15. Hubble Expansion For Pedestrians
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Snippets of Physics 15. Hubble Expansion for Pedestrians T Padmanabhan
M a n y fe a tu r e s o f th e e x p a n d in g u n iv e r se , w h ic h sh o u ld b e le g itim a te ly d isc u sse d u sin g g e n e r a l r e la tiv ity , c a n b e sn e a k e d in b y u sin g N e w to n ia n p h y sic s in a n e x p a n d in g c o o r d in a te sy ste m . In th is sp e c ia l issu e o n H u b b le , I d e sc r ib e s e v e r a l o f th e se fe a tu r e s a lo n g w ith so m e c a u tio n a r y c o m m e n ts. S in ce th e la rg e-sca le d y n a m ics o f th e u n iv erse is essen tia lly g ov ern ed b y g rav ity, a n y th eo retica l m o d el fo r g rav ity w ill h av e im p lica tio n s fo r th e la rg e-sca le p h y sics o f th e u n iv erse. T h is w a s k n ow n , o f co u rse, ev en to N ew to n w h o d id a ttem p t to d escrib e th e u n iv erse u sin g h is id ea s o f g rav ity. W e n ow k n ow , h ow ev er, th a t th e p ro p er d escrip tio n o f g rav ity sh o u ld b e b a sed o n E in stein 's g en era l rela tiv ity ra th er th a n o n N ew to n ia n id ea s. It w a s m en tio n ed in a p rev io u s in sta llm en t th a t th e d escrip tio n o f g rav ity in E in stein 's th eo ry is b a sed o n th e n o tio n o f cu rv ed sp a cetim e. In th e ca se o f th e u n iv erse, th is w ill in v o lv e trea tin g it a s a cu rv ed sp a cetim e w ith a g eo m etry d eterm in ed b y th e d istrib u tio n o f m a tter. T h e sim p lest o f su ch m o d els trea ts th e d istrib u tio n o f m a tter in th e u n iv erse a s u n ifo rm a n d iso tro p ic (a t su ± cien tly la rg e sca les) a n d tries to u n d ersta n d th e p ro p erties o f th e cu rv ed sp a cetim e p ro d u ced b y su ch a m a tter d istrib u tio n . W h ile a ll th ese m ig h t so u n d co m p lica ted , a su rp risin g fea tu re a b o u t su ch a m o d el o f th e u n iv erse is th a t m u ch o f its d y n a m ics ca n b e u n d ersto o d fa irly ea sily { w ith o u t in tro d u cin g co m p lica ted n o tio n s fro m g en era l rela tiv ity. N eed less to say, su ch a n a p p ro a ch is ¯ lled w ith p itfa lls a n d o n e n eed s to co n sta n tly v erify th a t o n e is n o t g ettin g ca rried aw ay b y th e sim p lify in g
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T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Keywords Cosmology, expansion of the universe, structure formation.
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It is not possible to introduce a set of coordinates on the surface of a sphere such that the 'line interval' reduces to the Pythagorean form. This is the difference between curved and flat space.
° av o u r o f N ew to n ia n p h y sics. In th is in sta llm en t, I w ill d escrib e h ow th ese id ea s w o rk . T h e d i® eren ce b etw een a ° a t sp a ce a n d a cu rv ed sp a ce ca n b e en co d ed in th e g en era liza tio n o f P y th a g o ra s th eo rem fo r in ¯ n itesim a lly sep a ra ted p o in ts. F o r ex a m p le, a ° a t 2 -d im en sio n a l su rfa ce (say, a p la in sh eet o f p a p er) a llow s u s to in tro d u ce sta n d a rd C a rtesia n co o rd in a tes (x ;y ) su ch th a t th e d ista n ce b etw een in ¯ n itesim a lly sep a ra ted p o in ts ca n b e ex p ressed in th e fo rm d l2 = d x 2 + d y 2 w h ich , o f co u rse, is ju st th e sta n d a rd P y th a g o ra s th eo rem . In co n tra st, co n sid er th e tw o -d im en sio n a l su rfa ce o n a sp h ere o f ra d iu s r o n w h ich w e h av e in tro d u ced tw o a n g u la r co o rd in a tes (µ ;Á ). T h e co rresp o n d in g fo rm u la w ill n ow rea d d l2 = r 2 d µ 2 + r 2 sin 2 µ d Á 2 . It is n o t p o ssib le to in tro d u ce a n y o th er set o f co o rd in a tes o n th e su rfa ce o f a sp h ere su ch th a t th is ex p ressio n { u su a lly ca lled th e `lin e in terva l' { red u ces to th e P y th a g o rea n fo rm . T h is is th e d i® eren ce b etw een a cu rv ed sp a ce a n d ° a t sp a ce. M ov e o n fro m sp a ce to sp a cetim e a n d fro m p o in ts to ev en ts. In ° a t sp a cetim e, w h ich w e u se in sp ecia l rela tiv ity, th e `P y th a g o ra s th eo rem ' g en era lizes to th e fo rm d s 2 = ¡ c 2 d t2 + d x 2 + d y 2 + d z 2 :
(1 )
T h e sp a tia l co o rd in a tes a p p ea r in th e sta n d a rd fo rm a n d th e in clu sio n o f tim e in tro d u ces th e a ll im p o rta n t m in u s sig n . B u t o n e ca n liv e w ith it an d trea t it a s a g en era liza tio n o f th e fo rm u la d l2 = d x 2 + d y 2 to 4 -d im en sio n s (w ith a n ex tra m in u s sig n ). B u t in a cu rv ed sp a cetim e, th is ex p ressio n w ill n o t h o ld a n d th e co o rd in a te d i® eren tia ls lik e c 2 d t2 ;d x 2 , etc., in th e in terva l w ill g et m u ltip lied b y fu n ctio n s o f sp a ce a n d tim e. T h is is ju st lik e o u r u sin g sin 2 µ d Á 2 ra th er th a n ju st d Á 2 to d escrib e th e cu rv ed 2 -d im en sio n a l su rfa ce o f a sp h ere. T h e p recise m a n n er in w h ich su ch a m o d i¯ ca tio n o ccu rs is d eterm in ed b y E in stein 's eq u a tio n a n d d ep en d s o n th e d istrib u tio n o f m a tter in sp a cetim e.
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W h ile th is ca n lea d to p retty co m p lica ted sp a cetim es in g en era l, th e la rg e-sca le u n iv erse tu rn s o u t to b e d escrib ed b y a rem a rka b ly sim p le g en era liza tio n o f th e lin e in terva l in (1 ). W e o n ly n eed to m o d ify it in to th e fo rm £ ¤ d s 2 = ¡ c 2 d t2 + a 2 (t) d x 2 + d y 2 + d z 2 ; (2 ) w h ere th e fu n ctio n a (t) is ca lled th e `ex p a n sio n fa cto r'. A ll th e in fo rm a tio n a b o u t th e b eh a v io u r o f th e u n iv erse is co n ta in ed in th is sin g le fu n ctio n w h ich { in tu rn { ca n b e d eterm in ed b y E in stein 's eq u a tio n if w e k n ow th e co n ten ts o f th e u n iv erse 1 . H ow ev er, ev en w ith o u t k n ow in g th e ex p licit fo rm o f a (t), o n e ca n ¯ g u re o u t a lo t o f th in g s a b o u t su ch a u n iv erse, a s w e sh a ll see. T h e k ey trick is to n o tice th a t, a t a n y g iv en tim e t, o n e ca n in tro d u ce a n ew sp a tia l co o rd in a te r(t) ´ a (t)x so th a t, a t th is in sta n t o f tim e, th e sp a ce lo o k s ju st lik e w h a t w e a re a ccu sto m ed to in sp ecia l rela tiv ity. T h e r is ca lled `p ro p er co o rd in a te' w h ile x is ca lled th e `co m ov in g co o rd in a te'. S in ce th e sp a ce lo o k s fa m ilia r in term s o f r , o n e u ses sta n d a rd law s o f p h y sics in term s o f th ese p ro p er co o rd in a tes a n d th en tra n sla tes th em b a ck to x , h o p in g fo r th e b est. A m a zin g ly, it w o rk s fo r m o st p u rp o ses.
1
As usual, we are simplifying the universe a little bit; it turns out that you actually need one more number characterizing the curvature of the space to describe the universe completely. But observations show that this number is quite close to zero and hence I will ignore it.
