LOG#132. Spacetime foam(II).

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The second post in this thread is completely speculative.

1) Suppose that Dark Energy is some kind of “fluid” made of some long wavelength “particles” or “exotic stuff”.

2) How different are those “Dark Energy particles” from the particles we know from the Standard Model?

A simple argument gives a stunning response:

N\sim \left(\dfrac{R_H}{L_p}\right)^2

plus the hypothesis of Boltzmann statistics in a cosmic volume V\sim R_H^3 and temperature T\sim R_H^{-1} provides the partition function

Z_N=\left(\dfrac{V}{\lambda^3}\right)^N\dfrac{1}{N!}

where 1/N! is due to the fact we expect the dark particles are indistinguible. Thus, it gives the entropy

S=N\left(\ln\left[\dfrac{V}{N\lambda^3}\right]+\dfrac{5}{2}\right)

But, if V\sim \lambda^3, then S (the entropy) becomes negative! This is absurd and the only alternative is that somehow N\sim 1 above, or equivalently, that the solution to this issue is not only that N must be absent but that the N dark energy particles are distinguishable particles! In that case, we obtain

S=N\left(\ln\left(\dfrac{V}{\lambda^3}\right)+\dfrac{3}{2}\right)+const

i.e.

S\sim N

This is a bit weird for a quantum statistics. The only consistent statistics in greater than 2 spacetime dimensions without Gibbs factor (1/N!) is the so-called Quantum Boltzmann statistics, and it is also named as infinite statistics.

The big idea of Jack Ng: particles of dark energy obey infinite statistics rahter than Bose or Fermi statistics!

In fact, not unsurprisingly, the M-theory approach to unification suggest something similar. In the paper http://arxiv.org/abs/0705.4581 , the authors have made a similar conjecture.

 Infinite statistics, quons and Cuntz algebra

A q-deformed Heisenberg algebra can be built from the relationships

a_ka^+_l-qa_l^+a_k=\delta_{kl}

or

\boxed{a_ia_j^+-qa_j^+a_i=\delta_{ij}}

This is the so-called quon statistics, or deformed Heisenberg algebra. Remarkably, you can recover the boson or fermion algebra if you plut q=+1 (bosons) or q=-1 (fermions).

The Cuntz algebra is the case with q=0, and it is also called infinite statistics or quantum Boltzmann statistics! It satisfies, hence

\boxed{a_ka^+_l=\delta_{kl}}

Any 2 states in infinite statistics act on vacuum \vert 0\rangle such as

\boxed{\langle 0\vert a_{i1}a_{i2}\cdots a_{iN}a^+_{jN}\cdots a^+_{j2}a^+_{j1}\vert 0\rangle =\delta_{i1,j1}\delta_{i2,j2}\cdots \delta_{iN,jN}}

That is, particles with infinite statistics are virtually distinguishable! The partition function is

Z_N=\sum e^{-\beta H}

and there is no Gibbs factor. In fact, all the representations of the particle permutation group can occur in infinite statistics!

Extra features of infinite statistics/Quantum Boltzmann statistics:

(1) The number operator \hat{N}, the hamiltonian \hat{H} operator, and any other observable are both “nonlocal” and “non polynomial” in the field operators.

(2) The TCP theorem and the cluster decomposition still hold, despite the nonlocal and non polynomial character of infinite statistics!

(3) QFT with infinite statistics are unitary, but it is highly non-trivial.

(4) Non-locality is a virtue from some Quantum Mechanics viewpoints, specially those regarding the issue of the Black Hole information paradox.

MONDian Dark Matter/Dark Energy

Suppose some test mass (m) due to some source (M). The Milgrom’s law of inertia reads

a=\begin{cases}a_N, \mbox{if}\;\; a>>a_c\\ \sqrt{a_Na_c},\mbox{if}\;\; a<<a_c\end{cases}

If we write

a_N=G_N\dfrac{M}{r^2}

then

a_c\sim \dfrac{cH_0}{(2\pi)}\sim 1.2\cdot 10^{-10}m/s^2

MOdified Newtonian Dynamics (MOND) has some “merits”:

1) It explains galactic rotation curves v\sim const when r>r_c.

2) It explains the Tully-Fisher relation (speed of stars are correlated with the galactic brightness as v^4prop M).

However, the MOND idea presents some “issues”:

1st. At cluster scales and cosmological scales, CDM (Cold Dark Matter hypothesis) seems to work better than it.

2nd. No MOND or more generally MOG (MOdified Gravity) field based theory predicts such a dynamics!

Idea: Could we reconcile CDM with MOND?

