Hinduism for physicists

Hinduism for Physicists! Or why Non-Abrahamic Eastern Religions do not have any conflict with Science. 

Kashyap Vasavada

            First of all, I thank Lubos for giving me an opportunity to express my views on this guest blog. I think, most of you, like me, read his blog to understand recent developments in physics. Lubos does an outstanding job in explaining these matters from a technical point of view and fills the gap between popular articles and original papers admirably.  Social and political issues are also discussed here often, but you may be wondering whether this blog belongs here. I am hoping that you will find it interesting. At the very outset, let me make it absolutely clear that there is no question about the great success of scientific method during the last few hundred years. As is well known, the scientific method consists of making observations with the sense organs (mainly eyes with the help of devices like telescopes, microscopes, electronics etc.); making models using our brains and checking if these models agree with the observations. This trivial statement will be important when we talk about the other method. By following the scientific method, we now know unbelievably large amount about the universe we live in. It will be foolish for anyone to suggest that scientists should abandon this method for investigating the universe. Thus I would be arguing for, not science or religion, but rather for science and religion.  In my opinion both science and religion have limitations and both are useful for the good of the mankind.

            The immediate reason for writing this blog is a recent survey that only about 21%of Americans believe in Big Bang theory, 27% in 4.5 Billion years old earth and 31% in theory of evolution. Assuming that the survey was correct, such a low rate of belief in science is probably based on literal interpretation by the participants, of their scriptures which were written thousands of years back. In the blog about that survey, I made a comment that eastern religions (in particular Hinduism and Buddhism) do not have any problem with science whatsoever. In the following I will try to explain this assertion. In fact you would have hard time
finding a single Indian (or someone from many other Asian countries) who believes in young earth creationism or is against Big Bang theory or theory of evolution.

            I will start with the concept of God in Hinduism. In some sense, Hinduism is the most misunderstood religion in the west, starting with its name! The real name is “Sanatana Dharma” meaning universal duty, responsibility or a prescribed way. The word “Hindu” was coined probably by Persians thousands of years back. They called people living near the river Sindhu (Indus) as Hindus! Unfortunately the word Hinduism is stuck like God particle for Higgs boson! So I will have to use it. The main scriptures are called “Vedas”, “Upanishads” and “Bhagvad Gita”. The religion did not originate from one prophet, but a number of sages contributed to the Vedic vision portrayed in the scriptures. Founding fathers of quantum theory like Schrodinger, Heisenberg and Bohr were familiar with these scriptures. These scriptures talk about (in Sanskrit) a formless, shapeless, omnipresent, omniscient God as Brahman. It is a universal, ultimate, super consciousness. It is supposed to be present in every particle of the universe. Thus the Hindu concept is: God is not like a king emperor who created the universe and rules arbitrarily from outside but it is an all pervading eternal entity. This consciousness may be present in every living and non-living matter i.e. in every single part of the universe and it would be synonymous with the laws of nature. The expression of consciousness in different systems may be different. So how do you realize it?  Well, the sages say that you have to prepare your mind to receive it.  Even in a scientific lab you have to prepare your system to observe any effect. In a sense this is similar to the fact that electromagnetic waves (including CMB) are present all around us and we do not realize their existence unless we have radio, TV, microwave detectors etc. Remember, Higgs field is all around us but we had to build a 10 Billion $ LHC to find Higgs particle! To realize this super consciousness, the prescription is to meditate, calm down your mind completely and only then you would realize it after a lot of hard effort. The scriptures say you do not have to believe it. If you are willing to put in effort you will verify it. This is a big hang up. It is not something which, like the result of a scientific experiment, can be seen by a crowd on a TV monitor. It is one on one. This is an experience outside our sense perception. How do I know it is real? Well, honestly, I do not have any personal experience. But I believe it is possible because I know some yogis have achieved it and they describe the experience.

            There is a beautiful interesting illustration in Vedas which Schrodinger mentions in his book on “what is life”. The question is why consciousness looks similar when our bodies look different. The answer in Vedas is that the source of consciousness is outside. We are merely reflecting it as multiple mirrors would reflect a single object!

