Holography of Little Inflation

Abstract

For several crucial microseconds of its early history, the Universe consisted
of a Quark-Gluon Plasma. As it cooled during this era, it traced out a
trajectory in the quark matter phase diagram. The form taken by this trajectory
is not known with certainty, but is of great importance: it determines, for
example, whether the cosmic plasma passed through a first-order phase change
during the transition to the hadron era, as has recently been suggested by
advocates of the "Little Inflation" model. Just before this transition, the
plasma was strongly coupled and therefore can be studied by holographic
techniques. We show that holography imposes a strong constraint (taking the
form of a bound on the baryonic chemical potential relative to the temperature)
on the domain through which the cosmic plasma could pass as it cooled, with
important consequences for Little Inflation. In fact, we find that holography
applied to Little Inflation implies that the cosmic plasma must have passed
quite close to the quark matter critical point, and might therefore have been
affected by the associated fluctuation phenomena.

Full text

Holography of Little Inflation
Brett McInnes
National University of Singapore
email: matmcinn@nus.edu.sg
ABSTRACT
For several crucial microseconds of its early history, the Universe consisted of a Quark-
Gluon Plasma. As it cooled during this era, it traced out a trajectory in the quark matter
phase diagram. The form taken by this trajectory is not known with certainty, but is of
great importance: it determines, for example, whether the cosmic plasma passed through
a first-order phase change during the transition to the hadron era, as has recently been
suggested by advocates of the “Little Inflation” model. Just before this transition, the
plasma was strongly coupled and therefore can be studied by holographic techniques.
We show that holography imposes a strong constraint (taking the form of a bound on the
baryonic chemical potential relative to the temperature) on the domain through which the
cosmic plasma could pass as it cooled, with important consequences for Little Inflation.
In fact, we find that holography applied to Little Inflation implies that the cosmic plasma
must have passed quite close to the quark matter critical point, and might therefore have
been affected by the associated fluctuation phenomena.
arXiv:1501.01759v1 [hep-th] 8 Jan 2015

1. Holography and Hadronization in the Early Universe
The description of a Quark-Gluon Plasma (QGP) is based on the quark matter phase
diagram [1, 2, 3], which specifies the state of the plasma in terms of the temperature T and
the baryonic chemical potential µ
B
. The plasma can take a very wide variety of different
forms, ranging from the high-T, low-µ
B
plasma explored by the ALICE experiment at
the LHC (see [4] for a recent overview with many references), to the less well-understood
relatively low-T , high-µ
B
environment being explored in the beam scan experiments at
the RHIC [5, 6, 7], or to be explored at such facilities as SHINE, NICA and FAIR [8, 9, 10].
The QGP is the dominant form of matter during an important phase of the evolution
of the early Universe, the plasma era which is thought to follow Inflationary reheating. It
was long believed that, during this era, the only relevant region of the quark matter phase
diagram is the low-µ
B
region: this is understandable in view of the generally accepted
value (η
B
≈ 10
−9
) of the net baryon density/entropy density ratio (which is related
to µ
B
/T ) at this point in cosmic history. Recently, however, a remarkable alternative
possibility has been pointed out by Boeckel et al. [11, 12, 13]: it has been suggested that
µ
B
might in fact have been very large (with µ
B
/T ranging from unity up to ≈ 100) during
the plasma era. This is compatible with the observed baryon asymmetry, since that is
generated during a short interval of “Little Inflation” associated with the decay of a false
QCD vacuum at the end of the plasma era.
In the conventional picture, the cosmic plasma hadronizes by passing through a smooth
crossover, as is now thought to describe the QGP at low values of µ
B
. In the Little In-
flation model, however, hadronization occurs beyond the much-discussed quark matter
critical point [14] (believed to be located at roughly T ≈ 150 MeV, µ
B
≈ 150 −300 MeV),
and therefore involves a first-order phase transition. This has many exciting consequences
for the theory of primordial density fluctuations, cosmic magnetogenesis, primordial grav-
itational waves, and much else (for example, the very interesting ideas of Kalaydzhyan
and Shuryak [15] regarding the acoustics of cosmic phase transitions seem to find their
most natural context in Little Inflation). In addition, the possibility of large values of
µ
B
during the plasma era has begun to play a role in investigations of the cosmic plasma
equation of state [16]. The two alternative trajectories of the cosmic plasma in the quark
matter phase diagram are shown, somewhat schematically, in Figure 1.
From the directly experimental point of view, such values of µ
B
in the cosmic plasma
could mean that the high-µ
B
facilities currently under construction will be the ones that
will directly probe (certain aspects of) conditions in the early Universe, back to the first
few microseconds; though this will only be true if the lower end of the µ
B
/T ≈ 1 − 100
estimated range is actually realised, since those facilities are unlikely to reach very far
beyond the critical point.
This region of the quark matter phase diagram is difficult to investigate theoretically.
One approach [17] uses the well-known “holographic” gauge-gravity duality; for the specific
application to heavy-ion collisions see [18, 19, 20, 21, 22]. This method attempts to throw
light on QCD-like thermal field theories by studying the dual, asymptotically anti-de
Sitter, black hole. Here the effects of large chemical potentials (and of the strong magnetic
fields arising generically in these collisions [23, 24], which can strongly affect the QGP
[25]) can be examined by endowing the black hole with electric and magnetic charges. It
2

