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Cosmic microwave background radiation ( CMB, CMBR, CBR) -, () - ( . . ), . , , , , , . , , . . 4000 , , ( , ), . z 1500 В 1000. , . , , , , . , , , , . , . , , , [5, 66, 65], [71, 92, 47, 58, 46]. [10] [13]. () . , , , , , . . § 1. () 1. . serendipitously. , , . - , , , , "" . , , - , . 1964 - "Crawford Hill Laboratory" (Holmdel ) 20 ( 21 ). . . ECHO, 100 , . , , , . , 12.5 , . , . . , ECHO [64], , . . , , , . [66]. , , -, . , 6 . 7.35 , , (). : 3.5 ± 1 .

1

. : , , , . , . , . , , . . , . , . : 1) , 2) , 3) 4) . , . - . , , . ( -, , - , - -), 30 , , = 3 . . , . . , . . MIT Radiation Laboratory ( ). , . CMBR 20 . 1964 , . . . . . . . , . . . : [68] [31] Astrophysical Journal. 3.2 , 3.0 ± 0.5 . 1972 15 0.27 73.5 . 1975 0.1 , . . 1978 . 2. . , , , . , , , , , . . , . 1941 . [17] CN , , , , 2.6 . - [57] , , 2.3 . - . . , ґ . , , . 1966 ( . [5]). 3.2 . . , . . . . , " " [16]. , . . 2004 . 2

. . [8, . 152153] . ", 3.7 ± 3.7 , 3.9 ± 4.2 . , . . ( 4 ) , , , 2 . , . . , -, . . . , 1957 . . . . , , . . . . , . . . . . , . . ." , . . . . [4] , , . , , . . , , , , . 2.3 ± 1 , , 1 1 . , 1 . , . . . . [5] , . . . : " , , (. . ), (. . . . ) ." 3. . . , . . 1999 COBE (COsmic Background Explorer) T0 = 2.7277 ± 0.002 ( 95 % ) . max = 1.604 · 1011 1/, 1.870 . , , max = 1.062 , ( max ) 2.822 · 1011 1/. , , 1/2 = 1.9910 · 1011 1/ 1/2 = c/1/2 = 1.506 . 8 2.404(T 0kB /ch)3 = 411.7 1/3, 4 () (8 5 kB /15h3 c3 )T0 = -13 3 3 -34 3 4.188 · 10 / ( 0.25 / ), 4.659 · 10 / . 2.047 · 10-13 /( 2 ). , [5], , , . . . 1. () (< 1 ) (< 1 ) (1 В 6 ) (> 10 ) , /3 10
-7

, 1/3 1 400 1 10-3 3 · 10
-8

0.25 10-2 3 · 10-3 10
-5

В 10
-4 -5 -5

-4

В 10
-9

-8

10

3 · 10

10 < 10

3 · 10 < 10

-12

-12

3

, , . , [30, 86], v T = T0 1 + cos 10-3 . c 3.343 ± 0.016 = 11.2h, = -7 l = (264.4 ± 0.3) , b = (48.4 ± 0.5) . , , 627 ± 22 / l = (276 ± 3) , b = (30 ± 3) . , , . 10-5 , T = (12.4 ± 2.8) . . .. .. [90] (. [82]). , - ( , ), . . . , , . § 2. 1. . , . . , , . , . 4000 В 2700 ( 105 ), z 1500 В 1000, , , . . , , . , , . , z 103 , . . , : T = T /T . . (., , [65]). , . . 10 -44 , . "" [20]. (. [5, 12, 65]). 1) ( , 1967). . -, , , T = /c2 . -, , . . t/t = /c 2 . , , R t2/3 . T 1/R t-2/3 , T = -(2/3)/c2 . : T = (1/3)/c2 . 2) , . . ni n . ( ) : n 1/R3 , ni 1/R3 , -: i 1/R3 T 3 , r 1/R4 T 4 . T = (1/3)/ = (1/4)r /r . 3) S , . , . T 3 , T = (1/3)S/S . 4) , , . T vr /c, vr , . . . 4

