Transcript Cosmological Inflation: a Personal Perspective

Cosmological Inflation: a
Personal Perspective
D. Kazanas
NASA/Goddard Space Flight
Center
Cosmology in the 70’s …
• Weinberg’s textbook
• Weinberg’s “The first 3 minutes”
– Open Problems outlined:
– 1. Entropy per baryon.
– 2. Horizons.
• Nucleosynthesis (Wagoner)
Several attempts to resolve the issue of the photons
per baryon through dissipative processes (Bulk
Viscosity – Weinberg; shear neutrino viscosity –
Misner).
• Two different views:
• Entropy/baryon too large (Weinberg)
• Entropy/baryon too small (Penrose)
– (It is too large if compared to a star too small if to a
black hole)
• (Chosen to take Weinberg’s view)
• Discussions with fellow graduate students for
ways to account for the entropy per baryon
(mainly bulk viscosity employing particles from
high energy physics zoo – totally unsuccessful).
• Never quite appreciated the magnitude of
entropy production needed from dissipative
processed to produce 109 photons/baryon.
Phase Transitions in Cosmology
• Searching for means to produce entropy by
processes involving the physics of High
Energies, came to the realization that the
linear potential between quarks could, in
principle, lead to the production of arbitrary
amounts of entropy (provided that this
transition were sufficiently delayed).
• This would apparently lead to production of
energy from the vacuum (latent heat).
The expansion of the universe could pull the quarks apart
producing particle – antiparticle pairs. If the transition does
not take place the quark tension could change the rate of
universal expansion.
u
d
d
u
d
u
proton
u
antiproton
u
u
Consider the quarks uud of a
Proton being pulled apart by
the expansion of the Universe.
• The potential energy between quarks increases linearly
with their distance. If we could pull quarks away from
each other using the expansion of the Universe, then we
could in principle produce large amounts of entropy.
• Could the linear tension between the expanding quarks
alter the dynamics of the Universe? (fruitless attempts;
interesting notion).
• The simplest assumption is that within a horizon size the
color forces almost cancel (but not exactly). Statistical
fluctuations of color of order N1/2, where N is the number
of quarks within the horizon (minimum entropy
production; Kazanas 1978).
• More favorable scenarios could lead to much larger
entropy (arbitrarily large! No constraints, no quantitative
measure).
Baryogenesis ….
• In 1978-79 the issue of h (entropy per baryon) is resolved in a totally
different way: Begin with roughly equal amounts of baryons and
antibaryons and produce a small excess of baryons!!
• The processes for producing this excess were set by Sakharov in
1967 but were rediscovered by Yoshimura (78), Dimopoulos &
Susskind (78), Ellis, Galliard & Nanopoulos (78) etc.
• So, what is one to do with all the phase transitions? Well, if they
cannot resolve the photon to baryon problem they should help
resolve the Horizon problem!!
On prouve tout ce qu’on veut, et la vrai difficulte’
est de savoir ce qu’on veut prouver.
Alain (Minerve ou de la Sagesse)
1979: Greek military service; postdoctoral offer by Floyd Stecker of
GSFC. Stopover in NORDITA, meeting with Sato; discussion of q-g
phase transitions; Sato was interested in working on another type of
phase transition!! Check the 1979 literature and find a paper by G.
Lasher who uses a strongly 1st order q-g transition to produce entropy.
• Brown & Stecker (1979) set out to examine
the possibility of matter-antimatter domains
in the Universe in models with soft CP
violation. The domain size is directly related
to the size of the horizon. New attempts to
use the quark tension to change the
expansion rate of the universe. References
to Frampton (76), Coleman (76) on the fate
of false vacuum.
Frampton, P. H. Phys. Rev D15, 2922 (76)
Soft CP violation;
walls; R~t2; Large
Horizon size but very
inhomogeneous
Zee, A. PRL, 44, 703 (1980)
Phase transitions again!!!!
Sato’s phase transition?
but for ev ~ f2T2, b = 2
expansion exponential
Inflation in “Reverse”?
• If the vacuum dominance in the early universe is simply
due to the high density of matter, it may be possible that
for sufficiently high density the vacuum energy may still
become important. Such a the situation may be
encountered in gravitational collapse (albeit inside the
horizon; Mbonye & Kazanas PRD, 72 024016 (05) ).
