Hello, my name is Tae Kim, I'm from the University of Notre Dame in U.S., not in Paris, nor France. So forgive me if I'm saying as a Notre Dame, not Notre Dame. So today I'm going to talk about the detection of pairwise hotspots on the CMB through deep neural network. So first, Marco gave a great introduction about cosmic inflation, and this is basically we think it's a common paradigm of explaining the seed that seeds the fluctuations on CMB and large scale structure. One thing that we are actually focusing on here is that inflation being the scale invariant. Actually we are going to violate that. So what it means is that there is a specific time frame there, we are going to produce this kind of heavy particle production during this inflation where mass is way larger than the Hubble scale. And as a result there will be the localized signals actually appear in the position space, and then this signal detection can be done either position space or in the power spectrum level, so both position or momentum space can be possible. And my main focus is that using machine learning as a tool so that we can pluck out those signals. This is a schematic of the particle production. As I mentioned, this pairwise massive particle gets produced at the specific time frame, and then this particle actually modifies and then creates the curvature of the perturbation as shown over here. Then later this particle actually gets, the perturbation gets frozen because it is created during the inflation, so the Hubble radius gets smaller and smaller during inflation, that's why it gets frozen until it re-enters the horizon. And then once it re-enters the horizon, it gets effects from the surrounding and then the profile gets modified. Then later on the CMB it actually creates some sort of hot spot or cold spot signatures. The way to incorporate this violation of the time variance is that using this specific model. Here we are actually opening up the portal between the inflatone and the heavy particle, and from here we can see basically that we are cooling that pairwise production is actually happening. Then this ultra-heavy particle is, since it is actually coupled, and then because as inflatone rolls the field value changes, this is actually the effective mass term gets minimized at some specific time frame. That's when we think as the maximum production of this heavy particle happens. I'll go more in slightly more detail in how this particle gets produced. The way to understand it is actually looking at this equation, the motion of this particle, sigma. With some proper variable substitution, which is you, we can see that this, once we clean out the mass, it gets simply becoming the simple harmonic oscillator. One thing to notice here is that the frequency actually is a time dependent. So when we look at the parabolic shape of the harmonic oscillator, it actually starts to deform, depending on the time. Then how much it deforms at a given time is depending on the, when we calculate the violation of the SAD of electricity, which is omega prime over omega square. It gets maximized when effective mass becomes minimized. Another way to look at it is because this potential, parabolic potential is time dependent, and then it gets maximally shifted at some specific time frame when the mass of the sigma gets minimized. We can think of it as, for the most of the time, the ground, the sigma is actually in the ground state, which is saying that there is no sigma present. But as the potential dramatically shifts at the specific time frame, the old ground state, it's actually not the new ground state. It is actually the linear combination of the excited and the ground state. So this actually a way to view how the particle production happens. Detailed particle production can be also expressed in Bogoliubov transformation, which I'm not gonna explain too much in detail because I don't have much time. But the point here is that at the specific time frame, the initial creation and annihilation operator can be expressed and as a later time creation and annihilation operator. And this can actually constrain, we can actually use this formulation to calculate the number density of the theory, number density of this heavy particle, and then we can put some back reaction constraints. Yeah, back reaction constraints because we don't want to stop the inflation because we produce so much, it's gonna basically stop the inflation from happening. So this is, I will basically skip all of those profile. So bottom line of this profile is that it actually gives the exponential shape. By doing that particle gets produced and then it get back reacts to the spacetime and then it curves. And then once we calculate, it gives a non-trivial rise to the one point function which later gets reduced into the exponential like. Okay. So the bottom line is that the primordial curvature perturbation from this heavy particle production looks like this. And then because it has two peaks because it produces a pair. This is not the end of the story. We have to use the sax wolf and then integrated sax wolf effects so that it can be projected into the CMB less scattering surface. The bottom line is it looks like this. There are two central peaks and then there are some decaying modes. The detection method, we are actually using convolutional neural network. This is mimics the how human sees things. For example, it enhances some specific features. Yeah, sorry for going over but what we are using is using, we are training the network based on the small patch of the background. Whether it has implanted signal or not. And the network is trained to capture those. It is actually worked on the thing that is really faint. We cannot even see very well, even though the signal is implanted. It actually yields a pretty good signal capturing. It actually at g goes forward, which is the inflectome coupling with the heavy particle is large and small, large. But the signal is faint. It yields a good signal capture. So that's all and we are actually trying to look into the actual plank data at some point to run this machine learning algorithm, scan over it to see if we can set the bounds on the. Training is done, actually. We are also trying a traditional method using matched filter technique to see if it's in, I think only one question. Do you need the mic? Yeah, so that's a good question. So right now, network-wise, since this is a well-controlled scenario right now, we are actually implanting the fake galaxies, faking the hotspot from the faint galaxy signal. That's also we have to implant it to make it more realistic. So we are actually in the middle of making it more real. But right now, if it is really idealistic, the background rejection is about 99%. So it's pretty good. But again, there is a robustness of the simulation versus the real. This? Okay. So primordial magnetic fields are magnetic fields that are originated in the early universe. And, sorry, how I can show a pointer? Sorry, that's... So this is the pointer. Okay, thank you. Wait a second. Okay. So primordial magnetic fields are magnetic fields that are originated in the early universe and then evolving with our expanding universe during different phases of its cosmic evolution. And since these fields are originated in the early universe, they could affect CMB, Big Bang-Miguel synthesis, or it could also produce gravitational waves. And at later times, they could also source as it could also induce earlier reunization of the universe. And for example, yes, I show here this sketch from the recent studies. We are shown that where the primordial magnetic field effects come comes in CMB angular power spectrum. Or if we see also the evolution of temperature of intergalactic medium, we would also see that with respect to standard scenario, primordial magnetic field would induce larger temperatures and therefore could also induce earlier reunization of the universe. So while talking about the importance of primordial, of the studies of primordial magnetic fields, we wonder where the hypothesis of primordial magnetic fields come from. And observations show that not only planets and stars, but also galaxy and galaxy clusters are permeated by large-scale coherent fields, which are of the order of microgauls in galaxy clusters and they are correlated at least on kilopyrus scales. And recent observation on laser spectral observations have shown that they are compatible with non-zero magnetic fields in cosmic whites. And for example, here is shown these black dotted lines from observations, which are compatible with non-zero magnetic fields. And actually, these type of observations put a lower bound on the strength of magnetic fields in cosmic whites. So there are different generation scenarios how primordial magnetic fields could be generated in the early universe and they could be generated during inflation or phase transitions. And in the inflationary scenario, their coherent scale would be unbound while in the phase transitional scenario, their coherent scale is bounded by the Hubble horizon scale at the moment of generation. And in our work, we study both inflation and phase transition-generated seed magnetic fields and we evolved these fields from redshift 50 using the cosmological MHD code ENSO. And we employ a lumped-CDM cosmology without cooling and feedback physics in order to solely focus on the evolution of primordial magnetic fields during structure formation and amplification of these fields due to adiabatic processes. A novel approach of our work is that we use the results of previous work, so we take into account evolution of these fields in the early universe and we take the outcome of the MHD simulations with the Pencil code as an initial conditions for our cosmological simulations. So in particular in our work, we ask a question when looking at the magnetized cosmic web, can we distinguish between different primordial magnetogenesis scenarios? And here are the initial conditions of our simulations. So we study inflationary-generated seed fields as a uniform magnetic field which has a spectrum, just it's a constant and it's just your delta function and it's just a peak that k equals your wave number. And in the inflationary scenario, we also study stochastic magnetic field which has a turbulent spectra because it develops turbulent spectra during the evolution in the early universe, so from scale in marine spectra, it develops turbulent spectra. And here I've shown this in the left two panels, our initial conditions, projected magnetic field strands. And we say that in the uniform case, it's just constant and directed along the diagonal while in the scale in marine case, it's just a stochastic distribution. And in the first transition-generated seed magnetic fields case, we study stochastic fields, but as we see, their power spectra is different, so they have characteristic