, , , . . . [14] . . , , . - g i j . e 1 T (e) = - 2
t
0

t

1 r v gi j i j e e dt + +e . t 4 r c

(1)

r

ei e, e0 = 0, tr , t0 . , [52]. 2. . , . : . ( 1 ) . 0 . , , "" "". "" = (x , y ). , , . . T () = T (). : 1 ~ T (K ) = 2 0 e
2 2 x +y 0

iK 2

1 ~ d T () = T (K ) = 2 0

2

0

d
0 0

de

i K cos

2 T () = 2 0

0

dT ()J0 (K ),
0

(2)

J0 (z )

2

1 J (z ) = 2

e
0

iz cos -i

d.

(3)

, , K .
2 0 T () = (2 )

2

e

-iK

~ T (K )d2 K = () =

2 0 (2 )

2 2 0

d
0

e

-iK cos

2 ~ T (K )K dK = 0 2

~ J0 (K )T (K )K dK. (4)
0

~ T (K ) , > 0 . 1 ~ eiK T ()d2 , (5) T (K ) = 2 0 . , (4) . . 0 : T = 1. (. [1, 6.561.5.]) 2 ~ T (K ) = 2 0

0

J0 (K )d =
0

2 J1 (K 0 ). K 0

(6)

5

6.512.3. [1] 2 T () = 2 0
2 0

0

. . ( ). 2 2 ~ T 2 (K ) = 0 K 2 T (K ) , (8) 2 . ,
C () = T ()T ( + ) =

1 1 J0 (K )J1 (K 0 )dK = 2 0

< 0 , = 0 , > 0 , (7)

1 2 0

d2 T ()T ( + ).

(9)

. , C () =
4 1 0 2 (2 ) 0 1 4 = 2 04 0 (2 ) 4

d2

e

-iK

~ T (K )d2 K
iK ~ T

e

iK (+) ~ T

(K )d2 K =

~ T (K )d2 K ~ T (K ) d2 K =
2

e

(K )d2 K (2 )2 (K - K ) =

0

=

2 0 (2 )

2

e

iK

2 0 (2 )

2 2 0

d
0
0

e

iK cos

~ T (K ) K dK = dK . K

2

=

2 0 2

0

~ T (K ) J0 (K )K dK =
0 0

2

T 2 (K )J0 (K )

(10)

3. . : T (n) =
l, m

a

l m Yl m

(, ),

(11)

n . C (µ) = T (n1 )T (n2 ) = 1 (4 )
2

d2 n

1

d2 n2 T (n1 )T (n2 )2 (n1 n2 - µ), lm

(12)

µ = n1 n2 , a C (µ) = 1 4
l

: |2 . (13)

a2 Pl (µ), l
l

a2 = l
m=-l

|a

lm

, (12) -

(µ - µ) =

l=0

2l + 1 Pl (µ)Pl (µ ) 2

(14)

(11), ( ). C (µ) = в 1 16 a
l2 , m
2

2 l=0
2

(2l + 1)Pl (µ) Y
l2 m
2

d2 n

1

d2 n

2 l1 , m
1

a

l1 m

1

Y

l1 , m

1

(1 , 1 )в (15)

l2 m

(2 , 2 )Pl (cos 1 cos 2 + sin 1 sin 2 cos(1 - 2 )). 6

, Pl (cos 1 cos 2 + sin 1 sin 2 cos(1 - 2 )) = 4 2l + 1
l

Y
m=-l

lm

(1 , 1 )Y

lm

(2 , 2 ).

(16)

, (15) (13). , . , , , . . Cl = |a
lm

|

2

=

1 |al |2 . 2l + 1

(17)

, ґ l. = 2 /l. 4. . Cl . , Cl , T2= l(l + 1)Cl , 2 T= l(l + 1)Cl . 2 (18)

l(l + 1) , l(l + 1)C l , . , l (18) (8). . 1 . l 1, 1, l + 2 w = Pl (cos ) d2 w cos dw + + l(l + 1)w = 0 (19) d2 sin d y = (2l + 1) sin : 2 1- y2 (2l + 1)
2

y2 d2 w + 1-3 dy 2 (2l + 1)

2

1 1 dw + 1- y dy (2l + 1)

2

w = 0.

