Modiﬁed Gravity - Montana State University...scale factor (from Planck to cosmic scale). is put back into perspective. It is not a problem of a ﬁne-tuned coincidence. It is rather

Modified Gravity

Gravitational Wave Tests of GR WorkshopMontana State University

5th April 2013

Jonathan Gair, IoA Cambridgewith liberal use of material from

Éanna Flanagan, Cornell

Thursday, 25 April 2013

Talk Outline

• What is modified gravity?

• Why study modified gravity?- cosmic acceleration and “dark energy”- galactic dynamics- new observational tools

• Effective field theory approach, as developed to explain cosmological observations

• Approaches to tests of GR using gravitational waves

• Example: gravitational wave tests of GR with space-based detectors


What is Modified Gravity?

• How do we decide if a given a modification of the laws of physics involves a modification of gravity? Perhaps ask which side of the Einstein equation is modified, or which term in action is modified:

Gµν = 8πGTµν

S =

d4x√−g

R

16πG+ Smatter[gµν ,Ψmatter]

• This criterion is in fact ambiguous.

Example:

S =

d4x√−g

R

16πG+ Smatter[eα(Φ)gµν ,Ψmatter]−

d4x√−g

12(∇Φ)2 − V (Φ)

gµν ≡ eα(Φ)gµν , Φ ≡ f(Φ),

S =

d4x√−g

A(Φ)R16πG

− 12(∇Φ)2 − V (Φ)

+ Smatter[gµν ,Ψmatter]


What is Modified Gravity (cont.)?

• How do we decide if a given a modification of the laws of physics involves a modification of gravity?

• Properties of gravity: (i) weak, (ii) long range (iii) universally coupled (iv) spin 2

• Operational definition: require corrections to long-range forces on all macroscopic bodies, eg in Solar System

• Do not require that interactions be spin 2 or that the number of dynamical degrees of freedom not be altered.


Why study modified gravity?

• Einstein’s modification of Newtonian gravity to develop general relativity was motivated in part by observations - anomalous perihelion precession of Mercury.• Do we have observational evidence for a violation of GR?

• Evidence from various cosmological probes including type IA supernovae, CMB, BAO etc. indicates that the rate of expansion of the Universe is increasing.• Need “dark energy” to explain acceleration - quintessence

14

16

18

20

22

24

0.0 0.2 0.4 0.6 0.8 1.01.0

0.5

0.0

0.5

1.0m

ag. r

esid

ual

from

em

pty

cosm

olog

y

0.25,0.750.25, 0 1, 0

0.25,0.75

0.25, 0

1, 0

redshift z

Supernova Cosmology ProjectKnop et al. (2003)

Calan/Tololo& CfA

SupernovaCosmologyProject

effe

ctiv

e m

B

ΩΜ , ΩΛ

ΩΜ , ΩΛ


Is dark energy motivation for modified gravity?• Can explain accelerated expansion within GR using a cosmological constant

• Three common arguments against a cosmological constant:- Historical: “Einstein’s greatest blunder” - Einstein rejected the cosmological constant. The blunder was trying to construct a static Universe. Requirement of second order, diffeomorphism invariant action naturally allows a cosmological constant

- Coincidence problem: at current epoch

- This has been true for most of Universe. Why is tH comparable to ? - Mild anthropic argument: we couldn’t exist at any other time.

4

or in time. Thus, there is a contradiction between theΛCDM model and such a cosmological principle: to be-lieve that the observed acceleration is caused by a cosmo-logical term in Einstein equations requires us to believealso that we are in a very special moment of the history ofthe universe. This is the “coincidence argument” againstthe cosmological constant scenario.

Probability arguments are often tricky and should behandled with care. We think the argument above is in-correct, for two reasons.

First, if the universe expands forever, as in the stan-dard ΛCDMmodel, then we cannot assume that we are ina random moment of the history of the universe, becauseall moments are “at the beginning” of a forever-lastingtime.

Hence we can only reasonably ask whether or not weare in a special point in a time span of the same orderof magnitude of our own cosmic time. Let us, say, triplethe current age tH of the universe, and ask whether weare in a special moment in the history of the universebetween t = 0 and t = 3tH . Common plots in cosmologyappear to indicate so, but these plots are usually in log-arithmic scale. Why should we use equiprobability on alogarithmic scale? More reasonable is to use equiproba-bility in proper time, or in the size of the scale factor a.Let’s choose the second (the conclusion is not affected).Let’s plot the values of ρb and ρΛ as they evolve witha. These are given in Figure 2 as functions of a and inFigure 3 as functions of ln a. At present, the ratio of thetwo is more than one order of magnitude and less thantwo orders of magnitudes. Consider the region where thetwo densities are within two orders of magnitude fromeach other. This is the interval of the a axis in which thedotted curve is within the grey band, namely the intervalof the a axis between the two dotted vertical lines. Thisinterval includes more than half of the total interval in aconsidered.