T
Standard
Cosmology
Critical Point
QGP
Ultra-High
Density Phase(s)
Plasma
Hadronizes
𝜇
𝐵
Figure 1: Possible Trajectories of the cosmic plasma in the Quark Matter Phase Diagram
(after Boeckel and Schaffner-Bielich [13])
is natural to ask whether this technique can be adapted to the cosmic case (where very
large magnetic fields are also to be expected [26, 27]).
It is of course clear that both heavy ion collisions and the early Universe are very
dynamic systems, whereas the black hole is static. One can take the point of view that
the holographic picture takes a “snapshot” of the system at a fixed time, but in the heavy-
ion case it is also possible, though difficult [28], to extend the theory so as to take the
dynamics into account.
One can also do this in the cosmic case, more simply [29]: one can follow the time
evolution of the system by exploiting the conformal flatness of FRW cosmologies (with
flat spatial sections, the only case we consider here). In such a spacetime, we have a
plasma of temperature T and a magnetic flux associated with a magnetic field B through
a (compact domain in a) fixed but arbitrary two-dimensional plane S. A conformal trans-
formation eliminates the time dependence, but this is irrelevant to the ratio B/T
2
, since
B and T
2
evolve at the same rate in standard cosmology: B/T
2
is a conformal invariant.
Such conformally invariant quantities can then be studied in the three-dimensional flat
spacetime with spatial sections modelled on S, which means that they can also be studied
in the four-dimensional holographic dual bulk spacetime. This approach to cosmic holog-
raphy is very limited: it only works for (spatially flat) FRW spacetimes, and it only allows
us to study a small range of physically interesting quantities, those which are conformally
invariant. Nevertheless it will prove to be useful.
The dual spacetime must be foliated by two-dimensional planes transverse to the radial
3

direction, so it is an (electrically and magnetically) charged black hole with a planar event
horizon, sometimes called a “black brane”. In [29] we studied such spacetimes from the
point of view of string theory; specifically, we asked whether it was consistent to assume
that string-theoretic objects, such as branes, can always be neglected in the bulk, even
under optimal conditions (the string coupling and the ratio of the string length scale
to the AdS curvature scale L are small). We found that this is not the case, because
under some circumstances the black hole itself begins to generate branes and radiate
them towards infinity. Requiring that this instability should not arise imposes a bound
on the conformally invariant ratio B/T
2
: one speaks of a holographic bound on the cosmic
magnetic field during the plasma era. It transpires that this bound is in fact satisfied,
though not by a large margin, in the current models of cosmic magnetogenesis.
However, in [29] we followed the standard assumption that the cosmic baryonic chemi-
cal potential is negligible throughout the plasma era, and so we have to revise those results
in the light of a possible episode of Little Inflation at the end of the plasma era. One can
see that there is an issue here, because a non-negligible chemical potential corresponds to
a non-zero electric charge on the dual black hole. Since the electric and magnetic charges
enter symmetrically into the black hole metric (as a result of the classical electromagnetic
duality of Maxwell’s equations), the chemical potential has a similar effect to a large mag-
netic field, and likewise threatens to trigger a “stringy” instability. This constrains Little
Inflation by bounding the (conformally invariant) ratio of the baryonic chemical potential
to the temperature.
In this work we extend the methods of [29] to study this holographic constraint. It
proves to be very stringent: the range µ
B
/T ≈ 1 − 100 discussed in [13] is greatly
narrowed to ≈ 1 ≤ µ
B
/T ≤ ≈ 2.35. Since the value of µ
B
/T at the quark matter
critical point provides the lower bound around 1, this result means that, if the cosmic
plasma does indeed undergo a first-order phase transition at the end of the plasma era, it
must pass close to the critical point. Thus the “race” to locate that point, and to identify
the physics associated with it [14], assumes cosmological significance
1
.
In short, if Little Inflation is to be compatible with holography, then it must occur
in precisely that part of the quark matter phase diagram where remarkable phenomena
associated with the quark matter critical point may soon be observed, but also where a
string-theoretic instability is not far off.
We begin with a brief review of the relevant bulk geometry and of its holographic
interpretation in this application.
1
One should however be aware that the cosmic plasma differs in some ways from the plasma produced
in heavy ion collisions: see below for a detailed discussion of this. On the other hand, certain properties
of the QGP will be important in both cases.
4