peak. And in helical cases, characteristic peak is towards smaller wave numbers because helical fields would have larger coherent scales after the evolution in the early universe. And yes, so here I've shown our initial coherent scales for different models. And here is the final picture, what we see after the evolution of these fields in the, during structure formation. And where we see actually that during structure formation, final distribution of magnetic field chose dependence on the initial topology of the magnetic field. So we see that the initially large-scale fields, such as uniform and scaling variant models would lead to larger, largest amplitude in cosmic web and also largest coherent scales when compared, for example, to helical and non-helical models, which have initially smaller coherent scales. And also, if we look at the power spectra, this is also obvious from the magnetic energy spectra. Here y axis show magnetic energy spectra and x axis wave number. And these dash dotted lines are initial spectra for stochastic models. And as we see on these very large scales, we see that initially inflation-generated uniform and scaling variant models lead to largest amplitude compared to phase transition-generated helical and non-helical models. And yeah, we can also study how the correlation length evolves during structure formation, which is actually defined here, and it's based on the magnetic energy spectra, which is EB. And if we look at the evolution of magnetic correlation length, here y axis show magnetic energy correlation length and x axis time, we see that at final rate chips, we would obtain larger coherence scales again from inflation-generated acid magnetic fields compared to phase transition-generated magnetic fields. And of course, we wonder also how we can observationally distinguish between these magnetic fields, between these primordial magnetic fields. And yes. And I will briefly show our results from Faraday rotation study. So Faraday rotation effect is an, so when polarized emission from a source passes through the magnetized plasma, its initial interesting polarization plane is rotated. And this is the Faraday rotation effect, and this change in the polarization plane is proportional to the so-called rotation measure and observed wavelength squared. And the rotation measure itself is an integrated quantity of electron number density and line-of-sight magnetic field strands, and therefore traces line-of-sight magnetic fields. And we also studied this rotation measure from our simulated models. And what do we find? We see that because of the initially uniformed scale invariant, initially Earth-scale fields would give rise to larger amplitude in the Kessmik web. They would also show largest and more correlated rotation measure values in the Kessmik web. Yeah, so then I will not say anything about conclusions and just leave this slide. Based on their effect on the rayonization history. Yeah, so actually I think where we can, like apart from the rotation measure, which we also study and compare with our observations from simulated, it's also on CMB power spectrum. How you can, if you can see, well actually I think that has not really been studied also, like it has not been studied what different topologies would lead in the CMB power spectra. Because usually people take either scaling variant model or just uniform model, and also in cosmological simulations. Yeah, it's the first time we did different topology of the magnetic field. But yeah, so that I think has not been done, but it would be interesting also. Sorry, if it's turbulence dominated, evolution in our simulations. Yeah, so I cannot say that we really resolve turbulence in our cosmological. So in our second project where we focus on galaxy clusters, there we employ higher resolution and then we focus on the amplification of magnetic fields from turbulence. But yeah, so here in the simulations we cannot really say what is the effect from the turbulence. Hello, okay. So there is a question from Zoom asking that are we choosing a special direction if we assume uniform field? Yeah, so yeah, as I showed, it's directed along diagonal and our simulations. And yeah, so it has special direction in simulations. Hello, is it better? Yeah, okay. So as she explained, the primordial magnetic fields in a great detail, so I won't go in discussing about the primordial magnetic fields. So what we have done in these, sorry, sorry. So this talk is based on these two works. So in this work, in these work, what we have done is we have tried to understand the problem of baryonyl symmetry by generating helical magnetic fields in the models, basically during inflation. So okay, so the observation suggests that there is a microgauss strength magnetic field which is coherence over a kiloparsec scale. And we don't have still any understanding of the origin of magnetic field in the sense that we do not have any compelling theoretical models to explain the origin of these magnetic fields of such a large coherence length. But a well-accepted paradigm suggests that these magnetic fields are generated during inflation due to the quantum fluctuations of the gauge field and which has been later amplified due to the dynamome mechanism or amplification process. And we know from that there is from these probes that from BBN and the CMB probes that there is a matter and antimatter asymmetry in the universe on large scale and that asymmetry is characterized by this quantity eta b, which is a baryon to photon number density. And this ratio is of the order of 10 to power minus 10. So Sakharov suggested that in this remarkable paper in 1967 that in order to create a baryon asymmetry in any particle theory model, one must satisfy these conditions independently like baryon number violation, charge and charge parity violation and departure from thermal equilibrium. So in this work what we have done we have tried to map these Sakharov's conditions with the symmetries of the universe in the presence of a magnetic field. So in this 1996 paper, Sasha Davidson pointed out an interesting relation between the primordial magnetic field and the Sakharov's condition. So due to the presence of primordial magnetic field there would be out of thermal equilibrium dynamics. And since magnetic field is odd under charge and charge parity violation, so charge parity symmetry. So there would be a CP violation due to the presence of magnetic field. And since it is a vector, so it will break the isotope also. But interesting thing to notice is that the first two conditions are same as the Sakharov's conditions. So what we, so this, the baryon number violation we cannot achieve in this model what in the symmetries of magnetic field in the presence of primordial non-helical magnetic field. So for that, in order to consistently map the symmetries of the universe we need a helical magnetic field and that is a missing key ingredient in her model. So what we have done is that, to understand the helical magnetic field, one can think about like if the electromagnetic field, the massless gauge field, U1 gauge field has two degrees of freedom. And these two degrees of freedom, if the propagation of these two degrees of freedom is different, then they lead to a non-helical electromagnetic field. Now this baryon photon number density is related to the, this transform number density by this relation one. And one can see that if A plus and A minus corresponds to the U1 gauge field with the left, right circular and left circular polarization. So if both modes are propagating differently, then we have this non-zero NCS, the transform number density. And hence that will create an imbalance between this baryon and anti-baryon. So we, in this way we can violate this baryon number, we can achieve this baryon number violation. So this is the missing ingredient in her model. So now with the helical magnetic field, one can consistently map the symmetries of the universe with the Sakharov's conditions. Now to achieve these things in the model, we have considered this interaction term. So due to the Riemann coupling, Riemann tensor, which breaks the conformal invariance of the action. And this, you can see that there is the alpha beta and f mu nu tilde, where this tilde is nothing but the dual of the electromagnetic field tensor, which breaks the parity symmetry of the action. So, and these two are the necessary conditions to generate the helical magnetic field. And in this capital M is the energy scale of this conformal breaking. So what we found is that the strength of the helical magnetic field on CMB scale is 10 to power minus 14 Gauss, which is well below the CMB constant on the magnetic field which is 10 to power minus nine Gauss. So, now as I have shown earlier that this gauge field is related to the, evolution of this gauge field is related to the baryon number density. So one can calculate from this relation, the eta B, which is the baryon asymmetry parameter. And we found that this baryon asymmetry parameter to, so to have this baryon asymmetry parameter of the order of 10 to power minus 10, these are the ranges of capital M and reheating temperature which are allowed. So, which is, which are within the constraints which we know on the reheating scale, basically from 10 to power 10 GeV to 10 to power 15 GeV. Yeah, that's, thanks. Thank you. No, no. CP. Yeah, so the presence of magnetic field will automatically, we have not looked at that problem which you are asking. In the sense, we have just tried to show that if there is a magnetic field, then it can create a baryon asymmetry, but those details we have not looked at. For example, the later in the, there could be some, is fell around wash out. So those things we have not looked at in this model. So it's a simple model in the sense we have, also we have assumed the instantaneous reheating. So yeah, so I don't know in this. Hi, what is the baryon symmetry violation? The source of baryon symmetry violation? The source of baryon symmetry violation is the presence of gauge field, the electromagnetic field. Okay. So if we have electromagnetic, the magnetic field or the gauge field which has a non-helical in nature, non-helical, for example, if we have in this relation one, so if we have a helical magnetic field, then a plus is equal to a minus. The both modes propagate, both modes have the same propagation. In that case, we have a zero chance to one number density and that since the chance to one number density is related to the baryon number density, that will be zero. And if that is zero, which means that n b a minus n bar, which is the difference between baryon number and anti-baryon number. That is zero. So with the non-helical field, we won't have any baryon number violation. Okay, thank you. But with the helical, we can achieve that violation in the... Hello? Okay. Well, thank you all. Today I'm going