous nature of scalar-tensor theories:

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Essentially all attempts to explain cosmic acceleration by modifying gravity involves introducing new long range forces. The simplest option is a scalar force i.e. these theories are some kind ofscalar-tensor theory (at least in appropriate limits).This includes theories that at first sight might not look like a scalar-tensor theory e.g. DGP, massive gravity.

Ubiquitous nature of scalar-tensor theories:

yx

By Weinberg/Deser theorem, at low energies, a Lorentz invariant theoryof a massless spin 2 particle must be general relativity (GR).This is why essentially all proposed long distance modifications to GRend up introducing a new particle, usually a scalar, mediating an extralong range force.


Also: if one is willing to consider some form of dark energy otherthan the cosmological constant e.g. some kind of light scalar (quintessence), it is very natural to think this scalar might coupleto matter. In fact, absent symmetries that forbid it, such a scalar-matter coupling is inevitable, i.e. a scalar-tensor theory.


What do these scalar-tensor theories look like?

S = SGR +

�d4x

�−1

2(∂ϕ)2 + Lint(ϕ)

�+

�d4x [hµνTm

µν + ϕTm]

graviton action:general relativity

scalar action:includes self-interaction

graviton-mattercoupling

scalar-mattercoupling

hµν = graviton

ϕ = scalar

Tmµν = matter energy −momentum

Graviton-matter coupling is universal i.e. graviton couples to all forms ofmatter with the same strength. This is forced upon us by consistency i.e. Einstein’s equivalence principle (Weinberg’s theorem).

Scalar-matter coupling is also universal. This is by construction.i.e. Let us consider the minimal models where the scalar is coupledto electrons, protons, dark matter, etc. at the same (grav.) strength,i.e. these theories are constructed to obey the equivalence principle.

Einstein frame

MP = 1


Despite the (microscopic) universal scalar-matter coupling, “equivalence principle” is violated anyhow.

S = SGR +

�d4x

�−1

2(∂ϕ)2 + Lint(ϕ)

�+

�d4x [hµνTm

µν + ϕTm]

Microscopic: scalar does not couple to photons whose ,i.e. photons bend in a gravitational field but not in a scalar field.This is why comparing lensing versus dynamical masses is useful(see Rachel Mandelbaum’s talk).

Tm = Tmµµ = 0


Despite the (microscopic) universal scalar-matter coupling, equivalence principle is violated anyhow.

S = SGR +

�d4x

�−1

2(∂ϕ)2 + Lint(ϕ)

�+

�d4x [hµνTm

µν + ϕTm]

Macroscopic: scalar charge gets renormalized in astronomical objects.

mass scalar charge

net force on an object = −M∇Φgrav. −Q∇ϕ

These scalar-tensor theories give Q/M = 1 for microscopic particles i.e. . Macroscopic objects, however, can have ; in fact Q/M ratio depends on the internal structure of the objects, thus leading to equivalence principle violations.

Chameleon/symmetron theories: non-derivative interaction e.g. . for objects with sufficiently deep GM/R.

Galileon theories: derivative interactions e.g. (DGP).The symmetry preserves Q/M = 1, because scalar equationtakes a Gauss law form: .

Lint(ϕ) ∼ 1/ϕ

Q ∼�

d3x [Tm + ∂ϕLint(ϕ)] 0

Q =

�d3xTm = M

Q/M �= 1

Lint(ϕ) ∼ (∂ϕ)2�ϕ

(ϕ → ϕ+ c+ b · x)∂ · J ∼ Tm A more careful derivation using Einstein, Infeld, Hofmann

(LH, Nicolis, Stubbs).


Chameleon screening.The scalar force has a range that depends on environment: in an object witha deep GM/R (e.g. the Sun!), the range becomes very small.

screened object Q/M = 0

ϕ

ϕTm

S = SGR +

�d4x

�−1

2(∂ϕ)2 + Lint(ϕ)

�+

�d4x [hµνTm

µν + ϕTm]

unscreened object Q/M = 1

Screened and unscreened object fall at rates that are O(1) different. Examples:

Observationally, we know objects with are screened e.g. Sun, Milky way.(Otherwise, we would have known about the scalar force already!)