            Then what about all these images and statues of hundreds of gods you see in a Hindu temple? The answer is:  everyone is not equipped to understand Brahman in the pure form. One needs some advancement. Suppose you are a physics teacher who is discussing electricity-magnetism. Your discussion  depends on whether the students are in  a primary school, high school, undergraduate program, graduate program or doing post-doctoral work. Similarly, everyone is not at the same spiritual level.  The suggestion is that, if it is too difficult for you to do meditation on the ultimate consciousness, it is OK to worship God in whatever deity form you are comfortable with. This gave rise to a concept of large number of different forms of God (Avatars or incarnations) in whom you can find solace and get guidance.  Over thousands of years a number of mythological stories also came up. Faithful believe that these deities are incarnations of God who came to earth in specific forms to achieve some purpose, typically to destroy evil and establish goodness. They may exist in a realm which is not directly accessible to our senses. Usually there are moral lessons also associated with Avatars. People may have their favorite deities to worship. The faith evolved over several thousand years as a kind of cafeteria system. You worship a deity depending on your comfort level and ability. Usually one’s upbringing also plays a role.  In the end it does not make any difference whom you worship. Everyone realizes what the icons represent symbolically. One strong point of Hinduism is the tolerance or (better) acceptance of different viewpoints. The belief is that there are thousand paths to realize God and everyone can choose his/her own path.  Some followers believe in miracles but not too strongly to challenge science! It is well known and even scientifically established that faith does work wonders and there is a close connection between body and mind.In my personal case, I find association with the temple activities very enriching. Not only that there is no anti-science talk, but also the prevalent view is that most or all activities are compatible with science. It does help that Hinduism is flexible and non-rigid. So I can continue my interest in learning and teaching physics and learn about metaphysics as part of religion too. One point I should emphasize to western readers: in Hinduism, philosophy, metaphysics and religious ritual practice are all mixed together. We think this is a strength rather than weakness. In a typical Hindu temple, you will find rituals going on all the time, together with philosophical talks. Rituals are prescribed for an average person because, at least for that time period he/she is engaged in thinking about God instead of mundane affairs of life and that would lead to cleansing  of mind  and get inspiration to lead a good life. Personally, I prefer philosophical, metaphysical discourses to rituals, but participating in rituals is also fine with me.

            As in other religions, Hinduism has commandments for leading a good life e.g., speaking truth, non-violence, love, compassion, ethics, morality etc. Hinduism believes that whatever one does, has consequences as a Karmic relationship. This would be similar to the law of action and reaction in physics. Admittedly, here, the mechanism for consequences remains unknown. But the belief is that one can wipe out bad Karmas with good Karmas. This may take several births. So Hinduism firmly believes in reincarnation, i.e. everyone has a soul (called Atman) which migrates from one body to another on death. Ultimately when one is completely free from Karmic bondage, he/she gets liberation called Moksha or Nirvana. Obviously, there is no scientific or material proof of these beliefs.

            Now let us consider two main issues in which some westerners see conflict with science: age of Universe and theory of evolution. On both of these issues, the Hindu sages got approximately correct ideas in agreement with science. Just by thought processes they realized that universe must be billions of years old, as noted by Carl Sagan in his book on cosmos. The other realization was that there must be some connection between animals and human beings. If human beings have souls, then animals too have souls. That gave rise to mythological stories that God came to earth in the form of first sea animals, then land animals and then human beings. These are ten Avatars of god Vishnu.

            How about origin of universe? Of course one cannot say that the ancient sages’ knowledge was anywhere comparable to the current knowledge.  But just see the astonishing description of origin from a scripture known as Vayupuran:

“In the beginning, there was nothing in the universe. The Brahman (the divine essence) alone was everywhere. The Brahman had neither color nor scent; it could not be felt or touched. It had no origin, no beginning or no end. The Brahman was constant and it was the origin of everything that was destined to be in the universe and the universe was shrouded in darkness.” Intersting! It was dark because visible light was not created yet!! In all Hindu scriptures, multiverse and cycles of creation and destruction of universe lasting billions and trillions of years are frequently mentioned.

            Another excerpt from Vedas: “The universe is brought about by the collapse of fullness in the transcendental field in which reside all the laws of nature responsible for the creation of the entire manifest universe. How is the transcendental level functioning? It is functioning from its unbounded nature to point to itself.  He who does not know that initial pure consciousness state, ultimate reality, what can the laws of nature accomplish for him? He who knows it, remains established in evenness, unity, wholeness of life”. Since Brahman was by itself, it is clear that it interacted with itself i.e. self-referral (like inflaton!!!) and eventually manifested in every particle of the universe. It is a very interesting parallel with modern cosmology. Strictly speaking the word “manifestation” rather than “creation” is used in Vedic cosmology with a subtle meaning.