(20)

l , . = 0 Pl (1) = 1, Pl (1 - 2 /2) = J0 ((l + 1/2)). 1 , l K = l + 1/2. C (cos ) = C 1 - 2 2 = 1 4 1 2
2 al P l=0 l

1-

2 2

1 4

a2 J0 ((l + 1/2))dl = l

1 4

(2l + 1)Cl J0 ((l + 1/2))dl = dK , K (21)

=

T2 (2l + 1)J0 ((l + 1/2))dl = l(l + 1)

T 2 (K )J0 (K )

(10) . , (20) , Pl (1 - 2 /2) 1/(l + 1/2)2 . , , , (18) l. Cl = 2 6 T2= C2 . l(l + 1) l(l + 1) (22)

C (µ) = 3C2 2

l=1

3C2 2l + 1 Pl (µ) = l(l + 1) 2

l=1

1 Pl (µ) + l

l=0

1 Pl (µ) . l+1

(23)

7

l. , 8.926.1. 8.926.2. [1]. , µ = -1 µ = 1: 2 3 C2 ln . (24) C (µ) = 2 1-µ

T , l 10. , T 2 T lg l, l. , l. . T 2 = l(l + 1)Cl /2 200300, . . 5. . . . , , . . T (n) T (r ), . , . . T (n) = T (r)e
-

d .

(25)

r , : e (25)
-

d = W (r)dr.

(26)

T (n) =
0

T (r)W (r)dr.

(27)

. z < 10 В 25 , . , . , . rrec , . , W (r) = ((r -r rec )/r ), , . . 2 2 1 e-(r-rrec ) /(2r ) . (28) W (r) = 2 r (., , [65]) r = 10 , r r(z ) = 2 c z . H0 1 + z ( 1 + z + 1)
rec

= r(z

rec

), (29)

, z = , . . r(zrec ) r() = 2c/H0 = 3 · 1028 = 104 . r. (26) , r = 0 = 0, :
r

1-e

-

=
0

W (r )dr .

(30)

, 1-e
-

1 = 2

r

e
r 0

-(r-r

rec

2 )2 /(2r )

dr =

r - rrec + r

rrec r

,

(31)

8

(x) . 1 . (32) = ln 1 - [((r - rrec )/r ) + (rrec /r )] . , . , ( ), § 4, ( ) [27]. . , , [2], [34, 24], LiH. , 10-8 В 10-6 , .

.

..... ...
LiH

k B Te n e , v

r

v

S

1. . . 1 , , , ( [65]). , , , . , . 6. . . . T < 10 %. : 10-2 [78] 10-3 [84]. [11]. T 2 · 10-5 В 10-4 [67, 3 ]. 1992 DMR (Differential Microwave Radiometer), COBE (COsmic Background Explorer). , T 30 > 7 . , , . 1994 [92] , . 1 T 10-5 . , 50 . , 1992 , , Monthly Notices. , 3, COBE, . 3 2006 , COBE, . George F.Smoot, [87], John C.Mather, , FIRAS (Far InfraRed Anisotropy Searches), [40]. . ( SP, Python, White Dish, Tenerife ., . , [77]) ( DASI, CAT, VSA, Jodrel Bank). : MAXIMA (Millimeter Anisotropy Experiment Imaging Array), TopHat . BOOMERang (Balloon Observations of Millimetric Extragalactic Radiation

9

and Geophysics) 10 . , , . , L2 , , MAP (Microwave Anisotropy Probe), 5 22 90 . 1 . , . WMAP. WMAP , . 2007 Planck. 9 20 800 , 1000 , COBE, 4 . [37]. . 2 T ( (18)) l. . 2, , : b = 0.02, CDM = 0.28 = 0.7. . 2, b = 0.05 , . CDM = 0.95 = 0, CDM = 0.35 = 0.60. , . : COBE [21], MAXIMA-1 [53], BOOMERANG [61], DASI (Degree Angular Scale Interferometer) [42], Tenerife [77]. .
30 25 20 15 10 5 0 0 200
COBE MAXIMA-1 BOOMERANG DASI