2. Planar AdS Black Holes and FRW Holography
The bulk geometry is described by a “Charged Planar AdS Black Hole” metric [30], a
solution of the AdS Einstein-Maxwell system
2
given by
g(CPAdSBH) = −
"
r
2
L
2
−
8πM
∗
r
+
4π(Q
∗2
+ P
∗2
)
r
2
#
dt
2
+
dr
2
r
2
L
2
−
8πM
∗
r
+
4π(Q
∗2
+P
∗2
)
r
2
+ r
2
h
dψ
2
+ dζ
2
i
. (1)
Here ψ and ζ are dimensionless coordinates on the planar sections transverse to the radial
coordinate r, L is the asymptotic AdS curvature radius, and M
∗
, Q
∗
, and P
∗
are geometric
parameters related respectively to the mass, electric charge, and magnetic charge per unit
horizon area. (See [29, 32] for the details.) In the usual way M
∗
, Q
∗
, and P
∗
determine
(for a fixed value of L) the value of r at the event horizon, r = r
h
: we have
r
2
h
L
2
−
8πM
∗
r
h
+
4π(Q
∗2
+ P
∗2
)
r
2
h
= 0. (2)
The potential for the electromagnetic field outside the black hole is
A =

1
r
h
−
1
r

Q
∗
L
dt +
P
∗
L
ψdζ, (3)
where the constant term in the coefficient of dt ensures that this one-form is regular. The
field strength is
F = −
Q
∗
r
2
L
dt ∧ dr +
P
∗
L
dψ ∧ dζ. (4)
Now we turn to the dual field theory on the boundary. The quark chemical potential
of this system is related holographically to the asymptotic value of the time component
of the potential form, while the magnetic field B of the dual system is related to the
asymptotic value of the field strength [33, 34, 35]. We therefore have (assuming that the
baryonic chemical potential µ
B
is thrice its quark counterpart)
µ
B
=
3Q
∗
r
h
L
(5)
and
B = P
∗
/L
3
. (6)
The temperature of the boundary system is that of the Hawking radiation of the black
hole, which is given by
T =
1
4πr
h
3r
2
h
L
2
−
4π(Q
∗2
+ P
∗2
)
r
2
h
!
, (7)
2
Note that apart from the trivial case with Q
∗
= P
∗
= 0, none of these metrics is an Einstein metric.
The effect with which we will be concerned below, in Section 3, has mostly been studied in the (Euclidean)
Einstein case; see for example [31] and references therein.
5