T o b eg in w ith , co n sid er tw o p a rticles lo ca ted a t x 1 a n d x 2 = x 1 + ±x w h ich a re in ¯ n itesim a lly sep a ra ted . T h e co o rd in a te d ista n ce b etw een th ese tw o p a rticles is j± x j w h ile th e p ro p er d ista n ce is j± l(t)j = a (t)j±x j. W e n ow n o te th a t, ev en if th e p a rticles d o n o t m ov e in term s o f x -co o rd in a te (i.e., ea ch p a rticle h a s a ¯ x ed x -co o rd in a te w h ich d o es n o t ch a n g e w ith tim e) th eir proper sep a ra tio n ch a n g es w ith tim e b eca u se o f th e a (t) fa cto r. T h e rela tiv e v elo city a t w h ich th ese p a rticles a re m ov in g fro m ea ch o th er is g iv en b y d ±l a_ = a_± x = ± l : (3 ) ±v = dt a T h is resu lt, a s w e ca n see, is essen tia lly H u b b le's law ! It sh ow s th a t th e tw o p a rticles a re m ov in g aw ay fro m
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ea ch o th er w ith a sp eed p ro p o rtio n a l to th eir sep a ra tio n w h en H (t) ´ (a_= a ) is p o sitiv e. G iv en th is resu lt, o n e ca n o b ta in sev era l o th er in terestin g co n seq u en ces. S u p p o se a n a rrow p en cil o f (n ea rly ) m o n o ch ro m a tic electro m a g n etic ra d ia tio n cro sses th ese tw o co m ov in g o b serv ers lo ca ted a t x 1 a n d x 2 = x 1 + ± x . W e w a n t to k n ow w h a t freq u en cy th ese tw o o b serv ers w ill a ttrib u te to th e electro m a g n etic ra d ia tio n . T h e tim e fo r th e electro m a g n etic ra d ia tio n to trav erse th e d ista n ce ± l w ill b e ± t = ± l= c. L et th e freq u en cy o f th e ra d ia tio n m ea su red b y th e ¯ rst o b serv er b e ! . S in ce th e ¯ rst o b serv er sees th e seco n d o n e to b e recedin g w ith v elo city ± v , sh e w ill ex p ect th e seco n d o b serv er to m ea su re a D o p p ler sh ifted freq u en cy (! + ± ! ), w h ere ±! ±v a_ ± l a_ ±a = ¡ = ¡ = ¡ ±t = ¡ : ! c a c a a
(4 )
H ow d o es o n e in terp ret th is rela tio n ? It sh ow s th a t th e freq u en cy o f electro m a g n etic ra d ia tio n a s m ea su red b y th e co m ov in g o b serv ers ch a n g es w h en th e ex p a n sio n fa cto r a (t) ch a n g es w ith tim e. If ± a is p o sitiv e (i.e., if th e u n iv erse is ex p a n d in g ), ± ! is n eg a tiv e in d ica tin g a red sh ift in th e freq u en cy o f ra d ia tio n . In fa ct, th e a b ov e eq u a tio n ca n b e im m ed ia tely in teg ra ted to g iv e ! (t)a (t) = co n sta n t :
The frequency of electromagnetic radiation changes due to expansion of the universe according to the law ω ∝ a–1.
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(5 )
W e th u s co n clu d e th a t th e freq u en cy o f electro m a g n etic ra d ia tio n ch a n g es d u e to ex p a n sio n o f th e u n iv erse a cco rd in g to th e law ! / a ¡1 . T h is a p p ro a ch w o rk s b eca u se, in a n in ¯ n itesim a l reg io n a ro u n d a n ev en t, o n e ca n a lw ay s u se th e law s o f sp ecia l rela tiv ity. (O n e ca n th in k o f it a s a va ria n t o f th e so -ca lled prin ciple of equ ivalen ce w h ich essen tia lly tells y o u th a t th e g en u in e e® ects o f sp a cetim e cu rva tu re a re seco n d o rd er in th e sep a ra tio n b etw een clo se ev en ts.) T h is is tru e in a n y sp a cetim e b u t w e w ill n o t u su a lly b e a b le to in teg ra te th e lo ca l resu lt a n d o b ta in a g lo b a l law in a g en era l sp a cetim e w h en
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th e g eo m etry va ries fro m p o in t to p o in t in sp a ce. W e co u ld a ch iev e it in th is p a rticu la r ca se b eca u se th e m o d i¯ ca tio n o f sp a cetim e in terva l fro m th e o n e in (1 ) to th e o n e in (2 ) d id n o t in v o lv e a n y fu n ctio n th a t d ep en d ed o n th e spatial co o rd in a tes. In fa ct, o n e ca n o b ta in a sim ila r resu lt fo r a n y p a rticle, n o t ju st p h o to n s. T o d o th is, let u s co n sid er a m a teria l p a rticle w h ich p a sses th e ¯ rst o b serv er w ith v elo city v . W h en it h a s cro ssed th e p ro p er d ista n ce ± l (in a tim e in terva l ± t), it p a sses th e seco n d o b serv er w h o se v elo city (rela tiv e to th e ¯ rst o n e) is ±u =
a_ a_ ±a ±l = vd t = v : a a a
(6 )
T h e v elo city a ttrib u ted to th is p a rticle b y th e seco n d o b serv er ca n b e o b ta in ed b y u sin g th e sp ecia l rela tiv istic law fo r th e a d d itio n o f v elo cities. T h is g iv es · ¸ v ¡ ±u (± u )2 v2 0 v = = v ¡ (1 ¡ 2 )± u + O 1 ¡ (v ± u = c 2 ) c c2 = v ¡ (1 ¡
v 2 ±a )v : c2 a
(7 )
v 2 ±a ) c2 a
(8 )
R ew ritin g th is eq u a tio n a s ± v = ¡ v (1 ¡ a n d in teg ra tin g , w e g et p = p
v co n sta n t = : a 1 ¡ (v 2 = c 2 )
(9 )
In o th er w o rd s, th e m a g n itu d e o f th e 3 -m o m en tu m d ecrea ses a s a ¡ 1 d u e to th e ex p a n sio n . If th e p a rticle is n o n -rela tiv istic, th en v / p a n d v elo city itself d eca y s a s a ¡1 .
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O n ce w e h av e th e sca lin g o f th e m o m en tu m o f p a rticles a n d p h o to n s, o n e ca n p ro ceed fu rth er a n d u n d ersta n d h ow th e en erg y d en sity o f ra d ia tio n ch a n g es in a n ex p a n d in g u n iv erse. T o d o th is, let u s co n sid er th e d escrip tio n o f a b u n ch o f p h o to n s in term s o f a d istrib u tio n fu n ctio n f (r ;p ;t) in p h a se sp a ce. A s u su a l, d N = f (r;p ;t)d 3 rd 3 p g iv es th e n u m b er o f p h o to n s in a sm a ll p h a se v o lu m e. In ex p a n d in g co o rd in a tes, th e sp a tia l v o lu m e in crea ses a s a 3 w h ile, fro m o u r p rev io u s resu lt, w e k n ow th a t th e m o m en tu m sp a ce v o lu m e d ecrea ses a s a ¡3 . S o th e p h a se sp a ce v o lu m e elem en t is in va ria n t a n d { sin ce d N is in va ria n t { th e d istrib u tio n fu n ctio n f rem a in s in va ria n t a s th e u n iv erse ex p a n d s. E x p ressin g th e m o m en tu m o f th e p h o to n a s p = (~! = c)^p , w h ere p^ is a u n it v ecto r in th e d irectio n o f p ro p a g a tio n , w e ¯ n d th a t th e m o m en tu m sp a ce v o lu m e is p ro p o rtio n a l to ! 2 d ! d − w h ere d − d en o tes th e so lid a n g le in th e d irectio n o f th e m o m en tu m p^ . S o w e ca n a lso w rite d N = f (r;p ;t)d 3 rd 3 p / f (r;! ; p^ ;t)! 2 d 3 r d ! d − : (1 0 )
Using our previous result that the distribution function remains invariant, we conclude that ρ (ω) /ω3 remains invariant as the universe expands.
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F u rth er, b eca u se f g iv es th e n u m ber d en sity o f p h o to n s p er u n it p h a se v o lu m e, th e co rresp o n d in g en erg y d en sity is g iv en b y (~! )f . It fo llow s th a t th e en erg y d en sity o f ra d ia tio n p er u n it ra n g e o f freq u en cy is p ro p o rtio n a l to ½ / ! 3 f . U sin g o u r p rev io u s resu lt th a t th e d istrib u tio n fu n ctio n rem a in s in va ria n t, w e co n clu d e th a t ½ (! )= ! 3 rem a in s in va ria n t a s th e u n iv erse ex p a n d s. T h is h a s a v ery in terestin g co n seq u en ce. S u p p o se th e u n iv erse is ¯ lled w ith a ra d ia tio n b a th , th e en erg y d en sity o f w h ich h a s th e fo rm ½ (! ) = ! 3 F (! = ® ), w h ere ® is so m e p a ra m eter a n d F is so m e a rb itra ry fu n ctio n o f its a rg u m en t. A s th e u n iv erse ex p a n d s, ½ = ! 3 rem a in s in va ria n t w h ile ! itself ch a n g es a s ! / a (t)¡1 . If w e d en o te th e va lu es m ea su red to d ay b y a su b scrip t 0 , th en ! 0 = ! (t)a (t)= a 0 . It fo llow s th a t fo r th e fa cto r ! 0 = ® , w e
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ca n w rite !0 ! (t)(a (t)= a 0 ) ! (t) = = : ® ® ® (a 0 = a (t))
As the universe
(1 1 )
T h is sh ow s th a t th e red sh iftin g o f th e freq u en cy ca n b e eq u iva len tly th o u g h t o f a s resca lin g o f th e p a ra m eter ® a s th e u n iv erse ex p a n d s w ith ® (t) / 1 = a (t). S o th e ra d ia tio n en erg y d en sity o f th e fo rm ½ (! ) = ! 3 F (! = ® ) reta in s its sh a p e a s th e u n iv erse ex p a n d s, ex cep t fo r a n ov era ll sca lin g .
expands, a Planck spectrum remains a Planck spectrum with the temperature redshifting according to the law T ∝ a–1.
