 Thank you. Can you guys hear me okay at the back? Can you hear me okay at the back? Okay. Thanks So I'd like to begin by thanking an anonymous donor for the power cable. I Found it over there. I'll put it back there when I'm done It's tough when you're Traveling and you're not very organized. I don't have power But I'm doing better than at breakfast because now I have some cash When I went for breakfast, I had five cents in my pocket and she didn't take credit cards But she let me eat for free. So that was nice So I put down some references so if you're interested in different parts of the Stuff yesterday or today then you can look up these papers Obviously the usual disclaimers. This is just some random sampling of pretty extensive literature Is there a way that we can send this information over to everybody or Okay, so we can do that at some point So this is what I want to cover today You're learning about some really amazing and pretty technical stuff So I'm gonna try and give you a lighter talk today Not a lot going on. So we'll begin with a review of what we did Then I'll show you the measurements That I barely got started on yesterday and then we'll go from Lensing to cosmology Just the state of the field a few slides I know David Weinberg showed that too, but a refresher wouldn't hurt and from there. We'll go to modified gravity motivation Screening which I know Rachel Rosen has covered But you'll get a slightly easier flavor on screening and then some Tests Mostly I'll talk about completely new kind of tests that you've probably not encountered so So those are the topics so So let's just do this lensing review. So yesterday we started with this metric And then we got to some deflection angle and then we switched to Projections on the sky And then we claimed that observable quantities involve one more derivative and we introduced these these particular combinations Where now derivatives are on the sky the two potentials have been set equal as in GR And these are all projected quantities Okay By the way, at least two people Attempted to come up with observables that don't involve second derivatives. So we just had a discussion about displacements So people are thinking about interesting ways to measure displacements the displacement field as well Right So then we noted that this guy Now going back to 3d Can be written as a redshift integral of some geometric factors times the density fluctuation Okay, so this kappa Which is the isotropic Magnification piece in the weak lensing limit is also the projection of the density So that makes interpretation of kappa really easy Okay, the amount of magnification you see is just related to the projected density The awkward thing is that in weak lensing by the way, that's the relationship to magnification is Linear only in the weak lensing regime and the strong lensing regime the shear also contributes to magnification The awkward thing though is that this is the guy that's related to the ellipticity of a galaxy and it's what we measure Okay, so we kind of avoided confronting the fact that for intuition and interpretation We think about kappa, but what we actually measure is gamma and Gamma is like a tidal field, right? These are these cross terms and differences It's non-local The relationship of gamma to the density is non-local which you expect intuitively because a density Fluctuation here can shear a galaxy here because gravity is non-local. So how do we interpret it? So I just thought I'd motivate that a little bit Because I showed you this mass map so this is what we are doing with With real data now so So as I mentioned We observed the shapes of these two million galaxies and made this mass map So the mass map refers to kappa So how did we get from? The gamma inferred from two million galaxies to kappa Well, if you write these equations in Fourier space Then you can see how So let's Fourier transform these and then what do you get? So in terms of this angular wave number L, let's define this Fourier space Variable so L is this angular wave number then this guy Okay, so you measure your shear At So you have this big field 150 square degrees you pixelize it You take all the galaxies in a pixel and use their Their ellipticities to measure the average shear within each pixel and then you interpolate onto the grid and Now you're in business because once you have the shear field represented on a grid You can do a Fourier transform And once you do a Fourier transform, sorry then You know you have an estimator of this guy And you can stick that in there And you can construct kappa you inverse Fourier transform and you get back that map So the actual estimators are a little bit more complicated because we have a finite field You have to deal with those masks in the middle but At least formally you can see that if you have a big enough field That the finite boundary or doesn't impact your Fourier modes too much. That's how you can do a reconstruction So that's for wide field lensing There are so this is called The Simple version of this is the Kaiser-Squire's algorithm. It's called Reconstruction of the mass field So that's the general case and it works as long as you have a wide enough field Good question The question is are we looking at a fixed redshift so This is us Galax the source galaxies are distributed in this case between redshift point six and one point three So the lensing kernel the weight Of the lensing mass I'll look something like this You know this w function So if you ask a given density fluctuation, how much is it contributing to the lensing Then it's multiplied by this guy, which I'm showing here So that means that if you're halfway between the source and the observer You make the maximum contribution and if you're too close to the source or too close to the observer You're not an effective lens so from that you can infer that the The peak is at about point four redshift point four is where the Lensing is happening Was that sort of your question? Yeah You know from this you get this using Poisson's equation so we've gone from side to the density field and Projected that you could other questions Okay, so we did two other special cases One was we looked at halos and For the spherical case We claimed or rather I claimed you may or may not buy it that this guy is The mean kappa inside that circle minus the kappa at that circle so this set of equations When you apply to a spherically symmetric case you get this simple relationship between gamma and kappa And and we applied this to voids as well, which you also treat as spherically symmetric We considered filaments just this picture that the shear field looks like this Does anybody have a two-line explanation for why? So just to remind you for the sphere it looks like this so you noticed Bruce Partridge showed you the