where we have used equation (2).
Combining equations (5), (6), and (7), we obtain
3r
4
h
− 4πT L
2
r
3
h
−
4π
9
µ
2
B
L
4
r
2
h
− 4πB
2
L
8
= 0. (8)
If the temperature is positive, then an event horizon exists and so this quartic can be
solved for r
h
, which can then be regarded as a function of the boundary parameters T ,
µ
B
, and B. In fact, given L and these three quantities, r
h
can be computed in this manner,
and then the black hole parameters M
∗
, Q
∗
, and P
∗
can be reconstructed from equations
(2), (5), and (6).
These last three quantities are of course constants, both in the bulk and in the obvious
(flat) boundary geometry. In the cosmological application, all of them must be promoted
to functions of cosmic time, since the dual quantities T, µ
B
, and B are such functions; but
from the bulk point of view, cosmic time is not a time coordinate but rather a parameter
along a curve in the abstract three-dimensional space of planar AdS black hole metrics
given in equation (1). We will see that the three “coordinates” (M
∗
, Q
∗
, P
∗
) depend
on this parameter through the FRW scale factor a(t); this makes it straightforward to
focus on conformally invariant quantities, which can be regarded as being defined on the
flat spacetime to which the FRW spacetime is conformally related, as explained in the
preceding Section. In detail, this works as follows.
First, for a plasma, T decreases according to 1/a(t). In a simple Boltzmann model
(like the one used in [36]), the antimatter/matter ratio is given by exp(−2 µ
B
/T ), so, in
any regime in which this ratio does not change rapidly, µ
B
likewise decreases in accordance
with 1/a(t). (As the temperature drops, massive particles annihilate and their entropy
is transferred to effectively massless particles, which implies that this naive model of
the particle populations can only be approximate. This approximation is nevertheless
adequate for our purposes; one might wish to apply it only to the plasma immediately
prior to the phase change.) The trajectory in the quark matter phase diagram is therefore
straight (see Figure 2 in [13] and Figure 1 above), and we have
µ
B
= ς
B
T, (9)
where ς
B
, the “specific baryonic chemical potential”, is a positive constant, the reciprocal
of the slope of the straight line
3
. Our principal objective in this work is in fact to constrain
ς
B
, by regarding it as a conformal invariant, in the sense discussed earlier.
Similarly, in conventional cosmology
4
, the magnetic field decreases according to 1/a(t)
2
,
so B/T
2
is a constant. Again, B/T
2
is the kind of conformally invariant quantity which
we can hope to constrain by means of holography, and that was done (when ς
B
= 0) in
[29].
3
Note that, because we are (for simplicity) not compactifying the planar sections here, there is no
Hawking-Page transition for these black holes [37], so we need not be concerned that such a transition
will interfere before the dual plasma hadronizes. The Hawking-Page transition can be restored, at any
desired temperature, by compactifying the planar event horizon to a torus [38]; in our case it would be
natural to choose it to occur at the temperature at which the cosmic plasma crosses the phase line.
4
Alternative possible evolution laws for B have been proposed [39, 40, 41], but are controversial
[42, 43, 44]. In any case, such “superadiabatic amplification” can be reconciled with a holographic bound
[29] only with difficulty; see below.
6

Now regard equation (8) as the definition of r
h
, which now becomes a function of
cosmic time in the FRW spacetime: that is, it is defined to be the (largest) solution of
this equation, given the coefficient functions T , µ
B
, and B. As the solution of a quartic
equation, it depends on these functions in a very complicated way. Remarkably enough,
however, its evolution with cosmic time is extremely simple: by inspecting equation (8),
given the above evolution laws for T , µ
B
, and B, one sees that r
h
decreases according
to 1/a(t). In the cosmological case one can then define Q
∗
by equation (5), and P
∗
by equation (6); because of the evolution law for r
h
, one finds that both evolve in the
same way (as should be the case, according to electromagnetic duality
5
), namely with
a(t)
−2
. Finally, equation (2) defines M
∗
for the FRW spacetime, and shows that it evolves
according to 1/a(t)
3
.
The important consequence of all this is that equations (2), (5), (6), and (8) can all be
interpreted either in the black hole bulk, or (by holography) in the flat space dual field
theory, or (by multiplying both sides of the equation by a suitable power of the scale factor
a(t)) in the FRW cosmological spacetime to which the latter is conformally equivalent.
This is certainly not a trivial statement: it is a consequence of the fact that the geometry
is “assembled” from components which are fundamentally planar. For example, if we had
used an asymptotically AdS black hole with a spherical event horizon, then equation (2)
would have taken the form
r
2
h
L
2
+ 1 −
2M
r
h
+
Q
2
+ P
2
4πr
2
h
= 0, (10)
where M, P , and Q are the usual (finite) mass and charge parameters; but clearly this
equation cannot transform in a homogeneous way under conformal transformations. The
formula for the Hawking temperature likewise acquires terms that rule out the above
procedure: it is unique to the planar case.
In the conventional picture of the evolution of the cosmic plasma, the specific baryonic
chemical potential ς
B
(equation (9)) is extremely small; whereas in Little Inflation it is
large, potentially as large as 100. Thus ς
B
is the central object of attention here, and the
sequel is devoted to explaining how holography constrains it.
3. The Brane Action
Our approach to FRW spacetimes focuses on two-dimensional planes embedded in the
spatial sections, and on the associated three-dimensional spacetimes. Each transverse
section r = constant in the bulk spacetime with metric given in equation (1) is a copy of
such a spacetime, and so it is natural to investigate the behaviour of these copies in the
bulk geometry.
In [29] we argued that these transverse sections can be studied by a simple function
S(r) defined (for four-dimensional asymptotically AdS black hole spacetimes with planar
sections) in the following manner. Let A
r
be the Lorentzian area of an (arbitrarily chosen)
compact domain
6
in the three-dimensional section (including the time axis) located at r,
5
This would not be the case if B evolved non-adiabatically, because then r
h
would evolve in a much
more complicated way.
6
The reader may prefer to transfer this discussion to the Euclidean domain, in which t is compactified,
7