            Now let us examine a frequent argument that science is based on logic and reason while religion is based strictly on faith. Well, our everyday logic and intuition are based on our life experiences with the world at the classical level. Modern physics has demolished this argument.  If you say that only thing physics should care for, is develop mathematically consistent models, (no matter how bizarre they appear to our intuition) and try to see if they are in agreement with experiments, then you are perfectly ok. In a sense, I agree with Lubos that as far as science is concerned, there is no need to look at the meaning of models. But the moment you look for meaning of the equations you get into mess (metaphysical if you will). As everyone on this blog knows, the world is made out of fuzzy wavelike dynamic stuff and not solid rigid objects we see around. The particles are in some sense both here and there at the same time and are described by a wave function, a superposition of seemingly contradictory properties. This closely parallels description of Brahman in Hindu scriptures “It moves and it moves not; it is far and it is near; it is within all this and it is also outside all this.”  The ultimate shock of quantum mechanics, for visualization in terms of our everyday life, came with Bell’s theorem and corresponding experiments on entanglements. Lubos has written about this topic because of its importance. One has to choose between locality and reality. I think most physicists choose to keep locality to save theory of relativity at the cost of reality i.e. the particles are believed to be in some kind of suspended state devoid of any specific properties until they are observed. It is well known that Einstein did not like this. Another basic finding of quantum theory is the involvement of the observer in the observed things. It is impossible to separate the effect of the measuring apparatus from the object measured. Detachment of the two is just not possible. As John Wheeler said “universe is indeed participatory!” Max Planck regarded consciousness as fundamental and matter as derivative from consciousness. At one time Wigner expressed his view that consciousness creates collapse of wave function. There have been debates for some 90 years about interpretation of quantum mechanics without any resolution in sight. Ideas about the entanglement of the observer and the object of observation are also emphasized in Upanishads. In the Hindu concept the observer (Brahman) is in the system itself and is time independent.

            There is a similar situation with theory of relativity, namely relativity of time for different observers, curving of space-time by matter in its neighborhood and possible singularity at the big bang. This theory has again challenged our intuition. Here also ancient Hindu sages did not have any problem. In certain scripture, you can find a statement that 100,000 human years is equivalent to 1 second of divine time!  They also talk about simultaneous visions of past, present and future and time travel!

            The readers of this blog already know about leading physicists discussing a multi-dimensional world of string theory. Hardly anyone would say that it is intuitive.  More recently, if BICEP2 experiment and its interpretation are verified, it would mean that our observable universe started with a size of smaller  than a proton  and grew by a factor of  e^80 or more by expansion of space. The conclusion is that our everyday logic just does not work in Modern Physics, although Mathematics works superbly.

            I am not saying that arguments of modern cosmology and modern physics are on the same footing as the metaphysical thoughts about universe and God in Hindu scriptures, but it sure should make one stop for a moment and think about the ultimate nature of reality in the two pictures. Again, all these intriguing developments are fine if you do not insist on visualizing by our poor little brains! I would say that one should not require higher standards of our intuitive understanding for religion than for science!

            My basic suggestion is that let us be modest. Although we can be proud of our achievement in understanding so much about the universe, just think for a moment. We are on a measly little planet bound to an average star in an average galaxy with more than 100 Billion stars. There are more than 100 Billion galaxies in our observable universe. There could be an infinite number of such universes. Our eyes and brains evolved in a specific manner on earth. Both of these have limitations. For example, our eyes are only sensitive to visible part of the electromagnetic spectrum only. Thus it would be height of arrogance and even stupidity to assume that what we can find with our sense organs and understand with our brains is all there is to it in the universe.  Although direct verification is hard at this point, it is not unreasonable to assume that there could be a world beyond our sensory perceptions.

