Hubble, Hubble's Law and the Expanding Universe

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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

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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.

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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

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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 measurements of up to 0:00100, increasing the number of stars with known parallax distances by a signi¯cant factor [3]. The upcoming Gaia mission [4] is likely to make further progress in this direction.

¤ The distance at which we get a parallax of 100 is called a parsec: 1 parsec= 3:08£ 1016 m. relation sh ip: r r L 1 lo g z =log ¡1 =log ¡1 : cH 0 4¼S cH 0 In sh ort, log z /¡0:5logS , w h ere th e con stant of p ro- portionalitydependson theluminosityofthesourceand th e value of th e H u b b le's con stant. A stron om ers u se an inverse logarithm ic scale called `m agn itud es' to q u an tify observed °uxes.T hesearede¯ned as:

m = ¡ 2:5logS=S0 :

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Here,S 0 is a reference °ux. T hus the relationship between redshiftand m agnitudefor a standard candleis: lo g z / 0:2m ) m / 5logz: T h e fu ll relation , w ith con stants, can b e w ritten as: r r m ¡ M =5log =5log ¡ 25 10 pc 1Mpc cH ¡1 =5logz +5log 0 ¡ 25 : 1Mpc Here M isthe absolute m agnitude that isde¯ned as th e m agn itud e if th e sou rce is located at a d istan ce of 10 pc. C learly, thisisrelated to the lum inosity and w e note that unless we know the luminosity of the source w e can n ot u se th e tw o ob servab les (m agn itud e m and redshift z ) to determ inetheH ubble'sconstant. H u b b le u sed a variety of w ay s to d eterm ine d istan ces to g a la x ie s [5 ]. U sin g th e 1 0 0 00H ooker telescop e, he w as ableto ¯nd w hat are called `Cepheid variables'in nearby galaxies. C ep h eid variab les are stars w h ose lum inosity varies w ith tim e in a p red ictab le m an n er an d th e tim e p eriod of variation is related to th e average lum inosity. T hisallowed H ubble to show that these galaxies w ere very far aw ay, certainly at d istan ces th at are m u ch larger th an th e size of ou r ow n galax y. T h is w as th e ¯ rst con vincing d eterm ination of d istan ces to oth er galax ies and thisisam ongstH ubble'sm ostsigni¯cant scienti¯c d iscoveries. Box3 illustrates the di±culties involved in m easuring distancestogalaxiesand them easurem entof H ubble'sconstant. H ubble then looked for potential standard candles in order to be able to determ ine distances to m ore dis- tan t galax ies. H e u sed th e b righ test stars, th e b righ test glob u lar clusters, an d m any oth er sou rces to d eterm ine d istan ces to ten s of galaxies. R ed sh ifts, an d h en ce recession velocities of a num b er of galaxies w ere already

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B o x 3 . W h y is M e a s u r in g H 0 Di±cult?

The main challenge in measuring the Hubble's constant is in the accurate determination of distances to galaxies¤ . This in turn requires us to use the distance ladder: the progres- sion of standard candles and parallax based methods of distance measurement through cross-calibration. Each step in the distance ladder involves some uncertainty and hence adds to the error in our knowledge of the luminosity of the standard candle that is ¯nally used. If each step in the distance ladder introduces a few percent error, and the standard candle used to determine distances to galaxies requires ¯ve steps of cross-calibration then one can see that the error in luminosity of the standard candle adds up to around ten percent. Another source of error are the observational uncertainties. If we are observing a star in a distant galaxy, then it is very di±cult to check if another star happens to be in the same direction. Finite resolution of imaging devices, the large surface density of stars in galaxies, and the large distances to galaxies makes blending of stars a very common phenomena. The problem becomes more acute for distant galaxies as the projected density of stars becomes large. This introduces errors in the measured °ux, and hence in the measured distance. Lastly, galaxies are not merely receding from us due to expansion of the universe; they move around in the gravitational ¯eld of other galaxies. This component of motion is called `peculiar velocity'. The total velocities are thus:

V = H 0 r + v :

Peculiar velocities are not expected to have any average value, these are expected to be random. If peculiar velocitiesp have a typical magnitude ¾ v , then these motions introduce an error ofp order ¾ v = ( 3H 0 r) in the determination of the Hubble's constant, where the factor of 3 arises from our use of only one component of the peculiar velocity. In order to appreciate the impact of this e®ect, let us consider some numbers.p In our universe, ¾ p » 300 km/s and H 0 ' 70 km/s/Mpc. Thus the factor ¾ v = ( 3H 0 r) » 25% at a distance of 10 Mpc, and the distance to the nearest large galaxy is less than 1 Mpc. One can reduce this factor by measuring distances to a large number of galaxies but this in itself is a fairly di±cult and challenging task. Therefore it is essential to use very distant galaxies for an accurate determination of Hubble's constant. On the other hand use of more distant galaxies increases errors due to the ¯rst two factors mentioned above.

¤ A noteworthy aside: It is apparent from the form of Hubble's law that it is possible to verify the relationship without any knowledge of the value of Hubble's constant. know n from thew orkdoneby V M Slipher[8].T hestage w as n ow set for th e d iscovery of th e d istan ce{velocity relation . H u b b le ap p roach ed th is issu e th ree years after hispapergivingdistances to galaxies,and showed that

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the relationship w as linear. B y thistim e, at least two relativists h ad alread y u sed th e d ata p u b lished by H u b - bleand Sliphertoarrive atthesam e conclusion (see Box 4 fordetails).H owever,itiseasy tosee thatH ubbleap- p roach ed th e p rob lem from a d i® eren t p ersp ective an d that hisdiscovery w as m ade independently.T he reason w e associate thisdiscovery w ith H ubblem ore than any- one elseisthathe continued tow orkon theproblem and re¯ n e m easu rem en ts in ord er to im p rove d eterm ination of distances to other galaxies and to m ake a convinc- ing case that the expansion w as cosm ological in origin

B ox 4. W ho D iscovered H ubble's L aw ?