to talk about our work we did last year with my PhD advisor. There. Diana Lopez-Nazil in collaboration with Federico Urbán. Okay, so probably what I'm going to show you here, what I'm going to talk, is about a model of ultralight dark matter, of spin two. Of spin two. Ultralight because its mass is very small. And if this model interacts with gravitational wave interferometers, we can detect its signal. So this is the task I'm going to show you now. Okay, so the spin two model. Are you all? The question of state for this model is like the question of motion, sorry, the question of motion for this field, is like the question of motion for the scalar one, except that now we have two indexes, but it's totally analogous. And we're going to focus on astrophysical cells and in this regime, in which the mass is much larger than the Hubble parameter. With this, we obtain a solution that is an oscillating one, in which the oscillation is given by mass. That means that the frequency of the field is given by the mass. Here what I want to remark is this epsilon Hj, which accounts for the bipolarizations of the tensile, which is massive. That's why we have bipolarizations. Okay, so two things I want to remark on this model is that as I said, the frequency is given by the mass of the field, but mostly it remains coherence, given that the coherence time is pretty big. But also this field is homogeneous, given that the Broglie wavelength is also pretty big compared to the scales we are going to be interested in. Okay, so I talked about the model, but I didn't talk about the gravitational waves we are going to look at. So most of us are most used to hear about the detection of gravitational waves coming from, for example, binary black hole merger, which are like strong events in the sense that they are very few in time, you know, like the chipmats are big, and they are very strong in the sense that the strength is of order 10 to the minus 21. But besides these events that can detect, there are also another signals that can be used to detect gravitational waves as these, as I don't know, as continuous gravitational waves. Instead of being very short in time and very strong, there are weaker, but they are coherent over a longer time. We are going to focus, and we are going to use these kinds of waves, continuous waves. Okay, so how do we expect the signal to be? In order to do that, okay, in order to do that, I'm going to present the interaction term that is going to account for the interaction between the field and the gravitational waves in the field meters, which is that way. Here, mp is the Planck mass, the reduced Planck mass, and alpha is the strength of the interaction. What this interesting to note is that we can make a change in the frame in order to absorb this interaction, no, in order to absorb this interaction. So now we redefine the metric, and we obtain no interaction at all, but instead now the propagation of the information is given in a metric that oscillates and that oscillation is given by the tensor field. So now the metric perturbation, which is this part, and I call Hij, is given by this term here. So now, what do we see or what does the detector of gravitational waves sees? Well, the detector sees the response function, which is defined this way, contracted with the oscillating of the metric perturbation. So what the detector sees is the, I mean, what is called the signal, is the contraction of the response function, the Hij, and the contraction with the metric perturbation. Okay, so in terms of this, we define the effective theoretical strain amplitude, which is defined as the average overpolarization angles and random phases, which came from the tensor field. And what we do now is to compare this theoretical signal that we expect with the sign sensitivity of different gravitational wave experiments. Okay, so what we get is this plot which I'm going to explain right now. So note that the signal that we expect is inversely proportional to the mass. So those lines over here are the theoretical strain that we expect, because we are in a log plot. So this signal is plotted by different values of alpha, 10 to the minus four, minus six, et cetera. And what we show here are different experiments, such as LIGO, Einstein Telecom, et cetera. And I'm not going to explain all of these, but I'm going to focus only on one, which is this one, which stands for Hanford, Limiton, and Virgo. Oops, spoiler. Okay, so what this plot says is the following. So take, for example, the strain we expect from the value of alpha, 10 to the minus four. For this value of alpha, and this mass, or this frequency, it's analogous, for what we have right now in the detections of Hanford, Limiton, and Virgo, we can detect a signal coming from this tensor field. For example, if we expect Liza, Liza, I don't know how to say it, Liza, Liza, to be this signal, then we are going to possibly detect or constrain darmador signals coming from this range of masses from these values of alpha, okay? So that is what this plot means. And now, yes, thank you. With respect to the qualification. Yeah, actually, here. Okay, okay, yes. I introduced it as a gap in strength, but actually it's a parameter of the theory itself. Yes, I can explain that this region over here is excluded by Shogawa feed force. So this is like the monstrengent constraint. We already used it. Yeah, in previous work, we used pulsar terminated to constrain this. So you showed the coupling, yeah? Between the matter sector and your spin tool, but what about the action for the spin tool only? How do you? The action for the spin tool. Yeah. Well, that was another question. The question of motion is this, which is similar to this color. But if you want to get that, you need to start from a theory. And this is where you can get it. It's like a first Pauli mass term, like this one. This tensor, epsilon hj, what? Mu nu have this form, which essentially is second derivatives of the field, like kinetic term, plus the mass term. So the structure is the same as the scalar field or vector field, et cetera. But it's different because we have more in it. Okay. But if you derive it from here, you can get the equation of motion. Okay. Well, thanks. Any questions from you all? Is some parameters space constrained by LIGO and Virgo? Something like this, I don't know if I get the question. LIGO? This is Stanford Livingston and Virgo. So this is LIGO, I believe. The red one or the green? The red one. The red one. Oh, okay. Stanford Livingston. I don't know if it refers to that, but. Okay. And then. Hello. I think. Okay, yeah. Thank you. I'm going to talk about this word. That is a collaboration with Matteo Vaggetti, Jorge Norena and Emiliano Cephusati, in which we study the theoretical covariance for device spectrum, and we are interested in the squeeze configurations. So before to start, I'm going to give an introduction. So one of the main problems in cosmology is have to do with the initial conditions that see the structure formation. So, yeah, sorry. So inflation provides a mechanism to generate the primordial perturbations. Sorry. And for example, single field inflation that predicts that the initial condition where gaussian, adiabatic, and almost scaly variance, so the perturbations are well described by the power spectrum. And this model is in well agreement, I mean it's consistent with the observations of the CMV. But there are also another models that predict that the primordial perturbations are no gaussian, that still are in agreement with observations. So we still don't know what is the model of inflation. So what we want to do is to use the statistics of the large scale structure to constrain primordial no gaussianity. For to do that, we use the bi-spectrum of the galaxy bi-spectrum that is a good observable to detect primordial no gaussianity. So we are interested in constrain primordial no gaussianity of the local type for which the bi-spectrum is sensitive in the limit when one of the modes is much smaller than the other two. So the idea is to improve the constraints from plan collaboration by using the data that is coming from the new, from the upcoming surveys. So in order to get information from the surveys, I mean to constrain the theory with the observations, we want to, we need to compute observables with the same precision of observations. So we need an accurate modeling for the bi-spectrum, but also we need an accurate modeling for the bi-spectrum covariance in order to extract the information from the bi-spectrum. So for correlation functions as the power spectrum and the bi-spectrum that depend on the scales, the covariance is given in terms of estimators of these observables. And in the case of the power spectrum, the covariance is estimated by using a large set of mocks. So many realizations are needed, but for the bi-spectrum, 1,000 of realizations are needed. So it is a computationally expensive. So we need an accurate description of the theoretical covariance. So for that, we need a de-joined power spectrum bi-spectrum covariance. So we need to compute this matrix. And to do that, so we consider that the measurements, that the measurements are made in a box of large, with a size of L. So we divide the box into bins and we compute this bi-spectrum and the power spectrum in Fourier space. H-bin is an spherical shell of centered at K and with radius delta K. So to compute the covariance, we use these estimators where the sum runs over all the modes that are into a bin, a K-bin. And this NK is the number of modes inside the bin. And this NT is the number of triangles, of fundamental triangles inside the bin. So we use these estimators to compute the covariance. And in this case, we are neglecting the connected part of the covariance. So we are only taking into account the disconnected part. Or for the power spectrum, for example, we only consider the terms that correlates modes with the same size. So in the previous slide, can you go to the... Yeah, so one of the sums is, one of the deltas is the Kroniker delta. Yeah, this is the Kroniker delta. We need three deltas for the device. Yeah. Yeah, and so for the cross covariance, we correlate a triangle and one mode that has the same size as one of the sides of this triangle. And for the wide spectrum, we consider the Gaussian part that is even only in terms of the power spectrum that correlates triangles with the same size. And this is the usual term that people use when you have a few number of simulations. But we are also here studying this term that is a no-gaussian term that correlates triangles that shares one of the modes. So usually, this no-gaussian term in the cross covariance and this no-gaussian term in the wide spectrum is neglected. But we are going to study what happened for these terms and the squeeze configurations. So to do that, we compare, we estimate the