Inflationary Universe:A possible solution to the horizon and flatness problems.: Inflationary Universe: A possible solution to the horizon and flatness problems. Alan H. Guth
1981 Sudeep Das
'Greatest Hits' Seminar October 11, 2004
Slide2: Plan of the Talk Rush through Big Bang Cosmology…
And stumble upon
Flatness , Horizon andamp; Monopole Problems.and#x13;
Enter Guth…
Inflation can solve these problems,
But it raises others…
New Inflation
Chaotic Inflation
Conclusions and Comments.
Slide3: The Standard Model of the Very Early Universe When does our story begin?
In the standard model, as time temperature .
At very high temperatures, quantum - (h) gravitational (G) effects become important and our knowledge of Physics fails.
The Natural energy scale at this time is the Plank Mass,
So, it is sensible to begin hot big-bang at a lower temperature, say
Slide4: The Standard Model of the Very Early Universeand#x3; Assume that the Universe is homogenous and isotropic, and therefore described by the Robertson-Walker Metric:
where k =+1, -1 or 0 for a closed, open and flat universe, respectively,
and an energy-momentum tensor:
I am assuming the speed of light c=1
Slide5: The Standard Model of the Very Early Universeand#x3; With these, the Einstein’s Equations governing the evolution of
the universe become: where, is the Hubble parameter. At any time, one can define a critical density Freidman Eqns.
Slide6: The Standard Model of the Very Early Universeand#x3; With the equation of state taken as:
where w = 1/3 for relativistic particles.
w = for NR matter.
and for a Cosmological Constant ,
We get, for a flat universe the following behavior: The early universe was radiation dominated.
Slide7: The Standard Model of the Very Early Universeand#x3; Thermodynamics of the ultra-relativistic plasma: The entropy density is defined as: Number density: Where,
Slide8: The Standard Model of the Very Early Universeand#x3; Thermodynamics of the ultra-relativistic plasma: The entropy density is defined as: Number density: and the following entropy conservation is assumed: implying For Tandgt;andgt;m, so that g(T) is a constant, we get
Slide9: The Standard Model of the Very Early Universeand#x3; Since , one can trade T for a in the Freidmann Equations.
In terms of T , the first of these, i.e. becomes where, with which is conserved.
Slide10: The Puzzles: Flatness Problemand#x3; Consider the Friedmann Eqn: Since, in the radiation or matter dominated epochs, Hence, and So, can only increase with time and hence has to be stupendously close to 1 in the beginning. HOW CLOSE? WMAP results give =1.02+/-0.02 today. positive
Slide11: The Puzzles: Flatness Problemand#x3;
Guth takes, today. He gets Since, S is assumed conserved, its value at early times can be estimated by its value today. today. Since, the bound on a leads to,
Slide12: The Puzzles: Flatness Problemand#x3; So, 1. Since, S is asuumed conserved, its value at early times can be estimated by its value today, with, and recalling, One finds,
Slide13: The Puzzles: Flatness Problemand#x3; So, 1. 2. Giving, and Taking One gets, So, by demanding to be within an order of magnitude of today,
we find and#x8;and#x8;and#x8;and#x8;and#x8;and#x8;and#x8;and#x8;and#x8;and#x8;and#x8;that has to be exceedingly fine tuned in the early universe.
Slide14: The Puzzles: Horizon Problemand#x3; Particle Horizon or Physical Horizon: Physical distance that light can travel starting from Big Bang ( t = 0 ).
Consider a ray of light traveling radially in a flat universe. Since, light travels along null geodesics: So, the coordinate distance traversed in time t (comoving horizon): The physical horizon distance is, therefore, Since , this turns out to be
Slide15: The Puzzles: Horizon Problemand#x3; Recall the Friedmann Equation written in terms of temperature As we have shown, the second term is very small and drops out. Then, the equation can be solved and solution has nature, This means that (hmm… this holds for
any geometry !!) So, the physical horizon distance is,
Slide16: The Puzzles: Horizon Problemand#x3; One can see the Horizon Problem vividly by considering the isotropy of the cosmic microwave background. At decoupling the comoving distance over which casual interactions occur is 180 h-1 Mpc. (subtends about 1 degree on the sky today).
This is much less than the comoving distance radiation travels after decoupling 5820 h-1 Mpc. How could these patches know that they had to be at the same temperature?
Slide17: The Puzzles: Horizon Problemand#x3; So, size of the region in causal contact at time t is Compare this with the size L(t) of a region that grows into our observable universe. Using conservation of entropy, We can take The ratio of the volumes is, Again with Meaning 10 83 causally disconnected
regions. Yet homogenous? Horizon Problem
Slide18: The Puzzles: Monopole Problemand#x3; If Grand Unification occurs with a simple guage group, any spontaneous breaking of the symmetry down to the Standard Model will lead to magnetic monopoles.
Tand#x13;he expected relic abundance of monopoles works out be, This is far too big. Greater by 14 orders of magnitude than observational constraints There are other unwanted model-dependent relics.
Slide19:
Slide20: A clue to the Solutions A crucial assumption that led us to the flatness and horizon problems was the conservation of entropy S. What if that was grossly wrong? What if ?
Slide21: A clue to the Solutions: Flatness Problem In establishing the Flatness Problem, we used : Which gave: Eventually giving, To obviate the flatness problem one needs
Slide22: A clue to the Solutions: Horizon Problem For the Horizon Problem we estimated the length that grows into the scale of the observed universe as: The length scale in causal contact (horizon) was leading to: So, the horizon problem can be obviated if
Slide23: Good ! But how does it work? How do we get such a huge entropy multiplication? A first order phase transition in the Early Universe.