• To treat this problem one needs an equation of state that
can interpolate between matter and vacuum.
n simpler. Thus Eq. 2.6 now t akes t he form
"
µ
¶ 2# µ
¶
½
½
pr (½) = ® ¡ (® + 1)
½;
½max
½max
a = 2.31, fixed by demanding dp/dr =1 at the inflection point d2p/dr2=0 and
rmax an arbitrary (but very high) density such that pmax = - rmax
• Use this equation of state in a static, spherically symmetric geometry
general form
2
º (r )
ds = ¡ e
2
¸ (r )
dt + e
2
dr + r
2
¡
2
2
dµ + sin µd'
2
¢
;
The field
equations
r the assumed spacetime symmet ry, Eqs.2 reduce t o
µ 0
¶
¸
1
1
¡ 2 + 2 = 8¼G½(r ) ;
e¡ ¸
r
r
r
µ
e¡
µ
e¡
¸
¸
0
º
1
+ 2
r
r
¶
¡
1
= 8¼Gpr (r ) ;
r2
¶
º 00 ¸ 0º 0 º 02 º 0 ¡ ¸ 0
¡
+
+
= 8¼p? (r ) :
2
4
4
2r
The solution
Eqs. 3.1 and 3.2 into Eq. 2.1 give
µ
¶
¡ 2
¢
2m (r ) ¡ (r ) 2
1
2
2
2
2
2
ds = ¡ 1 ¡
e dt +
dr + r dµ + sin µd' ;
2m(r
)
r
1¡ r
¡
¡
¡
¢
1¡
where
Z
¡ (r ) =
"
µ
8¼ ® ¡ (® + 1)
½
½max
¶ 2# µ
½
½max
¶µ
r
r ¡ 2m (r )
¶
½dr
l pressure p? = f pµ; p' g. On doing this one ¯nds (see
·
¸
3
r 0 1
Gm (r ) + 4¼Gr pr
p? = pr + pr + (pr + ½)
;
2
2
r ¡ 2Gm (r )
h
¡
ass m (r ) enclosed by a
at t he radial
coordinat
e r is ´t hi
h 2-sphere
i
h
³
Rr
r3
02
0
r3
y m (r ) = 0 ½max f exp ¡ r g (r 0 ) 2 gr dr = M 1 ¡ exp ¡ r g (r 0 ) 2
h
i
R
1
•
The solution interpolates between Schwarzshild at large r
R< r < 1
¡
µ
¶
¡
¢
¡
¢
2M
1
2
2
2
2
¢
¡
ds2 = ¡
1¡
dt 2 + ¡
dr
r
dµ
+
sin
µd'
:
2M
r
1¡ r
¢
R
•
and de Sitter near r = 0
(ii) T he spacet ime must be asympt ot ically de Sit t er
µ
¶
2
¡
r
1
2
2
2
´
ds2 = ¡
1 ¡ 2 dt 2 + ³
dr
+
r
dµ
+
sin
µd'
r2
r0
1¡ r2
2
0
³
•
with
´
¡
i
q
e, r 0 =
•
and
q
#
3
8¼G½m ax
'
= ½j
is t he u
ing on t he value of t he max
h
i
3
n ½= ½max exp ¡ r g (rr 0 ) 2
odel. In t he limit r ! 0, it
The radial den
d r g = 2M G.
¢
A non-singular BH
• Expressing the metric in null coordinates
metric invariant),
so that
Z
¶
ds2 = ¡ ©2 (u; v) dUdV + r 2 (U; V ) d- 2;
µ ¶
The t hree equat ions in Eq. (12)
1
1
§
U (u) = ¡ B e¡ 2 ° u and V (v) = B e2 ° v ,
p an
null coordinat e syst em
canEddingt
set t ¤oon-Finkelst
unity, andein-like
t he ° sdouble
(for t he
y u = t ¡ r ; ¡ 1 < u < 1 and v = t + r ¤ ; ¡ 1 < v < 1
12) t hat
A (r ) ¡
©2 =
e
4B 2 ° 2
°
2
(v¡ u)
A (r ) ¡
=
e
4B 2 ° 2
°r
¤
Z
e r¤ = §
dr
1¡
2m (r )
r
• We have computed the expansion q for ingoing (+) and
outgoing
geodesics
pansion µ(-)
= null
l is,
fo
esics, by
;®
2
@r
µ =
:
2
@
V
r © (r )
+
µ¡ =
2
@r
:
2
r © (r ) @U
• The expansion of the outgoing null geodesics initially converges and
then diverges passing through an inner horizon at r = 0.01 M to
reach 1 at r = 0, indicating the absence of a singularity.
