so we will talk about some key aspects of gravitational waves that will be necessary for doing cosmology then I will be talking about a bit about Bayesian parameter estimation I'll be abbreviating every now and then P here stands for a parameter estimation I'll try to talk about selection effects if I have time and I'm going to talk about some projection for the future and some open questions so this will be the topic of the first lecture and the these will be the topics of the second and third lecture so let us begin with let us begin with plot that all of you have seen this is not some plot that's too complicated that they cannot draw on the board let us draw this figure of the gravitational wave form we'll keep this interactive so let's begin with the question what do we see in this waveform this figure there has become famous that we do not see in astronomical observations what is what is different in this figure that we do not usually see in electromagnetic astronomy nee nee answers any sorry a time variation good what else time variation we do see in transient astronomy amplitude amplitude variation frequency modulation so the most important thing is we see an amplitude and a phase this is this phase is something that we are usually not sensitive to in electromagnetic astronomy not in all cases but in most cases we observe only the intensity while here we are sensitive independently to the amplitude and the phase it's going to turn out that this is going to be very very important for the kind of cosmology I am going to talk about so we are able to track the phase evolution as a result of which we are able to search for certain modeled signals so this is only one kind of sources that we are looking at I must have kept the pointer somewhere this is a this is slide from samaya who I'm replaced replacing this traces out some of the differences between optical Astronomy and gravitational wave astronomy one of the most important things is we are we because we are sensitive to phase we are looking for coherent sources while in optical Astronomy we usually look for incoherent sources from in current signals from different parts of the source so we see only the intensity and not the spectrum another important thing is we are here we are directly sense to the amplitude and the amplitude is something that falls off as one over R well in optical Astronomy via sensitive to the intensity amplitude it is false off as one over R when the intensity falls off as one over R square it falls off a lot faster this means that if I build an electromagnetic detector that is twice as sensitive then I can look square root of two times further well if I build a gravitational wave detector that is twice as sensitive we go not just square root of two times farther but twice as far so small improvement in the sensitivity takes us much farther away for gravitation gravitation protections [Music] going further over the overview these are not all the kinds of sources that we see but there are different kinds of sources so we can draw some kind of a chart here and instead of mod instead of weak and strong what I'll call here is something that our transients so transients are things that appear only for a short time and some things that a long duration or a background and we can also have sources that we can model and sources that are unmodeled of course we have these compact binary coil senses which we already talked about which shows signatures like this they are reasonably strong transients they stay only for a short duration for how long can someone fill in so from a few milliseconds let's say about a hundred milliseconds what we observe in the current sensitivity sensitivity band do maybe a few minutes they can also be transients that we cannot really model these are things like supernova explosions so it would just be some kind of an unmodeled burst signal signal it will be loud we'll be able to see it above the noise but it would be difficult to exactly expect what kind of a signal from any underlying physics the long-duration signals are some continuous signals almost sinusoidal with some modulation that comes from the rotation of the earth revolution of the earth and what can give out what can give out a continuous signal like this if you have a neutron star that has some kind of a mountain some kind of a small bump on top of it on top of it and if this neutron star is spinning really very fast then there's a change in quadrupole moment and as you probably know a change in quadrupole moment generates a gravitational generates gravitational waves we are able to see this at the at twice the frequency of rotation of the neutron star and that we observe has continuous signals if you have any questions then you should stop and stop me and ask me I'm not going to give a very thorough overview of gravitational waves because you might have heard other lectures and because I won't have time to come go into things in very in gory details so I'm going to just mention some key aspects and feel free to ask me to fill in if you find that I am missing something that you do not understand then there's the non-event then there's the unmodeled stochastic background unlike all these other sources which which which are coherent where we are very sensitive to the phase here we have lost the phase information because of superposition from a large number of sources so this is the stochastic background and the stochastic background can be the primordial background which is very very important in cosmology and also the background from all kinds of other sources that are happening later on so the primordial black background is very difficult to observe at least with the current and the near upcoming detections what we what we would expect to see before that is the stochastic background that comes from the superposition of all kinds of other sources like CBC supernova explosions continuous waves and this is also this is also something something that's very important for cosmology