Nucleation rate of new phaseandlt;andlt; expansion rate.
Universe supercools below the critical temperature Tc to a much lower temperature Ts , whence the transition takes place.
Latent heat is released, and temperature goes up again to Tr ~ Tc
Entropy density is multiplied by a factor with the scale factor remaining constant. So,
Slide24: Guth’s Scenario Figure from A. Albrecht and P. and#x13;Steinhardt, Phys. Rev. D, 48, 1220 (1981). 'False Vacuum' Universe lingers
in the so-called metastable with energy density As the universe cools, the true minimum gets deeper.
At T~Ts, the universe tunnels to the 'True Vacuum'.
Slide25: What is inflating here? Of course, entropy inflates…but wait… Consider again the old Freidmann Equation, but with the new energy density. As the universe, supercools the 2nd and 3rd terms fall off, leaving: which has soln., with This implies an exponentially inflating scale factor
Slide26: What is inflating here? We should be able to see this behavior directly from Freidmann Eqns. Stress-energy tensor for the 'false vacuum' is: This makes the Friedmann Eqn.:
Slide27: Summarizing Guth’s Inflation A strongly first order symmetry breaking phase transition exists.
The universe gets hung up behind a 'dimple' on the potential hill and supercools below the critical temperature.
During this time, the scale factor increases exponentially.
After cooling by about 28 orders of magnitude, the phase transition occurs.
Universe heats up by collision of the bubbles of the new phase, and entropy gets hugely scaled up.
This solves the horizon and flatness problem.
The monopole density gets diluted away by the expansion.
Slide28: Changing the point of view We saw that Guth attacked the problem from the entropy point of view. What if, we forget all about phase transitions and entropy and ask:
If we somehow create an accelerated expansion, will the problems of Big Bang theory be solved?
The answer is, as it should be, yes.
Slide29: Changing the point of view: Flatness Problem Recall that flatness problem emerged from the increasing nature of the RHS of: However, during inflation, Define Inflation to be an epoch with so that even if started from a value away from 0, it can be brought sufficiently close to zero by end of inflation so that
Flatness problem does not arise.
Slide30: Changing the point of view: Flatness Problem Curvature is flattened out by the huge expansion.
Slide31: Changing the point of view: Horizon Problem The physical horizon was shown to be, The comoving horizon is 1/(a H) and is a decreasing function of time during inflation. Therefore, the comoving horizon shrinks during inflation.
Slide32: start now end Solving Horizon Problem in Comoving Coordinates Smooth patch Liddle and Lyth, Cosmological Inflation and Large Scale Structure
Slide33: Inflation in the Abstract Inflation is an add-on to the standard Big Bang scenario. Following equivalent descriptions hold: An era of accelerated expansion, An era of shrinking horizon, An era of a non-standard equation of state,
Slide34: What went wrong with Guth’s Theory? Guth’s phase transition scenario surely conforms to these requirements. But there was a serious problem. The bubbles of true vacuum will form at various times in various places in the false vacuum.
They will have great difficulty merging because the space separating them will still be expanding exponentially.
The phase transition will never be completed even if the bubbles grow at the speed of light. Guth andamp; Weinberg (1983)
Slide35: What came after Guth’s Theory? New Inflation (Linde 1982, Albrecht andamp; Steinhardt 1982): Roll down a flat potential. Chaotic Inflation (Linde 1983) Slide through a viscous medium.
Slide36: New Inflation Linde (1982), Albrecht and Steinhardt (1982). Most of the interesting part of inflation takes place away from the false vacuum.
The field rolls down slowly.
Space inside bubbles of new phase expand exponentially. Still a phase transition. One bubble can grow to a size ~ 103000cm andgt;andgt; scale of obs. Universe~ 1028cm The whole universe can be accommodated in one such bubble.
Slide37: Chaotic Inflation (Linde 1983) Chaotic Inflation can occur even with simplest form of potentials like The equation of motion for the field has a 'drag' term due to expansion of the universe, This term slows down the velocity of the field. If ,the equation of state is that required for Inflation.
Slide38: Chaotic Inflation Simple!!
No supercooling/tunneling from false vacuum. No plateau.
No thermal equilibrium!
In 10-35 s, a Plank size region blows up to~and#x4;and#x4;and#x4;and#x4; cms ! Initial Universe may be thought of as having chaotically distributed values of field. Inflation took place only where was large. At the end of inflation, the field oscillates and decays into particle-pairs, which interact and thermalize at some temperature. Standard Big Bang takes over from here.
Slide39: Quantum Fluctuations can 'kick the rolling ball back up the hill' at some points.
These regions expand faster than the parent regions.
In many points within this region, the expansion decays into standard big bang evolution (universes as our own), quantum fluctuations again causes some other point to inflate even faster. This process goes on and on and the Universe reproduces itself. Eternal Inflation Essentially all inflationary models turn out to be 'eternal'. Still not fully understood !
Slide40: Conclusion andamp; Further Comments The inflationary paradigm is essential in theory of cosmic evolution because it is a single class of theories which consistently answers,
Why is the Universe so flat?
Why is the Universe so homogenous and isotropic?
Why aren’t there any magnetic monopoles?
How did the Hubble Expansion start?
How do the density perturbations and anisotropies in the CMB appear?. Scale invariance. (Katie’s Talk!!)
Guth’s 1981 paper is important because inflation was invented in it.
So far, observations have been consistent with predictions of the simplest inflationary models. WMAP: =1.02+/-0.02. (Flatness)
Spectral index n=0.93+/-0.03 ( 1 for scale invariance)