T h e P la n ck sp ectru m o f ra d ia tio n is o n e sp ecia l ca se in w h ich en erg y d en sity h a s th e a b o v e-m en tio n ed fu n ctio n a l fo rm w ith ½ / ! 3 [ex p (~! = k T ) ¡ 1 ]¡ 1 ´ ! 3 F (! = T ) :
(1 2 )
T h e releva n t p a ra m eter n ow is th e tem p era tu re o f th e ra d ia tio n . T h erefo re, a s th e u n iv erse ex p a n d s, a P la n ck sp ectru m rem a in s a P la n ck sp ectru m w ith th e tem p era tu re red sh iftin g a cco rd in g to th e law T / a ¡1 . It sh o u ld b e stressed th a t th is resu lt h a s n o th in g to d o w ith th erm a l eq u ilib riu m ! In fa ct, th e situ a tio n w e a re co n sid erin g is p recisely th e o th er ex trem e o f th erm a l eq u ilib riu m in w h ich th e ra d ia tio n h a s co m p letely d eco u p led fro m m a tter. T o cla rify th is p o in t, let m e b rie° y d escrib e w h a t h a p p en s in o u r rea l u n iv erse. V ery ea rly in th e ev o lu tio n o f th e u n iv erse, ch a rg ed p a rticles a n d p h o to n s w ere stro n g ly co u p led to ea ch o th er a n d ex isted in th e fo rm o f p la sm a in real th erm o d y n a m ic eq u ilib riu m . S u ch a stro n g co u p lin g im p lies th a t th e ra d ia tio n w ill b e th erm a lized a n d its sp ectra l d istrib u tio n w o u ld h av e th e P la n ck ia n fo rm . L et u s a ssu m e th a t a t so m e in sta n t o f tim e, w e sw itch o ® a ll th e in tera ctio n b etw een ra d ia tio n a n d m a tter (in o u r u n iv erse th is h a p p en ed w h en it w a s a b o u t o n e-th o u sa n d th o f th e p resen t size). F ro m th a t ep o ch o n w a rd s, ea ch o f th e p h o to n s h a s b een p ro p a g a tin g in th e ex p a n d in g u n iv erse w ith its freq u en cy red sh iftin g a cco rd in g to th e law ! / a ¡ 1 . T h e p h o to n s a re
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In Wheelerian language, we get “thermal equilibrium without thermal equilibrium”.
n o t in tera ctin g w ith m a tter: th ere is n o ex ch a n g e o f en erg y a n d th ere is n o p ro cess w h ich `m a in ta in s' th erm a l eq u ilib riu m . N ev erth eless, a t p resen t, th e p h o to n s w ill b e d escrib ed b y a P la n ck d istrib u tio n w ith a red sh ifted tem p era tu re, b eca u se th e fo rm o f th e P la n ck sp ectru m w ill b e p reserv ed w ith th e tem p era tu re o f th e ra d ia tio n d ecrea sin g a s T / a ¡1 u n d er co sm ic ex p a n sio n . T h is is a p u rely k in em a tic e® ect o ccu rrin g fo r p h o to n s w h ich a re p ro p a g a tin g freely th ro u g h th e u n iv erse. In W h eeleria n la n g u a g e, w e g et \ th erm a l eq u ilib riu m w ith o u t th erm a l eq u ilib riu m " . L et u s n ex t ta k e a clo ser lo o k a t th e d y n a m ics o f n o n rela tiv istic p a rticles in su ch a n ex p a n d in g u n iv erse. C o n sid er a p a rticle lo ca ted a t th e co m ov in g co o rd in a te x co rresp o n d in g to th e p ro p er co o rd in a te r (t) = a (t)x . E v en if x d o es n o t ch a n g e w ith tim e { i.e., ev en if th e p a rticle h a s co n sta n t co m o v in g co o rd in a te { its p ro p er co o rd in a te r w ill ch a n g e w ith tim e d u e to a (t). T h is w ill in d u ce a n a ccelera tio n o n th e p a rticle g iv en b y Är = (Äa = a )r. G iv en o u r u su a l p reju d ice th a t a ccelera tio n s a rise d u e to fo rces, it seem s n a tu ra l to a ttrib u te th is a ccelera tio n to th e ex isten ce o f a g lo b a l \ co sm ic p o ten tia l" © = ¡ (1= 2 )(Äa = a )r 2 , so th a t w e ca n w rite Är = ¡ r r © . (T h e su b scrip t r in r r is to rem in d o u rselv es th a t th e g ra d ien t is w ith resp ect to r a n d n o t x ; n o te th a t r r = a ¡ 1 r x .) W ith in th e co n tex t o f su ch N ew to n ia n co n sid era tio n s, w e ca n a ttrib u te th is p o ten tia l to a m a ss d en sity ½ b g (t) su ch th a t r 2r © = 4¼ G ½ b g (t). S im p le d ifferen tia tio n o f © g iv es aÄ 4¼ G = ¡ ½ b g (t) a 3
(1 3 )
w h ich rela tes th e ex p a n sio n fa cto r a (t) to a u n ifo rm b a ck g ro u n d d en sity ½ b g (t) o f m a tter in th e u n iv erse. It a ll seem s n a tu ra l to a ssu m e th a t th e to ta l n u m b er o f p a rticles w ith in a p ro p er v o lu m e sh o u ld n o t ch a n g e a s th e u n iv erse ex p a n d s, th ereb y su g g estin g ½ b g (t) / a ¡ 3 . If y o u su b stitu te th is in to (1 3), th en it is ea sy to sh ow
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th a t a (t) / t2 = 3 ev en th o u g h w e w ill n o t req u ire th is resu lt in o u r d iscu ssio n . In th is a p p ro a ch , w e th in k o f a la rg e n u m b er o f p a rticles b ein g d istrib u ted u n ifo rm ly th ro u g h o u t th e u n iv erse a n d a ttrib u te © to su ch a co llectio n o f p a rticles. T h is is, o f co u rse, co n cep tu a lly a d u b io u s p ro ced u re w ith in th e co n tex t o f N ew to n ia n g rav ity b u t it tu rn s o u t th a t { fo r th e sp eci¯ c ca se u n d er d iscu ssio n { g en era l rela tiv ity lea d s to th e sa m e resu lt. G iv en th is in terp reta tio n , o n e ca n a sk w h a t h a p p en s if w e p ertu rb th e d en sity in th e u n iv erse in a sp a ced ep en d en t m a n n er so th a t ½ b g (t) ! ½ b g (t)[1 + ± (t;x )]. T h e p o ten tia l w ill ch a n g e to ª ´ © + Á w ith th e ex tra b it Á p ro d u ced b y th e ex tra d en sity ½ b g (t)±(t;x ); th a t is: r 2r Á =
1 2 r Á = 4 ¼ G ½ b g (t)± (t;x ) : a2 x
(1 4 )
W e w o u ld lik e to in terp ret th is situ a tio n in term s o f a sy stem o f a la rg e n u m b er o f p a rticles w ith co m ov in g p o sitio n s x i, w h ere i = 1;2;::: la b els ea ch o f th e p a rticles. A s lo n g a s a ll th e p a rticles h av e co n sta n t va lu es fo r x i , a ll o f th em w ill m ov e aw ay fro m ea ch o th er d u e to th e ex p a n sio n w h ich w e a ttrib u te to th e p o ten tia l © . T h is is th e sp a tia lly u n ifo rm d en sity situ a tio n . B u t if th e p a rticles a re d istu rb ed fro m th eir co n sta n t x i va lu es, th en th e m a tter d en sity w ill b eco m e n o n u n ifo rm { w ith ± (t;x ) 6= 0 { a n d th e p o ten tia l © is m o d i¯ ed to ª ´ © + Á . O f co u rse, ea ch o f th e p a rticles w ill n ow feel th e fo rce d u e to th is to ta l p o ten tia l ª . T h e a ccelera tio n o f th e j -th p a rticle, n ow g iv en b y d 2 rj = aÄ x j + 2 aÄ x_ j + a xÄ j d t2
(1 5 )
a rises d u e to th e g ra d ien t o f th e m o d i¯ ed p o ten tia l ª ´ © + Á . W e n o te th a t th e ¯ rst term in th e rig h t-h a n d sid e o f (1 5 ) is (Äa = a )r w h ich is ju st ¡ r r © . T h erefo re, r r Á (= a ¡1 r x Á ) sh o u ld lea d to th e o th er tw o term s in
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Equation (16) is the key to understanding gravitational clustering in an expanding background.
