Helical magnetic fields in the early universe

so welcome to the 35th theoretical physics colloquium by professor axel brandenburg from nagita in stockholm he got his phd from the university of helsinki in 1990 then he had postdoctoral training for two years at nardina in copenhagen and then for two years at high altitude observatory at national center for atmospheric research in boulder he rece he got a faculty position nordic assistant professor at novita in 1994. he moved to to uh university of newcastle upon tyne in 1996 as a professor of applied mathematics and he returned to naradita in 2000 and was the professor there since then he is also an affiliate member at kth royal institute of technology and stockholm university uh between 2010 and 2015 he was a deputy director of nargita and between 2015 and 2018 a visiting professor at the university of colorado boulder in 2014 he was elected as a foreign member of the royal swedish academy of sciences in 2019 he became an honorary professor at ilia state university in belize georgia he served he serves i should say since 2010 as a member of the editorial board of astronomers and african uh his research interests are very wide generically astrophysical field dynamics astrobiology solar cell activity all kinds of applications of hydrodynamics he is one of the original authors of the famous pencil code and a member of the pencil called collaboration and today he will be talking about helical magnetic fields in the early universe and with that i'll give the microphone to oxo thanks very much eager for a very kind introduction and i'm happy to speak to you here today and it's a great opportunity to connect borders through zoom during corona times great so what i will be talking today is about is helical magnetic fields in the early universe magnetic helicity is an important quantity so let me first explain what it is magnetic helicity has also been quite a challenge to people in dynamo theory one of my basic strong interests before it came to understanding and studying the early universe was dynamo theory in various astrophysical settings and there was really for since the 1990s is myth about the about what is known as catastrophic quenching of the dynamo effect and so that again is related to magnetic helicity and telling you about this or showing you this really it gives you an idea of how strong a conserved quantity magnetic helicity really is in practice magnetic helicity also plays an important role in decaying turbulence and that obviously is relevant to the early universe because early universe mhd magnetic fields would only be decaying in the time after its generation one of the mechanisms that has been discussed for generating the magnetic field is the so-called chiral magnetic effects so i will be focusing on that one today as one of the mechanisms that could explain early universe magnetic fields and then i will turn to gravitational waves that could also be generated from the resulting reynolds and maxwell stresses from that turbulence so i will tell you a little bit about such signatures which also include signatures about helicity so the word helicity comes from the greek word elix and it really refers to curl-like structures like these beautiful ones these beautiful plants and also similarly for various types of life in on that in the microscopic world there's of course also a helicity at the micro physic microbiological world related to the question of homo chorality a very strong interest of myself also but i will not have time today to talk to you about that so the term helicity in connection with fluid dynamics and its topological underpinnings really go back to the work of keith moffat of cambridge of the year 1969 he was the first to realize that this integral u which is the velocity dotted into the vorticity the curl of you is a quantity that can be related to the linkage of vortex tubes vortex filaments and this is really the first paper that really put that may put that very clearly in writing and so one of the things you realize is that if you have a structure such as vortex structures and the same applies to magnetic fields if but if you have a vortex loop which uh because has to do with a vorticity which is aligned with a structure and then forms a tube-like structure once these structures are interlinked and therefore crossing here in projection it you have a linkage between these two tubes and only when you have that this integral that i showed you a moment ago is actually different from zero so if these loops were disconnected if they were not interlocked the magnet the kinetic energy would be zero you can have a variety of different not like structures here for example is a another complicated log not which has in this case a linkage number of three or strictly speaking is three halves but it leads to three times the amount the helicity in this case is three times um the the flux squared and so he was moffett was really the first to term them to coin the term helicity in the connection connection with hydrodynamics and also magneto hydrodynamics a bit later in fact he said in his 69 paper that the term magnetically is used in particle physics for the scalar product of the momentum and the spin of a particle and we'll come to that also in a moment and it would seem to be a natural candidate in the present context to describe the quantity u dot omega over the volume the quantity may then be described as c helicity per unit volume of the flow so this was the beginning of helicity in the studies of fluid dynamics and then also in connection with magneto in connection with magneto hydrodynamics which i'm talking about here the helicity the magnetic helicity is defined as the dot product of the magnetic field with its vector potential so a the curl of a gives again the magnetic field so it's somewhat different from the current illicit from the kinetic elasticity that i showed you a moment ago where we had the velocity and the curl of the velocity and in fact one should really say that the um that this quantity here is a quantity that describes the magnetic field so you have to take the inverse curl of the magnetic field and likewise in the previous case it was the helicity not of the velocity but it's the helicity of the vorticity and that's why i also talked about vortex structures so here we talk about magnetic structures and again we have a pair of interlocked flux rings flux is the quantity phi which can be calculated for a cross section of through the magnetic through a magnetic flux strength strand so this is a structure where individual magnetic field vectors would all be aligned with this structure and those all together would form a ring and this is indeed often seen also in in actual simulations these days and also in solar physics we see often magnetic flux structures here again these two structures are interlocked and we can now calculate this volume integral of a dot b by splitting it up into two different types of integrals one is the integral over a along a along the flux line so let's talk about this one first so this is the integral over the volume uh over the volume over the volume v2 here so it's actually this integral so we are looking talking about the these integral along the line um let me just make sure i'm getting this right here so we are talking about the line l1 so it is actually this one so it was slightly confused here so you're calculating the integral a dot l along this line so this has nothing to do with any magnetic flux quite yet but it is a line integral of a along this line and now using stoke's theorem you can turn that into a surface integral but now it would be a surface integral over the surface s2 so it's not explicitly shown here but this would be the surface that would be marked by this line l1 so it's this entire surface here now there is no magnetic field there and