THE THEORY OF DYNAMICAL FRICTION Andrey Katz
with A. Kurkela and A. Soloviev, to appear soon OUTLINE
Motivation: why is this important? What do we already know?
Description of the formalism
Calculation of the drag force:
ideal gas: recover the known cases, generic expression for arbitrary distribution and arbitrary “bullet” velocity
recovery of the ideal liquid results
viscous liquids
generic results of interacting gases in the relaxation time approximation, interpolation between the gas and the liquid
Comments on the boundary conditions effects
Outlook !2 Motion of black holes leaves a “wake” of stars and gas in their trail… the gravitational tug of this wake on the black hole then removes energy and angular momentum from blackhole orbit (dynamical friction)
WHAT IS DYNAMICAL FRICTION?
The problem is not new: A massive particles Dynamical friction propagates through a medium with a constant velocity. What is the gravitational drag force, induced by the wake?
First analysis: Chandrasekhar, 1942 !3 3 DYNAMICAL FRICTION AND MODERN HEP
Traditionally: used to calculate motion of bodies in the galactic dynamics — with classical NR results being perfectly adequate
More modern questions: effects of self-interacting DM (component) on the galactic dynamics — more complicated, self- interactions effects should be taken into account
Recently: calculate the the energy released by (primordial) black holes going through neutron stars and white dwarf — effects of matter in extreme conditions must be properly accounted for
Exotic objects in the accretion discs of the black holes
!4 the spatial derivatives. These two approaches are fully equivalent. Note that for feq we p0 assume a locally thermal distribution feq = f0 T (~v ~u)+ T/T . Of course we expect this ⌧ · P 1 e´ (144) assumption to break down for ⌧ , namely“ ´ in the free streaming case. However since the !1 1 importance of this term scales as ⌧ , we are not sensitive to this assumption in this limit. ! t 1 (145) When we know the perturbation f„we can„ O finallyp q calculate the correlator
T µ⌫ R R D d3p pµp⌫ (146) Gµ⌫↵ = withE „ TS µ⌫ = f (23) kin h (2⇡)3 p0 ↵ a Z ⌧ c P 1 e´ (144) where the dependence on h comes from y the Christo↵el symbols. Interestingly, the(147) object “ ´ ↵ À RS!
Gkin is not fully equivalent to the object that we have defined in the Eq. (7). Nonetheless, ! t 1 ! (145)t 1 (148) „ one„ canOp showq that they merely di↵er by contact„ terms, i.e. terms that do not contain new
poles in the kz complex plane and therefore the contribution of these terms will vanish in the RE RSD ⇢(146)r´ (149) „final answer. (Aleksi, make sure you agree„ with this explanation – ak) Finally, once a 3 ´1 wec have the object Gkin, we can go ahead andRE construct´ the dressed graviton propagator. y Reff R(147)E (150) À RS! 9 r ˆ 0 ˙
! t A.1 Free streaming scenario tˆ (148)a c (151) „ var „ s{
Let us first try to reproduce the results⇢ for⇢¯ the non-interacting gas in the non-relativistic ⇢ r´ ´(149) 1 (152) „ ” ⇢ „ Op q and ultra-relativistic cases. The force exerted on a body by a gas of non-interacting rela-
3 ´1 tivisticR particles´ using the results of [1] and [4] is: R R E (150)1 v2 c2 (153) eff 9 E r “ ´ { ˆ 0 ˙ 2 2 4⇡aG mb ⇢v WHAT DO WE KNOWF = 1ABOUTlog ⇤ . (24) W vE~2 D~ B~ H~ d3x (154) tˆ a c “ 8⇡ (151)p ¨ ` ¨ q var „ s{ ª THE DYNAMICAL FRICTION?e2 dr (Translate to the relativistic language – ak) In this expression ⇢ is the energy(155) density 2 r2 ⇢ ⇢¯ ª ´of a gas1 of non-relativistic particles, namely(152)⇢v = mn, where n is the number density of ” ⇢ „ Op q 2 2 v particles whose velocity does not4⇡ exceedG mb mp the bullet2 velocity,. In the opposite regime of the 1 1 1 Ideal non-relativistic gas F 2 dv v f v log ⇤ (156) “ v 0 p q ultrarelativistic1 v2 c2 gas the result is derived in(153)ª [10] was “ ´ { a 2 2 2 2 2 4⇡m G 164⇡v 2 ⇢02 P02 1 F F =p ` Gq mp ⇢`v qloglog ⇤ ✓. v cs (157) (25) Photons gas,W relativisticE~ D~ bulletB~ H~ d3x “ 3v2 (154)b p ´ q “ 8⇡ p ¨ ` ¨ q ª e2 dr –10– How do we interpolateNote that between the result in two the intermediate these regimes? case(155) and the interpolation between the two limiting 2 r2 ª cases to the best of our knowledge does not appearOstriker; in the 1998 literature and we present it here Ideal 2liquids:2 v 4⇡G mb mforp the first2 time. As we will shortly see, our techniques allows us to get to an analytic F dv1v1 f v1 (156) “ v2 p q Supersonic: expressionª0 that smoothly interpolates between10 two these limits.
4⇡m2G2 2 1 v2 2 ⇢ P F p ` q p 0 ` 0q log ⇤ ✓ v c (157) “ v2 p ´ sq Vanishes subsonically, though certain 11 boundary effects can change this conclusion
!5
Fig. 3.— Solid curves: Dynamical friction force in a gaseous medium as a function of Mach 10 number = V/c .Curvescorrespondtoln(c t/r )=4, 6, 8,...,16. Dashed curves: M s s min Corresponding dynamical friction force in a collisionless medium with particle velocity dispersion σ = c and r Vt= c t. s max ≡ M s CALCULATION OF DYNAMICAL FRICTION — HISTORICAL APPROACH
Chandrasekhar’s approach: calculate the probability function of occurrence of force F acting on a star and an average time during which the force acts
In the case of the ideal fluid (Rephaeli & Salpeter, Ostriker) one calculates the drag due to the asymmetry in the flow associated with the front of the shock wave.
