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4/4 Gravitational Waves and Binary Systems

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I have been delivered to the end of the game and then in um the government you he has a plan and so uh so I was explaining the 1st part of the effective embodied flood which was discussed summation of the dynamics and I want to emphasize to think so we start with proposed to expand the dynamics which is quite complicated and which is useless in the sense that if you try to predict what is for instance the last they will all be on the new tone up automation the first one PM the 2nd you get answers which do not converge anywhere we jump around and you don't know what would be the the good answer and also technically it is quite complicated days they are enormous simplifications India 51 but the presentation of the dynamics the 1st wisdom is that it uses only gage invariant constructs that is to say if these objects depend on the face space called in its use for them OK I can change them at will once you go to a general non-Newtonian which is expressing them until know a function of the action by Boston everything is gauging their does not depend on called admits is that when you think it is that's what I have to present at the site the quantum energy level 2 2 2 2 so this physics now the when you go I explained in detail how you do things at 1 thousand when you go at higher fuel bottled the simplification is that you 1st guests you 1st get the fact that the link between the effective energy now I would put the people want to know the old 1 and you find that the results you obtained at 1 PM and he's actually exactly at all and he and all cocaine and this was explicitly Poland although so the 1st element is there is a link between the real energy of the system and the energy of the affected problem which is given by this equation ones for which is saying that we be effective energy is my understand valiant as is leaner and understand where a sign that this is the year no no no no authority said 1 of the 2 it's the wheels which is not binding but this is the full royal including insist square so so this thing the square real images among thumb in and for the country's problems it's the and seized weapons binding energy square OK so you have this result that you find by calculation then what you find is that the dynamics of this dynamic is encoded in the 2 functions plus and in the functions that made define of the quantity all of expressions signed terms of GM overseas Quayle OK I don't want to Wenger was his grant GM over R is the Newtonian potential so GM where is the sum of the 2 masseuses square where on the UB just Yukon it's more you like the new talent and potential capital New it's a dimension less quantity now Schwab shielded his one-liners to you for years your 0 OK and that India will be what you find and yes I introduce the function but it's convenient as we know that interest shield the public 8 times need is 1 that you remember all the Foshan metric you in 2006 the function of the product in me prerecorded and you also introduced the interests of the magic on the behalf of just invitation to encode things In a simple way now I have the functions they and the function of the about these are the explained how do formations offshore option that is to say when you with these are functions of you and when you the especially find there is the old I most of these this is Russia and the macho condition is always what I've written by New Square price surgeon you knew it was written somewhere the National Commission for the effective public here OK now in Genoa you find this was found by others Shifrin myself at the pier in 2000 You need to modify the national condition New Square procedures whereby high although P that you expecting effective theories have misrepresented scrappers before those things like that and you call on the I although you golden focus is just a name that you need the formation of the national conditions so Q and let me define human New Square which is dimension less of that now acts at 1 PM and as I said at 1 PM you you don't need to add anything 1 kinda would be terms of of you square here and turns off all the up you here OK a journey distance absence in 0 and students is your tends to square this was a big thing in addition I was saying at 1 PM and I am sure Washington and New Square is just square OK at UPN 2 PM means these beasts OK here before 2 PM dynamics is equivalent to adding he wanted to win new and Cuba and here please 6 new New Square OK an NGO nothing so you think this still coefficients to a new 6 new is equivalent to on the think this is a big simplification at you find that you have you just need to add a provision to but the number at the piano they whereby square things OK in the West Rail physical they willingly so that the coefficient that appear the PAN will contain where and you find that everything is contained in 94 over the miners 41 or efforts to buy Square New you fault and then took a vision to which is plus 50 pool minus 6 new but new you killed and here you need here to add 2 times for minors you New New Square Peon over new to the fullest the term "quotation mark In a PR looking so the Fuller this is the coefficient this number this number and this number is equivalent to this Fabian thing but it's a big simplification which allows a technical simplification and at this stage and then we have computed at the next fall there are uh well there's an Excel Thurmond there is a real value them appearing at the next opportunity discuss at the console this is quickly and to and this visible goes diagonally uniform and thinking you can hear that the 1 thing the over this former you use different you embody to compute the dynamics of the dynamics I just erased is given by Tony and dynamics with the we're and press audition reaction force the Leonanie is what it is this quantity as a function of viable so I need to solve this thing in terms of the OK therefore I will get that these because it is a square would be a square root when you compute things you find that the real intelligence In new reading as a function of the free space coordinates both he equal work too capital and times Square because I have a screw to solve all these and disabilities 1 plus to new minus 1 plus plus this effective energy but different even satisfies the quality condition also is square minus square equal of New Square essentially and therefore the forgive energy is a square would itself therefore here I have a 2nd group which is squarely on the functioning of applying 1 prose he squared plus page about minus 1 and keep quiet square persists that the that feeling is this coefficient the new times the GM over To the survey did and Prime people for divided by Prime square OK this was near
square was to him so he had you already see the beginnings summations in the sense that the Internet is a little less clear what of career if I stop expanding this square I would generate many terms like prescriptive for 6 which I indeed part of the terms here but message in a new way so it is part of a of the good things but now what are the circular orbits I computer because I want to describe it as a sequence of circular orbits before being fully nonenzymatic just have idea because because I have been placed so and yet I should say new the formation parameter indeed effective embodiment of these never assuming that was a small parameter 1 not expanding this assuming 1 the 1 is once again is all that was uh only knew was that minus-5 you can say it is small but what is assumed is that we have a room the continues the formation between the smaller particles going along big black gold to the case where you have to equal must recalled that the Willetta dynamics is described by an effective matrix which when you see there is new in fact often with their new new new new New New Square you when you was was I go back to freshen this is the idea but no I haven't performed continuously trust you if you look at the effective potential for Circle you do the same thing as instructions but the difference is that in 1 minus 2 Emirati pacemaker major function it's a very simple calculations you take any textbook you like activity for instance the Book of what I'm sure the recent book you will find an effective potential for Circle bits and you in place of a function by the but then you have a surprise In the sense that if you look at this thing so they let us brought as a function of honor the 51 body this effective potential which is a function of our time someone throws angular momentum square over all square where I have this function if I do at this is to paean to remember that at 1 approximation have no modifications so so what is this function at the wanton approximation this is 1 minus to an overall by 1 person square overall square so the firm yeah let me from the fall of default putting this let me take this thing without this factor OK I will add this factor OK let me put the functions over just to and the 1 PM approximation idea of one-liners over all of which is the function like that this is dishwashing thing which is 0 2 with book if you computing it's at this is a Newtonian but it is also at 1 PM occured at 2 PM I have the 6 600 but this it's hard on you say it is too new Nuys is one-fourth of the maximum uh if I am not of the near you according to you to the cube is 1 of 2 to this Cuban swung over a it's 1 fault 1 32 it's not a big deal OK and then when you compute thinks that be you find that you have a function which is like that at 2 PM at the PAN I the 6 and that was the surprise at the time which is that this is numerically large is 18 points something and then when you compute the CPN expanded potential in this potential remember 1 minus to 2 is the essentially the gravitational potential is already the gravitational potential the guy with the new told McCutchen minus them over our ability to think at the if you put this there you find something which is isn't I that because all these are positive discussion positive decision positive so it is above the thing and then you find that you don't have like if something which courses 0 at this stage and this is what we stayed with the Alaskan Schaefer in 2000 we said No there is something that Goodier we want continuity between the Foshee Kissinger in US-Russian case we have a function which has analyze and because this thing to an is the warehousing was you is you always arises this loss the existence of an Verizon is what determines the existence of arrestable orbital for etc. can so that simple is that there isn't a license but to do that we just said we replace the Ethiopian even by these by the time they the simplest by the airport among which we'll have no rises and you say I want to but they are of the form 1 process and 1 you all over 1 of the 1 you firstly to square plus defeated killed that is to say but I did 1 thing we've won our however you up and the powers down to represent the putting on your folder in your dates a unique definition given this is a unique body and we said this would be the good be thing now if you brought this but there you find that it is the Fijian and by the you see is very close to the scene but it avoids this this thing which is the better speckled the Taylor expansion that something because focus so now at the beginning of something like that and up to date recently this was in a sense the best identical presentation 1st let us mention that the fact that all of those things are on the left the both means that in the In the gravitational attraction between 2 masters you collect the usual or Shoshie 1 overall by X which may keep less attractive less attractive in the following sense the stamp is minus to an overall it is in party potential but you had to minus him or 1st 2 an over all square pressed him about you so it means the extra forces you to the 1990 of gravity on the policy in the sense that they diminish their main attractive thing so it's a prediction