Gravitational interaction of the cosmic stringwith pointlike particles
Pavel Spirin
Moscow State U.

20th Lomonosov Conference
August 19-25, 2021, Moscow, RF

Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Table of contents• P.Spirin, Universe 6, (2020) no.10, 184• P.Spirin, Universe 7, (2021) no.7, 206Content:
Cosmic string: metric
Gravitational interaction
Particle’s dynamics
Cosmic String’s excitations
Transversal string’s perturbations
p- Longitudunal string’s perturbations
2nd order of PT: radiation
Restrictions
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Cosmic string’s geometry
Metric (cylindric coords):
ds2 = dt2 − dx2 − d%2 − β 2 %2 dϕ2 ,Geometry:
R = 2(1 − β)δ+ (%)/%,

0<β<1

δϕ = 2π(1 − β)Phase transition energy scale:1 − β22η =8πG
16For η = η GUT ∼ 10 GeV one has
1 − β ∼ 10−5d ∼ 10−29 cmComplement:
β0 ≡ 1 − β =δϕ2πPavel Spirin

β 0 = 4Gµ

Gravitational interaction of the cosmic string with pointlike parti

Cosmic string’s geometry

Conformally Euclidean (x = const) coords: % → r
β% = R0 (r/R0 ) β ,Metric (conformal coords):
ds2 = dt2 − dx2 − e−2(1−β) ln(r/R0 ) (dy 2 + dz 2 ) ,where r2 = y 2 + z 2 .

Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Gravitational interaction
Action: ZZ ppµ1
S =−Xaµ Xbν gµν γ ab γ̃ d2 σ − 2 R |g| d4 x−2Z κ21m−e gµν Ż µ Ż ν +ds2eEinstein equations:
R1Rµν − g µν = κ 2 T µν + T̄ µν (+Tscµν )22SourcesZp 2γ̃ d σ ,T µν = µ Xaµ Xbν γ ab δ 4 x − X(σ)
Z µ ν 4ŻŻ
δx−Z(s)T̄ µν = e1/2 p dsgλρ Ż λ Ż ρ|g|Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Perturbation Theory
Zeroth order: particle
0Z µ (s) = uµ s+bµ ,uµ = γ(1, 0, 0, v),bµ = (0, 0, b, 0)

String:0X µ = δaµ σ a00γab = ηabT µν = µ δaµ δbν η ab δ(y) δ(z) = µ δ(y) δ(z) diag(1, −1, 0, 0)First order: linearized fields
−1/2κ m1uµ uν − ηµν γ 2 (z − vt)2 + x2 + y 2h̄µν (x) = −4π222 0,1(2π) κµδ (q )
κµr
Σh(x)
=
Σln,hµν (q) = −µνµνµνδαβ q α q β2πR0
where Σµν ≡ diag (0, 0, 1, 1)
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Particle’s Dynamics
Integration of EoM with IC 1 Ż µ (s = 0) = 01Ż 0 = 1 Ż x = 01Ż y = −4Gµγv arctg

γvsbb2 + γ 2 v 2 s2Ż = 2Gµγv lnb2
Identification of R0 :1z

R0 = bScattering angle
αsc ' 4πGµ = πβ 0Trajectoryz bb2 + z 2y(z) = b − β 0 z arctg − lnb 2b2
Pavel SpirinGravitational interaction of the cosmic string with pointlike parti

Cosmic String’s excitations
String’s EoM∂2∂2 1 µ1− 2 X = κΠµν δaλ δbρ η abh̄λρ,ν − h̄νλ,ρ
2∂t∂x2y=z=0Gauge 1X a = 0 (a = t, x) : only the transverse
deflections survive!
22 Φα (t, x) = j α (t, x)Formal retarded solution (α = y, z)Z1e−iqx j α (ω, q)αΦ (t, x) = −dω dq(2π)2ω 2 − q 2 + 2iω
Sourcespαj (ω, q) ∼ exp − b q 2 + (ω/γv)2
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Formal retarded solution
Total solutions:z,yΦz,y = Φz,ycut + Φres
Partial solutions:zΦzcut = −Λsgn t IcutzΦzres = 2Λθ(t)IresyΦycut = ΛvIcutyΦyres = 2Λv θ(t)Ires