!6 and uµ = (1, 0, 0,v) is the four-velocity vector of the bullet. Note that hereafter we consider the simplest possible case, namely the bullet moving in a infinite medium with constant We also notice, that, maybe surprisingly, the dynamical friction in the interacting gas has velocity for su ciently long time. With this approach we will not be able to capture the never been properly calculated. In our paper we close this gap presenting a full calculation boundary sensitive e↵ects, like for example discussed in [7]. However, this will be mostly relaxation time approximation. The core observation of this paper is that the force exerted su oncient a bullet, for our propagating needs. We through will also gas, briefly can easily consider be calculated boundary from e↵ aects metric and perturbation. see how, for example,In order one to reproduces calculate the the latter, results it is of su [7] cient in our to formalism calculate the in Sec graviton XXX. propagator, dressed Theby the linear correlators perturbation of collective in the gravitationalmodes of matter. field The caused correlators by the bullet of the will collective cause modesa linear perturbationhave been in calculated the background in various medium, limits. For which example, will evolve in the in hydrodynamical time in a way that limit depends they have on thebeen material recently properties computed of the in [12]. medium. The For case now of the we interacting will not specify gas of further the ultra-relativistic the properties of theparticle medium. has been However, analyzed in general, in [13]. the In this medium work responsewe go further, of the calculating energy momentum the momentum tensor T µ⌫transport, which should correlations be thought functions about in the as a relaxation perturbation time to approximation the (3) to the of gravitational the interacting field
in genericgas of particles medium withcan be arbitrary characterized mass. This by the further response allows function us to computeGmedium dressed graviton propagators in momentum space and eventually the drag forced exerted on to a bullet, µ⌫ 4 µ⌫,↵ bullet Twake(x)= d x0Gmedium(x0,x)h↵ (x0). (6) propagating through the interacting gas. Z We willOur later paper slightly is structured abuse the as follows. notations In Sec and II drop we present , while our the formalism perturbation in great and detail the and un- perturbedexplain background the logic behind will the be calculation. distinguishable The fromcalculation the context. itself is performed This response in Sec function III. As a is nothingwarm but up, the we start retarded from propagator the free streaming of the gas energy regime, momentum reproducing tensor, the well or explicitly known literature
results of both the non-relativisticµ⌫,↵ and the ultra-relativisticµ⌫ ↵ gas approximation. We then Gmedium(x, x0)= i✓(t t0) [T (x),T (x0)] (7) switch to the the hydrodynamical case, analyzing both the well known inviscid case and the The response function depends in detail on the⌦ microscopic dynamical↵ properties of the slightly more complicated scenario with viscosity. Finally we perform the full calculation for mediumthe interacting and eventually gas that needs interpolates to be computed. between all While these cases. some In particular the last section cases, e.g. we drawthe our free- streamingconclusions gas and discuss the ideal the liquid, application are ofvery our well results known, for real-words the most problems. generic Thecases technical must be calculated,details are as relegatedwe show in to the appendices. next section and in the appendices. Finally, the wake itself will have its own gravitational field, which can again be computed usingAII. the DESCRIPTIONNEW Green function APPROACH of OF the THE linearized FORMALISM Einstein equation wake 4 grav ↵ h (x)= d x0G (x, x0)T (x0). (8) Our main objective isµ to⌫ calculate a force,µ⌫, exerted↵ onwake a bullet that propagates through Z Thatagasofparticles.Onecannoticethatinthelinearizedregime,theforceisexertedonthe is, hwake represents the gravitational field that was created by a wake that was created (Thermal) field theory approach + Gravity as an EFT by abullet gravitational by the gravitational field that was field created of its own by thewake bullet. and can Because be written the expressionsas that we got wake wake Definitionboth of for force the h µinduced⌫ and the Tµ⌫ aredp convolutionsµ in the coordinate space, we will further F µ = = m µ (hwake)v↵v , (1) proceedby the with wake: the calculation in the momentumdt space.↵ In Fourier space where the convolution becomeswhere a the simple zero product component the corresponds gravitational to field the rate of the of energy wake has loss aF particularly0 = dE . Of simplecourse this form The graviton field of the wake (Fourier space): dt approximation breaks downwake at short distances,dressed where non-linear↵ gravity kicks in, but this is hµ⌫ (!,k)=Gµ⌫,↵ (!,k)Tbullet(!,k)(9)
[T ,T ] Excitations of matter ⟨ ρσ αβ ⟩ G Full object to ⟨[ 4 ]⟩ G h αβ6 ρσ ,h calculate ,h
τω µν h [ ] ⟩ ⟨ bullet eikonal propagator
!7 expected to happen at distances comparable to the Schwartzschild radius, which is strictly smaller thatwhich the accretionhas a particularly radius. Therefore simple we will form go ahead along with the this approximation. worldline of the bullet
↵ 2 wake mu u d k dkz dressed A. The wake @ h (z = vt)=i ? k G (! = vk + i✏,~k) . (14) ⇢ µ⌫ (2⇡)2 2⇡ ⇢ µ⌫,↵ z which has a particularly simple form along the worldline of the bullet Z First, let us derived the formal expression for the gravitational field of the wake, produced Of course, if we are interested in the ⇢ =0index,whichisneededtocalculatetwooutof by the↵ bullet. For2 large separations, the bullet a↵ects the medium only through a linear wake mu u d k dkz dressed @ h (z = vt)=i ? k G (! = vk + i✏,~k) . 2 (14) ⇢ µ⌫ perturbation four of(2 independent the⇡)2 background2⇡ ⇢ µ terms gravitational⌫,↵ in Eq. field, (11),z which one it induces should use k⇢ = vkz. The +i✏ prescription chooses Z the appropriateg boundary= ⌘ + h¯bullet condition⌘ + h forbullet, the retarded propagator.(2) Of course, if we are interested in the ⇢ =0index,whichisneededtocalculatetwooutofµ⌫ µ⌫ µ⌫ ⌘ µ⌫ µ⌫ From Eq. (11) it is clear that the only relevant cases for are either ⇢ =0or⇢ = z. In both where 8⇡G. Note that in our conventions the metric fluctuation h is suppressed by four independent terms in Eq. (11), one⌘ should use k⇢ = vkz. The +i✏ prescription choosesµ⌫ the Newtonthese constant. cases Wethe also integral assume over that thedkz unperturbedin Eq. (13) medium can be is homogeneous further simplified and by dividing the left hand the appropriate boundary condition for the retarded propagator. isotropic with the unperturbed energy-momentum tensor of the medium side to two equal parts and changing the integration variable kz kz in the latter part. As From Eq. (11) it is clear that the only relevant cases for are either ⇢ =0or⇢ = z. In both ! long as the background over⇢ which000Gdressed is computed is time translation invariant (as it is these cases the integral over dkz in Eq. (13) can be further simplified by dividing the left hand medium 0 0 P 001 in all the cases weTµ are⌫ considering),= changing the sign of the(3) real part of the frequency turns B 00P 0 C side to two equal parts and changing the integration variable kz B kz inC the latter part. As B C a retarded Green function! B into00 0 advanced,P C which is given by the complex conjugate of the dressed B C long as the background over which G is computed is time translation@ A invariant (as it is with ⇢ theretarded energy density function. and P the As pressure. in all the The gravitational cases we are perturbation interested,hbullet thecaused retarded function is multiplied in all the cases we are considering), changing the sign of the real part of the frequency turnsµ⌫ FORMALby the bulletby a is single conveniently power found ofEXPRESSION throughkz, we a convolution get for the with integrand the Green function of the lin- a retarded Green function intoearized advanced, Einstein which equation is (hereafter given by we work the in complex the Lorenz conjugate gauge), i.e. graviton of the propagator with the energy momentum tensor of thedkz bullet: dressed dkz ikz dressed dressed retarded function. As in all the cases weFOR are interested, THE the retarded ikFORCEG function= is multiplied(G (G )⇤)(15) 2⇡ z µ⌫,↵ 2⇡ 2 µ⌫,↵ µ⌫,↵ z bullet Z 4 grav ↵ Z by a single power of k , we get for the integrand hµ⌫ (x)= d x0Gµ⌫,↵ (x, x0)Tbullet(x0). (4) dkz dressed Z = kzImGµ⌫,↵ . (16) Assuming that the bullet of mass m moves with a constant velocity2⇡ v through the dkz Letdressed us first dassumekz ikz thatdressed webullet knowdressed the excitationZ of matter. ik G = (G (G )⇤)(15) 2⇡ z µ⌫medium,,↵ the energy2⇡ momentum2 µ⌫,↵ tensor of theµ⌫, bullet↵ and its Fourier transform can be written Z TheZ fact that we are interested merely in the imaginary part as well as the +i✏ prescription Theas3 energy-momentumdk tensor of the bullet: z dressed z =give us a convenientkzImGµ⌫,↵ prescription. how the contour(16) should be closed in the dk integration. µ⌫ mbullet2⇡ µ ⌫ 2 ˜µ⌫ 2⇡mbullet µ ⌫ z Tbullet =Z u u (z vt) (x )= Tbullet = u u (! k v) (5) With a little bit of algebra? ) one can show that the force is then given simply by The fact that we are interested merely in the imaginary part as well as the +i✏ prescription From2 Ourhere metric we in this can paper use is “mostly the plus”,standard namely ( +++).machinery:m2 d2k calculatedk the 3 d4k ik x z z z dressed give us a convenient prescriptionWe how work the with contour the convention shouldf(x)= be closed4 e ·Ff(k).= in Further, the dk we willintegration. be interested? k in retardedIm G Green , (17) dressed propagator, convolve(2⇡) with the 2 energy 2 (2 ⇡momentum)2 2⇡ z functions, and therefore to pick the retardedR ordering, we must takeZ! to have a small positive imaginary With a little bit of algebratensor onepart can! of+ showi ✏.the thatbullet the and force further is then the given Christoffel simply by symbols.⇥ ⇤ with 2 2 z m d k dkz dressed dressed dressed ↵ µ ⌫ F = ? kzIm G 5, G G (17)u u u u , (18) 2 2 (2⇡)2 2⇡ ⌘ µ⌫,↵ Z ⇥ ⇤ ⇥ 2 1 ⇤ v2 where we additionally used thatStructure ( v! + v onekz)= could predictkz and the explicit structure of the with Qualities of the perturber and 2 2 µ from optical theorem the matterGdressed perfectlyvectorGdressed ufactorizeas itu↵ appearsu uµu⌫, below Eq (5). (18) ⌘ µ⌫,↵ !8 ⇥ 2 1 ⇤ v2 where we additionally used that ( v! + v 2 kz)= 2 kz and the explicit structure of the C. Scale separation and EFT validity range vector uµ as it appears below Eq (5).