Over here and now With this thing if I add this term if the angular momentum is Nunzio the curve so I Southknoll with this girl and I hadn't seen before good potential which means that I must notify by this term but that it was me curves of the foreign type they will be the always at this point because it is a product and then I realized things like that if the angular momentum is is high I will have a centrifugal barrier I will have a circular albeit stable secure obedience and because of the arises I have news OK if I diminish the value of their felt becomes smaller II really go to the region which is like that and then if it because even smaller I will this thing there would be an inflection point but and and therefore I dislocation and this is the location of the loss the secret albeit too I have these 2 objects that using a momentum and when the angular momentum gets to values such like that I have lost stable albeit and beyond that the thing will like people said the bus plunged in the sense that I like the ground falling under your feet something happens because no 1 can and that it secretly computer a we thing invasive prominently in Mathematica you know it takes 3 minutes to compute this OK I need to give to compute this but once you know it you'll be thing in and the minute you compute what is the location of the less stable orbit and you find that a the gravitational wave frequency add just to have now an idea let let's put number I said he'll be predicts what ways when the 2 Michaels get near there is a moment's whether to think we plunged to get there and after that as we will see its fast they would merge fast so it's an indication at which frequency you will see that additional ways for emerging black because up to now in nuclear things you have no skills you don't know what which scared something will happen now you have a scale that and I say because of the portable formula that additional frequencies twice the orbital frequency but with frequency means that's frequency in soccer Omega is the circular frequency 2 F is the frequency in hertz so there is a factor to on a guy you have to to buy new but Americans did did not frequency by the FA is useful something else so
it's nice to see if his frequent OK this Yukon AT T computer and then you find that this is 1 . 2 9 0 can you because of new when you recall 1 for uh the they said that when you equal one-fourth this frequency is 1 of the 6 to the where 6 is the 6 GM of the last table Robert case skippers lower tied 6 with additional medical factor which is GM the killed and which is despite here now the action due to you be defined that I have already was accepted in all that the push this year to shows the the last of the frequency of the last it up by 50 per cent it which is not subtle negligible being in this business community and the president of European it's creepy and you by the 87 the pianist was not known at the end of this year at 2 PM it is smaller OK it is Eiffel the nominees it's what we use we find this summer in our original paper we had only 2 PM and then we were going less high frequency OK but if you just its the year after and from the same year we computed the youngest issue for the thinking now and if you put numbers To compute numbers you use this formula G & Son is 5 microseconds OK and therefore I can convert the signal 6 to the thing off and things like that in their efforts and then you find that the series 5 thousand 700 parents divided by some mass units OK for instance let's assume that I have to masses 65 than the sum of the 2 muscles is 65 Salameh OK this gives the 87 . 7 so points and this thing predicts that the lasted all of it before the merger you which nearly 90 her when we find a salvo yesterday rejected on before you begin now which I will explain that that is to say when you include more information coming both from 180 coca Commission and from numerical rigidity actually it moves the scope of but it's a bit more and then it others have been shape here but it's not important tho it moves at a little bit more and then the frequency goes even higher in that case that is to say be that that we discuss these 97 . 6 7 soulful 92 nearly 1 and others in the final vote yet this is in the source said not not of the observers not our time this election fight over there because I am not saying where it is in the universe OK this is intrinsic frequency if I am there you know what worries about the condition of the I'm a a 2 year you replaced a year or 2 years earlier than the 1 led by 1 shot from over right the right behind you that yes yes value earlier extension so why you have to agree with you and knew they basically only with the support of all a case that so this is the same question that when with this was a prediction OK in 2000 we said we want something continues and then it fixes the by the you want because if you say that you can try all the bodies OK they're false choices and then you find that only 1 as the Nice properties so he said is the best so now but there is unique that is to say I give you the this expansion there is a unique presentation in this form whose pretty and expansion is equal to the seats but the poor this it failed to show you have as many questions as it was a very good 1 not having to add at this stage nobody knew whether I mean nobody took it seriously I a small dinner guests and it's not in all lot we had before knowing that made that let me insist on this because there is a paper by some of our American friends students offered keeps along with 1 of the authors of this Tokofsky and it is paper-based said on the efficiency of using but they published a book back I'm a member of this paper OK and we had used by the before and that move at the time there was no completely logical justification except the intuition that he'll be at to be in continuity with things OK then at this stage because this was 5 years before America to when you make a came on line there have shown that indeed this is a very good approximation of the thing this is now what with this but I did not want to describe the other part of the dynamics which is the addition reaction so I won't fight it again remember that you want the equations of motion as conservative American immigration present addition reactions and I said no additional action has to be the same in some way because as merger is very bad OK you cannot Hospital the 1st thing we did at the time it was to be used the ADI explained of refile in such a potash which is again to use by the information OK and at the beginning with what is on all of this is what we used cocaine and and actually this was good enough to show that yes to show similar things so in the proposed to find and so in the paper with 2 thousand please use of private information there are also the the flux which defines a about emission of the additional action on the white signed as as we were working at 2 PM and we did not need anybody this summation because we could use that appear in the CIA and does not need if you but they need you get the same thing it's the kind of it's a very small difference to buy the people not so it shows some consistency of doing bad things but then what did printed it predicts that when this 2 object's gets closer and closer they will reach moment like that where when the frequency which is something like that the 2 things we know we want to form OK and people at the time called the branch as it's as if you know the 2 things were going like that and then stopped and searched college but actually we found this was wrong what happens is when you you know what happens to the following when you are here and then at the next step you are like this you don't have a regular Vanessa divide momentum is very small at this stage so instead of suddenly plunging like that you continuing wrong because the angular momentum is the lounge and you have a little bit of legible mentally so what happens is that you have spiral which it's a little bit more inspiring so you reach the last able obedient and then you the prime she's also quasi I inspire but but then what happens because you you need so that the 1st prediction was that there was a broad of politician in the sense that the plunge is the smallest continuation of these files and in the signal in fact you don't see where is the plunge because the signal is still equivalent the sinusoidal OK but and and then you could pull but that's the during the plunge the ratio of the video momentum square and provided by the over and the longitudinal momentum P 5 square overall square was always a significantly smaller than what it was quite small even at the last table albeit which was showing you where quiz about but not quite because they were still effects because yes what points I want to say is that the approximation consisting saying that the point view is is sitting at the minimum of the things but during this is wrong because there is no additional reaction which poses the finally wants to push it in worlds and therefore the thing is always displays the deal was moving OK it's never really had but it was a direction for changes note you edit changes that you have a question yes of you sort of know you the question you sort of immigration products would trade your PL about you call nose deviation the and defied out to qualify for 5 you pulled by some reasoning that you don't need a major force if I don't care what I'm saying is the idiomatic approximation says let's take a minimum of H where are the 2 0 until that Israel this is not within days appeal that's in the piano which is known as you all the time but smaller and this is driving the system's PFI is diminishing but it's a function of our also amplified so you need a couple you the couple numerical e-mail but you have the right to integration at the time so
in 2004 with medicine we gave the 1st estimate of the complete reform it's published form inspire watering down the we also estimated the speed of the final black coal mine after the military at the time when we are using to PAN and OF ring on attachment that will described and then we predicted that the speed of the final recalled is 0 . 7 9 why modern shows it is . 7 . 6 9 record and the Bible which been found as it's been folder .period 7 years indeed so we where 14 per cent off but it's not too bad for something which was just to be in the sum to be an OK at the time and some of our colleagues In the School of keeps all that is to say in a paper of bride Creighton and fall on their where saying that there will be the gap between between 2 things both pseudonym they thought they had in mind that was before all these calculations so OK so they did not even know that they knew something like that would be true this said you cannot rely on this which is tall and therefore you can compute this cut additional waves only up to some stage because in their late teens by war this expansion becomes a bad and that when you form a final black Oracle record it will arms with the quiz and 1st discovered by the visual in 1970 so at the end you know that's the thing we indicates financially with some frequency and the main point of their fingers to say there's a big gap because this is like 12 AM and this is the relates the going on and then you don't know you absolutely need America at the time we said that and they say that that they were at least the 8th albeit the OK however aim refiners summed up most to you beat as a way of closing the gap as saying OK let's see some everything will be that animates we tossed you'll be at least qualitatively and submit quantitatively and then we can compute now the full we fall but in order to compute the full reform we need something all we need to know we we then the formalism at which point we will say because no I'm doing the the the waveform computation so during long inspire you described the system as I described by you skeletonized version which means that 2 black also but he plays like to point masses and they immediately additional ways the interact in somewhere