The Wikipedia article about inflationary cosmology gives a good summary of inflation theory and its problems. What is nice that inflation theory is a solution to real problems. What is not nice is that it leads to new problems.
 Horizon problem: temperature constant in extreme accuracy. No time to equilibrate.
 Flatness problem: 3space is unexpectedly flat suggesting critical 3space so that 3curvature vanishes. Here inflation theorists should have though a second though. Criticality suggests criticality in thermodynamical and even in quantum sense.
 Monopole problem of GUTs. Monopoles are not observed. This is a problem of GUTs and thus all theories that are based on GUTs as QFT limit.
This is a nice starting point. The inflationary solution of the problems would be a rapid expansion.
Inflation theory relies on GUT paradigm and proposes exponentially fast expansion due to a decay of false vacuumwith nonvanishing energy density decaying to particles. Horizon size becomes large. The fluctuations of the curvature of 3space get extremely small and 3space flattens. Monopole density goes to zero. Observed universe would correspond to the content of single horizon.
It has however turned out that there are problems  Steinhardt is one of the critical voices. In particular, one cannot avoid fine tuning after all. Inflation theory remains qualitative. There is no empirical evidence for inflaton fields, and Russian doll cosmology is strongly suggestive leading to multiverse.
TGD provides an alternative solution. First TGD very briefly.
 TGD emerges as a solution of energy problem of GRT and also as generalization of string models replacing string world sheets with spacetime surfaces. TGD provides a new view about spacetime. Spacetimes are 4D surfaces in M^{4}×CP_{2} . M^{4} and CP_{2} are fixed uniquely by the existence of twistor lift of TGD requiring that the twistor spaces involved have Kähler structure.
6D Kähler action dimensionally reduces to a sum of a volume term  cosmological constant having a spectrum and the analog of Maxwell action action. Preferred extremals are minimal surfaces  geometric analogs for massless fields  and extremals of also Maxwell action simultaneously but having 2D string world sheets as singularities at which there is charge transfer between the two action terms. String like objects thicken to magnetic fluxes as their M^{4} projection thickens. The action is essentially generalization of the action of point like charge in Maxwell field obtained by replacing point like particle with 3D surface.
 New view about quantum theory based on what I call zero energy ontology (ZEO) and number theoretical vision involving extension of physics to adelic physics with padic sectors of the theory describing cognition. Effective Planck constant is predicted to have hierarchy with values h_{eff}= n×h_{0}, n dimension of algebraic extension of rationals to which parameters of extremals belong. This hierarchy of phases has interpretation as dark matter. Quantum coherence in even astrophysical and cosmic scales becomes possible. In biology the implications are dramatic.
Consider now the TGD solution to problems motivating inflation and also many other problems.
 The problem of cosmological constant. Twistor lift replaces spacetime surface with the analog of its twistor space  an S^{2} bundle with twistor structure induced from the product of twistor spaces of M^{4} and CP_{2}. Length scale dependent cosmological constant is the outcome of dimensional reduction of 6D surfaces to S^{2} bundle with spacetime surface as basespace.
Cosmological constant decreases in stepwise during cosmological evolution in phase transitions. During the expansion periods following the reduction of cosmological constant the Kähler magnetic energy of expanding flux tube transforms to particles but eventually the expansion halts since volume term increases and one reaches energy minimum. This process produces matter.
 The solution to the problem of dark energy. The interpretation of the energy of monopole flux tubes having flux and return flux at different spacetime sheets connected by tiny wormhole contacs is as dark energy: the reason is that test particle between the sheets experiences no classical forces. Long range gravitational fields are however created. Also the recent accelerating phase in the recent cosmology might correspond to such a transition. What is amazing that there is direct connection with biological scales.
 What before possible rapidly expanding period leading to radiation dominated phase? There was no spacetime in the usual sense  that is as 4surfaces with 4D M^{4} projection  but cosmic strings with 2D M^{4} projections. String like objects dominated.
 Was there a period of rapid inflationary expansion? The stepwise variation of cosmological constant in phase transitions means increase of the thickness of the flux tubes assignable naturally to cosmological expansion at QFT limit. The transition from string dominated to radiation dominated phase creating spacetime in GRT sense would be one example. There would be entire series of this kind of fast expansions during which cosmological constant would be reduced. The recent accelerated expansion of the Universe would be also example of this kind of phase transition.
Quantum criticality provides second perspective. The phase transition from cosmic string dominated phase to radiation dominated phase is the first transition of this kind and one expects approximate flatness of 3space at QFT limit since criticality does not allow dimensional parameters. Vanishing 3 curvature is highly plausible at the quantum field theory limit in which spacetime sheets are replaced with a slightly curved deformation of M^{4} and fields are identified as sums of purely geometrically determined induced fields at various spacetime sheets. This requires very fast expansion and at the limit of vanishing cosmological constant infinitely large spacetime sheets  one theory allows vacuum extremals representing very fast expansion
 Why the constancy of the cosmic temperature? The cosmic string phase would have been quantum coherent in long scales or at least scales corresponding to the recent horizon size. The general prediction of quantum coherence even in astrophysical scales in the recent Universe and this revolutionizes the vision about astrophysics. The network formed by cosmic strings/flux tubes is like a nervous system and long range correlations between astrophysics of distant objects are predicted. The "AxisofEvil" one of the problems of inflation  is an excellent example of this. Flux tubes replace wormholes in EREPR correspondence and make possible long range quantum entanglement.
 Solution of the initial singularity problem. The energy density in cosmic string dominated phase behaved like 1/a^{2} (a the proper time coordinate of M^{4} lightcone). Energy per comoving volume went to zero. No initial singularity: silent whisper amplified to relatively big bang.
 Solution of the monopole problem. Cosmic strings can carry monopole flux but there are no magnetic charges. This is due to geometry of CP_{2} Kähler form is monopole field in homological sense.
 Magnetic field problem: how there can be magnetic fields in all scales in the recent cosmology? currents creating them are not possible in early cosmology. Monopole magnetic fields require no currents to create them. Monopole flux tubes are in central role in the model for the formation of galaxies, stars, even planets, even magnetic fields of Sun and Earth. Even in biology, and down to elementary particles physics. String like objects are accompanied by genuine fermionic strings associated with singularities of minimal surfaces seem to populate the entire physics in sharp contrast to superstrings.
See the chapter TGD inspired cosmology.