Polarization pattern around a temperature hotspot completely different physics to me. This is easier to see That is the spherical object can produce an Einstein ring which looks like this so the weak lensing shear field Looks like that polarization is harder But anyway from this how do you get this if you have a filament well if the filament is made up, sorry Anybody want to try? So so if you have a galaxy sitting here It'll be distorted like this if you have a galaxy sitting here. It'll be distorted like this the background galaxy that's the claim and the explanation is That if if you imagine a filament made up of a bunch of little halos Then you know if you're sitting between two halos Then you know that each halo is going to distort it like this this halo is going to distort like this There's as many here as they're here, so you have no choice, but to get a whisker this way If you have a finite filament and you're really far away from it Then it's just like a point mass, so it'll do this As you get closer You kind of end up here so So that's what a filament looks like Just to keep things interesting You know, this is the spherically symmetric piece in general halos are neither filaments nor spheres in General halos actually look like this. They're ellipses So if you wanted to figure out what the shear pattern due to an elliptical halo is How would you do it? You can decompose it into a monopole and a quadruple So let's look at this monopole a little bit The shear rotates Like this and as I claimed yesterday by the time you go to pi This is your angle By the time you come to pi the shears back to where you started, right? That's why the shear is like a spin-to field Or as if you had a velocity field or something you'd have to go all the way around to get back where you started So this is the shear field due to a monopole So a monopole has this cosine 2 theta variation What would the shear field you do a quadruple look like? Suppose my lens instead of the sphere was an elliptical object that was oriented this way So the long axis is along the x-axis If this is cosine 2 theta would a quadruple be cosine 3 theta 4 theta 5 theta sign 3 theta if it was sign 3 theta then Sorry, if it was cosine 3 theta then somewhere over here you would you would get back to this shear Now odd multiples don't quite work. So it would be so this is a cosine 2 theta a quadruple would be a cosine 4 theta So it's not even obvious that it has to be a cosine, but It could be either a cosine or a sign But it looks like this You can see it's not easy Okay So it's rotating twice as fast you're back here by the time you're at pi over four That's what it looks like The reason I'm spending time on all these void and filament and quadruple patterns is that I have a student who's now a postdoc at Ben who's Obsessed with these and he's made some beautiful measurements. So we're actually Writing up a paper on this quadruple measurement So, you know if you can measure the lensing around voids and the ellipticity of dark matter halos Then you can do a lot because with the monopole everything is degenerate gravity, you know the gravitational in fall the nature of dark matter the Yeah, you know the non-linear physics of gravitational in fall and the gravitational law itself are degenerate So you have to do careful simulations But with the quadruple you can do some pretty first-order tests of gravity and dark matter So so that's why we are trying to measure it from this loan survey Any questions? Yes, right That's a great point and that's because When you make a filament out of this this is just the quadruple piece But the filament also has a monopole piece and the monopole piece is five or ten times bigger So that's why Over here the monopole and quadruple align So there's no controversy that you end up with this pattern But over here the quadruple is actually got the wrong sign So it reduces the monopole, but this is what you this is what dominates here It's a good point Okay So the very first lensing paper I wrote we did like a little signal-to-noise estimate And I tried to reproduce it yesterday, and I failed spectacularly So it's a really simple thing. So it's like there's no way I need to prepare this because you know I know this in my sleep, and I was off only by a factor of hundred so So so I'll at least correct that with a few minutes. So the question is How many galaxies do you need to measure weak lensing? And I had claimed that the signal is About 1% and the noise is The intrinsic shape of a galaxy which is about 0.3 is the ellipticity divided by the square root of number of galaxies But I forgot something which is which I was emphasizing the rest of the talk that the signal depends on the coherence length you're looking at So it's 1% only if you're looking on arc minute scales But on arc minute scales you don't get the answer that we got then you don't get a thousand galaxies or 10,000 galaxies to make a 5 sigma measurement So you have to go a little bigger But when you go bigger the signal drops. So on scales of 10 arc minutes or larger The signal is actually more like 0.2% Whereas the noise just scales this way So when you work that out, then you do get Ng of 10 to the 5 to make a few sigma Measurement of lensing so that's kind of an important fact because You know that it's related to the development of technology and wide field cameras Which came online in the late 90s and so not surprisingly the year 2000 was when the first Detection of lensing was made so So this corresponds to something about the size of the moon So by the way, there's another subtlety to this that actually you need many patches Otherwise, you know, you're looking for an RMS signal. So just one patch you could be one sigma off easily So that makes the calculation actually more complicated But you need 10 to the 5 galaxies to make a few sigma detection And then if you want to make like a 5% measurement then you're Very quickly in the millions of galaxies regime Which is the state of the art of the field right now So So since you guys are mostly theorists As I appreciated last night when over dinner there was a 20-minute conversation Involving the spin of an electron in effective field theories I Think it was totally unresolved this question, right? So So So this may surprise you so so we're talking about the scale about the size of the moon on the sky So how many galaxies can you detect in an area of the sky? That's the size of the moon That's that's not a bad guess With a ground-based telescope, it would be like 10 to the 4 With the Hubble you could get to this number 10 to the 5 other questions alright Let's go to the slides and Look at some results Alright, this is the last piece So let's look at these equations So I talked about wide field. I talked about halos and these other