            The reader might say that all this sounds like new age pundits talking mysticism! Well it is but there may be an underlying subtle reality! A number of authors have written books comparing modern physics with metaphysics of east. There are mystical ideas floating around about quantum consciousness and unified field of consciousness.  There is a Harvard trained theoretical physicist John Hagelin who is a professor of physics at Maharishi University in Fairfield, Iowa. He has written mathematical equations for such fields (using Lagrangian formalism like a respectable conventional theoretical physics). While these attempts are fine, my feeling is that they are somewhat premature for theoretical physics. It is not clear how one can verify the solutions of these equations which can satisfy scientists.   Then there are also serious scientific models by Penrose-Hameroff, Stapp and some others looking for quantum mechanical processes in our brain which could explain consciousness. As of now they are inconclusive and have not been widely accepted. The main issue is that, as yet, neuroscientists have not understood consciousness. It is not clear how far down some kind of primitive consciousness goes in the tree of life. Suggestion that there could be some presence of consciousness in non-living systems would be called metaphysical at best, although some argue that the quantum dynamics already resembles a kind of consciousness. So consciousness may be much more subtle than our brain functions, thought processes etc. But we have to travel a long way to theoretical physics if indeed these ideas work out. I just want the readers to be aware that such models could be part of reality whatever it turns out to be.

            Although I do not wish to belabor the point, question of existence of God is similar to some statements in Gödel’s theorems. He has proved that within a mathematical system, there would always be some statements which cannot be proved or disproved.

            To summarize: I am not saying that Hinduism is based on statements which have material, scientific proof.  But at this point there is no direct contradiction with any facts which science can establish. Hinduism goes beyond a point where science stops. It has concentrated on inner (non-sensory) understanding of reality through methodology known as Yoga. In terms of our concepts of reality, it seems that a previously assumed rigid line between physics and metaphysics may be rapidly disappearing! Normally I stay on the physics side of the line. But some time I am not so sure!

            I have described Hinduism in some detail but not Buddhism. Buddhism arose in India as an off-shoot of Hinduism but became a completely independent religion in several other Asian countries. It has many similarities with Hinduism and believes in Karmic consequences. The belief is that humans can achieve ultimate liberation (Nirvana) by following the rightful behavior recommended by Buddha. According to Buddha, explaining the concept of God would be very difficult; hence Buddha neither rejected nor accepted the existence of a creator deity.

            Now I will make few remarks about religions in general. Personally I am comfortable with most religions at the commandments level. Most religions maintain that morality, ethics, love, compassion etc. are integral parts of good life. Religious people believe that these commandments came as revelations from God. Whether one believes in God or not, we have to accept that these are good principles for the society. Unfortunately, many people (believers and non-believers) do not follows them. While it is possible to be virtuous and spiritual without being religious, believers would have special reasons to follow these. I can sense an immediate response from the readers about some religious zealots who maintain that if you do not believe in their God, they would kill you. While this is true unfortunately (in a small number of cases) this is not that wide spread and we have to work to eliminate these tendencies by education. In addition, there are social problems of drugs, high crime rates, murders etc. for which religious institutions can play an important role. I do not have to remind this scientific readership that dangerous weapons of mass destruction created by science, namely nuclear, chemical and biological weapons, can bring an end to all human life and perhaps the entire life on this planet. Thus we have to find some way of living with each other in a peaceful manner. My suggestion is that religion if properly followed can play a useful role, without which there would be nothing to hang on.

            Although the main focus of this blog is to inform the readers about Hinduism and its lack of conflict with science, I conclude with some general remarks about science and religion.  I would say this to the general readership: debate about science and religion is not all black or white. There are many scientists who are believers. Some surveys indicate 30 to 50% of all scientists are believers to a certain extent. Some of these are high profile scientists, at least one Nobel Laureate in physics. They realize that there are some finer, subtle points about their particular religions. At the same time, scientists should protest against ideas of young earth creationism, intelligent design and anti-evolution propaganda. In addition, they should insist that they are the ones who decide what to teach in science classes in schools and they should not let non-scientists dictate it. Europeans may find this funny but this simple issue is becoming important in U.S.! There are frequent court battles about whether teachers should teach creationism as part of a science class! I routinely write in our local newspaper on these issues. On the other hand, I do not care also for tirades against religion in which some prominent physicists and other scientists have engaged. I think science and religion can have a peaceful coexistence and can enrich human life. In a way I am calling for moderation and acceptance of importance of each other by both sides. Let us have a balanced view of science and religion. 