they go undergo closer and closer but at some point you want to say this to backwards with and will form 1 black hole and then we made an assumption which at the time was not was a guest saying we can within the year before medicine determine at which moment there is a conversation between 2 0 that also and at the moment with these 2 records will emerge and become wonderful like all that we know what about mold and that's this marriage thing will immediately be be the presentable as the sum of quiz and 1 was this was the assumption of the year formalism and this is the way we computed in 2000 the 1st win form which was obtained by this thing in India before medicine you the domain of moment of Mildred that you compute Connecticut you before this moment the thing is described by 2 separate records that inmates by "quotation mark aboard plus Heileman to bolster Louisa May we think so computer display of evolution you study the dynamics using this and add the merger and the idea was to say after marriages the signal will be a sum of several quiz animals you need it's not cool to say 1 quiz along with his enough although in our initial proposal was set for simplicity let's put 1 liter we said that until we need to put more ended but the idea is if you know what the function is safe the idea looks trivial but actually it is addictive if you ever function and you know not only the value of the function but the value of its festivities and 2nd derivative you can match this function to a sum of result more quickly than 1 water which means some of CNN exponential miners are far and tea where the other in the complex numbers with positively about so they have the form if you want to have fired tests I didn't get it right this puts I only curing them so this spot is in the signatory part and this Spartans and then by you know these things because you can compute the final must have you can computes during the UB also what is the energy lost during the merger therefore you know what is the energy of the final black in the speed of the final therefore you can compute this thinks form and 1 in them to you know that this you don't know this but if you put it saved the coefficients he wanted to see the and if you impose that this function is continuous further narrative continues secondary to continues to give you the questions for the unknowns so you predict these things on the matching that they'll a smooth continuation he took 2nd totally bored think it works fine so and we are going to see much centered works the it is true was not all phases of India which is an example in the 1st paper we did the idea and we did what we think of him but it was just to show we could in principle computer before but if you do this that is state GOP waveform is finally made by joining 2 things 1 thing which is the inspire 1 motion nonenzymatic up to some moments that I was the final of merger which defines this thing and then you attach Tuen end the Ringling thing in this southern way if you look now at what it means for the gravitational wave frequencies the yes that's explode let's put half let me put the public frequency and when I don't have an orbital motion but the gravitational waves I mean we've these half the gravitational wave from circular frequency if you compute India being what is the orbital frequency as a function of time you find that a functions which "quotation mark and which a sharp maximum somewhere when you computing you beat you find that formerly the orbital frequency ceases to increase at some moment now what other various points here there is a point here which correspond to the last table and if you put numbers you find that only got a little multiplied by which is the dimension less number is a that at 2 begin Tuesday 0 . 0 0 7 the floor at the beginning of to 0 . 0 0 8 8 OK we will compare this with this 8 8 is what I was getting here produce like 92 60 405 60 almost and then there's this maximum know by looking at the European you'll find that this maximum happens when the the the related particle defective particle or in its brunch is costing the lights because before closing your eyes and there is another remarkable thing which is that if you consider for instance if you consider the function 8 over all square which is the question of a square OK as you know from textbooks if you formally consider the motion of a focus in order to the particle or in the authorities the particle in Fakenham symmetry metric you find that the effective potential yeah because here there was a new fuel OK the 1st 1 was New Square OK if you take a massless particles like a football news crazy analogy effective potential becomes being divided by all square and then disaffected potential there was a sudden shape because now the function of the overall square is is something which yeah so let me and used to be so at infinity is 1 of all-square that it has a maximum density 0 eyes and located in shield it is you want to end it is maximum at the OK which is the last it's the fault of it's OK he he'll be used said there is a light training which is not a court which is this thing but this is like case Pariser so it defines the end of this art is such that the city is the maximum frequency from the orbital this signal as a frequency given by the 1st quiz a nominal no from the studies of the Jewish
tribunal pies others you'll find that the frequency of the 1st presented him with is he is like a factor 2 up on this and therefore full because of this if you In tempering dysfunction in a smooth way by adding many quiz animal what happens is that the frequency will be something like that at the end of the be matching is saying this spot is seen by all of this is the wrath of all the days manager and then I will go to the 1st quiz 1 more OK and but but you define the merger by dislike 20 who came so it makes a prediction now what is the value of the flight training frequency he I gave the value of the gravitational wave frequency at the last stable if you me for this I think that it was the and that he used engine so I think 1 is saying well but this was a decision that would not be fit enough to play think but haven't you know what it is that faulty Percent of the speed of light and the other so 1 cannot say it's nearly the speed of light because it is at 57 per cent OK In the formalism it's not that you saying what fleet will be there you are saying I want to put the former England to the maximum to make a prediction suddenly I cannot go beyond this point so let's bored and say OK I will toss the formalism up to the absolute thing where it makes no sense beyond and then I say that 2 things are so close and that at this point I would guess that the 2 things with the 2 black cause we marriage like to Seoul bubbles and the idea was that the condition is so fast that it would be nearly that I can to continue to look and this is what has been a checked to to be told by numerical right into simulation 5 years later OK I don't know if we allow the fans to show you movies show this explicitly but the the it would OK so let me give also further numbers you find that the gravitational wave frequency for the report 24 more at merger In computer in he'll be and now which I will explain is a version of You'll be will now use comprise numerical ready to go beyond the CPN an analytical thing but finally it's it's it's too close to the city and it's not to whatever is just like a a few per cent of the adjustment you find that the field . 3 7 7 and their fault if I defined formerly the orbital frequency of acknowledges and this is the old point 19 I was quoting default which gave them 57 of the velocity of light but it's there and the guy and now if I computer things this is saying that you the merger of frequency which is this point along the thing is you can go clear this is the . 19 divided abide by GM over the killed you use these numerical formula and then you find that the city's 20 thousand 1 ability to don't withhold digits hurts divided by an informant and you should not believe that because you see 20 thousand Arabs we in the tank Europe thing because if you put 65 surround must hear you find that this is 1 of the 27 . 4 0 and 65 solemn Muslim guided by an all so I following his just because when you press your sometimes you don't you will occur L let me discuss because I really need this in the selectmen produced here 1 7 . 4 parents provided by and clean units of 60 finalists along with just 65 is taken from the property is just sitting here said was 65 press four-story OK let me discuss major liability was no I want to say what has been measure of how it is material that wine is an entity integral and useful for measurements and hold this type of predictions which was the first one is used really in the data not only icing on maybe like that but to extract things from the data for this I need to the color the basic work of Nobel about optimal filtering so what are we talking about we have an observer mission we have the output of being careful metals is something which was units including the difference this time is centered in the event that over and on the launch of the interferometry as a function of time this thing is the sum of 2 things Creek is the sum of the signal nature which is you know the TT way of accommodation when we leave but I'm not joking and this is a dimension with thing we are saying there is a 10 -minus 21 so we have a wave like that which has an amplitude really that minus 21 mentioned candidates which makes the thing today and then actually most of the time I had just of knowing and denies is something which is college OK coral noise means that it is not the white noise it is it is a does correlations in the I'm OK and I would define things now the coalition in times of war and of the 1 end of the 2 statistical average is a 2nd function of woman Institute before you transform of these as the 48 measure of the Fourier transform is called the noise OK said so I N T 1 and the 2 became please exit into a wall of ideologists and now here I am in principle to be careful In the year to fight if he was sent off In a sense of yes I think this formula is correct with the fastest so it defines the noises ascend the noise spectral density in their interferometry now in order to extract this thing except for sufficiently sponsored North and in the paper because 1 has already issued on the future of the date that you don't see the single OK in the discovery paper they have already futile that does so you can see but there is a signal because the signal was strong enough uh and with 4 smaller filtering you could see things if you don't know filtering it is lost in the noise all do you extracted from the noise this is the idea of nobody in there that you shouldn't do a scanner product between what you observed and he expected signal that is to say in order to extract a signal from the noise it's like extracting the needle former haystack you need to know all shape coral and precise structure of the needle and then you can find it because your ideas about the 2 mathematically you define the .period you define prohibits the space culture on the space of a function have to function of time OK you define it's Carol product division of product as the Internet or do you have divided by the inverse so-and-so it's the inverse noise today means the Fourier transform it's to the 1 pass page to the 2 complex conjugate S for your function it's a clear objects and now what you do is when you have of signal you might get products with the expected signal as a function of some parameters so for instance here this signaled depends on and the total mass and the amassed facial when you at spin it depends on 8 parameters so you have a signal which is you know an attitude which is a function of time which depends on the parameters you take its Fourier transform you time-shifted also tools say at which time stopped and things like that you had arbitrary phase because there's phrase projection and then you make this scandal product and then you look for which value of from that this thing is maximum most of the time this thing is when them and that is as a function of London and these will be the 1 signal of 1 of its own the those too 2 computes a good thing and in fact is good to divide it by the norm the squalor of the state-run oscillation embodies the unit Victoria of space and then
