GENERAL ⎜ ARTICLE Hubble, Hubble’s Law and the Expanding Universe J S Bagla H ubble'snam e isassociated closelyw ith theidea of an expanding universe as he discovered the relation b etw een th e recession velocity an d th e distances of galaxies. H ubble also did a lot of pioneering w ork on the distribution of galaxies in th e u n iverse. In th is article w e take a look Jasjeet Bagla works at the Harish-Chandra Research at H ubble'slaw and discuss how itrelates w ith Institute, Allahabad. His m odels of the universe. W e also give a histori- research is mainly on calperspectiveof the discoveries that led to the cosmology and he is H ubble'slaw . interested in all areas of physics. 1 . H u b b le 's L a w Edw in P H ubbleisbestknow n for hisdiscovery ofthe relation sh ip b etw een th e d istan ce an d rad ial velocities of galaxies. A llm odels of the universe are based on thisrelationship,now know n as `Hubble'slaw '.H ubble fou n d th at th e rate at w h ich galax ies are reced ing from u s is p rop ortion al to th e d istan ce, V / r, and used ob servation s to d eterm ine th e p rop ortion ality con stant. T hisconstant isnow called the `Hubble'sconstant'in hishonour. V = H 0 r: T h is form for th e relation sh ip h as im p ortan t im p lica- tion s (see Box1). W e step back a littleand ¯llinsom e background before continuingw ith our discussion oftheH ubble'slaw . T h e early p art of th e 20th cen tu ry saw con siderab le d is- Keywords: cussion and activityfocused on understan d ing th e stru c- Keywords Cosmology, Big Bang, expan- ture of our ow n galaxy. E ventually itw as understood sion of universe. th at ou r galax y is a fairly large sy stem w ith arou n d a 216 RESONANCE ⎜ March 2009 GENERAL ⎜ ARTICLE B o x 1 . H u b b le 's L a w a n d t h e C o sm o lo g ic a l P r in c ip le The Hubble's law is written as V = H 0 r; with V as the radial component of the velocity. The reason for writing down only the radial component is that this is the only component of the motion that we can observe through the shift of spectral lines. However, it is implicit in the form of the Hubble's law that the rate of expansion is independent of direction. In other words, the expansion is isotropic around us¤ . Clearly, the Hubble's law is consistent with a vectorial relationship between velocity and distance. V = H 0 r: In this form, it is easy to see that the Hubble's law retains its form if we shift the origin: as seen from the origin, the galaxy at r 1 recedes with velocity V 1 . Ifwenowtryto rewrite the recession law in the frame of this galaxy, we get: 0 r = r ¡ r 1 ; 0 0 V = V ¡ V 1 = H 0 (r ¡ r 1 )=H 0 r : As claimed, the Hubble's law retains its form in the frame of any other galaxy as well. Thus expansion appears the same in every direction, and from every place in the universe. Unless we are observing at a special moment in the history of the universe, this also means that the universe is homogeneous and isotropic. The cosmological principle [1] elevates and encapsulates this idea, and it is noteworthy that a homogeneous and isotropic model of the universe allows us to de¯ne a cosmic time. Most models of the universe are based on this principle. ¤ Exceptions are anisotropic models [2] where galaxies recede from us at di®erent rates in di®erent directions. However observations restrict the level of deviations from isotropy and one needs to construct models carefully in order to match observational data. hundred billion stars. O ur galaxy, or the G alaxy, is nearly80;000 ligh t years across. A stron om ers u se a d if- feren t u n it, a p arsec (1 p arsec = 3:26 lightyears) and the G alaxy isaround 25 kiloparsecs across. T he G alaxy is sh ap ed like a d isk w ith stars h igh ligh ting sp iral arm s inthedisk;thereisalsoa spheroidalbulge nearthecen- 1 tre of th e G alaxy . T he disk issurrounded by a faint 1 An appam is a good descrip- h alo of stars an d glob u lar clusters; each glob u lar cluster tion of the shape of disk and is a tigh t grou p of stars an d th ese m ay con tain 10 3 ¡ 106 bulge, though not in proportion. stars each . T h e S u n is arou n d 8 k p c from th e cen tre in th e d isk. RESONANCE ⎜ March 2009 217 GENERAL ⎜ ARTICLE M any other galaxies have been know n fora long tim e. Many galaxies have Howeveritwasnotveryclearwhethertheseareapart been known for a long ofour ow n galaxy or are sim ilarsystem s located very far time. Hubble provided away. H ubbleprovided the ¯rst reliabledeterm ination the first reliable of distances to these galaxies and convincingly proved determination of th at th ese are large sy stem s of stars in th eir ow n right. distances to these galaxies and N ow w e revertto our discussion oftheH ubble'slaw . In convincingly proved th e velocity {d istan ce relation , velocities are m easu red in that these are large k m / s, d ista n c e s in m illio n s o f p a rse c s (M p c ) a n d fo r th is systems of stars in reason the H ubble'sconstant isw ritten in com plicated their own right. looking units of km /s/M pc even though ithas dim en- sions of inverse time. W e can recast the H ubble'slaw an d w rite it in term s of d irect ob servab les. W e d o th is step b y step . W e ¯ rst rew rite th e recession velocity in term s of th e red sh ift of sp ectral lines th at is d eterm ined d irectly from sp ectra. V r z = = ¡1 : c cH 0 T he speed of light is denoted by the usual sym b ol c. Herez istheD oppler redshift;note that thisde¯nition o f re d sh ift is v a lid o n ly fo r jV j=c ¿ 1. D istances are often m easu red u sing referen ce stars, or oth er ob jects th at are kn ow n to h ave a given lum inosity (see Box2). In su ch a case, th e ° u x ob served from th e referen ce ob - ject is related to th e lum inosity an d th e d istan ce. T h e en ergy em itted in u n it tim e is rad iated u n iform ly in all directionsand eventuallyspreadsoutina sphericalshell ofradius r. T he energy observed perunitarea,perunit tim e can th en b e w ritten as L S = : 4¼r2 HereS isthe observed °ux and L is th e lu m in o sity . If w e ob serve a nu m b er of ligh t sou rces th en th e red sh ifts and observed °uxes are expected to have the follow ing 218 RESONANCE ⎜ March 2009 GENERAL ⎜ ARTICLE Box2.DistanceLadder Measuring distances to other galaxies is a challenging task as there is no direct method of ascertaining the distance. There are two basic methods that are combined for measuring distances to galaxies. ² Parallax: We measure the parallax of nearby stars across Earth's orbit around the Sun. The parallax angle is 100(3:08 £ 1016m=r) ¤ . Observations from the Earth can give reliable parallax measurements of up to 0:1 00. ² Standard Candle: If there is a source with known luminosity (energy output per unit time), then the observed °ux from such a source can be used to compute the distance if the radiation from the source has not been absorbed by intervening gas and dust. Luminosity L , °ux S and distance r are related as S = L=(4¼r2 ), assuming that the source radiates uniformly in all directions. There are no standard candles where the luminosity is known apriori, therefore one needs to do a calibration. In the absence of such a calibration we can only measure relative distances and not absolute distances. Calibration is done by matching with the distance to a group of stars measured using some other method, either parallax or some other standard candle. A chain of standard candles is used and calibrated against each other, with the nearest ones calibrated using the parallax method. This sequence of distance measurement methods is often referred to as the `distance ladder'. Each step in the distance ladder involves cross-calibration and introduces errors. The H ipparcosspace mission reduced errors by a signi¯cant amount by providing accurate parallax measure- ments of up to 0:00100, increasing the number of stars with known parallax distances by a signi¯cant factor [3].