T

30 25 20 15 10 5 0 0 200

T

COBE Tenerife BOOMERANG MAXIMA-1

400

600

800 1000

400

600

800 1000

l l . 2. T l 20002001 [37] () [77] (). , , , - . [33,51,58]. , . [92, 71]. R. Durrer [3537] , . [83] , . 2001 [58]. WMAP [22], , COBE. , http://map.gsfc.nasa.gov, . [88], , 10 , 6 , , , 4 . , . 10

T 2 l [43]. , [23]. - [79, 80]. § 3. 1. . . , , . . . [69]. 10%, 10-4 % , . DASI. , . , , WMAP [50]. . , , , . astro-ph , WMAP. , 0606511 , ( Q) . 0603450 WMAP. Plank . . 2. . , Q U . , , , : Q U ^ = (Ai,j ). (33) A= U -Q , ^ det(A) = -(Q2 + U 2 ) (34)

^ A. , . , , Q 2 + U 2 > 0. x p= I . r , r= I Q2 + U 2 , tg 2 = U , Q (35)

3. . , . , . , : dy = tg . dx tg = dy = dx Q2 + U 2 - Q . U (38) 1 + tg2 2 - 1 , tg 2 (37) (36)

4. . (38) , . , , 11

Q U . ( ) Q = q x x + q y y , U = u x x + uy y . (39)

uy Q - q y U ux Q - q x U , y=- , d = q x uy - q y ux . (40) d d u q , , uy = 0. uy = 0, ux = 0 x y . d = 0 , . p, I , . . r, : Q = r cos 2, U = r sin 2. (41) x= dQ = dr cos 2 - 2r sin 2d, dU = dr sin 2 + 2r cos 2d. (36) dy ux dQ - qx dU ux (dr cos 2 - 2r sin 2d) - qx (dr sin 2 + 2r cos 2d) =- = tg = - . dx uy dQ - qy dU uy (dr cos 2 - 2r sin 2d) - qy (dr sin 2 + 2r cos 2d) tg = t (43) , d ln r 2 qy t3 + (qx - 2uy )t2 - (qy + 2ux)t - qx N (t) = = . dt 1 + t2 uy t3 + (ux + 2qy )t2 - (uy - 2qx )t - ux D(t) 2 [qy t3 + (qx - 2uy )t2 - (qy + 2ux )t - qx ], uy ux ux + 2qy 2 uy - 2qx t- t- . D(t) = (1 + t2 ) t3 + uy uy uy N (t) = (45) (46) (44) (43) (42)

(44), ±i, . t1 , t2 , t3 . D(t) = (1 + t2 )(t - t1 )(t - t2 )(t - t3 ). (47)

5. . (44) , : N (t) 2t 1 2 3 = + + + . D(t) 1 + t2 t - t1 t - t2 t - t3 , , 1 = - (1 + t2 ) (1 + t2 ) (1 + t2 ) 1 2 3 (1 + t2 t3 ), 2 = - (1 + t1 t3 ), 3 = - (1 + t1 t2 ). (t1 - t2 )(t1 - t3 ) (t2 - t1 )(t2 - t3 ) (t3 - t1 )(t3 - t2 ) uy uy (1 + t1 t2 + t2 t3 + t3 t1 ), qy = - (t1 + t2 + t3 + t1 t2 t3 ); 2 2 (49) (48)

: (44) ux = u y t 1 t 2 t 3 , q x = N (t) = -(t1 + t2 + t3 + t1 t2 t3 )t3 + (t1 t2 + t2 t3 + t3 t1 - 3)t2 + (t1 + t2 + t3 - 3t1 t2 t3 )t - (t1 t2 + t2 t3 + t3 t1 + 1); (40) d = q x uy - q y ux = u2 y (1 + t1 t2 )(1 + t2 t3 )(1 + t3 t1 ). 2 12 (52) (51) (50)

, 1 + 2 + 3 = -2, t1 1 + t2 2 + t3 3 = -(t1 + t2 + t3 ) - t1 t2 t3 = 2 1 2 3 = (1 + t2 )(1 + t2 )(1 + t2 ) 3 1 2 (1 + t1 t2 )(1 + t2 t3 )(1 + t3 t1 ). (t1 - t2 )2 (t2 - t3 )2 (t3 - t1 )2 (54) qy . uy (53)

(48) (44) :
3

r = r0 (1 + t2 )
j =1

|t - tj |j .

