Tomislav Prokopec</a>

Welcome to the home page of

Tomislav Prokopec

I am associate professor (UHD) at the Institute for Theoretical Physics and Spinoza Institute at
the Faculty of Physics and Astronomy of Utrecht University.


My main research interest is in cosmology, in particular in cosmic inflation and formation of the large scale structure of our Universe, the origin of the matter-antimatter asymmetry and dark energy of the Universe.

Cosmic inflation is an epoch of a rapid and accelerated expansion of the very early Universe. By its rapid expansion, inflation can amplify quantum fluctuations of matter fields to produce a nearly scale invariant spectrum of cosmological pertubations, which get imprinted as temperature fluctuations on the cosmic microwave background radiation (CMBR) and in addition, they are believed to seed the large scale structre of our Universe. The question that interests me most is, what drives the primordial inflation, and to what extend can be determine the properties of cosmological perturbations from fundamental principles and physics. I find it equally fascinating to ponder on the origins of the matter-antimatter asymmetry, dark matter and the dark energy of the Universe. But more on these fascinating questions on my future website.

Here you will find a list of my publications from (IN-)SPIRE at My citation profile is here

Part of the lecture on cosmic inflation held on Aug 27 2009 at the Utrecht's Summer School.

The Cosmology Group of Tomislav Prokopec at the ITP, Spinoza Institute and EMMEPH

Current members:

Bogumila Swiezewska, postdoc (from Nov 2016).
Daan Marsmans, master student (from Feb 2016).
Yassir Awwad, master student (from Sep 2016).
Leonardo Rocha, master student (from Sep 2016).
Leihua Liu, PhD student (from Oct 2014).
Stefano Lucat, master student (May 2014-Jun 2015) and PhD student (Sep 2015 -).
Pavel Friedrich, PhD student (Dec 2015 -).
Roeland Koelewijn (master student, Sep 2015 - Jul 2016).

Former members:

Anja Marunovic, postdoc (Jun 2014-Aug 2016, funded by Marie Currie Actions, Croatian Ministry and by DITP).
Dražen Glavan, Ph.D. student (from May 1 2011-Aug 31 2015), visiting scholar (Sep 2015-Oct 2015); postdoc in Warshaw, Nov 2015-Oct 2018.
Zahra Zahrae, visiting from Sharif University, Iran (Sep 2015-Oct 2015)
Huibert het Lam, master student (master student, Sep 2015 - Jul 2016).
Francesco Fabian Gonzalez (master student, Sep 2015 - Jul 2016).
Julius Serbenta, master student (Feb 2015-Apr 2016).
Silvije Domazet, visiting from Rudjer Boskovic Institute, Zagreb (Nov 2015 and Apr 2016).
Xuewen Liu, master student (from Sep 2014-Jul 2015, co-supervised with Gerasimos Rigopoulos, Newcastle).
Angelos Tzetzias, master's student (from Sep 2013). Started PhD in Sep 2015 in Fuvan University, Shanghai, Sep 2015.
Pieter Hovenkamp, bachelor student (from Sep 2014, supervised by Anna Watts and Yuri Cavecchi from UvA).
Pablo Gregorian (supervised by Kostas Kokkotas in Tuebingen from Feb 2013).
Drian van der Woude, master's student (Aug 2013-Aug 2014), currently Enrico Pajer's PhD student
Yvette Welling (supervised by Ana Achucarro in Leiden, May 2013-Aug 2014), from Oct 1 2014, from Oct 2014 a PhD student at Leiden University
Jan J. Weenink, Ph.D. student (Nov 1 2009 - Oct 31 2013), working at ASML, Eindhoven
Shun-Pei Miao, postdoc (from Sep 1 2007 - Aug 31 2009) and a veni recipient (from Sep 1 2010 to Aug 2013), from Sep 2013 Assistant Professor in Taiwan.
Tomi Koivisto, postdoc (from Oct 1 2009 - Sep 30 2011). From Oct 2013 Assistant professor, NORDITA, Stockholm
Benoit Tabone,former master's student from Ecole Normal Superieur(Jan-Jul 2013).
Gianrocco Lazzari, former master's student (graduated in Aug 2013).
Dario Perricone, former master's student (graduated in Aug 2013).
Vasileios Prymidis, former master's student (graduated in Aug 2013), PhD student in physics at Utrecht.
Adriaan ter Braak, bachelor student (graduated 2011)
Jurjen Ferdinand Koksma Ph.D. student (from Sep 15 2007 - Sep 2011). As of Jan 2012 employed by McKinsey, Amsterdam
Valentina Dellaca master's student (graduated in Jan 2010)
Jeroen G. Burgers master's student (graduated in Nov 2009), working in oil industry
Gerasimos Rigopoulos, postdoc (Sep 2005 - Aug 2007), currently postdoc in Heidelberg, from 2015 faculty at Newcastle.
Tomas Janssen Ph.D. student (from Sep 1 2005 - Aug 31 2009) From Sep 2009 working in medical physics at the UMC in Amsterdam.
Christiaan Mantz, master student (graduated 2009), PhD student at VU, Amsterdam
Ante Bilandžić master's student (graduated in 2007), From Feb 2012 postdoc in the ALICE group of the Niels Bohr Institute, Copenhagen.
Mark Pinckers, master's student (graduated Feb 2012).
Wessel Valkenburg, master's student (graduated in 2006), PhD with Julien Lesgourgues (Annecy and CERN). Currently veni scholar in Leiden.