Now that we have developed the machinery to calculate the drag force when the retarded C. Scale separation and EFT validitytwo-point range correlation functions are known, let us return to the issue of scale separation
Now that we have developed the machinery to calculate the drag force when the retarded 8 two-point correlation functions are known, let us return to the issue of scale separation
8 RECIPE: HOW TO CALCULATE THE DRAG FORCE IN ANY REGIME
Find an expression for the excitation of matter: particle retarded propagator in the case of the ideal gas, sound modes for the ideal fluid in ideal liquid…
Use it to calculate the retarded propagator of the dressed graviton
Contract with the four-velocity of the bullet
Calculate the drag force as the Fourier transform of the imaginary part of the contracted dressed graviton SCALES OF THE PROBLEM
Relevant scales:
RIR — the largest relevant distance, the size of the cloud
RUV — the smallest distance, the size of the bullet (Schwarzschild radius)
The mean free path — at this scale the liquid EFT starts breaking down, must be replaced by full kinetic theory
What is the k-dependence in various regimes?
General dimensional considerations: F ∝ m2 G2
Comes from general expression Gravitational coupling2 from the dressed propagator !10 ⌧ P 1 e´ (144) “ ´
! t 1 (145) „ „ Op q
⌧ R R D P 1 e´ (146)(144) E „ S “ ´ a ! t 1 (145) c „ „ Op q y (147) À RS! R R D (146) E „ S ! t 1 a (148) „ c y (147) À RS! ⌧ P 1 e´ (144) ⇢ r´ ! t 1 “ ´ (149)(148) „ „ ! t 1 (145) 3 ´ „ „ Op q ⇢1 r´ (149) RE ´ „ Reff RE (150) 9 r0 RE RSD (146) 3 „ ˆ ˙ ´1 R ´ R R E a c (150) eff 9 E r ˆ 0 ˙ y (147) tˆ a c À RS! (151) var „ s{ tˆvar as c ! t 1 (151) (148) „ { „ ⇢ ⇢¯ ´ 1 ⇢ ⇢¯ (152) ” ⇢ „ O p q ´ 1 ⇢ r´ (152) (149) ” ⇢ „ Op q „ 3 ´1 R ´ R R E (150) 1 v2 c 2 1 v2 c2 eff 9 E r (153)(153) “ ´ { “ ´ { ˆ 0 ˙ a a 1 tˆvar as c (151) 1 W E~ D~ B~ H~ d3x „ { (154) ~ ~“ 8~⇡ p~ ¨ 3 ` ¨ q W E D B ª H d x (154) “ 8⇡ p ¨ ` e2 ¨ dr q ⇢ ⇢¯ ª ´ 1 (155) (152) 2 2 r2 ” ⇢ „ Op q STRUCTUREe dr ª OF THE DRAG 2 (155) 2 r 2 2 2 2 v 1 v c (153) FORCEª 4⇡G mb mEXPRESSIONp 2 1 1 1 “ ´ { F 2 dv v f v log ⇤ (156) “ v 0 p q 1 a 2 2 v ª ~ ~ ~ ~ 3 4⇡G mb mp 2 W E D B H d x (154) F dv1v1 f v1 log ⇤ “ 8⇡ p ¨ ` (156)¨ q 2 2 2 2 2 2 “ v 4⇡m G 1 pv q ⇢0 P0 2 ª Lowest Ffrequencies0 p ` —q largestp ` q log wavelengths:⇤ ✓ v c e dr ideal (157) “ ª v2 p ´ sq (155) liquid regime 2 r2 ª 2 2 2 2 2 2 2 4⇡m G Tµ1⌫ v⇢0 P⇢0 0uµu⌫P0 P0gµ⌫ F ⇢0 P20 G2 m v (158) F p “p` q` p q ` `q log ⇤ ùñ✓ v c9ps 4⇡`G mq b mp 2 (157) 2 1 1 1 “ v p ´F q 2 dv v f v log ⇤ (156) “ v 0 p q Already has dimensions of force. Namely the force is k-finite orª ∝ log k. Since 2 2 2 2 2 gravity is a long-range force, we would4⇡m expectG2 2 a 1divergencev ⇢0 log Pk.0 Tµ⌫ ⇢0 P0 uµu⌫ P0gµ⌫ F F⇢0 P0 G m p ` q p ` q(158)log ⇤ ✓ v c (157) “p ` q ` ùñ 9p10 “` q v2 p ´ sq Introduce first order corrections: viscous liquid. 2 2 Tµ⌫ ⇢0 P0 uµu⌫ P0gµ⌫ F ⇢0 P0 G m (158) ↵ “p ` q ` ùñ 9p ` q Tµ⌫ 2⌘ µ⌫ ⇣ ↵u µ⌫ (159) “´ ´ r 4 Both have one derivative — T 2⌘ ⇣ u↵ (159) viscositiesµ⌫ “´ areµ ⌫dim.´ 3r ↵ 4µ⌫ From dimensional analysis 10 F m2G2 ⌘ k (160) considerations: 9 s !11 F m2G2⇢ log k (161) 9 0 dF log k (162) d log k
T ↵ T ↵ ,Tµ⌫ contact terms (163) r s “ hµ⌫ ` @ D p m vp m v (164) ° p † p
10 ⌧ P 1 e´ (144) “ ´
! t 1 (145) „ „ Op q
RE RSD ⌧ (146) P„ 1 e´ (144) “a ´ c y (147) À! RSt! 1 (145) „ „ Op q ! t 1 (148) ⌧ P 1 e´ RE „ RSD (144) (146) “ ´ „ a ! t 1 ⇢ r´ c (145) (149) „ „ Op q y„ (147) À RS! 3 RE RSD ´(146)1 „ RE ´ a Reff R!E t 1 (150)(148) c 9 r0 y ˆ„ ˙ (147) À RS! ! t 1 tˆvar⇢ ars ´c (148) (151)(149) „ „„ {
3 ⇢ r´ ⇢ ⇢¯ (149) ´1 R ´ „ R ´R E1 (152)(150) 3 eff” ⇢ E„ Op q ´1 9 r RE ´ 0 Reff RE ˆ ˙(150) 9 r0 ˆ ˙ 1 v2 c2 (153) “tˆvar ´ as{ c (151) tˆvar as c „ { (151) „ { a 1 ⇢ ⇢¯ W ⇢ E~⇢¯ D~ B~ H~ d3x (154) ´ 1 “ 8⇡ ´p ¨ ` 1(152)¨ q (152) ” ⇢ „ Op q ” ª ⇢ „ Op q e2 dr (155) 1 v2 c2 2 r2 (153) “ ´ { ª 1 v2 c2 (153) a 1 3 “ ´ { W E~ 2 D~2 B~ H~ d x 2 2 v (154) “ 8⇡ F p m¨ G `⌘s k ¨ q 4⇡G mb mp a 2 (160) ª 9 F 1 dv1v1 f v1 log ⇤ 3 (156) e2 dr 2 ~ ~ ~ ~ “ W v 0 E D p Bq H d x (154) 2 8⇡ (155) 2 r 2 2 “ ª p ¨ ` ¨ q F m G ⇢0 log k ª (161) ª9 2 2 2 2 e 2 2 dr 2 2 v 4⇡m G 1 v ⇢ P 4⇡G mb mp 2 0 0 (155) F1 1 1 2 log ⇤ ✓ v c (157) F 2 dF dv v f v log ⇤ p `22 q pr ` q (156) s “ v 0 “logpk q v (162) p ´ q d logª k ª 2 2 2 2 2 2 2 v 4⇡m G 1 v ⇢0 P0 4⇡G m mp 2 2 2 F ↵ log ⇤ ✓ v c b (157) p `2 Tqµp ⌫T ` ⇢0q FP0 uµu⌫s P0gµ⌫ dv1v1 fF v1 ⇢0logP⇤0 G m (158)(156) “ T ↵ ,Tµv⌫ “pcontact` termsqp ´ q`2 ùñ (163)9p ` q STRUCTURE OF THE“ v DRAG0 p q r s “ hµ⌫ ` ª T ⇢ P @ u u PDg F ⇢ P G2m2 (158) µ⌫ “p 0 ` 0q µ ⌫ ` 0 µ⌫ ùñ 9p 0 ` 0q ↵ FORCE EXPRESSION 2Tµ2⌫ 2 2⌘ µ2⌫ 2 ⇣ ↵u µ⌫ (159) p mpvp 4m⇡pvm G 1 v ⇢0 P(164)0 ° † “´ ´ r 4 T 2⌘ F⇣ u↵ p `2 q p ` (159)q log ⇤ ✓ v cs (157) µ⌫ “´ µ⌫ ´ r“↵ 4µ⌫ v p ´ q The viscous fluid EFT breaks 2 2 2 3 The full theory2 2 should F m G ⌘sk a2k a3k ... F m G s k (165) (160) down at the scale of MFP: „ F p m2`G2 k` ` q give“ a finite result (160) 2 2 “Tµ⌫ s ⇢0 P0 uµu⌫ P0gµ⌫ F ⇢0 P0 G m (158) Another limit: very high frequencies, ideal gas.“p The` q ` ùñ 9p ` q 2 2 2 2 only parameter again is the energyF density,m G ⇢we0 log expect:k F m G ⇢0 log k (161) (161) 9 9 ↵ Tµ⌫ 2⌘ µ⌫ ⇣ ↵u µ⌫ (159) dF “´ ´ r 4 Viscous fluid is an EFT, where (162) we consider the first order d log k 10ideal gas 2 2 corrections to the low-k F m G s k (160) approximation. We also know 10 interacting gas,“ nonlinear, the high-k limit (ideal gas). EFT breaks down Naive interpolation between F m2G2⇢ log k (161) them: 9 0 dF log k (162) d log k ideal liquid 1st order correction, exponential rise (linear in k), viscous fluid EFT is valid !12 10
11 In these expression the coe cient cX (k) (which is obviously a function of k)standsforthe cross over regime that is calculated in the interacting kinetic theory. In fact, it represents n the resummation of all the terms of type ( skUV ) that clearly cannot be captured using merely the language of the EFT. We schematically illustrate this situation on Fig XXX. (Add the picture – ak) Of course, three di↵erent regimes presented in Eq. (20) should smoothly interpolate between one another. We will further analyze the interacting kinetic theory in the relaxation time approximation. While this is still an approximation in the vast majority of the systems we are interested in, it is su cient for most of our needs. With ⌧ being the relaxation time, we will be able to recover the viscid hydrodynamic regime by taking a limit ⌧!, ⌧ k 1. | | ⌧ By further setting to zero the viscosity we recover the very well known ideal liquid solution. In the opposite limit, ⌧!, ⌧ k 1 one of course recovers the free streaming regime. We | | will perform the explicit calculations in the next section.