special objects, which are going to be useful for measuring For testing gravity. The third thing is The fundamental thing which is this quantity the two-point correlation function of the shear as a function of separation angle theta So you take two galaxies you get an estimate of the shear you take the complex conjugate of the second one and So this product averaged over the whole sky and all pairs of galaxies is your two-point correlation function Okay, I'm going to write down something quite surprising that this is Identically equal to the two-point correlation function of kappa So for all this talk about kappa and gamma being really different kappa is local gamma is non-local This is convergence. This is title field when you actually take the two-point function They're identical and you can see that if you take the Fourier transform the power spectrum Then you can see that the square of this is The sum of the square of these two So it's not hard to prove So that's a very useful thing Doesn't hold for higher-order functions for higher-order functions. This guy has like a billion components very quickly But for the two-point function, this is a very handy thing. So what we're going to do is We're going to calculate the two-point function of the shear and we're going to do it in a bunch of redshift bins and that two-point function is just the projected power spectrum because kappa is just an integral over Delta so So that's this guy. So the thing you measure is actually just the two-point function of the density field times these things so So it's pretty easy to relate measurements to theory and by using this tomography Using these different redshift bins. We can prove sort of the evolution of the two-point function of the density field So that's sort of the lensing cosmology program. So let's go from these mass maps to these two-point functions So this was measured by the CFH legacy survey in a really nice paper. This is from the dark energy survey Can you see the preliminary thing? It's supposed to be a preliminary Bad thing that says prelimp- It's on my laptop. That's weird. It doesn't show up here. It's light gray. Anyway So this is showing the two-point function measured in three redshift bins You know point four to point six point six to point nine point nine to one point two So in the label zero is the lowest redshift bin and black is the highest But you get not just the three correlation functions, but also also the cross correlation So you end up with six correlation functions And you can see the range of the axes y-axis is going from ten to the minus four to ten to the minus six So that's one percent to point one percent shears squared and The predictions are lambda CDM This is not published The analysis is blinded by a modest factor So It's the relative spacing of the points that you can that's still meaningful And so you can see that this is something we are just barely able to do in three redshift bins We can make measurements. You can see the lowest redshift bin is pretty crap But if you look in an average sense the measurements scale with redshift The profile of the source gallon the light profile Yeah, the number density you just count the galaxies Yeah Mm-hmm, and that changes the error bars Okay, so, you know Bruce emphasized to you how hard CMB polarization is And I emphasize yesterday why deconvolving the PSF makes lensing really hard so this is one of the hardest measurements in cosmology and So in our survey the dark energy survey We've just reached this point with a pretty small fraction of the survey But it was an enormous amount of work with galaxy shapes and photometric redshifts to get to this point This is galaxy galaxy lensing. So this is that you know the monopole term of the mass profile So this is a somewhat easier to measure And you can see that this is a nfw profile prediction these are measurements So this is in arc minutes roughly speaking. This is one-tenth the virial radius This is about the virial radius and this is what's called the two halo term So we measure it reasonably well. We can model it using this one halo to halo decomposition We can even look for satellite contributions It works pretty well So I mentioned that we are measuring point one percent. She is comfortably. That's this level Point oh one percent. We are not yet there yet, but We are somewhere in between So this is the state of the art we can use it to get contours I'm not going to talk about this, but at least tell you the mass of the halo In which this galaxy sitting Sorry models for the offset of the center Right So, you know the way this measurement is made is that you you pick a lens galaxy and you measure the shear around it And you do that for in this case About a hundred thousand lens galaxies So each galaxy you just have the light and you're trying to measure the mass distribution on these huge scales So the light may not be sitting at the center of the halo And if that's the case then the inner points will show a flattening Because if the light is not at the center then it's not at the peak of the density and this can happen because galaxies merge And they may not have resettled back into the bottom of the potential well in this prediction We've not tried to model that When we look at galaxy clusters we see a clear effect of of off-centering Okay, so I've talked about voids and filaments, so I thought I would show you a measurement of void lensing questions and So this is halos we have a paper where we've shown a measurement from filaments. We took bright red galaxies in the Sloan survey and Just lined them up like this you took galaxies separated by a certain amount and then we looked for the shear in between Because you expect a filament to be connecting massive galaxies and we got a we got a measurement. Yeah For what is what I was just going to show you when you asked me to go back So yesterday I had shown you that a void profile doesn't peak at the center because the mass profile of the void is pretty flat And because the shear is given by the difference of the integrated and local mass It can go to zero at the center if the profile is completely flat. So this is a theoretical profile That fits the data. So this is the data. This is the projected density inferred from the shear And you can see it peaks at about one this is About 20 megaparsecs Because it's plotted in units of the void radius. So we identified under dense regions in the galaxy distribution We stacked them up. We measured the shear And so this is the measurement at a distance of 20 megaparsecs from the center of the void So this is this prediction using this best fit model and the black points are our measurements The pink magenta points is a test where we look at the b-mode component Which should be zero if all as well and you can see it's