Amplitudes:√√ R∞ R∞
2 −q 2−u|t|Jb(u/γv)
D−5[3]uecosqx
2/πz,y2Icut≡ D−5 dq duD−5[3]
2 + q2−2b 2 0u(γ v −2 u2 −q 2 )− 4qγv√∞
D−52/π Rcos qtz,y2 K D−5[3] bq/v
dqcosqxqIres ≡D−5sin qt2(bv) 2 0
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Seeking the solutions
Transverse amplitudes:1ht2 − x2 − b2 /v 2 π iyIres =+arctg,(t > 0)22bt/v
21v 2 t2 −v 2 x2 −(2v 2 −1)b2t2 −x2 −b2 /v 2yIcut = arctg √ 2 2 2 2 2 − arctg 2b|t|/v
2bv v t +(b +x )/γ2Both z-amplitudes diverge logarithmically!
Dimensional regularization: D = 4 + 2 ( > 0)v [v 2 (t + x)2 + b2 ][v 2 (t − x)2 + b2 ]zren Ires = − ln444rres√Cut-amplitude aftertheq-integration:(u=qγvũ2 + 1)∞
Γ()zreg Icut=2πXZ±2 hp
2 2 2 i−µ−2cut ũ
2dũ 2γv|t| ũ +1±ix −b ũũ + v −20Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Checking the solutions
Finally:p
22 2 22 2 222v
bx+γv
t+b+γv|t|γ 2 v 2 t2 + b2 + x2v
z
ren Icut = − ln2 (γ 2 v 2 t2 + b2 )2rcut

Properties of dalembertians:yz
22 Ires= 22 Ires=0Continuity requirements:
bvyyI˙cut(0+ , x) = −I˙cut(0− , x) = − 2 2v x + b2zzIres(0, x) = Icut
(0, x)
Length-scale factors relation:√rcut = 2rres ≡ r0
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Summarizing the renormalized solutions
The constructed solutions Φy and Φz satisfy:
continuity with respect to t and x;
satisfy the initial EoM;
reflect the causal structure;vanish in the limit v → 0;
their value has characteristic length parameter
2Gmγ = rE ;
Φz depends logarithmically upon the single scale
factor r0 .

Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Totally transverse string’s perurbation

Figure: Transverse string’s perurbation Φy (in units rE = 1) as a functionof time (normalized by b) at x/b = 0 (blue solid), x/b = 5 (greendashed), x/b = 10 (black dotted), x/b = 20 (red dashdotted) forv = 0.5

px2 + b2 ≡ tx

Relaxation: t = 2tx
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Totally transverse string’s perurbation

Figure: The profile of the transverse string’s perurbation Φy (in unitsrE = 1) at t/b = −5 (blue solid), t/b = 0 (green dashed), x/b = 10(black dotted), x/b = 50 (red dashdotted) for v = 0.5x1/2 (t) ≡pt2 − b2

V =Pavel Spirin∂x1/2t=√∂tt2 − b2

Gravitational interaction of the cosmic string with pointlike partip-Longitudinal string’s perurbation
From | ln(r/b)|  1/β 0 there exists rmax (or tmax )tmax ∼ β 0−2
Other restrictions:b  rg = 2Gm ,b  d = η −1

Hypothesis of the absence of interaction beyond tmax
r0 = 2γvtmax
Before the collision t < 0:zΦzbg}|{
2 2211v
x+bΦz (t, x) < Φz (0, x) = rE v 2 ln 0 + ln 2γ − ln
β2b2
Eventual valueΦz (tmax , x) ' rE v 2 ln γ + ln(1 + v) + ln 2 ≡ Φz∞
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike partip-Longitudinal string’s perurbation

Figure: Gravitational lensing by a string with angular deficit β 0 = 0.005
for b = 0.4 (blue solid), b = 0.6 (green dashed), b = 0.8 (black dotted),b = 1 (red dashdotted)
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike partip-Longitudinal string’s perurbation

Figure: String’s z-deflection (in units rE = 1) as a function of time (inunits b = 1) at x/b = 0 (blue solid), x/b = 5 (green dashed), x/b = 10(black dotted), x/b = 15 (red dashdotted) for v = 0.5 and β 0 = 10−4
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

String’s profile in z-direction

Figure: String’s z-profile on a length scale or order of b for v = 0.5√
Maxima: x = ± t2 − b2
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Evolution of string’s apex