Eva guest blog

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Let us assume that the BICEP2 result is confirmed as cosmological, and indicates primordial gravitational waves generated during inflation. Within the context of inflationary theory, this groundbreaking discovery has important implications for quantum gravity, for which string theory is our leading candidate.

String theory contains a rather simple mathematical structure -- monodromy -- which can generate a significant tensor signal. Here I'll explain that mechanism, and discuss its range of applicability as we currently understand it. String theory also contains a plethora of scalar fields, which in itself gives a realization of assisted inflation, N-flation, covered in and earlier blog post. (More generally, one has multiple directions along which monodromy operates, a combination of these two mechanisms.)

In two lines one can relate the number of e-foldings of inflation to the field range, assuming no strong variations in the slow roll parameters during the process. This relation, the so-called Lyth ``bound", implies a super-Planckian field range for the inflaton field \(\phi\) during the process. Inflation requires a slowly decreasing source of potential energy \(V(\phi)\) over this range \(\Delta \phi > M_{Planck}\).

From a low energy point of view, we may parameterize our ignorance of

Axion Monodromy Inflation guest post by E. Silverstein

Let us assume that the BICEP2 result is confirmed as cosmological, and indicates primordial gravitational waves generated during inflation. Within the context of inflationary theory, this groundbreaking discovery has important implications for quantum gravity, for which string theory is our leading candidate.

 String theory contains a rather simple mathematical structure -- monodromy -- which naturally generates a significant tensor signal.  In this guest post, I'll describe that mechanism, and discuss its range of applicability as we currently understand it. (String theory also contains multiple axion fields, which in itself gives an interesting realization of assisted inflation, N-flation, covered nicely in an earlier blog post. It was later realized that along each such direction the monodromy effect operates; in general, one may consider a combination of these two mechanisms.)

Before getting to inflation in string theory, it is important to understand the motivation for combining these subjects.  Using the chain rule, one can relate the number of e-foldings of inflation to the field range, assuming no strong variations in the slow roll parameters during the process.
\[
N_e = \int \frac{da}{a}=\int\frac{da}{dt}dt=\int\frac{HM_P}{\dot\phi}\frac{d\phi}{M_P}=8r^{-1/2}\frac{\Delta\phi}{M_P}
\]
where in the last step we used the slow-roll inflation result for the tensor/scalar ratio \(r\) and we assumed that \(\frac{HM_P}{\dot\phi}\) varies slowly, as in simple slow-roll inflationary models.  This relation, the so-called Lyth ``bound" http://arxiv.org/abs/hep-ph/9606387, combined with the BICEP2 result  \(r\gg .01\), implies a super-Planckian field range for the inflaton field \(\phi\) during the process.

Inflation requires a slowly decreasing source of potential energy \(V(\phi)\) over this range \(\Delta \phi > M_{Planck}\). Variation of the potential \(V(\phi)\) over ranges in \(\phi\) at (or below) the Planck mass scale is strongly constrained by the requirement that \(V\) generate enough e-foldings of inflation, and by CMB data on the power spectrum. Turning this around, the process of inflation and the observed perturbation spectrum are sensitive to an infinite number of corrections to the inflaton potential which are suppressed by the Planck mass scale.  We call this situation UV Sensitivity (although of course we're talking about much higher energies than the ultraviolet electromagnetic spectrum!), and it's a tremendous opportunity for getting a window into quantum gravity.

 Said differently, if we were to parameterize our ignorance of such effects from the point of view of low energy effective field theory, we would have to take into account the possibility that as the field \(\phi\) rolls along its field space, corrections to the potential arise via couplings to whatever degrees of freedom UV complete gravity.  Over a large range of field, the conditions could change dramatically, and it would seem miraculous to obtain the pristine conditions (slowly varying V) required for inflation.  As we will see, the structure of monodromy along axion directions in string theory produces a large field range, with an underlying softly broken discrete shift symmetry maintaining similar conditions all along the super-Planckian trajectory.  That is, the theory will naturally address this puzzle in a way that is tied to the structure of its gauge symmetries.

In general, an approximate symmetry under shifts of the field \(\phi\) can address this puzzle, even from the low energy field theory point of view.  As such, traditional large-field models of inflation, such as Chaotic Inflation and Natural Inflation are internally consistent and `natural' from the Wilsonian point of view -- the potential is protected from problematic quantum corrections.  