consistent with zero So we are measuring the radial elongation now are on voids and we are going out to three times the void radius We have measurements out to two two and a half. So these are pretty large distances So this is from a paper last year Any questions about the measurement? So, you know with voids the signal is much smaller, but because they're really big on the sky You get to use a lot of background galaxies This is from a paper that came out last week These guys tried to solve modified gravity Equations in two models some non-local model. They call it and a cubic Galilean model. I Haven't read the paper But you can see they choose these funny units to plot it But if you imagine our data lining up with the QCDM model, which it roughly does So that's a model with the same expansion history as the Galilean model, but general relativity And the claim is that this blue is The prediction of the Galilean model So this is already Ruled out by the measurements. I just showed you I Don't know what statistical significance, but you know, this is like a factor of two and you can see from the error bars just by eye that a factor of two You would have a few one sigma deviations But of course in the modeling of the void in gr itself It's a slightly complicated story because you identify a voyage using galaxies and so on So there's some uncertainty even in this gr prediction when you look at real voids. I Don't know what this feature is due to has anybody read this paper So so, you know, we had hoped that Voids would be a nice to test gravity theories because they are on screen regions and So you might expect First deviations to be larger and that's what their results seem to show Okay, so that's the first two parts I'm gonna Give you like a slightly Simple broad overview of the current state of cosmology measurements And the main goal is going to be to motivate the modified gravity discussion after that So before we do that, let's look at what we you know data sets so the measurements I've shown you are all from imaging surveys and There are three surveys currently in progress I showed you results using less than 200 square degrees of data each of these surveys is Covering more than a thousand square degrees. We are doing 5,000. They're doing 1500. They're doing maybe 2000 They're doing 1500 You know different pros and cons to different surveys, but in the next five years all of us hope to publish results So that's a huge advance in the statistical precision Related to imaging surveys, there are other surveys, right? You've heard about BAO measurements from boss So spectroscopic surveys are used to measure the 3d galaxy distribution And there's two at least two surveys that are going to come online in a in two or three years Of course CMB experiments are now measuring many things about the low redshift universe in addition to the CMB At redshift 1100 and so SPT and act now have next generation surveys That are a couple of years away And they'll be measuring CMB lensing and the sz effect which tell you a lot about structure at the same redshift range as these guys and you've heard about 21 centimeter surveys which hope to measure a 3d power spectrum using neutral hydrogen So these are you know four broad categories of surveys imaging spectroscopic CMB and 21 centimeter I'm sure this is not a comprehensive list. I apologize if you I've left your favorite survey out But these will have some activity in the next five years So I mean the last decade has been experimentally really rich With these spectacular results from W map and plank But the next decade all fronts are gonna be seeing major advances And then in the next decade at least on the imaging spectroscopic side of galaxy surveys There's LSST Euclid and W first So LSST is a ground-based imaging survey Euclid and W first are space-based surveys that have some imaging and some spectroscopy so So with these surveys We're gonna measure Geometry namely the distance redshift relation. So you can use the luminosity distance That supernova we do or angular diameter distance that be BAO surveys do And you can measure the expansion rate itself which the BAO measurements Give us you can also measure the growth of structure and you can measure it through fluctuations in the temperature of gas the mass distribution through lensing Sorry by this I'm in the CMB of the mass distribution gas and galaxies So these provide you many different ways of studying the growth of structure and this is just a picture illustrating how structure grows From the CMB epoch to the present So from this distribution of dark matter we have all these different traces that tell us how stuff is growing and Each of them is very imperfect Because we don't see the full 3d dark matter distribution directly So you need to work with everything available So, you know with plank we have much higher precision on the CMB and on the redshift 1100 universe than the late-time universe So we cannot take that as a given and then use this lever arm In scale and time to do all kinds of cosmological tests And so the one that I'm gonna talk about next is tests of gravity so this is one more slide on over viewing stuff that we have these different probes and This is what is observed and this is what it tells you so weak lensing you get out of an imaging survey You measure the shapes of galaxies you learn about some projection That equation right there in the upper left corner Involves this w of z which is distance factors angular diameter distances and the growth of structure You measure large-scale structure primarily through spectroscopic surveys the basic observable is the power spectrum of galaxies And if you look at the BO you get the geometry and if you look at the redshift space distortions you get the growth of structure Galaxy clusters are special objects. You see them in imaging. You can get Their velocity dispersions through spectroscopy and you can get other estimates of their mass distribution through sz and x-ray Their abundance is a cosmological probe that is sensitive to both geometry and growth Supanovi This is how dark energy was discovered Assume, you know the story that they get you the luminosity distance and then there's a whole bunch of other probes Starting with strong lensing in which the time delay gives you a Hubble constant Okay, so that's a one-slide overview of the primary probes of cosmology that I used for dark energy or gravity or anything else you care about And these are all the late-time probes in addition to the CMB Questions Okay, so this is the current State of the field. I haven't updated it with the new Planck results apologies So this is the distance redshift relation Which is really well measured between redshift zero and one and somewhat less well measured redshift above one So this is both