(1 5 ). W ith so m e rea rra n g em en t, th is lea d s to a_ 1 Äx j + 2 x_ j = ¡ 2 r x Á : (1 6 ) a a T h is eq u a tio n tells y o u h ow th e co m ov in g co o rd in a tes o f th e p a rticles ch a n g e th ereb y m a k in g th e d en sity d istrib u tio n o f p a rticles in th e u n iv erse n o n -u n ifo rm w h ich , in tu rn , g iv es rise to r x Á . T h e p o ten tia l Á ca n b e th o u g h t o f a s b ein g g en era ted b y th e p ertu rb a tio n s fro m th e u n ifo rm d en sity o f p a rticles. T h is eq u a tio n is th e k ey to u n d ersta n d in g g rav ita tio n a l clu sterin g in a n ex p a n d in g b a ck g ro u n d . E a ch o f th e term s in th is eq u a tio n h a s a n in terestin g in terp reta tio n . T h e ¯ rst term xÄ j is th e a ccelera tio n co rresp o n d in g to th e co m ov in g p o sitio n o f th e p a rticle, w h ich a rises ov er a n d a b ov e th e a ccelera tio n d u e to th e b a ck g ro u n d ex p a n sio n (th e ¯ rst term in (1 5 ) w h ich w e h av e a lrea d y a cco u n ted fo r b y th e g ra d ien t o f co sm ic p o ten tia l). T h e seco n d term is a d a m p in g (frictio n ) term w h ich tries to d ecrea se th e sp eed o f th e p a rticle. In fa ct, if th e rig h t-h a n d sid e o f (1 6 ) is zero , th e eq u a tio n ca n b e in teg ra ted to g iv e a 2 x_ = co n sta n t o r, a ltern a tiv ely a x_ = 1= a . T h is ca n b e rew ritten in th e fo rm 1 : (1 7 ) a T h e left-h a n d sid e is th e d ev ia tio n o f th e p ro p er v elo city r_ fro m th e H u b b le ex p a n sio n v elo city H r_. T h e q u a n tity a x_ is so m etim es ca lled `p ecu lia r v elo city ' fo r n o g o o d rea so n . T h e rig h t-h a n d sid e sh ow s th a t th is d i® eren ce d ecay s d ow n a 1= a so th a t { in th e a b sen ce o f th e r x Á term o n th e rig h t h a n d sid e o f (1 6 ) { p a rticles w ill ten d to a p p ro a ch th e co sm ic ex p a n sio n v elo city. T h is is th e k ey e® ect o f th e `frictio n ' term . a x_ = r_ ¡ H r =
F in a lly, th e g ra d ien t in th e rig h t h a n d sid e o f (1 6 ) is w h a t is k eep in g th e a ccelera tio n a liv e a n d lea d s to g rav ita tio n a l clu sterin g . H ere to o , th ere is o n e cu rio u s fea tu re. T h e p ertu rb a tio n o f th e b a ck g ro u n d d en sity, ±(t;x ),
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ca n b e eith er p o sitiv e o r n eg a tiv e in a n y g iv en reg io n (ex cep t fo r th e co n d itio n th a t ± > ¡ 1 to k eep ½ > 0 ). T h erefo re, th e so u rce fo r th e p o ten tia l Á in (1 4 ) ca n b e p o sitiv e d en sity o r n eg a tiv e d en sity ! (T h is is so m ew h a t lik e electro sta tics in w h ich th e ch a rg e d en sity ca n b e p o sitiv e o r n eg a tiv e.) S o th e g rav ita tio n a l fo rce p ro d u ced b y th is d istrib u tio n ca n b e a ttra ctiv e o r rep u lsiv e in th e rig h t-h a n d sid e o f (1 6 )! In fa ct a n u n d erd en se reg io n o f th e u n iv erse { u su a lly ca lled a `v o id ' in th e d istrib u tio n o f g a la x ies { ex erts a n e® ectiv e rep u lsiv e fo rce o n th e su rro u n d in g m a tter.
An underdense region of the universe – usually called a ‘void’ in the distribution of galaxies – exerts an effective repulsive force on the surrounding matter.
B y m u ltip ly in g (1 6 ) th ro u g h o u t b y a 2 w e ca n reca st it in th e fo rm d 2 (a x_ j ) = ¡ r x Á : dt
(1 8 )
O b v io u sly, th is eq u a tio n fo r ea ch o f th e p a rticles ca n b e o b ta in ed fro m a n ex p licitly tim e-d ep en d en t L a g ra n g ia n o f th e fo rm ¶ X µ1 2 2 L = m a x_j ¡ m Á ´ K ¡ U ; (1 9 ) 2 j w h ere w e h av e su m m ed ov er a ll th e p a rticles w ith th e u n d ersta n d in g th a t Á a t th e lo ca tio n o f ea ch p a rticle is p ro d u ced b y th e rest o f th e p a rticles. T h is L a g ra n g ia n , in tu rn , w ill lea d to a H a m ilto n ia n o f th e fo rm ¶ X µ p 2j H = + m Á ´ K + U: (2 0 ) 2 2m a j T h is a llow s u s to o b ta in a ra th er in terestin g a n d cu rio u s resu lt. W e ¯ rst reca ll th a t w h en ev er a H a m ilto n ia n d ep en d s ex p licitly o n tim e, w e h av e th e resu lt (d H = d t) = (@ H = @ t) w h ere th e rig h t-h a n d sid e is eva lu a ted k eep in g th e co o rd in a tes a n d m o m en ta co n sta n t. F ro m th e fo rm o f H , w e im m ed ia tely g et (@ K = @ t) = ¡ 2 (a_= a )K . O n th e o th er h a n d , th e p o ten tia l en erg y b etw een a n y tw o p a rticles va ries a s jr i ¡ r j j¡ 1 = a ¡1 jx i ¡ x j j¡1 . T h is im p lies
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th a t (@ U = @ t) = ¡ (a_= a )U . P u ttin g a ll th ese to g eth er, w e g et th e resu lt a_ d (K + U ) = ¡ (2K + U ): dt a
(2 1 )
T h is rela tio n g o es u n d er th e n a m e `C o sm ic V iria l th eo rem ' (o r `C o sm ic E n erg y eq u a tio n ') a n d is o n e o f th e few ex a ct resu lts y o u ca n o b ta in ra th er ea sily in th is p a rticu la r co n tex t. T h e left-h a n d sid e o f th e eq u a tio n rep resen ts th e ra te o f ch a n g e o f to ta l en erg y o f a co llectio n o f p a rticles. T h e rig h t-h a n d sid e tells y o u th a t (K + U ) is n o t co n serv ed fo r su ch a sy stem o f p a rticles ex cep t in tw o d i® eren t co n tex ts. T h e ¯ rst o n e is th e ra th er triv ia l ca se o f a_ = 0 w h ich is ju st sta n d a rd cla ssica l m ech a n ics w ith o u t a n y b a ck g ro u n d ex p a n sio n a n d th e to ta l en erg y is, o f co u rse, co n serv ed . T h e seco n d { a n d m o re cu rio u s situ a tio n { co rresp o n d s to 2 K + U = 0 w h ich y o u w ill reco g n ize is th e sta n d a rd v iria l eq u ilib riu m co n d itio n fo r a set o f p a rticles in tera ctin g v ia N ew to n ia n g rav ity. In p ra ctica l term s, th is resu lt im p lies th e fo llow in g . S u p p o se d u rin g th e ev o lu tio n o f th e u n iv erse a b u n ch o f p a rticles co m e to g eth er a n d fo rm a v iria lized self-g rav ita tin g clu ster. T h en , to th e ex ten t w e ca n ig n o re th e in tera ctio n o f th is clu ster w ith th e rest o f th e p a rticles in th e u n iv erse, its en erg y w ill b e co n serv ed . R o u g h ly sp ea k in g , su ch v iria lized clu sters d o n o t p a rticip a te in th e co sm ic d y n a m ics. F in a lly I w a n t to d escrib e a n in terestin g a n d ex a ct b o u n d o n th e k in etic en erg y o f p a rticles in su ch a clu ster fo rm ed in th e ex p a n d in g u n iv erse, w h ich ca n b e o b ta in ed fro m th e co sm ic v iria l th eo rem . T o d o th is, w e ¯ rst n o te th a t (2 1 ) ca n a lso b e w ritten a s: Virialized clusters do not participate in the cosmic dynamics.
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d a (K + U ) = ¡ K a_ < 0 dt
(2 2 )
S o w e k n ow th a t a (K + U ) is a d ecrea sin g fu n ctio n o f tim e. V ery ea rly o n , w h en n o sig n i¯ ca n t clu sterin g h a s
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o ccu rred , w e w ill h av e K ¼ 0 a n d U < 0 , m a k in g (K + U ) n eg a tiv e v ery ea rly in th e ev o lu tio n . It th en fo llo w s th a t w e m u st h av e K < ¡ U o r ra th er K < jU j a t a ll tim es. N ex t, n o te th a t (2 1 ) ca n a lso b e w ritten a s 1 d 2 a2 dU a (K + U ) = ¡ : dt 2 2 dt
(2 3 )
A s stru ctu res d ev elo p in th e u n iv erse, p o ten tia l w ells w ill g et d eep er a n d d eep er a n d h en ce d (¡ U )= d t > 0 m a k in g th e left-h a n d sid e p o sitiv e. In th e ea rly sta g es, sin ce U ¼ 0 , w e h av e (K + U = 2 ) > 0 . H en ce w e co n clu d e th a t, a t a n y la ter tim e K > ¡ (1= 2 )U o r K > (1 = 2 )jU j. C o m b in in g w ith th e p rev io u s resu lt, w e g et 1 jU j < K < jU j : 2
(2 4 )
T h is is a ra th er n ea t resu lt o n th e rela tio n sh ip b etw een k in etic a n d p o ten tia l en erg ies o f stru ctu res fo rm ed b y g rav ita tio n a l clu sterin g , w h ich m u st h o ld in d ep en d en t o f th e d eta ils o f th e p ro cess! Suggested Reading [1]
P J E Peebles, Large scale structure of the universe, Princeton University Press, 1980, section 24 .
[2]
T Padmanabhan, Structure formation in the universe, Cambridge University Press, 1992, Chapter 4.