therefore it reduces actually to this surface integral over the only place where there is a magnetic field which is here so you have therefore the flux which is the same along each position of the flux line for the second loop so therefore this first integral gives you the flux phi 2 for the second loop now the second integral is straightforwardly the loop the integral directly of of the of the b field uh everywhere a lot for this for the same volume and it is therefore the surface integral over the surface s2 so now we have calculated one part of the integral and we obtained the value of phi 1 phi 2 times phi 2 but only once but we have two structures we talked about this structure only but we have also a second structure but by the same argument we can calculate this product again and therefore we obtain twice the product of these two fluxes now if you were to interchange the orientation of one of these magnetic field directions the sign also of the elicity would change so it's a signed quantity and then you would have minus twice phi one phi two so it's a quantity that describes the linkage if they were not interlocked you wouldn't have gotten anything in between and the integral would really be zero so this is in a quantitatively exact calculation my derivation was a little bit uh hand waving because it assumed that this decomposition would be orthogonal but it is also a correct calculation if you if you do it properly so the unit of the magnetic helicity is of the flux squared so often solar physics when people tell you something about magnetic helicity they talk talk about maxwell squared so in the sun for example we know pretty much that the magnetic field of the sun involves something like 10 to the 46 maxwell squared now what produces magnetic helicity in astrophysical in contemporary astrophysical settings this i can best explain by looking at the geophysics where we have the earth where we have very much the same phenomenology ending then we have vorticity here we have no magnetic fields in this case but we do have different situations in the northern hemisphere and the southern hemisphere so this is something that will be different in the early universe where we have just one helicity if at all so in the in contemporary astrophysical and geophysical settings what causes here in the uh what is what causes helicity in geophysics and also solar physics is simply the fact that we have a gravity which points into the sun into the earth here and we have rotation now rotation is pointing uh along the to the towards the north pole whereas gravity points towards the center these are opposite directions in the northern hemisphere and that's why the kinetic helicity is negative in the northern hemisphere that's simply because also g dot omega is negative note that g is a proper vector omega the angular velocity is a pseudo-vector so it's that product is a pseudo scalar which then changes its sign in the southern hemisphere but both have the same direction at the equator the lcd would then be zero locally so let me say just one slide worth of catastrophic quenching which really made it very very clear already since the very early 2000s that magnetic helicity really does play a tremendously important role in numerical simulations when i say numerical simulations i should qualify this year by emphasizing that i'm talking about uh periodic boxes periodic boxes is a situation where magnetically really is fully conserved and that's because so a dot the integral over a dot b is really just minus twice the diffusivity times jrb there's no other term in this case and that's because it's a closed volume there's no surface terms here making the situation very simple that also of course explains why there is a difference between periodic boxes and real astrophysical systems where we do have fluxes of magnetic helicity but i will not talk about those here today so keep in mind that what i'm telling you here about catastrophic quenching which really has caused a headache too much of the dynamo community in the mid 90s it really has to do with periodic boxes so it does what it actually means in practice is quite nicely illustrated here by successful dynamo simulations in a periodic box with silicity so what it actually means in this case is the magnetic energy b squared here magnetic energy density can still grow in as a function of time but the time scale on which it can grow and actually to relatively large values is a micro physical molecular or atomic diffusive value so the time scale on which the magnetic field can grow into saturation becomes progressively longer slower as you make the micro physical diffusivity is smaller and smaller so in real astrophysical settings it is a small quantity and in fact already in in simulations of moderate size of even just 32 cubes so here's for example shown a simulation of 32 mesh points cubed very very cheap and nowadays it's ridiculously cheap but once you go to 128 cubed that's this one you begin to see that the saturation is not an immediate one and in fact the actual fit is a perfect fit to this kind of saturation behavior so this saturation of the large-scale magnetic field which is called b bar and this can be is given by the magnetic field energy at the small scales multiplied by the ratio of the wave number at which energy is being ejected the forcing wave number kf divided by the smallest wave number of the domain k1 which could be equal to 1 in a periodic box of size 2 pi but then this is not really reached in instantaneously it's actually only reached at a micro physical diffusive time scale and these are the fits which are shown as a dashed line and they are in all cases perfect fits so you see for example here in this simulation with a much higher reynolds number that it became extremely computationally expensive to actually reach the final saturation so this explains also why people when they do dynamo simulations and are not aware of this uh using immediately the highest possible resolution may not actually have a successful dynamo or may not have realized that there's a successful dynamo because the magnetic field would be very very small still shown on short time scales and it really needs to run for a ridiculously long time scale i should also say that in many numerical simulations one of astrophysical contemporary type people are actually ignoring magnetic diffusivity altogether and then you can never reach saturation so you see here how important it actually is to retain magnetic diffusivity so now i will come to the early universe if there are any questions at this point already let me know and please interrupt me if you want what is catastrophic catastrophic refers to the fact that the micro physical value plays a red decisive role and this value is very very small in astrophysics so it would be a catastrophe in that sense if eta was really equal to zero because if it was equal to zero then you have one minus one and you wouldn't see that the magnetic field is actually could actually ever be finite so it it was eric blackman back in 2000 or 2001 who called it catastrophic quenching before that time people have realized that or have have tried to calculate what is known as an alpha effect there is an analogy between alpha effect and the cardio magnetic effect i will be talking about later but this alpha effect is a micro it's a macro physical effect uh playing a role at the level of averaging over the scale of turbulent eddies and in that case this alpha effect which can explain the growth of a mean field a large-scale magnetic field would be quenched to zero if there is if and had a fully periodic domain does this answer your question and are there any other questions just let me know okay i will thank you so in the early universe we have somewhat uh different situations and it was my pleasure to be at nordita at the time when i had