Supanovi and BAO data And you can see that the best fit lambda CDM Works for nearly all the data points you can do a Cosmological analysis using this data and that's what gives you this best fit lambda CDM model with Omega lambda of point seven omega matter of point three You can measure the growth of structure I'm not attempting a review of the measurements of growth of structure. This is just giving you a flavor Galaxy clustering So this is the distribution of galaxies This is along the sky this is along the line of sight on large scales the in fall of galaxies Squashes this distribution along the line of sight and you can use that squashing or more generally the The dependence of the power spectrum on the cosine of this angle with respect to the line of sight you can use that to Learn about clustering and how it's evolving with redshift This is galaxy clustering measurements of the growth of structure at different redshifts and this is This galaxy lensing this is CMB lensing plotted in Using power spectra. So sorry these plots are not meant to give you some kind of Nice summary But just showing you the state of the field if you like you can look at the error bars and you can see that any given skill We are just barely getting we are not yet at the 10% level although the overall amplitudes are measured To the 10% level or better So this is so there's a few anomalies. So generally speaking lambda CDM is almost frustratingly Successful you can you know you measure parameters using the CMB and then you go out and measure other stuff And it seems to work pretty well But there's a few two sigma level anomalies that are worth watching and David Weinberg's lecture notes had some discussion of them. I don't know how much he got to So this is the amplitude of clustering The blue band is the CMB measurement extrapolate, you know Forecast forward in time using lambda CDM and These are measurements of galaxy clustering you can see they have pretty big error bars But they generally lie one sigma below the CMB The CFH legacy Lensing measurements come in right about here as well We are trying to measure this amplitude with DES and hope to have a paper out in a month and a bit So we'll see whether we come out here or here or elsewhere But you can see that the CMB extrapolation over predicts the growth of structure So that means that the extrapolation may not be correct In other words, maybe we should not use lambda CDM to do this extrapolation The other possibility is that these measurements are you know, they're within one sigma so they may fluctuate as you improve the data set and Move up to their one sigma Bounds and be consistent after all So an amusing fact that in general modified gravity would have predicted a deviation that way Because at least scalar tensor theories generally enhance forces. So you would expect Probes of structure using these kind of traces to give you a higher amplitude. So it's too early to To say anything conclusive, but it's worth noting So this is what happens if you take this seriously and try to look in a dark energy parameter space then the CMB and lower edge of probes drive you into different directions This is again a slightly outdated paper, but I like the color scheme so this is From Mark Wyman at all And so this is the amplitude of fluctuations now marginalized. So this is plank I'm sorry. This is plank and This is the low redshift value. I'm not sure why the Erebar is so tight But that's the direction of the discrepancy Similarly for H naught this discrepancy has evolved a little bit now be you and plank give similar measurements But they're still supernovae at moderate redshifts give a higher expansion rate Than the CMB Okay So Whether or not you believe you take seriously these anomalies in the data if you want to look beyond lambda CDM Then you want to look at the equation of state and how it departs possibly from minus one So this is the deviation from minus one what you can hope to do is Is that this is redshift zero almost the present so observations begin somewhere here And so you can look at different redshift ranges and just empirically try to measure it and you can ask these questions Is dark energy constant? You can go crazy and drop quintessence like models and ask is it spatially clustered or anisotropic You can look for couplings in the dark sector or dark sector and baryons and finally you can ask about modified gravity Okay, so these are the things that you can do with data and So now I'm gonna Get to this more this Motivation part of modified gravity. I'm sorry. I forgot what time did I start? 1115 I Started 1130 So I have 40 minutes Sorry Uh-huh Thank you any questions So you've heard I Wasn't there but by all accounts a very nice First lecture from Rachel Rosen about modified gravity So I'm gonna cover just some of the general ideas of screening and then I'll jump to observations As quickly as possible so that we don't have much overlap But some of these ideas are new so I'm sure it wouldn't hurt to hear them again so So as many of you know There's a theorem that the cosmological constant is the unique infrared modification to GR That does not introduce new degrees of freedom So the moment you allow yourself to look beyond lambda Dynamical dark energy or modified gravity you're talking about new degrees of freedom Which are also pretty generic in string theory high-dimensional theories motivated by string theory and so on so Although these things don't produce the kind of modified gravity theories We look at the idea that there's new degrees of freedom that Involve a scalar field couple non minimally to gravity Is just pretty common and unavoidable the moment you try to look beyond lambda CDM So if you're looking at theories that are or can be written as scalar tensor theories Then you generally expect Enhancements to the gravitational potential. There's a scalar field attractive forces and so When you have a gravity theory then you you can't be too choosy And so you end up with potentially observable effects on all scales Okay, so we're going to talk about gravity theories that are trying to reproduce cosmic acceleration So you imagine a theory that gives you exactly the expansion history of lambda CDM Then when you look at perturbations and interactions You can change things on all scales So that would be the signature of a difference from a dark energy model And the observational pursuits Also test the idea that you just have dark energy, but there's some couplings to standard model particles Okay So now I want to motivate the idea of screening. So let's consider the scalar field It has some fluctuations that couple to the energy density So the scalar field has to be light So it produces long-range scalar forces Which must be suppressed in order to pass solar system tests of GR So