(55)

: r0 r= cos2
3 j =1

| tg - tg j |j .

(56)

, , , = ± /2, , (53). 6. . (56) Q U . 0 2 . = . 2 , xy . x = cos , y = sin (57) r , (39) : r cos = qx cos + qy sin , r sin = ux cos + uy sin . = r |d| = tg = qx tg - ux , uy - qy tg tg = ux + uy tg . qx tg + qy (60)
2 2 ((u2 + u2 ) cos2 + (qx + qy ) sin2 - 2(ux qx + uy qy ) cos sin = x y

(58)

r
2 2 (qx + u2 ) cos2 + (qy + u2 ) sin2 + 2(qx qy + ux uy ) cos sin x y

(59)

, , , . , . 7. . . , tj . : t1 > t2 > t3 . 2 3 1 < 0, > 0, < 0, (61) 1 + t 2 t3 1 + t 3 t1 1 + t 1 t2 . . 2 1 + t3 t1 , 1 3 1 + t2 t3 1 + t1 t2 . . 1) 0 < t3 < t2 < t1 . (54) j , d > 0. (1 2 3 ) (- + -). 13

2) t3 < 0 < t2 < t1 . 1 + t1 t2 > 0 3 > 0. : ) 1 + t3 t1 > 0, 1 + t3 t2 > 0, d > 0 (- + -); ) 1 + t3 t1 < 0, 1 + t3 t2 > 0, d < 0 (- - -); ) 1 + t3 t1 < 0, 1 + t3 t2 < 0, d > 0 (+ - -).

3) t3 < t2 < 0 < t1 . 1 + t2 t3 > 0 1 < 0. : ) 1 + t1 t3 > 0 1 + t1 t2 > 0, d > 0 (- + -); ) 1 + t1 t3 < 0, 1 + t1 t2 > 0, d < 0, (- - -); ) 1 + t1 t3 < 0, 1 + t1 t2 < 0, d > 0 (- - +). 4) t3 < t2 < t1 < 0. , d > 0 (- + -). , d , , . , -. , , . . . , t1 = c, t2 = a + bi, t3 = a - bi, : : 1 + a 2 + b2 < 0, (62) 1 = -(1 + c2 ) (a - c)2 + b2 - , 2 + 3 1 (a2 + b2 - 1)(c2 - 1) + 4ac = , 2 2 (a - c)2 + b2 2 - 3 c[(a2 + b2 )2 - 1] - a(c2 - 1)(a2 + b2 + 1) M = -b = . i (a - c)2 + b2 R= (63) (64)

(44) r = r0 (t - t1 )1 (1 + t2 )[(t - a)2 + b2 ]R e (t) = b 1 arctg . b t-a
-M (t)

,

(65)

(66)

(65) , . , M , . . . [32], (. [10]), , . d < 0 "" (saddle) -. d > 0 "" (beak). , "" (cometa). , , , , . . 1) "" : ) y = y0 x2 , ) y = (y0 +ln |x|)x, ) y = y0 x. 2) "" : y = y0 /x. 3) "": r = r0 e . 4) "": r = r0 . . 3 . 8. . (39), Q = 2q q x, U = y , q = 0. D(t) = (1 + t2 ) t(t2 + 2q - 1), t1 = 0, 1 = , 1 - 2q 1-q . , q < 0 "", 0 < q < 1/2 "", 2 = 3 = - 1 - 2q q > 1/2 "". . q . r0 2 . 1) "" q = -1, t2 = 3, t3 = - 3, 1 = 2 = 3 = - , r = 3 | sin /2(sin2 /2 - 3 cos2 /2)|2/3 2) "" q = 1, t2 = i, t3 = -i. 1 = -2, r0 2 = 3 = 0 r = . 2 sin /2 14

. 3. : , "", "", "", "".