The velocity distance relation encapsulated in Hubble's law [6] is expected in all rela- tivistic cosmological models, the sole exception being Einstein's static model. Dynamical models with expansion were discovered by Friedmann, Lema^itre and Robertson in the decade preceeding Hubble's discovery. At the time when Friedmann [7] published his solutions of Einstein's equations, the size of the Galaxy was not known conclusively, nor was it known whether other galaxies are a part of the Galaxy or are similar systems situated at large distances. The issue was ¯nally settled by Hubble [5] who used Cepheid variables to measure distances to nearby galaxies and showed that these are independent systems in their own right and lie at very large distances. The recession velocities of a large number of galaxies had been measured painstakingly over the years by V M Slipher[8].ThustheHubble'slawcouldhavebeendiscoveredatanytimeafter1926. Lema^itre [9, 10] and Robertson [11] discovered cosmological solutions at this time; both realised that the recession of galaxies constitutes an observational evidence of models of an expanding universe. Both used the data from Slipher and Hubble to verify the velocity{distance relation and determine the proportionality constant. To them the form of the velocity{distance relation was natural and hence they did not highlight it in their papers. In an almost parallel e®ort, observers were trying to make sense of relativis- tic models. Lundmark [12] decided to ¯t a quadratic relationship between velocity and distance, postulating that there must be ¯nite maximum recession velocity. Hubble's paper [6] appears to be an attempt to demonstrate that the quadratic term is either notrequiredorthatitscoe±cientmustbeverysmall.BythetimeHubblefollowed up this work with more data [13], the language and the underlying paradigm underwent a signi¯cant shift: while the initial work by Hubble as well as Lundmark's work uses methods common in stellar and galactic astronomy, there was a sudden realisation of the cosmological scenario in all later work.

Hubble did indeed discover the Hubble's law, and did so independently, but he was not the ¯rst one to get there.

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[13,14].O n the otherhand,Lem a^itre [9,10] an d R ob ert- son [11] w ere ch eck ing w h eth er th e velocity {d istan ce relationship exp ected from theoretical m odels w as seen in n atu re or n ot. T he valuefor H ubble'sconstant determ ined by H ubble an d oth ers w as arou n d 500 k m /s/M p c. In th e con tex t ofcosm ologicalm odels,thisindicated an age oftheuni- 2 2 v e rse a ro u n d 2 b illio n y e a rs . H ow ever, radioactive dat- 1/H0 is a useful order of magni- ingshowedthatsomerocksonEarthweremucholder tude estimate of the age of the th an th is. T h is led to a crisis as th e u n iverse can n ot universe. A more precise value be older than allits contents. T he problem w as related requires knowledge of other pa- rameters but it can be shown to cross-calibration an d incorrect iden ti¯cation of som e that the order of magnitude esti- stan d ard can d les. It req u ired p ainstak ing w ork over th e mate is accurate to better than n ext qu arter of a cen tu ry to u n d erstan d th ese issues an d 50%. geta bettervalue for theH ubble'sconstant.A t present the valueofH ubble'sconstant isdeterm ined [15]to be closeto 70 km /s/M pc and thecorrespondingage ofthe universeis13:6 b illio n y e a rs. T he current challenge isto extend the redshift{m agni- tu d e relation sh ip to larger d istan ces as w e can ¯ n d ou t m ore about the universe. Indeed,thisaspect does not requiredeterm ination oftheH ubble'sconstantand hence th e errors d u e to cross-calibration are n ot relevan t. Fig- ure 1 show s the H ubblediagram for supernovae oftype Ia,the brighteststandard candleknow n to us. W e see clearly that at low redshifts (z ¿ 1),the data satis- ¯es the H ubble'slaw (m ¡ M / 5logz )verywell.At larger red sh ifts, th e e® ects of sp ace-tim e cu rvatu re m o d - 3 ifythisrelationshipand hence w e see som e deviations . 3 These deviations from Hubble’s F or th ese ob jects, w e d o n ot h ave a goo d calibration of law at high redshift are used to th e lum inosity an d th erefore w e can n ot u se th is d ata constrain the contents of the universe. The data shown here can to determ ine H ubble'sconstant,but we can verify the be used to demonstrate the ex- Hubble'slaw withthedata. istence of an exotic component called dark energy that has nega- W ith thisw e end the story ofH ubbleand H ubble'slaw . tive pressure, and is leading to W hat w e have outlined hereisonlyone aspect ofH ub- an accelerated expansion of the ble's contribution. H e contributed directly to m any universe.

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Figure 1. Hubble diagram for supernovae of type Ia. The data shown here corre- sponds to the Gold+Silver sample [16]. The line cor- responds to m – M∝ 5 log z, the relation expected from Hubble’s law.

asp ects of ex tra-galactic astron om y; ind eed h e started thisentire ¯eld.H ubbleconvinced hisem ployersabout the need for larger telescopes and the 10000Hookertele- scop e an d th e 200 00 H a le te le sc o p e a t th e P a lo m a r o b - servatories th at w ere set u p p rim arily for ex tra-galactic w ork h ave b een u sed for research in all areas of astron - om y. I can on ly refer you to oth er articles in th is volum e for som e detailsof other contributions m ade by E dw in P H ubble.