Hints: Verlinde (2010) proposed the entropic gravity scenario. Newton’s law of inertia F=ma follows from

i) F=T\dfrac{\Delta S}{\Delta x}

ii) \Delta S=2\pi k_B\dfrac{mc}{\lambda}\Delta x

iii) T_U=\dfrac{\hbar}{2\pi k_Bc}

Moreover, Verlinde also derived Newton’s universal gravitation from

i) F=T\dfrac{\Delta S}{\Delta x}

ii) The holographic screen with area A=4\pi r^2 and temperature T.

iii) The equipartition theorem E=\dfrac{N}{2}k_BT

iv) The holographic principle applied to the degrees of freedom on the holographic screen

N=\dfrac{Ac^3}{G_N\hbar}=\dfrac{A}{L_p^2}

and thus he could also derive F=G_N\dfrac{Mm}{r^2} in the same setting.

Therefore, we could modify the entropic argument to get Milgrom’s law in a de Sitter like Universe (dS) or dS spacetime. How could we do that? It is pretty simple, in a dS Universe, any inertial observer sees a temperature

T_U(dS)=\dfrac{a_0}{2\pi k_Bc}

For a non-inertial observer, we must substract this temperature to get the true acceleration for the non-inertial frame, i.e.,

\boxed{T_U(dS)'=T_U(ac)-T_U(dS)=\dfrac{1}{2\pi k_Bc}\left(\sqrt{a^2+a_0^2}-a_0\right)}

Now, we are ready to apply the Verlinde’s approach/argument:

F=T\nabla S=m\left(\sqrt{a^2+a_0^2}-a_0\right)

In the limits where a>>a_0 and a<<a_0 we obtain, respectively:

F_e\approx ma for a>>a_0

F_e\approx m\dfrac{a^2}{2a_0} for a<<a_0, and this is precisely Milgrom’s MOND law! Indeed, the fit is straightforward

m\sqrt{a_Na_c}=m\dfrac{a^2}{2a_0}

from which we get

a^4=4a_0^2a_Na_c

or

\boxed{a=\sqrt[4]{4a_0^2a_Na_c}=\left(2a_Na_0^3/\pi\right)^{1/4}}

Indeed, we can choose units with a_0\approx 2\pi a_c, and then

a_c=\dfrac{a_0}{(2\pi)}

Remember: a=G_N\dfrac{M}{r^2}=a_N for the purely newtonian gravity!

Indeed, we check that a<<a_0 implies the right behaviour for the galactic rotation curves, since

F_c=\dfrac{mv^2}{r}=m\dfrac{a^2}{a_0}

provides

v\sim constant

because

\dfrac{v^2}{r}=\dfrac{a^2}{a_0}\longrightarrow \dfrac{v^2}{r}=\dfrac{(2a_Na_0^3/\pi)^{1/2}}{a_0}\sim constant

We can also check this result “a la Verlinde”:

2\pi k_B\tilde{T}=2\pi k_B\left(\dfrac{2\tilde{E}}{Nk_B}\right)=4\pi\left(\dfrac{\tilde{M}}{A/G_N}\right)=\dfrac{G_N\tilde{M}}{r^2}

Here, \tilde{M} is the mass enclosed within a volume V=\dfrac{4}{3}\pi r^3, and \tilde{M}=M+M', where

M'=\dfrac{1}{\pi}\left(\dfrac{a_0}{a}\right)^2M

From the entropic force argument, we get

F_e=m\left(\sqrt{a^2+a_0^2}-a_0\right)=ma_N\left[1+\dfrac{1}{\pi}\left(\dfrac{a_0}{a}\right)^2\right]

And now, we have two limit cases:

1) Large accelerations a>>a_0, so

F_e\approx ma=a_N

and the equivalence principle holds since a=a_N.

2) Small accelerations a<<a_0, so

F_e\approx m\dfrac{a^2}{a_0}\approx ma_N\left(\dfrac{1}{\pi}\right)\left(\dfrac{a_0}{a}\right)^2

and thus, we get that

a=\left(2a_Na_0^3/\pi\right)^{1/4}

and there is some kind of equivalence principle inequivalence!

In the framework of this entropic MOND, there is no Dark Matter at all, but a MOND/MOG theory at small accelerations! It is interesting that there is some kind of correspondence between MOND and DM since we can write

F_e=G_N\dfrac{M+M'}{r^2}

and thus, some type of MOND/Dark Matter unification happens! Dark Matter of this type behaves as if there is NO dark matter but MOND, so it is called MOND dark matter theory. And as we have seen, F_e can be rewritten as a MOND law!

By the other hand, this theory can be also applied to Cosmology and Dark Energy! Let me show you how.