this thing is the sum of 2 terms because of socialism of present so I have this iteration of the atmosphere with a fictional the which from the process and H and not divided by his square French from the beach on the now the fact that you have to tear the invest noisier the noise means that this thing is that when the viable which is the noise is Gulston is will of random Gulston viable with the Sigma square with the variance 1 you just defined this way so here you have something which is plus or minus 1 and therefore you need this thing to be much higher than 1 to be able to say if you get this thing with something much Ireland once you know that you have detected something that's the idea you know OK I send you study all detector are lucky legality you have additional Wenzel along with them which means that the random things that you see most of the time is due to the noise not to To about about additional ways if there is about 1 of about additional waves anyway you introduce it in the noise you say I cannot distinguish it from the base of knowledge and things like that you measure it experimented OK we will see what is its value and then you define the signal-to-noise ratio square S and where also called all-square easy call actually to the square of this because this is 1 OK in economists and so waste is all square OK and and then so this thing is the scandal product it's page square divided by Chernomyrdin so that the the of the new is that taking a future which is the indisputable that is the the most accurate representation is the best way of extracting the signal from the noise you need to do a convolution weighted by the inverse noise of the output of the detector with what to expect and then this is the best stick to know when you have a signal that this is a fair and Kevina and therefore at the end if you use as optimal filter it shall not be equal to wage that is the if you have been able to compute the signal To be exacted a signal young mature in advance for instance the theory of gravity could be different from GI and what you compute but you will take this later OK if it turned out use court to which you find that great optimal the optimal easily call simply because now we have HH squarely address so you have it in each for the final result of of is that the optimal signaled to social is just the square of the age uh uh computed with this which means what this thing is the thing audience by the definition divided so it is the before transform square divided by the noise but this thing has for physicists something inconvenience which I don't know why I experimented is always "quotation mark this type for instance they say is the seat the spectral density of our detector or did the careful it's weakened payment of 23 don't conclude that you can measure of got evidence 20 clean because this is sister what perhaps the water also means that there if you want something which has the dimension of what you measure was just multiply has sent by now it becomes a strain that mention history if I'm elected by the IFI to modify by also this bias for but that also fully transform also is a bad object because the Fourier Transform is an integral role something dimension less times the the so it does mention of investor therefore if you want an object which is good for you transform withdrew mentioned you expect I to multiply also the Fourier transform biased but then too I need and explore have here but this is good because now the measure is the therefore is there mention unless it is the logarithmic measure so if I put things in Logar left the area I see is what you see is what you get so everything is dimension less and then in this form I of the love therefore this thing you can call the signals have squared and this thing can call the noise as square that is in this form is just a rewriting of the nose whistle but you have deleted the ratio of the signal in the units you like device squared divided by the noise in strain square integrated over and Octavian frequency nor liking origin who was working David frequency this is better and this you can compute you can compute analytically To some extent because of the following the uh "quotation mark the scenery is like that this is an Aussie laterally signals if I want to compute its Fourier transform I need to compute a function which is let's say the age of 2 people and amplitude any of the crimes because signs of a phase where the phase is this thing I want to compute the Fourier Transform but there is a very good approximation which is the stationary phase approximation when I computer and or 48 transform is the interwar something exponential I'd fly off the top of a tale of clean and then I have exponential minors to buy please I think the best this season as he led Table I know DOC that early in the war is the mandated by the central point point of stationary phase again so you can compute the identity could end this approximation is very good His Winograd it's not now you know a 10 per cent of automation is added to a part to the freed most of the time so anyway he gives a simple way of computing this where you can compute pitched in the office and then find for instance it is in the divided by Square would of have not where I have is that is the frequency so the frequency used to fight as he pulled 5 . that there until the faces of frequency the frequency is a function of time there is a time at which this frequency is equal to the frequency of the Fourier transform it's a little transformed funded so given the frequency of computer time I computer orders that and then it defines the formal OK so I can compute thinks this way and then you find out that this formula can be written so this is explained in many places but in policing the paper of mine with the idea and such a apart cashing in 2000 and then you find also that the the same old age as a is call square is according to the square of the amplitude as you expect because but the square of the amplitude is multiplied by a very important factor which is the number of use for cyclists contributing around frequency F to the signal this an event is the final and this is the time-dependent frequency is the dimension this number have divided by ever not divided the this is a dimension must number it says that if the frequency is being slowly so that I have a quiz I you know since signing waiver I should count many cyclists they help because I am a sign doing many cycles so it increases the effective measure ability of the signal by this quantity which is due to the cute factor of the way and anyway it's a contribution so so you can compute now you can compute everything with the multiple approximation of everything I say you can compute exactly in accordingly before from all you can compute an entity could be with some numerical all you can do the last thing which is to say I use the probable ranchland formula and I conclude that we think by hand just to see it's not very accurate but it is the main idea when you do that you find a very simple result the quadrupled from the last sale because at the end year get most of the numbers you know by hand "quotation mark formula says that the altitude of the gravitational waves is the 2nd derivative of the quotable moment available moment is an position position if I think to the limited I would love and velocity velocity an ecstatic with which is of the similar my which means the his 2nd innovative is proportional to velocity squared therefore the amplitude these velocity square a member of the velocity was defined drive Mister you an escape plan thank you as a junior and the orbital frequency .period OK see 1 here OK so from the frequency
the orbital frequency which are in terms of the gravitational with frequencies it would be fine GM s because I have a factor of 2 and the 2 parties divided by 2 1 thing so I can compute the the who what article velocity as a function of the gravitational wave frequency this is a function of frequency I use this and and this is what is the number of use for cycle before the thing change a lot but what is this window since lunchtime that's the that's the additional reaction In fact maybe not understand that then where is the oversee 5 OK this is why for instance this is at 2 . 5 PM at 2 . 5 yen Nova correction for the view the 5 pieces edition that addition being in GI is in effect that the 5th the but this is saying that the time it takes for the albeit to realize it does going to change because of edition that being in the universe of this and that therefore the useful effect ,comma instilled in fight that's all with a numerical coefficient when you put these 2 things together in this formula you find that you can compute their everything things at the political reports mission which is not good for but still which is an idea it's 1 of 15 by g square works of anyone improved the geometry can have wage of the 2 masses divided by the square of the distance at which the binaries In the binaries a signal goes down like 1 of the UN's computing the signal-to-noise ratio square OK and no I have been tabled d nonetheless by the way I should say sometimes I'm using intervals for minus Infinity's will press infinity sometimes you folded back on 0 to press Infiniti so there is effective to before but here everything is included Times 1 of the best this 1 of yes simply because of the square of the and I said the amplitude is the square probable the Square Square is before but I was invited by the 5 so I did before is 1 of the so I guess 1 of the and then have the noise and then what this is the signal that I can cope with up to the moment when the signal changes about the angles to 0 if we knew nothing we could send a signal goes like that to merger and after merger of impossible commission I could say I have effect function which is to say for frequencies above goes was you just to compute numbers and if you do that and I would show up with stop now for 15 years but that we just show the kind of thing I'm talking about here just maybe I can use this convertible I'm all for it as a function of love over alleged means natural log but in practice we will use based on mortgages to with what he's uh this thing so you see that there is a flight to inform you that date if nothing is more massive the signal is larger if the thing is that more distance the signal is larger so they this is nature of that give us that it is a number of that nobody can compute in advance of that but now all this we can compute OK what is the 1st you measure of the noise from the detector What is the noise uh the usual knows that you see things like that but because this is the noise and it's over OK this is saying the noises minimum around 150 arrests or something like that and then the noise increases at low frequencies and in cases of high-frequency here what you want is to take this noise multiplied by Esso is dimension less and take the inverse when you do that the noise is something like that which is saying no the maximum is the highest sensitivity OK endit yes so this is this this thing 1 over uh sorry this is a challenge the migrant it's 1 of region's square yeah and then there's so it's well over kitchens with over and what is this I just need to know the but the event is here it is to the 1 With the mask because the mass No is giving the conversion between but you measuring the detector that was 1 of the few tears is 1 of the few Dear OK but the source is emitting things which are frequented which depends on the mass so somewhere I need the master converts the 2 but there is a universal shape in the sense that this thing something for us it is the inverse power 1 failed and one-third of the use of his legs