but we are going to focus mainly on this sector CBC's and we will talk about how to do cosmology with compact binary coil senses yes from the stochastic background oh we can we can get the power law we can get the get the X we can get the power law and these are sources like cpc's will give it will give a certain power law while the primordial spectrum will have a different power law which is going to have a much lower amplitude but yes the the frequency dependence of the frequency dependence of expected the amplitude expected intensity this is your question so so that's that that's essentially the only only information that the power spectrum what's towards the how how strong is the signal at a certain frequency bin okay I'll very quickly mention some aspects of modeling again without going into gory detail so this waveform that I keep on drawing this waveform has three distinct parts the early part we call the in spiral this part we call the merger in this late part we call the ring down the in spiral waveform can we can be concur we can calculate the in spiral waveform using some something called post-newtonian expansion for the merger analytic calculations failed we really need numerical relativity to calculate the merger and for ring down we can do some kind of a black hole perturbation theory but we still need some fitting from numerical relativity and all this can even all this can be done if we know some parameters of the source some some properties of the source what are the properties over there the masses the two masses masses of the components what else the spins of the components usually we talk about spins in some dimensionless units where they're divided out by the appropriate power of the mass and this is this is all there is for black holes this is because the black the most Astrophysical black holes are not expected to be charged and if they're not charged then we expect them to be described only by their mass and spin this is a something called an no-hair theorem and due to this there are only six parameters for these six initial parameters for these black hole sources if they're if they're neutron stars then what happens there can be many additional parameters that come in because of the matter in the neutron star but to the reading order the what is important for gravitational waves is the fact that neutron stars in binary is they get tightly deformed one neutron star gets deformed in the presence of the other and because of this deformation there is a imprint on the gravitational wave form yes [Music] this is I made a mistake 2 plus 6 is 8 thanks for pointing this out not six but eight parameters so this is one more two masses three spins here three spins over there other questions yes the astrophysical black holes are expected to be reasonably neutral the access to physical black holes are not expected to be very highly charged so for all practical purposes we can neglect the charge effect Astrophysical black holes there is something else that I that I did not talk about that could be important here the a or the eight parameters are the mass of the first black hole three spin components of the first black hole X component Y component Z component similarly the mass of the second black hole three spin components x y&z of the second black hole no the these are the all the black holes that we are talking about our classical black holes so the spins are spins are classical and for black holes there's also a constraint in the dimensionless units in which we are talking about the total spin in these dimensionless units are less than one so if you write in terms of angular momentum then it would be j over m square but we are using dimensionless units here and there's a name for this there's a name for this theorem also oh what's the name for this theorem does anyone know what would happen if we what would happen if this dimensional spin were greater than one we would get something older what kind of a singularity naked singularity exactly so that the spin is less than 1 in this dimensionless units is something called the cosmic censorship neutron stars as I said there are other there there are many parameters describing the matter within the neutron star the most important parameter is something called the title deformability parameter which tells us how much a new pi neutron star deforms in the tidal field of the other neutron star and these title deformability parameters or simply tidal parameters lambda also then lambda 1 and lambda 2 for each of the two neutron stars they also enter the they also enter the set of parameters making it a cased 10 rather than 8 parameters this is this is simplification this is a huge simplification the physics of neutron stars is complex but this is the minimal way in which we can introduce some parameters for neutron stars yes the idle parameters of the neutron stars they are expected to depend on something called the neutron star equation of state and the equation of state of the neutron star predicts a relationship between the mass and the radius and once a neutron star becomes big it also becomes kind of easily deformable so that governs the how much the tidal deformability is this is a rough answer without going into the details I can give you references after the talk if you are more interested the other questions okay so quick question is it true that for neutron stars Oh what kind of is it true that for neutron stars what kind of what kind of dimensional spins do we expect for neutron stars zero-point-five much less than that much greater than that dimensionless spin is angular momentum over the square of the mass and this is dimensionless because we are working in I should have I should have mentioned earlier we will work in C equal to 1 J equal to 1 units so any any answers do we do we expect any what kind of dimensional spins do we expect for neutron stars the answer is it can be arbitrarily high there's no there's no there's no sense Shipman conjecture for neutron stars so