• Q: If N matter particles are confined in a small region R near r=0,
shouldn’t each have a momentum P~N1/3/R~[1/rmax]1/3? Should not
that gravitate? If ones wants to cancel this contribution by selfgravity (not necessarily possible) one is faced with a fine tuning
problem.
Which is the correct gravitational theory?
• Is there any room for an alternative to GR?
– “No, modulo quantum corrections”. If observation in
disagreement with theory, mistrust observations.!
• The success of the present theory leaves little
space for quick fixes. However…
• GR is the only theory of fundamental interactions
that is fixed not by a local invariance principle but
the requirement that the equations be 2nd order.
• In search for such a theory we have opted for local
scale invariance (Mannheim & Kazanas 90, 91, 94…)
• This can be achieved by introducing a new gauge
field (Weyl). However…
• the action
• is conformally invariant without the need of
introducing new fields (a is a dimensionless
constant).
• The static, spherically symmetric problem
leads to the (4th order) equations
and
or the simple exact (!!) relation
With
• With solution
Schwarschild
de Sitter
?
Quark potential?
b, g, k integration constants. The parameter g ~ 1/ RH , RH ~
Hubble radius. We thus expect significant deviations from
Newtonian dynamics when
For a galactic mass the corresponding radius is ~ 10 kpc !
• Galactic rotation curves without dark matter??
• The linear term in the solution produces a characteristic
acceleration of order cH0 (MOND?) from first principles.
• More solutions (non-vacuum):
The solution for a (conformally invariant) point-like charge
stress energy (the Reissner-Nordstrom problem).
is
i.e. in Weyl gravity (in all 4th order theories) the charge
contribution to the geometry has the same functional
form as that of the mass!!
• Demanding that the entire contribution of the charge to
the geometry is that of the mass demands that the
coupling constant be a ~ O (1062).
• The Newtonian Limit: One can integrate the Green’s
function of the
operator 8pG(r – r’) =
to get
the metric coefficient B(r)
The geometry
interior to a
spherical shell is
proportional to kr2
The Newtonian potential is the short distance limit to this theory
• So that
• In order for these to give the correct Newtonian behavior
f(r) must be the derivative of a d-function, i.e. the
elementary sources must be extended.
• We have concocted such a model source (Mannheim &
Kazanas 94)
• yielding
• The relation indicating the influence of the linear term in
the solution defines a Mass – Radius relation.
• It is of interest to note that virialized systems by and
large follow this relation including its normalization. In
combination with the virial relation v2 ~ M/R it implies a
relation between Mass and Velocity Dispersion in
virialized systems (M ~ v4). This relation is known as the
Tully-Fisher relation in galaxies or the Larson relation in
star forming regions. It is the same relation that extends
over 7 decades in radius or 14 decades in mass.
Larson 81
Solomon et al. 87
• The Tully – Fisher relation presumably is a result of the
inflationary perturbations and the non-linear interactions
of dark and luminous matter.
• There is a possibility that the same relation propagates
to smaller scales if the latter “condense” in pressure
equilibrium with the ambient pressure of galactic gas Pg
• This seems to provide a resolution to above correlation
except that experts tell me that this is not the way these
structures form and that….
… extrapolation by sixty (!!) orders of magnitude to the mass of the
electron gives a radius very close to the classical electron radius. The
electron and the Universe have the same column density.
Conclusions …
• The notions introduced by inflation in the early universe
may be also applicable to other settings.
• Observations apparently indicate the presence of a
characteristic acceleration that is not present in the
theory (but it is present in at least in one alternative to
GR).
• This may be the result of inflationary perturbations but
additional physics are needed to apply it to smaller
length scales.
•Speculation: These observations and those that imply the
presence of dark energy may indicate that inertial mass <
gravitational mass at small accelerations (finite size of the
Universe? R2 < c2/a0 ).