and let V
r
denote the Lorentzian volume of the four-dimensional bulk region between the
event horizon and that domain. Then we define
S(r) ≡ A
r
−
3
L
V
r
, (11)
with the understanding that this quantity is defined only up to an overall positive multi-
plicative constant, which we shall choose so that S(r) is dimensionless.
For planar submanifolds of AdS
4
itself (regarded as the r
h
→ 0 limit of the black
hole), S(r) vanishes identically; but that is not so for AdS black hole spacetimes, which
are merely asymptotically AdS. In that case, S(r) vanishes at the event horizon
7
, and it
is always positive nearby. Far from the event horizon, however, the situation is less clear,
since it is characteristic of asymptotically AdS geometries that areas and volumes grow at
much the same rate. In fact, S(r) can even become negative far from the event horizon:
eventually the volume can overcome the area.
That does not happen for the planar AdS-Schwarzschild geometry (see [29]); nor does
it happen for the charged planar AdS black holes studied in the preceding Section, as
long as the charges are fairly small. But, as we shall see, it can happen if the charges are
large, yet still sub-extremal
8
.
As we explained in [29], allowing S(r) to become negative, that is, smaller than its
value at the event horizon, has serious consequences. For it was shown by Seiberg and
Witten [45] (see also [46]), that S(r) is, up to a positive multiplicative factor proportional
to the tension of the brane, nothing but the action of a BPS 2-brane wrapping around
r = constant. Branes nucleating near to the event horizon, where this action is positive,
will tend to contract back into the event horizon, where the action vanishes. If however
there is a region beyond some value of r in which this action is lower than it is in the
vicinity of the event horizon, then a brane nucleating in that region will tend to escape to
infinity instead of contracting back into the black hole, and the system becomes unstable.
The dual phenomenon in the field theory is that a certain scalar field, even if suppressed
initially (on the grounds that there is no such field in QCD), begins to grow and quickly
dominates the gauge fields. Various aspects of such phenomena have been discussed
recently in [31] and [47].
We can evaluate S(r) for the metric in equation (1): it is
S
CPAdSBH
(r) =
r
2
L
2
r
r
2
L
2
−
8πM
∗
r
+
4π(P
∗2
+ Q
∗2
)
r
2
−
r
3
L
3
+
r
3
h
L
3
, (12)
where the last two terms correspond to the volume term in equation (11). This may be
written more usefully as
S
CPAdSBH
(r) =

−8πM
∗
+
4π(P
∗2
+Q
∗2
)
r

/L
1 +
q
1 −
8πM
∗
L
2
r
3
+
4π(P
∗2
+Q
∗2
)L
2
r
4
+
r
3
h
L
3
. (13)
and the “planar” coordinates ψ and ζ are naturally converted to coordinates on a torus. Then “area”
and “volume” have their conventional connotations and are automatically finite.
7
The Lorentzian area of the event horizon, including the time direction, is zero, since it is a null surface;
and of course V
r
also vanishes there, by its definition. One sees this more clearly in the Euclidean version,
where the event horizon becomes the origin of a polar coordinate system.
8
The black hole with metric (1) has extremal or sub-extremal charges if equation (2) has a positive
real solution: the condition for that is