magnitude of the no-gaussian terms we respect with the gaussians term and for the wide spectrum in this case. So we conclude that when the scales are comparable here in the wide spectrum, the no-gaussian term is irrelevant. We respect with the gaussian term as it is suppressed by the dimensional power spectrum. But when we consider a squeeze configuration, it can be of the same order as the gaussian contribution. When the mode that is shared here is a long wavelength mode. So for that squeeze configuration, the no-gaussian term is relevant. Okay, so then these no-gaussian terms are important for a squeeze configuration. So we compare with key-hot simulations with the gaussian initial conditions and then we compare with the covariance computed from simulations and using the model. So we can see that when we only use the gaussian term, there is a 100% error. While if we consider the full model with the no-gaussian terms, the error is between the 20%. So we can see also here that for a squeeze configuration, if we only consider the gaussian term, there is a one-order error. While for a normal triangle, it is enough to use the gaussian contribution. And we also want to know what happened for the off-diagonal terms. So we compute the correlation matrix and then we can see that there is agreement between simulations and the model. And also it's agreement for when the load mode is chaired. So we can conclude that these no-gaussian terms are important for squeeze configurations. So they are important for parameters that are sensitive to the squeeze configuration as the amplitude of local no-gaussianity. So here we use EO simulations that have no-gaussian initial conditions to compute the uncertainty in measure this parameter. So we can, we realize that, sorry, that there is a degradation of the constraints if we use the whole model with the no-gaussian terms. For that reason, it's important to think about. That's it. There is an important question in the Zoom. Is there any restriction on the number of modes? Yeah, yeah, actually, when we compute, yeah, because of the delta function that says that the triangle must be closed. So into the bins, there are most fundamental triangles that does not satisfy the triangle condition. So we need only to take into account the ones that satisfy the condition that there will be a closed triangle. So this... Just have a five minute break before the next speaker. Many thanks again. You said five minutes, right? Five minutes. All right, we can barely hear you. Alejandro, can you hear us? Yes, I can hear you. Can you speak a bit louder? Yeah, let me get closer to the microphone, maybe. Can you hear me better now? Yes, but could be even better. Okay, let me try to speak louder. Can you... Now it's better, right? Can you hear me now? Yes, yes, yes. Okay, so shall I start? Yes. Okay, so good afternoon. I'm Alejandro Pere Rodriguez, PhD student at Universidad Autónoma de Madrid, and I'm going to talk about the formation of primordial black holes and gravitational waves from the Zipati effects during infection, which is the topic I'm currently working on together with my PhD supervisor, Guillermo Ballesteros, and other collaborators. Before starting, let me give a brief recap on the basics of primordial black holes and gravitational wave formation. If we have a peak in the primordial power spectrum at a certain scale K, then we will have an enhancement in the density of primordial black holes at a certain mass one-to-one related to the scale of the peak and the density of the gravitational waves at a certain frequency again related to the scale of the peak. Further assuming a Gaussian distribution for the fluctuations, a certain value for the critical over density for the collapse and the reentry of perturbations during radiation domination. We know that for primordial black holes of a certain mass to account for the totality of black matter, we need a peak in the power spectrum of order 10 to the minus 2, which according to current constraints, is only possible in this window of masses here. And furthermore, the corresponding gravitational waves to this kind of peak have the right amplitude at the right frequencies to produce a signal potentially detectable by this. Right, so let me now move to the basics of dissipative effects during inflation, which arise when the inflaton field covers through radiation under some technical assumptions coming from thermal field field. At the level of the background, this coupling simply induces an extra friction term in the Friedman equation for the inflaton. However, at the level of the perturbations, things are more complicated. We get extra perturbations due to the radiation terms in the energy momentum tensor. And on top of that, the fluctuation dissipation theorem introduces stochastic transfer terms between a scalar and radiation perturbations, which at the level of the equations of motion translate into stochastic sources. Just for the record, there is a particular case of this dissipative effects, which is the warm inflation framework in which this kind of dynamics has half already been studied and I cite here two recent papers on the issue. So our objective is to compute this quantity here, which is the thermally average glumoria power spectrum for which essentially we need to know this, right? And we have to compute it from these equations here. And in order to do so, we have explored two mutually consistent approaches. The Fokker Planck approach in which we convert a system of stochastic differential equations like this one in the system of ordinary differential equations, not for the perturbations, but for the correlations of the perturbations. And after solving them, we can recast them into this thermal average with which to compute the power spectrum. And the other approach is a few Monte Carlo approach in which we randomize the stochastic source. We compute several times particular realizations of the perturbations and of the commuting curve at the perturbation and iterating and taking the average to estimate this thermal average and again compute the power spectrum. And we have found in our results that both methods are nicely consistent. So with these mathematical tools, we can study specific models and a class of models of particular interest for the production of primordial black holes are those in which dissipative effects are only relevant for a few defaults, which I have marked with this blue shade here, right? In which dissipative friction dominates over habit friction. At the level of the background, the phenomenology is rather intuitive. We have that the kinetic term of the inflaton is suppressed into the extra friction and the radiation energy density is enhanced into the extra dissipation. And at the level of perturbations, we get using either of the methods shown before, this kind of power spectrum, which has every feature we were looking for. It matches CMB amplitude of the right scales and it has a peak of over 10 to the minus two. And furthermore, we can tune the position of the peak by shifting sidewalls is a strongly dissipative region. Just a detail, this yellow or orange region here corresponds to the scales of the modes crossing the horizon during the strongly dissipative region. So we could have stopped here because we have a primordial power spectrum. We call them the features we were looking for, but we wanted to further understand what is the physical origin of this peak. And in order to do so, we perform an analytical approximation of the problem. Now I cannot enter in the technical details of how this approximation is performed, but it suffices to know that it allows to decompose the spectrum into two components, one of which is due to the homogeneous evolution of perturbations, that is the left-hand side of these equations, the way to zero and another contribution, which is a purely thermal contribution that comes from the stochastic noise introduced by thermal effects. If we compute the power spectrum in this analytical approximation, we get a result as this one here. You can see that it does not totally fit the numerical result, but that is to be expected because of the many simplifications that have been made in the middle. The relevant thing is that it catches the main qualitative features. And from here we can use some interesting physics, which is that, so these are the evolution of three particular modes for the perturbations for responding to these scales in the spectrum. And if we focus on this mode here, which crosses the horizon during the strong dissipative region, we see that the source term, that is the term due to the thermal noise, becomes dominant and it is heavily enhanced and it brings the modes to a thermal attractor which largely enhances the value of which they freeze outside the horizon. And this is precisely the physical origin of the peak in the primordial power spectrum. And it has nothing to do with the homogeneous dynamics of the perturbations. And well, in these papers here, they have performed particular cases of the calculations we have done and our results are consistent with theirs in the appropriate unit. So let me just finish with a summary of the main points. We have seen that this impact effects during inflation introduce quite interesting physics at the perturbation level. They introduce a degree of randomness due to stochastic dynamics, which again are due to the thermal noise. And this thermal noise leads to a thermal enhancement of the modes of perturbations. And if we design our model such that only certain modes are enhanced, for instance, having a short, strong dissipative period, we can produce a peak in the primordial power spectrum whose phenomenology is well-known again, the production of primordial black holes who could account for the mortality of black matter and a stochastic gravitational wave pattern potentially detectable by means. And I will stop here. Thank you very much. Thank you. So in the power spectrum, you have the peak around k is equal to one megapathic inverse, right? Well, the peak in the primordial power spectrum in our model, we can fit it freely because it's just a function of where we put the strong dissipative friction. So if I can share a screen again, let me show you. Yeah, so that will determine the mass range of the PBH. Exactly, exactly. There is a one-to-one relation between the scale of the peak. But if you have a peak around this scale, that cell, cell that have any effect on the CMB, temperature, temperature, anisotropy? No, no, no, because at CMB scales, we have taken care of being consistent with current observations by plant. So this is purely... So CMB scales are not here, where I signal them, they are like far... Roughly, okay, okay, right. Okay, so roughly what I want to understand is that, what is the allowed mass range for your model, for the PBHs? Yeah, so this one