09 (2012) 026 and 04 (2013) 010

astro-ph/1205.2384 and 1211.4045

Maxim Eingorn

Gravitational potentials anddynamics of galaxies againstthe cosmological background

North Carolina Central UniversityCREST and NASA Research Centers

Outline

– the Universe deep inside of the cell of uniformity

– inhomogeneously distributed discrete structures(galaxies and their groups / clusters)

– hyperbolic, flat and spherical spaces

– gravitational potentials

– motion of test bodies (for example, dwarf galaxies)

– gravitational attraction vs. cosmological expansion

– the Hubble flow at a few Mpc for our Local Group

– Milky Way and Andromeda destiny

0=K1−=K

1+=K

gravitational attractioncosmological expansion

(0;0)

Milky Way

Andromeda

An arbitrary dwarf galaxy

Introduction

less than 150 Mpc

?

inhomogeneous and anisotropic Universe

?

Hubble’s law

more than 150 Mpc

the “ΛCDM” model

homogeneous and isotropic Universe

hydrodynamical / continuous approach

Hubble’s law

[Labini, CQG, 2011]

Typical values of peculiar velocities~ 200-400 km/sec

The present value of the Hubble parameter~ 70 km/sec/Mpc

Distances at which the Hubble flow velocityis of the same order as peculiar velocities~ 3-6 Mpc

Hence, the Hubble flow can be observed at distances greater than this rough estimate.

It looks reasonable that EdwinHubble discovered his law in1929 after observing galaxieson scales less than~ 20 Mpc.

However, recent observations indicate thepresence of the Hubble flows at distanceseven less than3 Mpc!

[Sandage, Karachentsev, 1999-2012]

Theoretical substantiation?

At which distances from gravitating masses does cosmological expansion (the Hubble flow) overcome gravitational attraction?

What is the size of the region of local gravity where attraction prevails over expansion?

At which points does a free falling cosmic body have the zero acceleration?

Hydrodynamical approach = myopiaL.D. Landau & E.M. Lifshitz “Fluid Mechanics”

COSMOLOGICAL IDEAL FLUIDS

Any small volume element in thecosmologicalfluidis always supposed so large that it still contains avery great number of moleculesgalaxies.

“physically” infinitely small = very small comparedwith the volume of the body Universe, but largecompared with the distances between the moleculesgalaxies

“a cosmologicalfluid particle” ,“a point in a cosmologicalfluid”

Hydrodynamics Mechanics

( )2

1/2=

( ) [ ]

i kjik

jj

m c dx dx dsT

g t ds ds cdtδ −

−∑ r r

Energy-momentumtensor components:

Rest mass density:

( )jjj

m rr −∑ δγ

ρ21

1=

Final and irrevocable conclusion

Hydrodynamical / continuous approach isinapplicable at distances less than the cellof uniformity size ~ 150 Mpc.

The energy-momentum tensor choice

– The energy-momentum tensor of a system ofpoint-like particles is acceptable if the distancesbetween the bodies, modeled by them, are muchgreater than their sizes. This condition is satisfiedvery well in cosmology: cosmic bodies = galaxies.

– Exactly the same energy-momentum tensorchoice in astrophysics underlies the prevalentweak gravitational field approximation and givessure-fire results for the planets in the Solar systemincluding prediction of relativistic effects.

Mechanical / discrete approach

Background: the “ΛCDM” model(the homogeneous and isotropic Universe)

hyperbolicflatsphericalspaces

0=K

1−=K

1+=K

( )( )

+++

−−2

222

22222

41

1

==

zyxK

dxdxdadxdxdads

βααββα

αβδ

ηγη

openflatclosedUniverses

( ) Λ++Η 002

2

=3

Ta

K κ

Λ+Η+Η′=

22

2

a

K

The Friedmann equations

ηd

da

aa

a 1=′

=Η

4

8

c

GNπκ =

ikT are the average energy-momentum tensor

components of dust (pressureless matter).