1a

0.01

0.1

1

10

100

1000Ρb Ρ

FIG. 2: Dark energy (continuous line) and baryon energy(dashed line) as a function of the size a of the Universe. Thetwo are within two orders of magnitudes from one another (asthey are today (a = a(tH) = 1), for all the interval includedbetween the two vertical dashed lines, namely for the majorpart of the history of the universe, on our time scale.

From this point of view, the “coincidence problem”

1lna

0.01

0.1

1

10

100

1000Ρb Ρ

FIG. 3: Same as Figure 2, but in a logarithmic scale for thescale factor (from Planck to cosmic scale).

is put back into perspective. It is not a problem of afine-tuned coincidence. It is rather an issue of order ofmagnitudes: why is the order of magnitude tH compa-rable with the one determined by λ? That is, applyingthe cosmological principle to orders of magnitude, whydo we happen to live in an age of the universe which isnot many orders of magnitudes smaller or larger?Here comes the second objection to the coincidence

argument: the cosmological principle cannot be applieduncritically, as was pointed out by Robert Dicke [14] in1961 and convincingly argued by Brandon Carter [15] in1973. For instance, a rigorous application of the cosmo-logical principle would lead us to expect that the densityaround us must be of the same order of magnitude asthe average density of the universe (which is manifestlyfalse); or, to put it visually, that the Earth is mostly cov-ered by land and not oceans (most humans observe landand not water around them.) Humans do not live in arandom location on Earth. They leave on land and noton water. Our civilization is not located in a random lo-cation in the universe: it is located on a very high peakon the density fluctuations, far out of statistics. This ob-servation is a very mild form of anthropic principle. Thisis a form of anthropic principle that cannot be rejectedeven by those (like us) who most furiously oppose the useof anthropic-principle arguments in theoretical physics.In order for us to comfortably exist and observe it, wemust be on land and not on water, inside a galaxy andnot in intergalactic space, and in a period of the historyof the universe where a civilization like ours could exist.Namely when heavy elements abound, but the universeis not dead yet. Since the universe evolves, this periodis not too long, on a logarithmic scale: it cuts out scalesmuch shorter or much longer than ours.To summarize, in a universe with the of value of λ

like in our universe, it is quite reasonable that humansexist during those 10 or so billions years when Ωb and Ωλ

are within a few orders of magnitude from each other.Not much earlier, when there is nothing like the Earth,not much later when stars like ours will be dying. Ofcourse there is nothing rigorous in these arguments. Butthis is precisely the point: there is nothing rigorous or

Ωb ∼ 0.0227, ΩΛ ∼ 0.74

tΛ

Rµν −12gµνR + Λgµν = 8πGTµν

S[g] =1

16πG

(R[g]− 2Λ)

√g

Bianchi & Rovelli, arxiv:1002.3966Thursday, 25 April 2013

Is dark energy motivation for modified gravity?

- Vacuum energy density: quantisation of classical action introduces a correction to set by UV cut-off at Planck scale

- Need a mechanism to protect from this large correction. No convincing arguments known yet. But the nature of this mechanism is independent of low-energy gravitational theory.

Λ

- Compared to observed value

- Possible explanation: QFT expansion is usually about flat space, but presence of leads to curvature.

Λ ∼ c4M2P /2 ∼ 1087s−2

Λ ∼ 10−35s−2

Λ

- Cosmological acceleration is not in itself a good motivation for studying modifications to GR.