the opportunity to work together with people in high energy physics carl carrie inquist and and paul olsen who were at the time very concerned about the idea that if we had a mechanism that could produce magnetic fields in the early universe this would happen at very very small length scales um in those years i knew already about this concept of an inverse cascade and that what that means is that magnetic energy could actually and that requires helicity could actually inversely transfer cascade to larger and larger length scales as opposed to a direct cascade or a forward cascade where all the energy that is injected into the system is being forwardedly propagated into even smaller length scales so it is important to realize that in the early universe and these are the times after the electro weak phase transition that's when the universe was about 10 to the minus 10 10 11 seconds old the universe was perfectly conducting at the time highly collisional and therefore magneto hydrodynamics is actually applicable there is another enormously useful concept that was really only put forward in in our joint paper at the time and that is to say that the equations of magnetohydraminics in an expanding universe where you have the universe expanding at a rate a which is proportional to time to the one half can actually be rewritten in a form that is equivalent to those of the relativistic mhg equations in a non-expanding universe here you soon in these equations here you see no expansion factor no a or no r rb is the expansion term here and by using correspondingly rescaled quantities with a tilde tilde variables and you use conformal time which is dt integra divided by the scale factor integrated then these equations become completely independent of the scale factor this was a tremendous simplification because it meant that in those years already people could easily do simulations of the expanding universe at the time with all the standard mhd knowledge that was available in those years i mentioned just a moment ago the inverse cascade and i also briefly said that it should be contrasted against the forward cascade a forward cascade is what konogorov basically proposed so kolmogorov's concept of turbulence was that energy is ejected at some length scale and all the energy would propagate to smaller and smaller length scales or higher wave numbers that's simply because the interactions in the navier-stokes equation in the hydrodynamic equations are non-linear and any non-linearity produces energy at uh at a higher wave ray factor so the energy in k after a quadratic interaction would become twice k of course and so it would gradually step by step propagate to higher wave numbers the flux along that cascade that was another important realization is actually a constant independent of wave number and because of that you can even calculate the entire spectrum based on just dimensional arguments e is the spectrum the magnetic or the kinetic energy spectrum and it's defined in such a way that the integral over e is equal to the rms velocity squared times a half it therefore has the energy of centimeter square per second squared multiplied by a by another length scale squared so therefore because of one over k so it therefore has the units of centimeters cubed divided by second squared epsilon is the flux of energy and so it has energy it has the units of energy per unit time that means it has units of centimeters squared divided by a second cubed so by just putting in the arguments the dimensions here you find immediately that the exponent on epsilon must be equal to a plus two third and the exponent on k must be equal to minus five third multiplied by an unknown chromograph constant which turned out to be not so far away from unity it turned out to be approximately 1.6 so this gives you the famous kolmogorov k to the minus five third energy spectrum this should be contrasted to the situation of a magnetic helicity where it is in fact possible to have an inverse cascade as was realized by uriel fish since the 19 since 1975. it becomes most prominent this inverse cascade if you expose it to a circumstance of not driven turbulence but actually decaying turbulence so energy decaying turbulence means you just put in energy and this is this pink line here in the spectrum versus faith number initially and these are the simulations from our old paper with christensen and hindmarsh of 2001. you put in energy initially at small length scales so it has a peak at the at high wave numbers such that it's still resolved within the discoverization scheme it then turns out and that is simply the result of numerical simulations at the time 2001 that the energy then gradually changes in such that the spectrum changes in such a way that the peak propagates gradually to smaller and smaller wave numbers at the same time the magnetic energy at small wave numbers large length scales is indeed increasing so we have a real increase of the magnetic energy at high wave at a large length scale small wave numbers that is what is called an inverse cascade in the absence of helicity it you don't have that you have at most a very very small inverse cascade so it is actually real that we saw already back then but uh but it is much much smaller they will also not talk about this today an argument for the decay rate of the energy or the increase the rate at which if the wave number here decreases or the length scale increases can be given by uh what's given first by biscam and miller in 1999. so here the idea is that the helicity is is really a conserved quantity it's fully constant the rate of energy dissipation so the energy would not be constant but the energy will dissipate and it would dissipate at a turbulent rate so that means it's proportional to only macro physical quantities which again on dimensional arguments must be equal to the velocity to the third power divided by the length scale divided by the length square to the first power or a correspondingly magnetic energy to this three-halves power so energy can be expressed in units of an alvin speed and it would then have the units we would then need to raise it to the so it would have units of centimeters squared per second squared and to get the velocity of the unit of velocity you have to raise it to the power three halves to get a velocity unit so the rate at which the energy would change the e by dt it would decay that's why we have a minus sign would be equal to the energy to the three halves power divided by l and now we can actually use this law of conserved helicity again to substitute for e in terms of l or for l in terms of e so here we still have e to the three halves divided by l but now we can substitute for l again the ellicity which is constant multiplied by e to the minus one and therefore you have all together in e to the three halves power divided by helicity if you integrate this ordinary differential equation in time d by dt equals to minus e to the five third power you obtain that the energy must decay like t to the minus three halves what i haven't written down here is that l correspondingly must increase like e t to the plus two third power such that the product of uh e and l is again a constant any questions on this actually actually i do have a question about this yeah so this is the uh evolution when you have magnetic elasticity yeah what about if you have uh hydrodynamic uh velocity type of helicity is the same argument true no excellent question it is not true actually and the reason uh for that i need to go back to this page here so remember the in in this case the decay rate of magnetic helicity is proportional to the so-called current ellicity and this one has only has a factor in b which is like energy and one derivative of the b field so that means that if you make the diffusivity smaller and smaller and that i didn't write down if you make a