However, for the reason just discussed, such models make a strong assumption about the UV completion of gravity -- that its quantum (and classical) contributions to the potential not only respect a symmetry, but produce precisely the potential postulated in the field theory model.

Although the inflationary paradigm is compelling as it stands, and now well-tested, many theorists are not completely satisfied with purely ``bottom up" models (although needless to say some of these have been important and pioneering contributions).  Those of us in this category regard large-field inflation and its associated tensor signal as requiring, or at least strongly motivating, a treatment which accounts for quantum gravity effects.  Since string theory is a well-motivated candidate for quantum gravity (already passing many thought-experimental and mathematical consistency checks), it makes sense to analyze this question in that framework.

I will focus on one rather broad mechanism -- which has been realized by specific string theoretic models as proofs of principle.  Before continuing, let me address an issue that sometimes arises.  Various people have made comments along the lines that `most' string theory models are ruled out.  Certainly small-field models predicting tiny values of the tensor to scalar ratio are now falsified, a healthy part of science.  Those works, particularly http://arxiv.org/abs/hep-th/0308055,  played a crucial role in establishing a standard of theoretical control in the field, emphasizing the effect of Planck-suppressed operators; others such as http://arxiv.org/abs/hep-th/0404084 helped stimulate a more systematic, model-independent understanding of inflation, leading to a more complete analysis of non-Gaussianity in the CMB.  Although they played a useful role, these and many other models, at least in their original form, are dead given primordial B-modes from inflation.  However, there is no credible argument that string theoretic inflation is generically small-field; in fact to me (even before the BICEP2 announcement), it has always seemed quite possible that it goes the other way because of the plethora of axion fields in the low energy spectrum arising from string theory.  In any case, there are many works on both cases (large and small field); and needless to say the statistics of papers is a very different thing from the statistical distribution of string theory solutions.    

String theory contains many axion-like fields, descending from higher dimensional analogues of the electromagnetic potential field \(A_\mu\).  These include the 2-form potential field \(B_{MN}\) sourced by the fundamental string, and more general p-form potentials \(C^{(p)}_{M_1\dots M_p}\) sourced by the various branes of string theory.  Axion-like scalar fields in four dimensions arise from integrating these potential fields over the extra dimensions, for example
\[
 b(x)=\int_{2-cycle} B
\]
and its generalizations, some of which are related by string theory dualities.  There is a beautiful structure of inter-related gauge symmetries which are respected by the effective action, including terms of the form
\[
|\tilde F|^2 = |dC_p+ dC_{p-2}\wedge B+dC_{p-4}\wedge B\wedge B+\dots |^2
\]
generalizing a Stueckelberg type coupling \[(d\theta + A)^2\] familiar from symmetry breaking in quantum field theory.  In the latter case, the gauge symmetry \(A\to A+d\Lambda, \theta\to \theta -\Lambda\) is respected since it includes the transformation of \(\theta\).  Similarly, in the string theory effective action, the gauge symmetry under which \(B\to B+d\Lambda_1\) goes along with compensating shifts in the $C_n$, leaving the \(|\tilde F|^2\) term invariant.

The next step is to note that the fluxes obtained by integrating the field strengths  \(dC_n\) over internal cycles in the extra dimensions are quantized, taking integer values (appropriately normalized).  Also, the size and shape of the internal dimensions are dynamical `moduli', descending from higher-dimensional Einstein gravity (along with other fields, like the string coupling) plus string-theoretic corrections.  Altogether when we dimensionally reduce from higher dimensions to four dimensions, the potential is schematically of the form
\[
f_1(moduli) N_1^2  (b+Q_2)^2+ f_2(moduli) N_2^2(b+Q_2)^4 +\dots
\]
(and similarly for other types of axions).

We can read off several important features from this structure.

First, the theory as a whole has a periodicity: if we move from some value of the field \(b\), say \(b=b_0\), to \(b_0+1\), this is equivalent to shifting the flux quantum numbers \(Q_1\) and \(Q_2\).  However, with a given choice of flux quantum numbers -- i.e. on a given branch of the potential -- the field range of \(b\) is unbounded.  In particular, each branch of the potential is not periodic in \(b\), in the presence of generic fluxes.  This is a relative of the Witten effect in gauge theory.