Λ

- Problem is with the QFT. Why reject a simple explanation when a more complicated one does not work? (e.g., Lamb Shift)



• -CDM fails on galactic scalesΛ- Unpredicted observations:‣ Tully-Fisher relation

(Famaey & McGaugh arxiv:1112.3960)

Figure 3: The Baryonic Tully–Fisher (mass–rotation velocity) relation for galaxies with well mea-sured outer velocities Vf . The baryonic mass is the combination of observed stars and gas:Mb = M∗+Mg. Galaxies have been selected that have well observed, extended rotation curves from21 cm interferrometric observations providing a good measure of the outer, flat rotation velocity.The dark blue points are galaxies with M∗ > Mg [273]. The light blue points have M∗ < Mg [278]and are generally less precise in velocity, but more accurate in terms of the harmlessness on theresult of possible systematics on the stellar mass-to-light ratio. For a detailed discussion of thestellar mass-to-light ratios used here, see [273, 278]. The dotted line has slope 4 corresponding toa constant acceleration parameter, 1.2× 10−10 ms−2. The dashed line has slope 3 as expected inΛCDM with the normalization expected if all of the baryons associated with dark matter halosare detected. The difference between these two lines is the origin of the variation in the detectedbaryon fraction in Figure 2.

19



• -CDM fails on galactic scalesΛ- Unpredicted observations:‣ Tully-Fisher relation‣ Galactic rotation curves (Renzo’s rule)


Figure 13: Surface density profiles (top) and rotation curves (bottom) of two galaxies: the highsurface brightness spiral NGC 6946 (Figure 12, left) and the low surface brightness galaxy NGC1560 (right). The surface density of stars (blue circles) is estimated by azimuthal averaging inellipses fit to the K-band (2.2µm) light distribution. Similarly, the gas surface density (greencircles) is estimated by applying the same procedure to the 21 cm image. Note the different scalebetween low and high surface brightness galaxies. Also note features like the central bulge ofNGC 6946, which corresponds to a sharp increase in stellar surface density at small radius. In thelower panels, the observed rotation curves (data points) are shown together with the baryonic massmodels (lines) constructed from the observed distribution of baryons. Velocity data for NGC 6946include both HI data that define the outer, flat portion of the rotation curve [67] and Hα data fromtwo independent observations [55, 115] that define the shape of the inner rotation curve. Velocitydata for NGC 1560 come from two independent interferometric HI observations [29, 164]. Baryonicmass models are constructed from the surface density profiles by numerical solution of the Poissonequation using GIPSY [473]. The dashed blue line is the stellar disk, the red dot-dashed line is thecentral bulge, and the green dotted line is the gas. The solid black line is the sum of all baryoniccomponents. This provides a decent match to the rotation curve at small radii in the high surfacebrightness galaxy, but fails to explain the flat portion of the rotation curve at large radii. Thisdiscrepancy, and its systematic ubiquity in spiral galaxies, ranks as one of the primary motivationsfor dark matter. Note that the mass discrepancy is large at all radii in the low surface brightnessgalaxy.

33



• -CDM fails on galactic scalesΛ- Unpredicted observations:‣ Tully-Fisher relation‣ Galactic rotation curves (Renzo’s rule)‣ Mass discrepancy-acceleration relation


Figure 10: The mass discrepancy in spiral galaxies. The mass discrepancy is defined [271] as theratio V 2/V 2

b where V is the observed velocity and Vb is the velocity attributable to visible baryonicmatter. The ratio of squared velocities is equivalent to the ratio of total to baryonic enclosed massfor spherical systems. No dark matter is required when V = Vb, only when V > Vb. Manyhundreds of individual resolved measurements along the rotation curves of nearly one hundredspiral galaxies are plotted. The top panel plots the mass discrepancy as a function of radius. Noparticular linear scale is favored. Some galaxies exhibit mass discrepancies at small radii whileothers do not appear to need dark matter until quite large radii. The middle panel plots themass discrepancy as a function of centripetal acceleration a = V 2/r, while the bottom panel plotsit against the acceleration gN = V 2

b /r predicted by Newton from the observed baryonic surfacedensity Σb. Note that the correlation appears a little better with gN because the data are strecthedout over a wider range in gN than in a. Note also that systematics on the stellar mass-to-lightratios can make this relation slightly more blurred than shown here, but the relation is neverthelessalways present irrespective of the assumptions on stellar mass-to-light ratios [271]. There is thus aclear organization: the amplitude of the mass discrepancy increases systematically with decreasingacceleration and baryonic surface density.