eta smaller and smaller j increases but only to either to the plus one half power and so that means that as the product all together so this all together decays increases like either to the minus one half power so if eta becomes smaller it increases but because of this edap factor here this entire quantity still decreases if eta becomes smaller so therefore in the limit of small eta this is a conserved quantity by contrast the kinetic elasticity would be u dot omega and then you would have on the right hand side the velocity derivative instead of instead of the so you would have the velocity here where you would have the verticity here but you would have so this would then be the verticity like because this is what you see in this velocity but here instead you would have the nebula square del squared of the uh of the velocity and that again also increases but now it increases proportional to eta to the viscosity so you would have a viscosity factor here and then the vorticity itself increases with this what this decreasing kinetic viscosity and therefore this entire term would actually increase like the viscosity to the minus one half power the viscosity goes to zero this goes to infinity for the kinetic electricity and therefore kinetic elasticity even in the best limit would just never be conserved in practice unless it is exactly equal to zero in that case you would have zero uh multiplied by infinity which of course in reality is also going to infinity because it increases faster but that is a dramatic difference between kinetic energy and magnetic illusive yes thanks for the question and i didn't have a corresponding ludigra from this any other questions uh there were no others in the chat or anywhere so i'll let you know okay great just a brief comment here about the differences between decaying simulations which i of which i showed a moment ago some pictures and forced turbulence which also many people practice in numerical astrophysical astrophytics if you have this i explained already if you have a decaying a helical turbulence you get a inverse cascade quite nicely and it looks like this if you did the same thing with force turbulence you would have a forcing at a highway number you would get something like a converse spectrum at higher wave numbers and uh what then would happen is you would have a bit of a bump traveling to the left but it would then lead to a saturation at the smallest length scale people call it or sometimes a condensation at the smallest length scale and that's how for force turbulence the inverse cascade would look like decaying hydrodynamic simulations also non-helical energy simulations non-helical magnetic fields here would lead to a very big inverse cascade and the energy would decrease so here in in fact that's very clearly so that at a better resolution the bump would always stay strictly at the same height so this in foster uh decaying non-helical simulations it would decrease this bump and then in four simulations you would eventually have a steady state this is here for the kinetic energy you get a con corimoro spectrum if you drive the kinetic energy in this case um not a not a magnetic field and then you have a dynamo effect which means that also the magnetic energy m the solid line here increases as a function of time until it reaches saturation so this is the full spectrum of what people can see in periodic box simulations either forced or decaying or helical or non-helical so that's the full range of what people have explored so far so we will be talking about this situation here where we talk about decaying helical turbulence in fact it turns out this may have been already obvious to some people that the decay in decaying turbulence can actually be itself similar process and that what it means is that the magnetic energy spectrum which is of course the function of two parameters k entity can actually be written as a function of just t alone and this can be done by a scaling function phi which would be just equal to the spectrum but the spectrum multiplied by the correlation length psi sub m which is the weighted integral of the inverse wave number weighted with the energy spectrum divided by the divided by the total energy which is the integral over the spectrum so this is a quantity that would increase i just said a moment ago i called it capital l back then it would increase this time to proportion to t to the two third power so that means if you multiply it by k and the spectrum shifts to the right it you would reshift it back always to the position unity so this is a scaling function which is a non-dimensional function of this product here but then it can actually decay like in the helical case in non-helical cases and again you can compensate for that by introducing an exponent so in our non-helical case this exponent would be zero so if beta is equal to zero in the non-illegal case you have a decay of the energy p proportional to t to the minus exponent p to the two third power by contrast in in non-helical turbulence we have either a two or a one exponent here so the spectrum is decaying and then the exponents would be correspondingly different and those are all it can all be explained by certain concept quantities here in this case it's a magnetic helicity density in the hydrodynamic case it's the so-called luzon scale integral and this was goes all back to a paper by myself and tina kahne srila of 2017 where the details are explained and also in this case we can have a quantity which is the projected vector potential squared projected along local magnetic field lines which is a potentially conserved quantity so here are the corresponding plots but now at much higher resolution than what was possible back in 2001 here we see a 1024 cube simulation where we see that the pump really has strictly speaking the same height you can then rescale it so that the bump would always be at the same position at approximately unity and you can determine then very nicely is the spectral shapes which are proportional to k to the fourth power for the inverse cascade and you have a corresponding spectrum here and you see also that the exponents that i defined up here the exponent um on the energy which you can obtain as a function of time it's this instantaneous scaling exponent would then be given by the logarithmic derivative of the energy versus logarithmic time and likewise for the length scale so in fact i thought i got a minus sign here so p is defined positive and this one is decaying but here this is for the plus sign so that is correct so we see then that in the helical case we have an inverse cascade we get a universal scaling function phi and we also get universal scaling functions for non-healing mhg decay and non-helical hydrodynamic decay again we have a very reasonably well and certainly here very very very nice collapse of the data and furthermore we see that the instantaneous scaling exponents p and q in a parametric representation of p versus q obtain universal points in in this self-stimulating plot so for all the fully helical solutions end up at this point larger symbols mean later time all the non-helical ones end up at this point where beta is equal to one and the non-helical hydrodynamic simulations and also helical hydraulic simulations which doesn't make much of a difference and end up to be close to four but some often actually closer to three with the beta all right so now i will talk about the chiral magnetic effect the chiral magnetic effect also produces helical magnetic fields and it is related to the chirality of fermions chirality of famines is a property of of fermions that have been produced for example by the beta decay if you have a neutron in a neutron decaying on a time scale of every 10 minutes it sheds an electron which has its momentum in the opposite direction as the as a spin so i talked about that already