3200 /= acT ρThe average energy density

is the only non-zero mixed component.

quantity symbol dimension

scale factor a L (length)

rest mass density

in comoving frame

ρ M (mass)

average

rest mass density

in comoving frame

ρ M (mass)

and are the values of the scale factor and the Hubble parameter at present time

Ω+Ω+Ω−Λ+

Λ 2

20

3

302

02

22

3

422 =

33==

a

a

a

aH

a

Kcc

a

c

a

aH KM

ρκɺ

Ω+Ω−Λ+− Λ3

302

0

2

3

4

2

1=

36=

a

aH

c

a

c

a

aM

ρκɺɺ

0a 0H a

dt

da

aa

aH

1==ɺ

0=t

20

20

2

20

2

30

20

4

=,3

=,3 Ha

Kc

H

c

aH

cKM −ΩΛΩ=Ω Λ

ρκ1=KM Ω+Ω+Ω Λ

Contributionof radiation is

neglected.

H

11

= ≪cdt

dxa

d

dx αα

η0

00 || TT δδ β ≪

The slightly perturbed “ΛCDM” model(the weakly inhomogeneous and

anisotropic Universe)

The nonrelativistic limitin the comoving frame:

Indeed, the typical present value~ 300 km/secof peculiar velocities is only1/1000(0.1 %) ofthe speed of light.

00

2

2

1=3)(3 TaK δκΨ+ΗΦ+Ψ′Η−Ψ∆

0=2

1=)( 02

ββ δκ Tax

ΗΦ+Ψ′∂

∂

( )

( )

2

2

; ;

1(2 ) 2 ( )

2

1 1=

2 2

K

a T

αβ

ασ αβσ β

δ

γ κ δ

′′ ′ ′Ψ + Η Ψ + Φ + Η + Η Φ + ∆ Φ − Ψ − Ψ −

− Φ − Ψ −

LaplaceoperatorΨΦ = [Mukhanov, Rubakov]

βααβγη dxdxdads )2(1)2(1 222 Ψ−−Φ+≈

Scalar perturbations

0=)( ΗΦ+Φ′∂

∂βx )(

)(=),(

2 ηϕη

ac

rrΦ

00

2

2

1=3)(3 TaK δκΦ+ΗΦ+Φ′Η−Φ∆

2 2 20

0 rad rad3 3 3 4

3 3= =

c c cT

a a a a

δρ ρ δρ ρϕδ δε δεΦ+ + + +

)(4=3 ρρπϕϕ −+∆ NGK

( )2 2 2rad rad

1 13 2 =

2 6K a p aκ δ κ δε′′ ′ ′Φ + ΗΦ + Η + Η Φ − Φ =

ρρδρ −= (the difference between real andaverage rest mass densities)

rad 4

3

a

ρϕδε = −

The nonrelativistic gravitational potential isdefined by the positions of the galaxies, butnot by their velocities!

[Landau]

)(4= ρρπϕ −∆ NG

I. Flat space / Universe 0=K

ρ−The role of the term : the gravitational(Neumann-Seeliger) paradox is absent butspatial distribution of inhomogeneities isnot arbitrary! It is an essential drawback!

22

222 )(== ΩΣ+ dddxdxdl χχγ βααβ

−+∞∈+∞∈

+∈Σ

1=,)[0,,sinh

0=,)[0,,

1=,][0,,sin

=)(

K

K

K

χχχχ

πχχχ

The nonzero spatial curvature case

K

GN

3

4=

ρπϕφ +

0≠K

ρπφφ NGK 4=3+∆

The Helmholtz equation:

Spatial distribution of inhomogeneitiesis absolutely arbitrary!

0=3)()(

1 22

φχφχ

χχK

d

d

d

d +

Σ

Σ

One gravitating mass in the origin of coordinates:

the Newtonianlimit at

II. The spherical space= the closed Universe 1= +K

χχ sin=)(Σ

01 0

0

1= 2 cos 2sin

sinN

N

G mC G m

χφ χ χ

χ χ→

− − → −

0→χdivergent at

πχ →

denotes the geodesic distance between the i-th mass and the point of observation

III. The hyperbolic space= the open Universe 1= −K

χχ sinh=)(Σ

0

exp( 2 )= 0

sinhNG mχ

χφχ →+∞

−− →No gravitationalparadox!