here. It's a folder 10 to the 20 grams more. So they are like very, very light black holes. Okay. In primordial, in solar masses, it's 10 to the 15, 10 to the minus 15, so. Okay, so that will be quite a large K, I think. Not, okay, fine, understood. Are the scalar perturbations at the peak Gaussian? The sourced ones? The sourced one, okay, so. Here, right? The sourced one, you mean? We have not really studied the non-gaussianity of these models, but okay, so the thermal fluctuations we have considered are classical, so we have not really entered. So we have considered that the inflaton has, it's a standard quantum fluctuations with a bunch that is vacuum and everything and a similar Gaussian. And then we have introduced a classical thermal noise which we have not quantized in this calculation so far. Yeah, because I would imagine, perhaps it remains classical, but this is quite important as you probably know for the amount of primordial black holes. So the statistics of the perturbation is rather crucial for the total amount of PbH. Yes, what we have seen in one plot, I have not included in this presentation, is that when you compute, this is in the Monte Carlo approach, you can get like an empirical distribution of the perturbations for each value of the, of the commonness K. And in that case, you see that the distribution of the fluctuations is clearly non-Gaussian. But again, that distribution of fluctuation is taking into account both the quantum fluctuations of the inflaton and the, for the moment classical thermal fluctuations. So we have not really disentangled where the non-Gaussianic is coming from. So I have a question. In this plot, did I understand correctly that you are solving your stochastic equations in two different ways? Yes, to obtain this power spectrum here, we are using two numerical approaches, which is the Fokker-Planck approach and the Monte Carlo approach. So you are showing only one of them or what? No, because the result you obtain with both is the same, perfectly the same. And then there is this third method here, which is an analytical approximation, which you can see does not render the same result quantitatively, because it's an approximation. But qualitatively, it catches the main features and it allows to make this physical interpretation of what is the precise origin of the peak. Okay, thank you. Okay, so next speaker. Thank you. Thank you very much. Okay, do you hear me? Yeah. Yes? Okay. Okay, now it's better, I guess. Okay, well, hello everyone. Well, first, let me thank the organizers for this space for my work. Today, well, I'm Raquel Galasogarcia. I'm working at the IPHT at Paris-Saclet. And today I'm going to talk about self-similar solutions for a fuzzy dark matter. Okay, well, we have read evidence that dark matter exists, of course, mainly of its gravitational effects. And with all of these evidences, cosmologists said that the model called, land that called dark matter, to describe the universe. However, even if this model success, got a lot of success as a larger scales, it has some tensions at the smaller scales. And then the preferred scenario since the 80s, which was at WIMS, has still not been detected. So this has revived interesting alternative scenarios, including the possibility that dark matter could be associated with a scalar field. And the point of this scalar field dark matter models is that are able to form equilibrium configurations, say called solitons, that managed to solve some of the CDM tensions. Well, in terms of the solitons, here I mentioned, I plotted this plot just to illustrate the Kyrgyz problem, which is that in numerical simulations of, I call, dark matter, we predict that the dark matter, the density distribution inside the galaxies so Gospiguri's observation said that indeed it's so flat. So in the sense, you can think about this kind of solitons like with physics by hands, like kind of a interstatic equilibrium between gravity and another kind of pressure, which is the quantum pressure, if only we consider a scalar field dark matter model without any other interactions, which is with the model I'm going to talk about it today, which is fuzzy dark matter. So also in fuzzy dark matter, we consider that the mass of this scalar field is around 10 to the minus 22 electron volts. So a tiny mass, so I don't know. Ah, okay, here, like here. Okay, so, well, I'm going to focus here since I don't have too much time in that we can describe the fuzzy dark matter model as a fluid. Well, we start with an action, okay, with a field minimally optic with gravity and also we have the standard kinetic term that we need to, that this scalar field behaves like dark matter. So then the questions you see in one, well, the model transformation, we have the continuity, error and Poisson equation that indeed are so useful because in cold dark matter, we use these fluid equations to describe the dynamics of dark matter. But here I want to highlight that we have an extra term, which is the quantum pressure that is given by this expression that encodes this epsilon that in some how it mimics the H bar, which means that when we are near the semi-classical limit, we recover the cold dark matter distribution radius. If wave-like effects are important, we will have a different phenomenology. Okay, so here is just to illustrate which are the difference between fuzzy dark matter and cold dark matter. Well, here is in a large scale, so as you can see with fuzzy dark matter, we recover the success of the large scale distribution. Of course, the two panels are different because here we have quantum pressure and what happens is that we have a smooth distribution, whereas in cold dark matter, since there is nothing that can balance gravity, we have this more clumpy behavior. So now, the motivations of self-similar solutions. The idea is first, go beyond these static equilibrium configurations, which was the solitons, by investigating dynamical sub-similar solutions. Then the idea is to understand physical processes as a gravitational cooling. So here in this, well, here over there, it's a slice of one numerical simulation that I have done in the term of the phase distribution, well, excuse me, phase distribution, time round like this. And what we can see here is that if we started with a scalar cloud, which is a bit perturbed from the equilibrium, we can see that there is here with this kind of ejection of a scalar matter. And this is named as the gravitational cooling. So in other words, it's like a kind of mechanism that the system has to go to reach the equilibrium. So, well, in order to have, yes, this with analytical expressions, this is the motivation of this world. And of course, another motivation is to understand how are these sub-similar solutions infuse the matter compared to the sub-similar solutions that are in a cold-dark matter. Okay, so now I will go first now. So we will work in the fluid picture. So the objective is we use the sub-similar ansat, which is this one. And then we plug that into our equations of motion. And we describe our fields, well, we are going to study perturbations around the spanding background. We use the decision universe because it's convenient, because it's, well, the skin factor is Apollo. And also it describes, well, the matter here. So this is the scaling variable that we found that as you can see there, it has different behaviors from the link with the commubina and the physical coordinates. So I'm going to focus only in the non-linear regime of, I mean, the result for an overgenicity, which is 100 from the background, et cetera. Okay, well, just let me, so you hear that here in this, if we adopt a Lagrangian point of view, which is just to follow a mass shell, we can see that a mass shell inside the perturbation first start in this central peak. And then we have these kind of three well-distinguished velocity steps that we can see here in this field plot. And then it goes to the Hubble flow. So, and I let this slide of the comparison between cold-dark matter, sub-similar solutions and the Fusid-Dark matter wants to conclude my talk. Thank you. So on the slide that you've ended, so can you summarize in a sentence what the comparison between CDM and FDM is? Well, the point is that in Fusid-Dark matter, sub-similar solutions describes this gravitation and cooling effect, which is that we have the matter while it's inside the density perturbation and then it goes, well, it's a blow-up instead of the gravitational collapse that describes a sub-similar solution. Another point is that in cold-dark matter, we have a transition between the linear regime and the nonlinear regime. So we start with a small perturbation and because of the scaling, it can read the nonlinear regime. Whereas in Fusid-Dark matter, if we start in the linear regime, we cannot read the nonlinear regime. So in that sense, we cannot describe the structure formation with this kind of sub-similar solutions in Fusid-Dark matter because of this. So I would say it is to... So it's not as simple as saying the caustics in CDM are just washed out in FDM. So in CDM, there are caustics in the sub-similar solution where the density spikes. So it's not as simple as saying that only at the locations of the caustics I have some kind of smoothing because of the quantum nature and then in other places, things are more... It is not like that you're saying. No, it's not like that. I mean, here, I am so in you in this plot, how are the trajectory of a sub-similar solution in cold-dark matter compared to the trajectory in Fusid-Dark matter? Can I ask one more or? Maybe we can discuss later. He's the... Sorry, there is a question about magneto-hydrodynamics, but I'm not sure if you can... Or if you can't... Another question, can't generate cores at the smallest scale. I suppose you said that... Sorry? Can we generate cores at the smallest scale in this scenario? With these self-similar solutions? Well, the question is... Well, yeah, I mean, in Fusid-Dark matter, the idea is that we come from solitons and demand a core, a flat density profile at the center. So the answer is yes, that within this model, yes, we have a flat density at the center. So, yeah, these are cosmological self-similar solutions. So also we study if these cosmological self-similar solutions, it's a kind of... The soliton is kind of asymptotic of this, and the answer is no, I mean, we cannot describe this with... These similar solutions cannot describe the soliton. That doesn't convert. Thank you. From Sina Hoshangi, I don't know. Good afternoon. My name is Sina. I'm a PhD student at IPM Tehran, and I'm going to talk about this work. Okay. We know that large ray fluctuations that can be generated during inflation, and when reinterring the horizon after inflation during the radiation-dominated era, if the amplitude of fluctuation is greater than threshold, the mass