in the very beginning when this handedness of fermions was actually used as a description of a corresponding pseudoscaler in the hydrodynamics and in magneto hydrodynamics electrons typically have a left-handedness which means that the momentum and the spin point in opposite directions whereas positrons are right-handed and have an aligned arrangements between the two this is true for high energies and that's therefore true and at high temperatures are irrelevant to the early universe when the possibility of spin flipping is almost negligible spin flipping would be a possibility when the fermions would have would not have a very strong momentum you could imagine that the observer would overtake the family on and would obviously then see these spins being flipped that's simply a possible quantity result of the observer seeing the electron differently but this causes an important coupling also with electromagnetic fields and this is called the chiral magnetic effect i don't have the full his history and the literature here on this but it was of course discovered by vilenkin back in 1980s and it has been rediscovered in many different forms over the subsequent decades and there's many excellent review papers on this i will not go into the full details but let me just explain that it in particular causes an um proportionality between the current density and the magnetic field through a factor that is obviously a pseudo-scalar and it is related to the number difference of left-handed minus right-handed fermions once you have such an imbalance there would be a current which would be in a particular direction depending on whether you have uh left and or right-handed feminines so in this case if the if you have a left-handed one then the it would cause a current and believe that according to this definition there must be a minus sign here it would cause a a current which would then go in the opposite direction as a magnetic field if that's the case we have in a new term in the induction equation i should also emphasize that this term is often with a different normalization called mu5 for the people familiar with the chiromagnetic effect but here we have used a non-dimensional version of the of mu in such a way that the units of mu is that of a wave number k and so by after inserting this you can easily see that even in the absence of any velocities the induction equation which is here the uncurled version of the induction equation in terms of its vector potential it can become unstable if mu is large enough and in fact larger than k i forgot an either factor i just realized here outside so there should also be an either fact uh either factor here in our normalization and then when eta is larger than the wave number or in other words if the wave number is smaller than your chemical cairo potential mu then you would have a destabilization at all wave numbers smaller smaller than equal to mu and that we will see in just a moment where i will show the numerical simulations of of such a system before coming to that let me explain that there is a conservation law which is a modified conservation law of the usual magnetic helicity and it is obtained by dotting this equation the induction equation with the magnetic field then you obtain the usual conservation law for a dot b where this leads to the diffusion term which leads centered minus twice eta times j which would be the current density is equal to proportional to the curl of b dotted into this b but then there's another term namely the chemical chemical potential it itself obeys a evolution equation and depending on how much of a current elasticity you produce here's the current electricity or how much helicity magnetic helicity you produce this is the same term here and here the more you would begin to deplete the chiro-chemical potential so that means the more magnetic hilicity you produce by this destabilizing magnetic field the more you actually begin to deplete the chiro-chemical potential in such a way that the sum of the instantaneous magnetic helicity density and the averaged volume average cargo chemical potential is equal to a constant and equal to what we put in initially so if we put in initially a caro chemical potential then at all times regardless of the even the micro physical diffusivity and therefore in contrast to the previous uh case where i had to assume that the ca data was small here it is in fact valid for all almost all eaters and what that means is that the magnetic helicity it can be actually estimated in terms of the initial chirality and that now means in particular that the for a fully helical field which we will produce the magnetic energy can be estimated in terms of um magnetic energy magnetic energy density multiplied by the length scale which has the same dimensions and also the same almost the same value as a dot b must be equal to mu naught the initial hybrid chemical potential divided by lambda and it would be factors of two therefore a twiddle here so that means that the magnetic energy by this effect can never become larger than an upper limit that is given by how much cairo chemical potential we had initially and divided by this evolution by the speed of evolution which is this lambda factor here so this means that there is actually a limit on how much magnetic energy we can produce now we can even just look at this quantity here and estimate it based on based on dimensional arguments i think i had that um well let me just have a look here maybe i come to that actually in a moment yeah anyway so the full evolution system of course also involves the hydrodynamic equations and the continuity or energy equation which means we start a simulation by putting in energy through the cardio chemical potential which would produce through this instability it's similar to a dynamo instability it would produce magnetic energy it would also produce kinetic energy all of these would dissipate and it would also even produce a large scale magnetic field so there are different regimes i will only focus on and i will show both of these regimes that depends simply on how big the mu is and how big this lambda coefficient it was remember these are the two coefficients here it's actually my dimensional argument no it's not so anyway based on dimensional arguments we can estimate what the spectrum is so what we want to explain here is what happens when we inject energy at a wave number that that is less than mu remember all the wave numbers less than mu and the fastest growth would be mu divided by 2 would grow exponentially in time but then there would be a limit eventually when the energy can no longer grow exponentially and that's when the energy can only go to larger length scales because it must is forced to be inversely cascading and so you would be going to larger and larger energies in the spectrum according to a number of different simulations including our own earlier ones usually is a rather steep spectrum typically proportional to k to the minus two and if that's the case you can produce an a dimensional argument for the scaling law of the magnetic energy which is just dependent on mu and eta and nothing else in particular mu lambda does not appear yet because lambda would only lead to the final saturation of the energy once this entire process has finished and so this is a quantitative quantity a scaling law which numerically was confirmed and turns out to give a scaling factor here which is around 16. so it's larger than unity but it is of order of unity in that sense and uh it really provides an uh quantitative estimate of how this scaling would appear then at large uh large times you would have eventually saturation and it can it can be obtained by dimensional arguments but now the dimensional argument must involve lambda which is a quantity that leads to the feedback of the dilution of mu the vector potential