Another key point is that when we normalize the scalar field canonically, rescaling to form the inflaton field
\[
\phi = f b,
\]
the periodicity \(f\) in \(\phi\) is sub-Planckian, by a factor of \(1/L^2\), where \(L\) is the size of the internal dimensions in units of the string tension (see e.g. http://arxiv.org/abs/hep-th/0303252 ).  This is an example of the fact that despite the many solutions of string theory (the `landscape'), the theory has a lot of structure.  Not anything goes, even though it is true that the theory has many solutions.  In any case, despite this sub-Planckian underlying period, the fluxes generically unwrap the axion potential, leading to a super-Planckian field range.

Because of the underlying periodicity much of the physics remains similar along the whole super-Planckian excursion of the field.  This is in a nutshell how this mechanism in string theory addresses the original question raised by effective field theory above.

The same features arise in the presence of generic branes in string theory, something we can understand both directly, and using the AdS/CFT duality to relate the flux and brane descriptions.  On of our original realizations of this mechanism in a string compactification http://arxiv.org/abs/arXiv:0808.0706 (with McAllister and Westphal) arises in this way, with a linear potential built up via the direct coupling of axions to branes.   (See also http://arxiv.org/abs/arXiv:0907.2916 by Flauger et al as well interesting as field-theoretic treatments in e.g. Kaloper et al http://arxiv.org/abs/arXiv:1101.0026 and http://arxiv.org/abs/arXiv:1105.3740 and many other references.)
More simply, the potential is like a windup toy.  This has recently been generalized, with an interesting mechanism to start inflation, in http://arxiv.org/abs/arXiv:1211.4589.

A schematic picture of one example of axion monodromy in string theory (the direction around the circle being related by a local string duality to an axion direction, and with potential energy built up by the stretched `D4-brane').

Secondly, working for simplicity on the branch \(Q_1=Q_2=0\), the potential is analytic in \(b\) near the origin, but at large values (as are relevant for inflation), the other degrees of freedom such as the `moduli' (and also the internal configurations of fluxes etc.) can adjust in response to the built up potential energy.  As a result, we find a potential which near the origin is a simple power law, e.g. quadratic (or even quartic if \(N_2\) dominates -- something we are currently studying in light of the high BICEP2 central value for \(r\)).
But at large field values, the potential is typically flatter than the original integer power-law, as the additional degrees of freedom adjust.  This is a simple way of understanding the lower-than-quadratic power, \(V(\phi)\propto\phi\) that we obtained explicitly in the original example (as explained in http://arxiv.org/abs/arXiv:1011.4521 on this `flattening' effect).

Finally, although each branch of the potential goes out to large field range, the underlying periodicity leads to a residual sinusoidal modulation of the potential.  The amplitude and period of this modulation depends on the values of the moduli fields, and because those are dynamical they can themselves vary in time during the process.

Because of the moduli fields, the construction of complete string theory models realizing this (or any) mechanism for inflation is quite involved.    The top-down construction http://arxiv.org/abs/arXiv:0808.0706 provides a proof-of-principle, and has become a benchmark example in CMB studies.   But it is clear that the mechanism is much more general, as was emphasized in http://arxiv.org/abs/arXiv:1011.4521 as well as other works.  These include a useful paper by soon-to-be Stanford postdoc Guy Gur-Ari http://arxiv.org/abs/arXiv:1310.6787  which lays out some possible realizations on twisted tori, while pointing out an error in my original attempt to stabilize string theory on nilmanifolds (happily, this flaw was not uncovered until after the twisted tori suggested the monodromy mechanism, which transcends that particular compactification...).
  
Monodromy inflation is falsifiable on the basis of its gravity wave signature, and so given the BICEP2 result of nonzero \(r\) at high statistical significance (assuming it is indeed primordial), large-field inflation in general and monodromy inflation in particular has passed a significant observational test.  There are opportunities at a more model-dependent level for more detailed signatures, involving the residual modulation of the potential, see e.g. http://arxiv.org/abs/arXiv:1303.2616, http://arxiv.org/abs/arXiv:1308.3736, http://arxiv.org/abs/arXiv:1308.3705, http://arxiv.org/abs/arXiv:1308.3704 as well as the Planck 2013 release for interesting analyses putting limits on this possibility.  As mentioned previously, the dynamical nature of the amplitude and oscillation period make this a subtle analysis and we should try to develop a more systematic theoretical understanding.