inside the horizon will collapse to form a primordial black hole. Since in their nature, these fluctuations are large, we need non-perfective metals to explore the tail of the distribution of the fluctuations. In this talk, I will focus on probability distribution function for curvature perturbations, zeta. To calculate correctly the mass fraction or other related parameters for primordial black hole, you need the PDF for compaction function or density perturbation, as nicely discussed in the following references. Okay. Some non-perfective formalism were introduced to explore the tail non-perturbatively, such as stochastic inflation, where in this paper they showed... Sorry, the pointer doesn't work. Okay. Where in this paper they showed... The stochastic effects leads to exponential tail for the PDF of the curvature perturbation. Or in this paper, the first bullet point, with some quantum interaction and writing down the wave function of the universe, non-perturbatively, we have this behavior for the tail of the distribution. In our model, we use the classical delta formalism. Okay. Now we have the tool to explore the tail non-perturbatively. The natural question come up that can we have tails that decays more slowly than exponential? That's why I'm going to answer this question. Mathematicians call this distribution heavy-tailed distribution. There is a formal mathematical definition which intuitively says that this tail function, the capital F, decays more slowly than the exponential function. Since this... Checking this definition may have challenges for our numerical calculations. We provide a practical definition. We calculate this function d. And we said... We say a raw PDF is practically heavy-tailed if this d goes to zero for large values of zeta. Here, in this slide, I provide some examples of heavy-tailed distribution. The first one is log-normal distribution. This is levide distribution. And the distribution with this behavior at their tails like exponential of x to the p with the exponent p between zero and one or a power-level behavior according to both definitions are heavy-tailed PDFs. Okay. We assume... In our model, we assume that inflato undergoes at some period during inflation, undergoes this potential. We use the delta-informalism. We solve numerically without any approximation the Klein-Gordon equation governing the inflato with this potential and derive the number of defaults that takes the inflaton field from the initial perturbed field values to end of this phase with n as a function of delta-5. According to delta-informalism, we can calculate zeta as a function of delta-5 by this equation. Some people usually expand the delta-in relation up to linear order and second order which leads to Gaussian and exponential PDFs, exponential tail for their PDFs. To calculate the PDF, we use the conservation of probability relation I mentioned here. But here we keep the delta-in relation nonlinearly and we see that the behavior of the tail is power-love. Here is the result for some parameters we choose. The green plot is the Gaussian PDF derived from linear approximation. The red one is exponential derived from second-order approximation. And the blue one is fully nonlinear PDF calculated with these parameters. You see how the nonlinear changes the behavior of the tail. Here we calculate the function d versus zeta and we see the behavior of the tail goes like power-love for large values of zeta and here we have the parameter beta to estimate a mass fraction of zeta and you see orders of magnitude difference between the nonlinear beta calculated using the derivative treatment. Thank you. Can you go two slides back to the d? So I thought the d should be nonzero for heavy tail distribution, is that right? No, d should go to zero for heavy tail distribution. What motivates the particular potential that you took to do these calculations? The potential. We analytically show that we need a power-love tail. We require a power-love tail and we show analytically that this potential will give the power-love tail but we do it in the inverse direction and solve the client order equation and drive the power-love tail. I think you didn't get something simple. So it's a two-fill model or is the input only? No, one-fill. Five bar is the parameter. So it's a single-fill model. Then I have to ask you. I think they have a period of pastoral, what the ultra is the... Yes, this phase is the parameter chosen to be this phase to be non-attractive phase, yes. Is N of phi also polynomial or more complicated? No, this is a complicated function. Thank you. You can proceed with the last but not least presentation. Fly Rajul Akhet. So here you have your pointer. It takes a bit. Hi, everyone. Am I audible? Yeah. Yes, okay. So thank you to the organizers to give me the opportunity to talk here. So here is the topic that might offer, that the gravitational duty, which is based on this archive number, and this work is in collaboration with my supervisor, Dr. Dev Prasad Menthi. So first of all, I mainly point out two things, that how the present state or universe has been created. And in order to do that, we basically emphasize that why the reheating phase is important. And secondly, I'll talk about if there is any possibility to reheat the universe through only gravitational introduction. So what is the difficulty in proving the early phase? Because our knowledge about the cosmic history of the universe is based on the two major observations. One is the cosmic microwave background, and the other one is the BBN. The CMB basically predicts that there should be an early inflationary phase, and the energy scale is up to 10 to the power of 15 to 10 to the power of 16 Cb. And in the Big Bang ecosystem, this predicts the quantity such as the light element, and the energy scale is up to 1 AB. So there is a massive gap in terms of energy and the time scale between the inflation and the BBN, which is basically poor understood from both theoretical and the observational point of view. And here the reheating dynamics basically lies. And why the reheating phase is important, because after the end of the inflation, the universe is cold, dark, and dominated by the homogeneous inflaton field. And so we basically need to get a thermalized radiation-dominant state for nuclear synthesis. That's why we basically convert the inflaton energy density into the daughter fields to get a radiation-dominated part, and here the reheating dynamics basically lies. So this is the schematic diagram of this evolution of this co-moving Hubble radius, and this is the reheating phase interpreted between the inflation and the radiation-domination. So we need to understand how this modified expansion history inflates the prediction for the cosmological observables, like the CMB observables, the NS and R. Now it is important for doing this reheating dynamics, the inflationary parameters, which basically sets the initial condition for the reheating. So these are the overall parameters, the epsilon and eta, which is basically the first derivative and second derivative of the potential. And another important thing is the parameters, the unfolding number during this inflationary era, and the inflation energy scale. And these can be connected through the R and AS, or the tensile to scalar issue, and AS is the amplitude of the scalar part emission. And NS and R can be related with these overall parameters in this fashion. And the end of the inflation is set by epsilon equals to 1, and which gives the initial condition to the reheating dynamics. So go to the standard reheating phenomenology. In the standard reheating phenomenology, either we're doing the parametric resonance or the resonance production during basically the preheating era, or we can do the perturbative decay. We can consider any coupling between non-gapitational coupling between the inflaton and the dotted field in this way, and we can reheat the inhibitors. But here, in this work, we basically emphasize the fact that how only the gravitational interaction is sufficient to produce these to reheat the universe. And this is the gravitational interaction, the one of our MPH mu nu D mu nu term. And the initially thought as it is a plant mass suppression, so it is sort that it always thought to be the gravitational, this process would not enough to reheat our universe. But in our case, we see if we consider the higher question of state, if the inflaton equation of state is greater than wanted, then somehow we're able to achieve the radiation-dominated universe only considering the gravitational interaction. So this is our setup, the framework. We basically considered the S-channel scattering between the S-channel scattering and scattering production of the standard model or the dark meter from the inflaton through the exchange of gravitons. And this is the standard use of Boltzmann's equation describing the different energy components. This is the inflaton energy density, this is radiation, and this is the dark matter number density. So here, in our framework, there are only three parameters. One is the inflation energy scale. Another one, the inflaton equation of state, which is connected with the exponent of the potential. Because if you consider the 5 to the power 2n kind of potential, then using the Virial Theorem, if you average out those oscillations, then it's easily connected through this omega phi is n minus 1 by n plus 1 for 5 to the power 2n kind of potential. And then another parameter is the mass of the dark meter. And the total radiation decay width is dependent on the gamma s, gamma f, gamma i. This is basically the scattering rate. Scattering rate from phi phi to scalar, fermion and for gauge motions. Okay. So now to the next slide. So what are the initial condition and the constant? First of all, the initial condition of the inflator energy density is basically set by the inflation energy scale, which is the end of the inflation hn, basically set the rho phi in. And the rho r and rho dm, the radiation energy density and the dark matter energy at the initial time z. Then secondly, we consider the constant from the entropy conservation. If we consider the standard consideration at the end of the reheating, all the entropy are produced. So all the entropy are generated. So we can consider the co-moving entropy density is conserved from the end of the reheating to the present time. Then we can relate this reheating temperature to the present temperature in this way. And for a particular CMD scale, p-fold scale, 0.05 mega-persec inverse, reheating temperature with the present temperature 3 poc by poc. And another constant is coming from the present dark matter amines, which is 0.12. And there is the limit from the minimum value of the reheating temperature that's much greater than the BBN temperature. And the upper limit is set by the r and as bound, which is basically the 5 into 10 to the power 13 GB of the rho. So go to our results. So first we consider, without specifying any model, we consider only and this is the two important parameter to find out that they describe the reheating time range. One is the duration, this is the folding number and the reheating temperature, which is find out at the end of this reheating phase. And this expression for this co-moving number density of dark matter at the end of the reheating. And how this gravitational model basically predicts this dark matter sector that mass will be within 10 to the power 5 to 10 to the power 8 GB for thermally dark matter. And for post-heating dark matter of the order of 1,000 electron volt. And similarly for the inflatron sector, the inflatron equation must be lies between 0.6 to 0.9. So we have to consider the stiff equation of state and the energy scale is of the order of 10 to the power 9 to 10 to the power 13 GB. And so an infallibility number must be lies between 60 to the 63 60 to the power 62 to the power 6 to the power 3. So the two numbers are for the two choices of the equation of the state. No, no, no. Equation of can vary from 0 to 1. But we see if the equation of state from 0.6 to 0. 