the chemical potential which is of course the source of everything but it gets diluted or depleted and you obtain now a scaling for e which is equal to mu divided by lambda and independent or wave number independent of wave number means that in the spectrum here this is a sketch the bump would just be appear under a limit and this limit is given by mu divided by lambda in between you have the scaling law and so energy would really go and be injected here and would inversely cascade until it reaches saturation so now i show this you in action here here's a simulation in a color code of of the vertical component of the magnetic field here's the corresponding spectrum and you see that the energy increases first exponentially it's small length scale so it feels already fully helical but it's random at this point but eventually it reaches the scale of the domain once it reaches the scale of our cartesian domain it begins to feel the cartesian geometry and so it should not come as a surprise that in this particular simulation we end up with a magnetic field that has a sinusoidal length scale depend a sinusoidal dependence of one of the two components b z i think is b x here as a function of z and there would be a corresponding b component b y which is phase shifted by 90 degrees and is proportional to the cosine this would be proportional to the sine of z and so therefore the product of the two of the product of a and b would be equal to in other words the the derivative the curl of the magnetic field would be equal to the magnetic field itself uh all the magnetic helicity is fully helical the magnetic field is fully helical a and b are parallel to each other and also b and j are proportional to each other here you see the other limit where the magnetic energy where lambda is small and then the energy then there would hardly be an inverse cascade it would immediately go to the uh into the into the decaying phase where the length scale is still increasing and energy again decreasing whereas here energy was still increasing so now the dimensional argument really is that that you can explain that the product of magnetic field and length scale can just be explained by the temperature and that's all all the rest are just natural constants it would be the boltzmann constant stefan boltzmann constant and uh sorry what's my constant and and h-bar plum constant and speed of light based on this alone you obtain something which has the units of ergs per cubic centimeter multiplied by centimeter which is earths per square centimeter so that's the name of the game and if you put in the temperature of the universe as of today you would obtain a value which would correspond to a helicity of 10 to the minus 18 gauss even less gauss squared multiplied by megaparsec this g was a mistake here should have been deleted so this gives a relatively strong and stringent limit which is is relatively small compared to the so-called lower limits of the magnetic fields that have been inferred from the non-observation of secondary [Music] secondary gev emission from blazers from the halos of blazars this leads to a limit on the product of the magnetic field and the length scale here in x to the one half power so the product again has the same units so that would be a constant here along this line magnetic is square field squared multiplied by length scale being a conserved quantity our numerical limit was already slightly below or somewhat below a few or orders of magnitude below this limit so it would be perhaps it too weak to explain the magnetic the lower limits on the magnetic field but this is just receiving some caveats because this was a relatively simplified approach to this and what i will address in the remaining few minutes is whether this chiral chemical effect could also produce and perhaps obvious observationally relevant gravitational waves so i will not have much on this can i ask a question yes please this mu5 or whatever you assumed in other words that all electrons are left-handed yes yes yes that's right which is of course huge it's upper limit yes so the number of degrees of freedom uh was here assumed of course to be three in the early universe is larger and because you have many more families also that could lead to maybe a few orders of magnitude uh change but already when you assume a g of 100 here which we did for the other universe but then this is not this is not what i'm asking i'm asking why would electrons be left-handed in the early universe yes there should be special mechanism yes yes yes that is indeed another good question and it is a hypothesis to some extent which is uh related to the fact that we must have some kind of imbalance in the in the most likely in the chan simon's number in order to explain the imbalance between matter and antimatter true so we would be a small yes now it's not that small because it is exactly what it would actually be sufficient in order to explain the corresponding change in the magnetically city these are arguments that tanmay vacher's party who is either already in the audience or is still at his doctor's appointment has been making since over the last 20 years so there is a connection between changes in the transcendence number and baryon synthesis yeah sure and so that's what uh what one would be needing to allude to in this in this scenario there is apparently also a comment from kohei please here uh okay so i'd like to make a comment uh in in either case of the the way in the risk model of our universe the lightening electrons are relatively the less intellect less interactive particles so that it is easy easier to to remain remain conserved so that it would be not not left-handed electron but lighter hundred electrons would be better to be used and so for some some infl uh some some mechanism in the arena such as gut biogenesis with some some a bit bit technical implementation but it might be possible to generate light yes yes yeah thank you very much for a very important remark that's uh great and helping me out here so um the idea of a gravitational waves relevant to cosmology has a long history here in this case it was motivated by by inflation but what we are doing here and this is just a few more transparencies that i will be showing here we are looking at the magnetic stress the ibj and it would source gravitational waves the transverse tracer's projection of it and we calculated those in in recent work is my former ph.d student uh alberto and we found that there is a direct correspondence between the spectrum of the magnetic field and the spectrum of gravitational waves e gw which corresponds where we have a correspondence between a k to the minus five third energy spectrum for the magnetic field and uh would expect a corresponding spectrum for the gravitational waves of k to the minus eleventh third there is another important uh detail namely that the in the sub inertial range there is not a case uh k square k to the fourth multiplied by uh or divided by k squared as here so this is this eleventh third is a five third minus two uh here that is this is different and this spectrum would always be flat that's an important difference having to do with the fact that this one is not a white noise but it's actually a blue noise i will not go into such details here but but let me just say that that there is a fair chance that all these magnetic stresses could be well could lead to a spectra that are well above the lisa detective detectability limits sensitivity limits so here are cases where if the magnetic energy density is about 10 of the critical magnetic density and we call that spectrum omega per unit logarithmic frequency level uh would be clearly above the detectability limit and perhaps even larger than that so what we obtained back then was that there is a connection between gravitational wave energy and the input energy which is the