many pulsars that we observe I have this dimensional spin that are much higher of the order of 100 per hundred or so but for these neutron stars in in in binaries we to expect them to be reasonably so slowly spinning okay I've talked about the physics of New France s one important thing is for the waveform that we showed earlier is a binary black hole waveform for a neutron star we do not really know after the in spiral what happens what happens exactly the numerical relativity here also becomes very complicated because of the matter in the system and we get oh that's the in spiral then there's the merger and there's some kind of a post merger behavior here which is a rather complicated and again one has to rely on numerical relativity and these numerical relativity simulations are still in a rather rudimentary stage okay going on with parameters there are two polarizations for gravitational waves you have probably heard and it's easy to write it's useful to write them out as a plus component and as a cross component to the leading order this plus component goes as is this readable or value readable or not at all sorry they should be sign-off okay so the important things here are this leading dependence on the on the total on the total mass here I should define em as the total mass and eta is something called a symmetric mass ratio yes one plus cos square I owe Tahir is the inclination angle so this inclination angle tells us how the how the finally how the plane of the binary is rotated with respect to us so if we have a binary like this and this is this is this is the direction of the angular momentum then if you are looking if the gravitational wave is being emitted in this direction then this angle is the inclination angle Phi is the Phi over there is the orbital phase and the other things like if the frequency of the gravitational wave that can that can actually be obtained from the other parameter so this is not really a free parameter is there any other parameter that I haven't described yet the D what is D the distance of the source and a very important thing to note here is this this is this is quite expected that the amplitude is going to fall off as one over distance but this is also going to be important for the cosmology that we are going to do it is incomplete to tell you it's incomplete to talk about the polarizations unless I also talk about how the polarizations affect the actual measurement so what we measure is the strain in the decay what we measure is something called strain what is strain strain is the relative stretching as the gravitational wave passes through the detector and the detector strain which I will call H of T is depends on what the detector actually is like and this detector now is characterized by something called antenna pattern function or beam or beam pattern function for the particular detector so H of T is f plus of various angles times h plus plus if cross which is also a function of various angles times H cross I still have to write out the various angles now since we are running out of characters I'm going to introduce a Phi here which is not the same as the other Phi so maybe I'll write it as a different kind of fiight let's see what kind of Phi I put in there fine theta phi sy this is minus one plus cosine squared theta cosine 2 Phi cosine - sigh - cosine theta sine 2 Phi sine 2 sy while f cross as a function of these angles ie all these files are the other Phi Theta Phi sy is plus 1 plus this is a 1 plus cosine square theta 1 plus cosine square theta Oh sine Phi sine - sigh - cosine theta sine Phi cosine to say so here theta and Phi are just a theta and Phi for the spherical coordinate system and sy is something called the polarization angle and this polarization angle characterizes how much H+ we have over H cross so if the polarization angle is 0 for example then you see that this if the polarization angle is zero then there's a sign to Phi here is sine - Sai here you get only this term if the polarization angle is zero then you get contribution only from the h plus the other term gives you zero contribution if the polarization angle is PI if the polarization angle is PI over four then the other term cosine - then the other term vanishes cosine - sy so this entire term vanishes this vanishes and this vanishes so the entire contribution comes from the cross component fine and this oops I indicates that this these are again as opposed to x and a y polarization these are + and cross polarizations that are rotated by 45 degrees with respect to each other and not 90 degrees okay two more things since I introduced a theta and a fight is tit and a Phi which which describe the orientation of the source with respect to the detector usually it's more convenient instead of talking about theta and Phi to talk about positions in the sky of the object so these theta and this other Phi can be mapped to the right ascension and the declination of the source the sky look the sky localization and declination of the source and this is this it is not true that theta is Alpha Phi is Delta there's some there's some spherical transformation that's involved that you me worked out that depends on the how the interferometer is relocated with respect to the sky at that certain instead of type let's count the total number of parameters that we have now how many do we have now we had we had let's talk about black holes first let's forget the tidal parameters we had these eight intrinsic masses and spins then these are no new parameters these just came along from the masses we introduced two parameters here the distance and the inclination and also the overall phase so there are three parameters here and [Music] how many parameters do we have over here are a deck so the two sky localizations and the polarization angle three more parameters three and three six and eight fourteen we are missing one other parameter and that is the yes that comes F G W as a function of time comes from the comes from the modeling of the