III. CALCULATION OF THE DRAG FORCE
Now we apply the formalism that we have developed in the previous section to the actual problem of the dynamical friction. First, let us clarify the question, how do we compute the response function of the medium (7) that is further used in the calculation of the dressed graviton propagator (10). Note, that this function is known in certain limiting cases, e.g. itTHE has been calculated KINETIC for the massless THEORY gas in [13] and the viscid hydrodynamical case was analyzed in [12]. However, the generic case is not known and will be analyzed here (though partially numerically). To capture the effects of the interactions between the gas We will closely follow the prescription of [13] and will also work in the time relaxation particles, use relaxation time approximation. � is the time scale approximation. In thisof approximationthe interactions the Boltzmann equation for the distribution function is Boltzmann equation: p↵u pµ@ f ↵ p p (p)f = ↵ (f f ) . (21) µ r↵ ⌧ eq interaction kernel, with � is the The terminteractions on the right time hand scale side stands for the interactions between the particles of the
Idealmedium, liquid regime: and (k�)⌧ < for the distribution function feq, however we assumequantified that it is isotropic and depends only viscosities: ∽� on the energy of the particle p0, rather than on p~ and m separately, i.e. f = f(p0/T ). The Interacting gas — the “transitional regime”: (k�) ~ O(1) !13 10 ⌧ P 1 e´ (144) “ ´ ! t 1 (145) „ „ Op q R R D (146) E „ S a c y (147) À RS! ! t 1 (148) „ ⇢ r´ (149) „ 3 ´1 R ´ R R E (150) eff 9 E r ˆ 0 ˙ tˆ a c (151) var „ s{ ⇢ ⇢¯ ´ 1 (152) ” ⇢ „ Op q 1 v2 c2 (153) “ ´ { 1 a ↵ W E~ D~ B~ H~ d3x (154) Christo↵el symbols are calculated as a function of the metric perturbation hµ⌫.We“ 8⇡ p ¨ ` ¨ q ª 0 e2 dr have also introduced a notation ~vp p/p~ . Assuming a small correction f = f f0 due to (155) ⌘ MATTER EXCITATIONS 2 r2 the metric perturbation this equation can be formally solved in the Fourier space ª 2 2 v 4⇡G mb mp 2 AND KINETICF THEORYdv1v1 f v1 log ⇤ (156) ↵ (p) 2 1 feq/⌧ + p p ↵ f0 “ v 0 p q r ª f = 0 . (22) p ~ 2 2 2 2 2 i! + i~vp k +1Machinery:/⌧ how do we solve4⇡m theG kinetic1 v theory⇢0 P0 · F p ` q p ` q log ⇤ ✓ v cs (157) ↵ “ v2 p ´ q Christo↵0el symbols(technical) are calculated as a function of the metric perturbation hµ⌫.We Note that since the unperturbed distribution f0 depends only on p , one can calculate the T ⇢ P u u P g0 F ⇢ P G2m2 (158) haveBasic also assumptions: introducedµ⌫ “p a f notationeq0 is` isotropic,0q ~vµp ⌫ `p/p~ 0 µ.⌫ Assumingùñ 9p a small0 ` correction0q f = f f0 due to second term in the numerator either taking only the time derivative ↵ =0ortakingonly0 ⌘ the metricdepends perturbation only on this p equation can be formally solved in the Fourier space T 2⌘ ⇣ u↵ (159) the spatial derivatives. These two approaches are fully equivalent. Note that for feq weµ⌫ “´ µ⌫ ´ r↵ 4µ⌫ (p) p0 ↵ 1 feq/⌧ +2 2 p p ↵ f0 assume a locally thermal distribution feq = f0 T (~v ~u)+ T/T . Of course we expect this F m G k r (160) Formal· solution: f = 0 s . (22) p “i! + i~v ~k +1/⌧ assumption to break down for ⌧ , namely in the free streaming case. However since the p · !1 F m2G2⇢ log k (161) 1 Note that since the unperturbed distribution9 0 f depends only on p0, one can calculate the importance of this term scales as ⌧ , we areEnergy not sensitive momentum to this tensor assumption inMatter this limit. excitation (retarded0 correlator of dF second term in the numeratorenergy either momentum taking onlylog tensor):k the time derivative ↵ =0ortakingonly(162) When we know the perturbation f we can finally calculate the correlator d log k the spatial derivatives. These two approaches are fully equivalent. Note that for feq we T µ⌫ d3p pµp⌫ ↵ µ⌫↵ µ⌫ ↵ µ⌫ T p0 Gkin = with T = assume3 0 af locally thermal distributionT(23),T feq = f0 contact(~v ~u terms)+ T/T . Of course we expect(163) this h (2⇡) p r s “ hµ⌫ `T ↵ Z · assumption to break down@ for ⌧ D, namely in the free streaming case. However since the !1 where the dependence on h↵ comes from the Christo↵el symbols. Interestingly, the objectanalytic in10 k, importance of this term scales as ⌧ 1, we are not� sensitive to this assumption in this limit. !14 Gkin is not fully equivalent to the object that we have defined in the Eq. (7). Nonetheless, When we know the perturbation f we can finally calculate the correlator one can show that they merely di↵er by contact terms, i.e. terms that do not contain new T µ⌫ d3p pµp⌫ Gµ⌫↵ = with T µ⌫ = f (23) poles in the kz complex plane and therefore the contribution of these terms will vanishkin in theh (2⇡)3 p0 ↵ Z final answer. (Aleksi, make sure you agree with this explanation – ak) Finally, once where the dependence on h↵ comes from the Christo↵el symbols. Interestingly, the object we have the object Gkin, we can go ahead and construct the dressed graviton propagator. Gkin is not fully equivalent to the object that we have defined in the Eq. (7). Nonetheless, one can show that they merely di↵er by contact terms, i.e. terms that do not contain new A. Free streaming scenario poles in the kz complex plane and therefore the contribution of these terms will vanish in the final answer. (Aleksi, make sure you agree with this explanation – ak) Finally, once Let us first try to reproduce the results for the non-interacting gas in the non-relativistic we have the object Gkin, we can go ahead and construct the dressed graviton propagator. and ultra-relativistic cases. The force exerted on a body by a gas of non-interacting rela- tivistic particles using the results of [1] and [4] is: A. Free streaming scenario 4⇡G2m2⇢ F = b v log ⇤ . (24) v2 Let us first try to reproduce the results for the non-interacting gas in the non-relativistic (Translate to the relativistic language – ak) In this expressionand ultra-relativistic⇢ is the energy cases. density The force exerted on a body by a gas of non-interacting rela- tivistic particles using the results of [1] and [4] is: of a gas of non-relativistic particles, namely ⇢v = mn, where n is the number density of 4⇡G2m2⇢ particles whose velocity does not exceed the bullet velocity,. In the opposite regime of theF = b v log ⇤ . (24) v2 (Translate to the relativistic language – ak) In this expression ⇢ is the energy density 11 of a gas of non-relativistic particles, namely ⇢v = mn, where n is the number density of particles whose velocity does not exceed the bullet velocity,. In the opposite regime of the 11 ultrarelativistic gas the result is derived in [10] was 