Interpretation Hubble's law

1 interpretation

1.1 redshift velocity , recessional velocity

1.1.1 redshift velocity
1.1.2 recessional velocity


1.2 observability of parameters
1.3 expansion velocity vs relative velocity
1.4 idealized hubble s law
1.5 ultimate fate , age of universe
1.6 olbers paradox
1.7 dimensionless hubble parameter

interpretation

a variety of possible recessional velocity vs. redshift functions including simple linear relation v = cz; variety of possible shapes theories related general relativity; , curve not permit speeds faster light in accordance special relativity. curves linear @ low redshifts. see davis , lineweaver.

the discovery of linear relationship between redshift , distance, coupled supposed linear relation between recessional velocity , redshift, yields straightforward mathematical expression hubble s law follows:

v
=

h

0



d


{\displaystyle v=h_{0}\,d}

where

v


{\displaystyle v}

recessional velocity, typically expressed in km/s.
h0 hubble s constant , corresponds value of



h


{\displaystyle h}

(often termed hubble parameter value time dependent , can expressed in terms of scale factor) in friedmann equations taken @ time of observation denoted subscript 0. value same throughout universe given comoving time.




d


{\displaystyle d}

proper distance (which can change on time, unlike comoving distance, constant) galaxy observer, measured in mega parsecs (mpc), in 3-space defined given cosmological time. (recession velocity v = dd/dt).

hubble s law considered fundamental relation between recessional velocity , distance. however, relation between recessional velocity , redshift depends on cosmological model adopted, , not established except small redshifts.

for distances d larger radius of hubble sphere rhs , objects recede @ rate faster speed of light (see uses of proper distance discussion of significance of this):

r

h
s


=


c

h

0




 
.


{\displaystyle r_{hs}={\frac {c}{h_{0}}}\ .}

since hubble constant constant in space, not in time, radius of hubble sphere may increase or decrease on various time intervals. subscript 0 indicates value of hubble constant today. current evidence suggests expansion of universe accelerating (see accelerating universe), meaning that, given galaxy, recession velocity dd/dt increasing on time galaxy moves greater , greater distances; however, hubble parameter thought decreasing time, meaning if @ fixed distance d , watch series of different galaxies pass distance, later galaxies pass distance @ smaller velocity earlier ones.

redshift velocity , recessional velocity

redshift can measured determining wavelength of known transition, such hydrogen α-lines distant quasars, , finding fractional shift compared stationary reference. redshift quantity unambiguous experimental observation. relation of redshift recessional velocity matter. extensive discussion, see harrison.

redshift velocity

the redshift z described redshift velocity, recessional velocity produce same redshift if caused linear doppler effect (which, however, not case, shift caused in part cosmological expansion of space, , because velocities involved large use non-relativistic formula doppler shift). redshift velocity can exceed speed of light. in other words, determine redshift velocity vrs, relation:

v

r
s


≡
c
z
 
,


{\displaystyle v_{rs}\equiv cz\ ,}

is used. is, there no fundamental difference between redshift velocity , redshift: rigidly proportional, , not related theoretical reasoning. motivation behind redshift velocity terminology redshift velocity agrees velocity low-velocity simplification of so-called fizeau-doppler formula

z
=



λ

o



λ

e




−
1
=




1
+
v

/

c


1
−
v

/

c




−
1
≈


v
c


 
.


{\displaystyle z={\frac {\lambda _{o}}{\lambda _{e}}}-1={\sqrt {\frac {1+v/c}{1-v/c}}}-1\approx {\frac {v}{c}}\ .}

here, λo, λe observed , emitted wavelengths respectively. redshift velocity vrs not related real velocity @ larger velocities, however, , terminology leads confusion if interpreted real velocity. next, connection between redshift or redshift velocity , recessional velocity discussed. discussion based on sartori.

recessional velocity

suppose r(t) called scale factor of universe, , increases universe expands in manner depends upon cosmological model selected. meaning measured proper distances d(t) between co-moving points increase proportionally r. (the co-moving points not moving relative each other except result of expansion of space.) in other words:

d
(
t
)


d
(

t

0


)



=



r
(
t
)


r
(

t

0


)



 
,


{\displaystyle {\frac {d(t)}{d(t_{0})}}={\frac {r(t)}{r(t_{0})}}\ ,}

where t0 reference time. if light emitted galaxy @ time te , received @ t0, red shifted due expansion of space, , redshift z simply:

z
=



r
(

t

0


)


r
(

t

e


)



−
1
 
.


{\displaystyle z={\frac {r(t_{0})}{r(t_{e})}}-1\ .}

suppose galaxy @ distance d, , distance changes time @ rate dtd . call rate of recession recession velocity vr:

v

r


=

d

t


d
=




d

t


r

r


d
 
.


{\displaystyle v_{r}=d_{t}d={\frac {d_{t}r}{r}}d\ .}

we define hubble constant as

h
≡




d

t


r

r


 
,


{\displaystyle h\equiv {\frac {d_{t}r}{r}}\ ,}

and discover hubble law:

v

r


=
h
d
 
.


