Is cosmic inflation faster than the speed of light?

Cosmic inflation already works if the universe's volume doubles a mere 260 times. Every doubling takes 109 fundamental time steps tPlanck. If we were to imagine fundamental volumes, microscopic boxes of space multiplying like bacteria, every generation has 109 fundamental time units to reproduce just once. 109 human heartbeats are equivalent to about 109/(60*60*24*365) = 30 years. Not exactly bunny speed. Moreover, the duration of cosmic inflation (Δt ~ treheat) could have been much longer than we assumed, and then “30 years” would turn into many times the age of the universe instead.

But who knows what the size of the Planck time tPlanck really is?

No problem. Let us assume “TeV-gravity”, because it gives a whopping factor of 1017 to the
naively assumed Planck time tPlanck and thus may lead to black holes in the large hadron collider (LHC)! Still, the duration of cosmic inflation would be likely around Δt ~ treheat = 10-16 seconds, which is long before the limit from particle physics (a symmetry break at 10-12 s, remember). Thus, Δt /tPlanck is still bigger than 1010.

Straw-man: “Yeah, whatever, but bacteria? Are you some sort of crack pot or what? Cosmic expansion does not expand into an outside space like a culture of bacteria in a Petri dish! Give me string theory or something else I can pretend to be clever with in public!”

Well, think about the universe-on-a-membrane scenarios of string theory. Imagine the universe is a membrane 'woven' from fundamental strings.

Straw-man:“A three dimensional hyper-membrane in 9+1 dimensional space-time!”

Sock-puppet: “A fuzzy rug?”

Maybe and exactly, respectively. Somewhat like this:

Straw-man: “Planck worms making up the universe? Wrong on so many levels! They do not occupy a volume equal to their length cubed, they do not rotate at light velocity, they …”

Yes, I know! Anyway, fundamental strings are about one Planck length lPlanck long. The Planck length by the way is light velocity c times tPlanck. The strings thus could be considered to “occupy” (be responsible for, create, let emerge) an amount of volume V ~ lPlanck3. What we the last time imagined is basically that the membrane expands because the strings multiply like bacteria, doubling once in a while.

Straw-man: “Dude, I just told you: Cosmic expansion does not expand into an outside space like a culture of bacteria in a Petri dish!”

Yes, the membrane expanding is a misleading picture, but for more reasons: a certain point of the membrane would have to be a stationary fixed point inside the so called bulk space that contains the membrane universe. In other words, one of the strings would not move relative to the super-universe containing our membrane. This in turn would easily violate the cosmological principle and special relativity inside our membrane universe. Therefore, think of the strings just splitting in two halves once in a while.

Straw-man: “What?!? Now all the strings are more or less at rest relative to the bulk space around the membrane! Maybe the cosmological principle is fine with that, but relativity?”

Well, the emergent relativity valid inside the membrane is not violated by something being at rest to a hidden reference frame. With the bacteria picture, an emergent relativity would get into trouble far away from the region at rest, because there the bacteria would be pushed away from the center faster than light.

Anyway, back to the main point: If the strings just split, the volume relative to the space outside of the membrane does not increase. Only the average length of the strings becomes smaller over time. This is indistinguishable from expansion for those who are made out of large configurations of string interactions, that is you and me and our measurement instruments, and thus it is a valid dual description.

In the first blog entry we saw that cosmic inflation already works if the volume doubles a mere 260 times. Even with crazy assumptions like TeV-gravity “Δt /tPlanck is still bigger than 1010.” Putting it all together, the excitations of a fundamental string of the membrane, which travel at the speed of light c, travel forth and back along the length lPlanck of the string on average 1010/260 ~ 108 times before the string splits in half and produces just one more string. In conclusion:

Inflation is 100 million times slower than light!

Straw-man: “You are a douche bag! You had the factor of 108 a long time ago. All you added is a silly way to bring lPlanck into the story.”

Do you have a better way of explaining why cosmic expansion should be looked at on the Planck scale? I could do dimensional analysis claiming that the Planck length is thus important, but such would not be convincing anybody who is not already familiar with such matters.

So, in both the “rapid” hypothetical inflation scenarios, namely the Δt = 10-32 s one and the TeV-Gravity one, which both are in a sense 1020 times faster than sold to us as shockingly fast, the fundamental physical constituents may go leisurely through millions upon millions of cycles in order to reproduce or divide just once. In other words, inflation constitutes the most boring case of watching paint dry that I have ever encountered!

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The science behind this inflation trilogy is described in detail in:

S. Vongehr: Metric Expansion from Microscopic Dynamics in an Inhomogeneous Universe. Communications in Theoretical Physics 54(3), 477-483 (2010), arxiv.org/abs/1008.2810