64⇡ F = G2m2⇢v 2 log ⇤ . (25) 3 b Note that the result in the intermediate case and the interpolation between the two limiting cases to the best of our knowledge does not appear in the literature and we present it here for the first time. As we will shortly see, our techniques allows us to get to an analytic expression that smoothly interpolates between two these limits. Note that this caseultrarelativistic is relatively simple compared gas the to result the full is case, derived since the in expression [10] was for the variation of the distribution function reduces to i (p) 64⇡ 1 p p f0 2 2 2 f = ri F = G mb ⇢v (26)log ⇤ . (25) p0 i! + i~v ~k 3 p · and one solves this equationNote that straightforwardly the result without in the f intermediateeq. With these simplifications case and mul- the interpolation between the two limiting tiple integrals that we perform here become manageable and can be performed analytically. cases to the best0 of our knowledge does not appear in the literature and we present it here Notice that with our assumptions f0(p /T ),wehave for the first time. As we will shortly see, our techniques allows us to get to an analytic (p) pi 0 f = f 0 (p ). (27) i 0 p0 0 WARMexpression UP: thatr smoothly IDEAL interpolates between GAS two these limits. The rest of the calculation is pretty straightforward. First, we are interested in the Note that this case is relatively simple compared to the full case, since the expression for dressed graviton propagator. In our formalism the integral expression for this object reads 3 2 i µ ⌫ ! ⇠ 0 d p the variation of the distribution !⇠ p p p p functionf 0 (p ) reduces to Gdressed = I I 0 u u u⇢u (28) 3 2 2 ,µ⌫ ↵ ,⇢ 0 2 ~ p~ (2⇡) (! + i✏) + k h⇢ (p ) T i! + ik 0 Z ✓ ◆ · p i (p) The formal solution is much 1 p p f0 where as usual p0 = p~2 + m2. One can analytically perform the angularf = integration in theri (26) 0 ~ simpler than in the generic case: p i! + i~vp k momentum space andp project onto the four-velocities of the bullet to get an expression of · the following form: and one solves this equation straightforwardly without feq. With these simplifications mul- 0 4 dressed f 0(p )p kzG dkz = kzdkz dp 2 2 2 4 0 2 2 2 2 2 (29) = tiple= integrals that240⇡ we(k perform+ kz ) (p ) ( herek k becomez ( 1+v )) manageable⇥ and can be performed analytically. Z Z Z ? ? 0 2 2 0 0 kzp v +0 kz + k p NoticeR(k thatz,k ; p, with p ; v + ourS(k assumptionsz,k ; p, p ; v)log f0(p /T ),wehave? ? ? 0 2 2 kzp v pkz + k p! ? after we have substituted ! = k v. Functions R and S are fully analytic( inp) allp theirp variables.i 0 z 0 i f0 = 0 f0(p ). (27) twoBefore completely we proceed further, analytic we should be bit more careful with the analyticr structurep of our contributes to the branch integrals. A priori our integration contour goes slightly above the real axis. However, since functions of Thekz rest of the calculation is pretty straightforward. First, we are interested in the we are merely interested in the imaginary part of this expression,cut wealong can subtract the from real this axis integral its complexdressed conjugate, gravitonnamely allow propagator. the contour simply In to our circumvent formalism clockwise the the integral expression for this object reads 2 i For sufficiently big p > ɣmpv dthe3p log is always negative and the pµp⌫p!p⇠ f (p0) Gdressed = I I !⇠ 00 u u u⇢u (28) 12 3 2 2 ,µ⌫ ↵ ,⇢ 0 2 p~ entire imaginary part comes(2 from⇡) the(! jump+ i✏) +overk the branch cuth⇢ (p ) T i! + i~k Z ✓ ◆ · p0 !15 where as usual p0 = p~2 + m2. One can analytically perform the angular integration in the momentum space andp project onto the four-velocities of the bullet to get an expression of the following form: 0 4 dressed f 0(p )p kzG dkz = kzdkz dp 2 2 2 4 0 2 2 2 2 2 (29) = = 240⇡ (k + kz ) (p ) (k kz ( 1+v )) ⇥ Z Z Z ? ? 0 2 2 0 0 kzp v + kz + k p R(kz,k ; p, p ; v + S(kz,k ; p, p ; v)log ? ? ? 0 2 2 kzp v pkz + k p! ? after we have substituted ! = kzv. Functions R and S are fully analytic in allp their variables. Before we proceed further, we should be bit more careful with the analytic structure of our integrals. A priori our integration contour goes slightly above the real axis. However, since we are merely interested in the imaginary part of this expression, we can subtract from this integral its complex conjugate, namely allow the contour simply to circumvent clockwise the 12 ⌧ P 1 e´ (144) “ ´ ! t 1 (145) „ „ Op q R R D (146) E „ S a c (147) y ⌧ À RS! P 1 e´ (144) “ ´ ! t 1 ! t 1 (148) (145) „ „ „ Op q ⇢ r´ RE RSD (149) (146) „ „ 3 a c ´1 y (147) RE ´ À RS! Reff RE (150) 9 r0 ! t 1 (148) ˆ ˙ „ tˆ a c (151) var s ⇢ r´ (149) „ { „ 3 ⇢ ⇢¯ ´1 R ´ ´ 1 R R E (152) (150) ” ⇢ „ Op q eff 9 E r ˆ 0 ˙ 2 2 1 v c tˆvar as c (153) (151) “ ´ { „ { ⇢ ⇢¯ 1 a ´ 1 (152) W E~ D~ B~ H~ d3x ” ⇢ „ Op q (154) “ 8⇡ p ¨ ` ¨ q ª e2 dr 1 v2 c2 (153) “ ´ { (155) 2 r2 1 a ª W E~ D~ B~ H~ d3x (154) “ 8⇡ p ¨ ` ¨ q 2 2 v ª 4⇡G mb mp 2 e2 dr F dv1v1 f v1 log ⇤ (156) (155) “ v2 p q 2 r2 ª0 ª 2 2 v the spatial derivatives. These two approaches are fully equivalent. Note that for f we 2 2 2 2 2 4⇡G mb mp 2 eq 4⇡m G 1 v ⇢0 P0 F dv1v1 f v1 log ⇤ p0 (156) 2 assume a locally thermal distribution feq = f0 (~v ~u)+ T/T . Of course we expect this F p `2 q p ` q log ⇤ ✓ v cs“ v 0 p (157)q T · “ v p ´ q ª assumption to break down for ⌧ , namely in the free streaming case. However since the 2 2 2 2 2 !1 1 ⌧ 4⇡m G 1 v ⇢importance0 P0 of this term scales asP ⌧ 1, wee´ are not sensitive to this assumption in(144) this limit. 2 2 “ ´ Tµ⌫ ⇢0 P0 uµu⌫ P0gµ⌫ F ⇢0F P0 G m p `2 q p ` q log(158)⇤ ✓ v cs 2 2 (157) “ v When we know thep perturbation´ qF mf weG can⌘s finallyk calculate the correlator (160) “p ` q ` ùñ 9p ` q 9 T µ⌫ ! t 1 d3p pµp⌫ (145) Gµ⌫↵ = 2 with2 O T µ⌫ = f (23) T ⇢ P u u P g F ⇢ kin P G„m „ p q 3 (158)0 ↵ µ⌫ 0 0 µ ⌫ 0 µ⌫ 0 0 h↵ (2⇡) p Tµ⌫ 2⌘ µ⌫ IDEAL⇣ ↵u µ⌫ GAS:“p ` qANALYTIC` ùñ (159) 9pSTRUCTURE` q 2 2 Z F m G ⇢0 log k (161) “´ ´ r 4 where the dependence on h↵ comesR9E fromRSD the Christo↵el symbols. Interestingly,(146) the object ↵ „ OF THE DRESSED Tµ⌫ 2⌘ µ⌫ G ⇣PROPAGATORis↵ notu fullyµ equivalent⌫ to the object that we have defined(159) in the Eq. (7). Nonetheless, 2 2 “´ ´kinr 4 a F m G k (160) dF c s one can show that they merely di ↵yer by contactlog k terms, i.e. terms that do not contain(147) new (162) “ 2 2 À RS! F m Gpoles ink the k complex plane andd log thereforek the contribution of these(160) terms will vanish in the “ s z 2 2 final answer. (Aleksi, make sure! yout agree1 with this explanation – ak) Finally,(148) once F m G ⇢0 log k Im Im (161) „↵ kz 2 2 we have the object G , wekz can go ahead T and construct the dressed graviton propagator. 