28



• -CDM fails on galactic scalesΛ- Unpredicted observations:‣ Tully-Fisher relation‣ Galactic rotation curves (Renzo’s rule)‣ Mass discrepancy-acceleration relation‣ Fine-tuning in surface density


Figure 9: The dynamical acceleration ap = V 2p /Rp in units of a0 plotted against the characteristic

baryonic surface density [276]. Points as per Figure 5. The dotted line shows the relation ap =GΣb that would be obtained if the visible baryons sufficed to explain the observed velocities inNewtonian dynamics. Though the data do not follow this line, they do show a correlation (ap ∝Σ1/2

b ). This clearly indicates a dynamical role for the baryons, in contradiction to the simplestinterpretation [110] of Figure 5 that dark matter completely dominates the dynamics.

is unimportant. Nevertheless, this radius is advocated to be used by [110] since this maximizes thepossibility of perceiving the baryonic contribution in the plot of Figure 5. That this contributionis not present leads to the inference that Σb " ΣDM in all disk galaxies [110]. This is directlycontradicted by Figure 9, which shows a clear correlation between ap and Σb.

The higher the surface density of baryons is, the higher the observed acceleration. The slopeof the relation is not unity, ap ∝ Σb, as we would expect in the absence of a mass discrepancy,

but rather ap ∝ Σ1/2b . To simultaneously explain Figure 5 and Figure 9, there must be a strong

fine-tuning between dark and baryonic surface densities (i.e., Figure 6), a sort of repulsion betweenthem, a repulsion which is however contradicted by the correlations between baryonic and darkmatter bumps and wiggles in rotation curves (see Sect. 4.3.4).

4.3.3 Mass discrepancy-acceleration relation

So far we have discussed total quantities. For the BTFR, we use the total observed mass ofa galaxy and its characteristic rotation velocity. Similarly, the dynamical acceleration–baryonicsurface density relation uses a single characteristic value for each galaxy. These are not the onlyways in which the “magical” acceleration constant a0 appears in the data. In general, the massdiscrepancy only appears at very low accelerations a < a0 and not (much) above a0. Equivalently,the need for dark matter only becomes clear at very low baryonic surface densities Σ < Σ† = a0/G.Indeed, the amplitude of the mass discrepancy in galaxies anti-correlates with acceleration [271].

27

Mb = M∗ + Mg while the total baryonic mass available in a halo is fbMtot. The differencebetween these quantities implies a reservoir of dark baryons in some undetected form, Mother. Itis commonly speculated that the undetected baryons could be in a hard-to-detect hot, diffuse,ionized phase mixed in with the dark matter halo (and extending to comparable radius), or thatthe missing baryons have been entirely blown away by winds from supernovae. For the purposesof this argument, it does not matter which form the dark baryons take. All that matters is that asubstantial mass of them are required so that [283]

fd =Mb

fbMtot=

M∗ +Mg

M∗ +Mg +Mother. (3)

Since there is negligible intrinsic scatter in the observed BTFR, there must be effectively zeroscatter in fd. By inspection of Eq. 3, it is apparent that small scatter in fd can only be obtainednaturally in the limits M∗ +Mg ! Mother so that fd → 1 or M∗ +Mg # Mother so that fd → 0.Neither of these limits apply. We require not only an appreciable mass in dark baryons Mother,but we need the fractional mass of these missing baryons to vary in lockstep with the observedrotation velocity Vf . Put another way, for any given galaxy, we know not only how many baryonswe see, but also how many we do not see — a remarkable feat of non-observation.

Figure 5: Residuals (δ logVf ) from the baryonic Tully–Fisher relation as a function of a galaxy’scharacteristic baryonic surface density (Σb = 0.75Mb/R2

p [272], Rp being the radius at which thecontribution of baryons to the rotation curve peaks). Color differentiates between star (dark blue)and gas (light blue) dominated galaxies as in Figure 3, but not all galaxies there have sufficientdata (especially of Rp) to plot here. Stellar masses have been estimated with stellar populationsynthesis models [43]. More accurate data, with uncertainty on rotation velocity less than 5%,are shown as larger points; less accurate data are shown as smaller points. The rotation velocityof galaxies shows no dependence on the distribution of baryons as measured by Σb or Rp. Thisis puzzling in the conventional context, where V 2 = GM/r should lead to a strong systematicresidual [110].

Another remarkable fact about the BTFR is that it shows no residuals with variations inthe distribution of baryons [518, 444, 110, 272]. Figure 5 shows deviations from the BTFR asa function of the characteristic baryonic surface density of the galaxies, as defined in [272], i.e.,Σb = 0.75Mb/R2

p where Rp is the radius at which the rotation curve Vb(r) of baryons peaks. Overseveral decades in surface density, the BTFR is completely insensitive to variations in the massdistribution of the baryons. This is odd because, a priori, V 2 ∼ M/R, and thus V 4 ∼ MΣ.