In the Fall of 2017 I teach a D-ITP course NS-TP530M Advanced Topics in Theoretical Physics II.
The first part of the course will take place in Utrecht.
LECTURERS are: Tomislav Prokopec (Sep); Vadim Cheianov (UL) and Ben Freivogel (UvA, Oct)
LOCATION & TIME (in Utrecht): Sep 11, 18, 25, Oct 2:
Lectures: Mon 11:15-14:00 (with lunch break) in BBG 7.12. Exercises: 14:15-17:00 in BBG 7.12.
Assistants in the course (Utrecht): Peter Sterk, BBG 7.06, tel: +31 30 253 7558, email:
Pavel Friedrich offce BBG 7.65, tel. +31 30 253 3055, email:
Info about the course requirements can be found here.

updated on Oct 08, 2017


1. Problem set #1: Problem 3 is due at Sep 18, 2017
2. Problem set #2 Problem 6 is due at Sep 25, 2017
3. Problem set #3 Problem 8 is due at Oct 02, 2017
4. Problem set #4 Problem 11 is due at Oct 09, 2017

The final exam will take place on Mon Oct 9 in BBG 7.12 from 11:00-13:00. It will count for 40 percent of the grade.
The exam will consist of one theretical question (with 4-5 sub-questions) and one practical problem (similar to those you have done in the exercise classes).

Module 1: Introduction to non-equilibrium QFT
Tomislav Prokopec (UU)
Lectures: Sep 11, 18, 25, Oct 2 (from 11-14, incl. lunch break)
Exam: Oct 9 (11-17 hrs) in BBG 7.12
Location/Room: Princetonplein 5, Buys Ballot Building, room BBG 7.12, Utrecht

Abstract: In this mini-course we shall discuss basics of non-equilibrium quantum physics.
For simplicity, to a large extent we shall develop the relevant concepts in quantum mechanical systems and then indicate how our results generalize to quantum field-theoretic setting.
We shall cover the following topics:

* The density operator and entropy
* Propagators and two-point functions
* The Schwinger-Keldysh or the in-in formalism
* Perturbation theory
* Applications

Module 2
Vadim Cheianov (UL)
Lectures: Oct 16, 23, 30, Nov 6
Exam: Nov 13
Location/Room: Niels Bohrweg 2, Huygens Laboratory, room HL.226, Leiden

Abstract: tba

Module 3: Quantum field theory in curved spacetime
Ben Freivogel (UvA)
Lectures: Nov 20, 27, Dec 4, 11
Exam: Dec 18, Room G2.10
Location/Room: Nikhef, Science Park 105, room H.331, Amsterdam

Abstract: This module will focus on techniques and results from quantum field theory in curved spacetimes, such as the real world. Important applications we will cover include:
-Quantum fluctuations during inflation that generate the temperature fluctuations in the CMB
-Hawking radiation from black holes and the black hole information paradox.