um which is the magnetic energy most most in our cases depending on various scenarios for acoustic turbulence we obtain typically somewhat more efficient generation for the same energy input we found typically more gravitational energy and we are in the process of understanding what exactly is behind it and it has to do with the particular time dependence in acoustic turbulence hydrodynamic turbulence in this case so here just a few more experiments with different simulations that we are doing right at the moment uh where we calculate the corresponding gravitational wave energy for different scenarios with magnetic genesis magnetogenesis by the chiromagnetic effect so we can have some scenarios where depending on the diffusivity these are all lines where the diffusivity increases we have certain trajectories here or certain tracks in in this parametric representation where depending on the amount of energy so here is just one percent of the critical energy density you can reach already values of the gravitational wave energy that begin to become interesting and perhaps even more so as we continue this plot so the hope is not quite over that the chiral magnetic effect may also produce observationally relevant gravitational waves okay here let me advertise one paper with tina carnier just a few two two weeks ago where we calculated the circular polarization of these gravitational waves and we obtained almost hundred percent uh circular polarization for a fully helical magnetic field and we would expect a fully helical magnetic field and even facts effects of the inverse cascade as always all explained in detail in this recent archive paper so is that uh oh i have here a few more visualization actually this one is a nice one so normally uh when of course pod explains this polarization in terms of these two cross polarization and plus polarization circular polarization what it actually means is that the root a that the plane is actually rotating as a function of time so let me just show you this so this polarization plane is just rotating i think my animation is a bit slow sometimes but if i uh now also look at the so this is just a bubble and we would not be actually able to see the circular polarization in the universe until we were actually able to see the full spatial dependence because it positions different positions in the solar system if you had a lisa configuration a little bit away from the earth for example you would see a corresponding wobble but shifted in phase and with that information you could potentially observe circular polarization in the early universe so with that i have stretched enough of your time and i would like to conclude here by saying that magnetic helicity is an important quantity it's in nearly perfectly conserved in usual hydrodynamics it is at least so catastrophic quenching in periodic boxes which has led to a headache of some people in the hydrodynamics community back then but it also leads to inverse cascading in such a way that we can draw conclusions about the energy density even at the present time because the helicity that was produced early on would be exactly the same what we would have even nowadays and it can also lead to gravitational waves uh that would be fully circularly polarized hundred percent circular polarized at uh at certain frequencies uh and those would be detectable by lisa we would hope so with that i thank you very much for your attention and would be happy to answer additional questions thank you thank you very much for a very nice presentation now let's go to questions i think we have the first question uh from henry go ahead henry john hi uh i'd like to know uh what your research can say about the uh helicity of the magnetic fields in terms of like say the bipolar mass and energy ejections of quasars yeah um i have not looked into that in particular but it's just a little bit similar to the helicity that i've been discussing in connection with the sun so based on dimensional arguments i think one can just do a back of the envelope calculation by um by looking at the magnetic energy density and and a typical and i think that's actually it we would basically have a flux squared so we can calculate the the uh this is the flux in each hemisphere of a of the of the system that actually produces the quasar itself the accretion disk we can calculate in the northern hemisphere separately from the southern hemisphere the magnetic ellicity or the magnetic energy first of all and based on that we can estimate the flux which is just the modulus of b on in principle it's b but we haven't as a better estimate one can take the modulus and multiply this by the cross-sectional area of the accretion disk in latitude and based on that you can obtain a flux and um the magnetic helicity that you could be shed and would be shed and i would expect nearly perfect shedding of magnetic electricity that's why we are by real astrophysicists not in a periodic box of course magnetic illicit is being shed it could just be estimated by this taking this flux squared and so when we just need to estimate here what we have in terms of b and surface area you get a flux in maxwells and then you calculate maxwell squared and you have your quantity so i think it would be a quite a bit actually and of course it is a we know that it can already lead to seeding a cluster of galaxies by the by by the many different quasars that would be shedding magnetic energy and helicity but then of different signs of course so the cluster would not have enough helicity does that answer your question yes very nice thank you thank you okay next question sanjay already please all right thanks for a nice talk i really enjoyed it um i think i may have missed uh something you said about the time scale for the inverse cascade my understanding is that if you want to generate large scale magnetic fields starting with some mechanism that produces an initial chiral imbalance or gives you an initial mu5 there's a competition between the time scale associated with uh spin flips which is trying to get rid of the initial mu5 and the time scale for the inverse cascade to somehow take that initial mu5 and turn it so we have we have estimated that actually in our app j letter paper of 2017 the spin flip rate um in the early universe i mean it is comparable on the on the scale of the hubble horizon it would be uh comparable however the current magnetic effect really acts on small length scales to begin with and on those small length scales where the instability happens i think spin flipping there was a factor of either 700 or 7000 i forget know exactly what what number it was we would be safe so the spin flipping would not affect the early early part of the instability yeah and once uh once you have so you would be able to turn all this charity into into magnetic fields that would then propagate to larger and larger length scales and then you don't have to worry about spin anymore thanks i but i'm trying to understand what is the physics that sets the time scale for the inverse cascade i suspect i missed it yes these are the dynamical time scales so i remember i showed you a power law so it goes like t to the minus two thirds uh yes this it was um it was uh here energy is proportional to t to the minus two third power so that happens on a dynamical uh time scale so it would be a and it so it depends then actually another nice plot here if i go let's see if i go to a page oh i have lost here in my screen sharing uh sorry yeah let's go to page yeah this one so a useful way of explaining this would be just to plot the change of the magnetic field which would decay as a function of a length scale which would increase so as a function of time it would increase from small length scales to larger and larger length scales and we know roughly the time that we have namely approximately i think it's 12 orders