21



• -CDM also makes unobserved predictions- Bulk flow 200km/s cf 1000km/s observed- High redshift clusters- Local void hosts 3 galaxies versus 20 predicted- Missing satellites- Satellite phase-space distribution- Galaxy density profiles cusps vs observed cores- Angular momentum of baryons much smaller than observed- Cannot form bulgeless thin disks easily- Stability - low surface density discs should not be able to form bars and spirals- Missing baryons - observe only 5% of baryons that should be there.

• Motivated introduction of MOND.

Λ


• Observations of the Bullet cluster show an offset between the baryons and the dark matter.- Hard to explain within the context of MOND (but perhaps Einstein-Aether)?


• Clearly there is something we do not understand, but these observations may not provide good motivation for modified gravity. However, scale is different.


• The best motivation for studying modified gravity is that we are developing new tools that can probe unexplored regimes

• Cosmological probes, e.g., weak lensing, BAO, galaxy surveys• Gravitational waves‣ Observe dynamics in strong curvature regime near black hole horizons and for velocities comparable to c‣ New observations with Adv. LIGO, Pulsar Timing and eLISA.

Why study modified gravity?10-10

10-15

10-20

10-25

10-30

10-35

10-40

10-45

10-50

10-55

10-60

!=G

M/r3 c2 (c

m-2

)

10-15 10-10 10-5 100

"=GM/rc2

#

#

#

Figure 1: A parameter space for quantifying the strength of a gravitational field. The x-axismeasures the potential ε ≡ GM/rc2 and the y-axis measures the spacetime curvature ξ ≡ GM/r3c2

of the gravitational field at a radius r away from a central object of mass M . These two parametersprovide two different quantitative measures of the strength of the gravitational fields. The variouscurves, points, and legends are described in the text.

8

D Psaltis, Liv. Rev. Rel. (2008)

Potential

Cur

vatu

re


Effective Field Theory Paradigm: Review• A method of defining and parameterizing all approximate quantum field theories consistent with a given field content, set of symmetries, and energy cutoff Λ



• High energy physics assumed to be random, arbitrary

• Predictive and restrictive, excludes as fine-tuned models with



• Actually two cutoffs:



• Actually two cutoffs: • Framework more constraining for large cutoffs. Two cases:



• Why consider quantum issues in classical regimes?



• Why consider quantum issues in classical regimes?

• Greater discriminating power than classical analysis: ‣ Exclude strong coupling (loops), eg Palatini f(R), Chameleon, Galileon‣ Quantum instabilities (vacuum decay), eg Chern-Simons gravity‣ Fine tuning considerations give probability measure on space of theories


Effective Field Theories of Modified Gravity

• Which is better? Are they complementary?

• Type I: Covariant, independent of background cosmology Weinberg (2008); Burgess, Lee and Trott (2009); Park, Zurek and Watson (2010); Jiminez, Talavera and Verde (2011); Bloomfield and EF (2012); Jimenez et. al. (2012); Mueller, Bean and Watson (2012).

• Type II: Expands about given background cosmology Cheung et. al (2008); many others; Creminelli, d’Amico, Norena and Vernizzi (2009); Battye and Pearson (2012); Gubitosi, Piazza and Vernizzi (2012); Bloomfield, EF, Park and Watson (2012).

Applied to inflationary era

Applied to present cosmic acceleration

• How all-encompassing is effective field theory approach?


Background Independent, Covariant Approach

• Start with

• Define effect of higher derivative terms by reducing order

• Simplify using perturbative field redefinition freedom



Background Independent, Covariant Approach (cont)

• Final result

• Coefficients are arbitrary functions of field with specific scalings, eg


Background Dependent Approach

• Use 3+1 splitting:

• Expand:

• Specialize gauge so that

• Most general action preserving gauge choice

• Expand:


Effective field theory - summary

• The two methods that have been used in cosmology, background dependent and background independent, are complementary, but the background dependent variant is more useful.

• Approach provides a natural way to impose a probability measure on space of theories - theories that require fine tuning are excluded

- Coefficients can not be too large- Coefficients can not be too small- All terms at relevant order should be included


Gravitational wave tests of modified gravity

• Can use same or similar alternative theories to explore potential of gravitational wave observations.

• Two main approaches - Theory-led: consider specific alternatives to general relativity motivated by other considerations and ask which better explains the data.