15

20

40

10

20 0 -20 -40

0

-10

-20 -20

-10

0

10

20

-10

0

10

20

30

40

50

. 4. "" () "" ().
100

40 20 0 -20 -40 -40 -20 0 20
-100 -200 -150 -100 -50 0 50 100 150 200 50

0

-50

. 5. "" , r () , () .
40
50 40

20

30 20

0

10 0

-20

-10 -20

-40 -20

0

20

40

60

80

100

-30 -20

-10

0

10

20

. 6. "" () "" () .

16

. 4 . 5, , r. , . 4 , r , . "". . 5, , , . 5,. , . "" , , r0 , , , , . . . 9. . , (46) Q U . q = 0. r = r 0 /| cos(/2)|, , y . q = 1/2 . D(t) = (1 + t2 )t3 , N (t) = -3t2 - 1 r= r0 e sin (/2)
2
1 2

1 3 | sin /2| 1 3) "" q = 1/4, t2 = , t3 = - , 1 = 1, 2 = 3 = - r = r0 1 2 | sin2 /2 - 2 cos2 /2| 2 2

3/2

.

ctg(/2)

.

(67)

: , , , , 2 . "" (plough). . 6. "" , "" . ^ ^ 10. A. A , [10]. . . , : S= A xi xj
i,j

, , - () . "" (meteor). Q = (x - y )/2, U = y . D(t) = (1 + t 2 )t2 (t - 1), N (t) = -(t3 + 3t2 - t + 1), r0 e- ctg(/2) . (68) r= [cos(/2) - sin(/2)]2

=

2 2 -2 x2 y

Q+2

2 U. x y

(69)

P= . ^ 11. A [10]. : (Ai,j ) = 2 - i,j + i,k + j,k , (71) xi xj xk xj xk xi , i,k . i,k 1,1 = 2,2 = 0, 1,2 = -2,1 = 1. = ^ 01 -1 0 . (72)
k,j

A xk xi

i,j

=

2 2 -2 x2 y

U -2

2 Q x y

(70)

(71) : ^ A = (2

- ^ + ( 1) ^ 17

-

), ^

(73)

-, Q=

. U =2 2 2 2 - - . x y x2 y2 (74)

2 2 2 - +2 , 2 2 x y x y

S P S = 2 , P = 2 , (75) , . E -, B -. . § 4. 1. . , . . . . 1969 [95], , . , , [89, 7, 25, 73]. , . . : , , . , 3 · 10 14 1015 M . (1 = 3.086 · 1024 ) . . Te 108 . , , , . .
cc

( ) = n

2 e

k B Te 32 2 Z 2 e6 3 3 c3 (2 mkB Te )

g 3/2 cc

( ) exp(-xe ),

(76) (77)

xe = h /kB Te . , .
cc

=
0

cc

( )d = n

2 e

(kB Te )2 32 2 Z 2 e6 3 3 c3 h(2 mkB Te )

g 3/2 cc

.

(78) , (79)

(78) g 4
cc

cc

= 1.42 · 10

-27

Z2 T

1/2

n

2 e

. 3

, - 7 , . ( Z = 26) 13.6Z 2 = 9.0 , L , , (3/4)9.0 = 6.8 . , , , () : k B Te GM mp M 7 . 2Reff 3 · 1014 M Reff (80)