9 F m G ⇢0 log k T ↵ kin,Tµ⌫ contact terms(161) (163) 9 r s “⇢ hr´µ⌫ ` (149) dF dF „ log k A. Free(162) streaming@ scenarioD ◊ log ◊k 3 (162) d log k ´1 d log k RE ´ p Reff mpRvpE mpv (150) (164) Let us first try to reproduce° the9 resultsr0 for† the non-interacting gas in the non-relativistic ↵ Re ↵ Re ˆ ˙ T ↵ µ⌫ T and ultra-relativistic cases. The force exerted on a body by a gas of non-interacting rela- ↵ µFIG.⌫ 2: The dynamical friction of the free streaming gas. TheT figure,T on the left hand side showscontact terms (163) T ,T contact terms (163) 2 2 tˆvar as c 2 3 (151) r s “ hµ⌫ tivistic` particles usingF them resultsG ⌘ ofk [1]„ withouta and{k [4] is:a kany ... (165) r thes drag“ force ashµ a⌫ function` of the perturber velocity given the mass of the gas particle in units „ p s ` 2 ` 3 ` q @ D 4dissipation:⇡G2m2⇢ of temperature, while the figure on the right hand side shows the force as a function of the gas F⇢ =⇢¯ b v log ⇤ . (24) @ D ◊ p mpvp ◊mpv ´ 21 (164) (152) ” ⇢ „ Ovp q particles mass species for a given perturber mass. The inset figure on the° left hand side plot shows† dressed dressed p mpvp mpv (164)G ! i✏ G˚ ! i✏ (166) (Translate to the relativistic language – ak) In this expression ⇢ is the energy density °the same e↵ect on the linear† (rather than log) scale of the perturber velocities. p ` q“ p ´ q 1 v2 c2 (153) 10of a gas of non-relativistic particles,“ namely´ { ⇢v = mn, where n is the number density of a but is rather located between the points: particles whose velocity does not exceed the bullet velocity,. In the opposite regime of the 10 FIG. 3: Analytic structure of integrand in (29) as function of kz. TheCompare figure on the rightto the hand known1 results W E~ D~ B~ H~ d3x (154) ultrarelativistic gas the result“ is8 derived⇡ p ¨ in [10]` was¨ q sidek showsp the structure for high-momentak p p> mgv, where the branch cut (the blue wavy line) ª ? 15 11 IDEAL GAS: GENERIC SOLUTION Because of the analytic structure of the kz integral, the generic solution breaks down into two integrals over p, one of them running from p=0 to p = ɣmpv, and another running from p = ɣmpv to infinity. The integrands are continuous at the stitching point. Free streaming Free streaming 1 1 10 100 10 2 100 10 2 3 1 10 10 10 4 10 ? ? 2 k k 10 4 5 10 10 log log 3 6 10 d d 10 4 0.0 0.2 0.4 0.6 0.8 4 dF dF T 7 4 T 2 10 10 2 2 b 2 b 8 5 10 m 10 m =0 m v =0.9 9 10 6 10 m = T v =0.5 10 m =5T 10 v =0.1 7 10 11 m =8T 10 v =0.01 8 12 10 10 5 4 3 2 1 0 0 1 10 10 10 10 10 10 10 10 v/c mg/T The known limits (Chandrasekhar+photons gas) are easily reproduced from the general expression !17 real axis (of course, up to a factor of 2). With this deformation it is clear that R is analytic in the full complex plane and does not contribute to the integral. The function S however 22 22 F F mmGG ⌘sskk (160) (160) 99 multiplies a log, which contains a cut on the real axis. 2 2 F m G ⇢0 logTherek are two distinct structures(161) that the integral over kz can have depending on the F 9m2G2⇢ logINVISCIDk FLUID(161) 9 0 dF parameter p. For su ciently large p, namely log k (162) dFd log k log k (162) d log k p> mgv (30) Here the T ↵ energy momentum tensor and the propagators are T ↵ ,Tµ⌫ contact terms (163) r s “ hµ⌫ ` well known. Dressed propagator: @ D T ↵ (here m stands for the mass of the particle in the gas), the denominator of the log is always ↵ µ⌫ g 2 T ,T contact terms 2 (163)ideal ~ r ps “ m pvph ` mpv (164) f (k), ° Gdressedµ⌫ (!†= vkz + i✏)= sound . (38) negative, meaning that2 2 the~ 2 branch2 cut2 2 goes2~ 2 extends over the entire real axis. In this case @ D (1 v )kz + k ! (cs v )kz + csk i✏vkz ? ? 2 2 2 3 Fwithm G ⌘sk a2kthea3 contributionk ... due to the jump(165) over the branch cut is 2⇡i, and the rest (the S functions p„ mppvp` ` mp`v q (164) ° † 2 2 2 sound poles: on the imaginary axis for subsonic allideal the gravitational⇢0 + P0 16kz v poles! are on 2 2 2 2 2 2 f = and the prefactor)+ !(8kzv +( isv simply3)!) integratedk (v + 1) c from(v 3) minusv to1 plus(39) infinity. The final result reads dressedsound dressed 2 velocities, real otherwise.s The virtual phonons G ! i✏ 2G˚ " k! i✏ (166) # p ` q“imaginaryp axis´ q ⇣ ⇣ ⌘⌘ 2 2 2 3 4 2 become real F m G ⌘sk a2k a3k ... 1(165) p df „ p ` ` k`Gdressedq dk = dp 0 (31) k 2 cskz I Asz in the ideal2 2 gas case, without0 0 dissipation kz Becausei K (1 vk)z=> 0, itK is simple to see that96 the⇡k graviton(167)v propagatorsp dp will⇥ never go on “˘ ?1 v2 “ 2 2 mgv Z cs v there is no? contributionZ from the gravity dressedshell and´ the propagatordressed is´ an analytic function of kz for all v. Same is true for the sound G ! i✏ G˚ ! i✏ 4 (166)2 0 4 2 2 0 2 2 p ` q“ ap ´ q 2v 3p ( 5+3polesv )+(p ) ( 27 + 29v ) 6p (p ) ( 7+9v )+ mode as long as v csk p< mgv contributionkz = as? the, integrand is continuous.(40) Note that in the ultra-relativistic 2 2 ± v cs regime the branch cut always extends over the entire real axis and therefore the entire leading in to an imaginary part arising fromp the i✏ prescription contribution to the dynamical friction is given by (31) with the integration running from 1 1 2 2 2 2~ 2 = + i⇡ (cs v )kz + csk , (41) 2 2 2 112~ 2 P 2 2 2 2~ 2 ? (cs v )kz + cszerok toi✏vk infinity.z (cs v )kz + csk )! ? ? ⇣ ⌘ where denotes the real principal value integral. Inserting (41) to (38) and further to (59) P However, in the generic case this is not the only contribution. The contribution of the we get a simple form for the drag force integral over low-p is negligible in the case of ultra-relativistic particles, however becomes dF z (1 + v2)2(P + ⇢ ) = ✓(v c )m22 2 0 0 (42) dominantd log k in the non-relativistic s limit.32⇡v2 For small momenta, namely ? in agreement with [11], if we take into account that =8⇡G. That is, we see the energy loss arises from transferring energy fromp< the m bulletgv to the long- (33) lived asymptotic states—which in the case of fluid dynamics are the sound modes—that can 11 transport the energythe far branch away from cut is the located bullet. The on thecondition real (40) axis, equivalently however it reads does not extend all the way to infinity, cos ✓ = k / k = c /v (43) z | | s 13 17 F m2G2 ⌘ k (160) 9 s F m2G2⇢ log k (161) 9 0 dF log k (162) d log k MORE REALISTIC T ↵ PICTURE: T ↵ ,Tµ⌫ contact terms (163) r s “ hµ⌫ ` ADD@ THED VISCOSITY p mpvp mpv (164) In the system with dissipations° †the calculation is more involved and we cannot disregard the analytic structure F m2G2 ⌘ k a k2 a k3 ... (165) outside„ p sthe` real2 ` axis.3 ` q dressed dressed G ! i✏ G˚ ! i✏ (166) p ` q“ p ´ q The viscosity η=ζ=0 1/4 contributions