of magnitude in time that would correspond to eight orders of magnitude and length scale here which is what is plotted here eight order of magnitude and length scale between the time of the electronic phase transition and the time of recombination so that gives us 12 orders of magnitude in time and that corresponds to eight orders magnitude and length scale and so for a fully helical magnetic field that would be a decay which would be along this line and so it would decay in this case if it was fully helical and had was close to one micro gauss in co-moving units in the units of today it would decay down to nano nano gauss magnetic fields but not from the chiral magnetic effect this would be from a hypothetic magnetogenesis effect that is more efficient than the current magnetic effect the chiromagnetic effect would be approximately down here it would go along the same line but would be below that and this is here the family limit from the non-observations of secondary gv photons okay thank you thank you okay next question i'd like to arrive hello thank you i have a question concerning the gauging variants while hydro helicity velocity is are you breaking up [Music] we cannot hear you wow okay let me explain the engage uh gauge dependence here so yeah just a question about gauge dependence because for example felicity velocity is observable and so magnetic magnetic helicity is formerly gauge dependent but if the volume is an infinite one or a periodic periodic one is sketch independent yeah so periodicity plays a role yes yes so that indeed is a is a bit of a complication um in open domains and therefore in the case when we have elicited fluxes from domains it does indeed play a role and and indeed there are some fair questions to be answered i should also say that when we do numerical simulations we are always working with a so as far as i'm concerned um i'm evolving a in the numerical simulations and i have set boundary conditions on a and then a to b in that gauge of course is a well-defined thing then conceptually it's easy to work with that but i do understand that if you wanted to detect it observationally for example you run into problems of course and you are choosing some particular gauge right for for numerical simulations i'm using the wire gauge so that means that d a by dt is equal to minus e electric field that's a very simple cage and actually the best one i would think there's other gauges which can be uh can have conceptual advantages but yes it's a lot to be talked about and also just look at some of my literature there's a 2011 paper relevant to this question led by candelaria okay thanks thank you okay uh i think we had a comment from kohei canada so i'd like to give a comment on the spin flip introduction but uh so i think actually you consider the case saying the mu5 is order of the temperature but uh so comparing the spin flip interaction and have expanded late there it has been shown that the the the if then there is the initial initial in initial calculation potential mu five it means over t is larger than t minus four to a third then the the catalyst implementability can grow before the spin interaction become the effective that's the comment yes yes i uh that agrees i think with what we had in our letter paper you could perhaps look at look at that i think that we mentioned the number of 700 or even 7000 by which we will be safe as far as instability is concerned okay i see there's another question in the chat saying that the gauge dependence is physical but i couldn't already answer i mean if you if you add a gradient um then you buy then the term that you would get is equal to is is because the divergence of b is equal to zero and perhaps you know that of course because the diversion of b is zero is the integral becomes just equal to phi times the uh to the surface integral of phi times uh times b and so again for if there is no surface in the surface term this would be absent that's the best i can say about this so there for for homogeneous for homogeneous conditions i don't think there is a actually a problem it but it is dependent on the homogeneity so what you could of course imagine yes due to boundaries what you could imagine is of course a situation where the universe is uh non-uniform that you might have positive velocity negative velocity in different places then you would have illicit fluxes between the two those fluxes would still be physical and as arguments also for why they are physical but but indeed there is a formal engaged problem in those cases but this would only be the case if you're talking about uh large large scale homo inhomogeneous systems um okay i don't see any other questions but concerning this thing i'm a bit con confused uh some people seem to imply that you need to make a physical meaning out of helicity density that's not really required you really just need the total helicity as an integrated quantity and of course you could only define separately in the regions where the boundary conditions are such that magnetic fields for example go i think what is parallel to the surface or something like this so in that case you could define isolated regions but generically it's a topologically global quantity so you don't really need to worry about the density i assume yeah but we can actually still do that so when i do my numerical simulations of course uh then a dot b is actually especially also uniform uh at least uh it's a pretty but a pretty good approximation so that was that's what i meant once once however you have a system where you have a modulation a large scale modulation of of a local adobe in in any gauge then you would have fluxes and those fluxes wouldn't get themselves be gauge dependent and again it would lead to physical effects but if you want to describe them yeah you have a you have to make a decision about which gates you want to communicate in if you want to compare with somebody else so in solar physics there is a big body of literature since 1984 and 85 there's a paper by mitch berger and george field a very important one of 84 which developed certain concepts for calculating gauge invariant helicities that's is elicity based on some reference field so that again is a different concept i don't like that that much but um yeah look at some of my other papers where we do talk about magnetic helicity fluxes it turns out that um in cases where you can average in time um you can again be safe that in the sense that um in that case the time derivative of the helicity is gauge independent it's actually zero if you average over time and then you have a gauge independent term and the gauge dependent flux divergence but because one is gauge independent manifestly also the divergence of the flux helicity flux must be gauge independent under those circumstances either after time averaging or or the assumption of statistical stationarity so that's the concept that i like to talk about okay so let's see uh any other questions from the audience before we will wrap this up i see at least a number of um familiar faces here like uh like um gordon is still online of course and atanma is back from his appointment which is nice to see [Music] yeah thank you great to see you anyway i i cannot let this go before first thanking our speaker axel for a very nice presentation it was uh very nice of you to go into all these details and explain everything uh from basic principles so thank you very much i really appreciate it it was a real pleasure thank you very much igor for inviting me to this very wonderful seminar i've been looking and attending many of these already so it was always a great pleasure

Channel: Theoretical-Physics-Colloquium

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Rating: 5 out of 5

Keywords: theoretical, physics, science

Id: KXLaVf7-qIw

Length: 79min 35sec (4775 seconds)

Published: Wed Nov 25 2020