{\displaystyle v_{r}=hd\ .}

from perspective, hubble s law fundamental relation between (i) recessional velocity contributed expansion of space , (ii) distance object; connection between redshift , distance crutch used connect hubble s law observations. law can related redshift z approximately making taylor series expansion:

z
=



r
(

t

0


)


r
(

t

e


)



−
1
≈



r
(

t

0


)


r
(

t

0


)

(
1
+
(

t

e


−

t

0


)
h
(

t

0


)
)




−
1
≈
(

t

0


−

t

e


)
h
(

t

0


)
 
,


{\displaystyle z={\frac {r(t_{0})}{r(t_{e})}}-1\approx {\frac {r(t_{0})}{r(t_{0})\left(1+(t_{e}-t_{0})h(t_{0})\right)}}-1\approx (t_{0}-t_{e})h(t_{0})\ ,}

if distance not large, other complications of model become small corrections , time interval distance divided speed of light:

z
≈
(

t

0


−

t

e


)
h
(

t

0


)
≈


d
c


h
(

t

0


)
 
,


{\displaystyle z\approx (t_{0}-t_{e})h(t_{0})\approx {\frac {d}{c}}h(t_{0})\ ,}

or



c
z
≈
d
h
(

t

0


)
=

v

r


 
.


{\displaystyle cz\approx dh(t_{0})=v_{r}\ .}

according approach, relation cz = vr approximation valid @ low redshifts, replaced relation @ large redshifts model-dependent. see velocity-redshift figure.

observability of parameters

strictly speaking, neither v nor d in formula directly observable, because properties of galaxy, whereas our observations refer galaxy in past, @ time light see left it.

for relatively nearby galaxies (redshift z less unity), v , d not have changed much, , v can estimated using formula



v
=
z
c


{\displaystyle v=zc}

c speed of light. gives empirical relation found hubble.

for distant galaxies, v (or d) cannot calculated z without specifying detailed model how h changes time. redshift not directly related recession velocity @ time light set out, have simple interpretation: (1+z) factor universe has expanded while photon travelling towards observer.

expansion velocity vs relative velocity

in using hubble s law determine distances, velocity due expansion of universe can used. since gravitationally interacting galaxies move relative each other independent of expansion of universe, these relative velocities, called peculiar velocities, need accounted in application of hubble s law.

the finger of god effect 1 result of phenomenon. in systems gravitationally bound, such galaxies or our planetary system, expansion of space weaker effect attractive force of gravity.

idealized hubble s law

the mathematical derivation of idealized hubble s law uniformly expanding universe elementary theorem of geometry in 3-dimensional cartesian/newtonian coordinate space, which, considered metric space, entirely homogeneous , isotropic (properties not vary location or direction). stated theorem this:

any 2 points moving away origin, each along straight lines , speed proportional distance origin, moving away each other speed proportional distance apart.

in fact applies non-cartesian spaces long locally homogeneous , isotropic; negatively , positively curved spaces considered cosmological models (see shape of universe).

an observation stemming theorem seeing objects recede on earth not indication earth near center expansion occurring, rather every observer in expanding universe see objects receding them.

ultimate fate , age of universe

the age , ultimate fate of universe can determined measuring hubble constant today , extrapolating observed value of deceleration parameter, uniquely characterized values of density parameters (Ωm matter , ΩΛ dark energy). closed universe Ωm > 1 , ΩΛ = 0 comes end in big crunch , considerably younger hubble age. open universe Ωm ≤ 1 , ΩΛ = 0 expands forever , has age closer hubble age. accelerating universe nonzero ΩΛ inhabit, age of universe coincidentally close hubble age.

the value of hubble parameter changes on time, either increasing or decreasing depending on value of so-called deceleration parameter



q


{\displaystyle q}

, defined by

q
=
−

(
1
+




h
˙



h

2




)

.


{\displaystyle q=-\left(1+{\frac {\dot {h}}{h^{2}}}\right).}

in universe deceleration parameter equal zero, follows h = 1/t, t time since big bang. non-zero, time-dependent value of



q


{\displaystyle q}

requires integration of friedmann equations backwards present time time when comoving horizon size zero.

it long thought q positive, indicating expansion slowing down due gravitational attraction. imply age of universe less 1/h (which 14 billion years). instance, value q of 1/2 (once favoured theorists) give age of universe 2/(3h). discovery in 1998 q apparently negative means universe older 1/h. however, estimates of age of universe close 1/h.

olbers paradox

the expansion of space summarized big bang interpretation of hubble s law relevant old conundrum known olbers paradox: if universe infinite, static, , filled uniform distribution of stars, every line of sight in sky end on star, , sky bright surface of star. however, night sky largely dark. since 17th century, astronomers , other thinkers have proposed many possible ways resolve paradox, accepted resolution depends in part on big bang theory , in part on hubble expansion. in universe exists finite amount of time, light of finite number of stars has had chance reach yet, , paradox resolved. additionally, in expanding universe, distant objects recede us, causes light emanating them redshifted , diminished in brightness.

dimensionless hubble parameter

instead of working hubble s constant, common practice introduce dimensionless hubble parameter, denoted h, , write hubble s parameter h0 h × 100 km s mpc, uncertainty relative of value of h0 being relegated h. if subscript presented after h, refers value of h used in text s preceding calculation, , equal h0 / 100. h = 0.678, can represented h0.678. should not confused dimensionless value of hubble s constant, expressed in terms of planck units, current value of h0×tp = 1.18 × 10.