blow up at ρ0 η= (ρ0+P0),ζ=0 4 π 1/4 ρ0 ζ= (ρ0+P0),η=0 ⟂ 4 π the speed of sound. The z dk dF ⟂ k results are also linearly 2 1 γ sensitive to the cutoff of the EFT Perturber's velocity v !19 11 That is, the viscous correction is regular in the infrared as one could have expected. However, the correction to the force is seen to be linearly UV divergent. As shortest wavelengths receive largest corrections, and the leading order result depended without a bound on all frequencies, it is no surprise that the viscous correction comes with a linear UV divergence. This UV divergence turns the logarithmic dependence of the UV-cuto↵ kUV to a linear one. The corresponding length scale x 1/k is the shortest distance scale at which the That is, the viscousUV ⇠ correctionUV is regular in the infrared as one could have expected. However, approximations of our calculationthe correction are to the valid, force is and seen we to be see linearly that UV in a divergent. system As with shortest non-zero wavelengths receive largest corrections, and the leading order result depended without a bound on all viscosity, i.e., in all physical materials, the viscous correction is sensitive to the microscopic frequencies, it is no surprise that the viscous correction comes with a linear UV divergence. physics, as explained in Sec II. In order to know the behavior of the theory at distances below This UV divergence turns the logarithmic dependence of the UV-cuto↵ kUV to a linear one. xUV weThat approximate is, the viscous correction theThe full corresponding is theory regular in by the length a infrared kinetic scale as theory onexUV could1 in/k haveUV relaxation expected.is the shortest However, time distance approximations. scale at which the ⇠ The viscousthe correction fluid theory to theapproximations force will is emerge seen to be of(as our linearly expected) calculation UV divergent. as are a valid, low As energy and shortest we see limit wavelengths that of in this a system theory with with non-zero receive largest corrections,viscosity, andi.e. the, leading in all physical order result materials, depended the viscous without correction a bound ison sensitive all to the microscopic bulk shear viscosity as functions of mg and relaxation time ⌧. frequencies, it is no surprisephysics, that as explainedthe viscous in correction Sec II. In comes order to with know a linear the behavior UV divergence. of the theory at distances below This UV divergence turns the logarithmic dependence of the UV-cuto↵ k to a linear one. xUV we approximate the full theory by a kineticUV theory in relaxation time approximations. The corresponding length scale x 1/k is the shortest distance scale at which the C. The interactingThe gas viscous fluidUV ⇠ theoryUV will emerge (as expected) as a low energy limit of this theory with approximations of our calculation are valid, and we see that in a system with non-zero bulk shear viscosity as functions of mg and relaxation time ⌧. viscosity, i.e., in all physical materials, the viscous correction is sensitive to the microscopic Now we have generic results in the regimes of the non-interacting gas and viscous fluid, physics, as explained in Sec II. In order to know the behavior of the theory at distances below C. The interacting gas however x theseUV we results approximate looks the vastly full theory di↵ byerent a kinetic and theory one in would relaxation like time to approximations. have theory that smoothly interpolatesThe viscous between fluid theory two these willNow emerge we regimes, have (as generic expected) yielding results as a them in low the energy regimes as the limit of limiting of the this non-interacting theory cases. with gas and viscous fluid, bulk shear viscosity ashowever functions these of m resultsg and relaxation looks vastly time di↵⌧erent. and one would like to have theory that smoothly With the theoretical machinery that we have described in Sec II and assuming the fluc- interpolates between two these regimes, yielding them as the limiting cases. tuations in the equilibrium distribution function C. The interactingSOLVEWith gas the theoretical THE machinery FULL that we have KINETIC described in Sec II and assuming the fluc- ↵ tuations inChristo the equilibrium↵el symbols0 distribution are calculated function as a function of the metric perturbation hµ⌫.We 0 p T Now we have generic results f in= thef regimes(p ) of the~v non-interacting ~u + gas0 and viscous fluid, (50) eqhave also0 introducedTHEORYp a notation ~vp 0 p/p~ . Assuming a small correction f = f f0 due to T · 0 Tp⌘ ~ T however these results looks vastly di↵erent and one✓ would feq = likef0(p to) have◆~v theoryp u + that smoothly (50) the metric perturbation this equationT can· be formallyT solved in the Fourier space interpolates between two these regimes, yielding them as the limiting✓ cases. ◆ we get the following expression for the fluctuating energy-momentum↵ tensor (p) in the Fourier we get the following expression for the fluctuating1 feq/⌧ + energy-momentum p p ↵ f0 tensor in the Fourier With the theoretical machinery that we have described in f Sec= II and assuming ther fluc-. (22) Formal solution: 0 ~ space: p i! + i~vp k +1/⌧ tuations in the equilibriumspace: distribution function · 0 Note that since the unperturbed distribution f0 depends only on p , one can calculate the 0 0 Here~ closely following Romatschke, i(p) ↵ (p) 2p 0 ~ T f0(p ) ~vp u + T/Ti ↵ T ⌧ ↵ p p i f0 2 µ⌫ fp0 0dp(p ) d~v'~pd cos u✓+ T/T T ⌧ p p f0 p dp d' Tdfeqcossecond= ✓f0(p term) in~vp the u numerator+ either taking· only1512.02641 the↵ time (50) derivativei r↵ =0ortakingonly(51) µ⌫ 2⇡2TT · 4·⇡ T ⇣ ⌘ r~ T = 2 ✓ ◆" 1+⌧(ı! + i~vp k) (51)# theZ spatial derivatives.Z ⇣ These two approaches⌘ ~ are fully equivalent. Note that for feq we 2⇡ T 4⇡ " 1+⌧(ı! + i~vp k) · # we get the following expression for the fluctuating energy-momentum tensor in0 the Fourier Z Z · p µ⌫ again, closelyassume following a locally the thermal derivation distribution of [13]. Wefeq = canf0 interpretT (~v ~u)+ the T/T upper. Of row course of T we expectfrom this space: We have 4 unknown variables: �T, and three components· of µ⌫ again, closely following theEq. (51) derivation asassumption an equation of to[13]. break on WeT downand can for ~u. interpret⌧ In particular,, namely the we in upper the identify free streamingrow of case.T Howeverfrom since the u. Plugging this back into the expression!1 for energy � 0 1i ↵ (p) 2 importancef (p ) of~v this ~u term+ T/T scales asT⌧⌧ , wep arep notf sensitive to this assumption in this limit. Eq. (51) as an equationp dp ond 'Td cosand✓ ~0u. Inp particular, we