# Is the universe expanding faster than speed of light?

The celestial bodies like stars and galaxies are moving away from each other. How fast are they moving apart? Is that speed more than the speed of light?

According to Hubble's law

For [proper] distances $D$ larger than the radius of the Hubble sphere $r_{\mbox{HS}}$ , objects recede at a rate faster than the speed of light ... $$r_{\mbox{HS}}=\frac{c}{H_0}$$

With $H_0=67.8\mbox{ }(\mbox{km}/{\mbox{s}})/\mbox{Mpc}$ the Hubble constant according to the Planck mission, $c=299,792,458\mbox{ m}/\mbox{s}$ the speed of light, and one Megaparsec $1\mbox{ Mpc}=3.0857\cdot 10^{22}\mbox{ m}$, we get $$r_{\mbox{HS}}=\frac{c}{H_0}=\frac{299,792,458\mbox{ m}/\mbox{s}}{67.8\cdot 10^3\mbox{ }(\mbox{m}/{\mbox{s}})/3.0857\cdot 10^{22}\mbox{ m}}=1.3644\cdot 10^{26}\mbox{ m}.$$ That's $$1.3644\cdot 10^{26}\mbox{ m}/1.3644\cdot 10^{26}\mbox{ m}= 4,421 \mbox{ Mpc}$$ or $r_{\mbox{HS}}=14.422$ billion lightyears (1 pc = 3.26156 ly).

Hence objects further away than this proper distance (what you would measure with a chain of rulers) recede faster than the speed of light. The distance is also called Hubble length.

You may find 13.8 billion lightyears elsewhere. That's calculated with $H_0=70.4\mbox{ }(\mbox{km}/{\mbox{s}})/\mbox{Mpc},$ according to the estimates of 2010, based on WMAP data, or with even older data. The exact value is not known.

Since the Hubble constant isn't really constant over time, the universe is thought to be a little younger (13.8 billion years) than the 14.422 billion years you get by just dividing the Hubble distance by the speed of light.

• So you have said that the Hubble radius is 14.422 Gly, and that the age of the universe is 13.8 Gy. Are you implying that there exist no objects within our observational horizon which are traveling faster than the speed of light? Mar 10, 2014 at 15:53
• @astromax I think inflation blew it up a lot. In a non-linear way... Mar 10, 2014 at 23:01
• @draks I don't think that this answers my question to Gerald. Inflation blew up the coordinates of the universe, however, the fact still remains that he has quoted that the distance at which objects are moving faster than c is larger than the distance traveled by light since the big bang. Mar 11, 2014 at 13:31
• @astromax I've been calculating with data as available. It has been a little confusing in the first instant for me, too. Assuming the available data are correct, it seems to me, that after inflation a period of gravitational deceleration followed by the present acceleration phase could allow the present Hubble radius being larger than the light travel distance from the big bang. Due to present acceleration this doesn't imply, that there cannot be obects within our observational universe receding faster than the speed of light. Mar 11, 2014 at 14:54
• @Gerald: There is no need for such rationalizations; astromax is simply conflating different notions of distance. Also, if the Hubble parameter was really a constant in cosmological time, the universe would be infinitely old, because it would be the de Sitter $\Lambda$-vacuum. That the age of the universe is approximately $H_0^{-1}$ is pretty much a coincidence. Mar 12, 2014 at 8:19

The comoving radial distance, which is also the proper distance at current epoch, at redshift $z$ is given by $$D_\mathrm{C} = \frac{c}{H_0}\int_0^z\frac{\mathrm{d}x}{\sqrt{\Omega_\mathrm{M}(1+x)^3 + \Omega_k(1+x)^2+\Omega_\Lambda}}\text{.}$$ For a spatially flat cosmology, $\Omega_k = 0$, and so matter and dark energy density parameters are related through $\Omega_\mathrm{M} = 1-\Omega_\Lambda$.

From WMAP, $\Omega_\Lambda = 0.721\pm 0.025$ (and various refinements for measurements of dark energy density), from which it follows that every object comoving with the Hubble flow at redshift $z\gtrsim 1.4$ has superluminal recession velocity from us. This corresponds to a light travel time of $c^{-1}D_\mathrm{ltt}\gtrsim 9.1\,\mathrm{Gyr}$.

The most distant objects found by the Hubble telescope have $z\approx 12$, corresponding to a light travel time of approximately $13.4\,\mathrm{Gyr}$. Therefore, we do routinely see objects that have superluminal recession velocity from us.

I found the light travel times with the help of this cosmology calculator.

Clarification:

... however, the fact still remains that [Gerald] has quoted that the distance at which objects are moving faster than c is larger than the distance traveled by light since the big bang.

This is a conflation of distances. The light travel time distance $D_\mathrm{ltt}$ of objects with recession velocity of $c$ is only $\sim 9.1\,\mathrm{Gly}$. Intuitively, since the light we see now has been emitted in the distant past, the universe has expanded a lot in the time it took for light to get here, and therefore the object's recession velocity now is much higher. Assuming the object still exists.