the all OK so it is something which is very flat and they are which goes to 0 at manager so you have 1 of VF is this thing which is saying that but the manager frequency he is the 1 I computed it goes like 1 over and so depending on the mask I will and and my signal-to-noise ratio will be the politics the centerpiece integrated OK so the area but but this thing is a universal care of F minus 1 failed so if I incorporate the F minus 1 thing in this thing I can modify this group by multiplying by 2 the miners that's all I have again universal noise which is what is something like that and then if I do that I will have fight go as well as 1 fell to infinity I we love something like that and got to cut it off here so so at the end of my signal-to-noise ratio is given by the middle area is the area under this which I can compute numerically but this area depends a lot of ways the marriage of frequency and the bans on the factory-installed OK so now I can computer related areas that uh the we violence but 1 thing that you have to keep in mind is that if I consider that the the coalescence of 2 neutron stars the masses formerly of 2 merger there would be a folder of the settlements but the number the formula I had someone although this does not really applied formerly the merger frequency Indiana of they know if I take an equally please as I have fall thousands of for forking but my pick sensitivities along 150 or so when I consider a new twist on it and then I will have a single room which extends to the cure regions where actually my noise is very high which is saying that what I see in the signal is mostly from the inspired because for neutron star neutrons that most of the signal we come out when the 2 things are still farther apart and are in a diabetic chickens that for binary black coal webbing to show what 465 solemn as the figure is up ultimately this well that is to say uh that's viable on this 1 this thing as we received these these are I already said the please 187 . 4 this is the merger of frequency so the merger frequency and end this maximum actually when you put numbers is 56 service that is to say the signal is 50 per cent of the signal is accumulated before the merger there are up to something like the less stable and being of the plunged but after merger we would show a better figure out besides after the break a but they still a significant like 40 per cent of the signal-to-noise ratio which is obtained after the merger so it is indeed an excellent situation where because of this the masses 65 so long as you have roughly around the most sensitive part of the detector and you but you still need to know the signal from quite a few cycles before all up to us to imagine to if you want the full signal-to-noise ratio which is in that case 24 25 OK we have a break now and with the goal of 15 minutes some sites the tiny creations and he has we will discuss this city is both a numerical rating it a computation of the waveform and the and you'll be our reform tuned to America at and as we would see that 2 girls who propose and you cannot see I have a friend who used they had I would discuss them OK OK
so we go fast for so that the good paper where I'm staying really predicted thinks he's 19 18 he found to a transverse stressed was propagating and the velocity of lights let's play Our tribute to Joe Weber who as early as 1958 at a time where theorists some theories were confused about the reality of gravitational waves also because it did not know the result of what I'm sure cable not will pull the mathematically that there existed an additional is in clear mathematical sense anyway Joe Weber because he had discussed with and understood and doll of the romance and the title Jacoby immigration and things like
that and realize you could detect things OK OK I don't we do this thing that was the lecture while I was just mentioned again many people said binary pulsar is in the jury pool for gravitational waves but is just because they do in the indirect reasoning saying loses in we have shown with Natalie directive that the deal like the reasoning of the interaction of the equations of motion including the weight of the interaction between the 2 neutron stars contain damping effect from the next task forces at the oversee 5 that explain the observed 2 . binary puts so it is a direct quote binary south of proven that gravity propagates at the University of light and with the properties of Einstein's theory so we don't need to go so I've explained this morning and all the work on the dynamics the last time and explained that additional where generations have explained this morning this submission techniques in DUP formalism and as I said the references are contained in the weapons from them additional waves
on the ages Woodbridge OK now and they are not settled detector in the in the world the ones that are out of a functioning now although maybe they have stopped for a meeting of the where the 2 light goods textiles in Washington state and Louisiana therefore chemicals by Faulconer interferometry uh this is a 600 materials which does not live in Germany which of contribute the viewer growers should that been online would have seen the signal but was not online at the time but have you go to participated in the
analysis of the data this is the type of things over the years of the noise measurement and textiles and all its it went in noise went down In the frequency you see this is a 1 1 of the 15 here I have prompted the inverse effect is always there so I don't want to to go back to this day OK I already there so so I said at a time where a lot of people keep uncle claimed it is impossible to use of ways of estimating what happens during the merger of 2 black called with but we said No let's be bold and we use this new method and we predict this 5 years later the young postdoc uh from Canada France Pretorius was the first one to make America wanted to work and did the 1st computation of the greatest sense of 2 block called although you had as competent as big hopes with many people from this in corner after he found that by modifying the where fighting instead of using the title of publication monarchy reduced equations that even showcase polls in the existence of the solution of the news isn't all that but he he added the exact terms constant terms of white inside 2 of better properties of evolution of the understandably like Einstein Grecians and then although books soon we forget men will accompany Lee and others and John Baker and others they succeeded in computing the Qantas sense so this is our numerical rigidity computation of the waveforms as you can see each looks like that I must say that there are several countries in the world that are able to do numerical agility calculations of grossing records are lost friends does not belong to to this club rituals hampering factor for us no it's not a question of computer yes we have the computers but then you need people to the new people to ride the codes to because let's not enter into this this is the fact that this is a OK so the effective 1 body for medicine which was initiated in 1999 and 2000 is now we will see more numerical better things so now I want to I want to insist on the plan have I explained here the basics of U B which is the way it stopped soon it was expanded to the Fijian level which is this thing where 1 found the necessity of finding this modification of the massive condition then I suggest that our 2 extended because everything I've said this was a question of Natalie have said to block spinning black also OK and then you have the defacement thing and things when you are spinning records you need to extend the UB formalism and the energy that I suggested in 2001 is to say we would look forward the formation of a curb black or instead of a fortune by called with terms depending on you that people used the I'm Estonian where you add onto to these terms are it's been dependent on OK because there was no sign his last hold on yes No I'm talking about articles on the yes I'm talking about articles but fall for neutron stars actually there with the amount of spin the carries enough don't care about this and these are spilled the 1st the 1st generalization of the waveform financial for the month being encased was known for the spinning case in this paper we somebody Chen and analysts somewhat uh which due to the fact that the on Beijing kept the polls fall nearly 1 year was published after the numerical ratings in Waco close it was it was on the archival 6 months before the indictment so this was long before America entered into after numerical rating the reform came along then no 1 realize and even before that because there is a paper of 2002 With the then existing America rigidity friends around a few cocker Leonel and integral role where we call balls the philosophy of saying once that would be a numerical rating that you can consider that there is a flexibility in the effective on Monday forming effective on Monday formalism remember depended on Parliament as a 1 8 2 8 3 8 5 that appear in a potential and B-1 B-2 briefly that appeared in the the potential as a notice the motivation is due to the thing and that at the time we did not know the crucial at 5 and then the idea was suggested that sports an unknown coefficient a 5 and let's find by comparison to numerical writing today that what is the best value of the question if I that really .period used in Corryton into data that is as you use the framework and article framework of the and then D X top parameters that you have not yet computer uh you determined by comparison with numeric originated in the pocket and this philosophy was developed into what's in the hope of some of what in the States and we've assembled and and here at 8 years in the years starting with 77 we also use comprising work with so small mass ratio of the assimilation and I will explain that some of this and the new ID fall summing the reform which activities very important for the accuracy of the UB was invented here this is the factorize waveform that I will shoulder briefly later let me also mentioned case I don't have the time to come back to it that when discussing neutron stars neutron star corner cents merger actually why I said before the tidal effects of a small 5 there are 5 PM but the content of ideas to the 5 and 4 neutron star the wages is not equal to 2 gyms where there is a factor which is significant which to the power 5 makes it's actually not negligible so you need to include and they're quite important tidal interaction of neutron stars and compute them to take accuracy in that we have done with the that'll being from many people here let me show you so reminding you the basic philosophy of you'll be doing the inspire will thing you have to .period masses orbiting .period messes with this and that makes you define Argentine within the formalism and then you say that after all that's the signal is son office of the number of quasar animal molds which of the painting modes of a deformed call and at the point I want to say is that have 2 curves you see you have a record of you a black curls and 1 of the 2 the backer of the 1 this 1 is America unity and the record of his you'll be enough which means now I'm not so that you I agreeably but do you believe tuning parameters by what I said and as you can see there