some natural ways to realize this have been Have been Discussed or produced by theorists including some people in this room. So let's look at this So if you look at the linearized equation of Delta Phi, then it looks like this and There's three terms That can all be unconventional The kinetic term the mass term or the coupling to matter So I'm sure you guys have seen some of this before But any one of these terms You know they're non-linear and they can Accomplish the job of screening this force where we know GR is valid So if you look at the interaction between two bodies, then there's an enhanced force Which looks like the Newtonian force times this term So we'd like this term to to become really small inside the solar system And you can do that by making the coupling really small the mass really large So that the interaction ranges like sub millimeter or the kinetic term really large Which is the Weinstein mechanism that is the preferred one by By many theories under discussion like massive gravity or Galileo theories okay, so So this discussion has not involved a specific model or a detailed theory But just by looking at the way these different Mechanisms work you can get distinct observable effects as you go from modified gravity that you definitely need on large scales to produce cosmic acceleration To the regime You know where we sit where we want to recover GR So that's going from giga parsec scales to one AU scales where GR has been tested So there's a lot of ways that you can make this transition And so that leads to a pretty diverse observable effects and we know the parameters that we are after This coupling or this master So, you know as you know these parameters are going to vary Depending on something like the density or the depth of the potential or the mass of the object In which you're probing gravity So that's how cosmological effects can show up in galaxies because in unscreened environments You have This metric with the potentials not being equal and in particular the Newtonian potential that accelerates you know stars and galaxies gets a contribution from the scalar field and You know in the models that we've seen The force enhancements are you know a factor of a third in some theories and if you work out how they accelerate Galaxies you can get enhancements in the velocities of 10% or more now photons respond to the sum of these potentials and To the extent that these theories can be conformally transformed into an Einstein theory This lensing is unaltered and so you generally find that things that you learn from dynamics Give you a larger Answer for the mass distribution Than what you learn from lensing okay, so You know these are very simple arguments They often don't hold exactly for particular models, but they are quite useful for motivating tests of gravity So this lensing and dynamics comparison you can do on many different scales You can even do it with very non-linear systems that you don't understand Fully as long as you understand Your lensing measurement and your dynamics But you can also look at much smaller systems because stars and gas and dark matter can respond differently For chameleon theories it was realized by By these guys That stars would evolve differently and you know astronomy as we know it could be different You know they would evolve faster. They would get brighter. So the luminosity's colors ages could be a little different So there's one variant of that that I'll show you a little in a little bit that you can Get observable tests using pulsating stars The other general difference is that you can look at traces that are themselves gravitationally very different So, you know dark matter gas clouds stars and black holes Have a very wide range of compactness So if you look at a dark matter halo versus a gas cloud versus a star the surface potentials are very different and so they might Feel the scalar force Very differently because their screening levels can be different so, you know and in some theories a galaxy or cluster halo completely screens everything inside it and in that case you don't get a lot of lot to work with but As soon as you drop that That either smaller halos get unscreened or even big halos and not able to screen very compact objects Then you get all kinds of effects That things move at different rates and you get segregation effects that black holes and stars and gas Can can even start to split apart And these two really nice papers Alberto and lamb pointed out an effect that would operate on neutron stars and black holes Any questions? Sorry. I'm doing a lot of words and talking So I'm about to show you some results So these are you know without specifying models. I've tried to give you general motivations to look at what I call astrophysical tests of gravity as distinct from either cosmological or Solar system that the two traditional regimes Are you know primarily the solar system and lab have done tests of gr That are good to one part in ten to the four one part in ten to the five. It's really well tested Cosmological tests are still at the ten percent level So this is like a kind of new regime motivated by screening mechanisms So this is a one test No known theory None of the current theories predict a deviation in this regime Unfortunately, but the test is really nice that you can look at an Einstein ring produced by an elliptical galaxy And that gives you the lensing mass Okay, so you just measure the radius of the ring the redshift of the lens redshift of the source and it's like a 15-minute calculation to get the enclosed mass Then you can look at the stars in the elliptical galaxy and measure their velocity dispersion Okay, and that gives you the dynamical mass these stars are moving at hundreds of kilometers a second and It's like a really nice Maxwell-Boltzmann distribution of velocities that gives you the the the virial mass Inside the same radius and you can compare them and these guys get like statistical errors of five percent Overall, it's like a better than ten percent check And it gives you directly the ratio of these two metric potentials The same test can be repeated Using weak lensing On larger scales, so this is you know a few kilo parsecs, then you can go to 100 kilo parsecs One megaparsec and then you can go to even larger scales using power spectra that I'll show you in a second Okay, I'll spend a few minutes on this test that involves pulsating stars Because I really enjoyed this little exercise and got pretty tight constraints on chameleon theories so So nature has given us these pulsating stars sapphiods which for more than a century have been used to measure distances because Their pulsation period and luminosity are very tightly related and so You can use that to get a Distance because you measure the pulsation period you You use