Dr. Koenraad Schalm
Institute-Lorentz for Theoretical Physics
Leiden University
Niels Bohrweg 2
2335 CA Leiden
tel: +31 71 5275516

Dr. Enrico Pajer
Institute for Theoretical Physics
Utrecht University
tel: +31 30 253 5906

Prof. Wouter Waalewijn
Institute for Theoretical Physics | Institute of Physics
University of Amsterdam
Science Park 904
1098 XH Amsterdam
tel: +31 (0)20 525 5773

In the Fall semester of 2017 I teach General Relativity (NS-TP428M)

The rules and the program for the course can be found if you click here and here.
Tutorial (class) exercises for the course General Relativity 2016-17:

1. Problem set 1 for Sep 09, 2016: Carroll, Chapter 1, Problems 2,3, and 5.
2. Problem set 2 for Sep 16, 2016
3. Problem set 3 for Sep 23, 2016
4. Problem set 4 for Sep 30, 2016
5. Problem set 5 for Oct 07, 2016
6. Problem set 6 for Oct 14, 2016
7. Problem set 7 for Oct 21, 2016
8. Problem set 8 for Oct 28, 2016
9. Problem set 9 for Nov 04, 2016
10. Problem set 10 for Nov 18, 2016
11. Problem set 11 for Nov 25, 2016
12. Problem set 12 for Dec 02, 2016
13. Problem set 13 for Dec 09, 2016
14. Problem set 14 for Dec 16, 2016
15. Problem set 15 for Dec 23, 2016
16. Problem set 16 for Jan 13, 2017
17. Problem set 17 for Jan 20, 2017

Homeworks for the course General Relativity 2016-17:

1. Homework 1 due at 23 Sep 2016
2. Homework 2 due at 07 Oct 2016
3. Homework 3 due at 21 Oct 2016
4. Homework 4 due at 04 Nov 2016
5. Homework 5 due at 25 Nov 2016
6. Homework 6 due at 09 Dec 2016
7. Homework 7 due at 13 Jan 2017
8. Homework 8 due at 27 Jan 2017

The midterm for GR will take place on Friday, Nov 11 2016 from 13.30 - 16.30 in MEGARON, EDUCATORIUM. The exam is 3 hours, closed books.
You need to memorize the formulae given on the following sheets.
The final exam for GR will take place on Friday, on Fri Feb 03 2017 from 13:30-16:30 in BLAUW, RUPPERT. The exam is 3 hours, closed books.
You need to memorize the formulae given on the following sheets.

In the spring semester of 2015 I taught Observational and theoretical cosmology (NS-374B)
Tutorial (class) exercises and homeworks:



Tentamen 1 (midterm): Woensdag, May 20 2015, 09.00-12.45, BBG 061
Tentamen 2 (final): Maandag, Jun 29, 2015, 09.00-12.00, EDUC GAMMA
Hertentamen (retake): Woensdag, Jul 22, 2015, 09.00-12.00, EDUC BETA

In the Spring of 2014 taught a master level course Cosmology (NS-TP430M).
Lectures take place on Tuesdays from 15:15 - 17:00 in BBG 161.
The teaching assistant for the course is Watse Sybesma, office MG416, tel. 030 253 5922.
Exercise classes are on Thu from 9:00 - 12:45 at BBG 165.
Midterm (30%) will take place on Thu, 17 April 2013 at 09-12h (3 hours) in GAMMA of Educatorium.

Lecture notes for Cosmology (NS-TP430M):

Part I: An introduction to the Einstein theory of gravitation
Part II: The Standard Cosmological Model
Part III: Cosmic Inflation
Part IV: Cosmological Perturbations (part)

Homeworks for the cosmology course (NS-TP430M):
Problemset 1 is due at Feb 13, 2014
Problemset 2 is due at Feb 20, 2014
Problemset 3 is due at Feb 28, 2014
Problemset 4 is due at Mar 6, 2014
Problemset 5 is due at Mar 20, 2014
Problemset 6 is due at Mar 27, 2014
Problemset 7 is due at Apr 3, 2014
Problemset 8 is due at Apr 10, 2014
Problemset 9 is due at Apr 24, 2014
Problemset 10 is due at May 1, 2014
Problemset 11 is due at May 8, 2014
Problemset 12 is due at May 15, 2014
Problemset 13 is due at May 22, 2014
Problemset 14 is due at Jun 5, 2014
Problemset 15 is due at Jun 12, 2014
Problemset 16 is due at Jun 26, 2014
Problemset 17 is due at Jul 3, 2014

The rules for cosmology can be found here.
Some of the figures shown in the lectures can be found here.
The cosmology midterm is on Thu 17 Apr from 9am-12pm (3 hours) in the GAMMA lecture hall of Educatorium.
The formulae sheet that you need to memorize is given here.
The exam is closed books.

The cosmology final exam is on Thu 3 Jul from 9am-12pm (3 hours) in BBG 083.
The formulae sheet that you need to memorize is given here (odd) and here (even pages)
The exam is closed books.