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Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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Snippets of Physics 15. Hubble Expansion for Pedestrians T Padmanabhan
M a n y fe a tu r e s o f th e e x p a n d in g u n iv e r se , w h ic h sh o u ld b e le g itim a te ly d isc u sse d u sin g g e n e r a l r e la tiv ity , c a n b e sn e a k e d in b y u sin g N e w to n ia n p h y sic s in a n e x p a n d in g c o o r d in a te sy ste m . In th is sp e c ia l issu e o n H u b b le , I d e sc r ib e s e v e r a l o f th e se fe a tu r e s a lo n g w ith so m e c a u tio n a r y c o m m e n ts. S in ce th e la rg e-sca le d y n a m ics o f th e u n iv erse is essen tia lly g ov ern ed b y g rav ity, a n y th eo retica l m o d el fo r g rav ity w ill h av e im p lica tio n s fo r th e la rg e-sca le p h y sics o f th e u n iv erse. T h is w a s k n ow n , o f co u rse, ev en to N ew to n w h o d id a ttem p t to d escrib e th e u n iv erse u sin g h is id ea s o f g rav ity. W e n ow k n ow , h ow ev er, th a t th e p ro p er d escrip tio n o f g rav ity sh o u ld b e b a sed o n E in stein 's g en era l rela tiv ity ra th er th a n o n N ew to n ia n id ea s. It w a s m en tio n ed in a p rev io u s in sta llm en t th a t th e d escrip tio n o f g rav ity in E in stein 's th eo ry is b a sed o n th e n o tio n o f cu rv ed sp a cetim e. In th e ca se o f th e u n iv erse, th is w ill in v o lv e trea tin g it a s a cu rv ed sp a cetim e w ith a g eo m etry d eterm in ed b y th e d istrib u tio n o f m a tter. T h e sim p lest o f su ch m o d els trea ts th e d istrib u tio n o f m a tter in th e u n iv erse a s u n ifo rm a n d iso tro p ic (a t su ± cien tly la rg e sca les) a n d tries to u n d ersta n d th e p ro p erties o f th e cu rv ed sp a cetim e p ro d u ced b y su ch a m a tter d istrib u tio n . W h ile a ll th ese m ig h t so u n d co m p lica ted , a su rp risin g fea tu re a b o u t su ch a m o d el o f th e u n iv erse is th a t m u ch o f its d y n a m ics ca n b e u n d ersto o d fa irly ea sily { w ith o u t in tro d u cin g co m p lica ted n o tio n s fro m g en era l rela tiv ity. N eed less to say, su ch a n a p p ro a ch is ¯ lled w ith p itfa lls a n d o n e n eed s to co n sta n tly v erify th a t o n e is n o t g ettin g ca rried aw ay b y th e sim p lify in g
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T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.
Keywords Cosmology, expansion of the universe, structure formation.
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It is not possible to introduce a set of coordinates on the surface of a sphere such that the 'line interval' reduces to the Pythagorean form. This is the difference between curved and flat space.
° av o u r o f N ew to n ia n p h y sics. In th is in sta llm en t, I w ill d escrib e h ow th ese id ea s w o rk . T h e d i® eren ce b etw een a ° a t sp a ce a n d a cu rv ed sp a ce ca n b e en co d ed in th e g en era liza tio n o f P y th a g o ra s th eo rem fo r in ¯ n itesim a lly sep a ra ted p o in ts. F o r ex a m p le, a ° a t 2 -d im en sio n a l su rfa ce (say, a p la in sh eet o f p a p er) a llow s u s to in tro d u ce sta n d a rd C a rtesia n co o rd in a tes (x ;y ) su ch th a t th e d ista n ce b etw een in ¯ n itesim a lly sep a ra ted p o in ts ca n b e ex p ressed in th e fo rm d l2 = d x 2 + d y 2 w h ich , o f co u rse, is ju st th e sta n d a rd P y th a g o ra s th eo rem . In co n tra st, co n sid er th e tw o -d im en sio n a l su rfa ce o n a sp h ere o f ra d iu s r o n w h ich w e h av e in tro d u ced tw o a n g u la r co o rd in a tes (µ ;Á ). T h e co rresp o n d in g fo rm u la w ill n ow rea d d l2 = r 2 d µ 2 + r 2 sin 2 µ d Á 2 . It is n o t p o ssib le to in tro d u ce a n y o th er set o f co o rd in a tes o n th e su rfa ce o f a sp h ere su ch th a t th is ex p ressio n { u su a lly ca lled th e `lin e in terva l' { red u ces to th e P y th a g o rea n fo rm . T h is is th e d i® eren ce b etw een a cu rv ed sp a ce a n d ° a t sp a ce. M ov e o n fro m sp a ce to sp a cetim e a n d fro m p o in ts to ev en ts. In ° a t sp a cetim e, w h ich w e u se in sp ecia l rela tiv ity, th e `P y th a g o ra s th eo rem ' g en era lizes to th e fo rm d s 2 = ¡ c 2 d t2 + d x 2 + d y 2 + d z 2 :
(1 )
T h e sp a tia l co o rd in a tes a p p ea r in th e sta n d a rd fo rm a n d th e in clu sio n o f tim e in tro d u ces th e a ll im p o rta n t m in u s sig n . B u t o n e ca n liv e w ith it an d trea t it a s a g en era liza tio n o f th e fo rm u la d l2 = d x 2 + d y 2 to 4 -d im en sio n s (w ith a n ex tra m in u s sig n ). B u t in a cu rv ed sp a cetim e, th is ex p ressio n w ill n o t h o ld a n d th e co o rd in a te d i® eren tia ls lik e c 2 d t2 ;d x 2 , etc., in th e in terva l w ill g et m u ltip lied b y fu n ctio n s o f sp a ce a n d tim e. T h is is ju st lik e o u r u sin g sin 2 µ d Á 2 ra th er th a n ju st d Á 2 to d escrib e th e cu rv ed 2 -d im en sio n a l su rfa ce o f a sp h ere. T h e p recise m a n n er in w h ich su ch a m o d i¯ ca tio n o ccu rs is d eterm in ed b y E in stein 's eq u a tio n a n d d ep en d s o n th e d istrib u tio n o f m a tter in sp a cetim e.
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W h ile th is ca n lea d to p retty co m p lica ted sp a cetim es in g en era l, th e la rg e-sca le u n iv erse tu rn s o u t to b e d escrib ed b y a rem a rka b ly sim p le g en era liza tio n o f th e lin e in terva l in (1 ). W e o n ly n eed to m o d ify it in to th e fo rm £ ¤ d s 2 = ¡ c 2 d t2 + a 2 (t) d x 2 + d y 2 + d z 2 ; (2 ) w h ere th e fu n ctio n a (t) is ca lled th e `ex p a n sio n fa cto r'. A ll th e in fo rm a tio n a b o u t th e b eh a v io u r o f th e u n iv erse is co n ta in ed in th is sin g le fu n ctio n w h ich { in tu rn { ca n b e d eterm in ed b y E in stein 's eq u a tio n if w e k n ow th e co n ten ts o f th e u n iv erse 1 . H ow ev er, ev en w ith o u t k n ow in g th e ex p licit fo rm o f a (t), o n e ca n ¯ g u re o u t a lo t o f th in g s a b o u t su ch a u n iv erse, a s w e sh a ll see. T h e k ey trick is to n o tice th a t, a t a n y g iv en tim e t, o n e ca n in tro d u ce a n ew sp a tia l co o rd in a te r(t) ´ a (t)x so th a t, a t th is in sta n t o f tim e, th e sp a ce lo o k s ju st lik e w h a t w e a re a ccu sto m ed to in sp ecia l rela tiv ity. T h e r is ca lled `p ro p er co o rd in a te' w h ile x is ca lled th e `co m ov in g co o rd in a te'. S in ce th e sp a ce lo o k s fa m ilia r in term s o f r , o n e u ses sta n d a rd law s o f p h y sics in term s o f th ese p ro p er co o rd in a tes a n d th en tra n sla tes th em b a ck to x , h o p in g fo r th e b est. A m a zin g ly, it w o rk s fo r m o st p u rp o ses.
1
As usual, we are simplifying the universe a little bit; it turns out that you actually need one more number characterizing the curvature of the space to describe the universe completely. But observations show that this number is quite close to zero and hence I will ignore it.