that to get the luminosity you compare the luminosity to the measured flux and Then since flux times 4 pi d square is luminosity you can get d So this was a distance indicator It's part of the program to measure the Hubble constant But we turned it around as a test of gravity because the pulsation period goes like 1 over root g row You know every timescale goes like 1 over root g rows. So the pulsation period does too And so this relation actually holds when you solve this exactly and because the scalar force enhances g It lowers the pulsation period and messes up your distance estimate So if you could compare a distance obtained by this method with another method That's not sensitive to the gravity theory Then you can get a test of Gravity theories So this is just to show you that when you do these astronomical tests than everybody especially cosmologists tell you Oh, you don't want to mess with stars or galaxies. You know, nothing is measured. Well, nothing is understood. Well Well, these are sapphied variables. These are You know, four different observables their size their velocity and their brightness and the curve is Prediction of a numerical code stellar evolution plus pulsation code with just one or two free parameters And you can see how well it does with the data So we do understand some things about stars really well even though, you know, the data looks crap but This is these are the measurements that you get out of it. If you're careful So this is the reason that you might want to look at astrophysical systems Because the same theory that produces only a few percent deviations on large-scale structure produces Big enhancements. So this is the gravitational constant Inside a star. This is the radius of the star And so deep inside it gets creamed But in the envelope of the star For these kind of theories you can easily get 10 or 20 percent enhancements and even this one is allowed by cosmological tests so we used data on just 25 galaxies to get a constraint on the field value in chameleon theories Which translates into the range of this fifth force? So for kind of a natural cosmological Theory, you'd like this to be at least hundreds of megapar sacs. This is one megapar sac This is forgetting whether it goes like the square root or not Anyway, this is somewhat Between three and ten megapar sacs So so these were the previous constraints and we got constraints that were two orders of magnitude tighter than that So this was two years ago Since then there are other measurements that are approaching this level as well so so these theories are now, you know, extremely unnatural because They involve this fine-tuning of Of the field value You may have heard of f of our theories. They are a subset of these theories as well Sorry, I didn't explain what what this plot is. This is the range of the force This is the coupling constant the colored regions are excluded Sorry, I'm like so spaced out This is not our constraint forget that This is our constraint What, you know, once it one sigma two sigma, okay? So a summary then of how to look for deviations of gravity You know, so the cosmological regime is here hundred megapar sacs. These are all these astrophysical tests That I won't go over again But they span like a really wide range of scales because you can go all the way to the interior of stars Starting with the outskirts of galaxy halos. I haven't talked at all about lab and solar system tests But there's a number of interesting papers on all kinds of stuff you know the Test of the separation between the moon and the earth Do you know that somebody? astronauts put some mirrors on the moon when they went there decades ago and Now there's an experiment where you shine a laser beam back and forth from those mirrors and you can use that to measure the separation of the earth and the moon to about a millimeter or centimeter and you can do that over a lunar orbit and that Gives you a really nice test of gravity So there's all kinds of clever experiments Some of which are useful for these kind of Weinstein like theories and others The predicted deviations are really small. So you have to think carefully So the lesson I want you to take away is that when you're When you look at a new model then in addition to calculating cosmological stuff You should also think about special places Galaxies or voids or special objects black holes or Some very particular variants of lab and solar system tests It's not easy because if you're a theorist or a cosmologist, you're not familiar with the state of the art of measurements and You may not know some of the details of these systems But it's a pretty fun exercise Once you get Interested enough. Let's see how much more do I have? So once you start thinking about this and think about observations So remember I had this slide showing the four or five cosmological probes and the different kinds of surveys When you look at these astrophysical tests You get even deeper into astronomy because you have these different types of tests and they involve some really eccentric combinations of data spectroscopy to space imaging to radio observations So they really so all these observations exist the instruments exist You just need to put them together in interesting ways. And so we had some discussion About Designing some new, you know, mini surveys For tests of gravity, but of course even nicer if you can use existing data Okay, so let's go back to cosmology. So there's a metric And sorry for the current base job out of this review article, but let's look at a few equations so so now if you take that metric and look at The field equations for a scalar tensor theory Then you get a linearized Growth of structure equation that looks like this. So this is the density field so the second derivative of The density field is sourced by the potential and there's the Hubble drag term So this potential is the Newtonian potential If you write down the Poisson equation in this gauge, then it involves the other potential And Newton's constant So if you suppose you measure Lensing and you want to relate your shear to the mass distribution then when you've measured this sum of potentials So you probably want to introduce a new Gravitational constant so in GR these two are equal So that's why I've multiplied 4 by 2 to get 8 pi But then you can test whether the G twiddle that you Infer is Newton's constant or not So you can now Write the equation for the linear growth factor So you can write the density field as some its value at some fixed time times What's called the linear growth factor? And you can stick that into this linearized equation and you get something that looks like this Where this G twiddle divided by the ratio of metric potentials is just G Newton in GR But now you have two parameters The ratio of G twiddle to G