P
∗2
+ Q
∗2

3
≤ (27/4)πM
∗4
L
2
.
8

This function is non-negative if and only if its value as r → ∞ is non-negative: that is,
we need
−4πM
∗
L
2
+ r
3
h
≥ 0 (14)
if the bulk is to be stable. Notice that this inequality is well-defined, in the sense that
both terms on the left evolve according to a(t)
−3
when we transfer to the cosmological
spacetime. (The reader can verify that analogous statements hold for all of our subsequent
equations and inequalities.)
We conclude that the requirement that the holographic picture should be internally
consistent imposes a constraint on the black hole parameters in the bulk. Our next task is
to determine what this means for the boundary theory and the conformally related FRW
spacetime.
4. The Bound on µ
B
/T
Using equation (2), we can write (14) as
4π(P
∗2
+ Q
∗2
)L
2
≤ r
4
h
. (15)
Inserting this into equation (7) we obtain
2πT L
2
≥ r
h
. (16)
Combining (15) and (16) with equations (5) and (6), we obtain the fundamental inequal-
ity
9
B
2
+
µ
2
B
r
2
h
L
4
≤ 4π
3
T
4
. (17)
Thus we see that holography imposes a bound, given the temperature, on this combination
of the magnetic field and the baryonic chemical potential. Bear in mind, however, that
r
h
is to be regarded (via equation (8)) as a function of B, T , and µ
B
, obtained by solving
a quartic equation; so, expressed in terms of the physical parameters, this relation is
actually very complex. Furthermore, it involves L, which is not fixed in any obvious way
here: it would be preferable if our final conclusions were independent of that quantity. So
we need to examine (17) more carefully.
Our specific objective in this work is to constrain the physical parameters of the
cosmic plasma at the time when it hadronizes. As we know, there are two proposals for
the manner in which this happens: fortunately, both of them correspond to particularly
simple special cases of the inequality (17).
• In the conventional picture of the evolution of the cosmic plasma, the trajectory in
the quark matter phase plane is very close to the T axis, so that the cosmic plasma passes
through a smooth crossover on its route to hadronization: there is no first-order phase
transition and no Little Inflation. In that picture, then, µ
B
is negligible throughout the
plasma era, and (17) reduces to
B ≤ 2π
3/2
T
2
. (18)
9
In terms of the black hole parameters, the inequality (17) is expressed as

P
∗2
+ Q
∗2

3
≤ 4πM
∗4
L
2
.
Censorship (which we found earlier to demand

P
∗2
+ Q
∗2

3
≤ (27/4)πM
∗4
L
2
) is therefore always en-
sured here, though not by a very large margin.
9

This is the bound on cosmic magnetic fields explained in [29]; it implies a bound of ≈
3.6×10
18
gauss at the hadronization temperature. In this picture, cosmic magnetogenesis
may be associated with Inflation (see for example [48][49]), and the magnetic field energy
densities involved can be enormous, up to equipartition with the plasma density; so a
bound on B is of interest. Furthermore, this bound is important because it very strongly
constrains unconventional evolution laws for B, such as the one discussed in [44]; for,
in that case, B/T
2
would no longer be constant but would grow by a very large factor
(depending on the reheating temperature) during the plasma era, so it becomes difficult
to satisfy a bound like (18) at all times.
• In Little Inflation, µ
B
is far from negligible, so that the cosmic plasma does pass
through a first-order phase transition. But while this theory may possibly give a viable
account of cosmic magnetogenesis (see the discussion around Figure 18 in [27]), the mag-
netic fields involved are relatively small, about 10
−4
times the values typical of inflationary
magnetogenesis. (The magnetic field is generated along with the baryon asymmetry, so
the magnetic energy density is in the vicinity of equipartition with the baryonic, rather
than the plasma, energy density.) One can therefore assume that B is negligible for our
purposes
10
, and this greatly simplifies the situation because equation (8) is now quadratic
rather than quartic.
Before we proceed to the solution, we should stress that ignoring B would usually be
a very poor approximation in the case of the plasma produced in a heavy ion collision [23,
24, 25]. Furthermore, we are ignoring the effects of cosmic vorticity: that is the customary
assumption (though it might be desirable under some circumstances to reconsider it [50]);
but, again, the analogous assumption, that the angular momentum density is negligible,
is certainly not normally justified in the heavy-ion case, where the holographic dual is a
black hole endowed with angular momentum, as in [51, 32, 52, 53]. Again, as we have seen,
the time evolution of all physical parameters in the cosmic case is controlled in a simple
way by a single function, a(t); but it is not clear that any such simple description of the
dynamics is possible for a heavy-ion plasma. Finally, the time scales in the two cases are
very different: the heavy ion plasma exists for a time typical of strong-interaction physics
(measured in femtometres/c), while the cosmic plasma endures for several microseconds.
This is a crucial distinction for any discussion based, as ours is here, on the development
of an instability. Thus, our results do not extrapolate to the heavy-ion case in any
straightforward way. However, those features of the QGP (most importantly, its behaviour
near to the critical point) which are independent of the dynamics will be common to both
kinds of plasma.
Now solving (8) we have
r
h
=
2L
2
3

