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 photonsper 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

u

d

u

u d

u

d

u

antiproton

proton

Consider the quarks uud of aProton 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 (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.

Zee, A. PRL, 44, 703 (1980)

Frampton, P. H. Phys. Rev D15, 2922 (76) Soft CP violation; walls; R~t2; Large Horizon size but very inhomogeneous

Phase transitions again!!!!

Sato’s phase transition?

but for v ~ 2T2, 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 takes the form

pr (½) =

"

®¡ (®+1)µ

½½max

¶2#µ

½½max

¶½;

fixed by demanding dp/d at the inflection point d2p/d2=0

and max an arbitrary (but very high) density such that pmax = - max

• Use this equation of state in a static, spherically symmetric geometry

general form

ds2 = ¡ eº(r)dt2+e¸ (r)dr2+r2¡dµ2+sin2µd' 2

¢;

r theassumed spacetimesymmetry, Eqs.2 reduceto

e¡ ¸µ¸0

r¡

1r2

¶+

1r2=8¼G½(r) ;

e¡ ¸µº0

r+

1r2

¶¡

1r2=8¼Gpr (r) ;

e¡ ¸µº00

2¡¸0º0

4+º02

4+º0¡ ¸0

2r

¶= 8¼p? (r) :

Eqs. 3.1 and 3.2 into Eq. 2.1 give

ds2 = ¡µ1¡

2m(r)r

¶e¡ (r)dt2+

1

1¡ 2m(r)r

dr2+r2¡dµ2+sin2µd' 2

¢;

The field equations

The solution

¡µ

¡¶

1¡

¡ ¢

where

¡ (r) =Z8¼

"

®¡ (®+1)µ

½½max

¶2#µ

½½max

¶ µr

r ¡ 2m(r)

¶½dr

l pressurep? = fpµ;p' g. On doingthis one¯nds(see

p? =pr +r2p0r +

12(pr +½)

·Gm(r) +4¼Gr3pr

r ¡ 2Gm(r)

¸;

h¡

ass m(r) enclosed by a 2-sphere at the radial coordinate r is th

y m(r) =Rr0 ½maxf exp

h¡ r3

rg(r0)2

igr02dr0= M

h1¡ exp

³¡ r3

rg(r0)2

´i

R1 h i

• The solution interpolates between Schwarzshild at large r

• and de Sitter near r = 0

• with

• and

R < r < 1

ds2= ¡µ1¡

2Mr

¶dt2+

1¡1¡ 2M

r

¢dr2 ¡ r2¡dµ+sin2µd' 2

¢:

³¡

´

e, r0 =q

38¼G½max

=½j is the u

¡ ¢¡ ¢

R

(ii) Thespacetimemust beasymptotically deSitter

ds2 = ¡µ1¡

r2

r20

¶dt2+

1³1¡ r2

r20

´ dr2+r2¡dµ+sin2µd' 2

¢

q

i

'ingon thevalueof themax

n ½= ½max exph¡ r3

rg(r0)2

i

odel. In thelimit r ! 0, it

#

Theradial den

d rg =2MG.

A non-singular BH

• Expressing the metric in null coordinates

• We have computed the expansion for ingoing (+) and outgoing (-) null geodesics

metric invariant), so that

ds2 = ¡ ©2 (u;v)dUdV +r2 (U;V)d­ 2;The three equations in Eq. (12)

U (u) = ¡ Be¡12°u andV (v) = Be

12°v,

can set to unity, and the °s (for the

¶ Z

§panEddington-Finkelstein-likedoublenull coordinatey u = t ¡ r¤; ¡ 1 < u < 1 and v = t + r¤; ¡ 1 <

systemv < 1

µ ¶

e r¤= §Z

dr

1¡ 2m(r)r

12) that

©2 =A (r)4B2°2

e¡°2 (v¡ u) =

A (r)4B2°2

e¡ °r¤

pansion µ= l ;® is, fo

µ+ =2

r©(r)2@r@V

:

esics, by

µ¡ =2

r©(r)2@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/max]1/3? Should not that gravitate? If ones wants to cancel this contribution by self-gravity (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 is a dimensionless constant)

• The static, spherically symmetric problem

leads to the (4th order) equations

and

or the simple exact (!!) relation

With

• With solution

k integration constants. The parameterRHRH ~ Hubble radius. We thus expect significant deviations from Newtonian dynamics when

For a galactic mass the corresponding radius is ~ 10 kpc !

de Sitter Schwarschild

?Quark potential?

• 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 O (1062).

• The Newtonian Limit: One can integrate the Green’s function of the operator 8G(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 -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 ).