The topic of this article has interested many thinkers. One of them is the world renowned emeritus professor **Stephen Hawking** at Cambridge University who claims that the concept of time has no meaning before the beginning of the universe. He states that to ask what existed before the **Big Bang** is like asking what is South of the South Pole and that the universe has created itself as a one-shot product of quantum fluctuations. According to him, there was nothing before the big bang. However, **Sir Roger Penrose,** emeritus professor at the University of Oxford, winner of the Copley Medal and the Wolf Prize in Physics which he shared with Stephen Hawking, believes that the Big Bang is one of perhaps an endless succession of Big Bang events where quantum fluctuations in the remote future of a previous cosmic aeon has produced the Big Bang of our aeon.

Along with professor **Vahe Gurzadyan** of the Yerevan State University in Armenia, Penrose states that images of cosmic microwave background radiation received from NASA show imprints that are older than the Big Bang. According to the new theory of "**conformal cyclic cosmology**" described in his book **Cycles of Time**, black holes will eventually consume all the matter in the universe. When they have finished, all that will be left in the universe is energy which will trigger the next Big Bang and the new aeon.

Anyway, Big Bang is merely a name for a theory that describes but does not explain the cosmic expansion, like Dark Energy is a name for a theory describing but not explaining the apparent acceleration of the same expansion. **Sir Fred Hoyle** coined the term "Big Bang" to ridiculate the idea of a sudden beginning for the universe. He developed a theory of such a continuous creation that the density of matter in the expanding universe remains unchanged. While his "steady state theory" is now abandoned, the present Dark Energy hypotheses build on ideas of continuous creation and of unchanged density. Another version of a steady state theory is proposed by **C. J. Masreliez** at EST Foundation. According to him, time has no beginning and progresses in discrete increments of **the expanding spacetime**.

An eternal succession of discrete increments of time as proposed by Masreliez or of Big Bang events as proposed by Penrose represents thinking of time as moving only in one direction. It makes me remember the endless sequence of turtles in an old story incorporated in Hawking's book **A Brief History of Time**: A well-known scientist (some say it was **Bertrand Russell**) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise." The scientist gave a superior smile before replying, "What is the tortoise standing on?" "You're very clever, young man, very clever," said the old lady. "But it's turtles all the way down!"

Global sustainability seems to depend on whether we move fast enough from linear to circular thinking. A sustainable model of the universe should be circular as in **The Wave Theory of Time (WTT)**. Predicting the apparent acceleration of the cosmic expansion, WTT was proposed by me for publication in **Nature** before that discovery in 1998 but became first published by the Swedish Mathematical Society in the autumn of the The International Year of Astronomy 2009. It describes the beginning of the cosmic expansion as merely a zero crossing point in a continuous wave of time. **The Planck-Einstein equation** is assumed to connect a very low frequency continuous wave to the rest mass of an elementary particle for gravitation and the rest mass of the elementary charge to a very high frequency continuous wave. It is the huge ratio between these frequencies that determines the difference in strength between gravity and the electromagnetic force.

Hawking proposed once that the flow of time changes direction at a future peak of the cosmic expansion. He regretted that proposal later and called it a big blunder. According to WTT, time flows in opposite directions for subsequent half periods of the above-mentioned continuous waves. Our present direction of time is positive by definition so before the Big Bang existed "Negative Time". That is my short answer!

The figure above shows Leonardo da Vinci's iconic work "The Vitruvian Man" as modified by me to serve the purpose of illustrating my short answer to the deep question that is the topic of this text. Presently, **an explanatary text is available in Swedish** and under work in English to be included in this article. The Swedish Mathematical Society published in May this year an article in which a portion of the figure was included in a version provided with equations found in the above-mentioned text published 2009. The following lines are an extract of that article translated into English with reference made to the graph below. They describe the WTT circular model of the universe and of time flowing in alternating directions.

It is a generally accepted view that about 14 thousand million years ago the universe had a start 1 of expansion called the Big Bang, that the expansion is related to a variation in the metric of space and has started from nothing, and that this expansion has a positive acceleration at the present point of time 2 as shown by the graph rotated 90 degrees clockwise. Some scientists predict a prolonged phase of accelerated expansion 3 until all the elements of the universe break apart. Other scientists propose that quite soon the positive acceleration could cease and turn negative resulting in a phase of contraction 4.

The expansion and contraction phases 3 and 4 are marked with arrows between which a mathematical model is shown. The model describes the shift from positive to negative acceleration by means of exponential expressions using the natural base e raised to the power of x and to the power of -x. The contraction results in an end 5 as shown by the graph rotated another 90 degrees clockwise. A hasty interpretation would be that the birth of the universe in the Big Bang is followed by its death with everything disappearing in a Big Crunch.

It is a fact, however, that the above-mentioned mathematical model is derivable from a most fundamental mathematical model of repetitive phenomena, namely the sine wave model. Rotating the graph another 90 degrees clockwise, the shape of the sine wave is shown by the dashed line. If the variation in the metric of space is expressed as a ratio between a measure dt_{c} of cosmic time t_{c }and a measure dt_{r }of reference time t_{r }where dt_{r} is proportional to the second defined by atomic reference clocks and dt_{c} is proportional to the metric of space as determined by the constant speed c of light, then the equation dt_{c}/dt_{r} = sin (2π t_{c}/T) makes x in the above-mentioned exponential expressions equal to 2π t_{r/}T where T is the repetition period of the sine wave.

In the graph, the end 5 is followed by a new start in a phase that is shifted 180 degrees with respect to the start 1 and that results in new expansion and contraction phases. The two mathematical models have a common peak 6. According to the equation above, the measure dt_{c} alternates beween positive and negative values so the cosmic time t_{c} flows in alternating directions inside the dashed line. Curved spacetime describes gravitation and the primary cause behind the curvature can be interpreted by describing simply the cosmic time t_{c} as flowing in a closed loop. Its repetition period T determines the mass of omnipresent elementary particles of gravitation which provide an interpretation of Dark Matter. As to the positive acceleration in the expansion of space, it is here derived from the sine wave model without any assumption of Dark Energy as dominating in the universe and causing a stronger force than gravitation. That assumption can one day be viewed as the cause of the greatest bubble in the history of science.

A more detailed answer than given above to the question in the headline is that there were Dark Matter and Negative Time before the Big Bang.

## What Existed Before Big Bang?