To pass the course you need 55% points, which are distributed as:
1. 30% problem sets;
2. 30% midterm exam (min 30% of points required);
3. 40% final exam (min 30% of points required);
3. up to 5% student seminar (optional).

In the Spring semester 2013 (period IV) I tuaght:
Classical Field Theory (NS-364B).
The course was primarily aimed for third year bachelor students.


1. Gleb Arutyunov's lecture notes
2. Lev D. Landau and Evgeniy M. Lifshitz, Course of Theoretical Physics, Vol. 02,
"The Classical Theory of Fields", ISBN-10: 0750627689, ISBN-13: 978-0750627689.
3. Supplementary notes (under construction)

In class exercises and homeworks for Classical Field Theory:

1. In class exercises for Apr 23 and Homework 1 (to be handed in at the exercise class of May 14)
2. There is no exercise class on May 7 (Departement Day). Homework 2: Problems 4-8 from section 10.4. (to be handed in at the exercise class of May 21)
3. Exercises for May 14: Exercises to section 4, starting with exercise 4.9. (on p.150)
4. Exercises for May 21: Final exercises to section 4. (on p.150)
5. Exercises for May 28: Exercises 1-5 to section 5.
Homework 3 (to be handed in at the exercise class of Jun 12)
6. Exercises for Jun 04: Exercises 9 and 10 to section 5.
7. Exercises for Jun 11: Exercises 12-14 and 16 to section 5.
Homework 4 (last!): Exercises 11 and 15 to section 5 plus one more problem (to be handed in at the exercise class of Jun 25)
8. Exercises for Jun 18: Exercises 1-4 to section 6.

The rules for Classical Field Theory:

To pass the course you need 55% points, which are distributed as:
1. 10% problem sets;
2. 40% midterm exam (min 30% of points required);
3. 50% final exam (min 30% of points required);

Seminar topics for the course NS-TP501M (with an almost complete literature):

t o p i c s

Once completed, both student seminars and written reports will be posted on this website.

W r i t t e n . r e p o r t s & t a l k s:

1. Michael Agathos: Gravitational Waves in Cosmology, talk

2. Arjen G. Baarsma: Magnetic Monopoles, talk

3. Jildou Baarsma: Cosmic Microwave Background Radiation, talk

4. Erik van der Bijl: Weak Gravitational Lensing, talk

5. Michiel Bouwhuis: Black Hole Formation, talk

6. Jeroen Burgers: Gravitational Waves: Theory, Evidence and Possibilities, talk

7. Marcin Dukalski: Loop Quantum Gravity and Loop Quantum Cosmology, talk

8. Robert Knegjens: What Light Can Supersymmetry Shed on Dark Matter?, talk

9. Maciej Koch-Janusz: Electroweak Symmetry Breaking,Higgs and Technicolor,

10. Floris van Liere: Planet Formation, talk

11. Thomas Rot: Modified Newtonian Dynamics: a possible solution to the dark matter problem, talk (slide 1) talk (slide 2)

12. Mischa Spelt: Neutrino Oscillations, talk

13. Doru Sticlet: Phase Transitions in the Early Universe: Electroweak and QCD Phase Transitions, talk

14. Stijn J. van Tongeren: Rotating Black Holes, talk

15. Misha Veldhoen: Large Scale Structure Formation , talk

16. Maarten Verdult: Topological defects in cosmology: cosmic strings , talk

17. Jan G. Weenink: Baryogenesis, talk

18. Mathijs Wintraecken: Formation of Large Scale Structure, talk

19. Ori Yudilevich: Dark Energy and the Cosmological Constant, talk

Professional visits:

1. Sep 15 - Oct 15, 2012: Ruđer Bošković Institute, Zagreb, Croatia
2. Oct 28 - Dec 2, 2012: The University of Florida at Gainesville; Perimeter Institute, Canada; McGill, Canada
3. Mid Jan 2013 (1 week): Niels Bohr Institute, Denmark



Visiting address

Tomislav Prokopec
Institute for Theoretical Physics and Spinoza Institute
Utrecht University
Leuvenlaan 4
3584 CE Utrecht
The Netherlands

Postal address

Postbus 80195
3508 TD Utrecht
The Netherlands


Tel.: +31 30 253 2969
Fax: +31 30 253 5937
E-mail: t.prokopec at
Web page:


And this is Anastasia in Dec 2006, in Nov 2006. and in Jan 2007.