is an exquisite agreements including for the amplitude and especially the face except for the small each year but that so it shows that European there is a way to compute things so my integrity to our article from work and you just integrate them on the computer to get this now yeah goes the events so on 14 September 2015 that the 2 would I go detector also adds a handful of Washington state and Livingstone's Louisiana State's they they found signals this signals already filtered in the sense that if you add the broadband noise the 4 noise you would not see the single market but here there's a million more filtering in the sense that put the filter frequency between 35 and my father like village 50 or something like that so you puts in all the high-frequency shooter and low-frequency shooter and now we have limited bandwidth for noise and with this by with here you see the noise you hear its knowledge it's mind but if you start seeing on top of the noise there is something which looks like this thing which was formed in in 2000 and if you use the mother In pulled the Newark either purely majority or you'll be and which means you'll be imported with numerical rigidity folded last we are talking about you have this thing OK and while assuring that there are several levels of filtering what happens is um because you need to go faster when when I go you need to a filters that in a few milliseconds can tell something up on some beach in the detector ought to be able to to tell observers in excess inaccessible Megan I astronomy term All-Rookie if there is something in the sky going on so doubt there are several layers of filters uh very fast 1 that use decomposing in bins the the frequency at each moment and seeing in essentially what happens is here you see in the frequency band of these frequencies that is more than was OK I in there is more signal so you go the saying there is probably something going on then you use a second-level filters saying OK let's try to to feed that to some we've let that time plates where the frequencies chirping to Libya to say I haven't been Witcher and then at the level where you do it and then you do the full leaving filtering which is to say what I want to do is what I've explained before I take the noisy signal and I do this convolution weighted by the inverse noise waste the Fourier transformed from these but these computer for various values of M 1 and 2 and also spends up to the moment where as I explained before the scandal pulled between the 2 shows a peak for some values of the masses and then you say after we have seen something because the signal-to-noise ratio is 25 after much during and at the maximum of the signal-to-noise ratio Apple for masses equaled to 65 totals for Larsen at this stage in order not to come back to it I should say that actually the frequencies used to here well talking about events which are billion light years away moments and therefore the cosmological chief is not negligible it is like 9 per cent which is saying that you had shifted so the frequencies I had shifted the the the times our large things like that so over the frequency you see the detector or is it has to be no brochures stood if you want to say what is the frequency in the soles OK and if you excited the chop mass directly the chop must you get is not that your mustard is shifted by the 1 prison and then you need to know more analysis to extract the distance and the shift in order to compute the mass in the detective said yes so there is the yeah it adds to the uncertainty but finally not so much it is the fact that its 1 . 0 wait 5 or 1 . 1 9 uh it's it's a loaded TV small effect OK it's a it's a 9 per cent or 8 . 5 per cent you have uncertainties which are larger than that so it adds a little bit but it's not now let's discuss that's why I introduce the signal-to-noise ratio so here I have the function that I was discussing here which is it's the end this is the integrin so remember I said the square of the signal-to-noise ratio often matched filtering is invincible overall guys frequency these taxes he is the logo of the frequency of so I won't know to represent the curve of such that the area of the law this girl is equal to the signal-to-noise ratio square OK and and this is this girl this curve is what I was showing here except that you know there is more information putting because of the noise of the detective would would not have this bumpy but when I said that the lowest approximation of because he I computed using the quadruple formula of what was there the signals OK after multiplying divided by 5 was 1 of the but actually this 1 over the is doing these filed doing merger glitch happens there's something which happens and therefore this growth is is is modified it is something like that and then does that and also the the post-merger is is that most moves to get OK so these bonds when you went to bribery the noise yes this is it's not his seat this season in the OK now this is it uh so this thing which not multiplied by things should be left to the minus 7 6 anyway Indians by 1 it's a lying in the rubble of plot but you have the 6 I think due to the merger which opens here uh we find Osama we work this out to computing from UBS and the noise of Flegle 465 totals from us what he's so this is the integral rogue ice-making take 1 for the signal-to-noise ratio so the area on the this is the total Signora civilization square which is 25 square OK so that's why it's slouchy using their 800 and here you know guys make you have to do 1 you need and 1 unity of OK so you're over 25 square and should be consistent with this area now we have plotted the various curves a few days a 35 it's curve which actually should have been initiative is not important for this which is just to show that what they showed in the probe before was obtained by filtering out everything above here and everything while there so but still you were saying something because you keep most of the signal but I I discuss that from the theoretical point of view there is the 2 things go well then they reach a moment where they have their last
stable circular orbit but you costs smoothie without noticing something happens with steely gives marker of it that it's definitely the end of the spiral care in the sense that fall that we I a new brand so you can call it that although it's agrees style and as you see it easier which is saying that actually most of the signal-to-noise ratio square columns following the inspite the plot somewhere saying it comes before the most orbit a but still like you know a lot of the signal it is also after the last stable now weighs merger in the dubious science manager a is this white thing it is and in their paper of this said that introduce the composition frequencies which is now and not defined in a physical way they say we put 1 of the 32 works just to say it some part of the inspire actually what I think they did is they tried to look something to be able to say that after this frequency this frequencies after doing the plunge and before the merger that you still have enough signal-to-noise ratio to be able to say that you see something wrong Rogers separately on the inspire so that you can test that Jiang is OK during the spiral and geology our duty with the same masses is also valid during merger and bring down put together OK so it's a choice if you really ask what is the amount of signal-to-noise ratio after merger you see is the area compared to all this so this is soaring that the signal-to-noise ratio square after manager is not negligible and it's not nothing in this area but it is small compared most we did not compute numbers but you know like I don't know 85 per cent of the signal-to-noise ratio Square is before merger and then after manager you don't have much signal-to-noise ratios script that's why they cannot claim on a separate statement that 1 can claim that this thing is consistent with the same masses ingenuity as this but which is a good test of general activity it's not absolutely appalled that the quiz animals are there because you cannot extract really reflect curacy in and 1 modes what is the 1st was unknown now another picture out there show is that this widened signal which also they use them the user intelligent traits which is that of the already is this should great thing here OK if the question was put in black you would see the signal is very noisy but you have to add fuel you guide the eye by putting the theoretical predictions 101 from European and 1 the combination of wavelets and a widening in what widening it's in 1920 was made by the is that if you of corralled noise there exists only in absence formation about space which transformed coral noise in white noise and this transformation in 4 years space it is before you transform you divide by the square which square of SN the noise spectral density you go back to the time demand this is this season the transform from its according to Colonel OK T minus the primal which is defined by the inverse square of the knowledge but the poverty is when you apply these to the fullest signal noise press signals it changes the noise which is correlated in time in the noise which is uncoordinated in time and therefore which in a sense is adding noise at each time you know separated so it makes it clear what is the noise but it affects the signal because now the signal is like in old future held by the noise OK so you see now that we have signaled about what's you all know glasses sea OK that's why this shape is not the same but before but still it shows again the fact that you see in the mirror the merger is around at the fact that you see these and that the noise Cedeno's here is comparable to the signal here but here it is more of selected for the 1st Reading down you see McCain you now something surprising is 4 years people said that OK binary black costs this means and must only be 1515 why did people said that all of the Galactic in the Milky Way our galaxy all of the Black Course of masses between 4 and 14 or 15 farmers you don't know masses before being bigger than this even the largest extragalactic evidence .period as black or do something like 15 Thomas's 1 did not know in the universe In this sense of sources systems with 13 sword must calls but bitching sky and his collaborators in 2010 at predicted from the data of the throne survey that we know that half of the galaxies in the have low metallicity which means they are the daughter of polluting material that well-made off but mainly and the young and they predicted that this change a lot the history of formation of black loss binary black coats and this said 1st the Black Course would be more massive sold between 30 and so so in galaxies like that the records will be between 30 and 80 and they predicted also that it increases the number of such binary no records by big fractals thousand 10 thousand and therefore in 2010 they predicted that as as soon as my goal you go we'll have some summary advanced thing the 1st signal efficiency should be formed 30 30 by Morocco which it was a nice prediction because this is what has been seen saluted the original and 1 of the largest in the following years yes I have OK now yeah so that I have already discussed the I want to go to other things so that by the way this is the full and Estonian Jimmy on have written it was in the center of mass vis-a-vis the 2 PM and the pianist onions not in the sentiments of many Malta on OK I I never showed the explicit formulas as I said that the brochure them Ohio formalism which and being pushed to great extremes by brochure and and I have computed for instance the quadruple a Order of the gun waving by circular binary tool it should be there is the term which is known here this is the CPN level and this is a 3 . 5 PM level so these are the type of things that I said they are necesary this is very important but if you apply them bluntly and put that on the right inside this is not good because this area is larger than the 1st so you need a way to disarm them
and now let me explain some of the summation because I III quickly mentioned the you formalism which as these equations of motion together with that 1 does not see much of this is audition reaction I think 1 important thing is to the fact of life the way this is something and idea we introduce we've had assembled again and then we developed with Bala Iyer an understandable and the idea is that is that when you look at this fall realized 1 has put together someone effects because you expanding powers of the overseas and you put that with thing at some older but you can come back to to the physics behind that and say actually there physics where you can increase son to all all those a liking Q with the where you can certain all leading logs OK by solving actually but ,comma wave equation because you find that there are still effects due to the propagation of the vine additional ways in the coast time and if you solve exactly this thing and actually using the quarrel with function you can some uh the terms which due to the Kuril I'm over exactly it's uh some gamma function and then you can satirize this and after that you do a 2nd