Figure: Trajectory of a string’s apex in the particle’s scattering plane forv = 0.7 and β 0 = 0.005
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Scalar field addenda
Scalar part of the total action:Zp1Ssc =g µν ∂µ φ ∂ν φ |g| d4 x − φj2Z p
δ 4 (x − Z(s))µν2pŻdsg ∇µ ∇ν φ = −jj=f|g|(m + f φ)Z̈ µ = −mΓµνλ Ż ν Ż λ + f Πµν φ,ν
Zeroth order: scalar fieldfδ(x) δ(y − b) δ(z − vt)γFirst order: field equation ∂ 2 1 φ = 0 j−1/2f  21 µν1
Tsc = 0φ(x) = −γ (z − vt)2 + x2 + y 24π0

φ=00j(t, x) =Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

2nd order: radiation
Scalar correction to the particle’s trajectoryµm 1Z̈sc= f 0 Πµν 1 φ,ν ,µhhhh
Zsc= 0 (selfh−haction)
hhRadiation formula for the scalar fielddE
ω2=|j(k)|2 ,ω = k 0 = |k|3dωdΩ 16π
Source: j = 0 j + 1 j + ... ( 0 j – uniform motion)
22 φ = − 1 j(x)1j(x) ≡ ρ(x) + σ(x)Partial sources
Z (‘’local” and ‘’non-local”):
1 µρ(x) = −fZ (s) ∂µ δ 4 x − 0 Z(s) ds111 µν
1µν
1σ(x) = κ ∂µ h (x) ∂ν φ(x) − h(x) η ∂µ φ(x)2Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Radiation amplitudes
Local: (ψ ≡ 1 − v cos ϑ)κ 2 f µv i(kb) −(ku)b/γv 1
ρ(k) = −e ecos ϑ + i sin ϑ sin ϕ
4ωγψ 2
Charact. frequencies/angles: ω ∼ γ 2 /b ,
ϑ ∼ 1/γNon-local: 1 σ(k) = ... ei(kb) e−(ku)b/γv + ... e−ωbRpR ≡ γ −2 v −2 + sin2 ϑ cos2 ϕTotal source:κ2f µ(1 − sin2 ϑ cos2 ϕ) exp(−ωbR)
1j(k) = −4γωR ψ cos ϑ − v sin2 ϑ sin2 ϕ − ivR sin ϑ sin ϕCharact. frequencies/angles: ωbR . 1γv
hωi ∼hϑi ∼ γ −1hϕi ∼ 1b
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Emitted energy – angular distributionK(u)γv sin ϑdE 2G2 f 2 µ2 up=E(u)−,u=dϑπbψ 2γ21 + γ 2 v 2 sin2 ϑFigure: Plots of ϑ-distribution of the scalar bremsstrahlung, normalizedby overall factor 27 (π/γv)3 , in units G = µ = b = f = 1: v = 0.1 (red),v = 0.3 (green), v = 0.5 (blue), v = 0.7 (black), v = 0.9 (magenta).
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Emitted energy – frequency distributiondE  dE ∼
e−2bhRiω ,dωdω ω=0Figure: Plots of the frequency distribution of the scalar bremsstrahlung,normalized by the overall factor γ/v 2 , in units G = µ = b = f = 1:v = 0.2 (red), v = 0.4 (green), v = 0.6 (blue), v = 0.8 (black).
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Emitted energy – ultrarelativistic regime
ϕ-distribution: 31+sin ϕdE G2 f 2 µ2 γ 322=sin ϕ 1+2 cos ϕ − cos ϕ lndϕ21−sin ϕπb sin5 ϕTotal emitted energy:
3π G2 f 2 µ2 3γ
E=8bGeneric caseG2 f 2 µ2 3 3
E=γ v Ξ(v 2 )bwhere the introduced Ξ(v 2 ) satisfies
3π41.18 ≈6 Ξ(v 2 ) 6 ≈ 1.33
83Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Conclusions - Interesting features
Dimensional regularization was performed inside
the integral directly
A particle attracts string in the lateral direction(y-axis) and repels it along own motion (z-axis)
Two induced transverse waves along the string(Possible) Identification of the logarithmic
lenghth factor with the large-scale parameter
involves the conical angular deficit
The destructive interference (i) is complete,valid in any phase-volume domain; (ii) takes
place for any velocity
Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti

Acknowledgment for attention

Theorists’ lives matter!
Thank you!

Pavel Spirin

Gravitational interaction of the cosmic string with pointlike parti