An arbitrary number of gravitating masses:

3

4

sinh

)2(exp= 0

ρπϕ N

i

ii

iN

G

l

lmG +

−− ∑

il

im0

χaR =The physical distance

1≪≪ χ⇒aR

320

20

2

1= 108.4<= −− ×Ω

Ha

cK [Komatsu, 2011]

Mpca 50

41= 10410 ×≈⇒≈Ω −

−K

1105≪≪ χ⇒MpcR

χφ 0mGN−≈

1180 102.3// 70 −−×≈≈ secMpcseckmH

0.27≈ΩM0.73≈ΩΛ

Experimental data:

2=

22vma

a

mL +− ϕ

The Lagrange function:

The Lagrange equation:

The physical distance

The physical velocity

Motion of a nonrelativistic test bodywith the mass m

( )r

v∂∂− ϕ

aa

dt

d 1=2

a=R r

( )vR

rRV a

a

a

dt

ad

dt

d +===ɺ

the physical peculiar velocity

The reason for the Hubble flow is the global cosmological expansion of the Universe.

22

0

11 1

=,= ccRc

RV adta

t

aaa

a

t

++ ∫ɺ

22

0

121 1

=,= ccrc

v +∫ dta

t

at

physical peculiarvelocity

Hubblevelocity

The case of the negligible gravitational potential

The Lagrange function in spherical coordinates :ψπθ /2,=,r

Lagrange equations:

( )2222

2= ψϕ

ɺɺ rrma

a

mL ++−

( )2 22 2

= 0 =d M

ma rdt ma r

ψ ψ⇒ɺ ɺ

( )322

22 1

=ram

M

rara

dt

d +∂∂− ϕ

ɺ

The case of spherical symmetry of the problem

32

2

20

3

302

032

2

20

2

1==

Rm

M

R

mGR

a

aH

Rm

M

R

mGR

a

aR N

MN +−

Ω+Ω−+− Λ

ɺɺɺɺ

|/2|=

|/| 20

0

0=

03

0=

32

0

M

N

aa

NH

aa

N

H

mG

aa

mGRR

R

mGR

a

a

Ω−Ω==⇒=

Λɺɺ

ɺɺ

cosmological expansion,responsible for the Hubble flow

gravitationalattraction

?=

1.4HR Mpc≈

The zero-acceleration sphere

This value is close to theobserved one ~ 1 Mpc!

2/3

1/21/2

1/21/21/3

~2

3cosh~

2

3sinh1=~

Ω

ΩΩ+

Ω

ΩΩ+

ΩΩ

ΛΛ

ΛΛ

Λtta

MM

M

gMm Sun4512

0 103.8101.9 ×≈×≈

[Karachentsev, 2012]

Dimensionless units:

4220

22

00

=~

,=~

,=~,=~

HH RmH

MM

R

RRtHt

a

aa

3

2

232

2

~

~

~/2~

~2=~

~

R

M

RR

atd

Rd MM +Ω−Ω−

Ω+Ω− ΛΛ

0=~

M

(1;1)

Solid and dashed lines correspond to presence and absence of the gravitational

attraction respectively.

sec100~

kmVV ×≈MpcRR 1.4

~ ×≈

VHRtd

RdV

H 0

1=~

~=

~

1=~

M

2=~

M

3=~

M

0~ ≠M

0=~

M

(1;0)

the dashed black line = the pure Hubble flow

The average gravitational potential is zero, as it should be!