T o b eg in w ith , co n sid er tw o p a rticles lo ca ted a t x 1 a n d x 2 = x 1 + ±x w h ich a re in ¯ n itesim a lly sep a ra ted . T h e co o rd in a te d ista n ce b etw een th ese tw o p a rticles is j± x j w h ile th e p ro p er d ista n ce is j± l(t)j = a (t)j±x j. W e n ow n o te th a t, ev en if th e p a rticles d o n o t m ov e in term s o f x -co o rd in a te (i.e., ea ch p a rticle h a s a ¯ x ed x -co o rd in a te w h ich d o es n o t ch a n g e w ith tim e) th eir proper sep a ra tio n ch a n g es w ith tim e b eca u se o f th e a (t) fa cto r. T h e rela tiv e v elo city a t w h ich th ese p a rticles a re m ov in g fro m ea ch o th er is g iv en b y d ±l a_ = a_± x = ± l : (3 ) ±v = dt a T h is resu lt, a s w e ca n see, is essen tia lly H u b b le's law ! It sh ow s th a t th e tw o p a rticles a re m ov in g aw ay fro m
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ea ch o th er w ith a sp eed p ro p o rtio n a l to th eir sep a ra tio n w h en H (t) ´ (a_= a ) is p o sitiv e. G iv en th is resu lt, o n e ca n o b ta in sev era l o th er in terestin g co n seq u en ces. S u p p o se a n a rrow p en cil o f (n ea rly ) m o n o ch ro m a tic electro m a g n etic ra d ia tio n cro sses th ese tw o co m ov in g o b serv ers lo ca ted a t x 1 a n d x 2 = x 1 + ± x . W e w a n t to k n ow w h a t freq u en cy th ese tw o o b serv ers w ill a ttrib u te to th e electro m a g n etic ra d ia tio n . T h e tim e fo r th e electro m a g n etic ra d ia tio n to trav erse th e d ista n ce ± l w ill b e ± t = ± l= c. L et th e freq u en cy o f th e ra d ia tio n m ea su red b y th e ¯ rst o b serv er b e ! . S in ce th e ¯ rst o b serv er sees th e seco n d o n e to b e recedin g w ith v elo city ± v , sh e w ill ex p ect th e seco n d o b serv er to m ea su re a D o p p ler sh ifted freq u en cy (! + ± ! ), w h ere ±! ±v a_ ± l a_ ±a = ¡ = ¡ = ¡ ±t = ¡ : ! c a c a a
(4 )
H ow d o es o n e in terp ret th is rela tio n ? It sh ow s th a t th e freq u en cy o f electro m a g n etic ra d ia tio n a s m ea su red b y th e co m ov in g o b serv ers ch a n g es w h en th e ex p a n sio n fa cto r a (t) ch a n g es w ith tim e. If ± a is p o sitiv e (i.e., if th e u n iv erse is ex p a n d in g ), ± ! is n eg a tiv e in d ica tin g a red sh ift in th e freq u en cy o f ra d ia tio n . In fa ct, th e a b ov e eq u a tio n ca n b e im m ed ia tely in teg ra ted to g iv e ! (t)a (t) = co n sta n t :
The frequency of electromagnetic radiation changes due to expansion of the universe according to the law ω ∝ a–1.
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(5 )
W e th u s co n clu d e th a t th e freq u en cy o f electro m a g n etic ra d ia tio n ch a n g es d u e to ex p a n sio n o f th e u n iv erse a cco rd in g to th e law ! / a ¡1 . T h is a p p ro a ch w o rk s b eca u se, in a n in ¯ n itesim a l reg io n a ro u n d a n ev en t, o n e ca n a lw ay s u se th e law s o f sp ecia l rela tiv ity. (O n e ca n th in k o f it a s a va ria n t o f th e so -ca lled prin ciple of equ ivalen ce w h ich essen tia lly tells y o u th a t th e g en u in e e® ects o f sp a cetim e cu rva tu re a re seco n d o rd er in th e sep a ra tio n b etw een clo se ev en ts.) T h is is tru e in a n y sp a cetim e b u t w e w ill n o t u su a lly b e a b le to in teg ra te th e lo ca l resu lt a n d o b ta in a g lo b a l law in a g en era l sp a cetim e w h en
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th e g eo m etry va ries fro m p o in t to p o in t in sp a ce. W e co u ld a ch iev e it in th is p a rticu la r ca se b eca u se th e m o d i¯ ca tio n o f sp a cetim e in terva l fro m th e o n e in (1 ) to th e o n e in (2 ) d id n o t in v o lv e a n y fu n ctio n th a t d ep en d ed o n th e spatial co o rd in a tes. In fa ct, o n e ca n o b ta in a sim ila r resu lt fo r a n y p a rticle, n o t ju st p h o to n s. T o d o th is, let u s co n sid er a m a teria l p a rticle w h ich p a sses th e ¯ rst o b serv er w ith v elo city v . W h en it h a s cro ssed th e p ro p er d ista n ce ± l (in a tim e in terva l ± t), it p a sses th e seco n d o b serv er w h o se v elo city (rela tiv e to th e ¯ rst o n e) is ±u =
a_ a_ ±a ±l = vd t = v : a a a
(6 )
T h e v elo city a ttrib u ted to th is p a rticle b y th e seco n d o b serv er ca n b e o b ta in ed b y u sin g th e sp ecia l rela tiv istic law fo r th e a d d itio n o f v elo cities. T h is g iv es · ¸ v ¡ ±u (± u )2 v2 0 v = = v ¡ (1 ¡ 2 )± u + O 1 ¡ (v ± u = c 2 ) c c2 = v ¡ (1 ¡
v 2 ±a )v : c2 a
(7 )
v 2 ±a ) c2 a
(8 )
R ew ritin g th is eq u a tio n a s ± v = ¡ v (1 ¡ a n d in teg ra tin g , w e g et p = p
v co n sta n t = : a 1 ¡ (v 2 = c 2 )
(9 )
In o th er w o rd s, th e m a g n itu d e o f th e 3 -m o m en tu m d ecrea ses a s a ¡ 1 d u e to th e ex p a n sio n . If th e p a rticle is n o n -rela tiv istic, th en v / p a n d v elo city itself d eca y s a s a ¡1 .
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O n ce w e h av e th e sca lin g o f th e m o m en tu m o f p a rticles a n d p h o to n s, o n e ca n p ro ceed fu rth er a n d u n d ersta n d h ow th e en erg y d en sity o f ra d ia tio n ch a n g es in a n ex p a n d in g u n iv erse. T o d o th is, let u s co n sid er th e d escrip tio n o f a b u n ch o f p h o to n s in term s o f a d istrib u tio n fu n ctio n f (r ;p ;t) in p h a se sp a ce. A s u su a l, d N = f (r;p ;t)d 3 rd 3 p g iv es th e n u m b er o f p h o to n s in a sm a ll p h a se v o lu m e. In ex p a n d in g co o rd in a tes, th e sp a tia l v o lu m e in crea ses a s a 3 w h ile, fro m o u r p rev io u s resu lt, w e k n ow th a t th e m o m en tu m sp a ce v o lu m e d ecrea ses a s a ¡3 . S o th e p h a se sp a ce v o lu m e elem en t is in va ria n t a n d { sin ce d N is in va ria n t { th e d istrib u tio n fu n ctio n f rem a in s in va ria n t a s th e u n iv erse ex p a n d s. E x p ressin g th e m o m en tu m o f th e p h o to n a s p = (~! = c)^p , w h ere p^ is a u n it v ecto r in th e d irectio n o f p ro p a g a tio n , w e ¯ n d th a t th e m o m en tu m sp a ce v o lu m e is p ro p o rtio n a l to ! 2 d ! d − w h ere d − d en o tes th e so lid a n g le in th e d irectio n o f th e m o m en tu m p^ . S o w e ca n a lso w rite d N = f (r;p ;t)d 3 rd 3 p / f (r;! ; p^ ;t)! 2 d 3 r d ! d − : (1 0 )
Using our previous result that the distribution function remains invariant, we conclude that ρ (ω) /ω3 remains invariant as the universe expands.
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F u rth er, b eca u se f g iv es th e n u m ber d en sity o f p h o to n s p er u n it p h a se v o lu m e, th e co rresp o n d in g en erg y d en sity is g iv en b y (~! )f . It fo llow s th a t th e en erg y d en sity o f ra d ia tio n p er u n it ra n g e o f freq u en cy is p ro p o rtio n a l to ½ / ! 3 f . U sin g o u r p rev io u s resu lt th a t th e d istrib u tio n fu n ctio n rem a in s in va ria n t, w e co n clu d e th a t ½ (! )= ! 3 rem a in s in va ria n t a s th e u n iv erse ex p a n d s. T h is h a s a v ery in terestin g co n seq u en ce. S u p p o se th e u n iv erse is ¯ lled w ith a ra d ia tio n b a th , th e en erg y d en sity o f w h ich h a s th e fo rm ½ (! ) = ! 3 F (! = ® ), w h ere ® is so m e p a ra m eter a n d F is so m e a rb itra ry fu n ctio n o f its a rg u m en t. A s th e u n iv erse ex p a n d s, ½ = ! 3 rem a in s in va ria n t w h ile ! itself ch a n g es a s ! / a (t)¡1 . If w e d en o te th e va lu es m ea su red to d ay b y a su b scrip t 0 , th en ! 0 = ! (t)a (t)= a 0 . It fo llow s th a t fo r th e fa cto r ! 0 = ® , w e
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ca n w rite !0 ! (t)(a (t)= a 0 ) ! (t) = = : ® ® ® (a 0 = a (t))
As the universe
(1 1 )
T h is sh ow s th a t th e red sh iftin g o f th e freq u en cy ca n b e eq u iva len tly th o u g h t o f a s resca lin g o f th e p a ra m eter ® a s th e u n iv erse ex p a n d s w ith ® (t) / 1 = a (t). S o th e ra d ia tio n en erg y d en sity o f th e fo rm ½ (! ) = ! 3 F (! = ® ) reta in s its sh a p e a s th e u n iv erse ex p a n d s, ex cep t fo r a n ov era ll sca lin g .
expands, a Planck spectrum remains a Planck spectrum with the temperature redshifting according to the law T ∝ a–1.