wave form so that's not not really a free parameter it depends on the on the masses and the spins and when when exactly the merger is going to occur and since I mentioned when exactly the merger is going to occur there's a fourth parameter here the time at which the merger occurs time of wireless sense and that is one other parameter so that makes it 15 parameters for binary black holes and at least 17 parameters to describe a binary neutron star system questions did I go too fast did I go too slow yes the orbital phase is a function of time so here two useful parameters are the phase at a certain time so one can talk about the phase at the time of coalescence if coalescence is well-defined for example for a binary black hole the coalescence can be well-defined or one can talk about the orbital phase at a certain frequency one can see what is the orbital phase at 200 Hertz or what's orbital phase at 10 Hertz so we usually talk about orbital phase at coalescence you if you know the orbital phase at 200 Hertz and you can extrapolate what the orbital phase is going to be at the time of coalescence and again these are two parameters that kind of go hand-in-hand yes at least 17 so then then for the black hole then for the then what's what's going to change our any of the extrinsic parameters going to are any of the angles going to become different no no this part is going to stay the same are any of these parameters distance inclination orbital phase time of quality is going to change no what's going to happen in this sector only there's going to be only one tidal parameter so there is going to be at least not 15 but at least 16 parameters [Music] are the questions those functions are specific specific to the detector so what I didn't mention here is that this found these functions that I wrote down is for an interferometric detector for which the arms are at right angles to each other and then this these are for an interferometric detector for which the arm at right angles to each other I'll try to draw it in some way let's let's say this one arm is along the x direction and other arm is along the Y Direction this is the interferometric detector then this beam pattern functions depend on this is this this is the theta angle and the Phi angle is the Phi is the angle along the Phi is the angle along the x XY plane XY plane bin I'll try to I should have put this in on my presentation already but I do not have it now I'll try to put it in by the end of the day and also show you what these beam pattern functions look like for these interferometric detectors if you change the technique of detection then these functions are going to change other than one common other interferometric detector or one common other interferometric detector that is not at right angles is this triangular configuration for a future detector call the einstein telescope if you have a triangular configuration then what happens is you just get a factor of this angle course you get factors of trigonometric functions of this angle so cosine PI over 3 and sine PI over 3 multiplying each of these two terms other questions if not let's see what I have in my slides I already have a list of this is something that I have already talked about the parameters that are out there now we are getting to something interesting I mentioned that for our gravitational waves we are sensitive to both the amplitude and the phase and I also mentioned that this is going to be very important to us how so if we now want to measure these various parameters from the gravitational wave data it turns out that just from the phase we can measure the masses quite well okay the phase evolution depends on the masses of the system from the phase we are able to determine the mass quite well so phase already gives us the para already gives us various parameters of the system including masses while now we can eat these parameters that we have obtained from the face and plug them back into the amplitude and what's the dependence of the amplitude here the amplitude depends the amplitude to the leading order goes as the total mass function of some angles over distance so I've already obtained that there are pictures to this but to the leading order this is a dependence I can take these parameters including the mass put them in and it's it's important it's important here that the distance does not really enter the phase the distance enters only the amplitude so let's let me take a step backwards and tell you what are the parameters that enter the phase and what are the parameters that do not enter the phase all these intrinsic parameters over here eight or ten or nine whatever number you have these govern so let's distinguish between the intrinsic parameters intrinsic parameters the parameters there are properties of the binary itself so these these things there are properties of the binary itself they enter the enter the phasing because you can model the phase evolution of the binary with these parameters the other parameters the extrinsic parameters they are not parameters of the binary itself but they are rather a function of how we are located with respect to the binary so that would be the various angles and the distance especially the two angles in the sky in the distance and things like the a really the polarization and at what time the detection occurs so these are the extrinsic parameters these extrinsic parameters do not directly enter the face some of them enter as angles in the beam pattern some of them enter as angles while between the two polarizations so from the phase we can obtain these intrinsic parameters and that includes the masses we can take the mass plug it back into the amplitude and this now tells me the distance we are able to measure amplitude independently so we are able to measure distance directly to this source we are able to measure distance independent of any kind of a distance ladder while for in electromagnetic observations we need a distance ladder to go and measure distance so