In any case, given at least the successful prediction for gravity waves, we are now very interested in the range of possible values of the tensor to scalar ratio \(r\) (and other observables) that this mechanism (and related ideas such as N-flation http://arxiv.org/abs/hep-th/0507205) covers.  As a concrete first step, in ongoing work we have constructed additional examples of the `flattening' mechanism, but starting from quartic, \(|F\wedge B\wedge B|^2\) terms which can generate larger  values of \(r\).  The next step is to incorporate these into string compactifications with fully stabilized moduli, which would provide a proof-of-principle for relatively large values of the tensor signal.  Beyond that, we would like to understand the range of \(r\) values in theory space (UV complete) as systematically as possible.                

Incidentally, another mechanism for large values of \(r\) that we had previously considered (with Senatore and Zaldarriaga http://arxiv.org/abs/arXiv:1109.0542 ) involves another aspect of the quasi-periodic structure.  This structure raises the possibility of periodically repeated  and hence approximately scale-invariant production of particles or strings, which can themselves emit gravity waves.  That still requires inflation, but can easily enhance the tensor signal by up to a few orders of magnitude (and easily by an order one factor).  This mechanism is likely distinguishable by further data on the B modes (which will constrain their power spectrum and non-Gaussianity).

Altogether, I view monodromy inflation as more than just some random model but less than a complete theory.   It is tied to the structure of string theory and its symmetries in a way that seems pretty robust.  But at our current level of understanding, it is far from a complete theory -- if it were, we could write a computer program to generate the `discretuum' of values of observables like \(r\) that it UV completes.  We are far from that systematic an understanding, and even if we had it the data will not allow us to `invert' the problem and deduce a very particular model, even given smaller error bars to come with future data on the tensor to scalar ratio, its tilt,  and higher correlators.

Fortunately, some of the most important distinctions -- such as large versus small field -- are rather directly probed by the observations.  This breakthrough is somewhat analogous to the cosmological constant discovery 15 years ago -- even a single number can be extremely significant.  In the case of the B-modes, we theorists are a little better prepared than in the case of the cosmological constant, but there is much that remains to do.  We have entered an era of genuinely data-driven string theory research.  Exciting times!        

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BICEP2: Primordial Gravitational Waves!

Guest blog by Liam McAllister, Cornell University.

The BICEP2 team has just announced a remarkable discovery (FAQ): they argue that they have detected, at very high significance, the imprint of primordial gravitational waves on the polarization of the cosmic microwave background.  Moreover, the signal they see is very strong.  If they are right, this is The Big One.

BICEP on site at the South Pole

BICEP's map of the CMB:

Vorticity in the CMB according to BICEP

How should we interpret this result, and what are its implications?

If the BICEP measurement really is a detection of primordial gravitational waves, and if we interpret this finding in the context of the overwhelmingly favored theory for producing primordial gravitational waves — namely, inflation — then the implications of this finding are staggering.  I find it hard to imagine a more powerful, more transformative experimental result anywhere in fundamental physics, short of a discovery of extra dimensions or of a violation of quantum mechanics.

Let me now temper this excitement with a number of cautionary remarks, and then explain why a detection of inflationary gravitational waves would be so important.

Euler characteristic, reposted from TRF

Topology is an important branch of mathematics that studies all the "qualitative" or "discrete" properties of continuous objects such as manifolds, i.e. all the properties that aren't changed by any continuous transformations except for the singular (infinitely extreme) ones.



In this sense, topology is a vital arbiter in the "discrete vs continuous" wars. The very existence of topology as a discipline shows that "discrete properties" always exist even if you only work with continuous objects. On the other hand, topology always assumes that these features are "derived" – they're some of the properties of objects and these objects are deeper and that may have many other, continuous properties, too. The topological, discrete properties of these objects are just projections or caricatures of the "whole truth".

The sphere – the surface of a ball – can't be continuously deformed to a torus – the idealized two-dimensional inner tube inside a tire. The torus has a hole in the middle. So they're topologically distinct two-dimensional manifolds. We may prove that they're topologically different if we find a "topological invariant" – a number or a more complicated quantity that doesn't change by any continuous deformations – that has different values for both manifolds. Of course, the "number of holes" (known as genus) is a way to distinguish a sphere from a torus.