T h e P la n ck sp ectru m o f ra d ia tio n is o n e sp ecia l ca se in w h ich en erg y d en sity h a s th e a b o v e-m en tio n ed fu n ctio n a l fo rm w ith ½ / ! 3 [ex p (~! = k T ) ¡ 1 ]¡ 1 ´ ! 3 F (! = T ) :
(1 2 )
T h e releva n t p a ra m eter n ow is th e tem p era tu re o f th e ra d ia tio n . T h erefo re, a s th e u n iv erse ex p a n d s, a P la n ck sp ectru m rem a in s a P la n ck sp ectru m w ith th e tem p era tu re red sh iftin g a cco rd in g to th e law T / a ¡1 . It sh o u ld b e stressed th a t th is resu lt h a s n o th in g to d o w ith th erm a l eq u ilib riu m ! In fa ct, th e situ a tio n w e a re co n sid erin g is p recisely th e o th er ex trem e o f th erm a l eq u ilib riu m in w h ich th e ra d ia tio n h a s co m p letely d eco u p led fro m m a tter. T o cla rify th is p o in t, let m e b rie° y d escrib e w h a t h a p p en s in o u r rea l u n iv erse. V ery ea rly in th e ev o lu tio n o f th e u n iv erse, ch a rg ed p a rticles a n d p h o to n s w ere stro n g ly co u p led to ea ch o th er a n d ex isted in th e fo rm o f p la sm a in real th erm o d y n a m ic eq u ilib riu m . S u ch a stro n g co u p lin g im p lies th a t th e ra d ia tio n w ill b e th erm a lized a n d its sp ectra l d istrib u tio n w o u ld h av e th e P la n ck ia n fo rm . L et u s a ssu m e th a t a t so m e in sta n t o f tim e, w e sw itch o ® a ll th e in tera ctio n b etw een ra d ia tio n a n d m a tter (in o u r u n iv erse th is h a p p en ed w h en it w a s a b o u t o n e-th o u sa n d th o f th e p resen t size). F ro m th a t ep o ch o n w a rd s, ea ch o f th e p h o to n s h a s b een p ro p a g a tin g in th e ex p a n d in g u n iv erse w ith its freq u en cy red sh iftin g a cco rd in g to th e law ! / a ¡ 1 . T h e p h o to n s a re
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In Wheelerian language, we get “thermal equilibrium without thermal equilibrium”.
n o t in tera ctin g w ith m a tter: th ere is n o ex ch a n g e o f en erg y a n d th ere is n o p ro cess w h ich `m a in ta in s' th erm a l eq u ilib riu m . N ev erth eless, a t p resen t, th e p h o to n s w ill b e d escrib ed b y a P la n ck d istrib u tio n w ith a red sh ifted tem p era tu re, b eca u se th e fo rm o f th e P la n ck sp ectru m w ill b e p reserv ed w ith th e tem p era tu re o f th e ra d ia tio n d ecrea sin g a s T / a ¡1 u n d er co sm ic ex p a n sio n . T h is is a p u rely k in em a tic e® ect o ccu rrin g fo r p h o to n s w h ich a re p ro p a g a tin g freely th ro u g h th e u n iv erse. In W h eeleria n la n g u a g e, w e g et \ th erm a l eq u ilib riu m w ith o u t th erm a l eq u ilib riu m " . L et u s n ex t ta k e a clo ser lo o k a t th e d y n a m ics o f n o n rela tiv istic p a rticles in su ch a n ex p a n d in g u n iv erse. C o n sid er a p a rticle lo ca ted a t th e co m ov in g co o rd in a te x co rresp o n d in g to th e p ro p er co o rd in a te r (t) = a (t)x . E v en if x d o es n o t ch a n g e w ith tim e { i.e., ev en if th e p a rticle h a s co n sta n t co m o v in g co o rd in a te { its p ro p er co o rd in a te r w ill ch a n g e w ith tim e d u e to a (t). T h is w ill in d u ce a n a ccelera tio n o n th e p a rticle g iv en b y Är = (Äa = a )r. G iv en o u r u su a l p reju d ice th a t a ccelera tio n s a rise d u e to fo rces, it seem s n a tu ra l to a ttrib u te th is a ccelera tio n to th e ex isten ce o f a g lo b a l \ co sm ic p o ten tia l" © = ¡ (1= 2 )(Äa = a )r 2 , so th a t w e ca n w rite Är = ¡ r r © . (T h e su b scrip t r in r r is to rem in d o u rselv es th a t th e g ra d ien t is w ith resp ect to r a n d n o t x ; n o te th a t r r = a ¡ 1 r x .) W ith in th e co n tex t o f su ch N ew to n ia n co n sid era tio n s, w e ca n a ttrib u te th is p o ten tia l to a m a ss d en sity ½ b g (t) su ch th a t r 2r © = 4¼ G ½ b g (t). S im p le d ifferen tia tio n o f © g iv es aÄ 4¼ G = ¡ ½ b g (t) a 3
(1 3 )
w h ich rela tes th e ex p a n sio n fa cto r a (t) to a u n ifo rm b a ck g ro u n d d en sity ½ b g (t) o f m a tter in th e u n iv erse. It a ll seem s n a tu ra l to a ssu m e th a t th e to ta l n u m b er o f p a rticles w ith in a p ro p er v o lu m e sh o u ld n o t ch a n g e a s th e u n iv erse ex p a n d s, th ereb y su g g estin g ½ b g (t) / a ¡ 3 . If y o u su b stitu te th is in to (1 3), th en it is ea sy to sh ow
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th a t a (t) / t2 = 3 ev en th o u g h w e w ill n o t req u ire th is resu lt in o u r d iscu ssio n . In th is a p p ro a ch , w e th in k o f a la rg e n u m b er o f p a rticles b ein g d istrib u ted u n ifo rm ly th ro u g h o u t th e u n iv erse a n d a ttrib u te © to su ch a co llectio n o f p a rticles. T h is is, o f co u rse, co n cep tu a lly a d u b io u s p ro ced u re w ith in th e co n tex t o f N ew to n ia n g rav ity b u t it tu rn s o u t th a t { fo r th e sp eci¯ c ca se u n d er d iscu ssio n { g en era l rela tiv ity lea d s to th e sa m e resu lt. G iv en th is in terp reta tio n , o n e ca n a sk w h a t h a p p en s if w e p ertu rb th e d en sity in th e u n iv erse in a sp a ced ep en d en t m a n n er so th a t ½ b g (t) ! ½ b g (t)[1 + ± (t;x )]. T h e p o ten tia l w ill ch a n g e to ª ´ © + Á w ith th e ex tra b it Á p ro d u ced b y th e ex tra d en sity ½ b g (t)±(t;x ); th a t is: r 2r Á =
1 2 r Á = 4 ¼ G ½ b g (t)± (t;x ) : a2 x
(1 4 )
W e w o u ld lik e to in terp ret th is situ a tio n in term s o f a sy stem o f a la rg e n u m b er o f p a rticles w ith co m ov in g p o sitio n s x i, w h ere i = 1;2;::: la b els ea ch o f th e p a rticles. A s lo n g a s a ll th e p a rticles h av e co n sta n t va lu es fo r x i , a ll o f th em w ill m ov e aw ay fro m ea ch o th er d u e to th e ex p a n sio n w h ich w e a ttrib u te to th e p o ten tia l © . T h is is th e sp a tia lly u n ifo rm d en sity situ a tio n . B u t if th e p a rticles a re d istu rb ed fro m th eir co n sta n t x i va lu es, th en th e m a tter d en sity w ill b eco m e n o n u n ifo rm { w ith ± (t;x ) 6= 0 { a n d th e p o ten tia l © is m o d i¯ ed to ª ´ © + Á . O f co u rse, ea ch o f th e p a rticles w ill n ow feel th e fo rce d u e to th is to ta l p o ten tia l ª . T h e a ccelera tio n o f th e j -th p a rticle, n ow g iv en b y d 2 rj = aÄ x j + 2 aÄ x_ j + a xÄ j d t2
(1 5 )
a rises d u e to th e g ra d ien t o f th e m o d i¯ ed p o ten tia l ª ´ © + Á . W e n o te th a t th e ¯ rst term in th e rig h t-h a n d sid e o f (1 5 ) is (Äa = a )r w h ich is ju st ¡ r r © . T h erefo re, r r Á (= a ¡1 r x Á ) sh o u ld lea d to th e o th er tw o term s in
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Equation (16) is the key to understanding gravitational clustering in an expanding background.