gravitational waves provide direct measurement of the distance independent of any distance ladder good I just mention a few things that are going to be useful for us later later on google not lot - yes question [Music] yes what are the title bonuses and wine her why do it not my lambda is something that I do not know that I don't have an answer to but these title parameters they tell us how they tell us if we have a if we have a feel if we have a tidal okay mmm what's what's the what's tidal effect what so what's the title field a title field is do you know or you have a basic idea right so if you come close to a heavy object then the nearby essentially the two nearby geodesics have the RO chaga it's not not directly so the Roche nope is the I mean the region around the neutron star which yeah it's not very likely it's not very directly related so that these title deformability parameters are something like if you have if this is the title if this is the title deformation field idle field then the quadrupole deformation the quadrupole deformation is proportional to the title gravitational field and here I wrote a capital lambda because again between the capital lambda and the small lambda there's some scaling and now I am forgetting there's a mass over there's a mass over radius to the fifth which we go which way with this plus five or minus five I'm forgetting right now but we can easily look up and see how this lambda maps to the dimension is lambda I think small lambda is defined as capital a capital lambda divided by M over R to the five or it could be minus five over here other questions so this is a measurement let me see if I can if I can see if I can see things if I can see things better so with gravitational wave observations alone we are able to measure the we are able to measure the distance because it's only the amplitude that that the distance affects only the amplitude and not the phase of the waveform since the phase of the waveform also affects the intrinsic parameters we can measure the we can measure the intrinsic parameters from the phase and from the extrinsic from from this amplitude and on all the other parameters that we that we have obtained we can then estimate so example for example if you now you know or now you know all the intrinsic parameters like the like the intrinsic masses of the system you can say okay these are the masses if I put this binary at a distance of one hundred megaparsecs what is the amplitude of the strain going to be and you can calculate that number but you see not that number but a different number and this tells you because because this amplitude falls off as 1 over distance this tells you how far the source should be in order that the amplitude is not what it would be if the source was at one at hundred megaparsecs but it is whatever it is a bit better or yeah yeah so the observed the observed strain amplitude is dependent on the distance and you know that if you place this object if you place the objects with these intrinsic parameters at a certain distance what the expected strain should be but you observe a certain strain from this you can infer what distance this object is that for example if you take the intrinsic parameters and then you do not know you do not know the distance you say okay maybe this object is at 100 mega power 6 you place this object at 100 mega power 6 and then you calculate the strain and you see that the strain is some 4 into n to the minus 21 the strain at some particular frequency or some particular time is 4 into 10 to the minus 21 but your observed strain is only 2 into 10 to the minus 21 then this this would tell you that this object is not at a distance of hundred megaparsecs but must be farther away since you see only half the strain so this has to be at 200 mega power 6 instead yeah but for supernova I mean with supernova exactly but then to go to the expected amplitude you need to put in some you need to put in some astrophysics where there are uncertainties and which are also calibrated by observing nearby supernovae so that's how you use the standard that's how you use the distance ladder you calibrate the supernovas for supernovae first and then use them as standard candles right here such a calibration is such a calibration is not necessary because the phasing comes very facing comes very directly out of some gr calculations and some in our simulations that is that is the that is one of the main differences here okay other questions I'll show you some so one slide about how that it I like the details this works so in the details is there are there are a few steps so the first step that goes on is the searches so this is this is a mock of gravitational wave data and this is a template that we are searching with and so we actually search for the signal in the data and if and if a certain signal is present in the data we get we get some kind of a peak in this in this cross correlation function and we repeat this search with a number of different templates so that we are sensitive to all kinds of search we are sensitive to most kinds of systems this this happens this happens very fast I mean almost as soon as the detector data is coming in now there is a more rigorous step once you're done with the search is the search is just these searches are just to indicate to you which parts of the data are interesting this is this part Oh and they do not really tell you a lot about the system they do not tell you the exact masses they tell you almost nothing about the spin components so for that you need to do something called parameter estimation when you do a more rigorous analysis and you try to come up with all these you try to obtain estimates of all these parameters that I that I mentioned I mean 15 or 17 and this parameter estimation also there are various ways of doing it there are quick and dirty ways of doing it and there are rigorous and accurate ways of doing it and both of these are important because in the end you need to know very you you