factorization which what I had explained before which is this idea that we had introduced with fire and set up a cash that delays in the wide divergence of convergence of this thing is bad it that there is
a singularity somewhere if you wear considering the sticker albeit as an exact mathematical function white is Guosen Securities you are going to use it in jeans by wage and that analytical you can say the formula these forces could all be it's down to the level where there exists a good and the level where there exists a Roberts is beyond the massive also could all be the right wing because this is the last unstable but if you consider this mathematical formula you analyze that and the right-wing there will be a single item and probably this is the 1st single in the complex plane and therefore he defines the edges of convergence and trucks that is fewest amid edges of convergence you see that it works with the goal of the great vision so what do you do you factorizing as I explained fall by the few factorize the 1st in writing it will be there single writing and therefore this is what is known in in this
factorize we which OK which may be useful in little so this I'll explained for the I don't need to explain that yeah so here about the various factorization that me and only 2 bids in the details you that there is any infuriates some of of real value them that you can be certain by this function which contain you see an exponential like the group phase exponential-of to IQ not so it takes away a lot of bad terms then there is the source effective use the effective 1 body for formalism isn't to factorize what would be infinite if you were going to the right training and then impose the convergence of the after you have the best you do allow us to summation which is your writing as the end spyware or find other posted tone and series and now would oppose you Tennessee which is the end "quotation mark looks better than before OK and then and then this way but this is the way where you do so at the end the temple formula using you is is this type of things you write H as the positive various factor also where most of the fighters tells of 82 computer than his son but when we use this initially we find a sample to compare to numerical went into data we found that these formulas were very good a up to the lady spiral and beginning of the plunge but at manager there was like a 15 per cent difference OK and then we said all think it's clear there is a 50 percent difference but we are working with simple albeit CIA their effects which are months could so let's add then we know analytically out there should be there should be 1 plus the momentum behind your momentum square something like that which is the all for the securities and the coefficient you cannot computer that it secretly so you computed numerically by comparison numerical data and these are the end users so it is nothing you started using information from Munich eradicated beyond an identical thing but after the on something which is mainly analytical This is this anxiously factors which are allowed to be so good up to manager OK this was an example of the new information when we tested the it's uh which was here purely analytical reports coinciding with numerical data for the test particle limit for instance so this was a check that now all the mother actually we included it in fact there is also a lot which is you have to be this transparency this coefficient 85 which for many years was unknown is a full pin coefficients and thanks to recent work uh with that'll be a week with analytically computed and I should say that here we have by UN experts in the world and I shot on another type of technique that we have used to import DUP which is called gravitational set forth technique which I will mention attributes but which is a different way of using the academicians theory on black or to compute the afternoon so this time activities-known known and there is a lot about an hour but this term in that case is also known but you prefer to say I don't know if whites the 8 function as a function of Hispanic the iPad and now I 2 in the parliament of a 6 2 numerical went into data instead of including an infinite series you encourages the next there then you compute the by then you tuned in encouraging the and this gives you an effective value of 6 which is maybe not written here in the best way and this is the way you can get a numerical you begin now for now the Blase said that the goal all fighters some of what I know of has got in In a different version of which used many of the improvements we haven't would here and which was tuned to American immigrant hinted that because we did not have access for many years because of what I said to Americans Numico entered into the let's not spend time on this this let me give all the although aspects of you because you'll be was conceived to compute America already reforms for the crisis of 2 black holes but he can do more because this is the way of transforming the badly defined posted to expand the dynamics fall 2 bodies in something that can described the interaction of 2 comparable must by courts in this country conditions and in particular here we could compare various uh you'll be predictions to gage invariant quantities in binary black or dynamics like when you have a system of 2 black holes and you computed numerically the 2 articles go on they and got additional ways at infinity so they lose energy and angular momentum but you may continue can compute the losses of energy and angular momentum and therefore it gives an estimate of what is the remaining energy and the remaining angle momentum of 2 black was when they're close by subtracting the From the honor the dead zone is now a DNS integral momentum the losses to the additional weight so this way you can compute the curve giving the longer numerical activities D the energy this is the energy now it is a binding energy divided by a new so of dimension missing OK versus angular momentum divided also to mention and here you have numerical data which are in black OK and you have the you'll be analytical data which I'm right and that you have forced Newtonian because also the relation between energy and angular momentum attorney we have computed the and with the on shivered so when it's a formula you can compute and therefore on the same plot you can compute post Newtonian and he'll be numerical ITT and as you can see you ITT starts here amid some so-called junk condition and then goes along this coast of which is that you and it's a good thing and this is merger OK so up to manager this brought shows that the description of dynamics of you energy vessels which is addition thing local ordinances .period agenda there is a very nice agreement Alex on the check and others by of other cities under some also a newspaper have also compared UB and other things to pay as compensation of 2 black although this is less clear what is gage invariant there because they do that in gold in that and it's not completely clear but and and they were also in the regime which was not very close to merger but still a you'll be easier and you may data that are here for the is again a very good agreement for the spare yes 1 possession in the comparable mass case now I not have the time to discuss this is an outdated thing at the the extension as I said the extension of you'll to spin was initiated here in 2001 and then was pursued with me always keen Schiffer and then we find a sum of 1 and other people like in Enrico wells and others has seen an innocently century we uh we have a new thing which is still under development so I just want to show that for the neutron star cases although this is not just for the neutron star cases that new mentioned as I said there was this idea that 2 black costs can be replaced by 2 . message OK but as I said the neutron star as a low number and now as computed longer will this love number has in effect which is equivalent to having quadruple interaction between the title thing and the title of the formation if you express it in a in effective field theory of language and has emphasized Michael Berrigan Alstyne this means that you must either term to the effective action which is of this type thing so now instead of appointments you and to the point must culture which is this tense square contractor development and social contract the electric fence surrounds square which is an exotic locale and when you do this you can augment the UB interaction by this interaction and exchanges to the a full potential deeper polenta is the most important because it's the interaction and engine OK so it adds to the black coughing the title thing which had lost all the it's 1 of our 6 1 of 6 is just the Newtonian interaction of 2 portable self-induced quadrupled OK but you need the Aborigines the corrections that would know that I've been computed by his disabled people and particular of this thing term was computed by the metal myself and you find and and these are important because because they stem from a small entities they play an important and got additional snowfall since another thing that I want to mention Is that an effective 1 but the theory can also discussed the interaction of black also an actor Woody projected turns up to now I have been discussing the motion of 2 black also agrees a circular to murder but you can't really do like in this Matrix series have to black gold I should them against each other then they go away on each other and then they can escape at infinity can it as matrix and the investments exterior Yukon you have a gage invariant gage invariant is the scattering angle as a function of energy and angular momentum and therefore you can compare the predictions from you for what is the scattering angle as a function of energy maguro momentum with numerical activity because you can in America activity although it is in a sense more tricky than the notoriety of 2 buckles going along and you can do that and this we have done in the paper we forget sitting in an onion there said opera understandable Sunderland which underlies all up in 2014 and we studied the scattering so here you have the trajectories of the 2 black and you see you goal 2 very large cutting and other scattering angle is the angle between the initial thing and the final thing OK but when you go here you go more than by 5 get you go back from where you come and in these things the this is the impact parameter at infinity but when you ask what is the clauses distance of a portion of the 2 black cost you find at the end so it means here yet using will be stomped his country you know the 2 also they they the nearly management because the goal is to not very fast actually you can still X some information actually what you do is you compare because here you upholding the effective potential near the top of the burial of the centrifugal barrier if you if you make them if the impact parameter is reduced a little bit more than this the tilting marriage and then they don't come out so you have to stop at some level and here there is a comparison Of all this gage invariant function kind in degrees versus angle moment to at 6 energy you change angle a momentum which means you change the impact parameter the numerical data are here these are posted on your predictions like at 1 PM 2 PM 3 PM which are good In the legions of flouted by parameter but they lose value when they get cooler Why did you be keeps being good except really at the last thing where you start seeing that probably you should improve the thing let me so as I said that to walks developed