GM/R > 10−6

Diffuse gas (e.g. HI) and stars could yield discrepant dynamical masses fromrotation curves of dwarf galaxies. (Beware: asymmetric drift.)

Small galaxies could stream out of voids faster than large galaxies.

Red giants in dwarf galaxies can have a screened core but unscreened envelope,i.e. effective G variation within the star, giving at tip of RG branch.∆T ∼ 100K

Lensing mass = dynamical mass for screened galaxy, but not for unscreened galaxy.

Chamleleon: Khoury & Weltman, see Justin’s talk.

Lint(ϕ)

GM/R > ϕextGM/R < ϕext


A generic test of scalar-tensor theories.Idea: assuming black holes have no scalar hair/charge, while normal stars do, let’s check for the difference in their rate of free fall (Nordvedt).(More generally, compact objects should have Q/M < 1, because Mis dominated by gravitational binding energy rather than .)

This effect would be hopeless to detect, for classic Brans-Dicke theory,because solar system tests already tell us the Brans-Dicke scalarmust be very weakly coupled i.e. the scalar force is much weaker thangravity.

Recent versions of scalar-tensor theories offer a hope: they pass solar system tests, yet have interesting O(1) effects elsewhere (e.g. in cosmology).

S = SGR +

�d4x

�−1

2(∂ϕ)2 + Lint(ϕ)

�+

�d4x [hµνTm

µν + ϕTm]

Tmµν


Vainshtein screening - galileon theories, massive gravity.

Large scale structure produces an unscreened scalar that can penetrategalaxies (LH, Nicolis).

(The galileon symmetry means: given any nonlinear solution, adding a linear gradient gives another solution.)

ϕextthe linear galaxy

falls

Galileon: Nicolis, Rattazzi & Trincherini; Massive gravity: de Rahm, Tolley & Gabadadze; also gives self-acceleration.Lint(ϕ)


The idea is to look for the offset of massive black holes from thecenters of galaxies.The offset should be correlated with the direction of the streamingmotion. The massive black holes can take the form of quasars orlow luminosity galactic nuclei i.e. Seyferts.

The offset is estimated to be up to 0.1 kpc, for small galaxies.

Beware: other astrophysical effects (e.g. case of M87), compact star cluster.


Weber’s Bar

Resonance detector for gravitational waves


Hulse-Taylor binary pulsar 1913+16

periastron sep.1.1R⊙

orbital eccentricity: 0.6orbital peroid P : 0.3 daypulsar spin period: 59 milliseconds


No. 2, 2010 RELATIVISTIC BINARY PULSAR PSR B1913+16 1033

Figure 1. Timing residuals for PSR B1913+16. (a) Residuals from a fit fordata before mid-1992. The glitch in 2003 May can be recognized by a distinctchange in the slope of the residuals vs. time. The apparent change in mid-1992is much smaller and may or may not involve a discrete event. (b) Residuals froma fit of all data, holding astrometric and orbital parameters fixed at the values inTables 2 and 3; fitting for pulsar frequency and spin-down rate, f and f ; and notallowing for higher-order frequency derivatives or glitches. The glitch in 2003May is evident as a sharp discontinuity. (c) Residuals from the full timing fit,including higher-order frequency derivatives and the glitch.

1992), but in coming years the propagation delay should startto become observable. Damour & Deruelle (1986) characterizethe measurable quantities as range r = (Gm2/c

3) and shapes ! sin i of the Shapiro delay, where i is the orbital inclination.As orbital precession carries our line of sight deeper intothe companion’s gravitational well, future observations shouldpermit the robust measurement of these two parameters, andhence two additional tests of relativistic theories of gravity(Damour 2009; Esposito-Farese 2009).

5. SYSTEMIC VELOCITY

Our pulsar’s proper motion measurement (Section 3.1), com-bined with the distance estimate discussed in Section 3.3, corre-sponds to a transverse velocity (with respect to the solar systembarycenter) of 75 km s"1 with a galactic position angle of 306#,i.e., directed 36# above the galactic plane. The $30% distanceuncertainty places similar limits on velocity accuracies.