πT +
q
π
2
T
2
+ (π/3)µ
2
B

. (19)
It is clear that when (19) is substituted into (17), L drops out, and, in the absence of B,
10
However, in view of the uncertainties currently attending all theories of cosmic magnetogenesis, one
should consider the possibility that magnetic fields are larger in Little Inflation than expected — for
example, relics of inflationary magnetogenesis might be important. If that were the case, the effect would
be to strengthen our conclusions, in the sense that a detailed analysis of (17) shows that the upper bound
on µ
B
/T we are about to deduce would be lowered — though only to a small extent, even for very large
fields. These facts are discussed in the Appendix to this paper.
10

µ
B
and T are the only quantities remaining. Since they are proportional to each other,
(17) can in this case be reduced to an expression involving ς
B
(equation (9)) only: we find
πς
B
+ ς
B
q
π
2
+ (π/3)ς
2
B
≤ 3π
3/2
. (20)
Some algebra simplifies this to
ς
4
B
+ 18π
3/2
ς
B
− 27π
2
≤ 0. (21)
The quartic here is strictly increasing for ς
B
≥ 0, so its sole positive root yields an upper
bound on ς
B
. This root is exactly

1 − 2
1/3
+ 2
2/3

√
π, and so we have finally, restoring
µ
B
and T ,
µ
B
/T ≤

1 − 2
1/3
+ 2
2/3

√
π ≈ 2.353. (22)
The key datum now is the location of the quark matter critical point. To see why this
is so, refer to Figure 1: the phase transition line slopes downwards from the critical point,
into regions of larger µ
B
but smaller T . Therefore, if the trajectory of the cosmic plasma
in the phase diagram intercepts the transition line away from the critical point, it does so
at larger values of µ
B
/T than the value at the critical point: in short, this value puts a
lower bound on µ
B
/T at the point where the cosmic plasma hadronizes. Thus, requiring
Little Inflation to be compatible with holography constrains this parameter from both
sides.
Now in fact the precise location of the critical point is a matter of intense interest [14],
and there is reason to hope that it will be settled in the near future. Theoretical estimates,
for example from lattice theory, have become considerably more precise in recent years
[54]. (There is a growing consensus that the critical temperature is around 150 MeV; it is
the critical value of the baryonic chemical potential that is most difficult to compute.) It
is interesting to note that in the past (see for example [55]), lattice-theoretical estimates
of the critical value of µ
B
were in the 350-450 MeV range, threatening a conflict with
our inequality (22); but, more recently [14], a value for µ
B
/T at the critical point around
1− 2 has come to be favoured. More recently still, however, a sigma-model approximation
approach [56] has indicated that higher values may be possible, while an analysis of the
most recent experimental data apparently suggests a value below unity [57].
To be definite, let us settle on the range given in [14]; then we can summarize the
situation by stating that the holographic version of Little Inflation requires that, for an
interval of time
11
immediately before the cosmic plasma underwent a first-order phase
transition to the hadronic state, µ
B
/T must have satisfied
≈ 1 ≤ µ
B
/T ≤ ≈ 2.35. (23)
This is indeed a remarkable refinement of the range given in [13], 1 − 100.
If, for example, we put the critical point at µ
B
= 300 MeV, T = 150 MeV, and
assume for definiteness that the transition line near to the critical point makes an angle
of (very roughly) 45 degrees with the horizontal, then a simple calculation shows that
11
Following [13], and as discussed above, we are assuming here that µ
B
/T is constant during this time.
Note also that the holographic picture of the plasma only describes it when it is strongly coupled, which
may only have been the case during the late plasma era; so we do not claim that our bound applies at
all times.
11