he'll be enough of versions of you being they will both of banana with those last years the whereby on attacking the air and sea here in this brought have compared what is the final result of years of work on the UB description of the dynamics of records as I said most of the dynamic or information in these things is summarized in this area function because the function is the interaction energy for Monfils circular binaries OK the the function is linked to non circular wineries will fall the most interesting events you are most interested in the 8 functions OK here I have prompted various a function in the trusted limits this evolution is 1 one-liners to win over all of it is this thing and as I said when you include all the complicated for piano and finally it amounts to moving this thing out which was the energy prediction a member of the Order the terms were positive so they they were making less attractive OK and and here I have compared to girls this care which comes from a different way of tuning finding of representing the function by factoring in the blue thing is from packing the women were and the right thing is I where I had the biggest that thing and as you can see this still things have to another .period that's why it's not very surprising that notoriety when the merger is happens here so what this function is not use beyond this so here you see that the 2 growth of a close so when you use analytical information post-embargo ago activity in the new converge to the same year the description this is what this product is saying by contrast and in a sense describes what I had said when I said that the difference between neutron stars and black gold to this number OK to which is a 5 year term and in 1980 to 5 PM was beyond the arise and sold we were fighting with 2 prehensile IPN was 0 OK now here when you include this this feisty and correction is the root the tidal interaction so it modifies the function by adding at term but now the stamp it is negative you see this is the 1st a function one-liners to overall if you add that the effects of the new the formation which means that the number now gravity of 2 black also you have something which is less attractive he pushes the potential up so instead of minus 1 of which is attractive yield minus 1 of our press something repulsive but if you take into account that the title effects you add to the affection because the 2 things suck each other OK by the time of the formation and and therefore you get this thing although it looks very dramatic here because we have please some things in a certain way and actually the merger of the 2 annuitants here so in this region the effect is small because is smaller but it is on the other side that is to say that finally the after action due to the fact that the formation of the 2 neutral stance is more important than the 1 I forget about it but you need both because earlier Indians by all the gravity the thing more important and the cases the question of what is but what happened said the merger OK at 1st it depends on the mass ratio of the 2 things because if 1 is larger than the other 1 can disappointed with this walked 1 and things like that you there at some point when they're out of the same as the former disk complicated objects which we have about mold type thing immediately and then if you are beyond if 1 . 5 1 . 5 a few equation of state is such that like animals in world said Cesar Ramos is a bright than at the end it will create a black gold but in the meantime you will have a lot of matches going wrong in meeting also gamma-rays and things like that probably so he gets beyond what I need to go at it again in the black court case he'll be can't predict the signal for minus infinity to press infinity for the binary neutron star 1 is happy if you predict things to Roger and his last name the chain of the shortest which nobody nobody messes you cannot have a neutron star with the parties are masters at hand in hand with him yes the shape except that when you could invent you know without you trust us extremely cold I mean if you boson styles which wages again it's not the polls OK Is this I hope that this is now this is a good thing this is I said there is a signal-to-noise ratio OK now I said there is an optimal filter that is to say if you want to you want to approximate the real signal by some analytical signals now the measure of all goods you are he's the Venus scandal public because as we said at the end of the the maximum information you can extract is obtained by filtering the the thing by the your future and and then if your filter is a good is perfect like the presentation of the signal noise in the wrongs and lost in the mountains just to I show will be H H scandal public square of age if your filter is H. pines slightly different for marriage your maximum a signal-to-noise ratio will be smaller than H H by the cost signed all of the tools like Dole's H & H Prime inhibits space because we are in need of space where was Carol product and to also there is a Koshy Hsu asked Makovsky in the quantity of whatever which says that if I have to victims the the product of 2 vectors is length pants because science more than 1 again without positive different so you always lose in signal-to-noise ratio now the loss was in S uh which is called the faithfulness is the scandal product normalize of 2 things if it is a to 1 it means the signal to use easy quarter to lower the exact thing if it is more than 1 it means you lose something and this loss means you can lose some events and you won't have to be as closer to 1 in other words you compute the unfaithful unless of your complaints which is 1 minus the cost side of the 2 victims in about space and you want this 1 minus F to be small and and now all this is a representation it's a comparison of the faithful nights of you be in the version we developed recently fall black also and and now you see I have not explained how how we discuss with a spinning records but this is the speed parameters of the Black called 9 9 fall in that case so this means that nearly extreme spinning and A-minus 9 5 so that extremely spinning with spinning the other thing so here you are really using you know storm generative the EU be extended with being like and you compare how good you are compared to 4 numerical relativity OK and the smartest feminists to that is saying all of those things are better than 99 per cent but actually you see that most of them aren't around to see them and this fall so it means you have with 219 thousand . 9 9 per cent OK and this is also a histogram which is the maximum unfaithfulness and you see the median value of the unfaithfulness is around here it's so they minus-3 so either way this is better than what is needed for several years so this is saying that these analytical you before unforced spinning records with not that produce information used to June some parameters paralysis to have a very good presentation of the food numerical relating to the signal I should say to be complete that I said that the analysis of the of the politician with events uh I said the use of some of the of course filters and at the end of the EU's you'll be enough if there's actually at the end they used 2 types of higher accuracy from turns the he'll be enough details in the Bank of templates built by unlistenable Lebanon will using the new theory and they also used because there is a slight problem but for years we said it's not important than computers would be fast enough but apparently down that fast enough is that the UB equations of motion I want them for you you can write on the blackboard OK these equations but in the right hand side and they have additional reaction force which contains gamma function which contains you guide them all this and that you cannot solve this Ford dynamics by head OK you need to solved by solving and solving by all these won the right hand side isn't it a bit complicated takes a little bit of time OK it takes what's 30 seconds or something like that if you make eradicating takes 1 month to compute 1 wistful so the great advantage of having an integral to the additional with form is that you can compute many many complaints and thousand complaints and that a good but if each them played takes 30 seconds you Indiana Table 1 if you want to compute millions of templates this is too long this thing and the reason why they want to compute millions of complaint is they want to do a Bayesian analysis that is to say if you want to put this in fact all probability 5 tells of road that you measure of this and this quantity and to do this you need to estimate poverty distribution function in parameter space form computing many many things in vision and analysis and that is something takes 30 seconds it's too long so you can go to a 2nd stage which has been performed in that era by people like Michael grew up in Canada and in nominal and others so that we simply because of this where you take the you'll be enough things and then you look at them and and then even in analytical form effectively is a U decompose the face and the frequency in the four-year demand and you say OK I can feed them to something and so you fix something which was fitted to an hour and thing and then give something which is now completely analytical and which no text less than a minute 2nd to compute for each because it's an explicit for four-year demented and this you can do vision analysis with millions of something so I guess that's the and yes so that the conclusion is that yet because I did not say that at the end I did not show up I had all that transparency is that there somewhere else showing gravitational sent false analysis smuggled from the division theory analysis to excite parameters of new but this is like more advanced opaque the reason playfully on them so recent now and also the main points and I wanted to convey is that you cannot use it is wrong to say that's a you use numerical ready to predict the things OK you need analytical methods unity inferior to have a basis of medical things you need to do something in theory in you'll be and you'll be in form which allows 2 to predict many things you need more layers you need also filters so there is a lot going in the data analysis of the tablet somewhere in the middle of that defied that this will be farmers them exists and that there is a complementarity between America duty of here it was written is likely to be essential but as being essential to compute things because this has been done now and you be was part of the game OK thank you for your attention theft and right
Resultante
Vektorpotenzial
Gewichtete Summe
Hausdorff-Dimension
Physikalismus
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Physikalische Theorie
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Körper <Physik>
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Physikalische Theorie
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Physikalische Theorie
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Physikalisches System
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Physikalische Theorie
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Integral
Objekt <Kategorie>
Singularität <Mathematik>
Energiedichte
Quadratzahl
Minimalgrad
Flächeninhalt
Basisvektor
Mereologie
Ruhmasse
Normalvektor
Wellenform

Metadaten

Formale Metadaten

Titel 4/4 Gravitational Waves and Binary Systems
Serientitel Thibault Damour - Gravitational Waves and Binary Systems
Teil 4
Anzahl der Teile 4
Autor Damour, Thibault
Lizenz CC-Namensnennung 3.0 Unported:
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DOI 10.5446/20546
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2016
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik
Abstract This crash course will review the theory of the generation of gravitational waves, as well as the theory of the motion and radiation of the premier expected source for gravitational wave interferometric detectors: binary systems.

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