, 1 = 1.60207 · 10-9 11.6 · 106 (7 8 · 107 ) 12 A (7 1.8 A). . 18

ne , , 10-3 В10-2 1/3. Mgas = mp ne (4 /3)(3 · 1024)3 = 1014 В 1015M , (1/3 В 1/4) . , . , , , (76) (78). , -3/2 -3/2 9 re ye 1 ye 1 = , (81) tCoul = 6 3 2 ne cT ln 4 2 c ln ne re 3 ye 8 2 e2 = . T = re = 6.65 · 10-25 2 , re = = 2.818 · 10-13 3 3 mc2 4 n e re mc2 , ye = . tCoul = 1013 В 1015 . k B Te 1017 . , . § 1, , 1 2h 3 (82) I0 = B (Tr ) = 2 h /k T B r -1 ce Tr = T0 = 2.7277 , ( ) 1.87 . , , , , . , ґ . , . . 2. . ( ) ( [9]). k p. k1 p1 . ck , cp0 , ck1 , cp0 1 . p0 = m 2 c2 + p 2 , p
01 -3/2

=

m 2 c2 + p 2 . 1

(83)

p0 + k = p
01

+ k1 ,

p + k = p 1 + k1 .

(84)

, : p0 + k - k 1 = m2 c2 + (p + k - k1 )2 . (85)

, k p 0 - k p = k 1 p0 - k1 p + k k 1 - k k1 . (86)

k k1 , k p, k1 p µ, 1 . (86) k (p0 - p ) = k1 [p0 - p1 + k (1 - µ)]. k1 = k p0 - p . p0 - p1 + k (1 - µ) (88) (87)

, , . . p = 0. . p0 = mc mc 0 . (89) k1 = k 0 mc + k0 (1 - µ0 ) 19

, k =

h h = , (89) c 0 = 0 + C (1 - µ0 ). 1 (90)

C =
h = 0.024A. mc

(91)

. , , , , , . , , , , . . . , . , . , , , , . . 3. . , , . , . . . . , , 1 k B Te = ye mc2 1, h mc2 1. (92)

. , ( ), , n k B Te n T n e h 4 = + n + n2 . (93) 2 t mc h () n, . , 2h 3 (94) I = 2 n, c (93) , , . (77), , n = n(xe , t) : cT ne 1 n n (95) = x4 + n + n2 . t y e x2 x e e x e e : n , , n 2 , . . , , . 4. . , . , 20

, , , . . . , . (95) (82), n0 (xr ) = e
x
r

1 , -1

(96)

Te h = xe . (97) k B Tr Tr xe , xr , xr /xe = Te /Tr 108 /2.7. , (95) xr = n(xe , t) cT ne 1 = t y e x2 x e x
e 4 e

n(xe , t) xe

.

(98)

teff , Reff = cteff , . cteff ne T = Reff ne T = e . , n = e 1 y e x2 x e x
e 4 e

n0 (xr ) xe

=

e 1 d ye x2 dx r

x
r

4 r

dn0 (xr ) dxr

=

e xr ye (exr - 1)

2

xr -4 . th xr /2

(99)

(94) I = F (x) = x (101) : x22 sx (tx - 4). (103) 2 , x = 0, x , -4.1177 x = xmin = 2.2665, x = x0 = 3.8300, 6.7823 x = xmax = 6.5113, x . kB Tr /h, = hc/(xkB Tr ): min = 2.327 , 0 = 1.377 , max = 0.8101 . 1.6471. , , F (xr ) (100). , F (x) =
3

2h c2

kB T h

r

3

e F (xr ), ye x -4 . th x/2

(100)

1d dn0 (x) x4 x4 = ex x x2 dx dx (e - 1) sx =

2

(101) (102)

x x , tx = sh(x/2) th(x/2)

dx F (x) = x

d dn0 (x) dn0 (x) x4 dx = x4 dx dx dx

= 0.
0

(104)

0

0

. .
x
0

F (x)dx - =
0

F (x)dx = -0.33844, + =
x
0

4 4 /15

4 4 /15

= 1.33844.