Little Inflation can be compatible with holography only if the cosmic plasma hadronizes
between T ≈ 140 − 150 MeV, µ
B
≈ 300 − 315 MeV. This is very interesting, for two
reasons. First, it means that the cosmic plasma hadronizes at a point well within the range
probably accessible to near-future facilities such as SHINE, NICA and FAIR [8, 9, 10].
Second, it means that the trajectory of the cosmic plasma in the quark matter phase
diagram must have passed very near to the critical point itself; this means that the plasma
might possibly have experienced the characteristic fluctuation phenomena associated with
critical points, such as the QCD version of critical opalescence [58]. That could have all
manner of important consequences.
5. Conclusion: A Universe on the Brink
Little Inflation presents a version of cosmic history which differs very distinctly from the
conventional picture. Perhaps its most exciting feature is that it brings the cosmology of
the plasma era into the domain of quark physics with large values of the baryonic chemical
potential, where a number of remarkable phenomena may be observed experimentally in
the near future, in facilities currently under construction. However, Little Inflation itself
is compatible with values of µ
B
/T well beyond those accessible to those facilities. It
is therefore very interesting that, when holography is applied to this theory, one finds
that µ
B
/T is constrained to a very narrow range (the inequalities (23) above); this much
narrower range will indeed probably be reached by the experiments we mentioned.
If this picture is correct, then those experiments will be examining a system which
(in some important aspects, though not all) closely resembles the early Universe during
a brief but crucial period: the time when it was about to hadronize through a first-order
phase transition. In short, we could soon be witnessing “experimental early-Universe
cosmology” in a very non-trivial sense. (On the other hand, holography indicates that
a still more remarkable possibility compatible with Little Inflation, that hadronization
might take place near the quark matter triple point [4], is very unlikely.)
If phenomena like chromodynamic critical opalescence are actually observed in these
experiments, it will be important to consider whether such effects are compatible with
established cosmological observations and theory, if the cosmic plasma passed very near
to the quark matter critical point on its passage through the quark matter phase diagram.
One may well find that these fluctuation phenomena, which can be quite dramatic, are
ruled out by the observational data. If so, the implication would be that the Little
Inflation trajectory shown in Figure 1 stays well away from the critical point. As we have
seen, holography implies that there is very little leeway for that, meaning that the plasma
must have hadronized at the extreme upper end of the range given in (23) above. In other
words, one would conclude that the Universe, at that crucial point in its history, was on
the brink of becoming unstable from a strictly “stringy” effect — one involving branes in
the dual bulk. The significance of this would need to be considered very carefully.
Acknowledgements
The author is grateful to Prof Soon Wanmei for technical assistance, and to Cate Yawen
McInnes for encouraging him to complete this work expeditiously.
12

Appendix: The Effect of a Magnetic Field
As we explained above, Little Inflation provides a theory of cosmic magnetogenesis, but
the magnetic fields involved are not enormously large; so we approximated B by zero in
the inequality (17). One should however consider the consequences if that should prove
to be incorrect.
Since, in the conventional picture adopted here, B and T
2
evolve in the same way
during the plasma era, it is natural to set
B = β T
2
, (24)
where β is a positive constant, the value of which will be considered below. It will be
convenient also to express this parameter in a different way,
α
2
≡ 4π
3
− β
2
; (25)
it is clear from (17) that α can be assumed real and positive. Using this parameter, we
can now express (17) as
r
h
≤
α T
2
L
2
µ
B
. (26)
Now the quartic on the left in equation (8) is an increasing function at and beyond r
h
,
its largest real root, so substituting the right side of (26) into it we must obtain a non-
negative expression. With some simple manipulations (in the course of which L once
again drops out) one then finds

4π
3
−
8α
2
9

ς
4
B
+ α
3
ς
B
−
3α
4
4π
≤ 0. (27)
This is of course a quartic in ς
B
of the same kind as the one in the inequality (21); it
reduces to the latter when β = 0. Again, therefore, ς
B
is bounded above by the positive
root. It is elementary to show that, if one thinks of this root as a function of α, it is an
increasing function: that is, it is a decreasing function of β. Hence our claim that the
inclusion of a magnetic field would only serve to strengthen our bound, inequality (22).
In practice, however, the extent of this strengthening is negligible, as we now show.
The largest value of B/T
2
considered in theories of cosmic baryogenesis arises [27]
when one considers the possibility of equipartition between the magnetic field energy
density and the energy density of the plasma. We stress that such large values do not
normally arise in Little Inflation magnetogenesis, so the situation we are considering now
is very much an over-estimate of the effect. In any case, the Stefan-Boltzmann law implies
that, at equipartition,
B ≈
r
2
15
πT
2
; (28)
this corresponds to about 3.7 × 10
17
gauss at the phase transition: but it only translates
to β ≈ 1.15. Computing the corresponding value of α, inserting it into the left side of
(27), and solving numerically, one obtains
µ
B
/T ≤ ≈ 2.324. (29)
Comparing this with (22), one sees that, even in the most extreme case, the inclusion of
a magnetic field does not materially affect our conclusions.
13

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