0.6 to 1, then it successfully reheat the universe. Otherwise the reheating is always the sub, reheating and gravitational production always subdominant for the equation of state less than 0.6. Okay. And this is the prediction from the primordial gravitational, if we consider the primordial gravitational wave prediction and if we consider gravitational produced during this inflation and if we evolve the tensor perturbation during the different eras and calculate at the present time, then you can see for those modes which entering during this reheating phase basically from 1 to 10 to the power 11 this frequency range, this motor entering during the phase, for equation of state greater than one third, there is a tilt in the spectrum. And this is the interest of the spectrum. And our gravitational in the model only satisfied if this is nearly equal to cohesion. Otherwise it violated the BBN constraint just one minute. Okay. Then then you constrain the specific model. Here you constrain the alpha actor model and in the alpha actor model you change the alpha parameter and this is the result for different alpha value. Then we see the how the alpha actor model lies in the gravitational heating scenario in the NSR plane. And you see only in the one sigma range only alpha is equal to 10 marginally satisfied result. So for alpha is greater than 10, it is disfavored from the Planck observation. And here this is the last slide. Here we compare the gravitational heating scenario with the case where the explicit coupling is present. And if you consider different explicit coupling between the inflate on to the dotter field sector and you see that for a sufficient range of the coupling parameter, this gravitational heating model actually dominates. But if we consider the sufficient high coupling parameter then it take over this region. So these are the main points. So main points are that we model an independent approach and we get a precise cosmological prediction and here we switch up all the unknown parameters, all the coupling between inflate on sector to other sector, only consider the gravitational introduction. And here this gravitational heating predicts the very narrow range of dark matter must low temperature, low heating temperature and the gravitational spectrum which indicates the stiff reheating equation and this scenario basically discards the large number of impression in models also. Look to me that you are looking 2 into 2 scattering. I do 2 scattering process and exchange of gravitational. Can you just go back to the start? Here right, when you look at the process, you don't have a process where a single inflaton decays? No, no, no. But then you cannot get rid of all the inflatons, right? No, no, here if we consider the equation of state higher than 0.6 then it manages to decay. Because the equation of state is inflaton equation of state is higher than the reduction equation of state. So some amount produced during the initial time of reheating but it dilutes first so we at some point we get that addition. Okay, I see, I see. Thank you. There is a question in Zoom which is one of the initial conditions that you consider was rho dark matter equals 0. So while rho present is a finite number, does this mean that reheating process synthesized dark matter? Actually dark matter produced from the scattering of the inflaton. So initially the dark matter is no dark matter. Dark matter is 0 and finally the present we have to produce the sufficient amount of dark matter to satisfy the present okay. See you tomorrow.


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What's going on, everyone? Sam here with Layton Sports Cards, ripping 23 tops, definitive baseball hobby. Three box, case number six, pick your team style. We offer lists for names in our teams. Alex C. Cubs, Brent G. Braves, Brent C. with the twins. Brian H. Angels, Brian C. Cardinals, KCS Reds, Charles F. Brewers, Charlie C. Athletics, Daniel S. Pirates. Evan L. Tigers, Gregory J. with the Phillies. Howard Z. Astros, Blue Jays Padres. James J. Mariners, John K. White Sox, Joseph S. Guardians, Joshua B. Giants. Kevin Y. Metz, Christian M. Yankees, Matthew K. with the Dodgers. We've got Nick P. with the Rangers, Paul B. Royals, Peyton H. with the Marlins, Peter M. Orioles, Red Sox, D-Backs, Sam S. Nationals, Theodore H. Rockies, and Wayne P. with the Rays. So good luck, everybody. Let's do it. Because for us with a 10 box spot, thank you, buddy, and a hoop spot tomorrow. I'll hit the court. There's a lot more chill than everybody else's. What's up? Our court's a lot more chill than most. Yeah, I know. Is this playing? Is this the playing game, I'm assuming? Yeah. Two and one in the play-in right now. Damn right, Mark. Yeah, nobody else would buy in, because nobody else would have a chance to buy in, so that's why I would just grab those spots so quickly. Scoop it. Scoop it right up. Yep. All right. Ready for cotton? Yeah. I blessed it, so. True. Very true. He's soggy. All right, it's starting off with a Shohei. Nice one there to 40. 20 of 40. We've got 12 of 30. Matt Olson game used bat for the Braves. That's one there, Bravos. Just the definitive thing, filler thing. Starting off, nice one here. Austin Riley, 15 of 50. Jersey Auto for the Braves. Congrats, Bravos. That is Green. 16 of 25. The Over Piquero. Going to the Pirates. Congrats, Pittsburgh. Patrick Rye with Certified Spot and Bolt Thing, buddy. We've got a nice one here, is a Bo Jackson. 19 of 40. Game used bat auto for the Royals. Congrats, Casey. Cuero is everywhere. He is in this. He is. Really nice one here. Defining Images Auto, going to the Mariners. Jay Rod, congrats on that one. There you go, Seattle. That's awesome. That's really nice, dude. We'll get this one into a mag. Congrats on that. That's pretty nifty card as well. Four of 10. Four colored patch auto is going to be Tom Glavin. Nice one for the Mets. Congrats, Mets. This one's nice, dude. I like this one. Mets, Glavin. And then a little duel to end it. Two of five. Barry Larkin and Joey Vado. Congrats on that one. They're Sinsey. Two of five. Larkin and Voto. Great Larkin sig is always and Voto as well. Two of five. Hum, that's Glavin. We'll say I knew Candler wouldn't be happy about that. New Candler would not be happy about that. We got one of the Jumbo ones coming in. Either the helmet or the protectors of the plate. Let's go see what it will be. Starting off with a George Kirby. That is going to be two of 15. Two colored patch for the Mariners. The third one of those. Yeah, that's so weird, dude. Going to the Mars. Eight of 50 is Sandi Alcantra. Nice one there. Congrats on that slap. We'll take it. That is a batting helmet. Game used helmet of Sandi Alcantra. It's known batter, Sandi Alcantra. They must have won. Confirmed bomb hitter, Sandi Alcantra. Oh, that's nasty. RPA. Going to the Cardinals. Jordan Walker. Six of 50. Congrats on that. J-Walk, baby. J-Walk. RPA. Nice one there, Cardinals. Very nice. Before we continue on, 21 to go in flux, everybody. 21 in flux. It is, PB. It is, dude. 21 to go in flux, everyone. 21 to go breaking soon. Magnus Walker. Yeah, I would. All right. Rookie Otto. Miguel Vargas. That is going to be going to the Dodgers for 46 of 50. Miguel Vargas. Congrats on that, Dodgers. We've got Green Gersiotto of Pedro Martinez. Eight of 10 for the Red Sox. Red Sox Pedro. Never a bad thing at all. Red Sox Pedro. We've got the Big Hurt, Frank Thomas. Congrats on that one there. White Sox. I haven't seen him in this. Him? Yeah. That's cool. Yeah, this one's awesome. Going to the Yankees. 18 of 30. Framed Godzilla. Congrats on that one there. Yankees. Did you random that? Yeah, I think so. Dual Autograph Relic Collection. Vladdie Jr. and Ronald Acunha. Congrats on that one there. Going to do a random for this one. I'll throw a little break credit. Whoever doesn't win. Between the Blue Jays and the Braves. Nice one there. Dual Autograph Relic Collection. It's not like a parallel. So it's going to be one of the higher numbered ones. I guess it would be around 50 or so. Between the one that's not numbered or the one that's going to be like 50. We'll random that. Another thick one. That was a good box overall. Last box here. A good old defense. These are going to be few and far between. Max Scherzer, 38 of 40. As a Met. Met's Scherzer. Magic up 12-7. Let's go baby. First half team. We've got protectors at the plate to 25. That is going to be at Teosca Hernandez. Nice one there. So that'll be a catcher relic. Protectors at the plate. Teosca. Very nice, very nice. 15-7. Let's go Suggs. 4-25. Alex Bregman. Nice one there for the Astros on the Patch Auto. Alex Bregman. Yeah, they're the catcher relics. There's the protectors at the plate. Which are pretty sweet actually. Teosca a catcher? I guess. I don't think he is, right? No, I don't think so. I think he's an outfielder, right? Got Garrett Mitchell. 41-50. Rookie Auto for the for the brew crew. Yeah, I don't know what's happening with that one. That's weird. I was going to say I don't think he's not a catcher. No. Was he ever a catcher? No, he's an outfielder. 340, Andy Pettit. Game used Jersey Auto. Yeah, it could be an outfielder. That's fair. No, I think we'll smiths on it, right? Be right here, Joshua. Will Clark. 