The Friedmann equations read

1st Friedmann equation:

\dfrac{\ddot{R}}{R}=-\dfrac{4\pi G_N}{3}(\rho+3P)+\dfrac{\Lambda}{3}

and

2nd Friedmann equation:

H^2=\dfrac{8\pi G_N}{3}+\dfrac{\Lambda}{3}

Suppose that \tilde{M} is the active gravitational mass and we apply Verlinde’s approach to the Tolman-Komar mass

\displaystyle{\mathcal{M}=\dfrac{1}{4\pi G_N}\int dV R_{\mu\nu}u^\mu u^\nu}

We get

\displaystyle{\mathcal{M}=2\int dV(T_{\mu\nu}-\dfrac{1}{2}Tg_{\mu\nu}+\dfrac{\Lambda g_{\mu\nu}}{8\pi G_N})u^\mu u^\nu=\left(\dfrac{4}{3}\pi r^3\right)\left[(\rho +3P)-\dfrac{\Lambda}{4\pi G_N}\right]}

Einstein gravity plus the Verlinde approach and the MOND dark matter ide departure from usual MOND when \tilde{M}\longrightarrow \mathcal{M}. Why? It is simple. From the non relativistic source versus relativistic source correspondence we observe that

\sqrt{a^2+a_0^2}-a_0=\dfrac{G_N\mathcal{M}}{\tilde{r}^2}

for \tilde{r}=rR(t)

and

\sqrt{a^2+a_0^2}-a_0=\dfrac{G_N(M+M')}{\tilde{r}^2}+4\pi G_N\rho\tilde{r}-\dfrac{\Lambda}{3}\tilde{r}

Some comments about this last equation:

1) Using a naive MOND at clusters misses two additional terms, 4\pi G_N\rho \tilde{r} and -\dfrac{\Lambda}{3}\tilde{r}. It could explain why MOND fails at cluster (or larger) scales.

2) At galactic scales, MOND dark matter quanta are “massless” and they reproduce Milgrom’s MOND.

3) At cluster/cosmological scales, MOND dark matter becomes massive!

A gravitational Born-Infeld theory idea

We can feel that the dS Unruh temperature is not explained in the previous arguments. I mean, where does the Unruh temperature for a dS Universe

T_U'(dS)=\sqrt{a^2+a_0^2}-a_0

comes from? Well, it can be explained with a relatively simple field theory. There are some nonlinear electromagnetic-like lagrangians called Dirac-Born-Infeld theory, or simply Born-Infeld theory (BIT) for short. The lagrangian is defined as follows

\boxed{L_{BIT}=b^2\left(1-\sqrt{1-\dfrac{(E^2-B^2)}{b^2}-\dfrac{(E\cdot B)^2}{b^4}}\right)}

There is a simplified version of this lagrangian, if we write impose that B=0, that we call L_{BIT}^{II}:

L_{BIT}^{II}=b^2\left(1-\sqrt{1-\dfrac{E^2}{b^2}}\right)

and if we impose E=0, we get

L_{BIT}^{III}=b^2\left(1-\sqrt{1+\dfrac{B^2}{b^2}}\right)

These lagrangians are nonlinear and describes a nonlinear electrodynamics similar to the relativistic particle lagrangian (nonlinear too):

L_{SR}=mc^2\left(1-\sqrt{1-\left(\dfrac{v}{c}\right)^2}\right)

From this analogy, we observe that the maximal velocity in the relativistic particle is translated into a maximal force b in the BIT. Now, let us assume that the non-linear electromagnetism is adopted by modified gravity. Take L_{BIT}(gravity)\times (1/4\pi), due to the spin of gravity (remember than in the case of electrodynamics, D=\epsilon E). Therefore, the BIT for gravity has a lagrangian (we take the BIT III version without loss of generality)

\boxed{H_g=-\dfrac{L_{BIT}^{III}(grav)}{4\pi}=\dfrac{b^2}{4\pi}\left(\sqrt{1+\dfrac{D_g^2}{b^2}}-1\right)}

Now, write A_0=b^2 and A=bD_g, and it yields after substitution

\boxed{H_g=\dfrac{1}{4\pi}\left(\sqrt{A^2+A_0^2}-A_0\right)}

If we apply the equipartition theorem, such as H_g=\dfrac{k_BT_{eff}}{2} per degree of freedom (DOF), the entropic force argument provides that

\boxed{a_{eff}=\sqrt{A^2+A_0^2}-A_0}

and from the equivalence principle between acceleration and gravity, we get

a_{eff}=\sqrt{a^2+a_0^2}-a_0

and thus

\boxed{F_{BIT}(grav)=ma_{eff}=m\left(\sqrt{a^2+a_0^2}-a_0\right)}

But this equation is precisely the MOND dark matter ansatz we stated above! It seems that MOND has a nonlinear electrodynamical-like origin!

See you in my last spacetime foam post!

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