- Observation-led: look for evidence of a deviation from the predictions of GR in the data. If one is observed, attempt to explain it.


Observation-led approach

• Difficulty is that we will use matched filtering to detect signals. Need to have a model if we are to see anything.

• Can use theory to motivate a parameterized deviation framework. Ensure that all known alternatives are included. Different frameworks for different regimes

- Parameterised post-Newtonian framework (Solar System)

- Parameterised post-Keplerian framework (Binary Pulsars)

- Parameterised post-Friedmannian framework (Cosmological)

- Parameterised post-Einsteinian framework (Gravitational Waves)

• Or, focus on what your observations can measure well.


Extreme-mass-ratio inspirals

• An extreme mass ratio inspiral (EMRI) is the inspiral of a compact object (a white dwarf, neutron star or black hole) into a SMBH. Not main sequence stars, as these will be tidally disrupted before gravitational radiation becomes significant.

• Originate in dense stellar clusters through direct capture, binary splitting, tidal stripping of giant stars or star formation in a disc.

• For black holes with mass in the range , EMRIs will generate gravitational waves detectable by eLISA.

• eLISA should see a few tens of events, at redshift . Vast majority likely to be BH inspirals.

• Complex gravitational waveforms include three fundamental frequencies - orbital frequency, perihelion precession frequency and orbital plane precession frequency.

104M⊙ M 107M⊙

z 0.5


Extreme-mass-ratio inspirals• Extreme mass-ratio ensures orbit proceeds slowly- Many waveform cycles generated in the strong-field- Instantaneous emission at harmonics of three fundamental frequencies - tracks orbital motion- Over inspiral, waveform encodes spacetime structure- EMRIs are good for “mapping the metric”. Compare to Kerr to test “no-hair” theorem

• Therefore, attempt to map metric and place constraints on “hairy-ness”. If observed, look for explanations

• Astrophysical perturbations, e.g., accretion disc, distant perturber• Exotic central object, e.g., a boson star• Central object is a black-hole, but not a Kerr black hole, due to a violation of the assumptions of uniqueness theorem• GR is not the correct theory of gravity

Ml + iSl = M(ia)l


Spacetime mapping

• In GR, spacetime multipole moments are encoded in gravitational wave observables - precession frequencies and time spent in each frequency band (Ryan 1995).

Ωp

Ω= 3(MΩ)

23 − 4

S1

M2(MΩ) +

92− 3

2M2

M3

(MΩ)

43 + · · ·

Ryan 1997

• Expect to be able to measure the quadrupole moment to ~1% while simultaneously measuring mass and spin to ~0.1%.


• Initial work focussed within GR.- “Bumpy black holes” suggested by Collins & Hughes (2004) - better than multipole moment decomposition.- Various studies, e.g., Ryan (1997), Collins & Hughes (2004), Kesden et al. (2004), Glampedakis & Babak (2006), Li & Lovelace (2007), Gair et al. (2008).

Spacetime mapping

• Effect of astrophysical perturbations also considered, e.g., Barausse et al. (2007, 2008), Yunes et al. (2011).


• Tests against certain specific alternative theories were explored• Scalar-Tensor gravity (Brans-Dicke). Gravity coupled to a scalar field. Sources produce dipole radiation that modifies inspiral phase

• Massive Graviton. No consistent theory exists. Mimic effect by changing propagation speed of gravitational waves.

• Chern-Simons Modified Gravity. Parity-violating correction to GR inspired by string-theory (Sopuerta & Yunes (2009), Canizares et al. (2012)).• f(R) gravity. Obtained by replacing R with f(R) in Einstein-Hilbert action. EMRI constraints not competitive with laboratory tests of inverse-square law (Berry & Gair (2011)).

Spacetime mapping

δψ(f) = − 5S2

3584ωBDη2/5 (πMf)−7/3

δψ(f) = − π2DMλ2

g(1 + z)(πMf)−1


• Better if we can do something generic. Barack & Cutler (2007) considered effect of arbitrary perturbation to quadrupole moment

Spacetime mapping

• Gair & Yunes (2011) - want framework for generic deviations from relativity or arbitrary “bumpy” black holes. Try to do this without attempting to fully specify an alternative theory.• Make a minimal set of assumptions (Vigeland et al. 2011)- Metric theory of gravity. Test particles follow geodesics of a background metric, which is written as a small perturbation to Kerr