We can now estimate two components of the pulsar systemicvelocity in its own standard of rest by combining the measuredpulsar transverse velocity and distance, the solar motion withrespect to our local standard of rest (Schonrich et al. 2010),and galactic quantities R0 and !0. The third component ofmotion, which is inaccessible via proper motion measurements,lies close to the direction of Galactic rotation at the pulsar’sposition.

The pulsar’s galactic planar and polar velocity componentsrelative to its standard of rest are 247 km s"1 almost directlyaway from the galactic center and 51 km s"1 toward thegalactic North Pole, respectively. (This is significantly largerthan the measured velocity in the solar system barycenter framebecause the pulsar’s standard of rest velocity fortuitously cancelsmuch of the pulsar’s peculiar velocity with respect to it.) The

Figure 2. Orbital decay caused by the loss of energy by gravitational radiation.The parabola depicts the expected shift of periastron time relative to anunchanging orbit, according to general relativity. Data points represent ourmeasurements, with error bars mostly too small to see.

systemic velocity of B1913+16 is significantly larger than otherwell-measured double neutron star binary system velocities,including the J0737"3037 (transverse velocity 10 km s"1; Stairset al. 2006), J1518+4905 (transverse velocity 25 km s"1; Janssenet al. 2008), and B1534+12 (transverse velocity 122 km s"1;Thorsett et al. 2005) systems.

6. CONCLUSIONS

We have analyzed the full set of Arecibo timing data on pulsarB1913+16 to derive the best values of all measurable quantities.A significant proper motion has finally been determined. Asmall glitch was observed in the pulsar’s timing behavior, thesecond known glitch in the population of recycled pulsars. Themeasured rate of orbital period decay continues to be almostprecisely the value predicted by general relativity, providingconclusive evidence for the existence of gravitational radiation.Uncertainties in galactic accelerations now dominate the errorbudget in Pb and are likely to do so until the pulsar distancecan be measured more accurately. We expect that the Shapirogravitational propagation delay will yield additional tests ofrelativistic gravity within a few more years.

The three authors gratefully acknowledge financial supportfrom the US National Science Foundation. Arecibo Observatoryis operated by Cornell University under cooperative agreementwith the NSF. We thank Joseph Swiggum for assistance withanalyses of glitches in the pulsar population; and C. M. Ewers,4A. de la Fuente, J. T. Green, and Z. Pei for assistance withobservations.

REFERENCES

Arzoumanian, Z., Nice, D. J., Taylor, J. H., & Thorsett, S. E. 1994, ApJ, 422,671

Barker, B. M., & O’Connell, R. F. 1975a, Phys. Rev. D, 12, 329

4 Deceased.

Weisberg, Nice, Taylor 2010


Idea: focus instead on the scattering of a gravitational wave (GW) background by the binary. Order of magnitude estimates:

Gravitational wave background causes the binary period to random walk.∆P/P per period ∼ 10h

Strain most effective at harmonics of the orbital period, h f ∼ 4× 10−5

Hz

Over duration , accumulated rms ∆P/P ∼ 10h�

Ttot/PTtot

Accumulated rms periastron time shift ∆T ∼ 10hTtot

�Ttot/P

Ttot/P ∼ 4× 104Useful numbers: , Ttot ∼ 104 days

Periastron time measurement accuracy

Constrain strain by

∼ 10−7 day

∆T < 10−7 day

hNote: here represents the square root of the tensor power spectrum per log freq i.e. is the rms strain.h

h < 5× 10−15

Early work: Mashhoon; Carr, Hu & Mashhoon


Some technical details:

This is a stochastic process: compute two point correlation of orbitalchange, and relate it to the two point correlation of GW background.

Optimal data weighting.

�h(t)h(t�)� =�

df

fh2rmse

2πif(t−t�)

Remarks:

How much better can we do?

What other systems?

From upper limits to detection: issues.


Summary:

1. Scalar-tensor theories, despite their universal microscopic coupling to matter, contain seeds of O(1) equivalence principle violations, due to macroscopic renormalization of the scalar charge. Manifestations: stellar versus gaseous rotation curves in chameleon theories, off-centered black holes in Vainshtein/galileon theories, etc.

2. Binary random walk can be used to constrain/detectgravitational waves.


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