The average gravitational potential

3

4

sinh

)2(exp= 0

ρπϕ N

i

ii

iN

G

l

lmG +

−− ∑

this constant term does not affect motion ofnonrelativistic bodies, but it contributes to themetrics, and at present time this contribution is

32

20

20

20

102=3

8×

Ω∼

c

aH

ca

G MN ρπ

There is no any contradiction,because

3

4=total

ρπφ NG−

0=ϕ

For the sake of revolution

less than 150 Mpc

the slightly perturbed “ΛCDM” model

weaklyinhomogeneous and anisotropic Universe

mechanical /discrete approach

Hubble’s law

, ,i i i i i iX ax Y ay Z az= = =

( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )

( )( ) ( ) ( )

( )

3 22 2 2

3 22 2 2

3 22 2 2

1

1

1

j i j

N i ij i

i j i j i j

j i j

N i ij i

i j i j i j

j i j

N i ij i

i j i j i j

m X XG X a aX

aX X Y Y Z Z

m Y YG Y a aY

aX X Y Y Z Z

m Z ZG Z a aZ

aX X Y Y Z Z

≠

≠

≠

−− = −

− + − + −

−− = −

− + − + −

−− = −

− + − + −

∑

∑

∑

ɺɺ ɺɺ

ɺɺ ɺɺ

ɺɺ ɺɺ

Physical coordinates:

The Lagrange equations for the i-th mass:

( )( ) ( )

( )( ) ( )

2 2

3 22 22 2

2 2

3 22 22 2

1 1, , 1,...,

1 1, , 1,...,

j i jii

j ii j i j

j i jii

j ii j i j

m X Xd X d aX i j N

dt m a dtX X Y Y

m Y Yd Y d aY i j N

dt m a dtX X Y Y

≠

≠

−= − + =

− + −

−= − + =

− + −

∑

∑

ɶ ɶɶ ɶɶ

ɶ ɶɶɶ ɶ ɶ ɶ

ɶ ɶɶ ɶɶ

ɶ ɶɶɶ ɶ ɶ ɶ

Motion on the plane 0Z =1

1 N

ii

m mN =

= ∑

1 31 3 1220

1 31 3 1220

10

0.95 Mpc

10

0.95 Mpc

ii i

N

ii i

N

MH XX X

G m m

MH YY Y

G m m

= =

= =

⊙

⊙

ɶ

ɶ

Dynamics of three gravitating masseswith zero initial velocities

0t =ɶ 1t =ɶ 2t =ɶ

Four masses, nonzero initial velocities

0; 1; 2t =ɶ 0; 0.3; 0.6t =ɶ

both attraction and expansionexpansion only

The collision betweenMilky Way and Andromeda

Separation distance ~ 0.78 Mpc

Longitudinal velocity ~ 120 km/s

Transversal velocity ~ 100 km/s

Merger distance ~ 100 Kpc

Mass of MW ~ 1012 M

Mass of M31 ~ 1.6 × 1012 M

[Cox and Loeb, 2008]

The separation distance as a functionof time in the free fall approximation

without dynamical friction

both attraction and expansionattraction only

zero transversalvelocity

nonzerotransversalvelocity

no collisioncollisionin 3.68 Gyr

290 Kpc4.44 Gyr

Chandrasekhar’s dynamical friction

( ) ( )2

ph, 23

4 2erf expN mM

MM

QG Md

dt V

π ρ χχ χπ

= − − −

VV

MV Mis the physical velocity of the mass

ph,mρ is the physical rest mass density of the intragroup media

2MVχσ

= ( )21ln 1

2Q λ= +

is the Coulomb logarithm defined by the largest impact parameter , the initial relative velocity and the masses andMm

maxb0V

pec,

2i

i

vχ

σ=ɶ

ɶɶ

1 320

0 N

H

H G m

σσ

=

ɶ

20

ph, cr

3, 10

8mN

H

Gρ αρ α α

π= = ∼

The physical rest mass density begins todecrease at scales where cosmological expansiondominates over gravitational attraction, that isapproximately at 1 Mpc. Here . Therefore,1 .