- Integrable motion. Geodesics have three conserved integrals of motion, i.e., there is a generalised Carter constant.- Asymptotic flatness. Require that perturbation to metric tends to zero at spatial infinity.- Peeling. Require that .- Occam’s Razor. Set coefficients to zero where they are not required to recover known alternative theories of gravity.

hµν ∼ r−2, as r →∞

gµν = gKerrµν + hµν


• Better if we can do something generic. Barack & Cutler (2007) considered effect of arbitrary perturbation to quadrupole moment

Spacetime mapping

• Gair & Yunes (2011) - want framework for generic deviations from relativity or arbitrary “bumpy” black holes. Try to do this without attempting to fully specify an alternative theory.• Make a minimal set of assumptions (Vigeland et al. 2011)- Metric theory of gravity. Test particles follow geodesics of a background metric, which is written as a small perturbation to Kerr

- Integrable motion. Geodesics have three conserved integrals of motion, i.e., there is a generalised Carter constant.- Asymptotic flatness. Require that perturbation to metric tends to zero at spatial infinity.- Peeling. Require that .- Occam’s Razor. Set coefficients to zero where they are not required to recover known alternative theories of gravity.

hµν ∼ r−2, as r →∞

gµν = gKerrµν + hµν


Spacetime mapping• Resulting metric components are complicated.• Simplify further by expanding the free functions of radius as power series in 1/r

• The parameter N indicates the leading-order radial dependence of the perturbation, .• Consider special cases, , in which the leading coefficients are

• Also consider Chern-Simons modified gravity, which is given by

hµν ∼ 1/rN as r →∞

γA =∞

n=0

γA,n

M

r

n

, γ3 =1r

∞

n=0

γ3,n

M

r

n

B2 = γ1,2, γ3,1, γ3,3, γ4,2 BN = γ1,N , γ3,N+1, γ4,N ∀N ≥ 3

γ3,5 = −58aζ , γ3,6 = −65

28aζ . γ3,7 = −709

112aζ

BN


Spacetime mapping• Resulting metric components are complicated.• Simplify further by expanding the free functions of radius as power series in 1/r

• The parameter N indicates the leading-order radial dependence of the perturbation, .• Consider special cases, , in which the leading coefficients are

• Also consider Chern-Simons modified gravity, which is given by

hµν ∼ 1/rN as r →∞

γA =∞

n=0

γA,n

M

r

n

, γ3 =1r

∞

n=0

γ3,n

M

r

n

B2 = γ1,2, γ3,1, γ3,3, γ4,2 BN = γ1,N , γ3,N+1, γ4,N ∀N ≥ 3

γ3,5 = −58aζ , γ3,6 = −65

28aζ . γ3,7 = −709

112aζ

BN


• Construct EMRI waveform model based on “Analytic Kludge” model of Barack and Cutler (2004) - quadrupole waveforms from Keplerian orbits, with relativistic precessions and inspiral imposed.

• Adapt to modified gravity spacetimes by

- Correcting the precession frequencies. The geodesic equations separate, due to the existence of the Carter-like constant. From these equations, we can compute the leading order post-Newtonian and linear in correction to the radial frequency, the orbital plane precession frequency and the perihelion precession frequency.

- Correcting the inspiral trajectory. Estimate leading order in correction to the fluxes of energy and angular momentum that arise indirectly, i.e., from changes in the orbit. Do this by using the modified geodesic orbits in the quadrupole formula. Use a constant inclination assumption to compute the correction to the rate of change of the Carter constant.

Spacetime mapping


• Corrections to the kludge models for the special cases, , described earlier take the form

Modified Gravity EMRIs

BN

Mδ ˙γBN =(2πMν)(2N+1)/3

(1− e2)N−1gγ,N (e)(γ1,N + 2γ4,N )

MδαBN = −a(2πMν)2(N+1)/3

(1− e2)N−1/2gα,N (e)(γ1,N + 2γ4,N )

2πM2δνBN =

165

η(2πMν)2N/3+3

(1− e2)N+5/2gν,N (e)(γ1,N + 2γ4,N )

MδeBN = −165

η(2πMν)2N/3+2

(1− e2)N+3/2ge,N (e)(γ1,N + 2γ4,N )




BN

Mδ ˙γBN =(2πMν)(2N+1)/3

(1− e2)N−1gγ,N (e)(γ1,N + 2γ4,N )


(1− e2)N−1/2gα,N (e)(γ1,N + 2γ4,N )

2πM2δνBN =

165

η(2πMν)2N/3+3

(1− e2)N+5/2gν,N (e)(γ1,N + 2γ4,N )