(105)

- + + = 1. :

F (x)dx =
0 0

d dn0 (x) x x4 dx = dx dx

dn0 (x) x dx = 4 dx
4

x3 n0 (x)dx = 4
0

4 . 15

(106)

0

21

. , Mgas 0.33844kBTe T cEr = 1.5 · 1042 mc2 mp 1.33844kBTe Mgas L+ = 4 T cEr = 5.7 · 1042 mc2 mp L- = 4 k B Te 5.11 k B Te 5.11 Mgas 1014 M Mgas 1014 M (1 + z )4 /, (1 + z )4 /. (107) (108)

Er = 4.188 · 10-12 /3 , z . L+ /L- = 3.9547. . 5. . . . . , v , , Tr (v ) = Tr (1 - r ), (109)

r = vr /c, vr . , vr /c. , . . vr < 0, , vr > 0 . [15]: I = - B (Tr ) 2(kB Tr )3 T r r e = - r G(xr )e , Tr h2 c 2 G(x) = (110)

x22 x4 e x sx . (111) = 2 (e - 1) 2 . , (99) (100), , (110) ( ) .
x

8 6 4 2 0 -2 -4 -6 0 5 10 15 20

12 10

G(x) F (x)

8 6 4 2 0

r

r

x

. 7. F (x) G(x).

xr 20 . 8. xr .

0

5

10

15

. 7 F (x) G(x). . 8 xr , : r = kB Tr xr /h, r = hc/(kB Tr xr ). 1011 , . . ne (r ) Te (r ), , ye (r ) = mc2 /kB Te (r ) . , (100) F , Y =
T

ne (r ) dl. ye (r ) 22

(112)

, . , x, . 6. . . [26, 62] (. [75]). [74], 1/y e = kB Te /mc2 . , , 1/y e 0.04 : x0 = 3.83 1 + 1.13 ye , x
max

= 6.51 1 +

2.15 ye

.

(113)

( ). y e , . . ґ , ye . [48] 1/ye. [90]. [81, 62]. [81] : I 47 3 11 tx- 4 2t2 + s 2 r 10 - tx + 7 x = G(x) r + + tx- 1 2+ tx- 1 r + e ye 20 20 ye 5 10 (102), , = , xr x. I /
0.25 0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20
e

42 3 tx- 3 2 1 47 tx- 10 - t2+ t3 + 7 s. 2 ye 2 5 x 10 x 5x (114) 2 2 r + t , t = vt /c, vt +

2 x

I /
0.25
2 · 10
8 8

e

0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25

+0.04 +0.02 0

1.5 · 10 1 · 10 5 · 10
8 7

-0.04 -0.02

xr 20 . 9. I /e Te ( ).

0

5

10

15

xr 20 . 10. I /e r .
3

0

5

10

15

2h kB Tr c2 h . , (100), (114), , , . . = 0. , . 10 Te = 108 , . , . , , . , 2 2 , vt ( 2 = r + t ). , , 1/ye . , , , , . , . 2 [63] , e . . 9 I /e 23

, , . , . : 1) ( ) ; 2) ; 3) [49]; 4) . , , . , , . 2 2 e /ye r e . ( e /ye e r ) , (. . ). . , , , 2 e t . 4) . , , , , . . . , [6, 7], . , , [82]. , , , z = 0.5 В 3, . , - 5e . , , . : 1) [60] [94]; 2) ; 4) , ; , ; 5) , - - [54]; 6) , [18]; 7) [38]. (. [25]). , , . 7. . . , , . , [97], [93, 41, 59], [70] . , [19], [72] [28], [29, 55]. , , . . . H0 [75, 39]. . , . . EM = n2 dl. e ne dl.

, A =

, A2 /EM . , A2 /(EM ).

24

c q0 z + (q0 - 1)( 1 + 2q0 z - 1) , (115) dad = 2 H0 q 0 (1 + z )2 q0 , z . , , , [56]. [76]. (115) , q0 , , , . , . , , [91]. Plank . z . [45]. 8. . : ( , , ), ( ) . 1972 [11] . 0.5 ( ). 1, 3.83. , , () [25]. . , 1689 vr = 170+760 /. -570 , . , H0 . ( 24 82 (/)/), 47 (/)/ , , [25]. 30 , . , , , , . , 0.1 2 10 В 100 1 [44]. [85] , Plank 25000 , , 10000, . § 5. . . . - .

25

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§ 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 . . . . 12 13 13 14 17 18 . 19 20 20 22 23 24 25 25

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