need to know very precisely what these parameters would be if you want to go ahead and study populations of you or of you want to go ahead and see if these things are consistent with generativity so you have these so-called high latency high latency means analysis that takes a long time so they are accurate and they take a long time so for that you have high latency parameter estimation but is also very important for Astrophysics is this low latency parameter estimation so you try to do things as quickly as possible to get an estimate of Oh what are what are the rough masses which which part of the sky can this be coming from so if you know to a very rough accuracy which part of the sky these things can be coming from then you can go ahead and point telescopes at that part of the sky to see if there is a counterpart associated with this so this this low latency 30 parameter estimation is also important in that regard let's see this is for later yeah yeah so the sky localization the sky localization comes a bit from the fact that that to various detectors see the different signals at different times so that gives you the triangulation gives you a very basic estimate of the sky localization then the different what happens is the because you do not know these you do not know these angles the distance is not very accurately determined but there is some kind of a spread in the distance what happens is instead of getting instead of getting a very sharp distance you get the distance correlated with something with the inclination angle because this is also this is also an important thing that's going to be important for us even even to the leading order the amplitude has this dependence on distance as well as the inclination angle this one is a cosines one plus cosine squared iota one of one of them is the cosine outer but it's it's still in sun angle and it's not very easy to measure it so very often if we see a estimate of the parameters these are the contours of the probability distribution which I'm going to talk about later a bit more they depend if this is the distance and the inclination then you would not see instead of seeing something like something very sharp and when localized and not correlated like this we see a instead instead of something like this we see something like a banana and this indicates that these parameters are correlated and I wrote all the leading order waveforms if I write the full waveforms then almost every parameter is correlated with almost every other parameter and it's not not every other parameter but to a large extent there are correlations among various parameters okay I have I am running out of time so how much I have two more lectures but video suggests type was porn things until evening five minutes okay in five minutes let me answer the question that someone asked why why is this why is this red shift sitting here and that I didn't talk about so he for gravitational waves as I mentioned this [Music] leading order amplitude depends on the depends on the mass I I didn't say what what mass it's I said it's a total mass but I didn't say it's a total mass it's the total mass in which oh which coordinates so it's the total mass in which coordinates if the total it's the total mass in the coordinates of the system that's emitting these waves so is this is the total mass of the system that is out there if we observe the system over here then how is that total mass going to be affected what what mass are we going to see because the system is because the system is over there how is this this is because because the system is at a large distance well there is a cosmological redshift let's say that the cosmological redshift of the system is Z let's sit is this not very important for us so let's introduce the a small RG redshift Z so this is one more parameter let's really see how this how this affects things now how is it going to affect a mass a cosmological redshift how does how does redshift affect anything how is the redshift going to affect a certain time interval t he would go to 1 1 plus Z times T so all times would go to 1 plus Z time times all frequencies would go to any frequency over 1 plus Z and in these dimensionless units mass has the same units of that mass as the same dimensions as time I mean we can go and also check that maybe you can do it as a short exercise during the break so this mass times frequency is actually a dimensionless combination so a mass is going to go to one plus set times the same mass if you if you take this object and place it at a distance so far away that it gets a cosmological redshift then instead of seeing the mass M you would see the mass one plus Z times M and this quantity has been defined as the redshifted mass on these slides over here so from this gravitational wave phase from the staff here wave phase alone we cannot really see whether it's a redshift whether it's a object of total mass 60 at Z equal to 1 or is it 60 times 2 is 120 so it's the object of total mass what that said equal to 2 what should it this be 60 60 times 2 is 120 40 times 320 so from the gravitation wave phase alone we can we cannot see whether the total mass is 60 and the object is at a redshift of 1 or the or if the total mass is 40 and the object is that redshift up to this would this would give us the same phase okay however the amplitude is going to be different because the amplitude is going to be going to fall off as one over and the amplitude is going to depend on the distance as well and that is going to be weaker much weaker for this system and for the other system okay this brings us to a point where we can stop I was about three-quarters through the stuff that I wanted to cover in this lecture but I'll begin the next lecture talking about cosmology and then we'll go over to gravitationally of cosmology thank you for your attention and I can take some questions to ask questions during the break and we'll come back and discuss gravity of cosmology in the afternoon [Applause] [Music]