~ 1Q

ph,mρ

kT

mσ =

m is the mass of the intragroup media particle

5α ∼

510 KT ∼

max ~ 1 Mpcb

( )2 2

max 0 max 0

N N

b V b V

G M m G Mλ = ≈

+

( )( ) ( )

( ) ( )

( )( ) ( )

2 2

3 22 22 2

23pec,

2 2

3 22 2 2

1 1

3 2 1erf exp , , , ;

2

1 1

j i jii

i j i j

i i ii i i

i

j i ji

i j i j

m X Xd X d aX

dt m a dtX X Y Y

Qm dX daX i j A B i j

mv dt a dt

m Y Yd Y d a

dt m a dtX X Y Y

α χχ χπ

−= − +

− + −

− − − − = ≠

−= − +

− + −

ɶ ɶɶ ɶɶ

ɶ ɶɶɶ ɶ ɶ ɶ

ɶɶ ɶɶɶ ɶ

ɶ ɶɶ ɶ

ɶ ɶɶ ɶ

ɶ ɶɶɶ ɶ ɶ ɶ

( ) ( )

2

23pec,

3 2 1erf exp , , , ;

2

i

i i ii i i

i

Y

Qm dY daY i j A B i j

mv dt a dt

α χχ χπ

− − − − = ≠

ɶ

ɶɶ ɶɶɶ ɶ

ɶ ɶɶ ɶ

pec,

1 22 2

pec,

1 1, , ,

1 1

i ii i i

i ii i

dX dYda daX Y i A B

dt a dt dt a dt

dX dYda dav X Y

dt a dt dt a dt

= − − =

= − + −

vɶ ɶɶ ɶ

ɶ ɶɶɶ ɶ ɶ ɶɶ ɶ

ɶ ɶɶ ɶɶ ɶɶ

ɶ ɶ ɶ ɶɶ ɶ

( ) ( )2

0

2erf exp

x

x dξ ξπ

= −∫

The separation distance

67 MeVIGMm >

The touch of the galaxies will take place during the first passage for the intragroup media particle masses

100 Kpc

1.17σ =ɶ

The corresponding trajectoriesof MW (blue) and M31 (green)

collision in 6 Gyr

The merger willtake place for

17 MeVIGMm >

The separation distance

1 Mpc

2.31σ =ɶ

The corresponding trajectories of MW

(blue) and M31 (green)

collision in 37 Gyr

The dwarf galaxy acceleration

2

2

1 1d d a

dt a dt a

ϕ∂= = −∂

VW R

R

ɶ ɶɶɶ ɶ

ɶɶ ɶɶ ɶ

( )2 3

0 0

1

Na H G mϕ ϕ=ɶ

Right =Milky Way

Left = Andromeda

Vector field

0yW =ɶ0xW =ɶ

The contour plot of the acceleration absolute value,if the cosmological expansion is disregarded

0x yW W= =ɶ ɶ

0=Wɶ

The real contour plot

The exact zero acceleration surface is absent.

The mechanical approach in modern cosmologyis consistently developed and substantiated.

The late Universe inside the cell of uniformity(less than 150 Mpc) is described by the slightlyperturbed “ ΛCDM” model:

– the metrics

– the gravitational potential

Results and conclusions

βααβγη dxdxdads )2(1)2(1 222 Φ−−Φ+≈

)(

)(=),(

2 ηϕη

ac

rrΦ

)(4=3 ρρπϕϕ −+∆ NGK

In the case (the hyperbolic space = theopen Universe):

– inhomogeneities may be distributed completely randomly;

– the gravitational potential is finite at any vacuum point and its average value is zero;

– at present time the distance from our LocalGroup of galaxies, at which the cosmologicalexpansion ‘=’ the gravitational attraction, or thezero acceleration sphere radius, ~ 1.4 Mpc, andthis theoretical value is close to the observedone, namely 1 Mpc.

1−=K

The observations of the Hubbleflows even at a few Mpc mayreveal presence of dark energyin the Universe.

The mechanical approach gives a possibility tosimulate the dynamical behavior of an arbitrarynumber of randomly distributed inhomogeneitiesinside the cell of uniformity, including the MilkyWay and Andromeda galaxies together with thedwarf galaxies, taking into account gravitationalinteraction between them as well as cosmologicalexpansion of the Universe.

THANK YOU FOR ATTENTION !