identify↵ @⇢i 0 T µ⌫ = momentum tensor we ·have 400 different equationsr to solve:(51) 2 When we⇣ ⇢ know= the T perturbation⌘with ~ ⇢f we= can finallyT calculate the correlator (52) 2⇡ T 4⇡ " 1+⌧(ı! + i~vp k) @T # Z Z 0i · 3 i T @⇢ µ⌫ Use the unperturbedd p 3f 0to µ ⌫ 00 u µ⌫↵ T ⇢ fµ⌫ pdpp pp p again, closely following ⇢ the= derivation T ofwith [13]. We= ⇢ can= interpret withT the upper= row ofµ⌫ T3 0(from) ( ) (52) (53) ⇢G+kinP = calculatewith the T(2 energy⇡=) density3 0 f (23) @T h↵ (2⇡) p ~ Z Z Eq. (51) as an equation on T and0i u. In particular, we!20 identify3 i T d p 0 u = where thewith dependence⇢ on= h↵ comes fromf (p the)p Christo(p) ↵el symbols. Interestingly,(53) the object @⇢ 3 0 ⇢ = ⇢ T+00GP withis not fully ⇢ = equivalent T to(2 the⇡ object)20 that we have defined(52) in the Eq. (7). Nonetheless, kin @T Z T 0onei can show that theyd merely3p di↵er by contact terms, i.e. terms that do not contain new ui = with ⇢ = f (p)p0(p) (53) ⇢ + P (2⇡)3 0 poles in the kz complexZ plane and therefore the contribution of these terms will vanish in the final answer. (Aleksi,20 make sure you agree with this explanation – ak) Finally, once we have the20 object Gkin, we can go ahead and construct the dressed graviton propagator. A. Free streaming scenario Let us first try to reproduce the results for the non-interacting gas in the non-relativistic and ultra-relativistic cases. The force exerted on a body by a gas of non-interacting rela- tivistic particles using the results of [1] and [4] is: 4⇡G2m2⇢ F = b v log ⇤ . (24) v2 (Translate to the relativistic language – ak) In this expression ⇢ is the energy density of a gas of non-relativistic particles, namely ⇢v = mn, where n is the number density of particles whose velocity does not exceed the bullet velocity,. In the opposite regime of the 11 RESULTS FROM THE FULL KINETIC THEORY Smooth interpolation: 0.04 m≠0 0.10 m=0 0.03 0.08 Numerical result Numerical result ⟂ 0.06 subsonic z z 0.02 dF dF Hydro limit dlogk dlogk 0.04 Viscous fluid Kinetic theory limit 0.01 0.02 Ideal fluid Viscous fluid Kinetic theory limit 0.00 0.00 -4 10-4 0.01 1 100 10 0.01 1 100 k k ⟂ 1.50 1.5 1.45 m≠0 m=0 supersonic1.40 1.4 1.35 Numerical result Numerical result ⟂ z z 1.3 1.30 dF Viscous fluid dF dlogk dlogk Viscous fluid 1.25 1.2 Ideal fluid Ideal fluid 1.20 Kinetic theory limit Kinetic theory limit 1.15 1.1 1.10 0.001 0.010 0.100 1 10 100 1000 0.001 0.010 0.100 1 10 100 1000 k k !21 ⟂ MAPPING THE HYDRO EFT ONTO THE FULL THEORY The viscosities can be extracted from Kubo relations: Shear viscosity Viscosities for the Boltzmann 0.0200 0.0175 = 1/T, and 0.0150 � distribution. In the conformal 0.0125 3 0.0100 linear in ⌘ /T � limit T /5. 0.0075 �→ � 0.0050 0.0025 0.0000 1 0 1 10 10 10 m/T Bulk viscosity Bulk viscosity Bulk viscosity 0.06 1 0.25 m = T ⌧ = T 1 m =2T ⌧ =2T 0.05 mapped onto m 1 0.20 m =0.7 T ⌧ =0.4 T 0.04 3 3 0.15 0.03 ⇣ /T and maximized ⇣ /T � 0.10 0.02 0.05 0.01 at m~ T 0.00 0.00 0 1 100 101 10 10 ⌧T m/T F m2G2 ⌘ k (160) 9 s F m2G2⇢ log k (161) 9 0 dF log k (162) d log k T ↵ T ↵ ,Tµ⌫ contact terms (163) r s “ hµ⌫ ` @ D IV. COMMENTS ON BOUNDARY EFFECTS p m vp m v (164) ° p † p COMMENTSIn the previous section we considered ON a stationary “OSTRKIER’S situation where the medium extends over an infinite2 2 volume and2 the bullet3 has traveled in the medium for an infinite time. F m G ⌘CORRECTIONS”sk a2k a3k ... (165) „ p ` ` ` q While these are exactly the assumptions of the classical dynamical friction–10– problem, some variations are known. While the explicit calculations in more intricate situation is somewhat dressed dressed G ! i✏ G˚ ! i✏ (166) In contrastbeyond thep scope with` ofq“ our this our results work,p we´ and willq showthe how one can attack non-trivial boundary resultsconditions of in Rephaeli-Salpeter space and time with our formalism. the Of course, it would be interesting to see k csk explicitkz calculationsi K along thesekz line. K (167) force does not vanish2 identically2 2 in “˘ ?1 v “ cs v We firstthe follow subsonic´ the setup regime set by Ostriker ´ in [7], where the medium still has an infinite extent but the bullet appears ina the unperturbed medium at time t =0.Thisframework 4⇡G2m2⇢ 1 was originallyF consideredb 0 log to resolve` M the “paradox” of the subsonic bullet moving(168) in the ideal “ v2 1 liquid, and according to theˆ previous´ M calculation˙ does not feel any force at all (of course, as Finitewe have (but shown, might this is not be the numerically case in the viscous fluid). important),In order to boundary analyze this frame, effect one in follows time exactly the same steps we went through in Fig. 3.— Solid curves: Dynamical friction force in a gaseous medium as a function of Mach number = V/c .Curvescorrespondtoln(c t/r )=4, 6, 8,...,16. Dashed curves: M speeds ofs soundmin Sec II, however the energy-momentum tensor of the bulletCorresponding looks dynamical di friction↵erent force in a now: collisionless medium with particle velocity dispersion σ = c and r Vt= c t. s max ≡ M s µ⌫ 2 µ ⌫ And proceed Way to reproduce Tbullet(~x, t)= m (z vt) (~x )v v ✓(t)(56) ? ⇥ with a similar ivµv⌫ in our formalism: T µ⌫ (~k, !)= m calculation(57) bullet ! + kzv i✏ The (derivative of) the gravitational field!23 of the wake then reads 4 µ ⌫ wake 2 ↵ d k it(! ikzv) dressed iv v @⇢hµ⌫ (x =0,z = vt > 0) = i mv v e k⇢Gµ⌫,↵ (k) , ? (2⇡)4 ! + kzv i✏ Z (58) and the force 2 2 2 z dressed 2 z ˜dressed z m d k dk z G 2 2 d k dk z G F = ? k + m ? (! k v) 2 (2⇡)2 2⇡ ! + kzv i✏ (2⇡)2 2⇡ ! + kzv i✏ Z Z (59) with G˜dressed = Gdressedv˜µv↵v , v˜µ = 1, 0, 0, v (60) zµ↵ { } 11 This is pretty di↵erent from what one gets in (17), and moreover one cannot write it down as an imaginary part of a closed expression. In principle, after evaluated, these integrals should yield a relativistic generalization of results of [7], that we leave for future studies. 22 CONCLUSIONS Proposed a completely generic method to calculate the classical dynamical friction in any regime Showed a completely generic analytic expression for the ideal gas in any regime of temperatures/ velocities Developed the expression for viscous fluids within the EFT that make important contributions in the subsonic regime Showed the smooth interpolations between the low and high-k regimes within the full kinetic theory These improved results can now be used in plethora of physical problems: interacting DM, propagating of the BHs through the compact objects, physics of the accretion discs etc. Extension of the existing results to include the boundary effects, post-Newtonian corrections is straightforward (though laborious) using our formalism !24