MδeBN = −165

η(2πMν)2N/3+2

(1− e2)N+3/2ge,N (e)(γ1,N + 2γ4,N )

ge,2(e) =934

+674

e3 +14e5, gν,2(e) = 18 + 78e2 +

994

e4

gγ,2(e) =12, gα,2(e) = 1

ge,3(e) =72512

e +3838

e3 +14348

e5 , gν,3(e) = 42 +4072

e2 +3454

e4 +3316

e6 ,

gγ,3(e) =32

, gα,3(e) =34

+14e2 ,

ge,4(e) =3163

e +12713

96e3 +

246764

e5 +38e7 + (1− e2)1/2e

9518− 305

96e2 − 605

288e4

,

gν,4(e) = 69 +3191

8e2 +

503516

e4 +303964

e6 − 148

1− e2

1/2 48− 358e2 + 147e4 + 163e6

,

gγ,4(e) = 3 +34e2 , gα,4(e) = 1 + e2 ,

ge,5(e) = 155e +31565

96e3 +

30749192

e5 +5305768

e7 +596

e1− e2

1/2 304− 183e2 − 121e4

,

gν,5(e) = 99 +5601

8e2 + 905e4 +

1561764

e6 +1137256

e8 − 116

1− e2

1/2 144− 466e2 + 75e4 + 247e6

gγ,5(e) = 5 +154

e2 , gα,5(e) =54

+52e2 +

14e4




• The constant inclination assumption breaks down for the CS case. Circular to circular condition gives corrections for e=0

BN

Mδ ˙γBN =(2πMν)(2N+1)/3

(1− e2)N−1gγ,N (e)(γ1,N + 2γ4,N )


(1− e2)N−1/2gα,N (e)(γ1,N + 2γ4,N )

2πM2δνBN =

165

η(2πMν)2N/3+3

(1− e2)N+5/2gν,N (e)(γ1,N + 2γ4,N )

MδeBN = −165

η(2πMν)2N/3+2

(1− e2)N+3/2ge,N (e)(γ1,N + 2γ4,N )

δeCS = 0, 2πM2δνCS = 360(2πMν)20/3ηaζ sin θtp

Mδ ˙γCS =758

(2πMν)4aζ sin θtp, MδαCS = −58aζ(2πMν)4


-6e-23

-4e-23

-2e-23

0

2e-23

4e-23

6e-23

0 5000 10000 15000 20000 25000 30000

h I

t (s)

GR B2, !=0.1



-6e-23

-4e-23

-2e-23

0

2e-23

4e-23

6e-23

0 10000 20000 30000 40000 50000 60000 70000 80000

h I

t (s)

GR B2, !=0.1



• Estimate time to accumulate one cycle phase shift, given initially matched frequencies, for circular, equatorial orbits.


0.1

1

10

100

1000

10000

100000

1e+06

0.01 0.1 1 10

T/yr

M !"

B2B3B4B5CS

(γ1,N + 2γ4,N ) = 0.1

S/M2 = 0.5

η = 1× 10−6

M = 1× 106M⊙


Modified Gravity EMRIs• Use Fisher Matrix analysis to allow for parameter correlations


Summary• Current evidence for modifications to gravity is inconclusive

- Cosmological acceleration adequately explained by cosmological constant.- Galactic dynamics possibly explained by dark matter. Some evidence points away from MOND.

• New observations that probe different regimes of gravity is motivation enough.• Effective field theory gives one way to construct new theories, and provides a natural way to assign relative probabilities through need for fine tuning. Not all models equally probable.• Gravitational wave observations could be theory-led or observation-led.• Generic models allow for observation-led analysis, but should be informed by theory.


Some questions for discussion• Is it better to be observation-led or theory-led?

• Which models should we be considering?- Are cosmological models relevant? These are designed to explain very different phenomena.- Should we adherer to the chameleon mechanism, i.e., a theory that has been ruled out on one scale could still apply on a different scale?

• Can we further adapt our tests to what the observations are best able to tell us? (optimal directions in theory space)

• What do we lose from using generic models if we are interested in specific alternatives? What if the true theory is something that we haven’t considered and lies outside generic model space? Can we ever know? Does it matter?


Documents

Modiﬁed Gravity - Montana State University...scale factor (from Planck to cosmic scale). is put back into perspective. It is not a problem of a ﬁne-tuned coincidence. It is rather