Newton and the ratio of the metric potentials that you can try to measure So I did something a little impolite to these guys I'm introducing them, but I'm placing a question mark before they're you know before I've even mentioned them So why do I do that? It's because I'm not sure this parameterization is incredibly useful so So there's been all this work on You know parametrized post Friedman formulation of gravity theories these How many parameters do you need and what's the most general form and that's useful because if you don't know anything about the theories It's nice to have something to work with But they're really functions Because you know any time any theory that we've seen these guys depend on K. They depend on scale Their time evolution is is unconstrained so You know from suppose you were to do a cosmological parameter analysis And you thought you could add two numbers to your standard You know five or six cosmological parameters and try to measure them You could try to do that and you would get some constraints, but It would be a very a very weak exercise in testing gravity You'd be much better off if you actually had some handle on this space and time dependence and you could go after that specifically So I think we really will want to take Particular models and work out their particular predictions Even though you know all the theorists tell us. Yeah, this is the model I came up with don't take it too seriously, please but But you know There's no easy option. There's no very general test you can do Without losing a lot of precision because if you let these guys be functions of scale and time You're gonna need some survey that won't be done till 2050 to really measure it. Well So Sustaining general it's hard to be powerful Whereas if you are try to do a powerful test of a particular model, you're just testing that particular model so that's my depressing conclusion, but The good news for you guys is that there's new things to do from the theoretical side To just figure out how to formulate the predictions in a way that can be tested Okay I'm tired. You're tired. Luckily. We're out of time So I'm gonna have just one slide on this idea that You know, you can test gravity by look comparing growth and geometry But let's look a little bit closer at what we mean by growth Because we've already motivated the idea that given a mass distribution if you measure its properties using light deflection or Velocities or the distribution itself you will get different answers if You do your interpretation assuming GR But in fact GR is not right on those scales So to do that on cosmological scales You know, you got to work with power spectra. So this is the power spectrum of the density field This is the power spectrum of the lensing field where we've generalized Kappa to You know have different red shifts and So this is now given by a projection of this density power spectrum But we've introduced a function that lets you break GR. So this guy is given by some combination of G and phi and psi And you can calculate how different it is When you measure the galaxy distribution in redshift space, then you're sensitive to a different growth factor Namely the growth factor involved in velocities and Because velocities come out of flows out of a mass distribution You can use the continuity equation to relate the velocity growth factor to the time evolution of the density growth factor And these are some conventional symbols used to represent it So For observational purposes, we have three growth factors if we could measure the density field directly And you know the bias distribution of galaxies gives you that Then that's the density growth factor D Then the lensing growth factor is D squared times this guy and the velocity growth factor is this particular derivative so more specifically if you measure the galaxy power spectrum as a functional wave number and The angle of the line of sight with respect the line of sight and that wave number Then it's it can be decomposed into these three terms So this guy is the transverse direction And this guy is the line of sight direction Where you're entirely sensitive to velocities because you're only measuring redshifts and this is the cross term for some intermediate angle And so you get a combination of this velocity growth factor and the density growth factor So you can use all these power spectra and try to get at these different growth factors separately And that's kind of fun because if you're just Committed to dark energy plus GR then you only need one of these observables and you've got the growth factor So whichever gives you a tighter error bar. Well, too bad for the other guys, right? But if you're trying to test gravity theories, you really want them all independently So that makes a life as a observer more fun. All right last slide So I hope I've introduced it to the idea of many different experimental tests and although I didn't say it On every slide. These are also tests of some peculiar dark energy model And So let's classify them again. There's the idea of testing growth versus expansion it operates on tens or hundred megaparsecs to gigaparsec scales and It tests the hypothesis that our universe is described by GR plus some smooth dark energy okay, the current accuracy is at the 10% level and in Five or seven years, we'll be at the two to four percent level then this concept of different growth factors is another variant of lensing versus dynamical masses that you can measure inside galaxies or all the way to these large-scale power spectra and This is a test of GR Currently, it's not very accurate. We can hope for five percent accuracy So these numbers are now Gonna be dominated by systematic errors in the different observations. So just as Bruce Partridge emphasized That you know if you're interested in Primordial gravity waves and b-mode polarization You probably want to learn a little bit about CMB detectors and scan strategies So similarly this error budget is going to be dominated by some hardcore Experimental things that that you might want to look at if you're interested in one of these tests So then I mentioned these astrophysical tests, which is like, you know a dozen different things depending on the theory that go from the interior of stars to the outer parts of halos and they kind of rely on the qualitative behavior of screening mechanisms So if you come up with a new tracer or a new test you could get a Massive improvements and then there's the century old program of testing GR the PPN program that Parts of it can be powerful tests of gravity as well So that's the whole landscape if you want to learn more about the theory or the experiment There's a number of really good reviews. This is the one I was involved in so that's the one I'd recommend Thank you very much