3 of 25. On-Card Auto for the Giants. Who's that? Bless it for Joshua. Yes, sir. There you go. Will Clark. Try my best, Joshua. This one's nasty. The big, big-time Card 2 of 10 for the Orioles. Adley-Ruchman. Congrats on that one there, Baltimore. Two huge cards to end it here, I have to say, with this one, too. Yeah, yeah. That's nasty. I may have thought you saw the pod when I blessed, to be honest. Yeah, fair. Fair. Nasty Adley right there. Congrats, Baltimore. Orioles was Peter M. Good stuff, Peter. And then 6 of 10. Fernando Tautis Jr. And one Soto. Going to the Padres. Dual Relic Auto. Purple. Three color for Tautis. And, well, I guess one color technically for Soto. With the Camouflage jerseys. Yeah, that was awesome, dude. That's an awesome card. Very, very nice duel as well. Tautis and Soto. Before I do the random, Steven W. With 3 in certified. I think we're almost sold out on certified, basically. So, here's your link again. 21 to go on Flux. 21 to go on Flux. Go on Flux. Let's fill that up. 30th spot. I'm going to get the random set up for the Acunia and Vlad duel. Braves. Braves. Blue Jays. And blue Jays. All right. Again, going to do the random for the Vladimir Garot Jr. And Ronald Acunia. Dual Relic Auto. I'm showing it and it's not there. Future tide is right there. Let's go Spencer. I'll throw a $20 break credit. Whoever doesn't win this one between the Blue Jays or the Braves. So we've got going to roll the dice whatever seems on top. After six times, we'll get it. One, two, three, four, five, and six. Going to the Braves. Congrats, Atlanta. Picking that one up. Blue Jays. You will be getting $20 break credit. Okay. We've got our... Make sure I'm all the way on that. Yes, six and six. Max Scherzer. George Kirby. Matt Olson. And Shohei. As well as the two thick Jumbo Relics. Tasker Hernandez and Sandiel Contra. Yeah, we got protectors of the plate of a noncatcher and then a helmet relic of a picture. Here you go. Geez, man. No, they didn't do it this year. Will Clark. Andy Pettit. Garrett Mitchell. Alex Bregman. Frank Thomas. Hideki Matsui. That is Pedro. Miguel Vargas. Tom Glavin. That is Barry Larkin and Joey Vado. Duelado. Bo Jackson. Lee Overbeguero. And Austin Riley. And then Maggards. We had the J-Rod Defining Images Auto. We had the RPA of Jordan Walker to 50. We had Adley to 10, two of 10. On the Rookie Auto. And then we had the Juan Soto and Fernando Tatis Jr. So that will do it for the break. Everybody thanks again for the filp. Up next is 10 Box Tuesday.


Welcome back, in the last segment we saw implementations of two assignment operators for our class string. In this segment we are going to look at implementation of other operators. So first the square bracket or the indexing operator. So if we wish to access the individual characters of the stored character string we need to define this. So as we know if what has been written in the program is A1 this is read by C++ as A dot operator square bracket, operator square bracket operating on one. So this operator must be overloaded and that overloading is quite simple. So the receiver has a certain string and we want character 1 from it and where is the string of the receiver it is at A dot PTR. So we just have to get character 1 from A dot PTR that is about it. So we do not have to write A in this because the receiver is implicit. So this is going to be the receiver's PTR and we are going to take a displacement of I from it. So that is about it and notice that we are returning a reference. So we are returning a reference because that way we can change the character as well. So we would be able to write something like suppose we have A and A is PQR then we can write A of 0 equal to A of 1. So this A of 0 will also be translated or will also be interpreted by this operator. So this will return a reference to A0, so A0 is a variable and it will be a reference so it will mean that variable itself, it will not mean the value and so that variable itself will appear here so to say and therefore we can store into it. So that is why we are returning a reference rather than a value. So if we do this then if we have this defined and if we write this and we are allowed to write this then this will cause A to become QQR. Now there is an interesting operator concatenation for which we are going to overload our plus. So we would like A plus B to mean the concatenation of A and B. So we must create a new string first of all on the heap and store the concatenation into it. So how does that work? So we are going to return a string this time, so the result is going to be a string and our operator plus is going to be overloaded. So again let me just remind you A plus B becomes A dot operator plus of B, so therefore we need to redefine or we need to overload this function operator plus. So first we will create this string which is going to eventually contain the result and into that string we are going to store the result but this string has its own pointer and that pointer must point to the heap and how many elements, how long an array does it need on the heap? Well this array must contain the string from A as well as the string from B or looking at it here it should contain the string from the receiver as well as the string from the right hand side. So this space needed must be the length of the string pointed to by PTR as well as the length of the string pointed to by RHS dot PTR and a one so that we can append the null character to it. So we have allocated the space needed by writing this new thing and now we just have to copy. So this S copy that we had earlier is going to copy whatever is in the receiver into this result, the result area that we have created. This is created on the heap of course and then we have to copy the RHS string as well. So that same S copy is going to be used but we have to tell S copy that look I do not want it copied from the very beginning as what this would be doing but we want to copy it from index length of PTR. So we want to copy from RHS dot PTR index length of PTR, length of the string that we just copied. We are going to call S copy but this time we are going to give three arguments. So this is the destination, this is the source what we want copied and this is the displacement in the destination from where the copying has to begin and you can write this but this function is also given in the book in that same chapter 15 and finally res will be returned so the result of this will effectively be replacing this expression. So there will be a temporary which will be replacing this and then you can do whatever you want with it. It has c equal to a plus b so that then that temporary will be assigned to c but this evaluation will result in a temporary which contains the concatenation. So that is it, so that is the concatenation operator. Now we need the destructor as well and we talked about it but let us just do this again. So suppose we have this code which we had in the program that we had written. So at this point string b is created and at this point string b goes out of scope. So when string b goes out of scope c++ is automatically going to call the destructor tilde string. So tilde, well I should really say the destructor in the string class. So that will get called and it will be called on the receiver. So it will be tilde b actually the call will look like tilde b. So what do we want happening over here? So by the way there is a default destructor and the default destructor does nothing and we are going to change it. So we should delete p.ptr to prevent memory leaks. So how do we do this? So our class is going to contain this, our destructor is going to have delete p.tr inside it that is about it. So this is what will happen. If b goes out of the scope over here then it is as good as calling tilde b and this is the call that is going to happen. So this is the body of the call and b is going to be the receiver and so p.tr the p.tr of b is going to be deleted. So whatever memory b was pointing to will get deleted exactly as we wanted. And as I said earlier this works even if p.tr is null and in this case the delete does nothing. We also need to write the copy constructor. So remember that in our program we were calling some function f with a string object a as the argument. So how does c++ handle it? Well the copy constructor is called by c++ to copy a string object to the parameter. So this string object is copied to the parameter in f and that copy is a little bit of a special copy. It is not just like an ordinary assignment. It is a little bit more special than that and we will see in what way it is special and therefore c++ has a notion of a copy constructor. So the copy constructor looks like this. So it is a constructor and therefore there is no return type and it is constructing. So here we are going to use it to construct that parameter in f and so it is going to take one argument which is the object from which we are supposed to make the copy. So in this case argument is going to be a and rhs is really going to be a when this code gets executed and the receiver will be the variable which is the corresponding parameter in f. So what happens? Well ptr which is the receiver's ptr is going to be assigned, is going to be made to point to the string that rhs contains. So first it must allocate, we must allocate some memory and that is what we are doing over here and its length should be adequate for getting a copy of this. So rhs.ptr length plus 1 for the sentinel null. After that we are going to make a copy, the actual copy and that is about it. Notice that we did not delete ptr as we deleted in the case of the other assignment operations because this ptr is the ptr of the parameter of f which is of type string. So that is why we are copying this argument to that parameter. But this parameter is just being constructed as we speak and therefore it is not pointing to anything at all. So we do not have to delete ptr. And likewise there is nothing to be returned because this is a constructor and so in the assignment we were returning something. So the code looked a little bit more complex but here there is no chaining that we could be, that might be needed and therefore it is just going to be copying from this a to the parameter inside f and just this is necessary. So this is the core code of the assignment statement but before that we had to delete ptr in the assignment statement and then we had to return this object itself. But both of these things are not necessary as far as this constructor is concerned. And yes, so this is the code which is used to copy an argument to a parameter. But C++ uses the same code to copy a result back from the called function to the calling function and that same copy constructor is used and this code will work for that purpose as well. So what have we discussed in this segment? So we have given the definitions of all member functions needed to perform assignment, passing and returning from functions, concatenation, etc for our string objects. And these pieces of code should be inserted into the definition of the string. So next we will use this class to solve the problem of storing names and we will have concluding remarks for our entire lecture. But before that we will take a short break.