(1 5 ). W ith so m e rea rra n g em en t, th is lea d s to a_ 1 Äx j + 2 x_ j = ¡ 2 r x Á : (1 6 ) a a T h is eq u a tio n tells y o u h ow th e co m ov in g co o rd in a tes o f th e p a rticles ch a n g e th ereb y m a k in g th e d en sity d istrib u tio n o f p a rticles in th e u n iv erse n o n -u n ifo rm w h ich , in tu rn , g iv es rise to r x Á . T h e p o ten tia l Á ca n b e th o u g h t o f a s b ein g g en era ted b y th e p ertu rb a tio n s fro m th e u n ifo rm d en sity o f p a rticles. T h is eq u a tio n is th e k ey to u n d ersta n d in g g rav ita tio n a l clu sterin g in a n ex p a n d in g b a ck g ro u n d . E a ch o f th e term s in th is eq u a tio n h a s a n in terestin g in terp reta tio n . T h e ¯ rst term xÄ j is th e a ccelera tio n co rresp o n d in g to th e co m ov in g p o sitio n o f th e p a rticle, w h ich a rises ov er a n d a b ov e th e a ccelera tio n d u e to th e b a ck g ro u n d ex p a n sio n (th e ¯ rst term in (1 5 ) w h ich w e h av e a lrea d y a cco u n ted fo r b y th e g ra d ien t o f co sm ic p o ten tia l). T h e seco n d term is a d a m p in g (frictio n ) term w h ich tries to d ecrea se th e sp eed o f th e p a rticle. In fa ct, if th e rig h t-h a n d sid e o f (1 6 ) is zero , th e eq u a tio n ca n b e in teg ra ted to g iv e a 2 x_ = co n sta n t o r, a ltern a tiv ely a x_ = 1= a . T h is ca n b e rew ritten in th e fo rm 1 : (1 7 ) a T h e left-h a n d sid e is th e d ev ia tio n o f th e p ro p er v elo city r_ fro m th e H u b b le ex p a n sio n v elo city H r_. T h e q u a n tity a x_ is so m etim es ca lled `p ecu lia r v elo city ' fo r n o g o o d rea so n . T h e rig h t-h a n d sid e sh ow s th a t th is d i® eren ce d ecay s d ow n a 1= a so th a t { in th e a b sen ce o f th e r x Á term o n th e rig h t h a n d sid e o f (1 6 ) { p a rticles w ill ten d to a p p ro a ch th e co sm ic ex p a n sio n v elo city. T h is is th e k ey e® ect o f th e `frictio n ' term . a x_ = r_ ¡ H r =
F in a lly, th e g ra d ien t in th e rig h t h a n d sid e o f (1 6 ) is w h a t is k eep in g th e a ccelera tio n a liv e a n d lea d s to g rav ita tio n a l clu sterin g . H ere to o , th ere is o n e cu rio u s fea tu re. T h e p ertu rb a tio n o f th e b a ck g ro u n d d en sity, ±(t;x ),
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ca n b e eith er p o sitiv e o r n eg a tiv e in a n y g iv en reg io n (ex cep t fo r th e co n d itio n th a t ± > ¡ 1 to k eep ½ > 0 ). T h erefo re, th e so u rce fo r th e p o ten tia l Á in (1 4 ) ca n b e p o sitiv e d en sity o r n eg a tiv e d en sity ! (T h is is so m ew h a t lik e electro sta tics in w h ich th e ch a rg e d en sity ca n b e p o sitiv e o r n eg a tiv e.) S o th e g rav ita tio n a l fo rce p ro d u ced b y th is d istrib u tio n ca n b e a ttra ctiv e o r rep u lsiv e in th e rig h t-h a n d sid e o f (1 6 )! In fa ct a n u n d erd en se reg io n o f th e u n iv erse { u su a lly ca lled a `v o id ' in th e d istrib u tio n o f g a la x ies { ex erts a n e® ectiv e rep u lsiv e fo rce o n th e su rro u n d in g m a tter.
An underdense region of the universe – usually called a ‘void’ in the distribution of galaxies – exerts an effective repulsive force on the surrounding matter.
B y m u ltip ly in g (1 6 ) th ro u g h o u t b y a 2 w e ca n reca st it in th e fo rm d 2 (a x_ j ) = ¡ r x Á : dt
(1 8 )
O b v io u sly, th is eq u a tio n fo r ea ch o f th e p a rticles ca n b e o b ta in ed fro m a n ex p licitly tim e-d ep en d en t L a g ra n g ia n o f th e fo rm ¶ X µ1 2 2 L = m a x_j ¡ m Á ´ K ¡ U ; (1 9 ) 2 j w h ere w e h av e su m m ed ov er a ll th e p a rticles w ith th e u n d ersta n d in g th a t Á a t th e lo ca tio n o f ea ch p a rticle is p ro d u ced b y th e rest o f th e p a rticles. T h is L a g ra n g ia n , in tu rn , w ill lea d to a H a m ilto n ia n o f th e fo rm ¶ X µ p 2j H = + m Á ´ K + U: (2 0 ) 2 2m a j T h is a llow s u s to o b ta in a ra th er in terestin g a n d cu rio u s resu lt. W e ¯ rst reca ll th a t w h en ev er a H a m ilto n ia n d ep en d s ex p licitly o n tim e, w e h av e th e resu lt (d H = d t) = (@ H = @ t) w h ere th e rig h t-h a n d sid e is eva lu a ted k eep in g th e co o rd in a tes a n d m o m en ta co n sta n t. F ro m th e fo rm o f H , w e im m ed ia tely g et (@ K = @ t) = ¡ 2 (a_= a )K . O n th e o th er h a n d , th e p o ten tia l en erg y b etw een a n y tw o p a rticles va ries a s jr i ¡ r j j¡ 1 = a ¡1 jx i ¡ x j j¡1 . T h is im p lies
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th a t (@ U = @ t) = ¡ (a_= a )U . P u ttin g a ll th ese to g eth er, w e g et th e resu lt a_ d (K + U ) = ¡ (2K + U ): dt a
(2 1 )
T h is rela tio n g o es u n d er th e n a m e `C o sm ic V iria l th eo rem ' (o r `C o sm ic E n erg y eq u a tio n ') a n d is o n e o f th e few ex a ct resu lts y o u ca n o b ta in ra th er ea sily in th is p a rticu la r co n tex t. T h e left-h a n d sid e o f th e eq u a tio n rep resen ts th e ra te o f ch a n g e o f to ta l en erg y o f a co llectio n o f p a rticles. T h e rig h t-h a n d sid e tells y o u th a t (K + U ) is n o t co n serv ed fo r su ch a sy stem o f p a rticles ex cep t in tw o d i® eren t co n tex ts. T h e ¯ rst o n e is th e ra th er triv ia l ca se o f a_ = 0 w h ich is ju st sta n d a rd cla ssica l m ech a n ics w ith o u t a n y b a ck g ro u n d ex p a n sio n a n d th e to ta l en erg y is, o f co u rse, co n serv ed . T h e seco n d { a n d m o re cu rio u s situ a tio n { co rresp o n d s to 2 K + U = 0 w h ich y o u w ill reco g n ize is th e sta n d a rd v iria l eq u ilib riu m co n d itio n fo r a set o f p a rticles in tera ctin g v ia N ew to n ia n g rav ity. In p ra ctica l term s, th is resu lt im p lies th e fo llow in g . S u p p o se d u rin g th e ev o lu tio n o f th e u n iv erse a b u n ch o f p a rticles co m e to g eth er a n d fo rm a v iria lized self-g rav ita tin g clu ster. T h en , to th e ex ten t w e ca n ig n o re th e in tera ctio n o f th is clu ster w ith th e rest o f th e p a rticles in th e u n iv erse, its en erg y w ill b e co n serv ed . R o u g h ly sp ea k in g , su ch v iria lized clu sters d o n o t p a rticip a te in th e co sm ic d y n a m ics. F in a lly I w a n t to d escrib e a n in terestin g a n d ex a ct b o u n d o n th e k in etic en erg y o f p a rticles in su ch a clu ster fo rm ed in th e ex p a n d in g u n iv erse, w h ich ca n b e o b ta in ed fro m th e co sm ic v iria l th eo rem . T o d o th is, w e ¯ rst n o te th a t (2 1 ) ca n a lso b e w ritten a s: Virialized clusters do not participate in the cosmic dynamics.
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d a (K + U ) = ¡ K a_ < 0 dt
(2 2 )
S o w e k n ow th a t a (K + U ) is a d ecrea sin g fu n ctio n o f tim e. V ery ea rly o n , w h en n o sig n i¯ ca n t clu sterin g h a s
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o ccu rred , w e w ill h av e K ¼ 0 a n d U < 0 , m a k in g (K + U ) n eg a tiv e v ery ea rly in th e ev o lu tio n . It th en fo llo w s th a t w e m u st h av e K < ¡ U o r ra th er K < jU j a t a ll tim es. N ex t, n o te th a t (2 1 ) ca n a lso b e w ritten a s 1 d 2 a2 dU a (K + U ) = ¡ : dt 2 2 dt
(2 3 )
A s stru ctu res d ev elo p in th e u n iv erse, p o ten tia l w ells w ill g et d eep er a n d d eep er a n d h en ce d (¡ U )= d t > 0 m a k in g th e left-h a n d sid e p o sitiv e. In th e ea rly sta g es, sin ce U ¼ 0 , w e h av e (K + U = 2 ) > 0 . H en ce w e co n clu d e th a t, a t a n y la ter tim e K > ¡ (1= 2 )U o r K > (1 = 2 )jU j. C o m b in in g w ith th e p rev io u s resu lt, w e g et 1 jU j < K < jU j : 2
(2 4 )
T h is is a ra th er n ea t resu lt o n th e rela tio n sh ip b etw een k in etic a n d p o ten tia l en erg ies o f stru ctu res fo rm ed b y g rav ita tio n a l clu sterin g , w h ich m u st h o ld in d ep en d en t o f th e d eta ils o f th e p ro cess! Suggested Reading [1]
P J E Peebles, Large scale structure of the universe, Princeton University Press, 1980, section 24 .
[2]
T Padmanabhan, Structure formation in the universe, Cambridge University Press, 1992, Chapter 4.
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Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]
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